diff -r de505273c971 -r 00d4a435777f src/HOL/Bali/Table.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Bali/Table.thy	Mon Jan 28 17:00:19 2002 +0100
@@ -0,0 +1,445 @@
+(*  Title:      isabelle/Bali/Table.thy
+    ID:         $Id$
+    Author:     David von Oheimb
+    Copyright   1997 Technische Universitaet Muenchen
+*)
+header {* Abstract tables and their implementation as lists *}
+
+theory Table = Basis:
+
+text {*
+design issues:
+\begin{itemize}
+\item definition of table: infinite map vs. list vs. finite set
+      list chosen, because:
+  \begin{itemize} 
+  \item[+]  a priori finite
+  \item[+]  lookup is more operational than for finite set
+  \item[-]  not very abstract, but function table converts it to abstract 
+            mapping
+  \end{itemize}
+\item coding of lookup result: Some/None vs. value/arbitrary
+   Some/None chosen, because:
+  \begin{itemize}
+  \item[++] makes definedness check possible (applies also to finite set),
+     which is important for the type standard, hiding/overriding, etc.
+     (though it may perhaps be possible at least for the operational semantics
+      to treat programs as infinite, i.e. where classes, fields, methods etc.
+      of any name are considered to be defined)
+  \item[-]  sometimes awkward case distinctions, alleviated by operator 'the'
+  \end{itemize}
+\end{itemize}
+*}
+
+
+types ('a, 'b) table    (* table with key type 'a and contents type 'b *)
+      = "'a \<leadsto> 'b"
+      ('a, 'b) tables   (* non-unique table with key 'a and contents 'b *)
+      = "'a \<Rightarrow> 'b set"
+
+
+section "map of / table of"
+
+syntax
+  table_of      :: "('a \<times> 'b) list \<Rightarrow> ('a, 'b) table"    (* concrete table *)
+
+translations
+  "table_of" == "map_of"
+  
+  (type)"'a \<leadsto> 'b"       <= (type)"'a \<Rightarrow> 'b Option.option"
+  (type)"('a, 'b) table" <= (type)"'a \<leadsto> 'b"
+
+(* ### To map *)
+lemma override_find_left[simp]:
+"n k = None \<Longrightarrow> (m ++ n) k = m k"
+by (simp add: override_def)
+
+section {* Conditional Override *}
+constdefs
+cond_override:: 
+  "('b \<Rightarrow>'b \<Rightarrow> bool) \<Rightarrow> ('a, 'b)table \<Rightarrow> ('a, 'b)table \<Rightarrow> ('a, 'b) table"
+
+(* when merging tables old and new, only override an entry of table old when  
+   the condition cond holds *)
+"cond_override cond old new \<equiv>
+ \<lambda> k.
+  (case new k of
+     None         \<Rightarrow> old k                       
+   | Some new_val \<Rightarrow> (case old k of
+                        None         \<Rightarrow> Some new_val          
+                      | Some old_val \<Rightarrow> (if cond new_val old_val
+                                         then Some new_val     
+                                         else Some old_val)))"
+
+lemma cond_override_empty1[simp]: "cond_override c empty t = t"
+by (simp add: cond_override_def expand_fun_eq)
+
+lemma cond_override_empty2[simp]: "cond_override c t empty = t"
+by (simp add: cond_override_def expand_fun_eq)
+
+lemma cond_override_None[simp]:
+ "old k = None \<Longrightarrow> (cond_override c old new) k = new k"
+by (simp add: cond_override_def)
+
+lemma cond_override_override:
+ "\<lbrakk>old k = Some ov;new k = Some nv; C nv ov\<rbrakk> 
+  \<Longrightarrow> (cond_override C old new) k = Some nv"
