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/relations.ma".
14 (********** relations **********)
16 definition subR ≝ λA.λR,S:relation A.(∀a,b. R a b → S a b).
18 definition inv ≝ λA.λR:relation A.λa,b.R b a.
20 (* transitive closcure (plus) *)
22 inductive TC (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
23 |inj: ∀c. R a c → TC A R a c
24 |step : ∀b,c.TC A R a b → R b c → TC A R a c.
26 theorem trans_TC: ∀A,R,a,b,c.
27 TC A R a b → TC A R b c → TC A R a c.
28 #A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
31 theorem TC_idem: ∀A,R. exteqR … (TC A R) (TC A (TC A R)).
32 #A #R #a #b % /2/ #H (elim H) /2/
35 lemma monotonic_TC: ∀A,R,S. subR A R S → subR A (TC A R) (TC A S).
36 #A #R #S #subRS #a #b #H (elim H) /3/
39 lemma sub_TC: ∀A,R,S. subR A R (TC A S) → subR A (TC A R) (TC A S).
40 #A #R #S #Hsub #a #b #H (elim H) /3/
43 theorem sub_TC_to_eq: ∀A,R,S. subR A R S → subR A S (TC A R) →
44 exteqR … (TC A R) (TC A S).
45 #A #R #S #sub1 #sub2 #a #b % /2/
48 theorem TC_inv: ∀A,R. exteqR ?? (TC A (inv A R)) (inv A (TC A R)).
50 #H (elim H) /2/ normalize #c #d #H1 #H2 #H3 @(trans_TC … H3) /2/
54 inductive star (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
55 |inj: ∀b,c.star A R a b → R b c → star A R a c
58 lemma R_to_star: ∀A,R,a,b. R a b → star A R a b.
62 theorem trans_star: ∀A,R,a,b,c.
63 star A R a b → star A R b c → star A R a c.
64 #A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
67 theorem star_star: ∀A,R. exteqR … (star A R) (star A (star A R)).
68 #A #R #a #b % /2/ #H (elim H) /2/
71 lemma monotonic_star: ∀A,R,S. subR A R S → subR A (star A R) (star A S).
72 #A #R #S #subRS #a #b #H (elim H) /3/
75 lemma sub_star: ∀A,R,S. subR A R (star A S) →
76 subR A (star A R) (star A S).
77 #A #R #S #Hsub #a #b #H (elim H) /3/
80 theorem sub_star_to_eq: ∀A,R,S. subR A R S → subR A S (star A R) →
81 exteqR … (star A R) (star A S).
82 #A #R #S #sub1 #sub2 #a #b % /2/
85 theorem star_inv: ∀A,R.
86 exteqR ?? (star A (inv A R)) (inv A (star A R)).
88 #H (elim H) /2/ normalize #c #d #H1 #H2 #H3 @(trans_star … H3) /2/
92 ∀A,R,x,y.star A R x y → x = y ∨ ∃z.R x z ∧ star A R z y.
93 #A #R #x #y #Hstar elim Hstar
94 [ #b #c #Hleft #Hright *
95 [ #H1 %2 @(ex_intro ?? c) % //
96 | * #x0 * #H1 #H2 %2 @(ex_intro ?? x0) % /2/ ]
101 ∀A:Type[0].∀R:relation A.∀Q:A → A → Prop.
103 (∀a,b,c.R a b → star A R b c → Q b c → Q a c) →
104 ∀x,y.star A R x y → Q x y.
105 (* #A #R #Q #H1 #H2 #x #y #H0 elim H0
106 [ #b #c #Hleft #Hright #IH
107 cases (star_decomp_l ???? Hleft)
109 | * #z * #H3 #H4 @(H2 … H3) /2/
112 generalize in match (λb.H2 x b y); elim H0
113 [#b #c #Hleft #Hright #H2' #H3 @H3
116 [ #b #c #Hleft #Hright #IH //
121 lemma TC_to_star: ∀A,R,a,b.TC A R a b → star A R a b.
122 #R #A #a #b #TCH (elim TCH) /2/
125 lemma star_case: ∀A,R,a,b. star A R a b → a = b ∨ TC A R a b.
126 #A #R #a #b #H (elim H) /2/ #c #d #star_ac #Rcd * #H1 %2 /2/.
129 (* equiv -- smallest equivalence relation containing R *)
131 inductive equiv (A:Type[0]) (R:relation A) : A → A → Prop ≝
132 |inje: ∀a,b,c.equiv A R a b → R b c → equiv A R a c
133 |refle: ∀a,b.equiv A R a b
134 |syme: ∀a,b.equiv A R a b → equiv A R b a.
