src/HOL/Bali/Table.thy
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(*  Title:      HOL/Bali/Table.thy
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    ID:         $Id$
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    Author:     David von Oheimb
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    License:    GPL (GNU GENERAL PUBLIC LICENSE)
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*)
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header {* Abstract tables and their implementation as lists *}
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theory Table = Basis:
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text {*
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design issues:
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\begin{itemize}
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\item definition of table: infinite map vs. list vs. finite set
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      list chosen, because:
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  \begin{itemize} 
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  \item[+]  a priori finite
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  \item[+]  lookup is more operational than for finite set
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  \item[-]  not very abstract, but function table converts it to abstract 
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            mapping
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  \end{itemize}
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\item coding of lookup result: Some/None vs. value/arbitrary
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   Some/None chosen, because:
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  \begin{itemize}
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  \item[++] makes definedness check possible (applies also to finite set),
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     which is important for the type standard, hiding/overriding, etc.
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     (though it may perhaps be possible at least for the operational semantics
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      to treat programs as infinite, i.e. where classes, fields, methods etc.
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      of any name are considered to be defined)
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  \item[-]  sometimes awkward case distinctions, alleviated by operator 'the'
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  \end{itemize}
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\end{itemize}
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*}
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types ('a, 'b) table    --{* table with key type 'a and contents type 'b *}
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      = "'a \<rightharpoonup> 'b"
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      ('a, 'b) tables   --{* non-unique table with key 'a and contents 'b *}
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      = "'a \<Rightarrow> 'b set"
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section "map of / table of"
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syntax
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  table_of      :: "('a \<times> 'b) list \<Rightarrow> ('a, 'b) table"   --{* concrete table *}
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translations
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  "table_of" == "map_of"
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  (type)"'a \<rightharpoonup> 'b"       <= (type)"'a \<Rightarrow> 'b Option.option"
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  (type)"('a, 'b) table" <= (type)"'a \<rightharpoonup> 'b"
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(* ### To map *)
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lemma map_add_find_left[simp]:
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"n k = None \<Longrightarrow> (m ++ n) k = m k"
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by (simp add: map_add_def)
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section {* Conditional Override *}
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constdefs
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cond_override:: 
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  "('b \<Rightarrow>'b \<Rightarrow> bool) \<Rightarrow> ('a, 'b)table \<Rightarrow> ('a, 'b)table \<Rightarrow> ('a, 'b) table"
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--{* when merging tables old and new, only override an entry of table old when  
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   the condition cond holds *}
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"cond_override cond old new \<equiv>
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 \<lambda> k.
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  (case new k of
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     None         \<Rightarrow> old k                       
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   | Some new_val \<Rightarrow> (case old k of
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                        None         \<Rightarrow> Some new_val          
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                      | Some old_val \<Rightarrow> (if cond new_val old_val
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                                         then Some new_val     
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                                         else Some old_val)))"
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lemma cond_override_empty1[simp]: "cond_override c empty t = t"
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by (simp add: cond_override_def expand_fun_eq)
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lemma cond_override_empty2[simp]: "cond_override c t empty = t"
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by (simp add: cond_override_def expand_fun_eq)
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lemma cond_override_None[simp]:
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 "old k = None \<Longrightarrow> (cond_override c old new) k = new k"
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by (simp add: cond_override_def)
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lemma cond_override_override:
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 "\<lbrakk>old k = Some ov;new k = Some nv; C nv ov\<rbrakk> 
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  \<Longrightarrow> (cond_override C old new) k = Some nv"
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by (auto simp add: cond_override_def)
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lemma cond_override_noOverride:
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 "\<lbrakk>old k = Some ov;new k = Some nv; \<not> (C nv ov)\<rbrakk> 
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  \<Longrightarrow> (cond_override C old new) k = Some ov"
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by (auto simp add: cond_override_def)
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lemma dom_cond_override: "dom (cond_override C s t) \<subseteq> dom s \<union> dom t"
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by (auto simp add: cond_override_def dom_def)
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lemma finite_dom_cond_override:
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 "\<lbrakk> finite (dom s); finite (dom t) \<rbrakk> \<Longrightarrow> finite (dom (cond_override C s t))"
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apply (rule_tac B="dom s \<union> dom t" in finite_subset)
