diff options
Diffstat (limited to 'theories')
90 files changed, 596 insertions, 354 deletions
diff --git a/theories/Arith/Between.v b/theories/Arith/Between.v index f998e86192..58d3a2b38c 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 206fc0ab5d..f998c19fc7 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 dfd5769464..67c94fdf64 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 ceb91187ba..0fbcec5721 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 f824ee6fbe..bfc2b91a9f 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 65534b2e3c..49152549a4 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 bc3a318cf3..1fc8f7907a 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 9658124320..a173efc10a 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 340a796891..f8020a50e0 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 3b823da6f3..600e5e518d 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 *) diff --git a/theories/Bool/Bool.v b/theories/Bool/Bool.v index 721ab69328..06096c66ac 100644 --- a/theories/Bool/Bool.v +++ b/theories/Bool/Bool.v @@ -39,13 +39,13 @@ Lemma diff_true_false : true <> false. Proof. discriminate. Qed. -Hint Resolve diff_true_false : bool v62. +Hint Resolve diff_true_false : bool. Lemma diff_false_true : false <> true. Proof. discriminate. Qed. -Hint Resolve diff_false_true : bool v62. +Hint Resolve diff_false_true : bool. Hint Extern 1 (false <> true) => exact diff_false_true. Lemma eq_true_false_abs : forall b:bool, b = true -> b = false -> False. @@ -82,7 +82,7 @@ Definition leb (b1 b2:bool) := | true => b2 = true | false => True end. -Hint Unfold leb: bool v62. +Hint Unfold leb: bool. Lemma leb_implb : forall b1 b2, leb b1 b2 <-> implb b1 b2 = true. Proof. @@ -242,14 +242,14 @@ Lemma orb_true_intro : Proof. intros; apply orb_true_iff; trivial. Qed. -Hint Resolve orb_true_intro: bool v62. +Hint Resolve orb_true_intro: bool. Lemma orb_false_intro : forall b1 b2:bool, b1 = false -> b2 = false -> b1 || b2 = false. Proof. intros. subst. reflexivity. Qed. -Hint Resolve orb_false_intro: bool v62. +Hint Resolve orb_false_intro: bool. Lemma orb_false_elim : forall b1 b2:bool, b1 || b2 = false -> b1 = false /\ b2 = false. @@ -268,7 +268,7 @@ Lemma orb_true_r : forall b:bool, b || true = true. Proof. destr_bool. Qed. -Hint Resolve orb_true_r: bool v62. +Hint Resolve orb_true_r: bool. Lemma orb_true_l : forall b:bool, true || b = true. Proof. @@ -284,13 +284,13 @@ Lemma orb_false_r : forall b:bool, b || false = b. Proof. destr_bool. Qed. -Hint Resolve orb_false_r: bool v62. +Hint Resolve orb_false_r: bool. Lemma orb_false_l : forall b:bool, false || b = b. Proof. destr_bool. Qed. -Hint Resolve orb_false_l: bool v62. +Hint Resolve orb_false_l: bool. Notation orb_b_false := orb_false_r (only parsing). Notation orb_false_b := orb_false_l (only parsing). @@ -301,7 +301,7 @@ Lemma orb_negb_r : forall b:bool, b || negb b = true. Proof. destr_bool. Qed. -Hint Resolve orb_negb_r: bool v62. +Hint Resolve orb_negb_r: bool. Notation orb_neg_b := orb_negb_r (only parsing). @@ -318,7 +318,7 @@ Lemma orb_assoc : forall b1 b2 b3:bool, b1 || (b2 || b3) = b1 || b2 || b3. Proof. destr_bool. Qed. -Hint Resolve orb_comm orb_assoc: bool v62. +Hint Resolve orb_comm orb_assoc: bool. (*******************************) (** * Properties of [andb] *) @@ -392,7 +392,7 @@ Lemma andb_false_elim : Proof. destruct b1; simpl; auto. Defined. -Hint Resolve andb_false_elim: bool v62. +Hint Resolve andb_false_elim: bool. (** Complementation *) @@ -400,7 +400,7 @@ Lemma andb_negb_r : forall b:bool, b && negb b = false. Proof. destr_bool. Qed. -Hint Resolve andb_negb_r: bool v62. +Hint Resolve andb_negb_r: bool. Notation andb_neg_b := andb_negb_r (only parsing). @@ -418,7 +418,7 @@ Proof. destr_bool. Qed. -Hint Resolve andb_comm andb_assoc: bool v62. +Hint Resolve andb_comm andb_assoc: bool. (*******************************************) (** * Properties mixing [andb] and [orb] *) @@ -688,7 +688,7 @@ Lemma andb_prop_intro : Proof. destr_bool; tauto. Qed. -Hint Resolve andb_prop_intro: bool v62. +Hint Resolve andb_prop_intro: bool. Notation andb_true_intro2 := (fun b1 b2 H1 H2 => andb_prop_intro b1 b2 (conj H1 H2)) @@ -699,7 +699,7 @@ Lemma andb_prop_elim : Proof. destr_bool; auto. Qed. -Hint Resolve andb_prop_elim: bool v62. +Hint Resolve andb_prop_elim: bool. Notation andb_prop2 := andb_prop_elim (only parsing). diff --git a/theories/Bool/IfProp.v b/theories/Bool/IfProp.v index 11f3d1d6fe..4257b4bc1a 100644 --- a/theories/Bool/IfProp.v +++ b/theories/Bool/IfProp.v @@ -12,7 +12,7 @@ Inductive IfProp (A B:Prop) : bool -> Prop := | Iftrue : A -> IfProp A B true | Iffalse : B -> IfProp A B false. -Hint Resolve Iftrue Iffalse: bool v62. +Hint Resolve Iftrue Iffalse: bool. Lemma Iftrue_inv : forall (A B:Prop) (b:bool), IfProp A B b -> b = true -> A. destruct 1; intros; auto with bool. diff --git a/theories/Classes/Equivalence.v b/theories/Classes/Equivalence.v index c458894795..80b0b7e4c6 100644 --- a/theories/Classes/Equivalence.v +++ b/theories/Classes/Equivalence.v @@ -49,11 +49,11 @@ Infix "=~=" := pequiv (at level 70, no associativity) : equiv_scope. (** Shortcuts to make proof search easier. *) -Program Instance equiv_reflexive `(sa : Equivalence A) : Reflexive equiv. +Program Instance equiv_reflexive `(sa : Equivalence A) : Reflexive equiv | 1. -Program Instance equiv_symmetric `(sa : Equivalence A) : Symmetric equiv. +Program Instance equiv_symmetric `(sa : Equivalence A) : Symmetric equiv | 1. -Program Instance equiv_transitive `(sa : Equivalence A) : Transitive equiv. +Program Instance equiv_transitive `(sa : Equivalence A) : Transitive equiv | 1. Next Obligation. Proof. intros A R sa x y z Hxy Hyz. @@ -123,7 +123,7 @@ Section Respecting. End Respecting. -(** The default equivalence on function spaces, with higher-priority than [eq]. *) +(** The default equivalence on function spaces, with higher priority than [eq]. *) Instance pointwise_reflexive {A} `(reflb : Reflexive B eqB) : Reflexive (pointwise_relation A eqB) | 9. diff --git a/theories/Classes/Morphisms.v b/theories/Classes/Morphisms.v index 81b31d783f..06511ace57 100644 --- a/theories/Classes/Morphisms.v +++ b/theories/Classes/Morphisms.v @@ -470,7 +470,7 @@ Ltac partial_application_tactic := end in let rec do_partial H ar m := - match ar with + lazymatch ar with | 0%nat => do_partial_apps H m ltac:(fail 1) | S ?n' => match m with @@ -483,7 +483,7 @@ Ltac partial_application_tactic := let n := fresh in evar (n:nat) ; let v := eval compute in n in clear n ; let H := fresh in - assert(H:Params m' v) by typeclasses eauto ; + assert(H:Params m' v) by (subst m'; once typeclasses eauto) ; let v' := eval compute in v in subst m'; (sk H v' || fail 1)) || fk diff --git a/theories/Classes/RelationPairs.v b/theories/Classes/RelationPairs.v index cbde5f9ab5..8d1c49822b 100644 --- a/theories/Classes/RelationPairs.v +++ b/theories/Classes/RelationPairs.v @@ -43,6 +43,9 @@ Generalizable Variables A B RA RB Ri Ro f. Definition RelCompFun {A} {B : Type}(R:relation B)(f:A->B) : relation A := fun a a' => R (f a) (f a'). +(** Instances on RelCompFun must match syntactically *) +Typeclasses Opaque RelCompFun. + Infix "@@" := RelCompFun (at level 30, right associativity) : signature_scope. Notation "R @@1" := (R @@ Fst)%signature (at level 30) : signature_scope. @@ -65,6 +68,8 @@ Instance snd_measure : @Measure (A * B) B Snd. Definition RelProd {A : Type} {B : Type} (RA:relation A)(RB:relation B) : relation (A*B) := relation_conjunction (@RelCompFun (A * B) A RA fst) (RB @@2). +Typeclasses Opaque RelProd. + Infix "*" := RelProd : signature_scope. Section RelCompFun_Instances. diff --git a/theories/Compat/Coq84.v b/theories/Compat/Coq84.v index e46a556a98..a3e23f91c9 100644 --- a/theories/Compat/Coq84.v +++ b/theories/Compat/Coq84.v @@ -7,6 +7,10 @@ (************************************************************************) (** Compatibility file for making Coq act similar to Coq v8.4 *) + +(** Any compatibility changes to make future versions of Coq behave like Coq 8.5 are likely needed to make them behave like Coq 8.4. *) +Require Export Coq.Compat.Coq85. + (** See https://coq.inria.fr/bugs/show_bug.cgi?id=4319 for updates *) (** This is required in Coq 8.5 to use the [omega] tactic; in Coq 8.4, it's automatically available. But ZArith_base puts infix ~ at level 7, and we don't want that, so we don't [Import] it. *) Require Coq.omega.Omega. @@ -21,6 +25,9 @@ Global Set Nonrecursive Elimination Schemes. (** See bug 3545 *) Global Set Universal Lemma Under Conjunction. +(** Feature introduced in 8.5, disabled by default and configurable since 8.6. *) +Global Unset Refolding Reduction. + (** In 8.4, [constructor (tac)] allowed backtracking across the use of [constructor]; it has been subsumed by [constructor; tac]. *) Ltac constructor_84_n n := constructor n. Ltac constructor_84_tac tac := once (constructor; tac). @@ -56,8 +63,6 @@ Tactic Notation "symmetry" := symmetry. Tactic Notation "split" := split. Tactic Notation "esplit" := esplit. -Global Set Regular Subst Tactic. - (** Many things now import [PeanoNat] rather than [NPeano], so we require it so that the old absolute names in [NPeano.Nat] are available. *) Require Coq.Numbers.Natural.Peano.NPeano. @@ -68,3 +73,7 @@ Coercion sigT_of_sig : sig >-> sigT. Coercion sig_of_sigT : sigT >-> sig. Coercion sigT2_of_sig2 : sig2 >-> sigT2. Coercion sig2_of_sigT2 : sigT2 >-> sig2. + +(** In 8.4, the statement of admitted lemmas did not depend on the section + variables. *) +Unset Keep Admitted Variables. diff --git a/theories/Compat/Coq85.v b/theories/Compat/Coq85.v index af2e04f88b..64ba6b1e30 100644 --- a/theories/Compat/Coq85.v +++ b/theories/Compat/Coq85.v @@ -8,9 +8,29 @@ (** Compatibility file for making Coq act similar to Coq v8.5 *) +(** Any compatibility changes to make future versions of Coq behave like Coq 8.6 + are likely needed to make them behave like Coq 8.5. *) +Require Export Coq.Compat.Coq86. + +(** We use some deprecated options in this file, so we disable the + corresponding warning, to silence the build of this file. *) +Local Set Warnings "-deprecated-option". + (* In 8.5, "intros [|]", taken e.g. on a goal "A\/B->C", does not behave as "intros [H|H]" but leave instead hypotheses quantified in the goal, here producing subgoals A->C and B->C. *) -Unset Bracketing Last Introduction Pattern. +Global Unset Bracketing Last Introduction Pattern. +Global Unset Regular Subst Tactic. +Global Unset Structural Injection. +Global Unset Shrink Abstract. +Global Unset Shrink Obligations. +Global Set Refolding Reduction. + +(** The resolution algorithm for type classes has changed. *) +Global Set Typeclasses Legacy Resolution. +Global Set Typeclasses Limit Intros. +Global Unset Typeclasses Filtered Unification. +(** Allow silently letting unification constraints float after a "." *) +Global Unset Solve Unification Constraints. diff --git a/theories/Compat/Coq86.v b/theories/Compat/Coq86.v new file mode 100644 index 0000000000..6952fdf199 --- /dev/null +++ b/theories/Compat/Coq86.v @@ -0,0 +1,9 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(** Compatibility file for making Coq act similar to Coq v8.6 *)
