implement code generation for finite, card, op = and op <= for sets always via finite_UNIV and card_UNIV, as fragile rewrites based on sorts are hard to find and debug

--- a/src/HOL/Codegenerator_Test/RBT_Set_Test.thy	Thu Feb 14 12:24:56 2013 +0100
+++ b/src/HOL/Codegenerator_Test/RBT_Set_Test.thy	Thu Feb 14 16:01:28 2013 +0100
@@ -22,23 +22,13 @@
 lemma [code, code del]:
   "pred_of_set = pred_of_set" ..
 
-
-lemma [code, code del]:
-  "Cardinality.card' = Cardinality.card'" ..
-
-lemma [code, code del]:
-  "Cardinality.finite' = Cardinality.finite'" ..
-
-lemma [code, code del]:
-  "Cardinality.subset' = Cardinality.subset'" ..
-
-lemma [code, code del]:
-  "Cardinality.eq_set = Cardinality.eq_set" ..
-
-
 lemma [code, code del]:
   "acc = acc" ..
 
+lemmas [code del] =
+  finite_set_code finite_coset_code 
+  equal_set_code
+
 (*
   If the code generation ends with an exception with the following message:
   '"List.set" is not a constructor, on left hand side of equation: ...',

--- a/src/HOL/Library/Cardinality.thy	Thu Feb 14 12:24:56 2013 +0100
+++ b/src/HOL/Library/Cardinality.thy	Thu Feb 14 16:01:28 2013 +0100
@@ -199,7 +199,7 @@
 subsection {* A type class for computing the cardinality of types *}
 
 definition is_list_UNIV :: "'a list \<Rightarrow> bool"
-where [code del]: "is_list_UNIV xs = (let c = CARD('a) in if c = 0 then False else size (remdups xs) = c)"
+where "is_list_UNIV xs = (let c = CARD('a) in if c = 0 then False else size (remdups xs) = c)"
 
 lemma is_list_UNIV_iff: "is_list_UNIV xs \<longleftrightarrow> set xs = UNIV"
 by(auto simp add: is_list_UNIV_def Let_def card_eq_0_iff List.card_set[symmetric] 
@@ -211,15 +211,6 @@
   fixes card_UNIV :: "'a card_UNIV"
   assumes card_UNIV: "card_UNIV = Phantom('a) CARD('a)"
 
-lemma card_UNIV_code [code_unfold]: 
-  "CARD('a :: card_UNIV) = of_phantom (card_UNIV :: 'a card_UNIV)"
-by(simp add: card_UNIV)
-
-lemma is_list_UNIV_code [code]:
-  "is_list_UNIV (xs :: 'a list) = 
-  (let c = CARD('a :: card_UNIV) in if c = 0 then False else size (remdups xs) = c)"
-by(rule is_list_UNIV_def)
-
 subsection {* Instantiations for @{text "card_UNIV"} *}
 
 instantiation nat :: card_UNIV begin
@@ -396,80 +387,66 @@
 
 subsection {* Code setup for sets *}
 
+text {*
+  Implement operations @{term "finite"}, @{term "card"}, @{term "op \<subseteq>"}, and @{term "op ="} 
+  for sets using @{term "finite_UNIV"} and @{term "card_UNIV"}.
+*}
+
 lemma card_Compl:
   "finite A \<Longrightarrow> card (- A) = card (UNIV :: 'a set) - card (A :: 'a set)"
 by (metis Compl_eq_Diff_UNIV card_Diff_subset top_greatest)
 
-context fixes xs :: "'a :: card_UNIV list"
-begin
+lemma card_coset_code [code]:
+  fixes xs :: "'a :: card_UNIV list" 
+  shows "card (List.coset xs) = of_phantom (card_UNIV :: 'a card_UNIV) - length (remdups xs)"
+by(simp add: List.card_set card_Compl card_UNIV)
 
-definition card' :: "'a set \<Rightarrow> nat" 
-where [simp, code del, code_abbrev]: "card' = card"
-
-lemma card'_code [code]:
-  "card' (set xs) = length (remdups xs)"
-  "card' (List.coset xs) = of_phantom (card_UNIV :: 'a card_UNIV) - length (remdups xs)"
-by(simp_all add: List.card_set card_Compl card_UNIV)
+lemma [code, code del]: "finite = finite" ..
 