136 theorem trans_equiv: ∀A,R,a,b,c.
137 equiv A R a b → equiv A R b c → equiv A R a c.
138 #A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
141 theorem equiv_equiv: ∀A,R. exteqR … (equiv A R) (equiv A (equiv A R)).
145 lemma monotonic_equiv: ∀A,R,S. subR A R S → subR A (equiv A R) (equiv A S).
146 #A #R #S #subRS #a #b #H (elim H) /3/
149 lemma sub_equiv: ∀A,R,S. subR A R (equiv A S) →
150 subR A (equiv A R) (equiv A S).
151 #A #R #S #Hsub #a #b #H (elim H) /2/
154 theorem sub_equiv_to_eq: ∀A,R,S. subR A R S → subR A S (equiv A R) →
155 exteqR … (equiv A R) (equiv A S).
156 #A #R #S #sub1 #sub2 #a #b % /2/
159 (* well founded part of a relation *)
161 inductive WF (A:Type[0]) (R:relation A) : A → Prop ≝
162 | wf : ∀b.(∀a. R a b → WF A R a) → WF A R b.
164 lemma WF_antimonotonic: ∀A,R,S. subR A R S →
165 ∀a. WF A S a → WF A R a.
166 #A #R #S #subRS #a #HWF (elim HWF) #b
167 #H #Hind % #c #Rcb @Hind @subRS //
170 (* added from lambda_delta *)
172 lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2.
173 R a1 a → TC … R a a2 → TC … R a1 a2.
176 lemma TC_reflexive: ∀A,R. reflexive A R → reflexive A (TC … R).
179 lemma TC_star_ind: ∀A,R. reflexive A R → ∀a1. ∀P:predicate A.
180 P a1 → (∀a,a2. TC … R a1 a → R a a2 → P a → P a2) →
181 ∀a2. TC … R a1 a2 → P a2.
182 #A #R #H #a1 #P #Ha1 #IHa1 #a2 #Ha12 elim Ha12 -a2 /3 width=4/
185 inductive TC_dx (A:Type[0]) (R:relation A): A → A → Prop ≝
186 |inj_dx: ∀a,c. R a c → TC_dx A R a c
187 |step_dx : ∀a,b,c. R a b → TC_dx A R b c → TC_dx A R a c.
189 lemma TC_dx_strap: ∀A. ∀R: relation A.
190 ∀a,b,c. TC_dx A R a b → R b c → TC_dx A R a c.
191 #A #R #a #b #c #Hab elim Hab -a -b /3 width=3/
194 lemma TC_to_TC_dx: ∀A. ∀R: relation A.
195 ∀a1,a2. TC … R a1 a2 → TC_dx … R a1 a2.
196 #A #R #a1 #a2 #Ha12 elim Ha12 -a2 /2 width=3/
199 lemma TC_dx_to_TC: ∀A. ∀R: relation A.
200 ∀a1,a2. TC_dx … R a1 a2 → TC … R a1 a2.
201 #A #R #a1 #a2 #Ha12 elim Ha12 -a1 -a2 /2 width=3/
204 fact TC_star_ind_dx_aux: ∀A,R. reflexive A R →
205 ∀a2. ∀P:predicate A. P a2 →
206 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
207 ∀a1,a. TC … R a1 a → a = a2 → P a1.
208 #A #R #HR #a2 #P #Ha2 #H #a1 #a #Ha1
209 elim (TC_to_TC_dx ???? Ha1) -a1 -a
210 [ #a #c #Hac #H destruct /3 width=4/
211 | #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
215 lemma TC_star_ind_dx: ∀A,R. reflexive A R →
216 ∀a2. ∀P:predicate A. P a2 →
217 (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) →
218 ∀a1. TC … R a1 a2 → P a1.
219 #A #R #HR #a2 #P #Ha2 #H #a1 #Ha12
220 @(TC_star_ind_dx_aux … HR … Ha2 H … Ha12) //
223 definition Conf3: ∀A,B. relation2 A B → relation A → Prop ≝ λA,B,S,R.
224 ∀b,a1. S a1 b → ∀a2. R a1 a2 → S a2 b.
226 lemma TC_Conf3: ∀A,B,S,R. Conf3 A B S R → Conf3 A B S (TC … R).
227 #A #B #S #R #HSR #b #a1 #Ha1 #a2 #H elim H -a2 /2 width=3/