src/HOL/NanoJava/TypeRel.thy
author oheimb
Fri, 21 Sep 2001 18:23:15 +0200
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(*  Title:      HOL/NanoJava/TypeRel.thy
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    ID:         $Id$
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    Author:     David von Oheimb
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    Copyright   2001 Technische Universitaet Muenchen
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*)
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header "Type relations"
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theory TypeRel = Decl:
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consts
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  widen   :: "(ty    \<times> ty   ) set"  --{* widening *}
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  subcls1 :: "(cname \<times> cname) set"  --{* subclass *}
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syntax (xsymbols)
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  widen   :: "[ty   , ty   ] => bool" ("_ \<preceq> _"    [71,71] 70)
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  subcls1 :: "[cname, cname] => bool" ("_ \<prec>C1 _"  [71,71] 70)
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  subcls  :: "[cname, cname] => bool" ("_ \<preceq>C _"   [71,71] 70)
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syntax
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  widen   :: "[ty   , ty   ] => bool" ("_ <= _"   [71,71] 70)
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  subcls1 :: "[cname, cname] => bool" ("_ <=C1 _" [71,71] 70)
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  subcls  :: "[cname, cname] => bool" ("_ <=C _"  [71,71] 70)
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translations
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  "C \<prec>C1 D" == "(C,D) \<in> subcls1"
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  "C \<preceq>C  D" == "(C,D) \<in> subcls1^*"
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  "S \<preceq>   T" == "(S,T) \<in> widen"
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consts
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  method :: "cname => (mname \<leadsto> methd)"
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  field  :: "cname => (fname \<leadsto> ty)"
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subsection "Declarations and properties not used in the meta theory"
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text{* Direct subclass relation *}
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defs
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 subcls1_def: "subcls1 \<equiv> {(C,D). C\<noteq>Object \<and> (\<exists>c. class C = Some c \<and> super c=D)}"
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text{* Widening, viz. method invocation conversion *}
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inductive widen intros 
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  refl   [intro!, simp]:           "T \<preceq> T"
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  subcls         : "C\<preceq>C D \<Longrightarrow> Class C \<preceq> Class D"
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  null   [intro!]:                "NT \<preceq> R"
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lemma subcls1D: 
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  "C\<prec>C1D \<Longrightarrow> C \<noteq> Object \<and> (\<exists>c. class C = Some c \<and> super c=D)"
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apply (unfold subcls1_def)
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apply auto
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done
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lemma subcls1I: "\<lbrakk>class C = Some m; super m = D; C \<noteq> Object\<rbrakk> \<Longrightarrow> C\<prec>C1D"
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apply (unfold subcls1_def)
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apply auto
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done
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lemma subcls1_def2: 
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"subcls1 = (\<Sigma>C\<in>{C. is_class C} . {D. C\<noteq>Object \<and> super (the (class C)) = D})"
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apply (unfold subcls1_def is_class_def)
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apply auto
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done
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lemma finite_subcls1: "finite subcls1"
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apply(subst subcls1_def2)
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apply(rule finite_SigmaI [OF finite_is_class])
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apply(rule_tac B = "{super (the (class C))}" in finite_subset)
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apply  auto
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done
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constdefs
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  ws_prog  :: "bool"
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 "ws_prog \<equiv> \<forall>(C,c)\<in>set Prog. C\<noteq>Object \<longrightarrow> 
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                              is_class (super c) \<and> (super c,C)\<notin>subcls1^+"
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lemma ws_progD: "\<lbrakk>class C = Some c; C\<noteq>Object; ws_prog\<rbrakk> \<Longrightarrow>  
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  is_class (super c) \<and> (super c,C)\<notin>subcls1^+"
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apply (unfold ws_prog_def class_def)
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apply (drule_tac map_of_SomeD)
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apply auto
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done
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lemma subcls1_irrefl_lemma1: "ws_prog \<Longrightarrow> subcls1^-1 \<inter> subcls1^+ = {}"
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by (fast dest: subcls1D ws_progD)
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(* context (theory "Transitive_Closure") *)
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lemma irrefl_tranclI': "r^-1 Int r^+ = {} ==> !x. (x, x) ~: r^+"
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apply (rule allI)
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apply (erule irrefl_tranclI)
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done
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lemmas subcls1_irrefl_lemma2 = subcls1_irrefl_lemma1 [THEN irrefl_tranclI']
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lemma subcls1_irrefl: "\<lbrakk>(x, y) \<in> subcls1; ws_prog\<rbrakk> \<Longrightarrow> x \<noteq> y"
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apply (rule irrefl_trancl_rD)
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apply (rule subcls1_irrefl_lemma2)
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apply auto
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done
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lemmas subcls1_acyclic = subcls1_irrefl_lemma2 [THEN acyclicI, standard]
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lemma wf_subcls1: "ws_prog \<Longrightarrow> wf (subcls1\<inverse>)"
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by (auto intro: finite_acyclic_wf_converse finite_subcls1 subcls1_acyclic)
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consts class_rec ::"cname \<Rightarrow> (class \<Rightarrow> ('a \<times> 'b) list) \<Rightarrow> ('a \<leadsto> 'b)"
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recdef class_rec "subcls1\<inverse>"
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      "class_rec C = (\<lambda>f. case class C of None   \<Rightarrow> arbitrary 
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                                        | Some m \<Rightarrow> if wf (subcls1\<inverse>) 
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       then (if C=Object then empty else class_rec (super m) f) ++ map_of (f m)
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       else arbitrary)"
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(hints intro: subcls1I)
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lemma class_rec: "\<lbrakk>class C = Some m;  ws_prog\<rbrakk> \<Longrightarrow>
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 class_rec C f = (if C = Object then empty else class_rec (super m) f) ++ 
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                 map_of (f m)";
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apply (drule wf_subcls1)
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apply (rule class_rec.simps [THEN trans [THEN fun_cong]])
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apply  assumption
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apply simp
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done
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--{* Methods of a class, with inheritance and hiding *}
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defs method_def: "method C \<equiv> class_rec C methods"
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lemma method_rec: "\<lbrakk>class C = Some m; ws_prog\<rbrakk> \<Longrightarrow>
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method C = (if C=Object then empty else method (super m)) ++ map_of (methods m)"
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apply (unfold method_def)
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apply (erule (1) class_rec [THEN trans]);
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apply simp
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done
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--{* Fields of a class, with inheritance and hiding *}
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defs field_def: "field C \<equiv> class_rec C fields"
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lemma fields_rec: "\<lbrakk>class C = Some m; ws_prog\<rbrakk> \<Longrightarrow>
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field C = (if C=Object then empty else field (super m)) ++ map_of (fields m)"
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apply (unfold field_def)
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apply (erule (1) class_rec [THEN trans]);
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apply simp
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done
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end