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apply (rule dom_cond_override)
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by (rule finite_UnI)
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section {* Filter on Tables *}
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constdefs
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filter_tab:: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a, 'b) table \<Rightarrow> ('a, 'b) table"
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"filter_tab c t \<equiv> \<lambda> k. (case t k of 
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                           None   \<Rightarrow> None
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                         | Some x \<Rightarrow> if c k x then Some x else None)"
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lemma filter_tab_empty[simp]: "filter_tab c empty = empty"
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by (simp add: filter_tab_def empty_def)
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lemma filter_tab_True[simp]: "filter_tab (\<lambda>x y. True) t = t"
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by (simp add: expand_fun_eq filter_tab_def)
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lemma filter_tab_False[simp]: "filter_tab (\<lambda>x y. False) t = empty"
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by (simp add: expand_fun_eq filter_tab_def empty_def)
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lemma filter_tab_ran_subset: "ran (filter_tab c t) \<subseteq> ran t"
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by (auto simp add: filter_tab_def ran_def)
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lemma filter_tab_range_subset: "range (filter_tab c t) \<subseteq> range t \<union> {None}"
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apply (auto simp add: filter_tab_def)
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apply (drule sym, blast)
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done
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lemma finite_range_filter_tab:
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  "finite (range t) \<Longrightarrow> finite (range (filter_tab c t))"
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apply (rule_tac B="range t \<union> {None} " in finite_subset)
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apply (rule filter_tab_range_subset)
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apply (auto intro: finite_UnI)
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done
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lemma filter_tab_SomeD[dest!]: 
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"filter_tab c t k = Some x \<Longrightarrow> (t k = Some x) \<and> c k x"
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by (auto simp add: filter_tab_def)
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lemma filter_tab_SomeI: "\<lbrakk>t k = Some x;C k x\<rbrakk> \<Longrightarrow>filter_tab C t k = Some x"
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by (simp add: filter_tab_def)
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lemma filter_tab_all_True: 
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 "\<forall> k y. t k = Some y \<longrightarrow> p k y \<Longrightarrow>filter_tab p t = t"
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apply (auto simp add: filter_tab_def expand_fun_eq)
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done
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lemma filter_tab_all_True_Some:
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 "\<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"
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by (auto simp add: filter_tab_def expand_fun_eq)
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lemma filter_tab_all_False: 
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 "\<forall> k y. t k = Some y \<longrightarrow> \<not> p k y \<Longrightarrow>filter_tab p t = empty"
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by (auto simp add: filter_tab_def expand_fun_eq)
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lemma filter_tab_None: "t k = None \<Longrightarrow> filter_tab p t k = None"
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apply (simp add: filter_tab_def expand_fun_eq)
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done
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lemma filter_tab_dom_subset: "dom (filter_tab C t) \<subseteq> dom t"
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by (auto simp add: filter_tab_def dom_def)
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lemma filter_tab_eq: "\<lbrakk>a=b\<rbrakk> \<Longrightarrow> filter_tab C a = filter_tab C b"
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by (auto simp add: expand_fun_eq filter_tab_def)
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lemma finite_dom_filter_tab:
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"finite (dom t) \<Longrightarrow> finite (dom (filter_tab C t))"
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apply (rule_tac B="dom t" in finite_subset)
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by (rule filter_tab_dom_subset)
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lemma filter_tab_weaken:
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"\<lbrakk>\<forall> a \<in> t k: \<exists> b \<in> s k: P a b; 
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  \<And> k x y. \<lbrakk>t k = Some x;s k = Some y\<rbrakk> \<Longrightarrow> cond k x \<longrightarrow> cond k y
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 \<rbrakk> \<Longrightarrow> \<forall> a \<in> filter_tab cond t k: \<exists> b \<in> filter_tab cond s k: P a b"
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apply (force simp add: filter_tab_def)
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done
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lemma cond_override_filter: 
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  "\<lbrakk>\<And> k old new. \<lbrakk>s k = Some new; t k = Some old\<rbrakk> 
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    \<Longrightarrow> (\<not> overC new old \<longrightarrow>  \<not> filterC k new) \<and> 
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        (overC new old \<longrightarrow> filterC k old \<longrightarrow> filterC k new)
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   \<rbrakk> \<Longrightarrow>
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    cond_override overC (filter_tab filterC t) (filter_tab filterC s) 
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    = filter_tab filterC (cond_override overC t s)"
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by (auto simp add: expand_fun_eq cond_override_def filter_tab_def )
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section {* Misc. *}
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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"
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apply (erule make_imp)
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apply (induct l)
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apply simp
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apply (simp (no_asm))
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apply auto
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done
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lemma Ball_set_tableD: 
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  "\<lbrakk>(\<forall> (x,y)\<in> set l. P x y); x \<in> o2s (table_of l xa)\<rbrakk> \<Longrightarrow> P xa x"
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apply (frule Ball_set_table)
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by auto
