--- a/src/HOL/Imperative_HOL/Array.thy Mon Jul 05 10:42:27 2010 +0200
+++ b/src/HOL/Imperative_HOL/Array.thy Mon Jul 05 14:34:28 2010 +0200
@@ -28,7 +28,7 @@
[code del]: "nth a i = (do len \<leftarrow> length a;
(if i < len
then Heap_Monad.heap (\<lambda>h. (get_array a h ! i, h))
- else raise (''array lookup: index out of range''))
+ else raise ''array lookup: index out of range'')
done)"
definition
@@ -37,18 +37,12 @@
[code del]: "upd i x a = (do len \<leftarrow> length a;
(if i < len
then Heap_Monad.heap (\<lambda>h. (a, Heap.upd a i x h))
- else raise (''array update: index out of range''))
+ else raise ''array update: index out of range'')
done)"
lemma upd_return:
"upd i x a \<guillemotright> return a = upd i x a"
-proof (rule Heap_eqI)
- fix h
- obtain len h' where "Heap_Monad.execute (Array.length a) h = (len, h')"
- by (cases "Heap_Monad.execute (Array.length a) h")
- then show "Heap_Monad.execute (upd i x a \<guillemotright> return a) h = Heap_Monad.execute (upd i x a) h"
- by (auto simp add: upd_def bindM_def split: sum.split)
-qed
+ by (rule Heap_eqI) (simp add: upd_def bindM_def split: option.split)
subsection {* Derivates *}
@@ -99,14 +93,11 @@
lemma array_make [code]:
"Array.new n x = make n (\<lambda>_. x)"
- by (induct n) (simp_all add: make_def new_def Heap_Monad.heap_def
- monad_simp array_of_list_replicate [symmetric]
- map_replicate_trivial replicate_append_same
- of_list_def)
+ by (rule Heap_eqI) (simp add: make_def new_def array_of_list_replicate map_replicate_trivial of_list_def)
lemma array_of_list_make [code]:
"of_list xs = make (List.length xs) (\<lambda>n. xs ! n)"
- unfolding make_def map_nth ..
+ by (rule Heap_eqI) (simp add: make_def map_nth)
subsection {* Code generator setup *}
@@ -135,13 +126,11 @@
by (simp add: make'_def o_def)
definition length' where
- [code del]: "length' = Array.length \<guillemotright>== liftM Code_Numeral.of_nat"
+ [code del]: "length' a = Array.length a \<guillemotright>= (\<lambda>n. return (Code_Numeral.of_nat n))"
hide_const (open) length'
lemma [code]:
- "Array.length = Array.length' \<guillemotright>== liftM Code_Numeral.nat_of"
- by (simp add: length'_def monad_simp',
- simp add: liftM_def comp_def monad_simp,
- simp add: monad_simp')
+ "Array.length a = Array.length' a \<guillemotright>= (\<lambda>i. return (Code_Numeral.nat_of i))"
+ by (simp add: length'_def)
definition nth' where
[code del]: "nth' a = Array.nth a o Code_Numeral.nat_of"
@@ -155,7 +144,7 @@
hide_const (open) upd'
lemma [code]:
"Array.upd i x a = Array.upd' a (Code_Numeral.of_nat i) x \<guillemotright> return a"
- by (simp add: upd'_def monad_simp upd_return)
+ by (simp add: upd'_def upd_return)
subsubsection {* SML *}
--- a/src/HOL/Imperative_HOL/Heap_Monad.thy Mon Jul 05 10:42:27 2010 +0200
+++ b/src/HOL/Imperative_HOL/Heap_Monad.thy Mon Jul 05 14:34:28 2010 +0200
@@ -12,16 +12,12 @@
subsubsection {* Monad combinators *}
-datatype exception = Exn
-
text {* Monadic heap actions either produce values
and transform the heap, or fail *}
-datatype 'a Heap = Heap "heap \<Rightarrow> ('a + exception) \<times> heap"
+datatype 'a Heap = Heap "heap \<Rightarrow> ('a \<times> heap) option"
-primrec
- execute :: "'a Heap \<Rightarrow> heap \<Rightarrow> ('a + exception) \<times> heap" where
- "execute (Heap f) = f"
-lemmas [code del] = execute.simps
+primrec execute :: "'a Heap \<Rightarrow> heap \<Rightarrow> ('a \<times> heap) option" where
+ [code del]: "execute (Heap f) = f"
lemma Heap_execute [simp]:
"Heap (execute f) = f" by (cases f) simp_all
@@ -34,58 +30,67 @@
"(\<And>h. (\<lambda>x. execute (f x) h) = (\<lambda>y. execute (g y) h)) \<Longrightarrow> f = g"
by (auto simp: expand_fun_eq intro: Heap_eqI)
-lemma Heap_strip: "(\<And>f. PROP P f) \<equiv> (\<And>g. PROP P (Heap g))"
-proof
- fix g :: "heap \<Rightarrow> ('a + exception) \<times> heap"
- assume "\<And>f. PROP P f"
- then show "PROP P (Heap g)" .
