include "ground/lib/star.ma".
include "ground/lib/ltc.ma".
-(* LABELLED TRANSITIVE CLOSURE **********************************************)
+(* LABELLED TRANSITIVE CLOSURE FOR RELATIONS ********************************)
-alias symbol "subseteq" = "relation inclusion". (**)
+alias symbol "subseteq" = "relation inclusion". (* * alias *)
-(* Constructions with contextual transitive closure *************************)
+(* Constructions with ctc ***************************************************)
lemma ltc_CTC (C) (A) (i) (f) (B) (R:relation4 C A B B):
- left_identity … f i →
- ∀c. CTC … (λc. R c i) c ⊆ ltc … f … (R c) i.
+ left_identity … f i →
+ ∀c. CTC … (λc. R c i) c ⊆ ltc … f … (R c) i.
#C #A #i #f #B #R #Hf #c #b1 #b2 #H elim H -b2 /2 width=1 by ltc_rc/
#b #b2 #_ #Hb2 #IH >(Hf i) -Hf /2 width=3 by ltc_dx/
qed.
-(* Inversions with contextual transitive closure ****************************)
+(* Inversions with ctc ******************************************************)
lemma ltc_inv_CTC (C) (A) (i) (f) (B) (R:relation4 C A B B):
- associative … f → annulment_2 … f i →
- ∀c. ltc … f … (R c) i ⊆ CTC … (λc. R c i) c.
+ associative … f → annulment_2 … f i →
+ ∀c. ltc … f … (R c) i ⊆ CTC … (λc. R c i) c.
#C #A #i #f #B #R #H1f #H2f #c #b1 #b2
@(insert_eq_1 … i) #a #H
@(ltc_ind_dx A f B … H) -a -b2 /2 width=1 by inj/ -H1f