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Diffstat (limited to 'theories/FSets/OrderedType.v')
| -rw-r--r-- | theories/FSets/OrderedType.v | 566 |
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diff --git a/theories/FSets/OrderedType.v b/theories/FSets/OrderedType.v new file mode 100644 index 00000000..2bf08dc7 --- /dev/null +++ b/theories/FSets/OrderedType.v @@ -0,0 +1,566 @@ +(***********************************************************************) +(* 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 *) +(***********************************************************************) + +(* $Id: OrderedType.v 8667 2006-03-28 11:59:44Z letouzey $ *) + +Require Export SetoidList. +Set Implicit Arguments. +Unset Strict Implicit. + +(* TODO concernant la tactique order: + * propagate_lt n'est sans doute pas complet + * un propagate_le + * exploiter les hypotheses negatives restant a la fin + * faire que ca marche meme quand une hypothese depend d'un eq ou lt. +*) + +(** * Ordered types *) + +Inductive Compare (X : Set) (lt eq : X -> X -> Prop) (x y : X) : Set := + | LT : lt x y -> Compare lt eq x y + | EQ : eq x y -> Compare lt eq x y + | GT : lt y x -> Compare lt eq x y. + +Module Type OrderedType. + + Parameter t : Set. + + Parameter eq : t -> t -> Prop. + Parameter lt : t -> t -> Prop. + + Axiom eq_refl : forall x : t, eq x x. + Axiom eq_sym : forall x y : t, eq x y -> eq y x. + Axiom eq_trans : forall x y z : t, eq x y -> eq y z -> eq x z. + + Axiom lt_trans : forall x y z : t, lt x y -> lt y z -> lt x z. + Axiom lt_not_eq : forall x y : t, lt x y -> ~ eq x y. + + Parameter compare : forall x y : t, Compare lt eq x y. + + Hint Immediate eq_sym. + Hint Resolve eq_refl eq_trans lt_not_eq lt_trans. + +End OrderedType. + +(** * Ordered types properties *) + +(** Additional properties that can be derived from signature + [OrderedType]. *) + +Module OrderedTypeFacts (O: OrderedType). + Import O. + + Lemma lt_antirefl : forall x, ~ lt x x. + Proof. + intros; intro; absurd (eq x x); auto. + Qed. + + Lemma lt_eq : forall x y z, lt x y -> eq y z -> lt x z. + Proof. + intros; destruct (compare x z); auto. + elim (lt_not_eq H); apply eq_trans with z; auto. + elim (lt_not_eq (lt_trans l H)); auto. + Qed. + + Lemma eq_lt : forall x y z, eq x y -> lt y z -> lt x z. + Proof. + intros; destruct (compare x z); auto. + elim (lt_not_eq H0); apply eq_trans with x; auto. + elim (lt_not_eq (lt_trans H0 l)); auto. + Qed. + + Lemma le_eq : forall x y z, ~lt x y -> eq y z -> ~lt x z. + Proof. + intros; intro; destruct H; apply lt_eq with z; auto. + Qed. + + Lemma eq_le : forall x y z, eq x y -> ~lt y z -> ~lt x z. + Proof. + intros; intro; destruct H0; apply eq_lt with x; auto. + Qed. + + Lemma neq_eq : forall x y z, ~eq x y -> eq y z -> ~eq x z. + Proof. + intros; intro; destruct H; apply eq_trans with z; auto. + Qed. + + Lemma eq_neq : forall x y z, eq x y -> ~eq y z -> ~eq x z. + Proof. + intros; intro; destruct H0; apply eq_trans with x; auto. + Qed. + + Hint Immediate eq_lt lt_eq le_eq eq_le neq_eq eq_neq. + + Lemma le_lt_trans : forall x y z, ~lt y x -> lt y z -> lt x z. + Proof. + intros; destruct (compare y x); auto. + elim (H l). + apply eq_lt with