src/HOL/Imperative_HOL/Mrec.thy
author huffman
Sun Apr 01 16:09:58 2012 +0200 (2012-04-01)
changeset 47255 30a1692557b0
parent 44890 22f665a2e91c
permissions -rw-r--r--
removed Nat_Numeral.thy, moving all theorems elsewhere
     1 theory Mrec
     2 imports Heap_Monad
     3 begin
     4 
     5 subsubsection {* A monadic combinator for simple recursive functions *}
     6 
     7 text {* Using a locale to fix arguments f and g of MREC *}
     8 
     9 locale mrec =
    10   fixes f :: "'a \<Rightarrow> ('b + 'a) Heap"
    11   and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b Heap"
    12 begin
    13 
    14 function (default "\<lambda>(x, h). None") mrec :: "'a \<Rightarrow> heap \<Rightarrow> ('b \<times> heap) option" where
    15   "mrec x h = (case execute (f x) h of
    16      Some (Inl r, h') \<Rightarrow> Some (r, h')
    17    | Some (Inr s, h') \<Rightarrow> (case mrec s h' of
    18              Some (z, h'') \<Rightarrow> execute (g x s z) h''
    19            | None \<Rightarrow> None)
    20    | None \<Rightarrow> None)"
    21 by auto
    22 
    23 lemma graph_implies_dom:
    24   "mrec_graph x y \<Longrightarrow> mrec_dom x"
    25 apply (induct rule:mrec_graph.induct) 
    26 apply (rule accpI)
    27 apply (erule mrec_rel.cases)
    28 by simp
    29 
    30 lemma mrec_default: "\<not> mrec_dom (x, h) \<Longrightarrow> mrec x h = None"
    31   unfolding mrec_def 
    32   by (rule fundef_default_value[OF mrec_sumC_def graph_implies_dom, of _ _ "(x, h)", simplified])
    33 
    34 lemma mrec_di_reverse: 
    35   assumes "\<not> mrec_dom (x, h)"
    36   shows "
    37    (case execute (f x) h of
    38      Some (Inl r, h') \<Rightarrow> False
    39    | Some (Inr s, h') \<Rightarrow> \<not> mrec_dom (s, h')
    40    | None \<Rightarrow> False
    41    )" 
    42 using assms apply (auto split: option.split sum.split)
    43 apply (rule ccontr)
    44 apply (erule notE, rule accpI, elim mrec_rel.cases, auto)+
    45 done
    46 
    47 lemma mrec_rule:
    48   "mrec x h = 
    49    (case execute (f x) h of
    50      Some (Inl r, h') \<Rightarrow> Some (r, h')
    51    | Some (Inr s, h') \<Rightarrow> 
    52           (case mrec s h' of
    53              Some (z, h'') \<Rightarrow> execute (g x s z) h''
    54            | None \<Rightarrow> None)
    55    | None \<Rightarrow> None
    56    )"
    57 apply (cases "mrec_dom (x,h)", simp add: mrec.psimps)
    58 apply (frule mrec_default)
    59 apply (frule mrec_di_reverse, simp)
    60 by (auto split: sum.split option.split simp: mrec_default)
    61 
    62 definition
    63   "MREC x = Heap_Monad.Heap (mrec x)"
    64 
    65 lemma MREC_rule:
    66   "MREC x = 
    67   do { y \<leftarrow> f x;
    68                 (case y of 
    69                 Inl r \<Rightarrow> return r
    70               | Inr s \<Rightarrow> 
    71                 do { z \<leftarrow> MREC s ;
    72                      g x s z })}"
    73   unfolding MREC_def
    74   unfolding bind_def return_def
    75   apply simp
    76   apply (rule ext)
    77   apply (unfold mrec_rule[of x])
    78   by (auto simp add: execute_simps split: option.splits prod.splits sum.splits)
    79 
    80 lemma MREC_pinduct:
    81   assumes "execute (MREC x) h = Some (r, h')"
    82   assumes non_rec_case: "\<And> x h h' r. execute (f x) h = Some (Inl r, h') \<Longrightarrow> P x h h' r"
    83   assumes rec_case: "\<And> x h h1 h2 h' s z r. execute (f x) h = Some (Inr s, h1) \<Longrightarrow> execute (MREC s) h1 = Some (z, h2) \<Longrightarrow> P s h1 h2 z
    84     \<Longrightarrow> execute (g x s z) h2 = Some (r, h') \<Longrightarrow> P x h h' r"
    85   shows "P x h h' r"
    86 proof -
    87   from assms(1) have mrec: "mrec x h = Some (r, h')"
    88     unfolding MREC_def execute.simps .
