(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/excedence/".
-
-include "higher_order_defs/relations.ma".
-include "nat/plus.ma".
-include "constructive_connectives.ma".
-include "constructive_higher_order_relations.ma".
-
-record excedence : Type ≝ {
- exc_carr:> Type;
- exc_relation: exc_carr → exc_carr → Prop;
- exc_coreflexive: coreflexive ? exc_relation;
- exc_cotransitive: cotransitive ? exc_relation
+set "baseuri" "cic:/matita/ordered_sets/".
+
+include "excedence.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
}.
-interpretation "excedence" 'nleq a b =
- (cic:/matita/excedence/exc_relation.con _ a b).
+record pordered_set: Type ≝ {
+ pos_carr:> excedence;
+ pos_order_relation_properties:> is_porder_relation ? (le pos_carr) (eq pos_carr)
+}.
-definition le ≝ λE:excedence.λa,b:E. ¬ (a ≰ b).
+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.
-interpretation "ordered sets less or equal than" 'leq a b =
- (cic:/matita/excedence/le.con _ a b).
+definition total_order : ∀E:excedence. Type ≝
+ λE:excedence. ∀a,b:E. a ≰ b → a < b.
-lemma le_reflexive: ∀E.reflexive ? (le E).
-intros (E); unfold; cases E; simplify; intros (x); apply (H x);
+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.
-lemma le_transitive: ∀E.transitive ? (le E).
-intros (E); unfold; cases E; simplify; unfold Not; intros (x y z Rxy Ryz H2);
-cases (c x z y H2) (H4 H5); clear H2; [exact (Rxy H4)|exact (Ryz H5)]
-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 apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
+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.
-notation "a # b" non associative with precedence 50 for @{ 'apart $a $b}.
-interpretation "apartness" 'apart a b = (cic:/matita/excedence/apart.con _ a b).
+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
+ }.
-lemma apart_coreflexive: ∀E.coreflexive ? (apart E).
-intros (E); unfold; cases E; simplify; clear E; intros (x); unfold;
-intros (H1); apply (H x); cases H1; assumption;
-qed.
+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
+ }.
-lemma apart_symmetric: ∀E.symmetric ? (apart E).
-intros (E); unfold; intros(x y H); cases H; clear H; [right|left] assumption;
+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.
-lemma apart_cotrans: ∀E. cotransitive ? (apart E).
-intros (E); unfold; cases E (T f _ cTf); simplify; intros (x y z Axy);
-cases Axy (H); lapply (cTf ? ? z H) as H1; cases H1; clear Axy H1;
-[left; left|right; left|right; right|left; right] assumption.
+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 eq ≝ λE:excedence.λa,b:E. ¬ (a # b).
+definition lt ≝ λO:ordered_set.λa,b:O.a ≤ b ∧ a ≠ b.
-notation "a ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.
-interpretation "alikeness" 'napart a b =
- (cic:/matita/excedence/eq.con _ a b).
+interpretation "Ordered set lt" 'lt a b =
+ (cic:/matita/ordered_sets/lt.con _ a b).
-lemma eq_reflexive:∀E. reflexive ? (eq E).
-intros (E); unfold; cases E (T f cRf _); simplify; unfold Not; intros (x H);
-apply (cRf x); cases H; assumption;
+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 eq_symmetric:∀E.symmetric ? (eq E).
-intros (E); unfold; unfold eq; unfold Not;
-intros (x y H1 H2); apply H1; cases H2; [right|left] assumption;
+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 eq_transitive: ∀E.transitive ? (eq E).
-intros (E); unfold; cases E (T f _ cTf); simplify; unfold Not;
-intros (x y z H1 H2 H3); cases H3 (H4 H4); clear E H3; lapply (cTf ? ? y H4) as H5;
-cases H5; clear H5 H4 cTf; [1,4: apply H1|*:apply H2] clear H1 H2;
-[1,3:left|*:right] assumption;
+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 le_antisymmetric: ∀E.antisymmetric ? (le E) (eq E).
-intros (E); unfold; intros (x y Lxy Lyx); unfold; unfold; intros (H);
-cases H; [apply Lxy;|apply Lyx] assumption;
+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.
-definition lt ≝ λE:excedence.λa,b:E. a ≤ b ∧ a # b.
+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.
-interpretation "ordered sets less than" 'lt a b =
- (cic:/matita/excedence/lt.con _ a b).
-lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
-intros (E); unfold; unfold Not; intros (x H); cases H (_ ABS);
-apply (apart_coreflexive ? x ABS);
+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 lt_transitive: ∀E.transitive ? (lt E).
-intros (E); unfold; intros (x y z H1 H2); cases H1 (Lxy Axy); cases H2 (Lyz Ayz);
-split; [apply (le_transitive ???? Lxy Lyz)] clear H1 H2;
-cases Axy (H1 H1); cases Ayz (H2 H2); [1:cases (Lxy H1)|3:cases (Lyz H2)]
-clear Axy Ayz;lapply (exc_cotransitive E) as c; unfold cotransitive in c;
-lapply (exc_coreflexive E) as r; unfold coreflexive in r;
-[1: lapply (c ?? y H1) as H3; cases H3 (H4 H4); [cases (Lxy H4)|cases (r ? H4)]
-|2: lapply (c ?? x H2) as H3; cases H3 (H4 H4); [right; assumption|cases (Lxy H4)]]
+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.
-theorem lt_to_excede: ∀E:excedence.∀a,b:E. (a < b) → (b ≰ a).
-intros (E a b Lab); cases Lab (LEab Aab);
-cases Aab (H H); [cases (LEab H)] fold normalize (b ≰ a); assumption; (* BUG *)
+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)
+ }.
projects / helm.git / blobdiff