src/HOL/MicroJava/BV/JVMType.thy
author haftmann
Mon Mar 01 13:40:23 2010 +0100 (2010-03-01)
changeset 35416 d8d7d1b785af
parent 33954 1bc3b688548c
child 35417 47ee18b6ae32
permissions -rw-r--r--
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
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(*  Title:      HOL/MicroJava/BV/JVM.thy
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    Author:     Gerwin Klein
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    Copyright   2000 TUM
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*)
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header {* \isaheader{The JVM Type System as Semilattice} *}
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theory JVMType
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imports JType
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begin
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types
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  locvars_type = "ty err list"
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  opstack_type = "ty list"
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  state_type   = "opstack_type \<times> locvars_type"
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  state        = "state_type option err"    -- "for Kildall"
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  method_type  = "state_type option list"   -- "for BVSpec"
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  class_type   = "sig \<Rightarrow> method_type"
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  prog_type    = "cname \<Rightarrow> class_type"
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definition stk_esl :: "'c prog \<Rightarrow> nat \<Rightarrow> ty list esl" where
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  "stk_esl S maxs == upto_esl maxs (JType.esl S)"
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definition reg_sl :: "'c prog \<Rightarrow> nat \<Rightarrow> ty err list sl" where
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  "reg_sl S maxr == Listn.sl maxr (Err.sl (JType.esl S))"
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definition sl :: "'c prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> state sl" where
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  "sl S maxs maxr ==
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  Err.sl(Opt.esl(Product.esl (stk_esl S maxs) (Err.esl(reg_sl S maxr))))"
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definition states :: "'c prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> state set" where
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  "states S maxs maxr == fst(sl S maxs maxr)"
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definition le :: "'c prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> state ord" where
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  "le S maxs maxr == fst(snd(sl S maxs maxr))"
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definition  sup :: "'c prog \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> state binop" where
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  "sup S maxs maxr == snd(snd(sl S maxs maxr))"
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definition sup_ty_opt :: "['code prog,ty err,ty err] \<Rightarrow> bool"
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                 ("_ |- _ <=o _" [71,71] 70) where 
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  "sup_ty_opt G == Err.le (subtype G)"
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definition sup_loc :: "['code prog,locvars_type,locvars_type] \<Rightarrow> bool" 
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              ("_ |- _ <=l _"  [71,71] 70) where
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  "sup_loc G == Listn.le (sup_ty_opt G)"
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definition sup_state :: "['code prog,state_type,state_type] \<Rightarrow> bool"   
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               ("_ |- _ <=s _"  [71,71] 70) where
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  "sup_state G == Product.le (Listn.le (subtype G)) (sup_loc G)"
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definition sup_state_opt :: "['code prog,state_type option,state_type option] \<Rightarrow> bool" 
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                   ("_ |- _ <=' _"  [71,71] 70) where
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  "sup_state_opt G == Opt.le (sup_state G)"
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syntax (xsymbols)
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  sup_ty_opt    :: "['code prog,ty err,ty err] \<Rightarrow> bool" 
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                   ("_ \<turnstile> _ <=o _" [71,71] 70)
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  sup_loc       :: "['code prog,locvars_type,locvars_type] \<Rightarrow> bool" 
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                   ("_ \<turnstile> _ <=l _" [71,71] 70)
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  sup_state     :: "['code prog,state_type,state_type] \<Rightarrow> bool" 
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                   ("_ \<turnstile> _ <=s _" [71,71] 70)
