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diff --git a/theories/Structures/GenericMinMax.v b/theories/Structures/GenericMinMax.v new file mode 100644 index 00000000..68f20189 --- /dev/null +++ b/theories/Structures/GenericMinMax.v @@ -0,0 +1,656 @@ +(***********************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * INRIA-Rocquencourt & LRI-CNRS-Orsay *) +(* \VV/ *************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(***********************************************************************) + +Require Import Orders OrdersTac OrdersFacts Setoid Morphisms Basics. + +(** * A Generic construction of min and max *) + +(** ** First, an interface for types with [max] and/or [min] *) + +Module Type HasMax (Import E:EqLe'). + Parameter Inline max : t -> t -> t. + Parameter max_l : forall x y, y<=x -> max x y == x. + Parameter max_r : forall x y, x<=y -> max x y == y. +End HasMax. + +Module Type HasMin (Import E:EqLe'). + Parameter Inline min : t -> t -> t. + Parameter min_l : forall x y, x<=y -> min x y == x. + Parameter min_r : forall x y, y<=x -> min x y == y. +End HasMin. + +Module Type HasMinMax (E:EqLe) := HasMax E <+ HasMin E. + + +(** ** Any [OrderedTypeFull] can be equipped by [max] and [min] + based on the compare function. *) + +Definition gmax {A} (cmp : A->A->comparison) x y := + match cmp x y with Lt => y | _ => x end. +Definition gmin {A} (cmp : A->A->comparison) x y := + match cmp x y with Gt => y | _ => x end. + +Module GenericMinMax (Import O:OrderedTypeFull') <: HasMinMax O. + + Definition max := gmax O.compare. + Definition min := gmin O.compare. + + Lemma ge_not_lt : forall x y, y<=x -> x<y -> False. + Proof. + intros x y H H'. + apply (StrictOrder_Irreflexive x). + rewrite le_lteq in *; destruct H as [H|H]. + transitivity y; auto. + rewrite H in H'; auto. + Qed. + + Lemma max_l : forall x y, y<=x -> max x y == x. + Proof. + intros. unfold max, gmax. case compare_spec; auto with relations. + intros; elim (ge_not_lt x y); auto. + Qed. + + Lemma max_r : forall x y, x<=y -> max x y == y. + Proof. + intros. unfold max, gmax. case compare_spec; auto with relations. + intros; elim (ge_not_lt y x); auto. + Qed. + + Lemma min_l : forall x y, x<=y -> min x y == x. + Proof. + intros. unfold min, gmin. case compare_spec; auto with relations. + intros; elim (ge_not_lt y x); auto. + Qed. + + Lemma min_r : forall x y, y<=x -> min x y == y. + Proof. + intros. unfold min, gmin. case compare_spec; auto with relations. + intros; elim (ge_not_lt x y); auto. + Qed. + +End GenericMinMax. + + +(** ** Consequences of the minimalist interface: facts about [max]. *) + +Module MaxLogicalProperties (Import O:TotalOrder')(Import M:HasMax O). + Module Import T := !MakeOrderTac O. + +(** An alternative caracterisation of [max], equivalent to + [max_l /\ max_r] *) + +Lemma max_spec : forall n m, + (n < m /\ max n m == m) \/ (m <= n /\ max n m == n). +Proof. + intros n m. + destruct (lt_total n m); [left|right]. + split; auto. apply max_r. rewrite le_lteq; auto. + assert (m <= n) by (rewrite le_lteq; intuition). + split; auto. apply max_l; auto. +Qed. + +(** A more symmetric version of [max_spec], based only on [le]. + Beware that left and right alternatives overlap. *) + +Lemma max_spec_le : forall n m, + (n <= m /\ max n m == m) \/ (m <= n /\ max n m == n). +Proof. + intros. destruct (max_spec n m); [left|right]; intuition; order. +Qed. + +Instance : Proper (eq==>eq==>iff) le. +Proof. repeat