1 (**************************************************************************)
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 = (Or x y).
23 inductive And (A,B:CProp) : CProp ≝
24 | Conj : A → B → And A B.
26 interpretation "constructive and" 'and x y = (And x y).
28 inductive exT (A:Type) (P:A→CProp) : CProp ≝
29 ex_introT: ∀w:A. P w → exT A P.
31 interpretation "CProp exists" 'exists \eta.x = (exT _ x).
33 definition Not : CProp → Prop ≝ λx:CProp.x → False.
35 interpretation "constructive not" 'not x = (Not x).
37 definition cotransitive ≝
38 λC:Type.λlt:C→C→CProp.∀x,y,z:C. lt x y → lt x z ∨ lt z y.
40 definition coreflexive ≝ λC:Type.λlt:C→C→CProp. ∀x:C. ¬ (lt x x).
42 definition symmetric ≝ λC:Type.λlt:C→C→CProp. ∀x,y:C.lt x y → lt y x.
44 definition antisymmetric ≝ λA:Type.λR:A→A→CProp.λeq:A→A→Prop.∀x:A.∀y:A.R x y→R y x→eq x y.
46 definition reflexive ≝ λA:Type.λR:A→A→CProp.∀x:A.R x x.
48 definition transitive ≝ λA:Type.λR:A→A→CProp.∀x,y,z:A.R x y → R y z → R x z.