+by (auto simp add: cond_override_def)
+
+lemma cond_override_noOverride:
+ "\<lbrakk>old k = Some ov;new k = Some nv; \<not> (C nv ov)\<rbrakk> 
+  \<Longrightarrow> (cond_override C old new) k = Some ov"
+by (auto simp add: cond_override_def)
+
+lemma dom_cond_override: "dom (cond_override C s t) \<subseteq> dom s \<union> dom t"
+by (auto simp add: cond_override_def dom_def)
+
+lemma finite_dom_cond_override:
+ "\<lbrakk> finite (dom s); finite (dom t) \<rbrakk> \<Longrightarrow> finite (dom (cond_override C s t))"
+apply (rule_tac B="dom s \<union> dom t" in finite_subset)
+apply (rule dom_cond_override)
+by (rule finite_UnI)
+
+section {* Filter on Tables *}
+
+constdefs
+filter_tab:: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a, 'b) table \<Rightarrow> ('a, 'b) table"
+"filter_tab c t \<equiv> \<lambda> k. (case t k of 
+                           None   \<Rightarrow> None
+                         | Some x \<Rightarrow> if c k x then Some x else None)"
+
+lemma filter_tab_empty[simp]: "filter_tab c empty = empty"
+by (simp add: filter_tab_def empty_def)
+
+lemma filter_tab_True[simp]: "filter_tab (\<lambda>x y. True) t = t"
+by (simp add: expand_fun_eq filter_tab_def)
+
+lemma filter_tab_False[simp]: "filter_tab (\<lambda>x y. False) t = empty"
+by (simp add: expand_fun_eq filter_tab_def empty_def)
+
+lemma filter_tab_ran_subset: "ran (filter_tab c t) \<subseteq> ran t"
+by (auto simp add: filter_tab_def ran_def)
+
+lemma filter_tab_range_subset: "range (filter_tab c t) \<subseteq> range t \<union> {None}"
+apply (auto simp add: filter_tab_def)
+apply (drule sym, blast)
+done
+
+lemma finite_range_filter_tab:
+  "finite (range t) \<Longrightarrow> finite (range (filter_tab c t))"
+apply (rule_tac B="range t \<union> {None} " in finite_subset)
+apply (rule filter_tab_range_subset)
+apply (auto intro: finite_UnI)
+done
+
+lemma filter_tab_SomeD[dest!]: 
+"filter_tab c t k = Some x \<Longrightarrow> (t k = Some x) \<and> c k x"
+by (auto simp add: filter_tab_def)
+
+lemma filter_tab_SomeI: "\<lbrakk>t k = Some x;C k x\<rbrakk> \<Longrightarrow>filter_tab C t k = Some x"
+by (simp add: filter_tab_def)
+
+lemma filter_tab_all_True: 
+ "\<forall> k y. t k = Some y \<longrightarrow> p k y \<Longrightarrow>filter_tab p t = t"
+apply (auto simp add: filter_tab_def expand_fun_eq)
+done
+
+lemma filter_tab_all_True_Some:
+ "\<lbrakk>\<forall> k y. t k = Some y \<longrightarrow> p k y; t k = Some v\<rbrakk> \<Longrightarrow> filter_tab p t k = Some v"
+by (auto simp add: filter_tab_def expand_fun_eq)
+
+lemma filter_tab_None: "t k = None \<Longrightarrow> filter_tab p t k = None"
+apply (simp add: filter_tab_def expand_fun_eq)
+done
+
+lemma filter_tab_dom_subset: "dom (filter_tab C t) \<subseteq> dom t"
+by (auto simp add: filter_tab_def dom_def)
+
+lemma filter_tab_eq: "\<lbrakk>a=b\<rbrakk> \<Longrightarrow> filter_tab C a = filter_tab C b"
+by (auto simp add: expand_fun_eq filter_tab_def)
+
+lemma finite_dom_filter_tab:
+"finite (dom t) \<Longrightarrow> finite (dom (filter_tab C t))"
+apply (rule_tac B="dom t" in finite_subset)
+by (rule filter_tab_dom_subset)
+
+
+lemma filter_tab_weaken:
+"\<lbrakk>\<forall> a \<in> t k: \<exists> b \<in> s k: P a b; 
+  \<And> k x y. \<lbrakk>t k = Some x;s k = Some y\<rbrakk> \<Longrightarrow> cond k x \<longrightarrow> cond k y
