From: Andrea Asperti Date: Mon, 28 Oct 2013 14:37:01 +0000 (+0000) Subject: Splitted star X-Git-Tag: make_still_working~1059 X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=4adf9860cd26175c4d73b73e8adbb3c6ceaa19c9;p=helm.git Splitted star --- diff --git a/matita/matita/lib/basics/star.ma b/matita/matita/lib/basics/star.ma index 00a23b633..391e08530 100644 --- a/matita/matita/lib/basics/star.ma +++ b/matita/matita/lib/basics/star.ma @@ -193,348 +193,3 @@ lemma WF_antimonotonic: ∀A,R,S. subR A R S → #H #Hind % #c #Rcb @Hind @subRS // qed. -(* added from λδ *) - -lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2. - R a1 a → TC … R a a2 → TC … R a1 a2. -/3 width=3/ qed. - -lemma TC_reflexive: ∀A,R. reflexive A R → reflexive A (TC … R). -/2 width=1/ qed. - -lemma TC_star_ind: ∀A,R. reflexive A R → ∀a1. ∀P:predicate A. - P a1 → (∀a,a2. TC … R a1 a → R a a2 → P a → P a2) → - ∀a2. TC … R a1 a2 → P a2. -#A #R #H #a1 #P #Ha1 #IHa1 #a2 #Ha12 elim Ha12 -a2 /3 width=4/ -qed-. - -inductive TC_dx (A:Type[0]) (R:relation A): A → A → Prop ≝ - |inj_dx: ∀a,c. R a c → TC_dx A R a c - |step_dx : ∀a,b,c. R a b → TC_dx A R b c → TC_dx A R a c. - -lemma TC_dx_strap: ∀A. ∀R: relation A. - ∀a,b,c. TC_dx A R a b → R b c → TC_dx A R a c. -#A #R #a #b #c #Hab elim Hab -a -b /3 width=3/ -qed. - -lemma TC_to_TC_dx: ∀A. ∀R: relation A. - ∀a1,a2. TC … R a1 a2 → TC_dx … R a1 a2. -#A #R #a1 #a2 #Ha12 elim Ha12 -a2 /2 width=3/ -qed. - -lemma TC_dx_to_TC: ∀A. ∀R: relation A. - ∀a1,a2. TC_dx … R a1 a2 → TC … R a1 a2. -#A #R #a1 #a2 #Ha12 elim Ha12 -a1 -a2 /2 width=3/ -qed. - -fact TC_ind_dx_aux: ∀A,R,a2. ∀P:predicate A. - (∀a1. R a1 a2 → P a1) → - (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) → - ∀a1,a. TC … R a1 a → a = a2 → P a1. -#A #R #a2 #P #H1 #H2 #a1 #a #Ha1 -elim (TC_to_TC_dx ???? Ha1) -a1 -a -[ #a #c #Hac #H destruct /2 width=1/ -| #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/ -] -qed-. - -lemma TC_ind_dx: ∀A,R,a2. ∀P:predicate A. - (∀a1. R a1 a2 → P a1) → - (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) → - ∀a1. TC … R a1 a2 → P a1. -#A #R #a2 #P #H1 #H2 #a1 #Ha12 -@(TC_ind_dx_aux … H1 H2 … Ha12) // -qed-. - -lemma TC_symmetric: ∀A,R. symmetric A R → symmetric A (TC … R). -#A #R #HR #x #y #H @(TC_ind_dx ??????? H) -x /3 width=1/ /3 width=3/ -qed. - -lemma TC_star_ind_dx: ∀A,R. reflexive A R → - ∀a2. ∀P:predicate A. P a2 → - (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) → - ∀a1. TC … R a1 a2 → P a1. -#A #R #HR #a2 #P #Ha2 #H #a1 #Ha12 -@(TC_ind_dx … P ? H … Ha12) /3 width=4/ -qed-. - -lemma TC_Conf3: ∀A,B,S,R. Conf3 A B S R → Conf3 A B S (TC … R). -#A #B #S #R #HSR #b #a1 #Ha1 #a2 #H elim H -a2 /2 width=3/ -qed. - -(* ************ confluence of star *****************) - -lemma star_strip: ∀A,R. confluent A R → - ∀a0,a1. star … R a0 a1 → ∀a2. R a0 a2 → - ∃∃a. R a1 a & star … R a2 a. -#A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/ -#a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 -elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20 -elim (HR … Ha1 … Ha0) -a /3 width=5/ -qed-. - -lemma star_confluent: ∀A,R. confluent A R → confluent A (star … R). -#A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/ -#a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 -elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20 -elim (star_strip … HR … Ha0 … Ha1) -a /3 width=5/ -qed-. - -(* relations on unboxed pairs ***********************************************) - -inductive bi_TC (A,B) (R:bi_relation A B) (a:A) (b:B): relation2 A B ≝ - |bi_inj : ∀c,d. R a b c d → bi_TC A B R a b c d - |bi_step: ∀c,d,e,f. bi_TC A B R a b c d → R c d e f → bi_TC A B R a b e f. - -lemma bi_TC_strap: ∀A,B. ∀R:bi_relation A B. ∀a1,a,a2,b1,b,b2. - R a1 b1 a b → bi_TC … R a b a2 b2 → bi_TC … R a1 b1 a2 b2. -#A #B #R #a1 #a #a2 #b1 #b #b2 #HR #H elim H -a2 -b2 /2 width=4/ /3 width=4/ -qed. - -lemma bi_TC_reflexive: ∀A,B,R. bi_reflexive A B R → - bi_reflexive … (bi_TC … R). -/2 width=1/ qed. - -inductive bi_TC_dx (A,B) (R:bi_relation A B): bi_relation A B ≝ - |bi_inj_dx : ∀a1,a2,b1,b2. R a1 b1 a2 b2 → bi_TC_dx A B R a1 b1 a2 b2 - |bi_step_dx : ∀a1,a,a2,b1,b,b2. R a1 b1 a b → bi_TC_dx A B R a b a2 b2 → - bi_TC_dx A B R a1 b1 a2 b2. - -lemma bi_TC_dx_strap: ∀A,B. ∀R: bi_relation A B. - ∀a1,a,a2,b1,b,b2. bi_TC_dx A B R a1 b1 a b → - R a b a2 b2 → bi_TC_dx A B R a1 b1 a2 b2. -#A #B #R #a1 #a #a2 #b1 #b #b2 #H1 elim H1 -a -b /3 width=4/ -qed. - -lemma bi_TC_to_bi_TC_dx: ∀A,B. ∀R: bi_relation A B. - ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 → - bi_TC_dx … R a1 b1 a2 b2. -#A #B #R #a1 #a2 #b1 #b2 #H12 elim H12 -a2 -b2 /2 width=4/ -qed. - -lemma bi_TC_dx_to_bi_TC: ∀A,B. ∀R: bi_relation A B. - ∀a1,a2,b1,b2. bi_TC_dx … R a1 b1 a2 b2 → - bi_TC … R a1 b1 a2 b2. -#A #B #R #a1 #a2 #b1 #b2 #H12 elim H12 -a1 -a2 -b1 -b2 /2 width=4/ -qed. - -fact bi_TC_ind_dx_aux: ∀A,B,R,a2,b2. ∀P:relation2 A B. - (∀a1,b1. R a1 b1 a2 b2 → P a1 b1) → - (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) → - ∀a1,a,b1,b. bi_TC … R a1 b1 a b → a = a2 → b = b2 → P a1 b1. -#A #B #R #a2 #b2 #P #H1 #H2 #a1 #a #b1 #b #H1 -elim (bi_TC_to_bi_TC_dx … a1 a b1 b H1) -a1 -a -b1 -b -[ #a1 #x #b1 #y #H1 #Hx #Hy destruct /2 width=1/ -| #a1 #a #x #b1 #b #y #H1 #H #IH #Hx #Hy destruct /3 width=5/ -] -qed-. - -lemma bi_TC_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B. - (∀a1,b1. R a1 b1 a2 b2 → P a1 b1) → - (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) → - ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1. -#A #B #R #a2 #b2 #P #H1 #H2 #a1 #b1 #H12 -@(bi_TC_ind_dx_aux ?????? H1 H2 … H12) // -qed-. - -lemma bi_TC_symmetric: ∀A,B,R. bi_symmetric A B R → - bi_symmetric A B (bi_TC … R). -#A #B #R #HR #a1 #a2 #b1 #b2 #H21 -@(bi_TC_ind_dx … a2 b2 H21) -a2 -b2 /3 width=1/ /3 width=4/ -qed. - -lemma bi_TC_transitive: ∀A,B,R. bi_transitive A B (bi_TC … R). -#A #B #R #a1 #a #b1 #b #H elim H -a -b /2 width=4/ /3 width=4/ -qed. - -definition bi_Conf3: ∀A,B,C. relation3 A B C → predicate (bi_relation A B) ≝ - λA,B,C,S,R. - ∀c,a1,b1. S a1 b1 c → ∀a2,b2. R a1 b1 a2 b2 → S a2 b2 c. - -lemma bi_TC_Conf3: ∀A,B,C,S,R. bi_Conf3 A B C S R → bi_Conf3 A B C S (bi_TC … R). -#A #B #C #S #R #HSR #c #a1 #b1 #Hab1 #a2 #b2 #H elim H -a2 -b2 /2 width=4/ -qed. - -lemma bi_TC_star_ind: ∀A,B,R. bi_reflexive A B R → ∀a1,b1. ∀P:relation2 A B. - P a1 b1 → (∀a,a2,b,b2. bi_TC … R a1 b1 a b → R a b a2 b2 → P a b → P a2 b2) → - ∀a2,b2. bi_TC … R a1 b1 a2 b2 → P a2 b2. -#A #B #R #HR #a1 #b1 #P #H1 #IH #a2 #b2 #H12 elim H12 -a2 -b2 /3 width=5/ -qed-. - -lemma bi_TC_star_ind_dx: ∀A,B,R. bi_reflexive A B R → - ∀a2,b2. ∀P:relation2 A B. P a2 b2 → - (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) → - ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1. -#A #B #R #HR #a2 #b2 #P #H2 #IH #a1 #b1 #H12 -@(bi_TC_ind_dx … IH … a1 b1 H12) /3 width=5/ -qed-. - -definition bi_star: ∀A,B,R. bi_relation A B ≝ - λA,B,R. bi_RC A B (bi_TC … R). - -lemma bi_star_bi_reflexive: ∀A,B,R. bi_reflexive A B (bi_star … R). -/2 width=1/ qed. - -lemma bi_TC_to_bi_star: ∀A,B,R,a1,b1,a2,b2. - bi_TC A B R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2. -/2 width=1/ qed. - -lemma bi_R_to_bi_star: ∀A,B,R,a1,b1,a2,b2. - R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2. -/3 width=1/ qed. - -lemma bi_star_strap1: ∀A,B,R,a1,a,a2,b1,b,b2. bi_star A B R a1 b1 a b → - R a b a2 b2 → bi_star A B R a1 b1 a2 b2. -#A #B #R #a1 #a #a2 #b1 #b #b2 * -[ /3 width=4/ -| * #H1 #H2 destruct /2 width=1/ -] -qed. - -lemma bi_star_strap2: ∀A,B,R,a1,a,a2,b1,b,b2. R a1 b1 a b → - bi_star A B R a b a2 b2 → bi_star A B R a1 b1 a2 b2. -#A #B #R #a1 #a #a2 #b1 #b #b2 #H * -[ /3 width=4/ -| * #H1 #H2 destruct /2 width=1/ -] -qed. - -lemma bi_star_to_bi_TC_to_bi_TC: ∀A,B,R,a1,a,a2,b1,b,b2. bi_star A B R a1 b1 a b → - bi_TC A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2. -#A #B #R #a1 #a #a2 #b1 #b #b2 * -[ /2 width=4/ -| * #H1 #H2 destruct /2 width=1/ -] -qed. - -lemma bi_TC_to_bi_star_to_bi_TC: ∀A,B,R,a1,a,a2,b1,b,b2. bi_TC A B R a1 b1 a b → - bi_star A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2. -#A #B #R #a1 #a #a2 #b1 #b #b2 #H * -[ /2 width=4/ -| * #H1 #H2 destruct /2 width=1/ -] -qed. - -lemma bi_tansitive_bi_star: ∀A,B,R. bi_transitive A B (bi_star … R). -#A #B #R #a1 #a #b1 #b #H #a2 #b2 * -[ /3 width=4/ -| * #H1 #H2 destruct /2 width=1/ -] -qed. - -lemma bi_star_ind: ∀A,B,R,a1,b1. ∀P:relation2 A B. P a1 b1 → - (∀a,a2,b,b2. bi_star … R a1 b1 a b → R a b a2 b2 → P a b → P a2 b2) → - ∀a2,b2. bi_star … R a1 b1 a2 b2 → P a2 b2. -#A #B #R #a1 #b1 #P #H #IH #a2 #b2 * -[ #H12 elim H12 -a2 -b2 /2 width=5/ -H /3 width=5/ -| * #H1 #H2 destruct // -] -qed-. - -lemma bi_star_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B. P a2 b2 → - (∀a1,a,b1,b. R a1 b1 a b → bi_star … R a b a2 b2 → P a b → P a1 b1) → - ∀a1,b1. bi_star … R a1 b1 a2 b2 → P a1 b1. -#A #B #R #a2 #b2 #P #H #IH #a1 #b1 * -[ #H12 @(bi_TC_ind_dx … a1 b1 H12) -a1 -b1 /2 width=5/ -H /3 width=5/ -| * #H1 #H2 destruct // -] -qed-. - -(* relations on unboxed triples *********************************************) - -inductive tri_TC (A,B,C) (R:tri_relation A B C) (a1:A) (b1:B) (c1:C): relation3 A B C ≝ - |tri_inj : ∀a2,b2,c2. R a1 b1 c1 a2 b2 c2 → tri_TC A B C R a1 b1 c1 a2 b2 c2 - |tri_step: ∀a,a2,b,b2,c,c2. - tri_TC A B C R a1 b1 c1 a b c → R a b c a2 b2 c2 → - tri_TC A B C R a1 b1 c1 a2 b2 c2. - -lemma tri_TC_strap: ∀A,B,C. ∀R:tri_relation A B C. ∀a1,a,a2,b1,b,b2,c1,c,c2. - R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → - tri_TC … R a1 b1 c1 a2 b2 c2. -#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #HR #H elim H -a2 -b2 -c2 /2 width=5/ /3 width=5/ -qed. - -lemma tri_TC_reflexive: ∀A,B,C,R. tri_reflexive A B C R → - tri_reflexive … (tri_TC … R). -/2 width=1/ qed. - -inductive tri_TC_dx (A,B,C) (R:tri_relation A B C): tri_relation A B C ≝ - |tri_inj_dx : ∀a1,a2,b1,b2,c1,c2. R a1 b1 c1 a2 b2 c2 → tri_TC_dx A B C R a1 b1 c1 a2 b2 c2 - |tri_step_dx : ∀a1,a,a2,b1,b,b2,c1,c,c2. - R a1 b1 c1 a b c → tri_TC_dx A B C R a b c a2 b2 c2 → - tri_TC_dx A B C R a1 b1 c1 a2 b2 c2. - -lemma tri_TC_dx_strap: ∀A,B,C. ∀R: tri_relation A B C. - ∀a1,a,a2,b1,b,b2,c1,c,c2. - tri_TC_dx A B C R a1 b1 c1 a b c → - R a b c a2 b2 c2 → tri_TC_dx A B C R a1 b1 c1 a2 b2 c2. -#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H1 elim H1 -a -b -c /3 width=5/ -qed. - -lemma tri_TC_to_tri_TC_dx: ∀A,B,C. ∀R: tri_relation A B C. - ∀a1,a2,b1,b2,c1,c2. tri_TC … R a1 b1 c1 a2 b2 c2 → - tri_TC_dx … R a1 b1 c1 a2 b2 c2. -#A #B #C #R #a1 #a2 #b1 #b2 #c1 #c2 #H12 elim H12 -a2 -b2 -c2 /2 width=1/ /2 width=5/ -qed. - -lemma tri_TC_dx_to_tri_TC: ∀A,B,C. ∀R: tri_relation A B C. - ∀a1,a2,b1,b2,c1,c2. tri_TC_dx … R a1 b1 c1 a2 b2 c2 → - tri_TC … R a1 b1 c1 a2 b2 c2. -#A #B #C #R #a1 #a2 #b1 #b2 #c1 #c2 #H12 elim H12 -a1 -a2 -b1 -b2 -c1 -c2 -/2 width=1/ /2 width=5/ -qed. - -fact tri_TC_ind_dx_aux: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C. - (∀a1,b1,c1. R a1 b1 c1 a2 b2 c2→ P a1 b1 c1) → - (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) → - ∀a1,a,b1,b,c1,c. tri_TC … R a1 b1 c1 a b c → a = a2 → b = b2 → c = c2 → P a1 b1 c1. -#A #B #C #R #a2 #b2 #c2 #P #H1 #H2 #a1 #a #b1 #b #c1 #c #H1 -elim (tri_TC_to_tri_TC_dx … a1 a b1 b c1 c H1) -a1 -a -b1 -b -c1 -c -[ #a1 #x #b1 #y #c1 #z #H1 #Hx #Hy #Hz destruct /2 width=1/ -| #a1 #a #x #b1 #b #y #c1 #c #z #H1 #H #IH #Hx #Hy #Hz destruct /3 width=6/ -] -qed-. - -lemma tri_TC_ind_dx: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C. - (∀a1,b1,c1. R a1 b1 c1 a2 b2 c2 → P a1 b1 c1) → - (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) → - ∀a1,b1,c1. tri_TC … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1. -#A #B #C #R #a2 #b2 #c2 #P #H1 #H2 #a1 #b1 #c1 #H12 -@(tri_TC_ind_dx_aux ???????? H1 H2 … H12) // -qed-. - -lemma tri_TC_symmetric: ∀A,B,C,R. tri_symmetric A B C R → - tri_symmetric … (tri_TC … R). -#A #B #C #R #HR #a1 #a2 #b1 #b2 #c1 #c2 #H21 -@(tri_TC_ind_dx … a2 b2 c2 H21) -a2 -b2 -c2 /3 width=1/ /3 width=5/ -qed. - -lemma tri_TC_transitive: ∀A,B,C,R. tri_transitive A B C (tri_TC … R). -#A #B #C #R #a1 #a #b1 #b #c1 #c #H elim H -a -b -c /2 width=5/ /3 width=5/ -qed. - -definition tri_Conf4: ∀A,B,C,D. relation4 A B C D → predicate (tri_relation A B C) ≝ - λA,B,C,D,S,R. - ∀d,a1,b1,c1. S a1 b1 c1 d → ∀a2,b2,c2. R a1 b1 c1 a2 b2 c2 → S a2 b2 c2 d. - -lemma tri_TC_Conf4: ∀A,B,C,D,S,R. - tri_Conf4 A B C D S R → tri_Conf4 A B C D S (tri_TC … R). -#A #B #C #D #S #R #HSR #d #a1 #b1 #c1 #Habc1 #a2 #b2 #c2 #H elim H -a2 -b2 -c2 -/2 width=5/ -qed. - -lemma tri_TC_star_ind: ∀A,B,C,R. tri_reflexive A B C R → - ∀a1,b1,c1. ∀P:relation3 A B C. - P a1 b1 c1 → (∀a,a2,b,b2,c,c2. tri_TC … R a1 b1 c1 a b c → R a b c a2 b2 c2 → P a b c → P a2 b2 c2) → - ∀a2,b2,c2. tri_TC … R a1 b1 c1 a2 b2 c2 → P a2 b2 c2. -#A #B #C #R #HR #a1 #b1 #c1 #P #H1 #IH #a2 #b2 #c2 #H12 elim H12 -a2 -b2 -c2 -/2 width=6/ /3 width=6/ -qed-. - -lemma tri_TC_star_ind_dx: ∀A,B,C,R. tri_reflexive A B C R → - ∀a2,b2,c2. ∀P:relation3 A B C. P a2 b2 c2 → - (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) → - ∀a1,b1,c1. tri_TC … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1. -#A #B #C #R #HR #a2 #b2 #c2 #P #H2 #IH #a1 #b1 #c1 #H12 -@(tri_TC_ind_dx … IH … a1 b1 c1 H12) /3 width=6/ -qed-. diff --git a/matita/matita/lib/basics/star1.ma b/matita/matita/lib/basics/star1.ma new file mode 100644 index 000000000..79da56c47 --- /dev/null +++ b/matita/matita/lib/basics/star1.ma @@ -0,0 +1,358 @@ +(* + ||M|| This file is part of HELM, an Hypertextual, Electronic + ||A|| Library of Mathematics, developed at the Computer Science + ||T|| Department of the University of Bologna, Italy. + ||I|| + ||T|| + ||A|| This file is distributed under the terms of the + \ / GNU General Public License Version 2 + \ / + V_______________________________________________________________ *) + +include "basics/star.ma". + +(* added from λδ *) + +lemma TC_strap: ∀A. ∀R:relation A. ∀a1,a,a2. + R a1 a → TC … R a a2 → TC … R a1 a2. +/3 width=3/ qed. + +lemma TC_reflexive: ∀A,R. reflexive A R → reflexive A (TC … R). +/2 width=1/ qed. + +lemma TC_star_ind: ∀A,R. reflexive A R → ∀a1. ∀P:predicate A. + P a1 → (∀a,a2. TC … R a1 a → R a a2 → P a → P a2) → + ∀a2. TC … R a1 a2 → P a2. +#A #R #H #a1 #P #Ha1 #IHa1 #a2 #Ha12 elim Ha12 -a2 /3 width=4/ +qed-. + +inductive TC_dx (A:Type[0]) (R:relation A): A → A → Prop ≝ + |inj_dx: ∀a,c. R a c → TC_dx A R a c + |step_dx : ∀a,b,c. R a b → TC_dx A R b c → TC_dx A R a c. + +lemma TC_dx_strap: ∀A. ∀R: relation A. + ∀a,b,c. TC_dx A R a b → R b c → TC_dx A R a c. +#A #R #a #b #c #Hab elim Hab -a -b /3 width=3/ +qed. + +lemma TC_to_TC_dx: ∀A. ∀R: relation A. + ∀a1,a2. TC … R a1 a2 → TC_dx … R a1 a2. +#A #R #a1 #a2 #Ha12 elim Ha12 -a2 /2 width=3/ +qed. + +lemma TC_dx_to_TC: ∀A. ∀R: relation A. + ∀a1,a2. TC_dx … R a1 a2 → TC … R a1 a2. +#A #R #a1 #a2 #Ha12 elim Ha12 -a1 -a2 /2 width=3/ +qed. + +fact TC_ind_dx_aux: ∀A,R,a2. ∀P:predicate A. + (∀a1. R a1 a2 → P a1) → + (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) → + ∀a1,a. TC … R a1 a → a = a2 → P a1. +#A #R #a2 #P #H1 #H2 #a1 #a #Ha1 +elim (TC_to_TC_dx ???? Ha1) -a1 -a +[ #a #c #Hac #H destruct /2 width=1/ +| #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/ +] +qed-. + +lemma TC_ind_dx: ∀A,R,a2. ∀P:predicate A. + (∀a1. R a1 a2 → P a1) → + (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) → + ∀a1. TC … R a1 a2 → P a1. +#A #R #a2 #P #H1 #H2 #a1 #Ha12 +@(TC_ind_dx_aux … H1 H2 … Ha12) // +qed-. + +lemma TC_symmetric: ∀A,R. symmetric A R → symmetric A (TC … R). +#A #R #HR #x #y #H @(TC_ind_dx ??????? H) -x /3 width=1/ /3 width=3/ +qed. + +lemma TC_star_ind_dx: ∀A,R. reflexive A R → + ∀a2. ∀P:predicate A. P a2 → + (∀a1,a. R a1 a → TC … R a a2 → P a → P a1) → + ∀a1. TC … R a1 a2 → P a1. +#A #R #HR #a2 #P #Ha2 #H #a1 #Ha12 +@(TC_ind_dx … P ? H … Ha12) /3 width=4/ +qed-. + +lemma TC_Conf3: ∀A,B,S,R. Conf3 A B S R → Conf3 A B S (TC … R). +#A #B #S #R #HSR #b #a1 #Ha1 #a2 #H elim H -a2 /2 width=3/ +qed. + +(* ************ confluence of star *****************) + +lemma star_strip: ∀A,R. confluent A R → + ∀a0,a1. star … R a0 a1 → ∀a2. R a0 a2 → + ∃∃a. R a1 a & star … R a2 a. +#A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/ +#a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 +elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20 +elim (HR … Ha1 … Ha0) -a /3 width=5/ +qed-. + +lemma star_confluent: ∀A,R. confluent A R → confluent A (star … R). +#A #R #HR #a0 #a1 #H elim H -a1 /2 width=3/ +#a #a1 #_ #Ha1 #IHa0 #a2 #Ha02 +elim (IHa0 … Ha02) -a0 #a0 #Ha0 #Ha20 +elim (star_strip … HR … Ha0 … Ha1) -a /3 width=5/ +qed-. + +(* relations on unboxed pairs ***********************************************) + +inductive bi_TC (A,B) (R:bi_relation A B) (a:A) (b:B): relation2 A B ≝ + |bi_inj : ∀c,d. R a b c d → bi_TC A B R a b c d + |bi_step: ∀c,d,e,f. bi_TC A B R a b c d → R c d e f → bi_TC A B R a b e f. + +lemma bi_TC_strap: ∀A,B. ∀R:bi_relation A B. ∀a1,a,a2,b1,b,b2. + R a1 b1 a b → bi_TC … R a b a2 b2 → bi_TC … R a1 b1 a2 b2. +#A #B #R #a1 #a #a2 #b1 #b #b2 #HR #H elim H -a2 -b2 /2 width=4/ /3 width=4/ +qed. + +lemma bi_TC_reflexive: ∀A,B,R. bi_reflexive A B R → + bi_reflexive … (bi_TC … R). +/2 width=1/ qed. + +inductive bi_TC_dx (A,B) (R:bi_relation A B): bi_relation A B ≝ + |bi_inj_dx : ∀a1,a2,b1,b2. R a1 b1 a2 b2 → bi_TC_dx A B R a1 b1 a2 b2 + |bi_step_dx : ∀a1,a,a2,b1,b,b2. R a1 b1 a b → bi_TC_dx A B R a b a2 b2 → + bi_TC_dx A B R a1 b1 a2 b2. + +lemma bi_TC_dx_strap: ∀A,B. ∀R: bi_relation A B. + ∀a1,a,a2,b1,b,b2. bi_TC_dx A B R a1 b1 a b → + R a b a2 b2 → bi_TC_dx A B R a1 b1 a2 b2. +#A #B #R #a1 #a #a2 #b1 #b #b2 #H1 elim H1 -a -b /3 width=4/ +qed. + +lemma bi_TC_to_bi_TC_dx: ∀A,B. ∀R: bi_relation A B. + ∀a1,a2,b1,b2. bi_TC … R a1 b1 a2 b2 → + bi_TC_dx … R a1 b1 a2 b2. +#A #B #R #a1 #a2 #b1 #b2 #H12 elim H12 -a2 -b2 /2 width=4/ +qed. + +lemma bi_TC_dx_to_bi_TC: ∀A,B. ∀R: bi_relation A B. + ∀a1,a2,b1,b2. bi_TC_dx … R a1 b1 a2 b2 → + bi_TC … R a1 b1 a2 b2. +#A #B #R #a1 #a2 #b1 #b2 #H12 elim H12 -a1 -a2 -b1 -b2 /2 width=4/ +qed. + +fact bi_TC_ind_dx_aux: ∀A,B,R,a2,b2. ∀P:relation2 A B. + (∀a1,b1. R a1 b1 a2 b2 → P a1 b1) → + (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) → + ∀a1,a,b1,b. bi_TC … R a1 b1 a b → a = a2 → b = b2 → P a1 b1. +#A #B #R #a2 #b2 #P #H1 #H2 #a1 #a #b1 #b #H1 +elim (bi_TC_to_bi_TC_dx … a1 a b1 b H1) -a1 -a -b1 -b +[ #a1 #x #b1 #y #H1 #Hx #Hy destruct /2 width=1/ +| #a1 #a #x #b1 #b #y #H1 #H #IH #Hx #Hy destruct /3 width=5/ +] +qed-. + +lemma bi_TC_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B. + (∀a1,b1. R a1 b1 a2 b2 → P a1 b1) → + (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) → + ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1. +#A #B #R #a2 #b2 #P #H1 #H2 #a1 #b1 #H12 +@(bi_TC_ind_dx_aux ?????? H1 H2 … H12) // +qed-. + +lemma bi_TC_symmetric: ∀A,B,R. bi_symmetric A B R → + bi_symmetric A B (bi_TC … R). +#A #B #R #HR #a1 #a2 #b1 #b2 #H21 +@(bi_TC_ind_dx … a2 b2 H21) -a2 -b2 /3 width=1/ /3 width=4/ +qed. + +lemma bi_TC_transitive: ∀A,B,R. bi_transitive A B (bi_TC … R). +#A #B #R #a1 #a #b1 #b #H elim H -a -b /2 width=4/ /3 width=4/ +qed. + +definition bi_Conf3: ∀A,B,C. relation3 A B C → predicate (bi_relation A B) ≝ + λA,B,C,S,R. + ∀c,a1,b1. S a1 b1 c → ∀a2,b2. R a1 b1 a2 b2 → S a2 b2 c. + +lemma bi_TC_Conf3: ∀A,B,C,S,R. bi_Conf3 A B C S R → bi_Conf3 A B C S (bi_TC … R). +#A #B #C #S #R #HSR #c #a1 #b1 #Hab1 #a2 #b2 #H elim H -a2 -b2 /2 width=4/ +qed. + +lemma bi_TC_star_ind: ∀A,B,R. bi_reflexive A B R → ∀a1,b1. ∀P:relation2 A B. + P a1 b1 → (∀a,a2,b,b2. bi_TC … R a1 b1 a b → R a b a2 b2 → P a b → P a2 b2) → + ∀a2,b2. bi_TC … R a1 b1 a2 b2 → P a2 b2. +#A #B #R #HR #a1 #b1 #P #H1 #IH #a2 #b2 #H12 elim H12 -a2 -b2 /3 width=5/ +qed-. + +lemma bi_TC_star_ind_dx: ∀A,B,R. bi_reflexive A B