
diff -r 9b1ad366827f -r f51e2b3e3149 QuotScript.thy
--- a/QuotScript.thy	Fri Dec 04 14:35:36 2009 +0100
+++ b/QuotScript.thy	Fri Dec 04 15:04:05 2009 +0100
@@ -3,102 +3,106 @@
 begin
 
 definition 
-  "EQUIV E \<equiv> \<forall>x y. E x y = (E x = E y)" 
+  "equivp E \<equiv> \<forall>x y. E x y = (E x = E y)" 
 
 definition
-  "REFL E \<equiv> \<forall>x. E x x"
+  "reflp E \<equiv> \<forall>x. E x x"
 
 definition 
-  "SYM E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"
+  "symp E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"
 
 definition
-  "TRANS E \<equiv> \<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z"
+  "transp E \<equiv> \<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z"
 
-lemma EQUIV_REFL_SYM_TRANS:
-  shows "EQUIV E = (REFL E \<and> SYM E \<and> TRANS E)"
-unfolding EQUIV_def REFL_def SYM_def TRANS_def expand_fun_eq
+lemma equivp_reflp_symp_transp:
+  shows "equivp E = (reflp E \<and> symp E \<and> transp E)"
+unfolding equivp_def reflp_def symp_def transp_def expand_fun_eq
 by (blast)
 
-lemma EQUIV_REFL:
-  shows "EQUIV E \<Longrightarrow> (\<And>x. E x x)"
-  by (simp add: EQUIV_REFL_SYM_TRANS REFL_def)
+lemma equivp_refl:
+  shows "equivp R \<Longrightarrow> (\<And>x. R x x)"
+  by (simp add: equivp_reflp_symp_transp reflp_def)
+
+lemma equivp_reflp:
+  shows "equivp E \<Longrightarrow> (\<And>x. E x x)"
+  by (simp add: equivp_reflp_symp_transp reflp_def)
 
 definition
-  "PART_EQUIV E \<equiv> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"
+  "PART_equivp E \<equiv> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"
 
-lemma EQUIV_IMP_PART_EQUIV:
-  assumes a: "EQUIV E"
-  shows "PART_EQUIV E"
-using a unfolding EQUIV_def PART_EQUIV_def
+lemma equivp_IMP_PART_equivp:
+  assumes a: "equivp E"
+  shows "PART_equivp E"
+using a unfolding equivp_def PART_equivp_def
 by auto
 
 definition
-  "QUOTIENT E Abs Rep \<equiv> (\<forall>a. Abs (Rep a) = a) \<and> 
+  "Quotient E Abs Rep \<equiv> (\<forall>a. Abs (Rep a) = a) \<and> 
                         (\<forall>a. E (Rep a) (Rep a)) \<and> 
                         (\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"
 
-lemma QUOTIENT_ABS_REP:
-  assumes a: "QUOTIENT E Abs Rep"
+lemma Quotient_ABS_REP:
+  assumes a: "Quotient E Abs Rep"
   shows "Abs (Rep a) = a" 
-using a unfolding QUOTIENT_def
+using a unfolding Quotient_def
 by simp
 
-lemma QUOTIENT_REP_REFL:
-  assumes a: "QUOTIENT E Abs Rep"
+lemma Quotient_REP_reflp:
+  assumes a: "Quotient E Abs Rep"
   shows "E (Rep a) (Rep a)" 
-using a unfolding QUOTIENT_def
+using a unfolding Quotient_def
 by blast
 
-lemma QUOTIENT_REL:
-  assumes a: "QUOTIENT E Abs Rep"
+lemma Quotient_REL:
+  assumes a: "Quotient E Abs Rep"
   shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"
-using a unfolding QUOTIENT_def
+using a unfolding Quotient_def
 by blast
 
-lemma QUOTIENT_REL_ABS:
-  assumes a: "QUOTIENT E Abs Rep"
+lemma Quotient_REL_ABS:
+  assumes a: "Quotient E Abs Rep"
   shows "E r s \<Longrightarrow> Abs r = Abs s"
-using a unfolding QUOTIENT_def
+using a unfolding Quotient_def
 by blast
 