+ \<rbrakk> \<Longrightarrow> \<forall> a \<in> filter_tab cond t k: \<exists> b \<in> filter_tab cond s k: P a b"
+apply (auto simp add: filter_tab_def)
+done
+
+lemma cond_override_filter: 
+  "\<lbrakk>\<And> k old new. \<lbrakk>s k = Some new; t k = Some old\<rbrakk> 
+    \<Longrightarrow> (\<not> overC new old \<longrightarrow>  \<not> filterC k new) \<and> 
+        (overC new old \<longrightarrow> filterC k old \<longrightarrow> filterC k new)
+   \<rbrakk> \<Longrightarrow>
+    cond_override overC (filter_tab filterC t) (filter_tab filterC s) 
+    = filter_tab filterC (cond_override overC t s)"
+by (auto simp add: expand_fun_eq cond_override_def filter_tab_def )
+
+section {* Misc. *}
+
+lemma Ball_set_table: "(\<forall> (x,y)\<in> set l. P x y) \<Longrightarrow> \<forall> x. \<forall> y\<in> map_of l x: P x y"
+apply (erule make_imp)
+apply (induct l)
+apply simp
+apply (simp (no_asm))
+apply auto
+done
+
+lemma Ball_set_tableD: 
+  "\<lbrakk>(\<forall> (x,y)\<in> set l. P x y); x \<in> o2s (table_of l xa)\<rbrakk> \<Longrightarrow> P xa x"
+apply (frule Ball_set_table)
+by auto
+
+declare map_of_SomeD [elim]
+
+lemma table_of_Some_in_set:
+"table_of l k = Some x \<Longrightarrow> (k,x) \<in> set l"
+by auto
+
+lemma set_get_eq: 
+  "unique l \<Longrightarrow> (k, the (table_of l k)) \<in> set l = (table_of l k \<noteq> None)"
+apply safe
+apply (fast dest!: weak_map_of_SomeI)
+apply auto
+done
+
+lemma inj_Pair_const2: "inj (\<lambda>k. (k, C))"
+apply (rule injI)
+apply auto
+done
+
+lemma table_of_mapconst_SomeI:
+  "\<lbrakk>table_of t k = Some y'; snd y=y'; fst y=c\<rbrakk> \<Longrightarrow>
+   table_of (map (\<lambda>(k,x). (k,c,x)) t) k = Some y"
+apply (induct t)
+apply auto
+done
+
+lemma table_of_mapconst_NoneI:
+  "\<lbrakk>table_of t k = None\<rbrakk> \<Longrightarrow>
+   table_of (map (\<lambda>(k,x). (k,c,x)) t) k = None"
+apply (induct t)
+apply auto
+done
+
+lemmas table_of_map2_SomeI = inj_Pair_const2 [THEN map_of_mapk_SomeI, standard]
+
+lemma table_of_map_SomeI [rule_format (no_asm)]: "table_of t k = Some x \<longrightarrow>
+   table_of (map (\<lambda>(k,x). (k, f x)) t) k = Some (f x)"
+apply (induct_tac "t")
+apply auto
+done
+
+lemma table_of_remap_SomeD [rule_format (no_asm)]: 
+  "table_of (map (\<lambda>((k,k'),x). (k,(k',x))) t) k = Some (k',x) \<longrightarrow>
+  table_of t (k, k') = Some x"
+apply (induct_tac "t")
+apply  auto
+done
+
+lemma table_of_mapf_Some [rule_format (no_asm)]: "\<forall>x y. f x = f y \<longrightarrow> x = y \<Longrightarrow> 
+  table_of (map (\<lambda>(k,x). (k,f x)) t) k = Some (f x) \<longrightarrow> table_of t k = Some x"
+apply (induct_tac "t")
+apply  auto
+done
+
+lemma table_of_mapf_SomeD [rule_format (no_asm), dest!]: 
+"table_of (map (\<lambda>(k,x). (k, f x)) t) k = Some z \<longrightarrow> (\<exists>y\<in>table_of t k: z=f y)"
+apply (induct_tac "t")
+apply  auto
+done
+
+lemma table_of_mapf_NoneD [rule_format (no_asm), dest!]: 
+"table_of (map (\<lambda>(k,x). (k, f x)) t) k = None \<longrightarrow> (table_of t k = None)"
+apply (induct_tac "t")
+apply auto
+done
+
+lemma table_of_mapkey_SomeD [rule_format (no_asm), dest!]: 
+  "table_of (map (\<lambda>(k,x). ((k,C),x)) t) (k,D) = Some x \<longrightarrow> C = D \<and> table_of t k = Some x"
+apply (induct_tac "t")
+apply  auto
+done
