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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 set "baseuri" "cic:/matita/constructive_higher_order_relations".
17 include "constructive_connectives.ma".
19 definition cotransitive ≝
20 λC:Type.λlt:C→C→Type.∀x,y,z:C. lt x y → lt x z ∨ lt z y.
22 definition coreflexive ≝ λC:Type.λlt:C→C→Type. ∀x:C. ¬ (lt x x).
24 definition antisymmetric ≝
25 λC:Type.λle:C→C→Type.λeq:C→C→Type.∀x,y:C.le x y → le y x → eq x y.
27 definition symmetric ≝
28 λC:Type.λle:C→C→Type.∀x,y:C.le x y → le y x.
30 definition transitive ≝
31 λC:Type.λle:C→C→Type.∀x,y,z:C.le x y → le y z → le x z.
33 definition associative ≝
34 λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y,z. eq (op x (op y z)) (op (op x y) z).
36 definition commutative ≝
37 λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y. eq (op x y) (op y x).
39 alias id "antisymmetric" = "cic:/matita/higher_order_defs/relations/antisymmetric.con".
40 theorem antisimmetric_to_cotransitive_to_transitive:
41 ∀C:Type.∀le:C→C→Prop. antisymmetric ? le → cotransitive ? le → transitive ? le.
42 intros (T f Af cT); unfold transitive; intros (x y z fxy fyz);
43 lapply (cT ??z fxy) as H; cases H; [assumption] cases (Af ? ? fyz H1);
46 lemma Or_symmetric: symmetric ? Or.
47 unfold; intros (x y H); cases H; [right|left] assumption;