\ No newline at end of file diff --git a/theories/Compat/vo.itarget b/theories/Compat/vo.itarget index 43b197004f..7ffb86ebbd 100644 --- a/theories/Compat/vo.itarget +++ b/theories/Compat/vo.itarget @@ -1,3 +1,4 @@ AdmitAxiom.vo Coq84.vo Coq85.vo +Coq86.vo diff --git a/theories/FSets/FMapFacts.v b/theories/FSets/FMapFacts.v index eaeb2914b3..3c5690a724 100644 --- a/theories/FSets/FMapFacts.v +++ b/theories/FSets/FMapFacts.v @@ -24,6 +24,8 @@ Hint Extern 1 (Equivalence _) => constructor; congruence. Module WFacts_fun (E:DecidableType)(Import M:WSfun E). +Notation option_map := option_map (compat "8.4"). + Notation eq_dec := E.eq_dec. Definition eqb x y := if eq_dec x y then true else false. @@ -1986,7 +1988,7 @@ Module OrdProperties (M:S). simpl; intros; try discriminate. intros. destruct a; destruct l; simpl in *. - injection H; clear H; intros; subst. + injection H as -> ->. inversion_clear H1. red in H; simpl in *; intuition. elim H0; eauto. @@ -2050,10 +2052,10 @@ Module OrdProperties (M:S). generalize (elements_3 m). destruct (elements m). try discriminate. - destruct p; injection H; intros; subst. - inversion_clear H1. + destruct p; injection H as -> ->; intros H4. + inversion_clear H1 as [? ? H2|? ? H2]. red in H2; destruct H2; simpl in *; ME.order. - inversion_clear H4. + inversion_clear H4. rename H1 into H3. rewrite (@InfA_alt _ eqke) in H3; eauto with *. apply (H3 (y,x0)); auto. Qed. diff --git a/theories/FSets/FMapList.v b/theories/FSets/FMapList.v index 13cb559b99..5acdb7eb7e 100644 --- a/theories/FSets/FMapList.v +++ b/theories/FSets/FMapList.v @@ -8,7 +8,7 @@ (** * Finite map library *) -(** This file proposes an implementation of the non-dependant interface +(** This file proposes an implementation of the non-dependent interface [FMapInterface.S] using lists of pairs ordered (increasing) with respect to left projection. *) diff --git a/theories/FSets/FMapPositive.v b/theories/FSets/FMapPositive.v index 9e59f0c505..b1c0fdaa2f 100644 --- a/theories/FSets/FMapPositive.v +++ b/theories/FSets/FMapPositive.v @@ -274,8 +274,8 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. rewrite append_assoc_1; apply in_or_app; right; apply in_cons; apply IHm2; auto. rewrite append_assoc_0; apply in_or_app; left; apply IHm1; auto. - rewrite append_neutral_r; apply in_or_app; injection H; - intro EQ; rewrite EQ; right; apply in_eq. + rewrite append_neutral_r; apply in_or_app; injection H as ->; + right; apply in_eq. rewrite append_assoc_1; apply in_or_app; right; apply IHm2; auto. rewrite append_assoc_0; apply in_or_app; left; apply IHm1; auto. congruence. @@ -315,7 +315,7 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. apply in_or_app. left; apply IHm1; auto. right; destruct (in_inv H0). - injection H1; intros Eq1 Eq2; rewrite Eq1; rewrite Eq2; apply in_eq. + injection H1 as -> ->; apply in_eq. apply in_cons; apply IHm2; auto. left; apply IHm1; auto. right; apply IHm2; auto. @@ -346,7 +346,7 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. apply in_or_app. left; apply IHm1; auto. right; destruct (in_inv H0). - injection H1; intros Eq1 Eq2; rewrite Eq1; rewrite Eq2; apply in_eq. + injection H1 as -> ->; apply in_eq. apply in_cons; apply IHm2; auto. left; apply IHm1; auto. right; apply IHm2; auto. @@ -689,7 +689,7 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. destruct y2; destruct y0; compute in Hy2; destruct Hy2; subst. red; red; simpl. destruct H0. - injection H0; clear H0; intros _ H0; subst. + injection H0 as H0 _; subst. eapply xelements_bits_lt_1; eauto. apply E.bits_lt_trans with p. eapply xelements_bits_lt_1; eauto. diff --git a/theories/FSets/FMapWeakList.v b/theories/FSets/FMapWeakList.v index 0f11dd7a53..130cbee871 100644 --- a/theories/FSets/FMapWeakList.v +++ b/theories/FSets/FMapWeakList.v @@ -8,7 +8,7 @@ (** * Finite map library *) -(** This file proposes an implementation of the non-dependant interface +(** This file proposes an implementation of the non-dependent interface [FMapInterface.WS] using lists of pairs, unordered but without redundancy. *) Require Import FMapInterface. diff --git a/theories/FSets/FSetDecide.v b/theories/FSets/FSetDecide.v index ad067eb3d8..1db6a71e81 100644 --- a/theories/FSets/FSetDecide.v +++ b/theories/FSets/FSetDecide.v @@ -357,17 +357,8 @@ the above form: | _ => idtac end. - (** [if t then t1 else t2] executes [t] and, if it does not - fail, then [t1] will be applied to all subgoals - produced. If [t] fails, then [t2] is executed. *) - Tactic Notation - "if" tactic(t) - "then" tactic(t1) - "else" tactic(t2) := - first [ t; first [ t1 | fail 2 ] | t2 ]. - Ltac abstract_term t := - if (is_var t) then fail "no need to abstract a variable" + tryif (is_var t) then fail "no need to abstract a variable" else (let x := fresh "x" in set (x := t) in *; try clearbody x). Ltac abstract_elements := @@ -478,11 +469,11 @@ the above form: repeat ( match goal with | H : context [ @Logic.eq ?T ?x ?y ] |- _ => - if (change T with E.t in H) then fail - else if (change T with t in H) then fail + tryif (change T with E.t in H) then fail + else tryif (change T with t in H) then fail else clear H | H : ?P |- _ => - if prop (FSet_Prop P) holds by + tryif prop (FSet_Prop P) holds by (auto 100 with FSet_Prop) then fail else clear H @@ -747,7 +738,7 @@ the above form: | H: (In ?x ?r) -> False |- (E.eq ?y ?x) -> False => contradict H; fsetdec_body | H: ?P -> False |- ?Q -> False => - if prop (FSet_elt_Prop P) holds by + tryif prop (FSet_elt_Prop P) holds by (auto 100 with FSet_Prop) then (contradict H; fsetdec_body) else fsetdec_body diff --git a/theories/FSets/FSetList.v b/theories/FSets/FSetList.v index 1f36306c34..9c3ec71ae3 100644 --- a/theories/FSets/FSetList.v +++ b/theories/FSets/FSetList.v @@ -8,7 +8,7 @@ (** * Finite sets library *) -(** This file proposes an implementation of the non-dependant +(** This file proposes an implementation of the non-dependent interface [FSetInterface.S] using strictly ordered list. *) Require Export FSetInterface. diff --git a/theories/FSets/FSetPositive.v b/theories/FSets/FSetPositive.v index 7398c6d654..507f1cda69 100644 --- a/theories/FSets/FSetPositive.v +++ b/theories/FSets/FSetPositive.v @@ -1007,10 +1007,10 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits. destruct o. intros x H. injection H; intros; subst. reflexivity. revert IHl. case choose. - intros p Hp x H. injection H; intros; subst; clear H. apply Hp. + intros p Hp x H. injection H as <-. apply Hp. reflexivity. intros _ x. revert IHr. case choose. - intros p Hp H. injection H; intros; subst; clear H. apply Hp. + intros p Hp H. injection H as <-. apply Hp. reflexivity. intros. discriminate. Qed. @@ -1066,11 +1066,11 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits. induction s as [| l IHl o r IHr]; simpl. intros. discriminate. intros x. destruct (min_elt l); intros. - injection H. intros <-. apply IHl. reflexivity. + injection H as <-. apply IHl. reflexivity. destruct o; simpl. - injection H. intros <-. reflexivity. + injection H as <-. reflexivity. destruct (min_elt r); simpl in *. - injection H. intros <-. apply IHr. reflexivity. + injection H as <-. apply IHr. reflexivity. discriminate. Qed. @@ -1094,15 +1094,15 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits. induction s as [|l IHl o r IHr]; intros x y H H'. discriminate. simpl in H. case_eq (min_elt l). - intros p Hp. rewrite Hp in H. injection H; intros <-. + intros p Hp. rewrite Hp in H. injection H as <-. destruct y as [z|z|]; simpl; intro; trivial. apply (IHl p z); trivial. intro Hp; rewrite Hp in H. apply min_elt_3 in Hp. destruct o. - injection H. intros <- Hl. clear H. + injection H as <-. intros Hl. destruct y as [z|z|]; simpl; trivial. elim (Hp _ H'). destruct (min_elt r). - injection H. intros <-. clear H. + injection H as <-. destruct y as [z|z|]. apply (IHr e z); trivial. elim (Hp _ H'). @@ -1119,11 +1119,11 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits. induction s as [| l IHl o r IHr]; simpl. intros. discriminate. intros x. destruct (max_elt r); intros. - injection H. intros <-. apply IHr. reflexivity. + injection H as <-. apply IHr. reflexivity. destruct o; simpl. - injection H. intros <-. reflexivity. + injection H as <-. reflexivity. destruct (max_elt l); simpl in *. - injection H. intros <-. apply IHl. reflexivity. + injection H as <-. apply IHl. reflexivity. discriminate. Qed. @@ -1147,15 +1147,15 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits. induction s as [|l IHl o r IHr]; intros x y H H'. discriminate. simpl in H. case_eq (max_elt r). - intros p Hp. rewrite Hp in H. injection H; intros <-. + intros p Hp. rewrite Hp in H. injection H as <-. destruct y as [z|z|]; simpl; intro; trivial. apply (IHr p z); trivial. intro Hp; rewrite Hp in H. apply max_elt_3 in Hp. destruct o. - injection H. intros <- Hl. clear H. + injection H as <-. intros Hl. destruct y as [z|z|]; simpl; trivial. elim (Hp _ H'). destruct (max_elt l). - injection H. intros <-. clear H. + injection H as <-. destruct y as [z|z|]. elim (Hp _ H'). apply (IHl e z); trivial. diff --git a/theories/FSets/FSetWeakList.v b/theories/FSets/FSetWeakList.v index 2ea32e97cb..9dbea88495 100644 --- a/theories/FSets/FSetWeakList.v +++ b/theories/FSets/FSetWeakList.v @@ -8,7 +8,7 @@ (** * Finite sets library *) -(** This file proposes an implementation