src/HOL/Statespace/StateFun.thy
author schirmer
Mon Nov 12 11:07:22 2007 +0100 (2007-11-12)
changeset 25408 156f6f7082b8
parent 25174 d70d6dbc3a60
child 35114 b1fd1d756e20
permissions -rw-r--r--
added signatures;
tuned
     1 (*  Title:      StateFun.thy
     2     ID:         $Id$
     3     Author:     Norbert Schirmer, TU Muenchen
     4 *)
     5 
     6 header {* State Space Representation as Function \label{sec:StateFun}*}
     7 
     8 theory StateFun imports DistinctTreeProver 
     9 begin
    10 
    11 
    12 text {* The state space is represented as a function from names to
    13 values. We neither fix the type of names nor the type of values. We
    14 define lookup and update functions and provide simprocs that simplify
    15 expressions containing these, similar to HOL-records.
    16 
    17 The lookup and update function get constructor/destructor functions as
    18 parameters. These are used to embed various HOL-types into the
    19 abstract value type. Conceptually the abstract value type is a sum of
    20 all types that we attempt to store in the state space.
    21 
    22 The update is actually generalized to a map function. The map supplies
    23 better compositionality, especially if you think of nested state
    24 spaces.  *} 
    25 
    26 constdefs K_statefun:: "'a \<Rightarrow> 'b \<Rightarrow> 'a" "K_statefun c x \<equiv> c"
    27 
    28 lemma K_statefun_apply [simp]: "K_statefun c x = c"
    29   by (simp add: K_statefun_def)
    30 
    31 lemma K_statefun_comp [simp]: "(K_statefun c \<circ> f) = K_statefun c"
    32   by (rule ext) (simp add: K_statefun_apply comp_def)
    33 
    34 lemma K_statefun_cong [cong]: "K_statefun c x = K_statefun c x"
    35   by (rule refl)
    36 
    37 constdefs lookup:: "('v \<Rightarrow> 'a) \<Rightarrow> 'n \<Rightarrow> ('n \<Rightarrow> 'v) \<Rightarrow> 'a"
    38 "lookup destr n s \<equiv> destr (s n)"
    39 
    40 constdefs update:: 
    41   "('v \<Rightarrow> 'a1) \<Rightarrow> ('a2 \<Rightarrow> 'v) \<Rightarrow> 'n \<Rightarrow> ('a1 \<Rightarrow> 'a2) \<Rightarrow> ('n \<Rightarrow> 'v) \<Rightarrow> ('n \<Rightarrow> 'v)"
    42 "update destr constr n f s \<equiv> s(n := constr (f (destr (s n))))"
    43 
    44 lemma lookup_update_same:
    45   "(\<And>v. destr (constr v) = v) \<Longrightarrow> lookup destr n (update destr constr n f s) = 
    46          f (destr (s n))"  
    47   by (simp add: lookup_def update_def)
    48 
    49 lemma lookup_update_id_same:
    50   "lookup destr n (update destr' id n (K_statefun (lookup id n s')) s) =                  
    51      lookup destr n s'"  
    52   by (simp add: lookup_def update_def)
    53 
    54 lemma lookup_update_other:
    55   "n\<noteq>m \<Longrightarrow> lookup destr n (update destr' constr m f s) = lookup destr n s"  
    56   by (simp add: lookup_def update_def)
    57 
    58 
    59 lemma id_id_cancel: "id (id x) = x" 
    60   by (simp add: id_def)
    61   
    62 lemma destr_contstr_comp_id:
    63 "(\<And>v. destr (constr v) = v) \<Longrightarrow> destr \<circ> constr = id"
    64   by (rule ext) simp
    65 
    66 
    67 
    68 lemma block_conj_cong: "(P \<and> Q) = (P \<and> Q)"
    69   by simp
    70 
    71 lemma conj1_False: "(P\<equiv>False) \<Longrightarrow> (P \<and> Q) \<equiv> False"
    72   by simp
    73 
    74 lemma conj2_False: "\<lbrakk>Q\<equiv>False\<rbrakk> \<Longrightarrow> (P \<and> Q) \<equiv> False"
    75   by simp
    76 
    77 lemma conj_True: "\<lbrakk>P\<equiv>True; Q\<equiv>True\<rbrakk> \<Longrightarrow> (P \<and> Q) \<equiv> True"
    78   by simp
    79 
    80 lemma conj_cong: "\<lbrakk>P\<equiv>P'; Q\<equiv>Q'\<rbrakk> \<Longrightarrow> (P \<and> Q) \<equiv> (P' \<and> Q')"
    81   by simp
    82 
    83 
    84 lemma update_apply: "(update destr constr n f s x) = 
    85      (if x=n then constr (f (destr (s n))) else s x)"
    86   by (simp add: update_def)
    87 
    88 lemma ex_id: "\<exists>x. id x = y"
    89   by (simp add: id_def)
    90 
    91 lemma swap_ex_eq: 
    92   "\<exists>s. f s = x \<equiv> True \<Longrightarrow>
    93    \<exists>s. x = f s \<equiv> True"
    94   apply (rule eq_reflection)
    95   apply auto
    96   done
    97 
    98 lemmas meta_ext = eq_reflection [OF ext]
    99 
   100 (* This lemma only works if the store is welltyped:
   101     "\<exists>x.  s ''n'' = (c x)" 
   102    or in general when c (d x) = x,
   103      (for example: c=id and d=id)
   104  *)
   105 lemma "update d c n (K_statespace (lookup d n s)) s = s"
   106   apply (simp add: update_def lookup_def)
   107   apply (rule ext)
   108   apply simp
   109   oops
   110 
   111 end