-next
- fix f :: "'a Heap"
- assume assm: "\<And>g. PROP P (Heap g)"
- then have "PROP P (Heap (execute f))" .
- then show "PROP P f" by simp
-qed
-
-definition
- heap :: "(heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where
- [code del]: "heap f = Heap (\<lambda>h. apfst Inl (f h))"
+definition heap :: "(heap \<Rightarrow> 'a \<times> heap) \<Rightarrow> 'a Heap" where
+ [code del]: "heap f = Heap (Some \<circ> f)"
lemma execute_heap [simp]:
- "execute (heap f) h = apfst Inl (f h)"
+ "execute (heap f) = Some \<circ> f"
by (simp add: heap_def)
-definition
- bindM :: "'a Heap \<Rightarrow> ('a \<Rightarrow> 'b Heap) \<Rightarrow> 'b Heap" (infixl ">>=" 54) where
- [code del]: "f >>= g = Heap (\<lambda>h. case execute f h of
- (Inl x, h') \<Rightarrow> execute (g x) h'
- | r \<Rightarrow> r)"
-
-notation
- bindM (infixl "\<guillemotright>=" 54)
+lemma heap_cases [case_names succeed fail]:
+ fixes f and h
+ assumes succeed: "\<And>x h'. execute f h = Some (x, h') \<Longrightarrow> P"
+ assumes fail: "execute f h = None \<Longrightarrow> P"
+ shows P
+ using assms by (cases "execute f h") auto
-abbreviation
- chainM :: "'a Heap \<Rightarrow> 'b Heap \<Rightarrow> 'b Heap" (infixl ">>" 54) where
- "f >> g \<equiv> f >>= (\<lambda>_. g)"
-
-notation
- chainM (infixl "\<guillemotright>" 54)
-
-definition
- return :: "'a \<Rightarrow> 'a Heap" where
+definition return :: "'a \<Rightarrow> 'a Heap" where
[code del]: "return x = heap (Pair x)"
lemma execute_return [simp]:
- "execute (return x) h = apfst Inl (x, h)"
+ "execute (return x) = Some \<circ> Pair x"
by (simp add: return_def)
-definition
- raise :: "string \<Rightarrow> 'a Heap" where -- {* the string is just decoration *}
- [code del]: "raise s = Heap (Pair (Inr Exn))"
+definition raise :: "string \<Rightarrow> 'a Heap" where -- {* the string is just decoration *}
+ [code del]: "raise s = Heap (\<lambda>_. None)"
lemma execute_raise [simp]:
- "execute (raise s) h = (Inr Exn, h)"
+ "execute (raise s) = (\<lambda>_. None)"
by (simp add: raise_def)
+definition bindM :: "'a Heap \<Rightarrow> ('a \<Rightarrow> 'b Heap) \<Rightarrow> 'b Heap" (infixl ">>=" 54) where
+ [code del]: "f >>= g = Heap (\<lambda>h. case execute f h of
+ Some (x, h') \<Rightarrow> execute (g x) h'
+ | None \<Rightarrow> None)"
+
+notation bindM (infixl "\<guillemotright>=" 54)
+
+lemma execute_bind [simp]:
+ "execute f h = Some (x, h') \<Longrightarrow> execute (f \<guillemotright>= g) h = execute (g x) h'"
+ "execute f h = None \<Longrightarrow> execute (f \<guillemotright>= g) h = None"
+ by (simp_all add: bindM_def)
+
+lemma execute_bind_heap [simp]:
+ "execute (heap f \<guillemotright>= g) h = execute (g (fst (f h))) (snd (f h))"
+ by (simp add: bindM_def split_def)
+
+lemma return_bind [simp]: "return x \<guillemotright>= f = f x"
+ by (rule Heap_eqI) simp
+
+lemma bind_return [simp]: "f \<guillemotright>= return = f"
+ by (rule Heap_eqI) (simp add: bindM_def split: option.splits)
+
+lemma bind_bind [simp]: "(f \<guillemotright>= g) \<guillemotright>= k = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= k)"
+ by (rule Heap_eqI) (simp add: bindM_def split: option.splits)
+
+lemma raise_bind [simp]: "raise e \<guillemotright>= f = raise e"
+ by (rule Heap_eqI) simp
+
+abbreviation chainM :: "'a Heap \<Rightarrow> 'b Heap \<Rightarrow> 'b Heap" (infixl ">>" 54) where
+ "f >> g \<equiv> f >>= (\<lambda>_. g)"
+
+notation chainM (infixl "\<guillemotright>" 54)
+
subsubsection {* do-syntax *}
@@ -170,88 +175,10 @@
subsection {* Monad properties *}
-subsubsection {* Monad laws *}
-
-lemma return_bind: "return x \<guillemotright>= f = f x"
- by (simp add: bindM_def return_def)
-
-lemma bind_return: "f \<guillemotright>= return = f"
-proof (rule Heap_eqI)
- fix h
- show "execute (f \<guillemotright>= return) h = execute f h"
- by (auto simp add: bindM_def return_def split: sum.splits prod.splits)
-qed
-
-lemma bind_bind: "(f \<guillemotright>= g) \<guillemotright>= h = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= h)"
- by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
-