y; auto. + apply lt_trans with y; auto. + Qed. + + Lemma lt_le_trans : forall x y z, lt x y -> ~lt z y -> lt x z. + Proof. + intros; destruct (compare z y); auto. + elim (H0 l). + apply lt_eq with y; auto. + apply lt_trans with y; auto. + Qed. + + Lemma le_neq : forall x y, ~lt x y -> ~eq x y -> lt y x. + Proof. + intros; destruct (compare x y); intuition. + Qed. + + Lemma neq_sym : forall x y, ~eq x y -> ~eq y x. + Proof. + intuition. + Qed. + +Ltac abstraction := match goal with + (* First, some obvious simplifications *) + | H : False |- _ => elim H + | H : lt ?x ?x |- _ => elim (lt_antirefl H) + | H : ~eq ?x ?x |- _ => elim (H (eq_refl x)) + | H : eq ?x ?x |- _ => clear H; abstraction + | H : ~lt ?x ?x |- _ => clear H; abstraction + | |- eq ?x ?x => exact (eq_refl x) + | |- lt ?x ?x => elimtype False; abstraction + | |- ~ _ => intro; abstraction + | H1: ~lt ?x ?y, H2: ~eq ?x ?y |- _ => + generalize (le_neq H1 H2); clear H1 H2; intro; abstraction + | H1: ~lt ?x ?y, H2: ~eq ?y ?x |- _ => + generalize (le_neq H1 (neq_sym H2)); clear H1 H2; intro; abstraction + (* Then, we generalize all interesting facts *) + | H : lt ?x ?y |- _ => revert H; abstraction + | H : ~lt ?x ?y |- _ => revert H; abstraction + | H : ~eq ?x ?y |- _ => revert H; abstraction + | H : eq ?x ?y |- _ => revert H; abstraction + | _ => idtac +end. + +Ltac do_eq a b EQ := match goal with + | |- lt ?x ?y -> _ => let H := fresh "H" in + (intro H; + (generalize (eq_lt (eq_sym EQ) H); clear H; intro H) || + (generalize (lt_eq H EQ); clear H; intro H) || + idtac); + do_eq a b EQ + | |- ~lt ?x ?y -> _ => let H := fresh "H" in + (intro H; + (generalize (eq_le (eq_sym EQ) H); clear H; intro H) || + (generalize (le_eq H EQ); clear H; intro H) || + idtac); + do_eq a b EQ + | |- eq ?x ?y -> _ => let H := fresh "H" in + (intro H; + (generalize (eq_trans (eq_sym EQ) H); clear H; intro H) || + (generalize (eq_trans H EQ); clear H; intro H) || + idtac); + do_eq a b EQ + | |- ~eq ?x ?y -> _ => let H := fresh "H" in + (intro H; + (generalize (eq_neq (eq_sym EQ) H); clear H; intro H) || + (generalize (neq_eq H EQ); clear H; intro H) || + idtac); + do_eq a b EQ + | |- lt a ?y => apply eq_lt with b; [exact EQ|] + | |- lt ?y a => apply lt_eq with b; [|exact (eq_sym EQ)] + | |- eq a ?y => apply eq_trans with b; [exact EQ|] + | |- eq ?y a => apply eq_trans with b; [|exact (eq_sym EQ)] + | _ => idtac + end. + +Ltac propagate_eq := abstraction; clear; match goal with + (* the abstraction tactic leaves equality facts in head position...*) + | |- eq ?a ?b -> _ => + let EQ := fresh "EQ" in (intro EQ; do_eq a b EQ; clear EQ); + propagate_eq + | _ => idtac +end. + +Ltac do_lt x y LT := match goal with + (* LT *) + | |- lt x y -> _ => intros _; do_lt x y LT + | |- lt y ?z -> _ => let H := fresh "H" in + (intro H; generalize (lt_trans LT H); intro); do_lt x y LT + | |- lt ?z x -> _ => let H := fresh "H" in + (intro H; generalize (lt_trans H LT); intro); do_lt x y LT + | |- lt _ _ -> _ => intro; do_lt x y LT + (* Ge *) + | |- ~lt y x -> _ => intros _; do_lt x y LT + | |- ~lt x ?z -> _ => let H := fresh "H" in + (intro H; generalize (le_lt_trans H LT); intro); do_lt x y LT + | |- ~lt ?z y -> _ => let H := fresh "H" in + (intro H; generalize (lt_le_trans LT H); intro); do_lt x y LT + | |- ~lt _ _ -> _ => intro; do_lt x y LT + | _ => idtac + end. + +Definition hide_lt := lt. + +Ltac propagate_lt := abstraction; match goal with + (* when no [=] remains, the abstraction tactic leaves [<] facts first. *) + | |- lt ?x ?y -> _ => + let LT := fresh "LT" in (intro LT; do_lt x y LT; + change (hide_lt x y) in LT); + propagate_lt + | _ => unfold hide_lt in * +end. + +Ltac order := + intros; + propagate_eq; + propagate_lt; + auto; + propagate_lt; + eauto. + +Ltac false_order := elimtype False; order. + + Lemma gt_not_eq : forall x y, lt y x -> ~ eq x y. + Proof. + order. + Qed. + + Lemma eq_not_lt : forall x y : t, eq x y -> ~ lt x y. + Proof. + order. + Qed. + + Hint Resolve gt_not_eq eq_not_lt. + + Lemma eq_not_gt : forall x y : t, eq x y -> ~ lt y x. + Proof. + order. + Qed. + + Lemma lt_not_gt : forall x y : t, lt x y -> ~ lt y x. + Proof. + order. + Qed. + + Hint Resolve eq_not_gt lt_antirefl lt_not_gt. + + Lemma elim_compare_eq : + forall x y : t, + eq x y -> exists H : eq x y, compare x y = EQ _ H. + Proof. + intros; case (compare x y); intros H'; try solve [false_order]. + exists H'; auto. + Qed. + + Lemma elim_compare_lt : + forall x y : t, + lt x y -> exists H : lt x y, compare x y = LT _ H. + Proof. + intros; case (compare x y); intros H'; try solve [false_order]. + exists H'; auto. + Qed. + + Lemma elim_compare_gt : + forall x y : t, + lt y x -> exists H : lt y x, compare x y = GT _ H. + Proof. + intros; case (compare x y); intros H'; try solve [false_order]. + exists H'; auto. + Qed. + + Ltac elim_comp := + match goal with + | |- ?e => match e with + | context ctx [ compare ?a ?b ] => + let H := fresh in + (destruct (compare a b) as [H|H|H]; + try solve [ intros; false_order]) + end + end. + + Ltac elim_comp_eq x y := + elim (elim_compare_eq (x:=x) (y:=y)); + [ intros _1 _2; rewrite _2; clear _1 _2 | auto ]. + + Ltac elim_comp_lt x y := + elim (elim_compare_lt (x:=x) (y:=y)); + [ intros _1 _2; rewrite _2; clear _1 _2 | auto ]. + + Ltac elim_comp_gt x y := + elim (elim_compare_gt (x:=x) (y:=y)); + [ intros _1 _2; rewrite _2; clear _1 _2 | auto ]. + + Lemma eq_dec : forall x y : t, {eq x y} + {~ eq x y}. + Proof. + intros; elim (compare x y); [ right | left | right ]; auto. + Qed. + + Lemma lt_dec : forall x y : t, {lt x y} + {~ lt x y}. + Proof. + intros; elim (compare x y); [ left | right | right ]; auto. + Qed. + + Definition eqb x y : bool := if eq_dec x y then true else false. + + Lemma eqb_alt : + forall x y, eqb x y = match compare x y with EQ _ => true | _ => false end. + Proof. + unfold eqb; intros; destruct (eq_dec x y); elim_comp; auto. + Qed. + +(* Specialization of resuts about lists modulo. *) + +Notation In:=(InA eq). +Notation Inf:=(lelistA lt). +Notation Sort:=(sort lt). +Notation NoDup:=(NoDupA eq). + +Lemma In_eq : forall l x y, eq x y -> In x l -> In y l. +Proof. exact (InA_eqA eq_sym eq_trans). Qed. + +Lemma ListIn_In : forall l x, List.In x l -> In x l. +Proof. exact (In_InA eq_refl). Qed. + +Lemma Inf_lt : forall l x y, lt x y -> Inf y l -> Inf x l. +Proof. exact (InfA_ltA lt_trans). Qed. + +Lemma Inf_eq : forall l x y, eq x y -> Inf y l -> Inf x