red. intuition order. Qed. + +Instance max_compat : Proper (eq==>eq==>eq) max. +Proof. +intros x x' Hx y y' Hy. +assert (H1 := max_spec x y). assert (H2 := max_spec x' y'). +set (m := max x y) in *; set (m' := max x' y') in *; clearbody m m'. +rewrite <- Hx, <- Hy in *. +destruct (lt_total x y); intuition order. +Qed. + + +(** A function satisfying the same specification is equal to [max]. *) + +Lemma max_unicity : forall n m p, + ((n < m /\ p == m) \/ (m <= n /\ p == n)) -> p == max n m. +Proof. + intros. assert (Hm := max_spec n m). + destruct (lt_total n m); intuition; order. +Qed. + +Lemma max_unicity_ext : forall f, + (forall n m, (n < m /\ f n m == m) \/ (m <= n /\ f n m == n)) -> + (forall n m, f n m == max n m). +Proof. + intros. apply max_unicity; auto. +Qed. + +(** [max] commutes with monotone functions. *) + +Lemma max_mono: forall f, + (Proper (eq ==> eq) f) -> + (Proper (le ==> le) f) -> + forall x y, max (f x) (f y) == f (max x y). +Proof. + intros f Eqf Lef x y. + destruct (max_spec x y) as [(H,E)|(H,E)]; rewrite E; + destruct (max_spec (f x) (f y)) as [(H',E')|(H',E')]; auto. + assert (f x <= f y) by (apply Lef; order). order. + assert (f y <= f x) by (apply Lef; order). order. +Qed. + +(** *** Semi-lattice algebraic properties of [max] *) + +Lemma max_id : forall n, max n n == n. +Proof. + intros. destruct (max_spec n n); intuition. +Qed. + +Notation max_idempotent := max_id (only parsing). + +Lemma max_assoc : forall m n p, max m (max n p) == max (max m n) p. +Proof. + intros. + destruct (max_spec n p) as [(H,Eq)|(H,Eq)]; rewrite Eq. + destruct (max_spec m n) as [(H',Eq')|(H',Eq')]; rewrite Eq'. + destruct (max_spec m p); intuition; order. order. + destruct (max_spec m n) as [(H',Eq')|(H',Eq')]; rewrite Eq'. order. + destruct (max_spec m p); intuition; order. +Qed. + +Lemma max_comm : forall n m, max n m == max m n. +Proof. + intros. + destruct (max_spec n m) as [(H,Eq)|(H,Eq)]; rewrite Eq. + destruct (max_spec m n) as [(H',Eq')|(H',Eq')]; rewrite Eq'; order. + destruct (max_spec m n) as [(H',Eq')|(H',Eq')]; rewrite Eq'; order. +Qed. + +(** *** Least-upper bound properties of [max] *) + +Lemma le_max_l : forall n m, n <= max n m. +Proof. + intros; destruct (max_spec n m); intuition; order. +Qed. + +Lemma le_max_r : forall n m, m <= max n m. +Proof. + intros; destruct (max_spec n m); intuition; order. +Qed. + +Lemma max_l_iff : forall n m, max n m == n <-> m <= n. +Proof. + split. intro H; rewrite <- H. apply le_max_r. apply max_l. +Qed. + +Lemma max_r_iff : forall n m, max n m == m <-> n <= m. +Proof. + split. intro H; rewrite <- H. apply le_max_l. apply max_r. +Qed. + +Lemma max_le : forall n m p, p <= max n m -> p <= n \/ p <= m. +Proof. + intros n m p H; destruct (max_spec n m); + [right|left]; intuition; order. +Qed. + +Lemma max_le_iff : forall n m p, p <= max n m <-> p <= n \/ p <= m. +Proof. + intros. split. apply max_le. + destruct (max_spec n m); intuition; order. +Qed. + +Lemma max_lt_iff : forall n m p, p < max n m <-> p < n \/ p < m. +Proof. + intros. destruct (max_spec n m); intuition; + order || (right; order) || (left; order). +Qed. + +Lemma max_lub_l : forall n m p, max n m <= p -> n <= p. +Proof. + intros; destruct (max_spec n m); intuition; order. +Qed. + +Lemma max_lub_r : forall n m p, max n m <= p -> m <= p. +Proof. + intros; destruct (max_spec n m); intuition; order. +Qed. + +Lemma max_lub : forall n m p, n <= p -> m <= p -> max n m <= p. +Proof. + intros; destruct (max_spec n m); intuition; order. +Qed. + +Lemma max_lub_iff : forall n m p, max n m <= p <-> n <= p /\ m <= p. +Proof. + intros; destruct (max_spec n m); intuition; order. +Qed. + +Lemma max_lub_lt : forall n m p, n < p -> m < p -> max n m < p. +Proof. + intros; destruct (max_spec n m); intuition; order. +Qed. + +Lemma max_lub_lt_iff : forall n m p, max n m < p <-> n < p /\ m < p. +Proof. + intros; destruct (max_spec n m); intuition; order. +Qed. + +Lemma max_le_compat_l : forall n m p, n <= m -> max p n <= max p m. +Proof. + intros. + destruct (max_spec p n) as [(LT,E)|(LE,E)]; rewrite E. + assert (LE' := le_max_r p m). order. + apply le_max_l. +Qed. + +Lemma max_le_compat_r : forall n m p, n <= m -> max n p <= max m p. +Proof. + intros. rewrite (max_comm n p), (max_comm m p). + auto using max_le_compat_l. +Qed. + +Lemma max_le_compat : forall n m p q, n <= m -> p <= q -> + max n p <= max m q. +Proof. + intros n m p q Hnm Hpq. + assert (LE := max_le_compat_l _ _ m Hpq). + assert (LE' := max_le_compat_r _ _ p Hnm). + order. +Qed. + +End MaxLogicalProperties. + + +(** ** Properties concernant [min], then both [min] and [max]. + + To avoid too much code duplication, we exploit that [min] can be + seen as a [max] of the reversed order. +*) + +Module MinMaxLogicalProperties (Import O:TotalOrder')(Import M:HasMinMax O). + Include MaxLogicalProperties O M. + Import T. + + Module ORev := TotalOrderRev O. + Module MRev <: HasMax ORev. + Definition max x y := M.min y x. + Definition max_l x y := M.min_r y x. + Definition max_r x y := M.min_l y x. + End MRev. + Module MPRev := MaxLogicalProperties ORev MRev. + +Instance min_compat : Proper (eq==>eq==>eq) min. +Proof. intros x x' Hx y y' Hy. apply MPRev.max_compat; assumption. Qed. + +Lemma min_spec : forall n m, + (n < m /\ min n m == n) \/ (m <= n /\ min n m == m). +Proof. intros. exact (MPRev.max_spec m n). Qed. + +Lemma min_spec_le : forall n m, + (n <= m /\ min n m == n) \/ (m <= n /\ min n m == m). +Proof. intros. exact (MPRev.max_spec_le m n). Qed. + +Lemma min_mono: forall f, + (Proper (eq ==> eq) f) -> + (Proper (le ==> le) f) -> + forall x y, min (f x) (f y) == f (min x y). +Proof. + intros. apply MPRev.max_mono; auto. compute in *; eauto. +Qed. + +Lemma min_unicity : forall n m p, + ((n < m /\ p == n) \/ (m <= n /\ p == m)) -> p == min n m. +Proof. intros n m p. apply MPRev.max_unicity. Qed. + +Lemma min_unicity_ext : forall f, + (forall n m, (n < m /\ f n m == n) \/ (m <= n /\ f n m == m)) -> + (forall n m, f n m == min n m). +Proof. intros f H n m. apply MPRev.max_unicity, H; auto. Qed. + +Lemma min_id : forall n, min n n == n. +Proof. intros. exact (MPRev.max_id n). Qed. + +Notation min_idempotent := min_id (only parsing). + +Lemma min_assoc : forall m n p, min m (min n p) == min (min m n) p. +Proof. intros. symmetry; apply MPRev.max_assoc. Qed. + +Lemma min_comm : forall n m, min n m == min m n. +Proof. intros. exact (MPRev.max_comm m n). Qed. + +Lemma le_min_r : forall n m, min n m <= m. +Proof. intros. exact (MPRev.le_max_l m n). Qed. + +Lemma le_min_l : forall n m, min n m <= n. +Proof. intros. exact (MPRev.le_max_r m n). Qed. + +Lemma min_l_iff : forall n m, min n m == n <-> n <= m. +Proof. intros n m. exact (MPRev.max_r_iff m n). Qed. + +Lemma min_r_iff : forall n m, min n m == m <-> m <= n. +Proof. intros n m. exact (MPRev.max_l_iff m n). Qed. + +Lemma min_le : forall n m p, min n m <= p -> n <= p \/ m <= p. +Proof. intros