    89   from mrec have dom: "mrec_dom (x, h)"
    90     apply -
    91     apply (rule ccontr)
    92     apply (drule mrec_default) by auto
    93   from mrec have h'_r: "h' = snd (the (mrec x h))" "r = fst (the (mrec x h))"
    94     by auto
    95   from mrec have "P x h (snd (the (mrec x h))) (fst (the (mrec x h)))"
    96   proof (induct arbitrary: r h' rule: mrec.pinduct[OF dom])
    97     case (1 x h)
    98     obtain rr h' where "the (mrec x h) = (rr, h')" by fastforce
    99     show ?case
   100     proof (cases "execute (f x) h")
   101       case (Some result)
   102       then obtain a h1 where exec_f: "execute (f x) h = Some (a, h1)" by fastforce
   103       note Inl' = this
   104       show ?thesis
   105       proof (cases a)
   106         case (Inl aa)
   107         from this Inl' 1(1) exec_f mrec non_rec_case show ?thesis
   108           by (auto simp: mrec.psimps)
   109       next
   110         case (Inr b)
   111         note Inr' = this
   112         show ?thesis
   113         proof (cases "mrec b h1")
   114           case (Some result)
   115           then obtain aaa h2 where mrec_rec: "mrec b h1 = Some (aaa, h2)" by fastforce
   116           moreover from this have "P b h1 (snd (the (mrec b h1))) (fst (the (mrec b h1)))"
   117             apply (intro 1(2))
   118             apply (auto simp add: Inr Inl')
   119             done
   120           moreover note mrec mrec_rec exec_f Inl' Inr' 1(1) 1(3)
   121           ultimately show ?thesis
   122             apply auto
   123             apply (rule rec_case)
   124             apply auto
   125             unfolding MREC_def by (auto simp: mrec.psimps)
   126         next
   127           case None
   128           from this 1(1) exec_f mrec Inr' 1(3) show ?thesis by (auto simp: mrec.psimps)
   129         qed
   130       qed
   131     next
   132       case None
   133       from this 1(1) mrec 1(3) show ?thesis by (simp add: mrec.psimps)
   134     qed
   135   qed
   136   from this h'_r show ?thesis by simp
   137 qed
   138 
   139 end
   140 
   141 text {* Providing global versions of the constant and the theorems *}
   142 
   143 abbreviation "MREC == mrec.MREC"
   144 lemmas MREC_rule = mrec.MREC_rule
   145 lemmas MREC_pinduct = mrec.MREC_pinduct
   146 
   147 lemma MREC_induct:
   148   assumes "effect (MREC f g x) h h' r"
   149   assumes "\<And> x h h' r. effect (f x) h h' (Inl r) \<Longrightarrow> P x h h' r"
   150   assumes "\<And> x h h1 h2 h' s z r. effect (f x) h h1 (Inr s) \<Longrightarrow> effect (MREC f g s) h1 h2 z \<Longrightarrow> P s h1 h2 z
   151     \<Longrightarrow> effect (g x s z) h2 h' r \<Longrightarrow> P x h h' r"
   152   shows "P x h h' r"
   153 proof (rule MREC_pinduct[OF assms(1) [unfolded effect_def]])
   154   fix x h h1 h2 h' s z r
   155   assume "Heap_Monad.execute (f x) h = Some (Inr s, h1)"
   156     "Heap_Monad.execute (MREC f g s) h1 = Some (z, h2)"
   157     "P s h1 h2 z"
   158     "Heap_Monad.execute (g x s z) h2 = Some (r, h')"
   159   from assms(3) [unfolded effect_def, OF this(1) this(2) this(3) this(4)]
   160   show "P x h h' r" .
   161 next
   162 qed (auto simp add: assms(2)[unfolded effect_def])
   163 
   164 end