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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 *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "logic/equality.ma".
17 inductive Or (A,B:CProp) : CProp ≝
21 interpretation "constructive or" 'or x y =
22 (cic:/matita/dama/cprop_connectives/Or.ind#xpointer(1/1) x y).
24 inductive And (A,B:CProp) : CProp ≝
25 | Conj : A → B → And A B.
27 interpretation "constructive and" 'and x y =
28 (cic:/matita/dama/cprop_connectives/And.ind#xpointer(1/1) x y).
30 inductive exT (A:Type) (P:A→CProp) : CProp ≝
31 ex_introT: ∀w:A. P w → exT A P.
33 interpretation "CProp exists" 'exists \eta.x =
34 (cic:/matita/dama/cprop_connectives/exT.ind#xpointer(1/1) _ x).
36 inductive False : CProp ≝ .
38 definition Not ≝ λx:CProp.x → False.
40 interpretation "constructive not" 'not x =
41 (cic:/matita/dama/cprop_connectives/Not.con x).
43 definition cotransitive ≝
44 λC:Type.λlt:C→C→CProp.∀x,y,z:C. lt x y → lt x z ∨ lt z y.
46 definition coreflexive ≝ λC:Type.λlt:C→C→CProp. ∀x:C. ¬ (lt x x).
48 definition symmetric ≝ λC:Type.λlt:C→C→CProp. ∀x,y:C.lt x y → lt y x.
50 definition antisymmetric ≝ λA:Type.λR:A→A→CProp.λeq:A→A→CProp.∀x:A.∀y:A.R x y→R y x→eq x y.
52 definition reflexive ≝ λA:Type.λR:A→A→CProp.∀x:A.R x x.
54 definition transitive ≝ λA:Type.λR:A→A→CProp.∀x,y,z:A.R x y → R y z → R x z.