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  sup_state_opt :: "['code prog,state_type option,state_type option] \<Rightarrow> bool"
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                   ("_ \<turnstile> _ <=' _" [71,71] 70)
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lemma JVM_states_unfold: 
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  "states S maxs maxr == err(opt((Union {list n (types S) |n. n <= maxs}) <*>
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                                  list maxr (err(types S))))"
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  apply (unfold states_def sl_def Opt.esl_def Err.sl_def
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         stk_esl_def reg_sl_def Product.esl_def
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         Listn.sl_def upto_esl_def JType.esl_def Err.esl_def)
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  by simp
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lemma JVM_le_unfold:
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 "le S m n == 
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  Err.le(Opt.le(Product.le(Listn.le(subtype S))(Listn.le(Err.le(subtype S)))))" 
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  apply (unfold le_def sl_def Opt.esl_def Err.sl_def
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         stk_esl_def reg_sl_def Product.esl_def  
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         Listn.sl_def upto_esl_def JType.esl_def Err.esl_def) 
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  by simp
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lemma JVM_le_convert:
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  "le G m n (OK t1) (OK t2) = G \<turnstile> t1 <=' t2"
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  by (simp add: JVM_le_unfold Err.le_def lesub_def sup_state_opt_def 
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                sup_state_def sup_loc_def sup_ty_opt_def)
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lemma JVM_le_Err_conv:
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  "le G m n = Err.le (sup_state_opt G)"
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  by (unfold sup_state_opt_def sup_state_def sup_loc_def  
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             sup_ty_opt_def JVM_le_unfold) simp
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lemma zip_map [rule_format]:
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  "\<forall>a. length a = length b \<longrightarrow> 
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  zip (map f a) (map g b) = map (\<lambda>(x,y). (f x, g y)) (zip a b)"
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  apply (induct b) 
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   apply simp
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  apply clarsimp
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  apply (case_tac aa)
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  apply simp+
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  done
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lemma [simp]: "Err.le r (OK a) (OK b) = r a b"
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  by (simp add: Err.le_def lesub_def)
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lemma stk_convert:
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  "Listn.le (subtype G) a b = G \<turnstile> map OK a <=l map OK b"
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proof 
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  assume "Listn.le (subtype G) a b"
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  hence le: "list_all2 (subtype G) a b"
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    by (unfold Listn.le_def lesub_def)
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  { fix x' y'
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    assume "length a = length b"
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           "(x',y') \<in> set (zip (map OK a) (map OK b))"
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    then
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    obtain x y where OK:
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      "x' = OK x" "y' = OK y" "(x,y) \<in> set (zip a b)"
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      by (auto simp add: zip_map)
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    with le
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    have "subtype G x y"
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      by (simp add: list_all2_def Ball_def)
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    with OK
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    have "G \<turnstile> x' <=o y'"
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      by (simp add: sup_ty_opt_def)
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  }
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  with le
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  show "G \<turnstile> map OK a <=l map OK b"
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    by (unfold sup_loc_def Listn.le_def lesub_def list_all2_def) auto
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next
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  assume "G \<turnstile> map OK a <=l map OK b"
