include "basics/relations.ma".
-(********** relations **********)
-
-definition subR ≝ λA.λR,S:relation A.(∀a,b. R a b → S a b).
-
-definition inv ≝ λA.λR:relation A.λa,b.R b a.
-
(* transitive closcure (plus) *)
inductive TC (A:Type[0]) (R:relation A) (a:A): A → Prop ≝
(* 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/
| /2/ ]
qed.
-axiom 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) →
- ∀x,y.star A R x y → Q x y.
-(* #A #R #Q #H1 #H2 #x #y #H0 elim H0
-[ #b #c #Hleft #Hright #IH
- cases (star_decomp_l ???? Hleft)
- [ #Hx @H2 //
- | * #z * #H3 #H4 @(H2 … H3) /2/
-[
-|
-generalize in match (λb.H2 x b y); elim H0
-[#b #c #Hleft #Hright #H2' #H3 @H3
- @(H3 b)
- elim H0
-[ #b #c #Hleft #Hright #IH //
-| *)
-
-(* RC and star *)
+(* 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.
+
+fact star_ind_l_aux: ∀A,R,a2. ∀P:predicate A.
+ P a2 →
+ (∀a1,a. R a1 a → star … R a a2 → P a → P a1) →
+ ∀a1,a. star … R a1 a → a = a2 → P a1.
+#A #R #a2 #P #H1 #H2 #a1 #a #Ha1
+elim (star_to_starl ???? Ha1) -a1 -a
+[ #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
+| #a #H destruct /2 width=1/
+]
+qed-.
+
+(* imporeved version of star_ind_l with "left_parameter" *)
+lemma star_ind_l: ∀A,R,a2. ∀P:predicate A.
+ P a2 →
+ (∀a1,a. R a1 a → star … R a a2 → P a → P a1) →
+ ∀a1. star … R a1 a2 → P a1.
+#A #R #a2 #P #H1 #H2 #a1 #Ha12
+@(star_ind_l_aux … H1 H2 … Ha12) //
+qed.
+
+(* TC 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/
#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_star_ind_dx_aux: ∀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,a. TC … R a1 a → a = a2 → P a1.
-#A #R #HR #a2 #P #Ha2 #H #a1 #a #Ha1
-elim (TC_to_TC_dx ???? Ha1) -a1 -a
-[ #a #c #Hac #H destruct /3 width=4/
-| #a #b #c #Hab #Hbc #IH #H destruct /3 width=4/
-]
-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_star_ind_dx_aux … HR … Ha2 H … Ha12) //
-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.