R → + ∀a2,b2. ∀P:relation2 A B. P a2 b2 → + (∀a1,a,b1,b. R a1 b1 a b → bi_TC … R a b a2 b2 → P a b → P a1 b1) → + ∀a1,b1. bi_TC … R a1 b1 a2 b2 → P a1 b1. +#A #B #R #HR #a2 #b2 #P #H2 #IH #a1 #b1 #H12 +@(bi_TC_ind_dx … IH … a1 b1 H12) /3 width=5/ +qed-. + +definition bi_star: ∀A,B,R. bi_relation A B ≝ + λA,B,R. bi_RC A B (bi_TC … R). + +lemma bi_star_bi_reflexive: ∀A,B,R. bi_reflexive A B (bi_star … R). +/2 width=1/ qed. + +lemma bi_TC_to_bi_star: ∀A,B,R,a1,b1,a2,b2. + bi_TC A B R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2. +/2 width=1/ qed. + +lemma bi_R_to_bi_star: ∀A,B,R,a1,b1,a2,b2. + R a1 b1 a2 b2 → bi_star A B R a1 b1 a2 b2. +/3 width=1/ qed. + +lemma bi_star_strap1: ∀A,B,R,a1,a,a2,b1,b,b2. bi_star A B R a1 b1 a b → + R a b a2 b2 → bi_star A B R a1 b1 a2 b2. +#A #B #R #a1 #a #a2 #b1 #b #b2 * +[ /3 width=4/ +| * #H1 #H2 destruct /2 width=1/ +] +qed. + +lemma bi_star_strap2: ∀A,B,R,a1,a,a2,b1,b,b2. R a1 b1 a b → + bi_star A B R a b a2 b2 → bi_star A B R a1 b1 a2 b2. +#A #B #R #a1 #a #a2 #b1 #b #b2 #H * +[ /3 width=4/ +| * #H1 #H2 destruct /2 width=1/ +] +qed. + +lemma bi_star_to_bi_TC_to_bi_TC: ∀A,B,R,a1,a,a2,b1,b,b2. bi_star A B R a1 b1 a b → + bi_TC A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2. +#A #B #R #a1 #a #a2 #b1 #b #b2 * +[ /2 width=4/ +| * #H1 #H2 destruct /2 width=1/ +] +qed. + +lemma bi_TC_to_bi_star_to_bi_TC: ∀A,B,R,a1,a,a2,b1,b,b2. bi_TC A B R a1 b1 a b → + bi_star A B R a b a2 b2 → bi_TC A B R a1 b1 a2 b2. +#A #B #R #a1 #a #a2 #b1 #b #b2 #H * +[ /2 width=4/ +| * #H1 #H2 destruct /2 width=1/ +] +qed. + +lemma bi_tansitive_bi_star: ∀A,B,R. bi_transitive A B (bi_star … R). +#A #B #R #a1 #a #b1 #b #H #a2 #b2 * +[ /3 width=4/ +| * #H1 #H2 destruct /2 width=1/ +] +qed. + +lemma bi_star_ind: ∀A,B,R,a1,b1. ∀P:relation2 A B. P a1 b1 → + (∀a,a2,b,b2. bi_star … R a1 b1 a b → R a b a2 b2 → P a b → P a2 b2) → + ∀a2,b2. bi_star … R a1 b1 a2 b2 → P a2 b2. +#A #B #R #a1 #b1 #P #H #IH #a2 #b2 * +[ #H12 elim H12 -a2 -b2 /2 width=5/ -H /3 width=5/ +| * #H1 #H2 destruct // +] +qed-. + +lemma bi_star_ind_dx: ∀A,B,R,a2,b2. ∀P:relation2 A B. P a2 b2 → + (∀a1,a,b1,b. R a1 b1 a b → bi_star … R a b a2 b2 → P a b → P a1 b1) → + ∀a1,b1. bi_star … R a1 b1 a2 b2 → P a1 b1. +#A #B #R #a2 #b2 #P #H #IH #a1 #b1 * +[ #H12 @(bi_TC_ind_dx … a1 b1 H12) -a1 -b1 /2 width=5/ -H /3 width=5/ +| * #H1 #H2 destruct // +] +qed-. + +(* relations on unboxed triples *********************************************) + +inductive tri_TC (A,B,C) (R:tri_relation A B C) (a1:A) (b1:B) (c1:C): relation3 A B C ≝ + |tri_inj : ∀a2,b2,c2. R a1 b1 c1 a2 b2 c2 → tri_TC A B C R a1 b1 c1 a2 b2 c2 + |tri_step: ∀a,a2,b,b2,c,c2. + tri_TC A B C R a1 b1 c1 a b c → R a b c a2 b2 c2 → + tri_TC A B C R a1 b1 c1 a2 b2 c2. + +lemma tri_TC_strap: ∀A,B,C. ∀R:tri_relation A B C. ∀a1,a,a2,b1,b,b2,c1,c,c2. + R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → + tri_TC … R a1 b1 c1 a2 b2 c2. +#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #HR #H elim H -a2 -b2 -c2 /2 width=5/ /3 width=5/ +qed. + +lemma tri_TC_reflexive: ∀A,B,C,R. tri_reflexive A B C R → + tri_reflexive … (tri_TC … R). +/2 width=1/ qed. + +inductive tri_TC_dx (A,B,C) (R:tri_relation A B C): tri_relation A B C ≝ + |tri_inj_dx : ∀a1,a2,b1,b2,c1,c2. R a1 b1 c1 a2 b2 c2 → tri_TC_dx A B C R a1 b1 c1 a2 b2 