+lemma table_of_mapkey_SomeD2 [rule_format (no_asm), dest!]: 
+  "table_of (map (\<lambda>(k,x). ((k,C),x)) t) ek = Some x 
+   \<longrightarrow> C = snd ek \<and> table_of t (fst ek) = Some x"
+apply (induct_tac "t")
+apply  auto
+done
+
+lemma table_append_Some_iff: "table_of (xs@ys) k = Some z = 
+ (table_of xs k = Some z \<or> (table_of xs k = None \<and> table_of ys k = Some z))"
+apply (simp only: map_of_override [THEN sym])
+apply (rule override_Some_iff)
+done
+
+lemma table_of_filter_unique_SomeD [rule_format (no_asm)]:
+  "table_of (filter P xs) k = Some z \<Longrightarrow> unique xs \<longrightarrow> table_of xs k = Some z"
+apply (induct xs)
+apply (auto del: map_of_SomeD intro!: map_of_SomeD)
+done
+
+
+consts
+  Un_tables      :: "('a, 'b) tables set \<Rightarrow> ('a, 'b) tables"
+  overrides_t    :: "('a, 'b) tables     \<Rightarrow> ('a, 'b) tables \<Rightarrow>
+                     ('a, 'b) tables"             (infixl "\<oplus>\<oplus>" 100)
+  hidings_entails:: "('a, 'b) tables \<Rightarrow> ('a, 'c) tables \<Rightarrow> 
+                     ('b \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> bool"   ("_ hidings _ entails _" 20)
+  (* variant for unique table: *)
+  hiding_entails :: "('a, 'b) table  \<Rightarrow> ('a, 'c) table  \<Rightarrow> 
+                     ('b \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> bool"   ("_ hiding _ entails _"  20)
+  (* variant for a unique table and conditional overriding: *)
+  cond_hiding_entails :: "('a, 'b) table  \<Rightarrow> ('a, 'c) table  
+                          \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> bool"  
+                          ("_ hiding _ under _ entails _"  20)
+
+defs
+  Un_tables_def:       "Un_tables ts\<spacespace>\<spacespace>\<equiv> \<lambda>k. \<Union>t\<in>ts. t k"
+  overrides_t_def:     "s \<oplus>\<oplus> t        \<equiv> \<lambda>k. if t k = {} then s k else t k"
+  hidings_entails_def: "t hidings s entails R \<equiv> \<forall>k. \<forall>x\<in>t k. \<forall>y\<in>s k. R x y"
+  hiding_entails_def:  "t hiding  s entails R \<equiv> \<forall>k. \<forall>x\<in>t k: \<forall>y\<in>s k: R x y"
+  cond_hiding_entails_def: "t hiding  s under C entails R
+                            \<equiv> \<forall>k. \<forall>x\<in>t k: \<forall>y\<in>s k: C x y \<longrightarrow> R x y"
+
+section "Untables"
+
+lemma Un_tablesI [intro]:  "\<And>x. \<lbrakk>t \<in> ts; x \<in> t k\<rbrakk> \<Longrightarrow> x \<in> Un_tables ts k"
+apply (simp add: Un_tables_def)
+apply auto
+done
+
+lemma Un_tablesD [dest!]: "\<And>x. x \<in> Un_tables ts k \<Longrightarrow> \<exists>t. t \<in> ts \<and> x \<in> t k"
+apply (simp add: Un_tables_def)
+apply auto
+done
+
+lemma Un_tables_empty [simp]: "Un_tables {} = (\<lambda>k. {})"
+apply (unfold Un_tables_def)
+apply (simp (no_asm))
+done
+
+
+section "overrides"
+
+lemma empty_overrides_t [simp]: "(\<lambda>k. {}) \<oplus>\<oplus> m = m"
+apply (unfold overrides_t_def)
+apply (simp (no_asm))
+done
+lemma overrides_empty_t [simp]: "m \<oplus>\<oplus> (\<lambda>k. {}) = m"
+apply (unfold overrides_t_def)
+apply (simp (no_asm))
+done
+
+lemma overrides_t_Some_iff: 
+ "(x \<in> (s \<oplus>\<oplus> t) k) = (x \<in> t k \<or> t k = {} \<and> x \<in> s k)"
+by (simp add: overrides_t_def)
+
+lemmas overrides_t_SomeD = overrides_t_Some_iff [THEN iffD1, dest!]
