theory Mrec
imports Heap_Monad
begin
subsubsection {* A monadic combinator for simple recursive functions *}
text {* Using a locale to fix arguments f and g of MREC *}
locale mrec =
fixes f :: "'a \<Rightarrow> ('b + 'a) Heap"
and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b Heap"
begin
function (default "\<lambda>(x, h). None") mrec :: "'a \<Rightarrow> heap \<Rightarrow> ('b \<times> heap) option" where
"mrec x h = (case execute (f x) h of
Some (Inl r, h') \<Rightarrow> Some (r, h')
| Some (Inr s, h') \<Rightarrow> (case mrec s h' of
Some (z, h'') \<Rightarrow> execute (g x s z) h''
| None \<Rightarrow> None)
| None \<Rightarrow> None)"
by auto
lemma graph_implies_dom:
"mrec_graph x y \<Longrightarrow> mrec_dom x"
apply (induct rule:mrec_graph.induct)
apply (rule accpI)
apply (erule mrec_rel.cases)
by simp
lemma mrec_default: "\<not> mrec_dom (x, h) \<Longrightarrow> mrec x h = None"
unfolding mrec_def
by (rule fundef_default_value[OF mrec_sumC_def graph_implies_dom, of _ _ "(x, h)", simplified])
lemma mrec_di_reverse:
assumes "\<not> mrec_dom (x, h)"
shows "
(case execute (f x) h of
Some (Inl r, h') \<Rightarrow> False
| Some (Inr s, h') \<Rightarrow> \<not> mrec_dom (s, h')
| None \<Rightarrow> False
)"
using assms apply (auto split: option.split sum.split)
apply (rule ccontr)
apply (erule notE, rule accpI, elim mrec_rel.cases, auto)+
done
lemma mrec_rule:
"mrec x h =
(case execute (f x) h of
Some (Inl r, h') \<Rightarrow> Some (r, h')
| Some (Inr s, h') \<Rightarrow>
(case mrec s h' of
Some (z, h'') \<Rightarrow> execute (g x s z) h''
| None \<Rightarrow> None)
| None \<Rightarrow> None
)"
apply (cases "mrec_dom (x,h)", simp)
apply (frule mrec_default)
apply (frule mrec_di_reverse, simp)
by (auto split: sum.split option.split simp: mrec_default)
definition
"MREC x = Heap_Monad.Heap (mrec x)"
lemma MREC_rule:
"MREC x =
do { y \<leftarrow> f x;
(case y of
Inl r \<Rightarrow> return r
| Inr s \<Rightarrow>
do { z \<leftarrow> MREC s ;
g x s z })}"
unfolding MREC_def
unfolding bind_def return_def
apply simp
apply (rule ext)
apply (unfold mrec_rule[of x])
by (auto simp add: execute_simps split: option.splits prod.splits sum.splits)
lemma MREC_pinduct:
assumes "execute (MREC x) h = Some (r, h')"
assumes non_rec_case: "\<And> x h h' r. execute (f x) h = Some (Inl r, h') \<Longrightarrow> P x h h' r"
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
\<Longrightarrow> execute (g x s z) h2 = Some (r, h') \<Longrightarrow> P x h h' r"
shows "P x h h' r"
proof -
from assms(1) have mrec: "mrec x h = Some (r, h')"
unfolding MREC_def execute.simps .
from mrec have dom: "mrec_dom (x, h)"
apply -
apply (rule ccontr)
apply (drule mrec_default) by auto
from mrec have h'_r: "h' = snd (the (mrec x h))" "r = fst (the (mrec x h))"
by auto
from mrec have "P x h (snd (the (mrec x h))) (fst (the (mrec x h)))"
proof (induct arbitrary: r h' rule: mrec.pinduct[OF dom])
case (1 x h)
obtain rr h' where "the (mrec x h) = (rr, h')" by fastsimp
show ?case
proof (cases "execute (f x) h")
case (Some result)
then obtain a h1 where exec_f: "execute (f x) h = Some (a, h1)" by fastsimp
note Inl' = this
show ?thesis
proof (cases a)
case (Inl aa)
from this Inl' 1(1) exec_f mrec non_rec_case show ?thesis
by auto
next
case (Inr b)
note Inr' = this
show ?thesis
proof (cases "mrec b h1")
case (Some result)
then obtain aaa h2 where mrec_rec: "mrec b h1 = Some (aaa, h2)" by fastsimp
moreover from this have "P b h1 (snd (the (mrec b h1))) (fst (the (mrec b h1)))"
apply (intro 1(2))
apply (auto simp add: Inr Inl')
done
moreover note mrec mrec_rec exec_f Inl' Inr' 1(1) 1(3)
ultimately show ?thesis
apply auto
apply (rule rec_case)
apply auto
unfolding MREC_def by auto
next
case None
from this 1(1) exec_f mrec Inr' 1(3) show ?thesis by auto
qed
qed
next
case None
from this 1(1) mrec 1(3) show ?thesis by simp
qed
qed
from this h'_r show ?thesis by simp
qed
end
text {* Providing global versions of the constant and the theorems *}
abbreviation "MREC == mrec.MREC"
lemmas MREC_rule = mrec.MREC_rule
lemmas MREC_pinduct = mrec.MREC_pinduct
lemma MREC_induct:
assumes "crel (MREC f g x) h h' r"
assumes "\<And> x h h' r. crel (f x) h h' (Inl r) \<Longrightarrow> P x h h' r"
assumes "\<And> x h h1 h2 h' s z r. crel (f x) h h1 (Inr s) \<Longrightarrow> crel (MREC f g s) h1 h2 z \<Longrightarrow> P s h1 h2 z
\<Longrightarrow> crel (g x s z) h2 h' r \<Longrightarrow> P x h h' r"
shows "P x h h' r"
proof (rule MREC_pinduct[OF assms(1) [unfolded crel_def]])
fix x h h1 h2 h' s z r
assume "Heap_Monad.execute (f x) h = Some (Inr s, h1)"
"Heap_Monad.execute (MREC f g s) h1 = Some (z, h2)"
"P s h1 h2 z"
"Heap_Monad.execute (g x s z) h2 = Some (r, h')"
from assms(3) [unfolded crel_def, OF this(1) this(2) this(3) this(4)]
show "P x h h' r" .
next
qed (auto simp add: assms(2)[unfolded crel_def])
end