-lemma card'_UNIV [code_unfold]: 
-  "card' (UNIV :: 'a :: card_UNIV set) = of_phantom (card_UNIV :: 'a card_UNIV)"
-by(simp add: card_UNIV)
-
-definition finite' :: "'a set \<Rightarrow> bool"
-where [simp, code del, code_abbrev]: "finite' = finite"
-
-lemma finite'_code [code]:
-  "finite' (set xs) \<longleftrightarrow> True"
-  "finite' (List.coset xs) \<longleftrightarrow> of_phantom (finite_UNIV :: 'a finite_UNIV)"
+lemma [code]:
+  fixes xs :: "'a :: card_UNIV list" 
+  shows finite_set_code:
+  "finite (set xs) = True" 
+  and finite_coset_code:
+  "finite (List.coset xs) \<longleftrightarrow> of_phantom (finite_UNIV :: 'a finite_UNIV)"
 by(simp_all add: card_gt_0_iff finite_UNIV)
 
-definition subset' :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
-where [simp, code del, code_abbrev]: "subset' = op \<subseteq>"
-
-lemma subset'_code [code]:
-  "subset' A (List.coset ys) \<longleftrightarrow> (\<forall>y \<in> set ys. y \<notin> A)"
-  "subset' (set ys) B \<longleftrightarrow> (\<forall>y \<in> set ys. y \<in> B)"
-  "subset' (List.coset xs) (set ys) \<longleftrightarrow> (let n = CARD('a) in n > 0 \<and> card(set (xs @ ys)) = n)"
+lemma coset_subset_code [code]:
+  fixes xs :: "'a list" shows
+  "List.coset xs \<subseteq> set ys \<longleftrightarrow> (let n = CARD('a) in n > 0 \<and> card (set (xs @ ys)) = n)"
 by(auto simp add: Let_def card_gt_0_iff dest: card_eq_UNIV_imp_eq_UNIV intro: arg_cong[where f=card])
   (metis finite_compl finite_set rev_finite_subset)
 
-definition eq_set :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"
-where [simp, code del, code_abbrev]: "eq_set = op ="
-
-lemma eq_set_code [code]:
-  fixes ys
+lemma equal_set_code [code]:
+  fixes xs ys :: "'a :: equal list"
   defines "rhs \<equiv> 
   let n = CARD('a)
   in if n = 0 then False else 
         let xs' = remdups xs; ys' = remdups ys 
         in length xs' + length ys' = n \<and> (\<forall>x \<in> set xs'. x \<notin> set ys') \<and> (\<forall>y \<in> set ys'. y \<notin> set xs')"
-  shows "eq_set (List.coset xs) (set ys) \<longleftrightarrow> rhs" (is ?thesis1)
-  and "eq_set (set ys) (List.coset xs) \<longleftrightarrow> rhs" (is ?thesis2)
-  and "eq_set (set xs) (set ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)" (is ?thesis3)
-  and "eq_set (List.coset xs) (List.coset ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)" (is ?thesis4)
+  shows "equal_class.equal (List.coset xs) (set ys) \<longleftrightarrow> rhs" (is ?thesis1)
+  and "equal_class.equal (set ys) (List.coset xs) \<longleftrightarrow> rhs" (is ?thesis2)
+  and "equal_class.equal (set xs) (set ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)" (is ?thesis3)
+  and "equal_class.equal (List.coset xs) (List.coset ys) \<longleftrightarrow> (\<forall>x \<in> set xs. x \<in> set ys) \<and> (\<forall>y \<in> set ys. y \<in> set xs)" (is ?thesis4)
 proof -
   show ?thesis1 (is "?lhs \<longleftrightarrow> ?rhs")
   proof
     assume ?lhs thus ?rhs
-      by(auto simp add: rhs_def Let_def List.card_set[symmetric] card_Un_Int[where A="set xs" and B="- set xs"] card_UNIV Compl_partition card_gt_0_iff dest: sym)(metis finite_compl finite_set)
+      by(auto simp add: equal_eq rhs_def Let_def List.card_set[symmetric] card_Un_Int[where A="set xs" and B="- set xs"] card_UNIV Compl_partition card_gt_0_iff dest: sym)(metis finite_compl finite_set)
   next
     assume ?rhs
     moreover have "\<lbrakk> \<forall>y\<in>set xs. y \<notin> set ys; \<forall>x\<in>set ys. x \<notin> set xs \<rbrakk> \<Longrightarrow> set xs \<inter> set ys = {}" by blast
     ultimately show ?lhs
-      by(auto simp add: rhs_def Let_def List.card_set[symmetric] card_UNIV card_gt_0_iff card_Un_Int[where A="set xs" and B="set ys"] dest: card_eq_UNIV_imp_eq_UNIV split: split_if_asm)
+      by(auto simp add: equal_eq rhs_def Let_def List.card_set[symmetric] card_UNIV card_gt_0_iff card_Un_Int[where A="set xs" and B="set ys"] dest: card_eq_UNIV_imp_eq_UNIV split: split_if_asm)
   qed
-  thus ?thesis2 unfolding eq_set_def by blast
-  show ?thesis3 ?thesis4 unfolding eq_set_def List.coset_def by blast+
+  thus ?thesis2 unfolding equal_eq by blast
+  show ?thesis3 ?thesis4 unfolding equal_eq List.coset_def by blast+
 qed
 
-end
-
 notepad begin (* test code setup *)
 have "List.coset [True] = set [False] \<and> List.coset [] \<subseteq> List.set [True, False] \<and> finite (List.coset [True])"
   by eval
 end
 
-hide_const (open) card' finite' subset' eq_set
-
 end