matita/contribs/dama/dama/constructive_higher_order_relations.ma

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  14
  15
  16
  17 include "constructive_connectives.ma".
  18 include "higher_order_defs/relations.ma".
  19
  20 definition cotransitive ≝
  21  λC:Type.λlt:C→C→Type.∀x,y,z:C. lt x y → lt x z ∨ lt z y.
  22
  23 definition coreflexive ≝ λC:Type.λlt:C→C→Type. ∀x:C. ¬ (lt x x).
  24
  25 definition antisymmetric ≝
  26  λC:Type.λle:C→C→Type.λeq:C→C→Type.∀x,y:C.le x y → le y x → eq x y.
  27
  28 definition symmetric ≝
  29  λC:Type.λle:C→C→Type.∀x,y:C.le x y → le y x.
  30
  31 definition transitive ≝
  32  λC:Type.λle:C→C→Type.∀x,y,z:C.le x y → le y z → le x z.
  33
  34 definition associative ≝
  35  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
  37 definition commutative ≝
  38  λC:Type.λop:C→C→C.λeq:C→C→Type.∀x,y. eq (op x y) (op y x).
  39
  40 alias id "antisymmetric" = "cic:/matita/higher_order_defs/relations/antisymmetric.con".
  41 theorem antisimmetric_to_cotransitive_to_transitive:
  42  ∀C:Type.∀le:C→C→Prop. antisymmetric ? le → cotransitive ? le → transitive ? le.
  43 intros (T f Af cT); unfold transitive; intros (x y z fxy fyz);
  44 lapply (cT ??z fxy) as H; cases H; [assumption] cases (Af ? ? fyz H1);
  45 qed.
  46
  47 lemma Or_symmetric: symmetric ? Or.
  48 unfold; intros (x y H); cases H; [right|left] assumption;
  49 qed.
  50
  51