include "higher_order_defs/relations.ma".
include "nat/plus.ma".
include "constructive_connectives.ma".
+include "constructive_higher_order_relations.ma".
-definition cotransitive ≝
- λC:Type.λle:C→C→Prop.∀x,y,z:C. le x y → le x z ∨ le z y.
+record excedence : Type ≝ {
+ exc_carr:> Type;
+ exc_relation: exc_carr → exc_carr → Prop;
+ exc_coreflexive: coreflexive ? exc_relation;
+ exc_cotransitive: cotransitive ? exc_relation
+}.
-definition antisimmetric ≝
- λC:Type.λle:C→C→Prop.∀x,y:C. le x y → le y x → x=y.
+interpretation "excedence" 'nleq a b =
+ (cic:/matita/ordered_sets/exc_relation.con _ a b).
-record is_order_relation (C:Type) (le:C→C→Prop) : Type ≝
- { or_reflexive: reflexive ? le;
- or_transitive: transitive ? le;
- or_antisimmetric: antisimmetric ? le
- }.
+definition le ≝ λE:excedence.λa,b:E. ¬ (a ≰ b).
-record ordered_set: Type ≝
- { os_carrier:> Type;
- os_le: os_carrier → os_carrier → Prop;
- os_order_relation_properties:> is_order_relation ? os_le
- }.
+interpretation "ordered sets less or equal than" 'leq a b =
+ (cic:/matita/ordered_sets/le.con _ a b).
-interpretation "Ordered Sets le" 'leq a b =
- (cic:/matita/ordered_sets/os_le.con _ a b).
-
-theorem antisimmetric_to_cotransitive_to_transitive:
- ∀C.∀le:relation C. antisimmetric ? le → cotransitive ? le →
- transitive ? le.
- intros;
- unfold transitive;
- intros;
- elim (c ? ? z H1);
- [ assumption
- | rewrite < (H ? ? H2 t);
- assumption
- ].
+lemma le_reflexive: ∀E.reflexive ? (le E).
+intros (E); unfold; cases E; simplify; intros (x); apply (H x);
qed.
-definition is_increasing ≝ λO:ordered_set.λa:nat→O.∀n:nat.a n ≤ a (S n).
-definition is_decreasing ≝ λO:ordered_set.λa:nat→O.∀n:nat.a (S n) ≤ a n.
-
-definition is_upper_bound ≝ λO:ordered_set.λa:nat→O.λu:O.∀n:nat.a n ≤ u.
-definition is_lower_bound ≝ λO:ordered_set.λa:nat→O.λu:O.∀n:nat.u ≤ a n.
-
-record is_sup (O:ordered_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:ordered_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:ordered_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:ordered_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:ordered_set) (a:nat→O) : Type ≝
- { ib_bounded_below:> is_bounded_below ? a;
- ib_bounded_above:> is_bounded_above ? a
- }.
-
-record bounded_below_sequence (O:ordered_set) : Type ≝
- { bbs_seq:1> nat→O;
- bbs_is_bounded_below:> is_bounded_below ? bbs_seq
- }.
-
-record bounded_above_sequence (O:ordered_set) : Type ≝
- { bas_seq:1> nat→O;
- bas_is_bounded_above:> is_bounded_above ? bas_seq
- }.
-
-record bounded_sequence (O:ordered_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:ordered_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:ordered_set.λb:bounded_sequence O.
- mk_bounded_above_sequence ? b (bs_is_bounded_above ? b).
+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.
-coercion cic:/matita/ordered_sets/bounded_above_sequence_of_bounded_sequence.con.
+definition apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
-definition lower_bound ≝
- λO:ordered_set.λb:bounded_below_sequence O.
- ib_lower_bound ? b (bbs_is_bounded_below ? b).
+notation "a # b" non associative with precedence 50 for @{ 'apart $a $b}.
+interpretation "apartness" 'apart a b = (cic:/matita/ordered_sets/apart.con _ a 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.
+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.
-definition upper_bound ≝
- λO:ordered_set.λb:bounded_above_sequence O.
- ib_upper_bound ? b (bas_is_bounded_above ? b).
+lemma apart_symmetric: ∀E.symmetric ? (apart E).
+intros (E); unfold; intros(x y H); cases H; clear H; [right|left] assumption;
+qed.
-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.
+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.
qed.
-definition lt ≝ λO:ordered_set.λa,b:O.a ≤ b ∧ a ≠ b.
+definition eq ≝ λE:excedence.λa,b:E. ¬ (a # b).
-interpretation "Ordered set lt" 'lt a b =
- (cic:/matita/ordered_sets/lt.con _ a b).
+notation "a ≈ b" non associative with precedence 50 for @{ 'napart $a $b}.
+interpretation "alikeness" 'napart a b =
+ (cic:/matita/ordered_sets/eq.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
- ]
- ].
+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;
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.
+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;
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.
+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;
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.
+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;
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.
+definition lt ≝ λE:excedence.λa,b:E. a ≤ b ∧ a # b.
+interpretation "ordered sets less than" 'lt a b =
+ (cic:/matita/ordered_sets/lt.con _ a b).
-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
- ].
+lemma lt_coreflexive: ∀E.coreflexive ? (lt E).
+intros (E); unfold; unfold Not; intros (x H); cases H (_ ABS);
+apply (apart_coreflexive ? x ABS);
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
- ].
+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)]]
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
- ].
+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 *)
qed.
-
-record cotransitively_ordered_set: Type :=
- { cos_ordered_set :> ordered_set;
- cos_cotransitive: cotransitive ? (os_le cos_ordered_set)
- }.
projects / helm.git / blobdiff