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declare map_of_SomeD [elim]
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lemma table_of_Some_in_set:
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"table_of l k = Some x \<Longrightarrow> (k,x) \<in> set l"
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by auto
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lemma set_get_eq: 
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  "unique l \<Longrightarrow> (k, the (table_of l k)) \<in> set l = (table_of l k \<noteq> None)"
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apply safe
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apply (fast dest!: weak_map_of_SomeI)
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apply auto
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done
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lemma inj_Pair_const2: "inj (\<lambda>k. (k, C))"
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apply (rule inj_onI)
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apply auto
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done
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lemma table_of_mapconst_SomeI:
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  "\<lbrakk>table_of t k = Some y'; snd y=y'; fst y=c\<rbrakk> \<Longrightarrow>
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   table_of (map (\<lambda>(k,x). (k,c,x)) t) k = Some y"
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apply (induct t)
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apply auto
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done
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lemma table_of_mapconst_NoneI:
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  "\<lbrakk>table_of t k = None\<rbrakk> \<Longrightarrow>
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   table_of (map (\<lambda>(k,x). (k,c,x)) t) k = None"
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apply (induct t)
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apply auto
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done
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lemmas table_of_map2_SomeI = inj_Pair_const2 [THEN map_of_mapk_SomeI, standard]
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lemma table_of_map_SomeI [rule_format (no_asm)]: "table_of t k = Some x \<longrightarrow>
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   table_of (map (\<lambda>(k,x). (k, f x)) t) k = Some (f x)"
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apply (induct_tac "t")
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apply auto
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done
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lemma table_of_remap_SomeD [rule_format (no_asm)]: 
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  "table_of (map (\<lambda>((k,k'),x). (k,(k',x))) t) k = Some (k',x) \<longrightarrow>
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  table_of t (k, k') = Some x"
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apply (induct_tac "t")
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apply  auto
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done
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lemma table_of_mapf_Some [rule_format (no_asm)]: "\<forall>x y. f x = f y \<longrightarrow> x = y \<Longrightarrow> 
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  table_of (map (\<lambda>(k,x). (k,f x)) t) k = Some (f x) \<longrightarrow> table_of t k = Some x"
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apply (induct_tac "t")
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apply  auto
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done
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lemma table_of_mapf_SomeD [rule_format (no_asm), dest!]: 
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"table_of (map (\<lambda>(k,x). (k, f x)) t) k = Some z \<longrightarrow> (\<exists>y\<in>table_of t k: z=f y)"
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apply (induct_tac "t")
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apply  auto
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done
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lemma table_of_mapf_NoneD [rule_format (no_asm), dest!]: 
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"table_of (map (\<lambda>(k,x). (k, f x)) t) k = None \<longrightarrow> (table_of t k = None)"
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apply (induct_tac "t")
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apply auto
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done
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lemma table_of_mapkey_SomeD [rule_format (no_asm), dest!]: 
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  "table_of (map (\<lambda>(k,x). ((k,C),x)) t) (k,D) = Some x \<longrightarrow> C = D \<and> table_of t k = Some x"
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apply (induct_tac "t")
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apply  auto
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done
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lemma table_of_mapkey_SomeD2 [rule_format (no_asm), dest!]: 
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  "table_of (map (\<lambda>(k,x). ((k,C),x)) t) ek = Some x 
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   \<longrightarrow> C = snd ek \<and> table_of t (fst ek) = Some x"
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apply (induct_tac "t")
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apply  auto
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done
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lemma table_append_Some_iff: "table_of (xs@ys) k = Some z = 
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 (table_of xs k = Some z \<or> (table_of xs k = None \<and> table_of ys k = Some z))"
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d9b155757dc8 *** empty log message ***
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apply (simp)
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apply (rule map_add_Some_iff)
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done
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lemma table_of_filter_unique_SomeD [rule_format (no_asm)]:
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  "table_of (filter P xs) k = Some z \<Longrightarrow> unique xs \<longrightarrow> table_of xs k = Some z"
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apply (induct xs)
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apply (auto del: map_of_SomeD intro!: map_of_SomeD)
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done
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consts
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  Un_tables      :: "('a, 'b) tables set \<Rightarrow> ('a, 'b) tables"
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  overrides_t    :: "('a, 'b) tables     \<Rightarrow> ('a, 'b) tables \<Rightarrow>
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                     ('a, 'b) tables"             (infixl "\<oplus>\<oplus>" 100)
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  hidings_entails:: "('a, 'b) tables \<Rightarrow> ('a, 'c) tables \<Rightarrow> 
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                     ('b \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> bool"   ("_ hidings _ entails _" 20)
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a0b16d42d489 "Definite Assignment Analysis" included, with proof of correctness. Large adjustments of type safety proof and soundness proof of the axiomatic semantics were necessary. Completeness proof of the loop rule of the axiomatic semantic was altered. So the additional polymorphic variants of some rules could be removed.