+
+lemma overrides_t_right_empty [simp]: "n k = {} \<Longrightarrow> (m \<oplus>\<oplus> n) k = m k"  
+by (simp add: overrides_t_def)
+
+lemma overrides_t_find_right [simp]: "n k \<noteq> {} \<Longrightarrow> (m \<oplus>\<oplus> n) k = n k"  
+by (simp add: overrides_t_def)
+
+section "hiding entails"
+
+lemma hiding_entailsD: 
+  "\<lbrakk>t hiding s entails R; t k = Some x; s k = Some y\<rbrakk> \<Longrightarrow> R x y"
+by (simp add: hiding_entails_def)
+
+lemma empty_hiding_entails: "empty hiding s entails R"
+by (simp add: hiding_entails_def)
+
+lemma hiding_empty_entails: "t hiding empty entails R"
+by (simp add: hiding_entails_def)
+declare empty_hiding_entails [simp] hiding_empty_entails [simp]
+
+section "cond hiding entails"
+
+lemma cond_hiding_entailsD: 
+"\<lbrakk>t hiding s under C entails R; t k = Some x; s k = Some y; C x y\<rbrakk> \<Longrightarrow> R x y"
+by (simp add: cond_hiding_entails_def)
+
+lemma empty_cond_hiding_entails[simp]: "empty hiding s under C entails R"
+by (simp add: cond_hiding_entails_def)
+
+lemma cond_hiding_empty_entails[simp]: "t hiding empty under C entails R"
+by (simp add: cond_hiding_entails_def)
+
+lemma hidings_entailsD: "\<lbrakk>t hidings s entails R; x \<in> t k; y \<in> s k\<rbrakk> \<Longrightarrow> R x y"
+by (simp add: hidings_entails_def)
+
+lemma hidings_empty_entails: "t hidings (\<lambda>k. {}) entails R"
+apply (unfold hidings_entails_def)
+apply (simp (no_asm))
+done
+
+lemma empty_hidings_entails: 
+  "(\<lambda>k. {}) hidings s entails R"apply (unfold hidings_entails_def)
+by (simp (no_asm))
+declare empty_hidings_entails [intro!] hidings_empty_entails [intro!]
+
+
+
+(*###TO Map?*)
+consts
+  atleast_free :: "('a ~=> 'b) => nat => bool"
+primrec
+ "atleast_free m 0       = True"
+ atleast_free_Suc: 
+ "atleast_free m (Suc n) = (? a. m a = None & (!b. atleast_free (m(a|->b)) n))"
+
+lemma atleast_free_weaken [rule_format (no_asm)]: 
+  "!m. atleast_free m (Suc n) \<longrightarrow> atleast_free m n"
+apply (induct_tac "n")
+apply (simp (no_asm))
+apply clarify
+apply (simp (no_asm))
+apply (drule atleast_free_Suc [THEN iffD1])
+apply fast
+done
+
+lemma atleast_free_SucI: 
+"[| h a = None; !obj. atleast_free (h(a|->obj)) n |] ==> atleast_free h (Suc n)"
+by force
+
+declare fun_upd_apply [simp del]
+lemma atleast_free_SucD_lemma [rule_format (no_asm)]: 
+" !m a. m a = None --> (!c. atleast_free (m(a|->c)) n) -->  
+  (!b d. a ~= b --> atleast_free (m(b|->d)) n)"
+apply (induct_tac "n")
+apply  auto
+apply (rule_tac x = "a" in exI)
+apply  (rule conjI)
+apply  (force simp add: fun_upd_apply)
+apply (erule_tac V = "m a = None" in thin_rl)
+apply clarify
+apply (subst fun_upd_twist)
+apply  (erule not_sym)
+apply (rename_tac "ba")
+apply (drule_tac x = "ba" in spec)
+apply clarify
+apply (tactic "smp_tac 2 1")
+apply (erule (1) notE impE)
+apply (case_tac "aa = b")
+apply fast+
+done
+declare fun_upd_apply [simp]
+
+lemma atleast_free_SucD [rule_format (no_asm)]: "atleast_free h (Suc n) ==> atleast_free (h(a|->b)) n"
+apply auto
+apply (case_tac "aa = a")
+apply auto
+apply (erule atleast_free_SucD_lemma)
+apply auto
+done
+
+declare atleast_free_Suc [simp del]
+end