From: Enrico Tassi Date: Mon, 12 Nov 2007 16:41:02 +0000 (+0000) Subject: added ordered sets X-Git-Tag: 0.4.95@7852~18 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=3789c6cc5ad8d155f9907edb60ec2f953fb7f682;p=helm.git added ordered sets --- diff --git a/matita/dama/ordered_sets.ma b/matita/dama/ordered_sets.ma new file mode 100644 index 000000000..5ae4f564a --- /dev/null +++ b/matita/dama/ordered_sets.ma @@ -0,0 +1,283 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +set "baseuri" "cic:/matita/ordered_sets/". + +include "ordered_sets.ma". + +record is_porder_relation (C:Type) (le:C→C→Prop) (eq:C→C→Prop) : Type ≝ { + por_reflexive: reflexive ? le; + por_transitive: transitive ? le; + por_antisimmetric: antisymmetric ? le eq +}. + +record pordered_set: Type ≝ { + pos_carr:> excedence; + pos_order_relation_properties:> is_porder_relation ? (le pos_carr) (eq pos_carr) +}. + +lemma pordered_set_of_excedence: excedence → pordered_set. +intros (E); apply (mk_pordered_set E); apply (mk_is_porder_relation); +[apply le_reflexive|apply le_transitive|apply le_antisymmetric] +qed. + +definition total_order : ∀E:excedence. Type ≝ + λE:excedence. ∀a,b:E. a ≰ b → a < b. + +alias id "transitive" = "cic:/matita/higher_order_defs/relations/transitive.con". +alias id "cotransitive" = "cic:/matita/higher_order_defs/relations/cotransitive.con". +alias id "antisymmetric" = "cic:/matita/higher_order_defs/relations/antisymmetric.con". +theorem antisimmetric_to_cotransitive_to_transitive: + ∀C:Type.∀le:C→C→Prop. antisymmetric ? le → cotransitive ? le → transitive ? le. +intros (T f Af cT); unfold transitive; intros (x y z fxy fyz); +lapply (cT ? ? fxy z) as H; cases H; [assumption] cases (Af ? ? fyz H1); +qed. + +definition is_increasing ≝ λO:pordered_set.λa:nat→O.∀n:nat.a n ≤ a (S n). +definition is_decreasing ≝ λO:pordered_set.λa:nat→O.∀n:nat.a (S n) ≤ a n. + +definition is_upper_bound ≝ λO:pordered_set.λa:nat→O.λu:O.∀n:nat.a n ≤ u. +definition is_lower_bound ≝ λO:pordered_set.λa:nat→O.λu:O.∀n:nat.u ≤ a n. + +record is_sup (O:pordered_set) (a:nat→O) (u:O) : Prop ≝ + { sup_upper_bound: is_upper_bound O a u; + sup_least_upper_bound: ∀v:O. is_upper_bound O a v → u≤v + }. + +record is_inf (O:pordered_set) (a:nat→O) (u:O) : Prop ≝ + { inf_lower_bound: is_lower_bound O a u; + inf_greatest_lower_bound: ∀v:O. is_lower_bound O a v → v≤u + }. + +record is_bounded_below (O:pordered_set) (a:nat→O) : Type ≝ + { ib_lower_bound: O; + ib_lower_bound_is_lower_bound: is_lower_bound ? a ib_lower_bound + }. + +record is_bounded_above (O:pordered_set) (a:nat→O) : Type ≝ + { ib_upper_bound: O; + ib_upper_bound_is_upper_bound: is_upper_bound ? a ib_upper_bound + }. + +record is_bounded (O:pordered_set) (a:nat→O) : Type ≝ + { ib_bounded_below:> is_bounded_below ? a; + ib_bounded_above:> is_bounded_above ? a + }. + +record bounded_below_sequence (O:pordered_set) : Type ≝ + { bbs_seq:1> nat→O; + bbs_is_bounded_below:> is_bounded_below ? bbs_seq + }. + +record bounded_above_sequence (O:pordered_set) : Type ≝ + { bas_seq:1> nat→O; + bas_is_bounded_above:> is_bounded_above ? bas_seq + }. + +record bounded_sequence (O:pordered_set) : Type ≝ + { bs_seq:1> nat → O; + bs_is_bounded_below: is_bounded_below ? bs_seq; + bs_is_bounded_above: is_bounded_above ? bs_seq + }. + +definition bounded_below_sequence_of_bounded_sequence ≝ + λO:pordered_set.λb:bounded_sequence O. + mk_bounded_below_sequence ? b (bs_is_bounded_below ? b). + +coercion cic:/matita/ordered_sets/bounded_below_sequence_of_bounded_sequence.con. + +definition bounded_above_sequence_of_bounded_sequence ≝ + λO:pordered_set.λb:bounded_sequence O. + mk_bounded_above_sequence ? b (bs_is_bounded_above ? b). + +coercion cic:/matita/ordered_sets/bounded_above_sequence_of_bounded_sequence.con. + +definition lower_bound ≝ + λO:ordered_set.λb:bounded_below_sequence O. + ib_lower_bound ? b (bbs_is_bounded_below ? b). + +lemma lower_bound_is_lower_bound: + ∀O:ordered_set.∀b:bounded_below_sequence O. + is_lower_bound ? b (lower_bound ? b). + intros; + unfold lower_bound; + apply ib_lower_bound_is_lower_bound. +qed. + +definition upper_bound ≝ + λO:ordered_set.λb:bounded_above_sequence O. + ib_upper_bound ? b (bas_is_bounded_above ? b). + +lemma upper_bound_is_upper_bound: + ∀O:ordered_set.