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.
+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
#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): A → B → Prop ≝
+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.
-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/
+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).
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-.