-lemma QUOTIENT_REL_ABS_EQ:
-  assumes a: "QUOTIENT E Abs Rep"
+lemma Quotient_REL_ABS_EQ:
+  assumes a: "Quotient E Abs Rep"
   shows "E r r \<Longrightarrow> E s s \<Longrightarrow> E r s = (Abs r = Abs s)"
-using a unfolding QUOTIENT_def
+using a unfolding Quotient_def
 by blast
 
-lemma QUOTIENT_REL_REP:
-  assumes a: "QUOTIENT R Abs Rep"
+lemma Quotient_REL_REP:
+  assumes a: "Quotient R Abs Rep"
   shows "R (Rep a) (Rep b) = (a = b)"
-using a unfolding QUOTIENT_def
+using a unfolding Quotient_def
 by metis
 
-lemma QUOTIENT_REP_ABS:
-  assumes a: "QUOTIENT R Abs Rep"
+lemma Quotient_REP_ABS:
+  assumes a: "Quotient R Abs Rep"
   shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
-using a unfolding QUOTIENT_def
+using a unfolding Quotient_def
 by blast
 
-lemma IDENTITY_EQUIV:
-  shows "EQUIV (op =)"
-unfolding EQUIV_def
+lemma IDENTITY_equivp:
+  shows "equivp (op =)"
+unfolding equivp_def
 by auto
 
-lemma IDENTITY_QUOTIENT:
-  shows "QUOTIENT (op =) id id"
-unfolding QUOTIENT_def id_def
+lemma IDENTITY_Quotient:
+  shows "Quotient (op =) id id"
+unfolding Quotient_def id_def
 by blast
 
-lemma QUOTIENT_SYM:
-  assumes a: "QUOTIENT E Abs Rep"
-  shows "SYM E"
-using a unfolding QUOTIENT_def SYM_def
+lemma Quotient_symp:
+  assumes a: "Quotient E Abs Rep"
+  shows "symp E"
+using a unfolding Quotient_def symp_def
 by metis
 
-lemma QUOTIENT_TRANS:
-  assumes a: "QUOTIENT E Abs Rep"
-  shows "TRANS E"
-using a unfolding QUOTIENT_def TRANS_def
+lemma Quotient_transp:
+  assumes a: "Quotient E Abs Rep"
+  shows "transp E"
+using a unfolding Quotient_def transp_def
 by metis
 
 fun
@@ -139,38 +143,38 @@
   "(op =) ===> (op =) \<equiv> (op =)"
 by (rule eq_reflection) (simp add: expand_fun_eq)
 
-lemma FUN_QUOTIENT:
-  assumes q1: "QUOTIENT R1 abs1 rep1"
-  and     q2: "QUOTIENT R2 abs2 rep2"
-  shows "QUOTIENT (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
+lemma FUN_Quotient:
+  assumes q1: "Quotient R1 abs1 rep1"
+  and     q2: "Quotient R2 abs2 rep2"
+  shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
 proof -
   have "\<forall>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
     apply(simp add: expand_fun_eq)
     using q1 q2
-    apply(simp add: QUOTIENT_def)
+    apply(simp add: Quotient_def)
     done
   moreover
   have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
     apply(auto)
-    using q1 q2 unfolding QUOTIENT_def
+    using q1 q2 unfolding Quotient_def
     apply(metis)
     done
   moreover
   have "\<forall>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and> 
         (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
     apply(auto simp add: expand_fun_eq)
-    using q1 q2 unfolding QUOTIENT_def
+    using q1 q2 unfolding Quotient_def
     apply(metis)
-    using q1 q2 unfolding QUOTIENT_def
+    using q1 q2 unfolding Quotient_def
     apply(metis)
-    using q1 q2 unfolding QUOTIENT_def
+    using q1 q2 unfolding Quotient_def
     apply(metis)
-    using q1 q2 unfolding QUOTIENT_def
+    using q1 q2 unfolding Quotient_def
     apply(metis)
     done
   ultimately
-  show "QUOTIENT (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
-    unfolding QUOTIENT_def by blast
+  show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
+    unfolding Quotient_def by blast
 qed
 