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  --{* variant for unique table: *}
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  hiding_entails :: "('a, 'b) table  \<Rightarrow> ('a, 'c) table  \<Rightarrow> 
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                     ('b \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> bool"   ("_ hiding _ entails _"  20)
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a0b16d42d489 "Definite Assignment Analysis" included, with proof of correctness. Large adjustments of type safety proof and soundness proof of the axiomatic semantics were necessary. Completeness proof of the loop rule of the axiomatic semantic was altered. So the additional polymorphic variants of some rules could be removed.
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  --{* variant for a unique table and conditional overriding: *}
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  cond_hiding_entails :: "('a, 'b) table  \<Rightarrow> ('a, 'c) table  
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                          \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> bool"  
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                          ("_ hiding _ under _ entails _"  20)
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defs
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  Un_tables_def:       "Un_tables ts\<spacespace>\<spacespace>\<equiv> \<lambda>k. \<Union>t\<in>ts. t k"
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  overrides_t_def:     "s \<oplus>\<oplus> t        \<equiv> \<lambda>k. if t k = {} then s k else t k"
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  hidings_entails_def: "t hidings s entails R \<equiv> \<forall>k. \<forall>x\<in>t k. \<forall>y\<in>s k. R x y"
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  hiding_entails_def:  "t hiding  s entails R \<equiv> \<forall>k. \<forall>x\<in>t k: \<forall>y\<in>s k: R x y"
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  cond_hiding_entails_def: "t hiding  s under C entails R
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                            \<equiv> \<forall>k. \<forall>x\<in>t k: \<forall>y\<in>s k: C x y \<longrightarrow> R x y"
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section "Untables"
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lemma Un_tablesI [intro]:  "\<And>x. \<lbrakk>t \<in> ts; x \<in> t k\<rbrakk> \<Longrightarrow> x \<in> Un_tables ts k"
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apply (simp add: Un_tables_def)
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apply auto
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done
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   321
lemma Un_tablesD [dest!]: "\<And>x. x \<in> Un_tables ts k \<Longrightarrow> \<exists>t. t \<in> ts \<and> x \<in> t k"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
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   322
apply (simp add: Un_tables_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   323
apply auto
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   324
done
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   325
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   326
lemma Un_tables_empty [simp]: "Un_tables {} = (\<lambda>k. {})"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   327
apply (unfold Un_tables_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   328
apply (simp (no_asm))
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   329
done
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   330
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   331
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   332
section "overrides"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   333
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   334
lemma empty_overrides_t [simp]: "(\<lambda>k. {}) \<oplus>\<oplus> m = m"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   335
apply (unfold overrides_t_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   336
apply (simp (no_asm))
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   337
done
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   338
lemma overrides_empty_t [simp]: "m \<oplus>\<oplus> (\<lambda>k. {}) = m"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   339
apply (unfold overrides_t_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   340
apply (simp (no_asm))
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   341
done
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   342
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   343
lemma overrides_t_Some_iff: 
00d4a435777f Isabelle/Bali sources;
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parents:
diff changeset
   344
 "(x \<in> (s \<oplus>\<oplus> t) k) = (x \<in> t k \<or> t k = {} \<and> x \<in> s k)"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   345
by (simp add: overrides_t_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   346
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   347
lemmas overrides_t_SomeD = overrides_t_Some_iff [THEN iffD1, dest!]
