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