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".
16 include "datatypes/constructors.ma".
18 inductive Or (A,B:CProp) : CProp ≝
22 interpretation "constructive or" 'or x y = (Or x y).
24 inductive And (A,B:CProp) : CProp ≝
25 | Conj : A → B → And A B.
27 interpretation "constructive and" 'and x y = (And x y).
29 inductive And3 (A,B,C:CProp) : CProp ≝
30 | Conj3 : A → B → C → And3 A B C.
32 notation < "a ∧ b ∧ c" with precedence 35 for @{'and3 $a $b $c}.
34 interpretation "constructive ternary and" 'and3 x y z = (Conj3 x y z).
36 inductive And4 (A,B,C,D:CProp) : CProp ≝
37 | Conj4 : A → B → C → D → And4 A B C D.
39 notation < "a ∧ b ∧ c ∧ d" with precedence 35 for @{'and4 $a $b $c $d}.
41 interpretation "constructive quaternary and" 'and4 x y z t = (Conj4 x y z t).
43 inductive exT (A:Type) (P:A→CProp) : CProp ≝
44 ex_introT: ∀w:A. P w → exT A P.
46 interpretation "CProp exists" 'exists \eta.x = (exT _ x).
48 notation "\ll term 19 a, break term 19 b \gg"
49 with precedence 90 for @{'dependent_pair $a $b}.
50 interpretation "dependent pair" 'dependent_pair a b =
54 definition pi1exT ≝ λA,P.λx:exT A P.match x with [ex_introT x _ ⇒ x].
56 λA,P.λx:exT A P.match x return λx.P (pi1exT ?? x) with [ex_introT _ p ⇒ p].
58 interpretation "exT \fst" 'pi1 = (pi1exT _ _).
59 interpretation "exT \fst" 'pi1a x = (pi1exT _ _ x).
60 interpretation "exT \fst" 'pi1b x y = (pi1exT _ _ x y).
61 interpretation "exT \snd" 'pi2 = (pi2exT _ _).
62 interpretation "exT \snd" 'pi2a x = (pi2exT _ _ x).
63 interpretation "exT \snd" 'pi2b x y = (pi2exT _ _ x y).
65 inductive exT23 (A:Type) (P:A→CProp) (Q:A→CProp) (R:A→A→CProp) : CProp ≝
66 ex_introT23: ∀w,p:A. P w → Q p → R w p → exT23 A P Q R.
69 λA,P,Q,R.λx:exT23 A P Q R.match x with [ex_introT23 x _ _ _ _ ⇒ x].
71 λA,P,Q,R.λx:exT23 A P Q R.match x with [ex_introT23 _ x _ _ _ ⇒ x].
73 interpretation "exT2 \fst" 'pi1 = (pi1exT23 _ _ _ _).
74 interpretation "exT2 \snd" 'pi2 = (pi2exT23 _ _ _ _).
75 interpretation "exT2 \fst" 'pi1a x = (pi1exT23 _ _ _ _ x).
76 interpretation "exT2 \snd" 'pi2a x = (pi2exT23 _ _ _ _ x).
77 interpretation "exT2 \fst" 'pi1b x y = (pi1exT23 _ _ _ _ x y).
78 interpretation "exT2 \snd" 'pi2b x y = (pi2exT23 _ _ _ _ x y).
80 definition Not : CProp → Prop ≝ λx:CProp.x → False.
82 interpretation "constructive not" 'not x = (Not x).
84 definition cotransitive ≝
85 λC:Type.λlt:C→C→CProp.∀x,y,z:C. lt x y → lt x z ∨ lt z y.
87 definition coreflexive ≝ λC:Type.λlt:C→C→CProp. ∀x:C. ¬ (lt x x).
89 definition symmetric ≝ λC:Type.λlt:C→C→CProp. ∀x,y:C.lt x y → lt y x.
91 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.
93 definition reflexive ≝ λA:Type.λR:A→A→CProp.∀x:A.R x x.
95 definition transitive ≝ λA:Type.λR:A→A→CProp.∀x,y,z:A.R x y → R y z → R x z.