 definition
@@ -195,11 +199,11 @@
 by simp
 
 lemma RESPECTS_REP_ABS:
-  assumes a: "QUOTIENT R1 Abs1 Rep1"
+  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
+using a b[simplified RESPECTS_THM] c unfolding Quotient_def
 by blast
 
 lemma RESPECTS_o:
@@ -216,18 +220,18 @@
 *)
 
 lemma FUN_REL_EQ_REL:
-  assumes q1: "QUOTIENT R1 Abs1 Rep1"
-  and     q2: "QUOTIENT R2 Abs2 Rep2"
+  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
+using FUN_Quotient[OF q1 q2] unfolding Respects_def Quotient_def expand_fun_eq
 by blast
 
 (* TODO: it is the same as APPLY_RSP *)
 (* q1 and q2 not used; see next lemma *)
 lemma FUN_REL_MP:
-  assumes q1: "QUOTIENT R1 Abs1 Rep1"
-  and     q2: "QUOTIENT R2 Abs2 Rep2"
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
   shows "(R1 ===> R2) f g \<Longrightarrow> R1 x y \<Longrightarrow> R2 (f x) (g y)"
 by simp
 
@@ -236,23 +240,23 @@
 by simp
 
 lemma FUN_REL_EQUALS:
-  assumes q1: "QUOTIENT R1 Abs1 Rep1"
-  and     q2: "QUOTIENT R2 Abs2 Rep2"
+  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)
-using FUN_QUOTIENT[OF q1 q2] r1 r2 unfolding QUOTIENT_def Respects_def
+using FUN_Quotient[OF q1 q2] r1 r2 unfolding Quotient_def Respects_def
 apply(metis FUN_REL_IMP)
 using r1 unfolding Respects_def expand_fun_eq
 apply(simp (no_asm_use))
-apply(metis QUOTIENT_REL[OF q2] QUOTIENT_REL_REP[OF q1])
+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"
+  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"
@@ -260,56 +264,56 @@
 using q1 q2 r1 r2 a
 by (simp add: FUN_REL_EQUALS)
 
-(* We don't use it, it is exactly the same as QUOTIENT_REL_REP but wrong way *)
+(* We don't use it, it is exactly the same as Quotient_REL_REP but wrong way *)
 lemma EQUALS_PRS:
-  assumes q: "QUOTIENT R Abs Rep"
+  assumes q: "Quotient R Abs Rep"
   shows "(x = y) = R (Rep x) (Rep y)"
-by (rule QUOTIENT_REL_REP[OF q, symmetric])
+by (rule Quotient_REL_REP[OF q, symmetric])
 
 lemma equals_rsp:
-  assumes q: "QUOTIENT R Abs Rep"
+  assumes q: "Quotient R Abs Rep"
   and     a: "R xa xb" "R ya yb"
   shows "R xa ya = R xb yb"
-using QUOTIENT_SYM[OF q] QUOTIENT_TRANS[OF q] unfolding SYM_def TRANS_def
+using Quotient_symp[OF q] Quotient_transp[OF q] unfolding symp_def transp_def
 using a by blast
 
 lemma lambda_prs:
-  assumes q1: "QUOTIENT R1 Abs1 Rep1"
-  and     q2: "QUOTIENT R2 Abs2 Rep2"
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
 unfolding expand_fun_eq
-using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2]
+using Quotient_ABS_REP[OF q1] Quotient_ABS_REP[OF q2]
 by simp
 
 lemma lambda_prs1:
-  assumes q1: "QUOTIENT R1 Abs1 Rep1"
-  and     q2: "QUOTIENT R2 Abs2 Rep2"
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
 unfolding expand_fun_eq
-using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2]
+using Quotient_ABS_REP[OF q1] Quotient_ABS_REP[OF q2]
 by simp
 
