751 (* TODO: The following lemmas can be moved somewhere... *)

   752 lemma split_rsp2[quot_respect]: "((R1 ===> R2 ===> prod_rel R1 R2 ===> op =) ===>

   753   prod_rel R1 R2 ===> prod_rel R1 R2 ===> op =) split split"

   754   by auto

   755 

   756 lemma split_prs2[quot_preserve]:

   757   assumes q1: "Quotient R1 Abs1 Rep1"

   758   and q2: "Quotient R2 Abs2 Rep2"

   759   shows "((Abs1 ---> Abs2 ---> prod_fun Abs1 Abs2 ---> id) ---> prod_fun Rep1 Rep2 ---> prod_fun Rep1 Rep2 ---> id) split = split"

   760   by (simp add: expand_fun_eq Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])

   761 

   762 lemma alpha_gen2:

   763   "(bs, x1, x2) \<approx>gen (\<lambda>(x1, y1) (x2, y2). R1 x1 x2 \<and> R2 y1 y2) (\<lambda>(a, b). f1 a \<union> f2 b) pi (cs, y1, y2) =

   764   (f1 x1 \<union> f2 x2 - bs = f1 y1 \<union> f2 y2 - cs \<and> (f1 x1 \<union> f2 x2 - bs) \<sharp>* pi \<and> R1 (pi \<bullet> x1) y1 \<and> R2 (pi \<bullet> x2) y2)"

   765 by (simp add: alpha_gen)

   766