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  thus "Listn.le (subtype G) a b"
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    apply (unfold sup_loc_def list_all2_def Listn.le_def lesub_def)
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    apply (clarsimp simp add: zip_map)
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    apply (drule bspec, assumption)
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    apply (auto simp add: sup_ty_opt_def subtype_def)
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    done
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qed
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lemma sup_state_conv:
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  "(G \<turnstile> s1 <=s s2) == 
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  (G \<turnstile> map OK (fst s1) <=l map OK (fst s2)) \<and> (G \<turnstile> snd s1 <=l snd s2)"
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  by (auto simp add: sup_state_def stk_convert lesub_def Product.le_def split_beta)
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lemma subtype_refl [simp]:
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  "subtype G t t"
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  by (simp add: subtype_def)
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theorem sup_ty_opt_refl [simp]:
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  "G \<turnstile> t <=o t"
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  by (simp add: sup_ty_opt_def Err.le_def lesub_def split: err.split)
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lemma le_list_refl2 [simp]: 
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  "(\<And>xs. r xs xs) \<Longrightarrow> Listn.le r xs xs"
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  by (induct xs, auto simp add: Listn.le_def lesub_def)
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theorem sup_loc_refl [simp]:
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  "G \<turnstile> t <=l t"
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  by (simp add: sup_loc_def)
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theorem sup_state_refl [simp]:
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  "G \<turnstile> s <=s s"
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  by (auto simp add: sup_state_def Product.le_def lesub_def)
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theorem sup_state_opt_refl [simp]:
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  "G \<turnstile> s <=' s"
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  by (simp add: sup_state_opt_def Opt.le_def lesub_def split: option.split)
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theorem anyConvErr [simp]:
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  "(G \<turnstile> Err <=o any) = (any = Err)"
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  by (simp add: sup_ty_opt_def Err.le_def split: err.split)
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theorem OKanyConvOK [simp]:
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  "(G \<turnstile> (OK ty') <=o (OK ty)) = (G \<turnstile> ty' \<preceq> ty)"
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  by (simp add: sup_ty_opt_def Err.le_def lesub_def subtype_def)
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theorem sup_ty_opt_OK:
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  "G \<turnstile> a <=o (OK b) \<Longrightarrow> \<exists> x. a = OK x"
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  by (clarsimp simp add: sup_ty_opt_def Err.le_def split: err.splits)
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lemma widen_PrimT_conv1 [simp]:
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  "\<lbrakk> G \<turnstile> S \<preceq> T; S = PrimT x\<rbrakk> \<Longrightarrow> T = PrimT x"
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  by (auto elim: widen.cases)
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theorem sup_PTS_eq:
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  "(G \<turnstile> OK (PrimT p) <=o X) = (X=Err \<or> X = OK (PrimT p))"
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  by (auto simp add: sup_ty_opt_def Err.le_def lesub_def subtype_def 
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              split: err.splits)
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theorem sup_loc_Nil [iff]:
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  "(G \<turnstile> [] <=l XT) = (XT=[])"
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  by (simp add: sup_loc_def Listn.le_def)
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theorem sup_loc_Cons [iff]:
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  "(G \<turnstile> (Y#YT) <=l XT) = (\<exists>X XT'. XT=X#XT' \<and> (G \<turnstile> Y <=o X) \<and> (G \<turnstile> YT <=l XT'))"
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  by (simp add: sup_loc_def Listn.le_def lesub_def list_all2_Cons1)
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theorem sup_loc_Cons2:
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  "(G \<turnstile> YT <=l (X#XT)) = (\<exists>Y YT'. YT=Y#YT' \<and> (G \<turnstile> Y <=o X) \<and> (G \<turnstile> YT' <=l XT))"
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  by (simp add: sup_loc_def Listn.le_def lesub_def list_all2_Cons2)
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lemma sup_state_Cons:
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  "(G \<turnstile> (x#xt, a) <=s (y#yt, b)) = 
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   ((G \<turnstile> x \<preceq> y) \<and> (G \<turnstile> (xt,a) <=s (yt,b)))"
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  by (auto simp add: sup_state_def stk_convert lesub_def Product.le_def)
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theorem sup_loc_length:
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  "G \<turnstile> a <=l b \<Longrightarrow> length a = length b"
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proof -
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  assume G: "G \<turnstile> a <=l b"
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  have "\<forall>b. (G \<turnstile> a <=l b) \<longrightarrow> length a = length b"
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    by (induct a, auto)
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  with G
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  show ?thesis by blast
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qed
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theorem sup_loc_nth:
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  "\<lbrakk> G \<turnstile> a <=l b; n < length a \<rbrakk> \<Longrightarrow> G \<turnstile> (a!n) <=o (b!n)"
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proof -
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  assume a: "G \<turnstile> a <=l b" "n < length a"
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  have "\<forall> n b. (G \<turnstile> a <=l b) \<longrightarrow> n < length a \<longrightarrow> (G \<turnstile> (a!n) <=o (b!n))"
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    (is "?P a")
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  proof (induct a)
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    show "?P []" by simp
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    fix x xs assume IH: "?P xs"
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    show "?P (x#xs)"
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    proof (intro strip)
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      fix n b
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      assume "G \<turnstile> (x # xs) <=l b" "n < length (x # xs)"
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      with IH
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      show "G \<turnstile> ((x # xs) ! n) <=o (b ! n)"
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        by - (cases n, auto)
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    qed
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  qed
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  with a
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  show ?thesis by blast
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qed
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theorem all_nth_sup_loc:
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  "\<forall>b. length a = length b \<longrightarrow> (\<forall> n. n < length a \<longrightarrow> (G \<turnstile> (a!n) <=o (b!n))) 
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  \<longrightarrow> (G \<turnstile> a <=l b)" (is "?P a")
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proof (induct a)
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  show "?P []" by simp
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  fix l ls assume IH: "?P ls"
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  show "?P (l#ls)"
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  proof (intro strip)
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    fix b
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    assume f: "\<forall>n. n < length (l # ls) \<longrightarrow> (G \<turnstile> ((l # ls) ! n) <=o (b ! n))"
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    assume l: "length (l#ls) = length b"
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    then obtain b' bs where b: "b = b'#bs"
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      by - (cases b, simp, simp add: neq_Nil_conv, rule that)
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    with f
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    have "\<forall>n. n < length ls \<longrightarrow> (G \<turnstile> (ls!n) <=o (bs!n))"
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      by auto
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    with f b l IH
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    show "G \<turnstile> (l # ls) <=l b"
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      by auto
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  qed
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qed
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theorem sup_loc_append:
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  "length a = length b \<Longrightarrow> 
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   (G \<turnstile> (a@x) <=l (b@y)) = ((G \<turnstile> a <=l b) \<and> (G \<turnstile> x <=l y))"
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proof -
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  assume l: "length a = length b"
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  have "\<forall>b. length a = length b \<longrightarrow> (G \<turnstile> (a@x) <=l (b@y)) = ((G \<turnstile> a <=l b) \<and> 