 (* Not used since applic_prs proves a version for an arbitrary number of arguments *)
 lemma APP_PRS:
-  assumes q1: "QUOTIENT R1 abs1 rep1"
-  and     q2: "QUOTIENT R2 abs2 rep2"
+  assumes q1: "Quotient R1 abs1 rep1"
+  and     q2: "Quotient R2 abs2 rep2"
   shows "abs2 ((abs1 ---> rep2) f (rep1 x)) = f x"
 unfolding expand_fun_eq
-using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2]
+using Quotient_ABS_REP[OF q1] Quotient_ABS_REP[OF q2]
 by simp
 
 (* Ask Peter: assumption q1 and q2 not used and lemma is the 'identity' *)
 lemma LAMBDA_RSP:
-  assumes q1: "QUOTIENT R1 Abs1 Rep1"
-  and     q2: "QUOTIENT R2 Abs2 Rep2"
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
   and     a: "(R1 ===> R2) f1 f2"
   shows "(R1 ===> R2) (\<lambda>x. f1 x) (\<lambda>y. f2 y)"
 by (rule a)
 
 (* ASK Peter about next four lemmas in quotientScript
 lemma ABSTRACT_PRS:
-  assumes q1: "QUOTIENT R1 Abs1 Rep1"
-  and     q2: "QUOTIENT R2 Abs2 Rep2"
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
   shows "f = (Rep1 ---> Abs2) ???"
 *)
 
@@ -320,17 +324,17 @@
 using r1 r2 by auto
 
 lemma REP_ABS_RSP:
-  assumes q: "QUOTIENT R Abs Rep"
+  assumes q: "Quotient R Abs Rep"
   and     a: "R x1 x2"
   shows "R x1 (Rep (Abs x2))"
-using q a by (metis QUOTIENT_REL[OF q] QUOTIENT_ABS_REP[OF q] QUOTIENT_REP_REFL[OF q])
+using q a by (metis Quotient_REL[OF q] Quotient_ABS_REP[OF q] Quotient_REP_reflp[OF q])
 
 (* Not used *)
 lemma REP_ABS_RSP_LEFT:
-  assumes q: "QUOTIENT R Abs Rep"
+  assumes q: "Quotient R Abs Rep"
   and     a: "R x1 x2"
   shows "R x1 (Rep (Abs x2))"
-using q a by (metis QUOTIENT_REL[OF q] QUOTIENT_ABS_REP[OF q] QUOTIENT_REP_REFL[OF q])
+using q a by (metis Quotient_REL[OF q] Quotient_ABS_REP[OF q] Quotient_REP_reflp[OF q])
 
 (* ----------------------------------------------------- *)
 (* Quantifiers: FORALL, EXISTS, EXISTS_UNIQUE,           *)
@@ -340,26 +344,26 @@
 (* bool theory: COND, LET *)
 
 lemma IF_PRS:
-  assumes q: "QUOTIENT R Abs Rep"
+  assumes q: "Quotient R Abs Rep"
   shows "If a b c = Abs (If a (Rep b) (Rep c))"
-using QUOTIENT_ABS_REP[OF q] by auto
+using Quotient_ABS_REP[OF q] by auto
 
 (* ask peter: no use of q *)
 lemma IF_RSP:
-  assumes q: "QUOTIENT R Abs Rep"
+  assumes q: "Quotient R Abs Rep"
   and     a: "a1 = a2" "R b1 b2" "R c1 c2"
   shows "R (If a1 b1 c1) (If a2 b2 c2)"
 using a by auto
 
 lemma LET_PRS:
-  assumes q1: "QUOTIENT R1 Abs1 Rep1"
-  and     q2: "QUOTIENT R2 Abs2 Rep2"
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
   shows "Let x f = Abs2 (Let (Rep1 x) ((Abs1 ---> Rep2) f))"
-using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2] by auto
+using Quotient_ABS_REP[OF q1] Quotient_ABS_REP[OF q2] by auto
 
 lemma LET_RSP:
-  assumes q1: "QUOTIENT R1 Abs1 Rep1"
-  and     q2: "QUOTIENT R2 Abs2 Rep2"
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
   and     a1: "(R1 ===> R2) f g"
   and     a2: "R1 x y"
   shows "R2 (Let x f) (Let y g)"
@@ -376,17 +380,17 @@
 
