include "basics/relations.ma".
-(********** relations **********)
+(* transitive closcure (plus) *)
+inductive TC (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
+ |inj: ∀c. R a c → TC A R a c
+ |step : ∀b,c.TC A R a b → R b c → TC A R a c.
+
+theorem trans_TC: ∀A,R,a,b,c.
+ TC A R a b → TC A R b c → TC A R a c.
+#A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
+qed.
+
+theorem TC_idem: ∀A,R. exteqR … (TC A R) (TC A (TC A R)).
+#A #R #a #b % /2/ #H (elim H) /2/
+qed.
+
+lemma monotonic_TC: ∀A,R,S. subR A R S → subR A (TC A R) (TC A S).
+#A #R #S #subRS #a #b #H (elim H) /3/
+qed.
+
+lemma sub_TC: ∀A,R,S. subR A R (TC A S) → subR A (TC A R) (TC A S).
+#A #R #S #Hsub #a #b #H (elim H) /3/
+qed.
+
+theorem sub_TC_to_eq: ∀A,R,S. subR A R S → subR A S (TC A R) →
+ exteqR … (TC A R) (TC A S).
+#A #R #S #sub1 #sub2 #a #b % /2/
+qed.
+
+theorem TC_inv: ∀A,R. exteqR ?? (TC A (inv A R)) (inv A (TC A R)).
+#A #R #a #b %
+#H (elim H) /2/ normalize #c #d #H1 #H2 #H3 @(trans_TC … H3) /2/
+qed.
+
+(* star *)
inductive star (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
- |inj: ∀b,c.star A R a b → R b c → star A R a c
- |refl: star A R a a.
+ |sstep: ∀b,c.star A R a b → R b c → star A R a c
+ |srefl: star A R a a.
+
+lemma R_to_star: ∀A,R,a,b. R a b → star A R a b.
+#A #R #a #b /2/
+qed.
theorem trans_star: ∀A,R,a,b,c.
star A R a b → star A R b c → star A R a c.
#A #R #a #b % /2/ #H (elim H) /2/
qed.
-definition subR ≝ λA.λR,S:relation A.(∀a,b. R a b → S a b).
-
lemma monotonic_star: ∀A,R,S. subR A R S → subR A (star A R) (star A S).
#A #R #S #subRS #a #b #H (elim H) /3/
qed.
#A #R #S #sub1 #sub2 #a #b % /2/
qed.
+theorem star_inv: ∀A,R.
+ exteqR ?? (star A (inv A R)) (inv A (star A R)).
+#A #R #a #b %
+#H (elim H) /2/ normalize #c #d #H1 #H2 #H3 @(trans_star … H3) /2/
+qed.
+
+lemma star_decomp_l :
+ ∀A,R,x,y.star A R x y → x = y ∨ ∃z.R x z ∧ star A R z y.
+#A #R #x #y #Hstar elim Hstar
+[ #b #c #Hleft #Hright *
+ [ #H1 %2 @(ex_intro ?? c) % //
+ | * #x0 * #H1 #H2 %2 @(ex_intro ?? x0) % /2/ ]
+| /2/ ]
+qed.
+
+(* right associative version of star *)
+inductive starl (A:Type[0]) (R:relation A) : A → A → Prop ≝
+ |sstepl: ∀a,b,c.R a b → starl A R b c → starl A R a c
+ |refll: ∀a.starl A R a a.
+
+lemma starl_comp : ∀A,R,a,b,c.
+ starl A R a b → R b c → starl A R a c.
+#A #R #a #b #c #Hstar elim Hstar
+ [#a1 #b1 #c1 #Rab #sbc #Hind #a1 @(sstepl … Rab) @Hind //
+ |#a1 #Rac @(sstepl … Rac) //
+ ]
+qed.
+
+lemma star_compl : ∀A,R,a,b,c.
+ R a b → star A R b c → star A R a c.
+#A #R #a #b #c #Rab #Hstar elim Hstar
+ [#b1 #c1 #sbb1 #Rb1c1 #Hind @(sstep … Rb1c1) @Hind
+ |@(sstep … Rab) //
+ ]
+qed.
