lemma in_fun:
  shows "x \<in> ((f ---> g) s) = g (f x \<in> s)"
  by (simp add: mem_def)

lemma respects_thm:
  shows "Respects (R1 ===> R2) f = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (f y))"
  unfolding Respects_def
  by (simp add: expand_fun_eq)

lemma respects_rep_abs:
  assumes a: "Quotient R1 Abs1 Rep1"
  and     b: "Respects (R1 ===> R2) f"
  and     c: "R1 x x"
  shows "R2 (f (Rep1 (Abs1 x))) (f x)"
  using a b[simplified respects_thm] c unfolding Quotient_def
  by blast

lemma respects_mp:
  assumes a: "Respects (R1 ===> R2) f"
  and     b: "R1 x y"
  shows "R2 (f x) (f y)"
  using a b unfolding Respects_def
  by simp

lemma respects_o:
  assumes a: "Respects (R2 ===> R3) f"
  and     b: "Respects (R1 ===> R2) g"
  shows "Respects (R1 ===> R3) (f o g)"
  using a b unfolding Respects_def
  by simp

lemma fun_rel_eq_rel:
  assumes q1: "Quotient R1 Abs1 Rep1"
  and     q2: "Quotient R2 Abs2 Rep2"
  shows "(R1 ===> R2) f g = ((Respects (R1 ===> R2) f) \<and> (Respects (R1 ===> R2) g)
                             \<and> ((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g))"
  using fun_quotient[OF q1 q2] unfolding Respects_def Quotient_def expand_fun_eq
  by blast

lemma let_babs:
  "v \<in> r \<Longrightarrow> Let v (Babs r lam) = Let v lam"
  by (simp add: Babs_def)

lemma fun_rel_equals:
  assumes q1: "Quotient R1 Abs1 Rep1"
  and     q2: "Quotient R2 Abs2 Rep2"
  and     r1: "Respects (R1 ===> R2) f"
  and     r2: "Respects (R1 ===> R2) g"
  shows "((Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g) = (\<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"
  apply(rule_tac iffI)
  apply(rule)+
  apply (rule apply_rsp'[of "R1" "R2"])
  apply(subst Quotient_rel[OF fun_quotient[OF q1 q2]])
  apply auto
  using fun_quotient[OF q1 q2] r1 r2 unfolding Quotient_def Respects_def
  apply (metis let_rsp q1)
  apply (metis fun_rel_eq_rel let_rsp q1 q2 r2)
  using r1 unfolding Respects_def expand_fun_eq
  apply(simp (no_asm_use))
  apply(metis Quotient_rel[OF q2] Quotient_rel_rep[OF q1])
  done

(* ask Peter: fun_rel_IMP used twice *)
lemma fun_rel_IMP2:
  assumes q1: "Quotient R1 Abs1 Rep1"
  and     q2: "Quotient R2 Abs2 Rep2"
  and     r1: "Respects (R1 ===> R2) f"
  and     r2: "Respects (R1 ===> R2) g"
  and     a:  "(Rep1 ---> Abs2) f = (Rep1 ---> Abs2) g"
  shows "R1 x y \<Longrightarrow> R2 (f x) (g y)"
  using q1 q2 r1 r2 a
  by (simp add: fun_rel_equals)

lemma lambda_rep_abs_rsp:
  assumes r1: "\<And>r r'. R1 r r' \<Longrightarrow>R1 r (Rep1 (Abs1 r'))"
  and     r2: "\<And>r r'. R2 r r' \<Longrightarrow>R2 r (Rep2 (Abs2 r'))"
  shows "(R1 ===> R2) f1 f2 \<Longrightarrow> (R1 ===> R2) f1 ((Abs1 ---> Rep2) ((Rep1 ---> Abs2) f2))"
  using r1 r2 by auto

(* We use id_simps which includes id_apply; so these 2 theorems can be removed *)
lemma id_prs:
  assumes q: "Quotient R Abs Rep"
  shows "Abs (id (Rep e)) = id e"
  using Quotient_abs_rep[OF q] by auto

lemma id_rsp:
  assumes q: "Quotient R Abs Rep"
  and     a: "R e1 e2"
  shows "R (id e1) (id e2)"
  using a by auto