 (* Not used *)
 lemma APPLY_PRS:
-  assumes q1: "QUOTIENT R1 Abs1 Rep1"
-  and     q2: "QUOTIENT R2 Abs2 Rep2"
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
   shows "f x = Abs2 (((Abs1 ---> Rep2) f) (Rep1 x))"
-using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2] by auto
+using Quotient_ABS_REP[OF q1] Quotient_ABS_REP[OF q2] by auto
 
 (* In the following theorem R1 can be instantiated with anything,
    but we know some of the types of the Rep and Abs functions;
-   so by solving QUOTIENT assumptions we can get a unique R2 that
+   so by solving Quotient assumptions we can get a unique R2 that
    will be provable; which is why we need to use APPLY_RSP *)
 lemma apply_rsp:
-  assumes q: "QUOTIENT R1 Abs1 Rep1"
+  assumes q: "Quotient R1 Abs1 Rep1"
   and     a: "(R1 ===> R2) f g" "R1 x y"
   shows "R2 ((f::'a\<Rightarrow>'c) x) ((g::'a\<Rightarrow>'c) y)"
 using a by (rule FUN_REL_IMP)
@@ -401,29 +405,29 @@
 
 (* We use id_simps which includes id_apply; so these 2 theorems can be removed *)
 lemma I_PRS:
-  assumes q: "QUOTIENT R Abs Rep"
+  assumes q: "Quotient R Abs Rep"
   shows "id e = Abs (id (Rep e))"
-using QUOTIENT_ABS_REP[OF q] by auto
+using Quotient_ABS_REP[OF q] by auto
 
 lemma I_RSP:
-  assumes q: "QUOTIENT R Abs Rep"
+  assumes q: "Quotient R Abs Rep"
   and     a: "R e1 e2"
   shows "R (id e1) (id e2)"
 using a by auto
 
 lemma o_PRS:
-  assumes q1: "QUOTIENT R1 Abs1 Rep1"
-  and     q2: "QUOTIENT R2 Abs2 Rep2"
-  and     q3: "QUOTIENT R3 Abs3 Rep3"
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
+  and     q3: "Quotient R3 Abs3 Rep3"
   shows "f o g = (Rep1 ---> Abs3) (((Abs2 ---> Rep3) f) o ((Abs1 ---> Rep2) g))"
-using QUOTIENT_ABS_REP[OF q1] QUOTIENT_ABS_REP[OF q2] QUOTIENT_ABS_REP[OF q3]
+using Quotient_ABS_REP[OF q1] Quotient_ABS_REP[OF q2] Quotient_ABS_REP[OF q3]
 unfolding o_def expand_fun_eq
 by simp
 
 lemma o_RSP:
-  assumes q1: "QUOTIENT R1 Abs1 Rep1"
-  and     q2: "QUOTIENT R2 Abs2 Rep2"
-  and     q3: "QUOTIENT R3 Abs3 Rep3"
+  assumes q1: "Quotient R1 Abs1 Rep1"
+  and     q2: "Quotient R2 Abs2 Rep2"
+  and     q3: "Quotient R3 Abs3 Rep3"
   and     a1: "(R2 ===> R3) f1 f2"
   and     a2: "(R1 ===> R2) g1 g2"
   shows "(R1 ===> R3) (f1 o g1) (f2 o g2)"
@@ -435,9 +439,9 @@
 
 
 lemma COND_PRS:
-  assumes a: "QUOTIENT R absf repf"
+  assumes a: "Quotient R absf repf"
   shows "(if a then b else c) = absf (if a then repf b else repf c)"
-  using a unfolding QUOTIENT_def by auto
+  using a unfolding Quotient_def by auto
 