of the non-dependant +(** This file proposes an implementation of the non-dependent interface [FSetInterface.WS] using lists without redundancy. *) Require Import FSetInterface. diff --git a/theories/Init/Logic_Type.v b/theories/Init/Logic_Type.v index 4a5f2ad696..4536dfc0fd 100644 --- a/theories/Init/Logic_Type.v +++ b/theories/Init/Logic_Type.v @@ -64,7 +64,7 @@ Definition identity_rect_r : intros A x P H y H0; case identity_sym with (1 := H0); trivial. Defined. -Hint Immediate identity_sym not_identity_sym: core v62. +Hint Immediate identity_sym not_identity_sym: core. Notation refl_id := identity_refl (compat "8.3"). Notation sym_id := identity_sym (compat "8.3"). diff --git a/theories/Init/Peano.v b/theories/Init/Peano.v index 3749baf61c..6c4a63501d 100644 --- a/theories/Init/Peano.v +++ b/theories/Init/Peano.v @@ -33,7 +33,6 @@ Open Scope nat_scope. Definition eq_S := f_equal S. Definition f_equal_nat := f_equal (A:=nat). -Hint Resolve eq_S: v62. Hint Resolve f_equal_nat: core. (** The predecessor function *) @@ -41,7 +40,6 @@ Hint Resolve f_equal_nat: core. Notation pred := Nat.pred (compat "8.4"). Definition f_equal_pred := f_equal pred. -Hint Resolve f_equal_pred: v62. Theorem pred_Sn : forall n:nat, n = pred (S n). Proof. @@ -85,7 +83,6 @@ Notation plus := Nat.add (compat "8.4"). Infix "+" := Nat.add : nat_scope. Definition f_equal2_plus := f_equal2 plus. -Hint Resolve f_equal2_plus: v62. Definition f_equal2_nat := f_equal2 (A1:=nat) (A2:=nat). Hint Resolve f_equal2_nat: core. diff --git a/theories/Init/Specif.v b/theories/Init/Specif.v index 6c0221856e..9fc00e80c1 100644 --- a/theories/Init/Specif.v +++ b/theories/Init/Specif.v @@ -9,6 +9,7 @@ (** Basic specifications : sets that may contain logical information *) Set Implicit Arguments. +Set Reversible Pattern Implicit. Require Import Notations. Require Import Datatypes. @@ -298,7 +299,7 @@ Proof. apply (h2 h1). Defined. -Hint Resolve left right inleft inright: core v62. +Hint Resolve left right inleft inright: core. Hint Resolve exist exist2 existT existT2: core. (* Compatibility *) diff --git a/theories/Init/Tactics.v b/theories/Init/Tactics.v index 59fdbb42f0..5d1e87ae0c 100644 --- a/theories/Init/Tactics.v +++ b/theories/Init/Tactics.v @@ -55,12 +55,6 @@ Ltac contradict H := | _ => (elim H;fail) || pos H end. -(* Transforming a negative goal [ H:~A |- ~B ] into a positive one [ B |- A ]*) - -Ltac swap H := - idtac "swap is OBSOLETE: use contradict instead."; - intro; apply H; clear H. - (* To contradict an hypothesis without copying its type. *) Ltac absurd_hyp H := @@ -79,7 +73,7 @@ Ltac case_eq x := generalize (eq_refl x); pattern x at -1; case x. (* use either discriminate or injection on a hypothesis *) -Ltac destr_eq H := discriminate H || (try (injection H; clear H; intro H)). +Ltac destr_eq H := discriminate H || (try (injection H as H)). (* Similar variants of destruct *) diff --git a/theories/Init/Wf.v b/theories/Init/Wf.v index 985ecaf279..b5b17e5e8d 100644 --- a/theories/Init/Wf.v +++ b/theories/Init/Wf.v @@ -34,7 +34,7 @@ Section Well_founded. destruct 1; trivial. Defined. - Global Implicit Arguments Acc_inv [x y] [x]. + Global Arguments Acc_inv [x] _ [y] _, [x] _ y _. (** A relation is well-founded if every element is accessible *) diff --git a/theories/Lists/List.v b/theories/Lists/List.v index b666992206..30f1dec22c 100644 --- a/theories/Lists/List.v +++ b/theories/Lists/List.v @@ -21,12 +21,13 @@ Set Implicit Arguments. Open Scope list_scope. -(** Standard notations for lists. +(** Standard notations for lists. In a special module to avoid conflicts. *) Module ListNotations. -Notation " [ ] " := nil (format "[ ]") : list_scope. -Notation " [ x ] " := (cons x nil) : list_scope. -Notation " [ x ; y ; .. ; z ] " := (cons x (cons y .. (cons z nil) ..)) : list_scope. +Notation "[ ]" := nil (format "[ ]") : list_scope. +Notation "[ x ]" := (cons x nil) : list_scope. +Notation "[ x ; y ; .. ; z ]" := (cons x (cons y .. (cons z nil) ..)) : list_scope. +Notation "[ x ; .. ; y ]" := (cons x .. (cons y nil) ..) (compat "8.4") : list_scope. End ListNotations. Import ListNotations. @@ -195,7 +196,7 @@ Section Facts. Qed. Theorem app_nil_r : forall l:list A, l ++ [] = l. - Proof. + Proof. induction l; simpl; f_equal; auto. Qed. @@ -248,8 +249,7 @@ Section Facts. generalize (app_nil_r l); intros E. rewrite -> E; auto. intros. - injection H. - intro. + injection H as H H0. assert ([] = l ++ a0 :: l0) by auto. apply app_cons_not_nil in H1 as []. Qed. @@ -335,16 +335,16 @@ Section Facts. absurd (length (x1 :: l1 ++ l) <= length l). simpl; rewrite app_length; auto with arith. rewrite H; auto with arith. - injection H; clear H; intros; f_equal; eauto. + injection H as H H0; f_equal; eauto. Qed. End Facts. -Hint Resolve app_assoc app_assoc_reverse: datatypes v62. -Hint Resolve app_comm_cons app_cons_not_nil: datatypes v62. -Hint Immediate app_eq_nil: datatypes v62. -Hint Resolve app_eq_unit app_inj_tail: datatypes v62. -Hint Resolve in_eq in_cons in_inv in_nil in_app_or in_or_app: datatypes v62. +Hint Resolve app_assoc app_assoc_reverse: datatypes. +Hint Resolve app_comm_cons app_cons_not_nil: datatypes. +Hint Immediate app_eq_nil: datatypes. +Hint Resolve app_eq_unit app_inj_tail: datatypes. +Hint Resolve in_eq in_cons in_inv in_nil in_app_or in_or_app: datatypes. @@ -518,7 +518,7 @@ Section Elts. Proof. revert l. induction n as [|n IH]; intros [|x l] H; simpl in *; try easy. - - exists nil; exists l. injection H; clear H; intros; now subst. + - exists nil; exists l. now injection H as ->. - destruct (IH _ H) as (l1 & l2 & H1 & H2). exists (x::l1); exists l2; simpl; split; now f_equal. Qed. @@ -1385,9 +1385,8 @@ End Fold_Right_Recursor. Lemma combine_split : forall (l:list A)(l':list B), length l = length l' -> split (combine l l') = (l,l'). Proof. - induction l; destruct l'; simpl; intros; auto; try discriminate. - injection H; clear H; intros. - rewrite IHl; auto. + induction l, l'; simpl; trivial; try discriminate. + now intros [= ->%IHl]. Qed. Lemma in_combine_l : forall (l:list A)(l':list B)(x:A)(y:B), @@ -1471,7 +1470,7 @@ End Fold_Right_Recursor. destruct (in_app_or _ _ _ H); clear H. destruct (in_map_iff (fun y : B => (a, y)) l' (x,y)) as (H1,_). destruct (H1 H0) as (z,(H2,H3)); clear H0 H1. - injection H2; clear H2; intros; subst; intuition. + injection H2 as -> ->; intuition. intuition. Qed. @@ -1545,7 +1544,7 @@ Section length_order. End length_order. Hint Resolve lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons: - datatypes v62. + datatypes. (******************************) @@ -1614,7 +1613,7 @@ Section SetIncl. End SetIncl. Hint Resolve incl_refl incl_tl incl_tran incl_appl incl_appr incl_cons - incl_app: datatypes v62. + incl_app: datatypes. (**************************************) @@ -2366,7 +2365,7 @@ Notation rev_acc := rev_append (only parsing). Notation rev_acc_rev := rev_append_rev (only parsing). Notation AllS := Forall (only parsing). (* was formerly in TheoryList *) -Hint Resolve app_nil_end : datatypes v62. +Hint Resolve app_nil_end : datatypes. (* end hide *) Section Repeat. diff --git a/theories/Lists/Streams.v b/theories/Lists/Streams.v index 7ec3d2503f..1c302b22f4 100644 --- a/theories/Lists/Streams.v +++ b/theories/Lists/Streams.v @@ -51,7 +51,7 @@ Lemma tl_nth_tl : Proof. simple induction n; simpl; auto. Qed. -Hint Resolve tl_nth_tl: datatypes v62. +Hint Resolve tl_nth_tl: datatypes. Lemma Str_nth_tl_plus : forall (n m:nat) (s:Stream), diff --git a/theories/Logic/Eqdep.v b/theories/Logic/Eqdep.v index f3a2783e1b..5ef86b8e77 100644 --- a/theories/Logic/Eqdep.v +++ b/theories/Logic/Eqdep.v @@ -33,5 +33,5 @@ Export EqdepTheory. (** Exported hints *) -Hint Resolve eq_dep_eq: eqdep v62. +Hint Resolve eq_dep_eq: eqdep. Hint Resolve inj_pair2 inj_pairT2: eqdep. diff --git a/theories/Logic/EqdepFacts.v b/theories/Logic/EqdepFacts.v index 30e26c7c67..bd59159bb5 100644 --- a/theories/Logic/EqdepFacts.v +++ b/theories/Logic/EqdepFacts.v @@ -164,7 +164,7 @@ Proof. split; auto using eq_sig_eq_dep, eq_dep_eq_sig. Qed. -(** Dependent equality is equivalent tco a dependent pair of equalities *) +(** Dependent equality is equivalent to a dependent pair of equalities *) Set Implicit Arguments. diff --git a/theories/Logic/FunctionalExtensionality.v b/theories/Logic/FunctionalExtensionality.v index 04d9a6704d..9551fea1ab 100644 --- a/theories/Logic/FunctionalExtensionality.v +++ b/theories/Logic/FunctionalExtensionality.v @@ -56,6 +56,78 @@ Proof. apply functional_extensionality in H. destruct H. reflexivity. Defined. +(** A version of [functional_extensionality_dep] which is provably + equal to [eq_refl] on [fun _ => eq_refl] *) +Definition functional_extensionality_dep_good + {A} {B : A -> Type} + (f g : forall x : A, B x) + (H : forall x, f x = g x) + : f = g + := eq_trans (eq_sym (functional_extensionality_dep f f (fun _ => eq_refl))) + (functional_extensionality_dep f g H). + +Lemma functional_extensionality_dep_good_refl {A B} f + : @functional_extensionality_dep_good A B f f (fun _ => eq_refl) = eq_refl. +Proof. + unfold functional_extensionality_dep_good; edestruct functional_extensionality_dep; reflexivity. +Defined. + +Opaque functional_extensionality_dep_good. + +Lemma forall_sig_eq_rect + {A B} (f : forall a : A, B a) + (P : { g : _ | (forall a, f a = g a) } -> Type) + (k : P (exist (fun g => forall a, f a = g a) f (fun a => eq_refl))) + g +: P g. +Proof. + destruct g as [g1 g2]. + set (g' := fun x => (exist _ (g1 x) (g2 x))). + change g2 with (fun x => proj2_sig (g' x)). + change g1 with (fun x => proj1_sig (g' x)). + clearbody g'; clear g1 g2. + cut (forall x, (exist _ (f x) eq_refl) = g' x). + { intro H'. + apply functional_extensionality_dep_good in H'. + destruct H'. + exact k. } + { intro x. + destruct (g' x) as [g'x1 g'x2]. + destruct g'x2. + reflexivity. } +Defined. + +Definition forall_eq_rect + {A B} (f : forall a : A, B a) + (P : forall g, (forall a, f a = g a) -> Type) + (k : P f (fun a => eq_refl)) + g H + : P g H + := @forall_sig_eq_rect A B f (fun g => P (proj1_sig g) (proj2_sig g)) k (exist _ g H). + +Definition forall_eq_rect_comp {A B} f P k + : @forall_eq_rect A B f P k f (fun _ => eq_refl) = k. +Proof. + unfold forall_eq_rect, forall_sig_eq_rect; simpl. + rewrite