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".
13 include "basics/core_notation/compose_2.ma".
15 (********** predicates *********)
17 definition predicate: Type[0] → Type[0]
20 (********** relations **********)
21 definition relation : Type[0] → Type[0]
24 definition relation2 : Type[0] → Type[0] → Type[0]
27 definition relation3 : Type[0] → Type[0] → Type[0] → Type[0]
30 definition relation4 : Type[0] → Type[0] → Type[0] → Type[0] → Type[0]
31 ≝ λA,B,C,D.A→B→C→D→Prop.
33 definition reflexive: ∀A.∀R :relation A.Prop
36 definition symmetric: ∀A.∀R: relation A.Prop
37 ≝ λA.λR.∀x,y:A.R x y → R y x.
39 definition transitive: ∀A.∀R:relation A.Prop
40 ≝ λA.λR.∀x,y,z:A.R x y → R y z → R x z.
42 definition irreflexive: ∀A.∀R:relation A.Prop
43 ≝ λA.λR.∀x:A.¬(R x x).
45 definition cotransitive: ∀A.∀R:relation A.Prop
46 ≝ λA.λR.∀x,y:A.R x y → ∀z:A. R x z ∨ R z y.
48 definition tight_apart: ∀A.∀eq,ap:relation A.Prop
49 ≝ λA.λeq,ap.∀x,y:A. (¬(ap x y) → eq x y) ∧
52 definition antisymmetric: ∀A.∀R:relation A.Prop
53 ≝ λA.λR.∀x,y:A. R x y → ¬(R y x).
55 definition singlevalued: ∀A,B. predicate (relation2 A B) ≝ λA,B,R.
56 ∀a,b1. R a b1 → ∀b2. R a b2 → b1 = b2.
58 definition confluent1: ∀A. relation A → predicate A ≝ λA,R,a0.
59 ∀a1. R a0 a1 → ∀a2. R a0 a2 →
62 definition confluent: ∀A. predicate (relation A) ≝ λA,R.
63 ∀a0. confluent1 … R a0.
65 (* triangular confluence of two relations *)
66 definition Conf3: ∀A,B. relation2 A B → relation A → Prop ≝ λA,B,S,R.
67 ∀b,a1. S a1 b → ∀a2. R a1 a2 → S a2 b.
69 (* Reflexive closure ************)
71 definition RC: ∀A:Type[0]. relation A → relation A ≝
72 λA,R,x,y. R … x y ∨ x = y.
74 lemma RC_reflexive: ∀A,R. reflexive A (RC … R).
77 (********** operations **********)
78 definition Rcomp ≝ λA.λR1,R2:relation A.λa1,a2.
79 ∃am.R1 a1 am ∧ R2 am a2.
80 interpretation "relation composition" 'compose R1 R2 = (Rcomp ? R1 R2).
82 definition Runion ≝ λA.λR1,R2:relation A.λa,b. R1 a b ∨ R2 a b.
83 interpretation "union of relations" 'union R1 R2 = (Runion ? R1 R2).
85 definition Rintersection ≝ λA.λR1,R2:relation A.λa,b.R1 a b ∧ R2 a b.
86 interpretation "interesecion of relations" 'intersects R1 R2 = (Rintersection ? R1 R2).
88 definition inv ≝ λA.λR:relation A.λa,b.R b a.
90 (*********** sub relation ***********)
91 definition subR ≝ λA.λR,S:relation A.(∀a,b. R a b → S a b).
92 interpretation "relation inclusion" 'subseteq R S = (subR ? R S).
95 ∀T.∀R:relation T.R ⊆ R.
99 lemma sub_comp_l: ∀A.∀R,R1,R2:relation A.
100 R1 ⊆ R2 → R1 ∘ R ⊆ R2 ∘ R.
101 #A #R #R1 #R2 #Hsub #a #b * #c * /4/
104 lemma sub_comp_r: ∀A.∀R,R1,R2:relation A.
105 R1 ⊆ R2 → R ∘ R1 ⊆ R ∘ R2.
106 #A #R #R1 #R2 #Hsub #a #b * #c * /4/
109 lemma sub_assoc_l: ∀A.∀R1,R2,R3:relation A.
110 R1 ∘ (R2 ∘ R3) ⊆ (R1 ∘ R2) ∘ R3.
111 #A #R1 #R2 #R3 #a #b * #c * #Hac * #d * /5/
114 lemma sub_assoc_r: ∀A.∀R1,R2,R3:relation A.
115 (R1 ∘ R2) ∘ R3 ⊆ R1 ∘ (R2 ∘ R3).
116 #A #R1 #R2 #R3 #a #b * #c * * #d * /5 width=5/
119 (************* functions ************)
122 λA,B,C:Type[0].λf:B→C.λg:A→B.λx:A.f (g x).