n m p H. destruct (MPRev.max_le _ _ _ H); auto. Qed. + +Lemma min_le_iff : forall n m p, min n m <= p <-> n <= p \/ m <= p. +Proof. intros n m p. rewrite (MPRev.max_le_iff m n p); intuition. Qed. + +Lemma min_lt_iff : forall n m p, min n m < p <-> n < p \/ m < p. +Proof. intros n m p. rewrite (MPRev.max_lt_iff m n p); intuition. Qed. + +Lemma min_glb_l : forall n m p, p <= min n m -> p <= n. +Proof. intros n m. exact (MPRev.max_lub_r m n). Qed. + +Lemma min_glb_r : forall n m p, p <= min n m -> p <= m. +Proof. intros n m. exact (MPRev.max_lub_l m n). Qed. + +Lemma min_glb : forall n m p, p <= n -> p <= m -> p <= min n m. +Proof. intros. apply MPRev.max_lub; auto. Qed. + +Lemma min_glb_iff : forall n m p, p <= min n m <-> p <= n /\ p <= m. +Proof. intros. rewrite (MPRev.max_lub_iff m n p); intuition. Qed. + +Lemma min_glb_lt : forall n m p, p < n -> p < m -> p < min n m. +Proof. intros. apply MPRev.max_lub_lt; auto. Qed. + +Lemma min_glb_lt_iff : forall n m p, p < min n m <-> p < n /\ p < m. +Proof. intros. rewrite (MPRev.max_lub_lt_iff m n p); intuition. Qed. + +Lemma min_le_compat_l : forall n m p, n <= m -> min p n <= min p m. +Proof. intros n m. exact (MPRev.max_le_compat_r m n). Qed. + +Lemma min_le_compat_r : forall n m p, n <= m -> min n p <= min m p. +Proof. intros n m. exact (MPRev.max_le_compat_l m n). Qed. + +Lemma min_le_compat : forall n m p q, n <= m -> p <= q -> + min n p <= min m q. +Proof. intros. apply MPRev.max_le_compat; auto. Qed. + + +(** *** Combined properties of min and max *) + +Lemma min_max_absorption : forall n m, max n (min n m) == n. +Proof. + intros. + destruct (min_spec n m) as [(C,E)|(C,E)]; rewrite E. + apply max_l. order. + destruct (max_spec n m); intuition; order. +Qed. + +Lemma max_min_absorption : forall n m, min n (max n m) == n. +Proof. + intros. + destruct (max_spec n m) as [(C,E)|(C,E)]; rewrite E. + destruct (min_spec n m) as [(C',E')|(C',E')]; auto. order. + apply min_l; auto. order. +Qed. + +(** Distributivity *) + +Lemma max_min_distr : forall n m p, + max n (min m p) == min (max n m) (max n p). +Proof. + intros. symmetry. apply min_mono. + eauto with *. + repeat red; intros. apply max_le_compat_l; auto. +Qed. + +Lemma min_max_distr : forall n m p, + min n (max m p) == max (min n m) (min n p). +Proof. + intros. symmetry. apply max_mono. + eauto with *. + repeat red; intros. apply min_le_compat_l; auto. +Qed. + +(** Modularity *) + +Lemma max_min_modular : forall n m p, + max n (min m (max n p)) == min (max n m) (max n p). +Proof. + intros. rewrite <- max_min_distr. + destruct (max_spec n p) as [(C,E)|(C,E)]; rewrite E; auto with *. + destruct (min_spec m n) as [(C',E')|(C',E')]; rewrite E'. + rewrite 2 max_l; try order. rewrite min_le_iff; auto. + rewrite 2 max_l; try order. rewrite min_le_iff; auto. +Qed. + +Lemma min_max_modular : forall n m p, + min n (max m (min n p)) == max (min n m) (min n p). +Proof. + intros. rewrite <- min_max_distr. + destruct (min_spec n p) as [(C,E)|(C,E)]; rewrite E; auto with *. + destruct (max_spec m n) as [(C',E')|(C',E')]; rewrite E'. + rewrite 2 min_l; try order. rewrite max_le_iff; right; order. + rewrite 2 min_l; try order. rewrite max_le_iff; auto. +Qed. + +(** Disassociativity *) + +Lemma max_min_disassoc : forall n m p, + min n (max m p) <= max (min n m) p. +Proof. + intros. rewrite min_max_distr. + auto using max_le_compat_l, le_min_r. +Qed. + +(** Anti-monotonicity swaps the role of [min] and [max] *) + +Lemma