functional_extensionality_dep_good_refl; reflexivity. +Qed. + +Definition f_equal__functional_extensionality_dep_good + {A B f g} H a + : f_equal (fun h => h a) (@functional_extensionality_dep_good A B f g H) = H a. +Proof. + apply forall_eq_rect with (H := H); clear H g. + change (eq_refl (f a)) with (f_equal (fun h => h a) (eq_refl f)). + apply f_equal, functional_extensionality_dep_good_refl. +Defined. + +Definition f_equal__functional_extensionality_dep_good__fun + {A B f g} H + : (fun a => f_equal (fun h => h a) (@functional_extensionality_dep_good A B f g H)) = H. +Proof. + apply functional_extensionality_dep_good; intro a; apply f_equal__functional_extensionality_dep_good. +Defined. + (** Apply [functional_extensionality], introducing variable x. *) Tactic Notation "extensionality" ident(x) := @@ -68,6 +140,87 @@ Tactic Notation "extensionality" ident(x) := apply forall_extensionality) ; intro x end. +(** Iteratively apply [functional_extensionality] on an hypothesis + until finding an equality statement *) +(* Note that you can write [Ltac extensionality_in_checker tac ::= tac tt.] to get a more informative error message. *) +Ltac extensionality_in_checker tac := + first [ tac tt | fail 1 "Anomaly: Unexpected error in extensionality tactic. Please report." ]. +Tactic Notation "extensionality" "in" hyp(H) := + let rec check_is_extensional_equality H := + lazymatch type of H with + | _ = _ => constr:(Prop) + | forall a : ?A, ?T + => let Ha := fresh in + constr:(forall a : A, match H a with Ha => ltac:(let v := check_is_extensional_equality Ha in exact v) end) + end in + let assert_is_extensional_equality H := + first [ let dummy := check_is_extensional_equality H in idtac + | fail 1 "Not an extensional equality" ] in + let assert_not_intensional_equality H := + lazymatch type of H with + | _ = _ => fail "Already an intensional equality" + | _ => idtac + end in + let enforce_no_body H := + (tryif (let dummy := (eval unfold H in H) in idtac) + then clearbody H + else idtac) in + let rec extensionality_step_make_type H := + lazymatch type of H with + | forall a : ?A, ?f = ?g + => constr:({ H' | (fun a => f_equal (fun h => h a) H') = H }) + | forall a : ?A, _ + => let H' := fresh in + constr:(forall a : A, match H a with H' => ltac:(let ret := extensionality_step_make_type H' in exact ret) end) + end in + let rec eta_contract T := + lazymatch (eval cbv beta in T) with + | context T'[fun a : ?A => ?f a] + => let T'' := context T'[f] in + eta_contract T'' + | ?T => T + end in + let rec lift_sig_extensionality H := + lazymatch type of H with + | sig _ => H + | forall a : ?A, _ + => let Ha := fresh in + let ret := constr:(fun a : A => match H a with Ha => ltac:(let v := lift_sig_extensionality Ha in exact v) end) in + lazymatch type of ret with + | forall a : ?A, sig (fun b : ?B => @?f a b = @?g a b) + => eta_contract (exist (fun b : (forall a : A, B) => (fun a : A => f a (b a)) = (fun a : A => g a (b a))) + (fun a : A => proj1_sig (ret a)) + (@functional_extensionality_dep_good _ _ _ _ (fun a : A => proj2_sig (ret a)))) + end + end in + let extensionality_pre_step H H_out Heq := + let T := extensionality_step_make_type H in + let H' := fresh in + assert (H' : T) by (intros; eexists; apply f_equal__functional_extensionality_dep_good__fun); + let H''b := lift_sig_extensionality H' in + case H''b; clear H'; + intros H_out Heq in + let rec extensionality_rec H H_out Heq := + lazymatch type of H with + | forall a, _ = _ + => extensionality_pre_step H H_out Heq + | _ + => let pre_H_out' := fresh H_out in + let H_out' := fresh pre_H_out' in + extensionality_pre_step H H_out' Heq; + let Heq' := fresh Heq in + extensionality_rec H_out' H_out Heq'; + subst H_out' + end in + first [ assert_is_extensional_equality H | fail 1 "Not an extensional equality" ]; + first [ assert_not_intensional_equality H | fail 1 "Already an intensional equality" ]; + (tryif enforce_no_body H then idtac else clearbody H); + let H_out := fresh in + let Heq := fresh "Heq" in + extensionality_in_checker ltac:(fun tt => extensionality_rec H H_out Heq); + (* If we [subst H], things break if we already have another equation of the form [_ = H] *) + destruct Heq; rename H_out into H. + (** Eta expansion follows from extensionality. *) Lemma eta_expansion_dep {A} {B : A -> Type} (f : forall x : A, B x) : diff --git a/theories/Logic/Hurkens.v b/theories/Logic/Hurkens.v index 841f843c07..56e03e965c 100644 --- a/theories/Logic/Hurkens.v +++ b/theories/Logic/Hurkens.v @@ -562,7 +562,7 @@ End Paradox. End NoRetractFromSmallPropositionToProp. -(** * Large universes are no retracts of [Prop]. *) +(** * Large universes are not retracts of [Prop]. *) (** The existence in the Calculus of Constructions with universes of a retract from some [Type] universe into [Prop] is inconsistent. *) diff --git a/theories/MSets/MSetAVL.v b/theories/MSets/MSetAVL.v index cc023cc3f8..a3c265a21f 100644 --- a/theories/MSets/MSetAVL.v +++ b/theories/MSets/MSetAVL.v @@ -417,6 +417,7 @@ Local Open Scope Int_scope. Let's do its job by hand: *) Ltac join_tac := + let l := fresh "l" in intro l; induction l as [| lh ll _ lx lr Hlr]; [ | intros x r; induction r as [| rh rl Hrl rx rr _]; unfold join; [ | destruct ((rh+2) <? lh) eqn:LT; diff --git a/theories/MSets/MSetDecide.v b/theories/MSets/MSetDecide.v index f2555791b2..9c622fd786 100644 --- a/theories/MSets/MSetDecide.v +++ b/theories/MSets/MSetDecide.v @@ -357,17 +357,8 @@ the above form: | _ => idtac end. - (** [if t then t1 else t2] executes [t] and, if it does not - fail, then [t1] will be applied to all subgoals - produced. If [t] fails, then [t2] is executed. *) - Tactic Notation - "if" tactic(t) - "then" tactic(t1) - "else" tactic(t2) := - first [ t; first [ t1 | fail 2 ] | t2 ]. - Ltac abstract_term t := - if (is_var t) then fail "no need to abstract a variable" + tryif (is_var t) then fail "no need to abstract a variable" else (let x := fresh "x" in set (x := t) in *; try clearbody x). Ltac abstract_elements := @@ -478,11 +469,11 @@ the above form: repeat ( match goal with | H : context [ @Logic.eq ?T ?x ?y ] |- _ => - if (change T with E.t in H) then fail - else if (change T with t in H) then fail + tryif (change T with E.t in H) then fail + else tryif (change T with t in H) then fail else clear H | H : ?P |- _ => - if prop (MSet_Prop P) holds by + tryif prop (MSet_Prop P) holds by (auto 100 with MSet_Prop) then fail else clear H @@ -747,7 +738,7 @@ the above form: | H: (In ?x ?r) -> False |- (E.eq ?y ?x) -> False => contradict H; fsetdec_body | H: ?P -> False |- ?Q -> False => - if prop (MSet_elt_Prop P) holds by + tryif prop (MSet_elt_Prop P) holds by (auto 100 with MSet_Prop) then (contradict H; fsetdec_body) else fsetdec_body diff --git a/theories/MSets/MSetInterface.v b/theories/MSets/MSetInterface.v index bd88116899..74a7f6df89 100644 --- a/theories/MSets/MSetInterface.v +++ b/theories/MSets/MSetInterface.v @@ -345,6 +345,9 @@ Module Type WRawSets (E : DecidableType). predicate [Ok]. If [Ok] isn't decidable, [isok] may be the always-false function. *) Parameter isok : t -> bool. + (** MS: + Dangerous instance, the [isok s = true] hypothesis cannot be discharged + with typeclass resolution. Is it really an instance? *) Declare Instance isok_Ok s `(isok s = true) : Ok s | 10. (** Logical predicates *) diff --git a/theories/MSets/MSetList.v b/theories/MSets/MSetList.v index fb0d1ad9df..05c20eb8fa 100644 --- a/theories/MSets/MSetList.v +++ b/theories/MSets/MSetList.v @@ -8,7 +8,7 @@ (** * Finite sets library *) -(** This file proposes an implementation of the non-dependant +(** This file proposes an implementation of the non-dependent interface [MSetInterface.S] using strictly ordered list. *) Require Export MSetInterface OrdersFacts OrdersLists. diff --git a/theories/MSets/MSetPositive.v b/theories/MSets/MSetPositive.v index 8dd240f46e..be95a03799 100644 --- a/theories/MSets/MSetPositive.v +++ b/theories/MSets/MSetPositive.v @@ -908,10 +908,10 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits. destruct o. intros x H. injection H; intros; subst. reflexivity. revert IHl. case choose. - intros p Hp x H. injection H; intros; subst; clear H. apply Hp. + intros p Hp x H. injection H as <-. apply Hp. reflexivity. intros _ x. revert IHr. case choose. - intros p Hp H. injection H; intros; subst; clear H. apply Hp. + intros p Hp H. injection H as <-. apply Hp. reflexivity. intros. discriminate. Qed. @@ -968,11 +968,11 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits. induction s as [| l IHl o r IHr]; simpl. intros. discriminate. intros x. destruct (min_elt l); intros. - injection H. intros <-. apply IHl. reflexivity. + injection H as <-. apply IHl. reflexivity. destruct o; simpl. - injection H. intros <-. reflexivity. + injection H as <-. reflexivity. destruct (min_elt r); simpl in *. - injection H. intros <-. apply IHr. reflexivity. + injection H as <-. apply IHr. reflexivity. discriminate. Qed. @@ -996,15 +996,15 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits. induction s as [|l IHl o r IHr]; intros x y H H'. discriminate. simpl in H. case_eq (min_elt l). - intros p Hp. rewrite Hp in H. injection H; intros <-. + intros p Hp. rewrite Hp in H. injection H as <-. destruct y as [z|z|]; simpl; intro; trivial. apply (IHl p z); trivial. intro Hp; rewrite Hp in H. apply min_elt_spec3 in Hp. destruct o. - injection H. intros <- Hl. clear H. + injection H as <-. intros Hl. destruct y as [z|z|]; simpl; trivial. elim (Hp _ H'). destruct (min_elt r). - injection H. intros <-. clear H. + injection H as <-. destruct y as [z|z|]. apply (IHr e z); trivial. elim (Hp _ H'). @@ -1021,11 +1021,11 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits. induction s as [| l IHl o r IHr]; simpl. intros. discriminate. intros x. destruct (max_elt r); intros. - injection H. intros <-. apply IHr. reflexivity. + injection H as <-. apply IHr. reflexivity. destruct o; simpl. - injection H. intros <-. reflexivity. + injection H as <-. reflexivity. destruct (max_elt l); simpl in *. - injection H. intros <-. apply IHl. reflexivity. + injection H as <-. apply IHl. reflexivity. discriminate. Qed. @@ -1049,15 +1049,15 @@ Module PositiveSet <: S with Module E:=PositiveOrderedTypeBits. induction s as [|l IHl o r IHr]; intros