l. +Proof. exact (InfA_eqA eq_lt). Qed. + +Lemma Sort_Inf_In : forall l x a, Sort l -> Inf a l -> In x l -> lt a x. +Proof. exact (SortA_InfA_InA eq_refl eq_sym lt_trans lt_eq eq_lt). Qed. + +Lemma ListIn_Inf : forall l x, (forall y, List.In y l -> lt x y) -> Inf x l. +Proof. exact (@In_InfA t lt). Qed. + +Lemma In_Inf : forall l x, (forall y, In y l -> lt x y) -> Inf x l. +Proof. exact (InA_InfA eq_refl (ltA:=lt)). Qed. + +Lemma Inf_alt : + forall l x, Sort l -> (Inf x l <-> (forall y, In y l -> lt x y)). +Proof. exact (InfA_alt eq_refl eq_sym lt_trans lt_eq eq_lt). Qed. + +Lemma Sort_NoDup : forall l, Sort l -> NoDup l. +Proof. exact (SortA_NoDupA eq_refl eq_sym lt_trans lt_not_eq lt_eq eq_lt) . Qed. + +Hint Resolve ListIn_In Sort_NoDup Inf_lt. +Hint Immediate In_eq Inf_lt. + +End OrderedTypeFacts. + +Module PairOrderedType(O:OrderedType). + Import O. + Module MO:=OrderedTypeFacts(O). + Import MO. + + Section Elt. + Variable elt : Set. + Notation key:=t. + + Definition eqk (p p':key*elt) := eq (fst p) (fst p'). + Definition eqke (p p':key*elt) := + eq (fst p) (fst p') /\ (snd p) = (snd p'). + Definition ltk (p p':key*elt) := lt (fst p) (fst p'). + + Hint Unfold eqk eqke ltk. + Hint Extern 2 (eqke ?a ?b) => split. + + (* eqke is stricter than eqk *) + + Lemma eqke_eqk : forall x x', eqke x x' -> eqk x x'. + Proof. + unfold eqk, eqke; intuition. + Qed. + + (* ltk ignore the second components *) + + Lemma ltk_right_r : forall x k e e', ltk x (k,e) -> ltk x (k,e'). + Proof. auto. Qed. + + Lemma ltk_right_l : forall x k e e', ltk (k,e) x -> ltk (k,e') x. + Proof. auto. Qed. + Hint Immediate ltk_right_r ltk_right_l. + + (* eqk, eqke are equalities, ltk is a strict order *) + + Lemma eqk_refl : forall e, eqk e e. + Proof. auto. Qed. + + Lemma eqke_refl : forall e, eqke e e. + Proof. auto. Qed. + + Lemma eqk_sym : forall e e', eqk e e' -> eqk e' e. + Proof. auto. Qed. + + Lemma eqke_sym : forall e e', eqke e e' -> eqke e' e. + Proof. unfold eqke; intuition. Qed. + + Lemma eqk_trans : forall e e' e'', eqk e e' -> eqk e' e'' -> eqk e e''. + Proof. eauto. Qed. + + Lemma eqke_trans : forall e e' e'', eqke e e' -> eqke e' e'' -> eqke e e''. + Proof. + unfold eqke; intuition; [ eauto | congruence ]. + Qed. + + Lemma ltk_trans : forall e e' e'', ltk e e' -> ltk e' e'' -> ltk e e''. + Proof. eauto. Qed. + + Lemma ltk_not_eqk : forall e e', ltk e e' -> ~ eqk e e'. + Proof. unfold eqk, ltk; auto. Qed. + + Lemma ltk_not_eqke : forall e e', ltk e e' -> ~eqke e e'. + Proof. + unfold eqke, ltk; intuition; simpl in *; subst. + exact (lt_not_eq H H1). + Qed. + + Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl. + Hint Resolve ltk_trans ltk_not_eqk ltk_not_eqke. + Hint Immediate eqk_sym eqke_sym. + + (* Additionnal facts *) + + Lemma eqk_not_ltk : forall x x', eqk x x' -> ~ltk x x'. + Proof. + unfold eqk, ltk; simpl; auto. + Qed. + + Lemma ltk_eqk : forall e e' e'', ltk e e' -> eqk e' e'' -> ltk e e''. + Proof. eauto. Qed. + + Lemma eqk_ltk : forall e e' e'', eqk e e' -> ltk e' e'' -> ltk e e''. + Proof. + intros (k,e) (k',e') (k'',e''). + unfold ltk, eqk; simpl; eauto. + Qed. + Hint Resolve eqk_not_ltk. + Hint Immediate ltk_eqk eqk_ltk. + + Lemma InA_eqke_eqk : + forall