c2 + |tri_step_dx : ∀a1,a,a2,b1,b,b2,c1,c,c2. + R a1 b1 c1 a b c → tri_TC_dx A B C R a b c a2 b2 c2 → + tri_TC_dx A B C R a1 b1 c1 a2 b2 c2. + +lemma tri_TC_dx_strap: ∀A,B,C. ∀R: tri_relation A B C. + ∀a1,a,a2,b1,b,b2,c1,c,c2. + tri_TC_dx A B C R a1 b1 c1 a b c → + R a b c a2 b2 c2 → tri_TC_dx A B C R a1 b1 c1 a2 b2 c2. +#A #B #C #R #a1 #a #a2 #b1 #b #b2 #c1 #c #c2 #H1 elim H1 -a -b -c /3 width=5/ +qed. + +lemma tri_TC_to_tri_TC_dx: ∀A,B,C. ∀R: tri_relation A B C. + ∀a1,a2,b1,b2,c1,c2. tri_TC … R a1 b1 c1 a2 b2 c2 → + tri_TC_dx … R a1 b1 c1 a2 b2 c2. +#A #B #C #R #a1 #a2 #b1 #b2 #c1 #c2 #H12 elim H12 -a2 -b2 -c2 /2 width=1/ /2 width=5/ +qed. + +lemma tri_TC_dx_to_tri_TC: ∀A,B,C. ∀R: tri_relation A B C. + ∀a1,a2,b1,b2,c1,c2. tri_TC_dx … R a1 b1 c1 a2 b2 c2 → + tri_TC … R a1 b1 c1 a2 b2 c2. +#A #B #C #R #a1 #a2 #b1 #b2 #c1 #c2 #H12 elim H12 -a1 -a2 -b1 -b2 -c1 -c2 +/2 width=1/ /2 width=5/ +qed. + +fact tri_TC_ind_dx_aux: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C. + (∀a1,b1,c1. R a1 b1 c1 a2 b2 c2→ P a1 b1 c1) → + (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) → + ∀a1,a,b1,b,c1,c. tri_TC … R a1 b1 c1 a b c → a = a2 → b = b2 → c = c2 → P a1 b1 c1. +#A #B #C #R #a2 #b2 #c2 #P #H1 #H2 #a1 #a #b1 #b #c1 #c #H1 +elim (tri_TC_to_tri_TC_dx … a1 a b1 b c1 c H1) -a1 -a -b1 -b -c1 -c +[ #a1 #x #b1 #y #c1 #z #H1 #Hx #Hy #Hz destruct /2 width=1/ +| #a1 #a #x #b1 #b #y #c1 #c #z #H1 #H #IH #Hx #Hy #Hz destruct /3 width=6/ +] +qed-. + +lemma tri_TC_ind_dx: ∀A,B,C,R,a2,b2,c2. ∀P:relation3 A B C. + (∀a1,b1,c1. R a1 b1 c1 a2 b2 c2 → P a1 b1 c1) → + (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) → + ∀a1,b1,c1. tri_TC … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1. +#A #B #C #R #a2 #b2 #c2 #P #H1 #H2 #a1 #b1 #c1 #H12 +@(tri_TC_ind_dx_aux ???????? H1 H2 … H12) // +qed-. + +lemma tri_TC_symmetric: ∀A,B,C,R. tri_symmetric A B C R → + tri_symmetric … (tri_TC … R). +#A #B #C #R #HR #a1 #a2 #b1 #b2 #c1 #c2 #H21 +@(tri_TC_ind_dx … a2 b2 c2 H21) -a2 -b2 -c2 /3 width=1/ /3 width=5/ +qed. + +lemma tri_TC_transitive: ∀A,B,C,R. tri_transitive A B C (tri_TC … R). +#A #B #C #R #a1 #a #b1 #b #c1 #c #H elim H -a -b -c /2 width=5/ /3 width=5/ +qed. + +definition tri_Conf4: ∀A,B,C,D. relation4 A B C D → predicate (tri_relation A B C) ≝ + λA,B,C,D,S,R. + ∀d,a1,b1,c1. S a1 b1 c1 d → ∀a2,b2,c2. R a1 b1 c1 a2 b2 c2 → S a2 b2 c2 d. + +lemma tri_TC_Conf4: ∀A,B,C,D,S,R. + tri_Conf4 A B C D S R → tri_Conf4 A B C D S (tri_TC … R). +#A #B #C #D #S #R #HSR #d #a1 #b1 #c1 #Habc1 #a2 #b2 #c2 #H elim H -a2 -b2 -c2 +/2 width=5/ +qed. + +lemma tri_TC_star_ind: ∀A,B,C,R. tri_reflexive A B C R → + ∀a1,b1,c1. ∀P:relation3 A B C. + P a1 b1 c1 → (∀a,a2,b,b2,c,c2. tri_TC … R a1 b1 c1 a b c → R a b c a2 b2 c2 → P a b c → P a2 b2 c2) → + ∀a2,b2,c2. tri_TC … R a1 b1 c1 a2 b2 c2 → P a2 b2 c2. +#A #B #C #R #HR #a1 #b1 #c1 #P #H1 #IH #a2 #b2 #c2 #H12 elim H12 -a2 -b2 -c2 +/2 width=6/ /3 width=6/ +qed-. + +lemma tri_TC_star_ind_dx: ∀A,B,C,R. tri_reflexive A B C R → + ∀a2,b2,c2. ∀P:relation3 A B C. P a2 b2 c2 → + (∀a1,a,b1,b,c1,c. R a1 b1 c1 a b c → tri_TC … R a b c a2 b2 c2 → P a b c → P a1 b1 c1) → + ∀a1,b1,c1. tri_TC … R a1 b1 c1 a2 b2 c2 → P a1 b1 c1. +#A #B #C #R #HR #a2 #b2 #c2 #P #H2 #IH #a1 #b1 #c1 #H12 +@(tri_TC_ind_dx … IH … a1 b1 c1 H12) /3 width=6/ +qed-.