x y H H'. discriminate. simpl in H. case_eq (max_elt r). - intros p Hp. rewrite Hp in H. injection H; intros <-. + intros p Hp. rewrite Hp in H. injection H as <-. destruct y as [z|z|]; simpl; intro; trivial. apply (IHr p z); trivial. intro Hp; rewrite Hp in H. apply max_elt_spec3 in Hp. destruct o. - injection H. intros <- Hl. clear H. + injection H as <-. intros Hl. destruct y as [z|z|]; simpl; trivial. elim (Hp _ H'). destruct (max_elt l). - injection H. intros <-. clear H. + injection H as <-. destruct y as [z|z|]. elim (Hp _ H'). apply (IHl e z); trivial. diff --git a/theories/MSets/MSetRBT.v b/theories/MSets/MSetRBT.v index 751d4f35c6..83a2343dd4 100644 --- a/theories/MSets/MSetRBT.v +++ b/theories/MSets/MSetRBT.v @@ -911,7 +911,7 @@ Proof. { inversion_clear O. assert (InT x l) by now apply min_elt_spec1. auto. } simpl. case X.compare_spec; try order. - destruct lc; injection E; clear E; intros; subst l s0; auto. + destruct lc; injection E; subst l s0; auto. Qed. Lemma remove_min_spec1 s x s' `{Ok s}: @@ -1948,7 +1948,7 @@ Module Make (X: Orders.OrderedType) <: generalize (fun x s' => @Raw.remove_min_spec1 s x s' Hs). set (P := Raw.remove_min_ok s). clearbody P. destruct (Raw.remove_min s) as [(x0,s0)|]; try easy. - intros H U. injection U. clear U; intros; subst. simpl. + intros H U. injection U as -> <-. simpl. destruct (H x s0); auto. subst; intuition. Qed. diff --git a/theories/MSets/MSetWeakList.v b/theories/MSets/MSetWeakList.v index 372acd56ad..2ac57a932b 100644 --- a/theories/MSets/MSetWeakList.v +++ b/theories/MSets/MSetWeakList.v @@ -8,7 +8,7 @@ (** * Finite sets library *) -(** This file proposes an implementation of the non-dependant +(** This file proposes an implementation of the non-dependent interface [MSetWeakInterface.S] using lists without redundancy. *) Require Import MSetInterface. diff --git a/theories/Numbers/BigNumPrelude.v b/theories/Numbers/BigNumPrelude.v index 45a7527c97..bd8930872c 100644 --- a/theories/Numbers/BigNumPrelude.v +++ b/theories/Numbers/BigNumPrelude.v @@ -10,7 +10,7 @@ (** * BigNumPrelude *) -(** Auxillary functions & theorems used for arbitrary precision efficient +(** Auxiliary functions & theorems used for arbitrary precision efficient numbers. *) @@ -22,7 +22,7 @@ Require Export Zpow_facts. Declare ML Module "numbers_syntax_plugin". (* *** Nota Bene *** - All results that were general enough has been moved in ZArith. + All results that were general enough have been moved in ZArith. Only remain here specialized lemmas and compatibility elements. (P.L. 5/11/2007). *) diff --git a/theories/Numbers/Cyclic/ZModulo/ZModulo.v b/theories/Numbers/Cyclic/ZModulo/ZModulo.v index c115a831d9..04fc5a8dfa 100644 --- a/theories/Numbers/Cyclic/ZModulo/ZModulo.v +++ b/theories/Numbers/Cyclic/ZModulo/ZModulo.v @@ -369,7 +369,7 @@ Section ZModulo. assert (Z.div_eucl ([|x|]*[|y|]) wB = (([|x|]*[|y|])/wB,([|x|]*[|y|]) mod wB)). unfold Z.modulo, Z.div; destruct Z.div_eucl; auto. generalize (Z_div_mod ([|x|]*[|y|]) wB wB_pos); destruct Z.div_eucl as (h,l). - destruct 1; injection H; clear H; intros. + destruct 1; injection H as ? ?. rewrite H0. assert ([|l|] = l). apply Zmod_small; auto. @@ -411,7 +411,7 @@ Section ZModulo. unfold Z.modulo, Z.div; destruct Z.div_eucl; auto. generalize (Z_div_mod [|a|] [|b|] H0). destruct Z.div_eucl as (q,r); destruct 1; intros. - injection H1; clear H1; intros. + injection H1 as ? ?. assert ([|r|]=r). apply Zmod_small; generalize (Z_mod_lt b wB wB_pos); fold [|b|]; auto with zarith. @@ -522,7 +522,7 @@ Section ZModulo. unfold Z.modulo, Z.div; destruct Z.div_eucl; auto. generalize (Z_div_mod a [|b|] H3). destruct Z.div_eucl as (q,r); destruct 1; intros. - injection H4; clear H4; intros. + injection H4 as ? ?. assert ([|r|]=r). apply Zmod_small; generalize (Z_mod_lt b wB wB_pos); fold [|b|]; auto with zarith. diff --git a/theories/Numbers/Integer/Abstract/ZDivEucl.v b/theories/Numbers/Integer/Abstract/ZDivEucl.v index 278e1bcffa..c2fa69e535 100644 --- a/theories/Numbers/Integer/Abstract/ZDivEucl.v +++ b/theories/Numbers/Integer/Abstract/ZDivEucl.v @@ -30,18 +30,23 @@ Require Import ZAxioms ZMulOrder ZSgnAbs NZDiv. We just ignore them here. *) -Module Type EuclidSpec (Import A : ZAxiomsSig')(Import B : DivMod' A). - Axiom mod_always_pos : forall a b, b ~= 0 -> 0 <= a mod b < abs b. +Module Type EuclidSpec (Import A : ZAxiomsSig')(Import B : DivMod A). + Axiom mod_always_pos : forall a b, b ~= 0 -> 0 <= B.modulo a b < abs b. End EuclidSpec. Module Type ZEuclid (Z:ZAxiomsSig) := NZDiv.NZDiv Z <+ EuclidSpec Z. -Module Type ZEuclid' (Z:ZAxiomsSig) := NZDiv.NZDiv' Z <+ EuclidSpec Z. Module ZEuclidProp (Import A : ZAxiomsSig') (Import B : ZMulOrderProp A) (Import C : ZSgnAbsProp A B) - (Import D : ZEuclid' A). + (Import D : ZEuclid A). + + (** We put notations in a scope, to avoid warnings about + redefinitions of notations *) + Infix "/" := D.div : euclid. + Infix "mod" := D.modulo : euclid. + Local Open Scope euclid. Module Import Private_NZDiv := Nop <+ NZDivProp A D B. @@ -615,4 +620,3 @@ intros (c,Hc). rewrite Hc. now apply mod_mul. Qed. End ZEuclidProp. - diff --git a/theories/Numbers/Integer/BigZ/BigZ.v b/theories/Numbers/Integer/BigZ/BigZ.v index 56cb9bbc2c..7c76011f21 100644 --- a/theories/Numbers/Integer/BigZ/BigZ.v +++ b/theories/Numbers/Integer/BigZ/BigZ.v @@ -138,7 +138,7 @@ intros NEQ. generalize (BigZ.spec_div_eucl a b). generalize (Z_div_mod_full [a] [b] NEQ). destruct BigZ.div_eucl as (q,r), Z.div_eucl as (q',r'). -intros (EQ,_). injection 1. intros EQr EQq. +intros (EQ,_). injection 1 as EQr EQq. BigZ.zify. rewrite EQr, EQq; auto. Qed. diff --git a/theories/Numbers/Integer/BigZ/ZMake.v b/theories/Numbers/Integer/BigZ/ZMake.v index 8673b8a58f..fec6e06837 100644 --- a/theories/Numbers/Integer/BigZ/ZMake.v +++ b/theories/Numbers/Integer/BigZ/ZMake.v @@ -147,7 +147,7 @@ Module Make (NN:NType) <: ZType. Proof. apply Bool.eq_iff_eq_true. rewrite Z.leb_le. unfold Z.le, leb. rewrite spec_compare. - destruct Z.compare; split; try easy. now destruct 1. + now destruct Z.compare; split. Qed. Definition min n m := match compare n m with Gt => m | _ => n end. @@ -427,13 +427,13 @@ Module Make (NN:NType) <: ZType. (* Pos Neg *) generalize (NN.spec_div_eucl x y); destruct (NN.div_eucl x y) as (q,r). break_nonneg x px EQx; break_nonneg y py EQy; - try (injection 1; intros Hr Hq; rewrite NN.spec_eqb, NN.spec_0, Hr; + try (injection 1 as Hq Hr; rewrite NN.spec_eqb, NN.spec_0, Hr; simpl; rewrite Hq, NN.spec_0; auto). change (- Zpos py) with (Zneg py). assert (GT : Zpos py > 0) by (compute; auto). generalize (Z_div_mod (Zpos px) (Zpos py) GT). unfold Z.div_eucl. destruct (Z.pos_div_eucl px (Zpos py)) as (q',r'). - intros (EQ,MOD). injection 1. intros Hr' Hq'. + intros (EQ,MOD). injection 1 as Hq' Hr'. rewrite NN.spec_eqb, NN.spec_0, Hr'. break_nonneg r pr EQr. subst; simpl. rewrite NN.spec_0; auto. @@ -442,13 +442,13 @@ Module Make (NN:NType) <: ZType. (* Neg Pos *) generalize (NN.spec_div_eucl x y); destruct (NN.div_eucl x y) as (q,r). break_nonneg x px EQx; break_nonneg y py EQy; - try (injection 1; intros Hr Hq; rewrite NN.spec_eqb, NN.spec_0, Hr; + try (injection 1 as Hq Hr; rewrite NN.spec_eqb, NN.spec_0, Hr; simpl; rewrite Hq, NN.spec_0; auto). change (- Zpos px) with (Zneg px). assert (GT : Zpos py > 0) by (compute; auto). generalize (Z_div_mod (Zpos px) (Zpos py) GT). unfold Z.div_eucl. destruct (Z.pos_div_eucl px (Zpos py)) as (q',r'). - intros (EQ,MOD). injection 1. intros Hr' Hq'. + intros (EQ,MOD). injection 1 as Hq' Hr'. rewrite NN.spec_eqb, NN.spec_0, Hr'. break_nonneg r pr EQr. subst; simpl. rewrite NN.spec_0; auto. @@ -457,7 +457,7 @@ Module Make (NN:NType) <: ZType. (* Neg Neg *) generalize (NN.spec_div_eucl x y); destruct (NN.div_eucl x y) as (q,r). break_nonneg x px EQx; break_nonneg y py EQy; - try (injection 1; intros Hr Hq; rewrite Hr, Hq; auto). + try (injection 1 as -> ->; auto). simpl. intros <-; auto. Qed. diff --git a/theories/Numbers/NatInt/NZGcd.v b/theories/Numbers/NatInt/NZGcd.v index 1d36729435..44088f8c4a 100644 --- a/theories/Numbers/NatInt/NZGcd.v +++ b/theories/Numbers/NatInt/NZGcd.v @@ -60,8 +60,6 @@ Proof. intros n. exists 0. now nzsimpl. Qed. -Hint Rewrite divide_1_l divide_0_r : nz. - Lemma divide_0_l : forall n, (0 | n) -> n==0. Proof. intros n (m,Hm). revert Hm. now nzsimpl. diff --git a/theories/Numbers/Natural/BigN/BigN.v b/theories/Numbers/Natural/BigN/BigN.v index ec1017f505..e8ff516f35 100644 --- a/theories/Numbers/Natural/BigN/BigN.v +++ b/theories/Numbers/Natural/BigN/BigN.v @@ -110,7 +110,7 @@ intros NEQ. generalize (BigN.spec_div_eucl a b). generalize (Z_div_mod_full [a] [b] NEQ). destruct BigN.div_eucl as (q,r), Z.div_eucl as (q',r'). -intros (EQ,_). injection 1. intros EQr EQq. +intros (EQ,_). injection 1 as EQr EQq. BigN.zify. rewrite EQr, EQq; auto. Qed. diff --git a/theories/Numbers/Natural/BigN/NMake.v b/theories/Numbers/Natural/BigN/NMake.v index 98949736cb..1425041a10 100644 --- a/theories/Numbers/Natural/BigN/NMake.v +++ b/theories/Numbers/Natural/BigN/NMake.v @@ -338,7 +338,7 @@ Module Make (W0:CyclicType) <: NType. Proof. apply eq_iff_eq_true. rewrite Z.leb_le. unfold Z.le, leb. rewrite spec_compare. - destruct Z.compare; split; try easy. now destruct 1. + now destruct Z.compare; split. Qed. Definition min (n m : t) : t := match compare n m with Gt => m | _ => n end. diff --git a/theories/Numbers/Natural/BigN/NMake_gen.ml b/theories/Numbers/Natural/BigN/NMake_gen.ml index 601fa108f9..5177fae65f 100644 --- a/theories/Numbers/Natural/BigN/NMake_gen.ml +++ b/theories/Numbers/Natural/BigN/NMake_gen.ml @@ -147,7 +147,7 @@ pr pr " Local Notation Size := (SizePlus O)."; pr ""; - pr " Tactic Notation \"do_size\" tactic(t) := do %i t." (size+1); + pr " Tactic Notation (at level 3) \"do_size\" tactic3(t) := do %i t." (size+1); pr ""; pr " Definition dom_t n := match n with"; diff --git a/theories/Numbers/Natural/Peano/NPeano.v b/theories/Numbers/Natural/Peano/NPeano.v index 58b1b01806..d0168bd9ec 100644 --- a/theories/Numbers/Natural/Peano/NPeano.v +++ b/theories/Numbers/Natural/Peano/NPeano.v @@ -8,8 +8,88 @@ (* Evgeny Makarov, INRIA, 2007 *) (************************************************************************) -Require Import PeanoNat NAxioms. +Require Import PeanoNat