124 interpretation "function composition" 'compose f g = (compose ? ? ? f g).
126 definition injective: ∀A,B:Type[0].∀ f:A→B.Prop
127 ≝ λA,B.λf.∀x,y:A.f x = f y → x=y.
129 definition surjective: ∀A,B:Type[0].∀f:A→B.Prop
130 ≝λA,B.λf.∀z:B.∃x:A.z = f x.
132 definition commutative: ∀A:Type[0].∀f:A→A→A.Prop
133 ≝ λA.λf.∀x,y.f x y = f y x.
135 definition commutative2: ∀A,B:Type[0].∀f:A→A→B.Prop
136 ≝ λA,B.λf.∀x,y.f x y = f y x.
138 definition associative: ∀A:Type[0].∀f:A→A→A.Prop
139 ≝ λA.λf.∀x,y,z.f (f x y) z = f x (f y z).
141 (* functions and relations *)
142 definition monotonic : ∀A:Type[0].∀R:A→A→Prop.
144 λA.λR.λf.∀x,y:A.R x y → R (f x) (f y).
146 (* functions and functions *)
147 definition distributive: ∀A:Type[0].∀f,g:A→A→A.Prop
148 ≝ λA.λf,g.∀x,y,z:A. f x (g y z) = g (f x y) (f x z).
150 definition distributive2: ∀A,B:Type[0].∀f:A→B→B.∀g:B→B→B.Prop
151 ≝ λA,B.λf,g.∀x:A.∀y,z:B. f x (g y z) = g (f x y) (f x z).
153 lemma injective_compose : ∀A,B,C,f,g.
154 injective A B f → injective B C g → injective A C (λx.g (f x)).
157 (* extensional equality *)
159 (* moved inside sets.ma
160 definition exteqP: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝
161 λA.λP,Q.∀a.iff (P a) (Q a). *)
163 definition exteqR: ∀A,B:Type[0].∀R,S:A→B→Prop.Prop ≝
164 λA,B.λR,S.∀a.∀b.iff (R a b) (S a b).
166 definition exteqF: ∀A,B:Type[0].∀f,g:A→B.Prop ≝
167 λA,B.λf,g.∀a.f a = g a.
170 notation " x \eqP y " non associative with precedence 45
173 interpretation "functional extentional equality"
174 'eqP A x y = (exteqP A x y). *)
176 notation "x \eqR y" non associative with precedence 45
179 interpretation "functional extentional equality"
180 'eqR A B R S = (exteqR A B R S).
182 notation " f \eqF g " non associative with precedence 45
185 interpretation "functional extentional equality"
186 'eqF A B f g = (exteqF A B f g).
188 (********** relations on unboxed pairs **********)
190 definition bi_relation: Type[0] → Type[0] → Type[0]
193 definition bi_reflexive: ∀A,B. ∀R:bi_relation A B. Prop
194 ≝ λA,B,R. ∀a,b. R a b a b.
196 definition bi_symmetric: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
197 ∀a1,a2,b1,b2. R a2 b2 a1 b1 → R a1 b1 a2 b2.
199 definition bi_transitive: ∀A,B. ∀R: bi_relation A B. Prop ≝ λA,B,R.
200 ∀a1,a,b1,b. R a1 b1 a b →
201 ∀a2,b2. R a b a2 b2 → R a1 b1 a2 b2.
203 definition bi_RC: ∀A,B:Type[0]. bi_relation A B → bi_relation A B ≝
204 λA,B,R,a1,b1,a2,b2. R … a1 b1 a2 b2 ∨ (a1 = a2 ∧ b1 = b2).
206 lemma bi_RC_reflexive: ∀A,B,R. bi_reflexive A B (bi_RC … R).
209 (********** relations on unboxed triples **********)
211 definition tri_relation: Type[0] → Type[0] → Type[0] → Type[0]
212 ≝ λA,B,C.A→B→C→A→B→C→Prop.
214 definition tri_reflexive: ∀A,B,C. ∀R:tri_relation A B C. Prop
215 ≝ λA,B,C,R. ∀a,b,c. R a b c a b c.
217 definition tri_symmetric: ∀A,B,C. ∀R: tri_relation A B C. Prop ≝ λA,B,C,R.
219 R a2 b2 c2 a1 b1 c1 → R a1 b1 c1 a2 b2 c2.
221 definition tri_transitive: ∀A,B,C. ∀R: tri_relation A B C. Prop ≝ λA,B,C,R.
222 ∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c →
223 ∀a2,b2,c2. R a b c a2 b2 c2 → R a1 b1 c1 a2 b2 c2.