max_min_antimono : forall f, + Proper (eq==>eq) f -> + Proper (le==>inverse le) f -> + forall x y, max (f x) (f y) == f (min x y). +Proof. + intros f Eqf Lef x y. + destruct (min_spec x y) as [(H,E)|(H,E)]; rewrite E; + destruct (max_spec (f x) (f y)) as [(H',E')|(H',E')]; auto. + assert (f y <= f x) by (apply Lef; order). order. + assert (f x <= f y) by (apply Lef; order). order. +Qed. + +Lemma min_max_antimono : forall f, + Proper (eq==>eq) f -> + Proper (le==>inverse le) f -> + forall x y, min (f x) (f y) == f (max x y). +Proof. + intros f Eqf Lef x y. + destruct (max_spec x y) as [(H,E)|(H,E)]; rewrite E; + destruct (min_spec (f x) (f y)) as [(H',E')|(H',E')]; auto. + assert (f y <= f x) by (apply Lef; order). order. + assert (f x <= f y) by (apply Lef; order). order. +Qed. + +End MinMaxLogicalProperties. + + +(** ** Properties requiring a decidable order *) + +Module MinMaxDecProperties (Import O:OrderedTypeFull')(Import M:HasMinMax O). + +(** Induction principles for [max]. *) + +Lemma max_case_strong : forall n m (P:t -> Type), + (forall x y, x==y -> P x -> P y) -> + (m<=n -> P n) -> (n<=m -> P m) -> P (max n m). +Proof. +intros n m P Compat Hl Hr. +destruct (CompSpec2Type (compare_spec n m)) as [EQ|LT|GT]. +assert (n<=m) by (rewrite le_lteq; auto). +apply (Compat m), Hr; auto. symmetry; apply max_r; auto. +assert (n<=m) by (rewrite le_lteq; auto). +apply (Compat m), Hr; auto. symmetry; apply max_r; auto. +assert (m<=n) by (rewrite le_lteq; auto). +apply (Compat n), Hl; auto. symmetry; apply max_l; auto. +Defined. + +Lemma max_case : forall n m (P:t -> Type), + (forall x y, x == y -> P x -> P y) -> + P n -> P m -> P (max n m). +Proof. intros. apply max_case_strong; auto. Defined. + +(** [max] returns one of its arguments. *) + +Lemma max_dec : forall n m, {max n m == n} + {max n m == m}. +Proof. + intros n m. apply max_case; auto with relations. + intros x y H [E|E]; [left|right]; rewrite <-H; auto. +Defined. + +(** Idem for [min] *) + +Lemma min_case_strong : forall n m (P:O.t -> Type), + (forall x y, x == y -> P x -> P y) -> + (n<=m -> P n) -> (m<=n -> P m) -> P (min n m). +Proof. +intros n m P Compat Hl Hr. +destruct (CompSpec2Type (compare_spec n m)) as [EQ|LT|GT]. +assert (n<=m) by (rewrite le_lteq; auto). +apply (Compat n), Hl; auto. symmetry; apply min_l; auto. +assert (n<=m) by (rewrite le_lteq; auto). +apply (Compat n), Hl; auto. symmetry; apply min_l; auto. +assert (m<=n) by (rewrite le_lteq; auto). +apply (Compat m), Hr; auto. symmetry; apply min_r; auto. +Defined. + +Lemma min_case : forall n m (P:O.t -> Type), + (forall x y, x == y -> P x -> P y) -> + P n -> P m -> P (min n m). +Proof. intros. apply min_case_strong; auto. Defined. + +Lemma min_dec : forall n m, {min n m == n} + {min n m == m}. +Proof. + intros. apply min_case; auto with relations. + intros x y H [E|E]; [left|right]; rewrite <- E; auto with relations. +Defined. + +End MinMaxDecProperties. + +Module MinMaxProperties (Import O:OrderedTypeFull')(Import M:HasMinMax O). + Module OT := OTF_to_TotalOrder O. + Include MinMaxLogicalProperties OT M. + Include MinMaxDecProperties O M. + Definition max_l := max_l. + Definition max_r := max_r. + Definition min_l := min_l. + Definition min_r := min_r. + Notation max_monotone := max_mono. + Notation min_monotone := min_mono. + Notation max_min_antimonotone := max_min_antimono. + Notation min_max_antimonotone := min_max_antimono. +End MinMaxProperties. + + +(** ** When the equality