Even NAxioms. -(** * [PeanoNat.Nat] already implements [NAxiomSig] *) +(** This file is DEPRECATED ! Use [PeanoNat] (or [Arith]) instead. *) + +(** [PeanoNat.Nat] already implements [NAxiomSig] *) Module Nat <: NAxiomsSig := Nat. + +(** Compat notations for stuff that used to be at the beginning of NPeano. *) + +Notation leb := Nat.leb (compat "8.4"). +Notation ltb := Nat.ltb (compat "8.4"). +Notation leb_le := Nat.leb_le (compat "8.4"). +Notation ltb_lt := Nat.ltb_lt (compat "8.4"). +Notation pow := Nat.pow (compat "8.4"). +Notation pow_0_r := Nat.pow_0_r (compat "8.4"). +Notation pow_succ_r := Nat.pow_succ_r (compat "8.4"). +Notation square := Nat.square (compat "8.4"). +Notation square_spec := Nat.square_spec (compat "8.4"). +Notation Even := Nat.Even (compat "8.4"). +Notation Odd := Nat.Odd (compat "8.4"). +Notation even := Nat.even (compat "8.4"). +Notation odd := Nat.odd (compat "8.4"). +Notation even_spec := Nat.even_spec (compat "8.4"). +Notation odd_spec := Nat.odd_spec (compat "8.4"). + +Lemma Even_equiv n : Even n <-> Even.even n. +Proof. symmetry. apply Even.even_equiv. Qed. +Lemma Odd_equiv n : Odd n <-> Even.odd n. +Proof. symmetry. apply Even.odd_equiv. Qed. + +Notation divmod := Nat.divmod (compat "8.4"). +Notation div := Nat.div (compat "8.4"). +Notation modulo := Nat.modulo (compat "8.4"). +Notation divmod_spec := Nat.divmod_spec (compat "8.4"). +Notation div_mod := Nat.div_mod (compat "8.4"). +Notation mod_bound_pos := Nat.mod_bound_pos (compat "8.4"). +Notation sqrt_iter := Nat.sqrt_iter (compat "8.4"). +Notation sqrt := Nat.sqrt (compat "8.4"). +Notation sqrt_iter_spec := Nat.sqrt_iter_spec (compat "8.4"). +Notation sqrt_spec := Nat.sqrt_spec (compat "8.4"). +Notation log2_iter := Nat.log2_iter (compat "8.4"). +Notation log2 := Nat.log2 (compat "8.4"). +Notation log2_iter_spec := Nat.log2_iter_spec (compat "8.4"). +Notation log2_spec := Nat.log2_spec (compat "8.4"). +Notation log2_nonpos := Nat.log2_nonpos (compat "8.4"). +Notation gcd := Nat.gcd (compat "8.4"). +Notation divide := Nat.divide (compat "8.4"). +Notation gcd_divide := Nat.gcd_divide (compat "8.4"). +Notation gcd_divide_l := Nat.gcd_divide_l (compat "8.4"). +Notation gcd_divide_r := Nat.gcd_divide_r (compat "8.4"). +Notation gcd_greatest := Nat.gcd_greatest (compat "8.4"). +Notation testbit := Nat.testbit (compat "8.4"). +Notation shiftl := Nat.shiftl (compat "8.4"). +Notation shiftr := Nat.shiftr (compat "8.4"). +Notation bitwise := Nat.bitwise (compat "8.4"). +Notation land := Nat.land (compat "8.4"). +Notation lor := Nat.lor (compat "8.4"). +Notation ldiff := Nat.ldiff (compat "8.4"). +Notation lxor := Nat.lxor (compat "8.4"). +Notation double_twice := Nat.double_twice (compat "8.4"). +Notation testbit_0_l := Nat.testbit_0_l (compat "8.4"). +Notation testbit_odd_0 := Nat.testbit_odd_0 (compat "8.4"). +Notation testbit_even_0 := Nat.testbit_even_0 (compat "8.4"). +Notation testbit_odd_succ := Nat.testbit_odd_succ (compat "8.4"). +Notation testbit_even_succ := Nat.testbit_even_succ (compat "8.4"). +Notation shiftr_spec := Nat.shiftr_spec (compat "8.4"). +Notation shiftl_spec_high := Nat.shiftl_spec_high (compat "8.4"). +Notation shiftl_spec_low := Nat.shiftl_spec_low (compat "8.4"). +Notation div2_bitwise := Nat.div2_bitwise (compat "8.4"). +Notation odd_bitwise := Nat.odd_bitwise (compat "8.4"). +Notation div2_decr := Nat.div2_decr (compat "8.4"). +Notation testbit_bitwise_1 := Nat.testbit_bitwise_1 (compat "8.4"). +Notation testbit_bitwise_2 := Nat.testbit_bitwise_2 (compat "8.4"). +Notation land_spec := Nat.land_spec (compat "8.4"). +Notation ldiff_spec := Nat.ldiff_spec (compat "8.4"). +Notation lor_spec := Nat.lor_spec (compat "8.4"). +Notation lxor_spec := Nat.lxor_spec (compat "8.4"). + +Infix "<=?" := Nat.leb (at level 70) : nat_scope. +Infix "<?" := Nat.ltb (at level 70) : nat_scope. +Infix "^" := Nat.pow : nat_scope. +Infix "/" := Nat.div : nat_scope. +Infix "mod" := Nat.modulo (at level 40, no associativity) : nat_scope. +Notation "( x | y )" := (Nat.divide x y) (at level 0) : nat_scope. diff --git a/theories/Numbers/Rational/SpecViaQ/QSig.v b/theories/Numbers/Rational/SpecViaQ/QSig.v index a40d940598..8e20fd0608 100644 --- a/theories/Numbers/Rational/SpecViaQ/QSig.v +++ b/theories/Numbers/Rational/SpecViaQ/QSig.v @@ -115,7 +115,10 @@ Ltac solve_wd2 := intros x x' Hx y y' Hy; qify; now rewrite Hx, Hy. Local Obligation Tactic := solve_wd2 || solve_wd1. Instance : Measure to_Q. -Instance eq_equiv : Equivalence eq := {}. +Instance eq_equiv : Equivalence eq. +Proof. + change eq with (RelCompFun Qeq to_Q); apply _. +Defined. Program Instance lt_wd : Proper (eq==>eq==>iff) lt. Program Instance le_wd : Proper (eq==>eq==>iff) le. @@ -141,7 +144,10 @@ Proof. intros. qify. destruct (Qcompare_spec [x] [y]); auto. Qed. (** Let's implement [TotalOrder] *) Definition lt_compat := lt_wd. -Instance lt_strorder : StrictOrder lt := {}. +Instance lt_strorder : StrictOrder lt. +Proof. + change lt with (RelCompFun Qlt to_Q); apply _. +Qed. Lemma le_lteq : forall x y, x<=y <-> x<y \/ x==y. Proof. intros. qify. apply Qle_lteq. Qed. diff --git a/theories/PArith/BinPos.v b/theories/PArith/BinPos.v index 0ccfad7b28..7baf102aaa 100644 --- a/theories/PArith/BinPos.v +++ b/theories/PArith/BinPos.v @@ -201,7 +201,6 @@ Proof. Qed. (** ** No neutral elements for addition *) - Lemma add_no_neutral p q : q + p <> p. Proof. revert q. @@ -508,7 +507,7 @@ Qed. Lemma mul_xO_discr p q : p~0 * q <> q. Proof. induction q; try discriminate. - rewrite mul_xO_r; injection; assumption. + rewrite mul_xO_r; injection; auto. Qed. (** ** Simplification properties of multiplication *) diff --git a/theories/Program/Equality.v b/theories/Program/Equality.v index 17f05c5113..d6f9bb9df0 100644 --- a/theories/Program/Equality.v +++ b/theories/Program/Equality.v @@ -8,7 +8,6 @@ (** Tactics related to (dependent) equality and proof irrelevance. *) -Require Export ProofIrrelevance. Require Export JMeq. Require Import Coq.Program.Tactics. @@ -143,7 +142,7 @@ Ltac pi_eq_proof_hyp p := | [ H : X = Y |- _ ] => match p with | H => fail 2 - | _ => rewrite (proof_irrelevance (X = Y) p H) + | _ => rewrite (UIP _ X Y p H) end | _ => fail " No hypothesis with same type " end @@ -166,7 +165,7 @@ Hint Rewrite <- eq_rect_eq : refl_id. [coerce_* t eq_refl = t]. *) Lemma JMeq_eq_refl {A} (x : A) : JMeq_eq (@JMeq_refl _ x) = eq_refl. -Proof. apply proof_irrelevance. Qed. +Proof. apply UIP. Qed. Lemma UIP_refl_refl A (x : A) : Eqdep.EqdepTheory.UIP_refl A x eq_refl = eq_refl. @@ -333,7 +332,7 @@ Ltac simplify_one_dep_elim_term c := (let hyp := fresh in intros hyp ; move hyp before y ; revert_until hyp ; generalize dependent y ; refine (solution_right _ _ _ _)(* ; intros until 0 *)) - | ?f ?x = ?g ?y -> _ => let H := fresh in progress (intros H ; injection H ; clear H) + | ?f ?x = ?g ?y -> _ => let H := fresh in progress (intros H ; simple injection H; clear H) | ?t = ?u -> _ => let hyp := fresh in intros hyp ; exfalso ; discriminate | ?x = ?y -> _ => let hyp := fresh in diff --git a/theories/Program/Subset.v b/theories/Program/Subset.v index c8f37318d1..2a3ec926b2 100644 --- a/theories/Program/Subset.v +++ b/theories/Program/Subset.v @@ -9,6 +9,7 @@ Require Import Coq.Program.Utils. Require Import Coq.Program.Equality. +Require Export ProofIrrelevance. Local Open Scope program_scope. diff --git a/theories/Program/Tactics.v b/theories/Program/Tactics.v index 7384790dae..dfd6b0eae0 100644 --- a/theories/Program/Tactics.v +++ b/theories/Program/Tactics.v @@ -264,7 +264,7 @@ Ltac bang := match goal with | |- ?x => match x with - | appcontext [False_rect _ ?p] => elim p + | context [False_rect _ ?p] => elim p end end. diff --git a/theories/Program/Wf.v b/theories/Program/Wf.v index a2fd05cd96..c490ea5166 100644 --- a/theories/Program/Wf.v +++ b/theories/Program/Wf.v @@ -211,7 +211,7 @@ Ltac fold_sub f := match goal with | [ |- ?T ] => match T with - appcontext C [ @Fix_sub _ _ _ _ _ ?arg ] => + context C [ @Fix_sub _ _ _ _ _ ?arg ] => let app := context C [ f arg ] in change app end diff --git a/theories/QArith/Qcanon.v b/theories/QArith/Qcanon.v index 6bfa47bc55..9f9651d84a 100644 --- a/theories/QArith/Qcanon.v +++ b/theories/QArith/Qcanon.v @@ -84,7 +84,7 @@ Arguments Q2Qc q%Q. Lemma Q2Qc_eq_iff (q q' : Q) : Q2Qc q = Q2Qc q' <-> q == q'. Proof. split; intro H. - - injection H. apply Qred_eq_iff. + - now injection H as H%Qred_eq_iff. - apply Qc_is_canon. simpl. now rewrite H. Qed. @@ -266,7 +266,7 @@ Theorem Qcmult_integral : forall x y, x*y=0 -> x=0 \/ y=0. Proof. intros. destruct (Qmult_integral x y); try qc; auto. - injection H; clear H; intros. + injection H as H. rewrite <- (Qred_correct (x*y)). rewrite <- (Qred_correct 0). rewrite H; auto with qarith. diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v index f26bac2bb4..379fee6f49 100644 --- a/theories/Reals/RIneq.v +++ b/theories/Reals/RIneq.v @@ -389,7 +389,7 @@ Lemma Rplus_ne : forall r, r + 0 = r /\ 0 + r = r. Proof. split; ring. Qed. -Hint Resolve Rplus_ne: real v62. +Hint Resolve Rplus_ne: real. (**********) @@ -425,7 +425,6 @@ Proof. apply (f_equal (fun v => v + r)). Qed. -(*i Old i*)Hint Resolve Rplus_eq_compat_l: v62. (**********) Lemma Rplus_eq_reg_l : forall r r1 r2, r + r1 = r + r2 -> r1 = r2. @@ -501,21 +500,21 @@ Lemma Rmult_0_r : forall r, r * 0 = 0. Proof. intro; ring. Qed. -Hint Resolve Rmult_0_r: real v62. +Hint Resolve Rmult_0_r: real. (**********) Lemma Rmult_0_l : forall r, 0 * r = 0. Proof. intro; ring. Qed. -Hint Resolve Rmult_0_l: real v62. +Hint Resolve Rmult_0_l: real. (**********) Lemma Rmult_ne : forall r, r * 1 = r /\ 1 * r = r. Proof. intro; split; ring. Qed. -Hint Resolve Rmult_ne: real v62. +Hint Resolve Rmult_ne: real. (**********) Lemma Rmult_1_r : forall r, r * 1 = r. @@ -530,7 +529,6 @@ Proof. auto with real. Qed. -(*i Old i*)Hint Resolve Rmult_eq_compat_l: v62. Lemma Rmult_eq_compat_r : forall r r1 r2, r1 = r2 -> r1 * r = r2 * r. Proof. @@ -646,7 +644,7 @@ Lemma Ropp_0 : -0 = 0. Proof. ring. Qed. -Hint Resolve Ropp_0: real v62. +Hint Resolve Ropp_0: real. (**********) Lemma Ropp_eq_0_compat : forall r, r = 0 -> - r = 0. diff --git a/theories/Reals/Ranalysis_reg.v b/theories/Reals/Ranalysis_reg.v index e57af7311f..0c27d407c3 100644 --- a/theories/Reals/Ranalysis_reg.v +++ b/theories/Reals/Ranalysis_reg.v @@ -109,6 +109,7 @@ Ltac intro_hyp_glob trm := | Rabs => idtac | ?X1 => let p := constr:(X1) in + let HYPPD := fresh "HYPPD" in match goal with | _:(derivable p) |- _ => idtac | |- (derivable p) => idtac @@ -250,6 +251,7 @@ Ltac intro_hyp_pt trm pt := end | ?X1 => let p := constr:(X1) in + let HYPPD := fresh "HYPPD" in match goal with | _:(derivable_pt p pt) |- _ => idtac | |- (derivable_pt p pt) => idtac @@ -341,8 +343,10 @@ Ltac is_diff_pt := | _:(derivable_pt ?X1 ?X2) |- (derivable_pt ?X1 ?X2) => assumption | _:(derivable ?X1) |- (derivable_pt ?X1 ?X2) => + let HypDDPT := fresh "HypDDPT" in cut (derivable X1); [ intro HypDDPT; apply HypDDPT | assumption ] | |- (True -> derivable_pt _ _) => + let HypTruE := fresh "HypTruE" in intro HypTruE; clear HypTruE; is_diff_pt | _ => try @@ -411,6 +415,7 @@ Ltac is_diff_glob := apply (derivable_comp X2 X1); is_diff_glob | _:(derivable ?X1) |- (derivable ?X1) => assumption | |- (True -> derivable _) => + let HypTruE := fresh "HypTruE" in intro HypTruE; clear HypTruE; is_diff_glob | _ => try @@ -490,14 +495,17 @@ Ltac is_cont_pt := | _:(continuity_pt ?X1 ?X2) |- (continuity_pt ?X1 ?X2) => assumption | _:(continuity ?X1) |- (continuity_pt ?X1 ?X2) => + let HypDDPT := fresh "HypDDPT" in cut (continuity X1); [ intro HypDDPT; apply HypDDPT | assumption ] | _:(derivable_pt ?X1 ?X2) |- (continuity_pt ?X1 ?X2) => apply derivable_continuous_pt; assumption | _:(derivable ?X1) |- (continuity_pt ?X1 ?X2) => + let HypDDPT := fresh "HypDDPT" in cut (continuity X1); [ intro HypDDPT; apply HypDDPT | apply derivable_continuous; assumption ] | |- (True -> continuity_pt _ _) => + let HypTruE := fresh "HypTruE" in intro HypTruE; clear HypTruE; is_cont_pt | _ => try @@ -567,6 +575,7 @@ Ltac is_cont_glob := apply (continuity_comp X2 X1); try is_cont_glob || assumption | _:(continuity ?X1) |- (continuity ?X1) => assumption | |- (True -> continuity _) => + let HypTruE := fresh "HypTruE" in intro HypTruE; clear HypTruE; is_cont_glob | _:(derivable ?X1) |- (continuity ?X1) => apply derivable_continuous; assumption diff --git a/theories/Reals/Raxioms.v b/theories/Reals/Raxioms.v index 9d55e4e639..9fbda92a2f 100644 --- a/theories/Reals/Raxioms.v +++ b/theories/Reals/Raxioms.v @@ -32,7 +32,7 @@ Hint Resolve Rplus_assoc: real. (**********) Axiom Rplus_opp_r : forall r:R, r + - r = 0. -Hint Resolve Rplus_opp_r: real v62. +Hint Resolve Rplus_opp_r: real. (**********) Axiom Rplus_0_l : forall r:R, 0 + r = r. @@ -44,11 +44,11 @@ Hint Resolve Rplus_0_l: real. (**********) Axiom Rmult_comm : forall r1 r2:R, r1 * r2 = r2 * r1. -Hint Resolve Rmult_comm: real v62. +Hint Resolve Rmult_comm: real. (**********) Axiom Rmult_assoc : forall r1 r2 r3:R, r1 * r2 * r3 = r1 * (r2 * r3). -Hint Resolve Rmult_assoc: real v62. +Hint Resolve Rmult_assoc: real. (**********) Axiom Rinv_l : forall r:R, r <> 0 -> / r * r = 1. @@ -69,7 +69,7 @@ Hint Resolve R1_neq_R0: real. (**********) Axiom Rmult_plus_distr_l : forall r1 r2 r3:R, r1 * (r2 + r3) = r1 * r2 + r1 * r3. -Hint Resolve Rmult_plus_distr_l: real v62. +Hint Resolve Rmult_plus_distr_l: real. (*********************************************************) (** * Order axioms *) diff --git a/theories/Reals/Sqrt_reg.v b/theories/Reals/Sqrt_reg.v index 10527442e8..d43baee8cd 100644 --- a/theories/Reals/Sqrt_reg.v +++ b/theories/Reals/Sqrt_reg.v @@ -339,7 +339,7 @@ Proof. rewrite <- H1; rewrite sqrt_0; unfold Rminus; rewrite Ropp_0; rewrite Rplus_0_r; rewrite <- H1 in H5; unfold Rminus in H5; rewrite Ropp_0 in H5; rewrite Rplus_0_r in H5. - destruct (Rcase_abs x0) as [Hlt|Hgt]_eqn:Heqs. + destruct (Rcase_abs x0) as [Hlt|Hgt] eqn:Heqs. unfold sqrt. rewrite Heqs. rewrite Rabs_R0; apply H2. rewrite Rabs_right. diff --git a/theories/Relations/Operators_Properties.v b/theories/Relations/Operators_Properties.v index 220cebeace..fe8a96acc7 100644 --- a/theories/Relations/Operators_Properties.v +++ b/theories/Relations/Operators_Properties.v @@ -36,7 +36,7 @@ Section Properties. Section Clos_Refl_Trans. Local Notation "R *" := (clos_refl_trans R) - (at level 8, left associativity, format "R *"). + (at level 8, no associativity, format "R *"). (** Correctness of the reflexive-transitive closure operator *) diff --git a/theories/Relations/Relation_Definitions.v b/theories/Relations/Relation_Definitions.v index b6005b9d11..9c98879cea 100644 --- a/theories/Relations/Relation_Definitions.v +++ b/theories/Relations/Relation_Definitions.v @@ -66,10 +66,10 @@ Section Relation_Definition. End Relation_Definition. -Hint Unfold reflexive transitive antisymmetric symmetric: sets v62. +Hint Unfold reflexive transitive antisymmetric symmetric: sets. Hint Resolve Build_preorder Build_order Build_equivalence Build_PER preord_refl preord_trans ord_refl ord_trans ord_antisym equiv_refl - equiv_trans equiv_sym per_sym per_trans: sets v62. + equiv_trans equiv_sym per_sym per_trans: sets. -Hint Unfold inclusion same_relation commut: sets v62. +Hint Unfold inclusion same_relation commut: sets. diff --git a/theories/Relations/Relation_Operators.v b/theories/Relations/Relation_Operators.v index ffd682d62c..88239475c3 100644 --- a/theories/Relations/Relation_Operators.v +++ b/theories/Relations/Relation_Operators.v @@ -226,9 +226,9 @@ Section Lexicographic_Exponentiation. End Lexicographic_Exponentiation. -Hint Unfold transp union: sets v62. -Hint Resolve t_step rt_step rt_refl rst_step rst_refl: sets v62. -Hint Immediate rst_sym: sets v62. +Hint Unfold transp union: sets. +Hint Resolve t_step rt_step rt_refl rst_step rst_refl: sets. +Hint Immediate rst_sym: sets. (* begin hide *) (* Compatibility *) diff --git a/theories/Sets/Classical_sets.v b/theories/Sets/Classical_sets.v index 8a4bb9f424..837437a227 100644 --- a/theories/Sets/Classical_sets.v +++ b/theories/Sets/Classical_sets.v @@ -122,4 +122,4 @@ Section Ensembles_classical. End Ensembles_classical. Hint Resolve Strict_super_set_contains_new_element Subtract_intro - not_SIncl_empty: sets v62. + not_SIncl_empty: sets. diff --git a/theories/Sets/Constructive_sets.v b/theories/Sets/Constructive_sets.v index 8d2344f931..6291248eb5 100644 --- a/theories/Sets/Constructive_sets.v +++ b/theories/Sets/Constructive_sets.v @@ -141,4 +141,4 @@ End Ensembles_facts. Hint Resolve Singleton_inv Singleton_intro Add_intro1 Add_intro2 Intersection_inv Couple_inv Setminus_intro Strict_Included_intro Strict_Included_strict Noone_in_empty Inhabited_not_empty Add_not_Empty - not_Empty_Add Inhabited_add Included_Empty: sets v62. + not_Empty_Add Inhabited_add Included_Empty: sets. diff --git a/theories/Sets/Ensembles.v b/theories/Sets/Ensembles.v index 8f579214ad..0fefb354b2 100644 --- a/theories/Sets/Ensembles.v +++ b/theories/Sets/Ensembles.v @@ -90,9 +90,8 @@ Section Ensembles. End Ensembles. -Hint Unfold In Included Same_set Strict_Included Add Setminus Subtract: sets - v62. +Hint Unfold In Included Same_set Strict_Included Add Setminus Subtract: sets. Hint Resolve Union_introl Union_intror Intersection_intro In_singleton Couple_l Couple_r Triple_l Triple_m Triple_r Disjoint_intro - Extensionality_Ensembles: sets v62. + Extensionality_Ensembles: sets. diff --git a/theories/Sets/Finite_sets.v b/theories/Sets/Finite_sets.v index f38dd6fdf7..edbc1efec0 100644 --- a/theories/Sets/Finite_sets.v +++ b/theories/Sets/Finite_sets.v @@ -43,8 +43,8 @@ Section Ensembles_finis. End Ensembles_finis. -Hint Resolve Empty_is_finite Union_is_finite: sets v62. -Hint Resolve card_empty card_add: sets v62. +Hint Resolve Empty_is_finite Union_is_finite: sets. +Hint Resolve card_empty card_add: sets. Require Import Constructive_sets. diff --git a/theories/Sets/Image.v b/theories/Sets/Image.v index 34ea857d17..e74ef41e4b 100644 --- a/theories/Sets/Image.v +++ b/theories/Sets/Image.v @@ -200,4 +200,4 @@ Section Image. End Image. -Hint Resolve Im_def image_empty finite_image: sets v62. +Hint Resolve Im_def image_empty finite_image: sets. diff --git a/theories/Sets/Multiset.v b/theories/Sets/Multiset.v index ec38b8923f..42d0c76dcc 100644 --- a/theories/Sets/Multiset.v +++ b/theories/Sets/Multiset.v @@ -187,7 +187,7 @@ End multiset_defs. Unset Implicit Arguments. -Hint Unfold meq multiplicity: v62 datatypes. +Hint Unfold meq multiplicity: datatypes. Hint Resolve munion_empty_right munion_comm munion_ass meq_left meq_right - munion_empty_left: v62 datatypes. -Hint Immediate meq_sym: v62 datatypes. + munion_empty_left: datatypes. +Hint Immediate meq_sym: datatypes. diff --git a/theories/Sets/Partial_Order.v b/theories/Sets/Partial_Order.v index 3610ebce6e..335fec5b09 100644 --- a/theories/Sets/Partial_Order.v +++ b/theories/Sets/Partial_Order.v @@ -51,8 +51,8 @@ Section Partial_orders. End Partial_orders. -Hint Unfold Carrier_of Rel_of Strict_Rel_of: sets v62. -Hint Resolve Definition_of_covers: sets v62. +Hint Unfold Carrier_of Rel_of Strict_Rel_of: sets. +Hint Resolve Definition_of_covers: sets. Section Partial_order_facts. diff --git a/theories/Sets/Powerset.v b/theories/Sets/Powerset.v index d636e0468a..7c2435da04 100644 --- a/theories/Sets/Powerset.v +++ b/theories/Sets/Powerset.v @@ -175,14 +175,14 @@ Qed. End The_power_set_partial_order. -Hint Resolve Empty_set_minimal: sets v62. -Hint Resolve Power_set_Inhabited: sets v62. -Hint Resolve Inclusion_is_an_order: sets v62. -Hint Resolve Inclusion_is_transitive: sets v62. -Hint Resolve Union_minimal: sets v62. -Hint Resolve Union_increases_l: sets v62. -Hint Resolve Union_increases_r: sets v62. -Hint Resolve Intersection_decreases_l: sets v62. -Hint Resolve Intersection_decreases_r: sets v62. -Hint Resolve Empty_set_is_Bottom: sets v62. -Hint Resolve Strict_inclusion_is_transitive: sets v62. +Hint Resolve Empty_set_minimal: sets. +Hint Resolve Power_set_Inhabited: sets. +Hint Resolve Inclusion_is_an_order: sets. +Hint Resolve Inclusion_is_transitive: sets. +Hint Resolve Union_minimal: sets. +Hint Resolve Union_increases_l: sets. +Hint Resolve Union_increases_r: sets. +Hint Resolve Intersection_decreases_l: sets. +Hint Resolve Intersection_decreases_r: sets. +Hint Resolve Empty_set_is_Bottom: sets. +Hint Resolve Strict_inclusion_is_transitive: sets. diff --git a/theories/Sets/Powerset_Classical_facts.v b/theories/Sets/Powerset_Classical_facts.v index 09c90506bc..e802beac9a 100644 --- a/theories/Sets/Powerset_Classical_facts.v +++ b/theories/Sets/Powerset_Classical_facts.v @@ -90,7 +90,7 @@ Section Sets_as_an_algebra. apply Subtract_intro; auto with sets. red; intro H'1; apply H'; rewrite H'1; auto with sets. Qed. - Hint Resolve incl_soustr_add_r: sets v62. + Hint Resolve incl_soustr_add_r: sets. Lemma add_soustr_2 : forall (X:Ensemble U) (x:U), @@ -328,9 +328,9 @@ Section Sets_as_an_algebra. End Sets_as_an_algebra. -Hint Resolve incl_soustr_in: sets v62. -Hint Resolve incl_soustr: sets v62. -Hint Resolve incl_soustr_add_l: sets v62. -Hint Resolve incl_soustr_add_r: sets v62. -Hint Resolve add_soustr_1 add_soustr_2: sets v62. -Hint Resolve add_soustr_xy: sets v62. +Hint Resolve incl_soustr_in: sets. +Hint Resolve incl_soustr: sets. +Hint Resolve incl_soustr_add_l: sets. +Hint Resolve incl_soustr_add_r: sets. +Hint Resolve add_soustr_1 add_soustr_2: sets. +Hint Resolve add_soustr_xy: sets. diff --git a/theories/Sets/Powerset_facts.v b/theories/Sets/Powerset_facts.v index 63e84199d6..e9696a1cad 100644 --- a/theories/Sets/Powerset_facts.v +++ b/theories/Sets/Powerset_facts.v @@ -254,5 +254,5 @@ Section Sets_as_an_algebra. End Sets_as_an_algebra. Hint Resolve Empty_set_zero Empty_set_zero' Union_associative Union_add - singlx incl_add: sets v62. + singlx incl_add: sets. diff --git a/theories/Sets/Relations_1.v b/theories/Sets/Relations_1.v index de96fa560e..45fb8134c8 100644 --- a/theories/Sets/Relations_1.v +++ b/theories/Sets/Relations_1.v @@ -60,6 +60,6 @@ Section Relations_1. End Relations_1. Hint Unfold Reflexive Transitive Antisymmetric Symmetric contains - same_relation: sets v62. + same_relation: sets. Hint Resolve Definition_of_preorder Definition_of_order - Definition_of_equivalence Definition_of_PER: sets v62. + Definition_of_equivalence Definition_of_PER: sets. diff --git a/theories/Sets/Relations_2.v b/theories/Sets/Relations_2.v index f1026e31a1..1e0b83fe51 100644 --- a/theories/Sets/Relations_2.v +++ b/theories/Sets/Relations_2.v @@ -48,7 +48,7 @@ Definition Strongly_confluent : Prop := End Relations_2. -Hint Resolve Rstar_0: sets v62. -Hint Resolve Rstar1_0: sets v62. -Hint Resolve Rstar1_1: sets v62. -Hint Resolve Rplus_0: sets v62. +Hint Resolve Rstar_0: sets. +Hint Resolve Rstar1_0: sets. +Hint Resolve Rstar1_1: sets. +Hint Resolve Rplus_0: sets. diff --git a/theories/Sets/Relations_3.v b/theories/Sets/Relations_3.v index 92b2998855..c05b5ee76a 100644 --- a/theories/Sets/Relations_3.v +++ b/theories/Sets/Relations_3.v @@ -51,10 +51,10 @@ Section Relations_3. Definition Noetherian : Prop := forall x:U, noetherian x. End Relations_3. -Hint Unfold coherent: sets v62. -Hint Unfold locally_confluent: sets v62. -Hint Unfold confluent: sets v62. -Hint Unfold Confluent: sets v62. -Hint Resolve definition_of_noetherian: sets v62. -Hint Unfold Noetherian: sets v62. +Hint Unfold coherent: sets. +Hint Unfold locally_confluent: sets. +Hint Unfold confluent: sets. +Hint Unfold Confluent: sets. +Hint Resolve definition_of_noetherian: sets. +Hint Unfold Noetherian: sets. diff --git a/theories/Sorting/Permutation.v b/theories/Sorting/Permutation.v index 8470b79550..44f8ff6a71 100644 --- a/theories/Sorting/Permutation.v +++ b/theories/Sorting/Permutation.v @@ -318,7 +318,7 @@ Lemma Permutation_length_2_inv : Proof. intros a1 a2 l H; remember [a1;a2] as m in H. revert a1 a2 Heqm. - induction H; intros; try (injection Heqm; intros; subst; clear Heqm); + induction H; intros; try (injection Heqm as ? ?; subst); discriminate || (try tauto). apply Permutation_length_1_inv in H as ->; left; auto. apply IHPermutation1 in Heqm as [H1|H1]; apply IHPermutation2 in H1 as []; diff --git a/theories/Strings/Ascii.v b/theories/Strings/Ascii.v index 97cb746f37..55a533c55a 100644 --- a/theories/Strings/Ascii.v +++ b/theories/Strings/Ascii.v @@ -40,7 +40,7 @@ Defined. (** * Conversion between natural numbers modulo 256 and ascii characters *) -(** Auxillary function that turns a positive into an ascii by +(** Auxiliary function that turns a positive into an ascii by looking at the last 8 bits, ie z mod 2^8 *) Definition ascii_of_pos : positive -> ascii := diff --git a/theories/Strings/String.v b/theories/Strings/String.v index 943bb48e92..451b65cbe9 100644 --- a/theories/Strings/String.v +++ b/theories/Strings/String.v @@ -83,7 +83,7 @@ intros H; generalize (H 0); simpl; intros H1; inversion H1. case (Rec s). intros H0; rewrite H0; auto. intros n; exact (H (S n)). -intros H; injection H; intros H1 H2 n; case n; auto. +intros H; injection H as H1 H2. rewrite H2; trivial. rewrite H1; auto. Qed. @@ -238,14 +238,14 @@ intros n m s1 s2; generalize n m s1; clear n m s1; elim s2; simpl; auto. intros n; case n; simpl; auto. intros m s1; case s1; simpl; auto. -intros H; injection H; intros H1; rewrite <- H1; auto. +intros H; injection H as <-; auto. intros; discriminate. intros; discriminate. intros b s2' Rec n m s1. case n; simpl; auto. generalize (prefix_correct s1 (String b s2')); case (prefix s1 (String b s2')). -intros H0 H; injection H; intros H1; rewrite <- H1; auto. +intros H0 H; injection H as <-; auto. case H0; simpl; auto. case m; simpl; auto. case (index 0 s1 s2'); intros; discriminate. @@ -271,7 +271,7 @@ intros n m s1 s2; generalize n m s1; clear n m s1; elim s2; simpl; auto. intros n; case n; simpl; auto. intros m s1; case s1; simpl; auto. -intros H; injection H; intros H1; rewrite <- H1. +intros H; injection H as <-. intros p H0 H2; inversion H2. intros; discriminate. intros; discriminate. @@ -279,7 +279,7 @@ intros b s2' Rec n m s1. case n; simpl; auto. generalize (prefix_correct s1 (String b s2')); case (prefix s1 (String b s2')). -intros H0 H; injection H; intros H1; rewrite <- H1; auto. +intros H0 H; injection H as <-; auto. intros p H2 H3; inversion H3. case m; simpl; auto. case (index 0 s1 s2'); intros; discriminate. diff --git a/theories/Structures/OrdersFacts.v b/theories/Structures/OrdersFacts.v index 954d3df203..0115d8a544 100644 --- a/theories/Structures/OrdersFacts.v +++ b/theories/Structures/OrdersFacts.v @@ -434,21 +434,21 @@ Lemma eqb_compare x y : (x =? y) = match compare x y with Eq => true | _ => false end. Proof. apply eq_true_iff_eq. rewrite eqb_eq, <- compare_eq_iff. -destruct compare; now split. +now destruct compare. Qed. Lemma ltb_compare x y : (x <? y) = match compare x y with Lt => true | _ => false end. Proof. apply eq_true_iff_eq. rewrite ltb_lt, <- compare_lt_iff. -destruct compare; now split. +now destruct compare. Qed. Lemma leb_compare x y : (x <=? y) = match compare x y with Gt => false | _ => true end. Proof. apply eq_true_iff_eq. rewrite leb_le, <- compare_le_iff. -destruct compare; split; try easy. now destruct 1. +now destruct compare. Qed. End BoolOrderFacts. diff --git a/theories/Vectors/VectorDef.v b/theories/Vectors/VectorDef.v index c692238041..1f8b76cb62 100644 --- a/theories/Vectors/VectorDef.v +++ b/theories/Vectors/VectorDef.v @@ -30,7 +30,7 @@ Inductive t A : nat -> Type := |nil : t A 0 |cons : forall (h:A) (n:nat), t A n -> t A (S n). -Local Notation "[]" := (nil _). +Local Notation "[ ]" := (nil _) (format "[ ]"). Local Notation "h :: t" := (cons _ h _ t) (at level 60, right associativity). Section SCHEMES. @@ -102,7 +102,7 @@ Definition const {A} (a:A) := nat_rect _ [] (fun n x => cons _ a n x). Computational behavior of this function should be the same as ocaml function. *) -Definition nth {A} := +Definition nth {A} := fix nth_fix {m} (v' : t A m) (p : Fin.t m) {struct v'} : A := match p in Fin.t m' return t A m' -> A with |Fin.F1 => caseS (fun n v' => A) (fun h n t => h) @@ -293,12 +293,14 @@ Eval cbv delta beta in fold_right (fun h H => Datatypes.cons h H) v Datatypes.ni End VECTORLIST. Module VectorNotations. -Notation "[]" := [] : vector_scope. +Delimit Scope vector_scope with vector. +Notation "[ ]" := [] (format "[ ]") : vector_scope. +Notation "[]" := [] (compat "8.5") : vector_scope. Notation "h :: t" := (h :: t) (at level 60, right associativity) : vector_scope. -Notation " [ x ] " := (x :: []) : vector_scope. -Notation " [ x ; y ; .. ; z ] " := (cons _ x _ (cons _ y _ .. (cons _ z _ (nil _)) ..)) : vector_scope -. +Notation "[ x ]" := (x :: []) : vector_scope. +Notation "[ x ; y ; .. ; z ]" := (cons _ x _ (cons _ y _ .. (cons _ z _ (nil _)) ..)) : vector_scope. +Notation "[ x ; .. ; y ]" := (cons _ x _ .. (cons _ y _ (nil _)) ..) (compat "8.4") : vector_scope. Notation "v [@ p ]" := (nth v p) (at level 1, format "v [@ p ]") : vector_scope. Open Scope vector_scope. End VectorNotations. diff --git a/theories/Wellfounded/Lexicographic_Product.v b/theories/Wellfounded/Lexicographic_Product.v index 4b8447f496..b2ada2f919 100644 --- a/theories/Wellfounded/Lexicographic_Product.v +++ b/theories/Wellfounded/Lexicographic_Product.v @@ -44,14 +44,11 @@ Section WfLexicographic_Product. apply H2. auto with sets. - injection H1. - destruct 2. - injection H3. - destruct 2; auto with sets. + injection H1 as <- _. + injection H3 as <- _; auto with sets. rewrite <- H1. - injection H3; intros _ Hx1. - subst x1. + injection H3 as -> H3. apply IHAcc0. elim inj_pair2 with A B x y' x0; assumption. Defined. diff --git a/theories/ZArith/Zwf.v b/theories/ZArith/Zwf.v index 1ac00bddd0..90754af3b4 100644 --- a/theories/ZArith/Zwf.v +++ b/theories/ZArith/Zwf.v @@ -56,7 +56,7 @@ Section wf_proof. End wf_proof. -Hint Resolve Zwf_well_founded: datatypes v62. +Hint Resolve Zwf_well_founded: datatypes. (** We also define the other family of relations: @@ -88,4 +88,4 @@ Section wf_proof_up. End wf_proof_up. -Hint Resolve Zwf_up_well_founded: datatypes v62. +Hint Resolve Zwf_up_well_founded: datatypes. diff --git a/theories/theories.itarget b/theories/theories.itarget deleted file mode 100644 index aacab2d978..0000000000 --- a/theories/theories.itarget +++ /dev/null @@ -1,25 +0,0 @@ -Arith/vo.otarget -Bool/vo.otarget -Classes/vo.otarget -Compat/vo.otarget -FSets/vo.otarget -MSets/vo.otarget -Structures/vo.otarget -Init/vo.otarget -Lists/vo.otarget -Vectors/vo.otarget -Logic/vo.otarget -PArith/vo.otarget -NArith/vo.otarget -Numbers/vo.otarget -Program/vo.otarget -QArith/vo.otarget -Reals/vo.otarget -Relations/vo.otarget -Setoids/vo.otarget -Sets/vo.otarget -Sorting/vo.otarget -Strings/vo.otarget -Unicode/vo.otarget -Wellfounded/vo.otarget -ZArith/vo.otarget |