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
3 begin
5 subsubsection {* A monadic combinator for simple recursive functions *}
7 text {* Using a locale to fix arguments f and g of MREC *}
9 locale mrec =
10   fixes f :: "'a \<Rightarrow> ('b + 'a) Heap"
11   and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b Heap"
12 begin
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
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
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])
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
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)
62 definition
63   "MREC x = Heap_Monad.Heap (mrec x)"
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)
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
139 end
141 text {* Providing global versions of the constant and the theorems *}
143 abbreviation "MREC == mrec.MREC"
144 lemmas MREC_rule = mrec.MREC_rule
145 lemmas MREC_pinduct = mrec.MREC_pinduct
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])
164 end