x m, InA eqke x m -> InA eqk x m. + Proof. + unfold eqke; induction 1; intuition. + Qed. + Hint Resolve InA_eqke_eqk. + + Definition MapsTo (k:key)(e:elt):= InA eqke (k,e). + Definition In k m := exists e:elt, MapsTo k e m. + Notation Sort := (sort ltk). + Notation Inf := (lelistA ltk). + + Hint Unfold MapsTo In. + + (* An alternative formulation for [In k l] is [exists e, InA eqk (k,e) l] *) + + Lemma In_alt : forall k l, In k l <-> exists e, InA eqk (k,e) l. + Proof. + firstorder. + exists x; auto. + induction H. + destruct y. + exists e; auto. + destruct IHInA as [e H0]. + exists e; auto. + Qed. + + Lemma MapsTo_eq : forall l x y e, eq x y -> MapsTo x e l -> MapsTo y e l. + Proof. + intros; unfold MapsTo in *; apply InA_eqA with (x,e); eauto. + Qed. + + Lemma In_eq : forall l x y, eq x y -> In x l -> In y l. + Proof. + destruct 2 as (e,E); exists e; eapply MapsTo_eq; eauto. + Qed. + + Lemma Inf_eq : forall l x x', eqk x x' -> Inf x' l -> Inf x l. + Proof. exact (InfA_eqA eqk_ltk). Qed. + + Lemma Inf_lt : forall l x x', ltk x x' -> Inf x' l -> Inf x l. + Proof. exact (InfA_ltA ltk_trans). Qed. + + Hint Immediate Inf_eq. + Hint Resolve Inf_lt. + + Lemma Sort_Inf_In : + forall l p q, Sort l -> Inf q l -> InA eqk p l -> ltk q p. + Proof. + exact (SortA_InfA_InA eqk_refl eqk_sym ltk_trans ltk_eqk eqk_ltk). + Qed. + + Lemma Sort_Inf_NotIn : + forall l k e, Sort l -> Inf (k,e) l -> ~In k l. + Proof. + intros; red; intros. + destruct H1 as [e' H2]. + elim (@ltk_not_eqk (k,e) (k,e')). + eapply Sort_Inf_In; eauto. + red; simpl; auto. + Qed. + + Lemma Sort_NoDupA: forall l, Sort l -> NoDupA eqk l. + Proof. + exact (SortA_NoDupA eqk_refl eqk_sym ltk_trans ltk_not_eqk ltk_eqk eqk_ltk). + Qed. + + Lemma Sort_In_cons_1 : forall e l e', Sort (e::l) -> InA eqk e' l -> ltk e e'. + Proof. + inversion 1; intros; eapply Sort_Inf_In; eauto. + Qed. + + Lemma Sort_In_cons_2 : forall l e e', Sort (e::l) -> InA eqk e' (e::l) -> + ltk e e' \/ eqk e e'. + Proof. + inversion_clear 2; auto. + left; apply Sort_In_cons_1 with l; auto. + Qed. + + Lemma Sort_In_cons_3 : + forall x l k e, Sort ((k,e)::l) -> In x l -> ~eq x k. + Proof. + inversion_clear 1; red; intros. + destruct (Sort_Inf_NotIn H0 H1 (In_eq H2 H)). + Qed. + + Lemma In_inv : forall k k' e l, In k ((k',e) :: l) -> eq k k' \/ In k l. + Proof. + inversion 1. + inversion_clear H0; eauto. + destruct H1; simpl in *; intuition. + Qed. + + Lemma In_inv_2 : forall k k' e e' l, + InA eqk (k, e) ((k', e') :: l) -> ~ eq k k' -> InA eqk (k, e) l. + Proof. + inversion_clear 1; compute in H0; intuition. + Qed. + + Lemma In_inv_3 : forall x x' l, + InA eqke x (x' :: l) -> ~ eqk x x' -> InA eqke x l. + Proof. + inversion_clear 1; compute in H0; intuition. + Qed. + + End Elt. + + Hint Unfold eqk eqke ltk. + Hint Extern 2 (eqke ?a ?b) => split. + Hint Resolve eqk_trans eqke_trans eqk_refl eqke_refl. + Hint Resolve ltk_trans ltk_not_eqk ltk_not_eqke. + Hint Immediate eqk_sym eqke_sym. + Hint Resolve eqk_not_ltk. + Hint Immediate ltk_eqk eqk_ltk. + Hint Resolve InA_eqke_eqk. + Hint Unfold MapsTo In. + Hint Immediate Inf_eq. + Hint Resolve Inf_lt. + Hint Resolve Sort_Inf_NotIn. + Hint Resolve In_inv_2 In_inv_3. + +End PairOrderedType. + + |