 
 
@@ -446,15 +450,15 @@
 (* Set of lemmas for regularisation of ball and bex *)
 lemma ball_reg_eqv:
   fixes P :: "'a \<Rightarrow> bool"
-  assumes a: "EQUIV R"
+  assumes a: "equivp R"
   shows "Ball (Respects R) P = (All P)"
-  by (metis EQUIV_def IN_RESPECTS a)
+  by (metis equivp_def IN_RESPECTS a)
 
 lemma bex_reg_eqv:
   fixes P :: "'a \<Rightarrow> bool"
-  assumes a: "EQUIV R"
+  assumes a: "equivp R"
   shows "Bex (Respects R) P = (Ex P)"
-  by (metis EQUIV_def IN_RESPECTS a)
+  by (metis equivp_def IN_RESPECTS a)
 
 lemma ball_reg_right:
   assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"
@@ -467,27 +471,27 @@
   by (metis COMBC_def Collect_def Collect_mem_eq a)
 
 lemma ball_reg_left:
-  assumes a: "EQUIV R"
+  assumes a: "equivp R"
   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
-  by (metis EQUIV_REFL IN_RESPECTS a)
+  by (metis equivp_reflp IN_RESPECTS a)
 
 lemma bex_reg_right:
-  assumes a: "EQUIV R"
+  assumes a: "equivp R"
   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
-  by (metis EQUIV_REFL IN_RESPECTS a)
+  by (metis equivp_reflp IN_RESPECTS a)
 
 lemma ball_reg_eqv_range:
   fixes P::"'a \<Rightarrow> bool"
   and x::"'a"
-  assumes a: "EQUIV R2"
+  assumes a: "equivp R2"
   shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
   apply(rule iffI)
   apply(rule allI)
   apply(drule_tac x="\<lambda>y. f x" in bspec)
   apply(simp add: Respects_def IN_RESPECTS)
   apply(rule impI)
-  using a EQUIV_REFL_SYM_TRANS[of "R2"]
-  apply(simp add: REFL_def)
+  using a equivp_reflp_symp_transp[of "R2"]
+  apply(simp add: reflp_def)
   apply(simp)
   apply(simp)
   done
@@ -495,15 +499,15 @@
 lemma bex_reg_eqv_range:
   fixes P::"'a \<Rightarrow> bool"
   and x::"'a"
-  assumes a: "EQUIV R2"
+  assumes a: "equivp R2"
   shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
   apply(auto)
   apply(rule_tac x="\<lambda>y. f x" in bexI)
   apply(simp)
   apply(simp add: Respects_def IN_RESPECTS)
   apply(rule impI)
-  using a EQUIV_REFL_SYM_TRANS[of "R2"]
-  apply(simp add: REFL_def)
+  using a equivp_reflp_symp_transp[of "R2"]
+  apply(simp add: reflp_def)
   done
 
 lemma all_reg:
@@ -551,15 +555,15 @@
 
 (* 2 lemmas needed for cleaning of quantifiers *)
 lemma all_prs:
-  assumes a: "QUOTIENT R absf repf"
+  assumes a: "Quotient R absf repf"
   shows "Ball (Respects R) ((absf ---> id) f) = All f"
-  using a unfolding QUOTIENT_def
+  using a unfolding Quotient_def
   by (metis IN_RESPECTS fun_map.simps id_apply)
 
 lemma ex_prs:
-  assumes a: "QUOTIENT R absf repf"
+  assumes a: "Quotient R absf repf"
   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
-  using a unfolding QUOTIENT_def
+  using a unfolding Quotient_def
   by (metis COMBC_def Collect_def Collect_mem_eq IN_RESPECTS fun_map.simps id_apply)
 
 end




