set "baseuri" "cic:/matita/constructive_connectives/".
-inductive or (A,B:Type) : Type \def
- Left : A → or A B
- | Right : B → or A B.
+inductive Or (A,B:Type) : Type ≝
+ Left : A → Or A B
+ | Right : B → Or A B.
interpretation "constructive or" 'or x y =
- (cic:/matita/constructive_connectives/or.ind#xpointer(1/1) x y).
+ (cic:/matita/constructive_connectives/Or.ind#xpointer(1/1) x y).
-inductive ex (A:Type) (P:A→Prop) : Type \def
+inductive And (A,B:Type) : Type ≝
+ | Conj : A → B → And A B.
+
+interpretation "constructive and" 'and x y =
+ (cic:/matita/constructive_connectives/And.ind#xpointer(1/1) x y).
+
+inductive ex (A:Type) (P:A→Prop) : Type ≝
ex_intro: ∀w:A. P w → ex A P.
notation < "hvbox(Σ ident i opt (: ty) break . p)"
(cic:/matita/constructive_connectives/ex.ind#xpointer(1/1) _ x).
alias id "False" = "cic:/matita/logic/connectives/False.ind#xpointer(1/1)".
-definition Not ≝ λx:Type.False.
+definition Not ≝ λx:Type.x → False.
interpretation "constructive not" 'not x =
(cic:/matita/constructive_connectives/Not.con x).
\ No newline at end of file
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).
+
+lemma le_reflexive: ∀E.reflexive ? (le E).
+intros (E); unfold; cases E; simplify; intros (x); apply (H x);
+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 apart ≝ λE:excedence.λa,b:E. a ≰ b ∨ b ≰ a.
+
+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 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.
+
+lemma apart_symmetric: ∀E.symmetric ? (apart E).
+intros (E); unfold; intros(x y H); cases H; clear H; [right|left] assumption;
+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.
+qed.
+
+definition eq ≝ λE:excedence.λa,b:E. ¬ (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).
+
+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.
+
+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 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 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.
+
+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 lt_coreflexive: ∀E.coreflexive ? (lt E).
+intros (E); unfold; unfold Not; intros (x H); cases H (_ ABS);
+apply (apart_coreflexive ? x ABS);
+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)]]
+qed.
+
+theorem mah: ∀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.
+
+-- altro file
+opposto TH è assioma per ordine totale.
+
+--
+
+
+
+
+
+
+
+record is_order_relation (C:Type) (le:C→C→Prop) (eq:C→C→Prop) : Type ≝ {
+ or_reflexive: reflexive ? le;
+ or_transitive: transitive ? le;
+ or_antisimmetric: antisymmetric ? le eq
+}.
+
+record ordered_set: Type ≝ {
+ os_carr:> excedence;
+ os_order_relation_properties:> is_order_relation ? (le os_carr) (apart os_carr)
+}.
+
+ordered_set.
+
+E
-interpretation "Ordered Sets le" 'leq a b =
- (cic:/matita/ordered_sets/os_le.con _ a b).
+E
theorem antisimmetric_to_cotransitive_to_transitive:
∀C.∀le:relation C. antisimmetric ? le → cotransitive ? le →
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