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   348
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   349
lemma overrides_t_right_empty [simp]: "n k = {} \<Longrightarrow> (m \<oplus>\<oplus> n) k = m k"  
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   350
by (simp add: overrides_t_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   351
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   352
lemma overrides_t_find_right [simp]: "n k \<noteq> {} \<Longrightarrow> (m \<oplus>\<oplus> n) k = n k"  
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   353
by (simp add: overrides_t_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   354
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   355
section "hiding entails"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   356
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   357
lemma hiding_entailsD: 
00d4a435777f Isabelle/Bali sources;
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parents:
diff changeset
   358
  "\<lbrakk>t hiding s entails R; t k = Some x; s k = Some y\<rbrakk> \<Longrightarrow> R x y"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   359
by (simp add: hiding_entails_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   360
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   361
lemma empty_hiding_entails: "empty hiding s entails R"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   362
by (simp add: hiding_entails_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   363
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   364
lemma hiding_empty_entails: "t hiding empty entails R"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   365
by (simp add: hiding_entails_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   366
declare empty_hiding_entails [simp] hiding_empty_entails [simp]
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   367
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   368
section "cond hiding entails"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   369
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   370
lemma cond_hiding_entailsD: 
00d4a435777f Isabelle/Bali sources;
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parents:
diff changeset
   371
"\<lbrakk>t hiding s under C entails R; t k = Some x; s k = Some y; C x y\<rbrakk> \<Longrightarrow> R x y"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   372
by (simp add: cond_hiding_entails_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   373
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   374
lemma empty_cond_hiding_entails[simp]: "empty hiding s under C entails R"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   375
by (simp add: cond_hiding_entails_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   376
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   377
lemma cond_hiding_empty_entails[simp]: "t hiding empty under C entails R"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   378
by (simp add: cond_hiding_entails_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   379
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   380
lemma hidings_entailsD: "\<lbrakk>t hidings s entails R; x \<in> t k; y \<in> s k\<rbrakk> \<Longrightarrow> R x y"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   381
by (simp add: hidings_entails_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   382
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   383
lemma hidings_empty_entails: "t hidings (\<lambda>k. {}) entails R"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   384
apply (unfold hidings_entails_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   385
apply (simp (no_asm))
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   386
done
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   387
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   388
lemma empty_hidings_entails: 
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   389
  "(\<lambda>k. {}) hidings s entails R"apply (unfold hidings_entails_def)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   390
by (simp (no_asm))
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   391
declare empty_hidings_entails [intro!] hidings_empty_entails [intro!]
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   392
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   393
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   394
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   395
(*###TO Map?*)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   396
consts
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   397
  atleast_free :: "('a ~=> 'b) => nat => bool"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   398
primrec
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   399
 "atleast_free m 0       = True"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   400
 atleast_free_Suc: 
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   401
 "atleast_free m (Suc n) = (? a. m a = None & (!b. atleast_free (m(a|->b)) n))"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   402
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   403
lemma atleast_free_weaken [rule_format (no_asm)]: 
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   404
  "!m. atleast_free m (Suc n) \<longrightarrow> atleast_free m n"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   405
apply (induct_tac "n")
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   406
apply (simp (no_asm))
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   407
apply clarify
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   408
apply (simp (no_asm))
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   409
apply (drule atleast_free_Suc [THEN iffD1])
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   410
apply fast
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   411
done
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   412
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   413
lemma atleast_free_SucI: 
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   414
"[| h a = None; !obj. atleast_free (h(a|->obj)) n |] ==> atleast_free h (Suc n)"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   415
by force
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   416
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   417
declare fun_upd_apply [simp del]
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   418
lemma atleast_free_SucD_lemma [rule_format (no_asm)]: 
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   419
" !m a. m a = None --> (!c. atleast_free (m(a|->c)) n) -->  
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   420
  (!b d. a ~= b --> atleast_free (m(b|->d)) n)"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   421
apply (induct_tac "n")
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   422
apply  auto
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   423
apply (rule_tac x = "a" in exI)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   424
apply  (rule conjI)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   425
apply  (force simp add: fun_upd_apply)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   426
apply (erule_tac V = "m a = None" in thin_rl)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   427
apply clarify
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   428
apply (subst fun_upd_twist)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   429
apply  (erule not_sym)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   430
apply (rename_tac "ba")
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   431
apply (drule_tac x = "ba" in spec)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   432
apply clarify
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   433
apply (tactic "smp_tac 2 1")
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   434
apply (erule (1) notE impE)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   435
apply (case_tac "aa = b")
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   436
apply fast+
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   437
done
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   438
declare fun_upd_apply [simp]
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   439
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   440
lemma atleast_free_SucD [rule_format (no_asm)]: "atleast_free h (Suc n) ==> atleast_free (h(a|->b)) n"
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   441
apply auto
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   442
apply (case_tac "aa = a")
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   443
apply auto
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   444
apply (erule atleast_free_SucD_lemma)
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   445
apply auto
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   446
done
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   447
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   448
declare atleast_free_Suc [simp del]
00d4a435777f Isabelle/Bali sources;
schirmer
parents:
diff changeset
   449
end