is Leibniz, we can skip a few [Proper] precondition. *) + +Module UsualMinMaxLogicalProperties + (Import O:UsualTotalOrder')(Import M:HasMinMax O). + + Include MinMaxLogicalProperties O M. + + Lemma max_monotone : forall f, Proper (le ==> le) f -> + forall x y, max (f x) (f y) = f (max x y). + Proof. intros; apply max_mono; auto. congruence. Qed. + + Lemma min_monotone : forall f, Proper (le ==> le) f -> + forall x y, min (f x) (f y) = f (min x y). + Proof. intros; apply min_mono; auto. congruence. Qed. + + Lemma min_max_antimonotone : forall f, Proper (le ==> inverse le) f -> + forall x y, min (f x) (f y) = f (max x y). + Proof. intros; apply min_max_antimono; auto. congruence. Qed. + + Lemma max_min_antimonotone : forall f, Proper (le ==> inverse le) f -> + forall x y, max (f x) (f y) = f (min x y). + Proof. intros; apply max_min_antimono; auto. congruence. Qed. + +End UsualMinMaxLogicalProperties. + + +Module UsualMinMaxDecProperties + (Import O:UsualOrderedTypeFull')(Import M:HasMinMax O). + + Module P := MinMaxDecProperties O M. + + Lemma max_case_strong : forall n m (P:t -> Type), + (m<=n -> P n) -> (n<=m -> P m) -> P (max n m). + Proof. intros; apply P.max_case_strong; auto. congruence. Defined. + + Lemma max_case : forall n m (P:t -> Type), + P n -> P m -> P (max n m). + Proof. intros; apply max_case_strong; auto. Defined. + + Lemma max_dec : forall n m, {max n m = n} + {max n m = m}. + Proof. exact P.max_dec. Defined. + + Lemma min_case_strong : forall n m (P:O.t -> Type), + (n<=m -> P n) -> (m<=n -> P m) -> P (min n m). + Proof. intros; apply P.min_case_strong; auto. congruence. Defined. + + Lemma min_case : forall n m (P:O.t -> Type), + P n -> P m -> P (min n m). + Proof. intros. apply min_case_strong; auto. Defined. + + Lemma min_dec : forall n m, {min n m = n} + {min n m = m}. + Proof. exact P.min_dec. Defined. + +End UsualMinMaxDecProperties. + +Module UsualMinMaxProperties + (Import O:UsualOrderedTypeFull')(Import M:HasMinMax O). + Module OT := OTF_to_TotalOrder O. + Include UsualMinMaxLogicalProperties OT M. + Include UsualMinMaxDecProperties O M. + Definition max_l := max_l. + Definition max_r := max_r. + Definition min_l := min_l. + Definition min_r := min_r. +End UsualMinMaxProperties. + + +(** From [TotalOrder] and [HasMax] and [HasEqDec], we can prove + that the order is decidable and build an [OrderedTypeFull]. *) + +Module TOMaxEqDec_to_Compare + (Import O:TotalOrder')(Import M:HasMax O)(Import E:HasEqDec O) <: HasCompare O. + + Definition compare x y := + if eq_dec x y then Eq + else if eq_dec (M.max x y) y then Lt else Gt. + + Lemma compare_spec : forall x y, CompSpec eq lt x y (compare x y). + Proof. + intros; unfold compare; repeat destruct eq_dec; auto; constructor. + destruct (lt_total x y); auto. + absurd (x==y); auto. transitivity (max x y); auto. + symmetry. apply max_l. rewrite le_lteq; intuition. + destruct (lt_total y x); auto. + absurd (max x y == y); auto. apply max_r; rewrite le_lteq; intuition. + Qed. + +End TOMaxEqDec_to_Compare. + +Module TOMaxEqDec_to_OTF (O:TotalOrder)(M:HasMax O)(E:HasEqDec O) + <: OrderedTypeFull + := O <+ E <+ TOMaxEqDec_to_Compare O M E. + + + +(** TODO: Some Remaining questions... + +--> Compare with a type-classes version ? + +--> Is max_unicity and max_unicity_ext really convenient to express + that any possible definition of max will in fact be equivalent ? + +--> Is it possible to avoid copy-paste about min even more ? + +*) |