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            (G \<turnstile> x <=l y))" (is "?P a") 
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  proof (induct a)
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    show "?P []" by simp
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    fix l ls assume IH: "?P ls"    
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    show "?P (l#ls)" 
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    proof (intro strip)
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      fix b
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      assume "length (l#ls) = length (b::ty err list)"
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      with IH
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      show "(G \<turnstile> ((l#ls)@x) <=l (b@y)) = ((G \<turnstile> (l#ls) <=l b) \<and> (G \<turnstile> x <=l y))"
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        by - (cases b, auto)
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    qed
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  qed
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  with l
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  show ?thesis by blast
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qed
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theorem sup_loc_rev [simp]:
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   305
  "(G \<turnstile> (rev a) <=l rev b) = (G \<turnstile> a <=l b)"
kleing@10812
   306
proof -
kleing@10812
   307
  have "\<forall>b. (G \<turnstile> (rev a) <=l rev b) = (G \<turnstile> a <=l b)" (is "\<forall>b. ?Q a b" is "?P a")
kleing@10812
   308
  proof (induct a)
kleing@10812
   309
    show "?P []" by simp
kleing@10812
   310
kleing@10812
   311
    fix l ls assume IH: "?P ls"
kleing@10812
   312
kleing@10812
   313
    { 
kleing@10812
   314
      fix b
kleing@10812
   315
      have "?Q (l#ls) b"
wenzelm@25362
   316
      proof (cases b)
kleing@10812
   317
        case Nil
kleing@10812
   318
        thus ?thesis by (auto dest: sup_loc_length)
kleing@10812
   319
      next
wenzelm@25362
   320
        case (Cons a list)
kleing@10812
   321
        show ?thesis
kleing@10812
   322
        proof
kleing@10812
   323
          assume "G \<turnstile> (l # ls) <=l b"
kleing@10812
   324
          thus "G \<turnstile> rev (l # ls) <=l rev b"
kleing@10812
   325
            by (clarsimp simp add: Cons IH sup_loc_length sup_loc_append)
kleing@10812
   326
        next
kleing@10812
   327
          assume "G \<turnstile> rev (l # ls) <=l rev b"
kleing@10812
   328
          hence G: "G \<turnstile> (rev ls @ [l]) <=l (rev list @ [a])"
kleing@10812
   329
            by (simp add: Cons)          
kleing@10812
   330
          
kleing@10812
   331
          hence "length (rev ls) = length (rev list)"
kleing@10812
   332
            by (auto dest: sup_loc_length)
kleing@10812
   333
kleing@10812
   334
          from this G
kleing@10812
   335
          obtain "G \<turnstile> rev ls <=l rev list" "G \<turnstile> l <=o a"
kleing@10812
   336
            by (simp add: sup_loc_append)
kleing@10812
   337
kleing@10812
   338
          thus "G \<turnstile> (l # ls) <=l b"
kleing@10812
   339
            by (simp add: Cons IH)
kleing@10812
   340
        qed
kleing@10812
   341
      qed    
kleing@10812
   342
    }
kleing@10812
   343
    thus "?P (l#ls)" by blast
kleing@10812
   344
  qed
kleing@10812
   345
kleing@10812
   346
  thus ?thesis by blast
kleing@10812
   347
qed
kleing@10812
   348
kleing@10812
   349
kleing@10812
   350
theorem sup_loc_update [rule_format]:
kleing@13006
   351
  "\<forall> n y. (G \<turnstile> a <=o b) \<longrightarrow> n < length y \<longrightarrow> (G \<turnstile> x <=l y) \<longrightarrow> 
kleing@10812
   352
          (G \<turnstile> x[n := a] <=l y[n := b])" (is "?P x")
kleing@10812
   353
proof (induct x)
kleing@10812
   354
  show "?P []" by simp
kleing@10812
   355
kleing@10812
   356
  fix l ls assume IH: "?P ls"
kleing@10812
   357
  show "?P (l#ls)"
kleing@10812
   358
  proof (intro strip)
kleing@10812
   359
    fix n y
kleing@10812
   360
    assume "G \<turnstile>a <=o b" "G \<turnstile> (l # ls) <=l y" "n < length y"
kleing@10812
   361
    with IH
kleing@10812
   362
    show "G \<turnstile> (l # ls)[n := a] <=l y[n := b]"
kleing@10812
   363
      by - (cases n, auto simp add: sup_loc_Cons2 list_all2_Cons1)
kleing@10812
   364
  qed
kleing@10812
   365
qed
kleing@10812
   366
kleing@10812
   367
kleing@10812
   368
theorem sup_state_length [simp]:
kleing@13006
   369
  "G \<turnstile> s2 <=s s1 \<Longrightarrow> 
kleing@10812
   370
   length (fst s2) = length (fst s1) \<and> length (snd s2) = length (snd s1)"
kleing@10812
   371
  by (auto dest: sup_loc_length simp add: sup_state_def stk_convert lesub_def Product.le_def);
kleing@10812
   372
kleing@10812
   373
theorem sup_state_append_snd:
kleing@13006
   374
  "length a = length b \<Longrightarrow> 
kleing@10812
   375
  (G \<turnstile> (i,a@x) <=s (j,b@y)) = ((G \<turnstile> (i,a) <=s (j,b)) \<and> (G \<turnstile> (i,x) <=s (j,y)))"
kleing@10812
   376
  by (auto simp add: sup_state_def stk_convert lesub_def Product.le_def sup_loc_append)
kleing@10812
   377
kleing@10812
   378
theorem sup_state_append_fst:
kleing@13006
   379
