2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
7 ||A|| This file is distributed under the terms of the
8 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "basics/logic.ma".
14 (********** predicates *********)
16 definition predicate: Type[0] → Type[0]
19 (********** relations **********)
20 definition relation : Type[0] → Type[0]
23 definition relation2 : Type[0] → Type[0] → Type[0]
26 definition reflexive: ∀A.∀R :relation A.Prop
29 definition symmetric: ∀A.∀R: relation A.Prop
30 ≝ λA.λR.∀x,y:A.R x y → R y x.
32 definition transitive: ∀A.∀R:relation A.Prop
33 ≝ λA.λR.∀x,y,z:A.R x y → R y z → R x z.
35 definition irreflexive: ∀A.∀R:relation A.Prop
36 ≝ λA.λR.∀x:A.¬(R x x).
38 definition cotransitive: ∀A.∀R:relation A.Prop
39 ≝ λA.λR.∀x,y:A.R x y → ∀z:A. R x z ∨ R z y.
41 definition tight_apart: ∀A.∀eq,ap:relation A.Prop
42 ≝ λA.λeq,ap.∀x,y:A. (¬(ap x y) → eq x y) ∧
45 definition antisymmetric: ∀A.∀R:relation A.Prop
46 ≝ λA.λR.∀x,y:A. R x y → ¬(R y x).
48 (********** operations **********)
49 definition Rcomp ≝ λA.λR1,R2:relation A.λa1,a2.
50 ∃am.R1 a1 am ∧ R2 am a2.
51 interpretation "relation composition" 'compose R1 R2 = (Rcomp ? R1 R2).
53 definition Runion ≝ λA.λR1,R2:relation A.λa,b. R1 a b ∨ R2 a b.
54 interpretation "union of relations" 'union R1 R2 = (Runion ? R1 R2).
56 definition Rintersection ≝ λA.λR1,R2:relation A.λa,b.R1 a b ∧ R2 a b.
57 interpretation "interesecion of relations" 'intersects R1 R2 = (Rintersection ? R1 R2).
59 definition inv ≝ λA.λR:relation A.λa,b.R b a.
61 (*********** sub relation ***********)
62 definition subR ≝ λA.λR,S:relation A.(∀a,b. R a b → S a b).
63 interpretation "relation inclusion" 'subseteq R S = (subR ? R S).
66 ∀T.∀R:relation T.R ⊆ R.
70 lemma sub_comp_l: ∀A.∀R,R1,R2:relation A.
71 R1 ⊆ R2 → R1 ∘ R ⊆ R2 ∘ R.
72 #A #R #R1 #R2 #Hsub #a #b * #c * /4/
75 lemma sub_comp_r: ∀A.∀R,R1,R2:relation A.
76 R1 ⊆ R2 → R ∘ R1 ⊆ R ∘ R2.
77 #A #R #R1 #R2 #Hsub #a #b * #c * /4/
80 lemma sub_assoc_l: ∀A.∀R1,R2,R3:relation A.
81 R1 ∘ (R2 ∘ R3) ⊆ (R1 ∘ R2) ∘ R3.
82 #A #R1 #R2 #R3 #a #b * #c * #Hac * #d * /5/
85 lemma sub_assoc_r: ∀A.∀R1,R2,R3:relation A.
86 (R1 ∘ R2) ∘ R3 ⊆ R1 ∘ (R2 ∘ R3).
87 #A #R1 #R2 #R3 #a #b * #c * * #d * /5 width=5/
90 (************* functions ************)
93 λA,B,C:Type[0].λf:B→C.λg:A→B.λx:A.f (g x).
95 interpretation "function composition" 'compose f g = (compose ? ? ? f g).
97 definition injective: ∀A,B:Type[0].∀ f:A→B.Prop
98 ≝ λA,B.λf.∀x,y:A.f x = f y → x=y.
100 definition surjective: ∀A,B:Type[0].∀f:A→B.Prop
101 ≝λA,B.λf.∀z:B.∃x:A.z = f x.
103 definition commutative: ∀A:Type[0].∀f:A→A→A.Prop
104 ≝ λA.λf.∀x,y.f x y = f y x.
106 definition commutative2: ∀A,B:Type[0].∀f:A→A→B.Prop
107 ≝ λA,B.λf.∀x,y.f x y = f y x.
109 definition associative: ∀A:Type[0].∀f:A→A→A.Prop
110 ≝ λA.λf.∀x,y,z.f (f x y) z = f x (f y z).
112 (* functions and relations *)
113 definition monotonic : ∀A:Type[0].∀R:A→A→Prop.
115 λA.λR.λf.∀x,y:A.R x y → R (f x) (f y).
117 (* functions and functions *)
118 definition distributive: ∀A:Type[0].∀f,g:A→A→A.Prop
119 ≝ λA.λf,g.∀x,y,z:A. f x (g y z) = g (f x y) (f x z).
121 definition distributive2: ∀A,B:Type[0].∀f:A→B→B.∀g:B→B→B.Prop
122 ≝ λA,B.λf,g.∀x:A.∀y,z:B. f x (g y z) = g (f x y) (f x z).
124 lemma injective_compose : ∀A,B,C,f,g.
125 injective A B f → injective B C g → injective A C (λx.g (f x)).
128 (* extensional equality *)
130 (* moved inside sets.ma
131 definition exteqP: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝
132 λA.λP,Q.∀a.iff (P a) (Q a). *)
134 definition exteqR: ∀A,B:Type[0].∀R,S:A→B→Prop.Prop ≝
135 λA,B.λR,S.∀a.∀b.iff (R a b) (S a b).
137 definition exteqF: ∀A,B:Type[0].∀f,g:A→B.Prop ≝
138 λA,B.λf,g.∀a.f a = g a.
141 notation " x \eqP y " non associative with precedence 45
144 interpretation "functional extentional equality"
145 'eqP A x y = (exteqP A x y). *)
147 notation "x \eqR y" non associative with precedence 45
150 interpretation "functional extentional equality"
151 'eqR A B R S = (exteqR A B R S).
153 notation " f \eqF g " non associative with precedence 45
156 interpretation "functional extentional equality"
157 'eqF A B f g = (exteqF A B f g).
159 (********** relations on unboxed pairs **********)
161 definition bi_relation: Type[0] → Type[0] → Type[0]
164 definition bi_reflexive: ∀A,B. ∀R :bi_relation A B. Prop
165 ≝ λA,B,R. ∀x,y. R x y x y.