∀b:bounded_above_sequence O. + is_upper_bound ? b (upper_bound ? b). + intros; + unfold upper_bound; + apply ib_upper_bound_is_upper_bound. +qed. + +definition lt ≝ λO:ordered_set.λa,b:O.a ≤ b ∧ a ≠ b. + +interpretation "Ordered set lt" 'lt a b = + (cic:/matita/ordered_sets/lt.con _ a b). + +definition reverse_ordered_set: ordered_set → ordered_set. + intros; + apply mk_ordered_set; + [2:apply (λx,y:o.y ≤ x) + | skip + | apply mk_is_order_relation; + [ simplify; + intros; + apply (or_reflexive ? ? o) + | simplify; + intros; + apply (or_transitive ? ? o); + [2: apply H1 + | skip + | assumption + ] + | simplify; + intros; + apply (or_antisimmetric ? ? o); + assumption + ] + ]. +qed. + +interpretation "Ordered set ge" 'geq a b = + (cic:/matita/ordered_sets/os_le.con _ + (cic:/matita/ordered_sets/os_pre_ordered_set.con _ + (cic:/matita/ordered_sets/reverse_ordered_set.con _ _)) a b). + +lemma is_lower_bound_reverse_is_upper_bound: + ∀O:ordered_set.∀a:nat→O.∀l:O. + is_lower_bound O a l → is_upper_bound (reverse_ordered_set O) a l. + intros; + unfold; + intro; + unfold; + unfold reverse_ordered_set; + simplify; + apply H. +qed. + +lemma is_upper_bound_reverse_is_lower_bound: + ∀O:ordered_set.∀a:nat→O.∀l:O. + is_upper_bound O a l → is_lower_bound (reverse_ordered_set O) a l. + intros; + unfold; + intro; + unfold; + unfold reverse_ordered_set; + simplify; + apply H. +qed. + +lemma reverse_is_lower_bound_is_upper_bound: + ∀O:ordered_set.∀a:nat→O.∀l:O. + is_lower_bound (reverse_ordered_set O) a l → is_upper_bound O a l. + intros; + unfold in H; + unfold reverse_ordered_set in H; + apply H. +qed. + +lemma reverse_is_upper_bound_is_lower_bound: + ∀O:ordered_set.∀a:nat→O.∀l:O. + is_upper_bound (reverse_ordered_set O) a l → is_lower_bound O a l. + intros; + unfold in H; + unfold reverse_ordered_set in H; + apply H. +qed. + + +lemma is_inf_to_reverse_is_sup: + ∀O:ordered_set.∀a:bounded_below_sequence O.∀l:O. + is_inf O a l → is_sup (reverse_ordered_set O) a l. + intros; + apply (mk_is_sup (reverse_ordered_set O)); + [ apply is_lower_bound_reverse_is_upper_bound; + apply inf_lower_bound; + assumption + | intros; + change in v with (os_carrier O); + change with (v ≤ l); + apply (inf_greatest_lower_bound ? ? ? H); + apply reverse_is_upper_bound_is_lower_bound; + assumption + ]. +qed. + +lemma is_sup_to_reverse_is_inf: + ∀O:ordered_set.∀a:bounded_above_sequence O.∀l:O. + is_sup O a l → is_inf (reverse_ordered_set O) a l. + intros; + apply (mk_is_inf (reverse_ordered_set O)); + [ apply is_upper_bound_reverse_is_lower_bound; + apply sup_upper_bound; + assumption + | intros; + change in v with (os_carrier O); + change with (l ≤ v); + apply (sup_least_upper_bound ? ? ? H); + apply reverse_is_lower_bound_is_upper_bound; + assumption + ]. +qed. + +lemma reverse_is_sup_to_is_inf: + ∀O:ordered_set.∀a:bounded_above_sequence O.∀l:O. + is_sup (reverse_ordered_set O) a l → is_inf O a l. + intros; + apply mk_is_inf; + [ apply reverse_is_upper_bound_is_lower_bound; + change in l with (os_carrier (reverse_ordered_set O)); + apply sup_upper_bound; + assumption + | intros; + change in l with (os_carrier (reverse_ordered_set O)); + change in v with (os_carrier (reverse_ordered_set O)); + change with (os_le (reverse_ordered_set O) l v); + apply (sup_least_upper_bound ? ? ? H); + change in v with (os_carrier O); + apply is_lower_bound_reverse_is_upper_bound; + assumption + ]. +qed. + +lemma reverse_is_inf_to_is_sup: + ∀O:ordered_set.∀a:bounded_above_sequence O.∀l:O. + is_inf (reverse_ordered_set O) a l → is_sup O a l. + intros; + apply mk_is_sup; + [ apply reverse_is_lower_bound_is_upper_bound; + change in l with (os_carrier (reverse_ordered_set O)); + apply (inf_lower_bound ? ? ? H) + | intros; + change in l with (os_carrier (reverse_ordered_set O)); + change in v with (os_carrier (reverse_ordered_set O)); + change with (os_le (reverse_ordered_set O) v l); + apply (inf_greatest_lower_bound ? ? ? H); + change in v with (os_carrier O); + apply is_upper_bound_reverse_is_lower_bound; + assumption + ]. +qed. + +record cotransitively_ordered_set: Type := + { cos_ordered_set :> ordered_set; + cos_cotransitive: cotransitive ? (os_le cos_ordered_set) + }.