+
+lemma star_to_starl: ∀A,R,a,b.star A R a b → starl A R a b.
+#A #R #a #b #Hs elim Hs //
+#d #c #sad #Rdc #sad @(starl_comp … Rdc) //
+qed.
+
+lemma starl_to_star: ∀A,R,a,b.starl A R a b → star A R a b.
+#A #R #a #b #Hs elim Hs // -Hs -b -a
+#a #b #c #Rab #sbc #sbc @(star_compl … Rab) //
+qed.
+
+lemma star_ind_l :
+ ∀A:Type[0].∀R:relation A.∀Q:A → A → Prop.
+ (∀a.Q a a) →
+ (∀a,b,c.R a b → star A R b c → Q b c → Q a c) →
+ ∀a,b.star A R a b → Q a b.
+#A #R #Q #H1 #H2 #a #b #H0
+elim (star_to_starl ???? H0) // -H0 -b -a
+#a #b #c #Rab #slbc @H2 // @starl_to_star //
+qed.
+
+(* RC and star *)
+
+lemma TC_to_star: ∀A,R,a,b.TC A R a b → star A R a b.
+#R #A #a #b #TCH (elim TCH) /2/
+qed.
+
+lemma star_case: ∀A,R,a,b. star A R a b → a = b ∨ TC A R a b.
+#A #R #a #b #H (elim H) /2/ #c #d #star_ac #Rcd * #H1 %2 /2/.
+qed.
+
(* equiv -- smallest equivalence relation containing R *)
inductive equiv (A:Type[0]) (R:relation A) : A → A → Prop ≝
theorem trans_equiv: ∀A,R,a,b,c.
equiv A R a b → equiv A R b c → equiv A R a c.
-#A #R #a #b #c #Hab #Hbc (inversion Hbc) /2/
+#A #R #a #b #c #Hab #Hbc (elim Hbc) /2/
qed.
theorem equiv_equiv: ∀A,R. exteqR … (equiv A R) (equiv A (equiv A R)).
#H #Hind % #c #Rcb @Hind @subRS //
qed.
+(* added from lambda_delta *)
+
+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-.
+
+definition Conf3: ∀A,B. relation2 A B → relation A → Prop ≝ λA,B,S,R.
+ ∀b,a1. S a1 b → ∀a2. R a1 a2 → S a2 b.
+
+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.
+
+inductive bi_TC (A,B:Type[0]) (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 A B (bi_TC … R).
+/2 width=1/ qed.
+
+inductive bi_TC_dx (A,B:Type[0]) (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 ??????? 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 ?????????? 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 → bi_relation A B → Prop ≝ 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 … P ? IH … H12) /3 width=5/
+qed-.
+
+definition bi_star: ∀A,B,R. bi_relation A B ≝ λA,B,R,a1,b1,a2,b2.
+ (a1 = a2 ∧ b1 = b2) ∨ bi_TC A B R a1 b1 a2 b2.
+
+lemma bi_star_bi_reflexive: ∀A,B,R. bi_reflexive A B (bi_star … R).
+/3 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 *
+[ * #H1 #H2 destruct /2 width=1/
+| /3 width=4/
+]
+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 *
+[ * #H1 #H2 destruct /2 width=1/
+| /3 width=4/
+]
+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 *
+[ * #H1 #H2 destruct /2 width=1/
+| /2 width=4/
+]
+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 *
+[ * #H1 #H2 destruct /2 width=1/
+| /2 width=4/
+]
+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 *
+[ * #H1 #H2 destruct /2 width=1/
+| /3 width=4/
+]
+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 *
+[ * #H1 #H2 destruct //
+| #H12 elim H12 -a2 -b2 /2 width=5/ -H /3 width=5/
+]
+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 *
+[ * #H1 #H2 destruct //
+| #H12 @(bi_TC_ind_dx ?????????? H12) -a1 -b1 /2 width=5/ -H /3 width=5/
+]
+qed-.
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