  "length a = length b \<Longrightarrow> 
kleing@10812
   380
  (G \<turnstile> (a@x,i) <=s (b@y,j)) = ((G \<turnstile> (a,i) <=s (b,j)) \<and> (G \<turnstile> (x,i) <=s (y,j)))"
kleing@10812
   381
  by (auto simp add: sup_state_def stk_convert lesub_def Product.le_def sup_loc_append)
kleing@10812
   382
kleing@10812
   383
theorem sup_state_Cons1:
kleing@10812
   384
  "(G \<turnstile> (x#xt, a) <=s (yt, b)) = 
kleing@10812
   385
   (\<exists>y yt'. yt=y#yt' \<and> (G \<turnstile> x \<preceq> y) \<and> (G \<turnstile> (xt,a) <=s (yt',b)))"
nipkow@14025
   386
  by (auto simp add: sup_state_def stk_convert lesub_def Product.le_def)
kleing@10812
   387
kleing@10812
   388
theorem sup_state_Cons2:
kleing@10812
   389
  "(G \<turnstile> (xt, a) <=s (y#yt, b)) = 
kleing@10812
   390
   (\<exists>x xt'. xt=x#xt' \<and> (G \<turnstile> x \<preceq> y) \<and> (G \<turnstile> (xt',a) <=s (yt,b)))"
nipkow@14025
   391
  by (auto simp add: sup_state_def stk_convert lesub_def Product.le_def sup_loc_Cons2)
kleing@10812
   392
kleing@10812
   393
theorem sup_state_ignore_fst:  
kleing@13006
   394
  "G \<turnstile> (a, x) <=s (b, y) \<Longrightarrow> G \<turnstile> (c, x) <=s (c, y)"
kleing@10812
   395
  by (simp add: sup_state_def lesub_def Product.le_def)
kleing@10812
   396
kleing@10812
   397
theorem sup_state_rev_fst:
kleing@10812
   398
  "(G \<turnstile> (rev a, x) <=s (rev b, y)) = (G \<turnstile> (a, x) <=s (b, y))"
kleing@10812
   399
proof -
kleing@13006
   400
  have m: "\<And>f x. map f (rev x) = rev (map f x)" by (simp add: rev_map)
kleing@10812
   401
  show ?thesis by (simp add: m sup_state_def stk_convert lesub_def Product.le_def)
kleing@10812
   402
qed
kleing@10812
   403
  
kleing@10812
   404
kleing@10812
   405
lemma sup_state_opt_None_any [iff]:
kleing@10812
   406
  "(G \<turnstile> None <=' any) = True"
kleing@10812
   407
  by (simp add: sup_state_opt_def Opt.le_def split: option.split)
kleing@10812
   408
kleing@10812
   409
lemma sup_state_opt_any_None [iff]:
kleing@10812
   410
  "(G \<turnstile> any <=' None) = (any = None)"
kleing@10812
   411
  by (simp add: sup_state_opt_def Opt.le_def split: option.split)
kleing@10812
   412
kleing@10812
   413
lemma sup_state_opt_Some_Some [iff]:
kleing@10812
   414
  "(G \<turnstile> (Some a) <=' (Some b)) = (G \<turnstile> a <=s b)"
kleing@10812
   415
  by (simp add: sup_state_opt_def Opt.le_def lesub_def del: split_paired_Ex)
kleing@10812
   416
kleing@10812
   417
lemma sup_state_opt_any_Some [iff]:
kleing@10812
   418
  "(G \<turnstile> (Some a) <=' any) = (\<exists>b. any = Some b \<and> G \<turnstile> a <=s b)"
kleing@10812
   419
  by (simp add: sup_state_opt_def Opt.le_def lesub_def split: option.split)
kleing@10812
   420
kleing@10812
   421
lemma sup_state_opt_Some_any:
kleing@10812
   422
  "(G \<turnstile> any <=' (Some b)) = (any = None \<or> (\<exists>a. any = Some a \<and> G \<turnstile> a <=s b))"
kleing@10812
   423
  by (simp add: sup_state_opt_def Opt.le_def lesub_def split: option.split)
kleing@10812
   424
kleing@10812
   425
kleing@10812
   426
theorem sup_ty_opt_trans [trans]:
kleing@13006
   427
  "\<lbrakk>G \<turnstile> a <=o b; G \<turnstile> b <=o c\<rbrakk> \<Longrightarrow> G \<turnstile> a <=o c"
kleing@10812
   428
  by (auto intro: widen_trans 
kleing@10812
   429
           simp add: sup_ty_opt_def Err.le_def lesub_def subtype_def 
kleing@10812
   430
           split: err.splits)
kleing@10812
   431
kleing@10812
   432
theorem sup_loc_trans [trans]:
kleing@13006
   433
  "\<lbrakk>G \<turnstile> a <=l b; G \<turnstile> b <=l c\<rbrakk> \<Longrightarrow> G \<turnstile> a <=l c"
kleing@10812
   434
proof -
kleing@10812
   435
  assume G: "G \<turnstile> a <=l b" "G \<turnstile> b <=l c"
kleing@10812
   436
 
kleing@13006
   437
  hence "\<forall> n. n < length a \<longrightarrow> (G \<turnstile> (a!n) <=o (c!n))"
kleing@10812
   438
  proof (intro strip)
kleing@10812
   439
    fix n 
kleing@10812
   440
    assume n: "n < length a"
kleing@10812
   441
    with G
kleing@10812
   442
    have "G \<turnstile> (a!n) <=o (b!n)"
kleing@10812
   443
      by - (rule sup_loc_nth)
kleing@10812
   444
    also 
kleing@10812
   445
    from n G
kleing@13006
   446
    have "G \<turnstile> \<dots> <=o (c!n)"
kleing@10812
   447
      by - (rule sup_loc_nth, auto dest: sup_loc_length)
kleing@10812
   448
    finally
kleing@10812
   449
    show "G \<turnstile> (a!n) <=o (c!n)" .
kleing@10812
   450
  qed
kleing@10812
   451
kleing@10812
   452
  with G
kleing@10812
   453
  show ?thesis 
kleing@10812
   454
    by (auto intro!: all_nth_sup_loc [rule_format] dest!: sup_loc_length) 
kleing@10812
   455
qed
kleing@10812
   456
  
kleing@10812
   457
kleing@10812
   458
theorem sup_state_trans [trans]:
kleing@13006
   459
  "\<lbrakk>G \<turnstile> a <=s b; G \<turnstile> b <=s c\<rbrakk> \<Longrightarrow> G \<turnstile> a <=s c"
kleing@10812
   460
  by (auto intro: sup_loc_trans simp add: sup_state_def stk_convert Product.le_def lesub_def)
kleing@10812
   461
kleing@10812
   462
theorem sup_state_opt_trans [trans]:
kleing@13006
   463
  "\<lbrakk>G \<turnstile> a <=' b; G \<turnstile> b <=' c\<rbrakk> \<Longrightarrow> G \<turnstile> a <=' c"
kleing@10812
   464
  by (auto intro: sup_state_trans 
kleing@10812
   465
           simp add: sup_state_opt_def Opt.le_def lesub_def 
kleing@10812
   466
           split: option.splits)
kleing@10812
   467
kleing@10812
   468
end