1 include "basics/logic.ma".
3 (********** relations **********)
4 definition relation : Type[0] → Type[0]
7 definition reflexive: ∀A.∀R :relation A.Prop
10 definition symmetric: ∀A.∀R: relation A.Prop
11 ≝ λA.λR.∀x,y:A.R x y → R y x.
13 definition transitive: ∀A.∀R:relation A.Prop
14 ≝ λA.λR.∀x,y,z:A.R x y → R y z → R x z.
16 definition irreflexive: ∀A.∀R:relation A.Prop
17 ≝ λA.λR.∀x:A.¬(R x x).
19 definition cotransitive: ∀A.∀R:relation A.Prop
20 ≝ λA.λR.∀x,y:A.R x y → ∀z:A. R x z ∨ R z y.
22 definition tight_apart: ∀A.∀eq,ap:relation A.Prop
23 ≝ λA.λeq,ap.∀x,y:A. (¬(ap x y) → eq x y) ∧
26 definition antisymmetric: ∀A.∀R:relation A.Prop
27 ≝ λA.λR.∀x,y:A. R x y → ¬(R y x).
29 (********** functions **********)
32 λA,B,C:Type[0].λf:B→C.λg:A→B.λx:A.f (g x).
34 interpretation "function composition" 'compose f g = (compose ? ? ? f g).
36 definition injective: ∀A,B:Type[0].∀ f:A→B.Prop
37 ≝ λA,B.λf.∀x,y:A.f x = f y → x=y.
39 definition surjective: ∀A,B:Type[0].∀f:A→B.Prop
40 ≝λA,B.λf.∀z:B.∃x:A.z = f x.
42 definition commutative: ∀A:Type[0].∀f:A→A→A.Prop
43 ≝ λA.λf.∀x,y.f x y = f y x.
45 definition commutative2: ∀A,B:Type[0].∀f:A→A→B.Prop
46 ≝ λA,B.λf.∀x,y.f x y = f y x.
48 definition associative: ∀A:Type[0].∀f:A→A→A.Prop
49 ≝ λA.λf.∀x,y,z.f (f x y) z = f x (f y z).
51 (* functions and relations *)
52 definition monotonic : ∀A:Type[0].∀R:A→A→Prop.
54 λA.λR.λf.∀x,y:A.R x y → R (f x) (f y).
56 (* functions and functions *)
57 definition distributive: ∀A:Type[0].∀f,g:A→A→A.Prop
58 ≝ λA.λf,g.∀x,y,z:A. f x (g y z) = g (f x y) (f x z).
60 definition distributive2: ∀A,B:Type[0].∀f:A→B→B.∀g:B→B→B.Prop
61 ≝ λA,B.λf,g.∀x:A.∀y,z:B. f x (g y z) = g (f x y) (f x z).
63 (* extensional equality *)
65 definition exteqR: ∀A,B:Type[0].∀R,S:A→B→Prop.Prop ≝
66 λA,B.λR,S.∀a.∀b.iff (R a b) (S a b).
68 definition exteqF: ∀A,B:Type[0].∀f,g:A→B.Prop ≝
69 λA,B.λf,g.∀a.f a = g a.
71 notation "x \eqR y" non associative with precedence 45
74 interpretation "functional extentional equality"
75 'eqR A B R S = (exteqR A B R S).
77 notation " f \eqF g " non associative with precedence 45
80 interpretation "functional extentional equality"
81 'eqF A B f g = (exteqF A B f g).