-lemma bind_bind': "f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= h x) = f \<guillemotright>= (\<lambda>x. g x \<guillemotright>= (\<lambda>y. return (x, y))) \<guillemotright>= (\<lambda>(x, y). h x y)"
- by (rule Heap_eqI) (auto simp add: bindM_def split: split: sum.splits prod.splits)
-
-lemma raise_bind: "raise e \<guillemotright>= f = raise e"
- by (simp add: raise_def bindM_def)
-
-
-lemmas monad_simp = return_bind bind_return bind_bind raise_bind
-
-
subsection {* Generic combinators *}
-definition
- liftM :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b Heap"
-where
- "liftM f = return o f"
-
-definition
- compM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> ('b \<Rightarrow> 'c Heap) \<Rightarrow> 'a \<Rightarrow> 'c Heap" (infixl ">>==" 54)
-where
- "(f >>== g) = (\<lambda>x. f x \<guillemotright>= g)"
-
-notation
- compM (infixl "\<guillemotright>==" 54)
-
-lemma liftM_collapse: "liftM f x = return (f x)"
- by (simp add: liftM_def)
-
-lemma liftM_compM: "liftM f \<guillemotright>== g = g o f"
- by (auto intro: Heap_eqI' simp add: expand_fun_eq liftM_def compM_def bindM_def)
-
-lemma compM_return: "f \<guillemotright>== return = f"
- by (simp add: compM_def monad_simp)
-
-lemma compM_compM: "(f \<guillemotright>== g) \<guillemotright>== h = f \<guillemotright>== (g \<guillemotright>== h)"
- by (simp add: compM_def monad_simp)
-
-lemma liftM_bind:
- "(\<lambda>x. liftM f x \<guillemotright>= liftM g) = liftM (\<lambda>x. g (f x))"
- by (rule Heap_eqI') (simp add: monad_simp liftM_def bindM_def)
-
-lemma liftM_comp:
- "liftM f o g = liftM (f o g)"
- by (rule Heap_eqI') (simp add: liftM_def)
-
-lemmas monad_simp' = monad_simp liftM_compM compM_return
- compM_compM liftM_bind liftM_comp
-
-primrec
- mapM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b list Heap"
-where
- "mapM f [] = return []"
- | "mapM f (x#xs) = do y \<leftarrow> f x;
- ys \<leftarrow> mapM f xs;
- return (y # ys)
- done"
-
-primrec
- foldM :: "('a \<Rightarrow> 'b \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b Heap"
-where
- "foldM f [] s = return s"
- | "foldM f (x#xs) s = f x s \<guillemotright>= foldM f xs"
-
-definition
- assert :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a Heap"
-where
- "assert P x = (if P x then return x else raise (''assert''))"
+definition assert :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a Heap" where
+ "assert P x = (if P x then return x else raise ''assert'')"
lemma assert_cong [fundef_cong]:
assumes "P = P'"
@@ -259,28 +186,42 @@
shows "(assert P x >>= f) = (assert P' x >>= f')"
using assms by (auto simp add: assert_def return_bind raise_bind)
+definition liftM :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b Heap" where
+ "liftM f = return o f"
+
+lemma liftM_collapse [simp]:
+ "liftM f x = return (f x)"
+ by (simp add: liftM_def)
+
+lemma bind_liftM:
+ "(f \<guillemotright>= liftM g) = (f \<guillemotright>= (\<lambda>x. return (g x)))"
+ by (simp add: liftM_def comp_def)
+
+primrec mapM :: "('a \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b list Heap" where
+ "mapM f [] = return []"
+| "mapM f (x#xs) = do
+ y \<leftarrow> f x;
+ ys \<leftarrow> mapM f xs;
+ return (y # ys)
+ done"
+
+
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 => ('b + 'a) Heap"
- and g :: "'a => 'a => 'b => 'b Heap"
+ fixes f :: "'a \<Rightarrow> ('b + 'a) Heap"
+ and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b Heap"
begin
-function (default "\<lambda>(x,h). (Inr Exn, undefined)")
- mrec
-where
- "mrec x h =
- (case Heap_Monad.execute (f x) h of
- (Inl (Inl r), h') \<Rightarrow> (Inl r, h')
- | (Inl (Inr s), h') \<Rightarrow>
- (case mrec s h' of
- (Inl z, h'') \<Rightarrow> Heap_Monad.execute (g x s z) h''
- | (Inr e, h'') \<Rightarrow> (Inr e, h''))
- | (Inr e, h') \<Rightarrow> (Inr e, h')
- )"
+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:
@@ -290,38 +231,37 @@
apply (erule mrec_rel.cases)
by simp
-lemma mrec_default: "\<not> mrec_dom (x, h) \<Longrightarrow> mrec x h = (Inr Exn, undefined)"
+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 Heap_Monad.execute (f x) h of
- (Inl (Inl r), h') \<Rightarrow> False
- | (Inl (Inr s), h') \<Rightarrow> \<not> mrec_dom (s, h')
- | (Inr e, h') \<Rightarrow> False
+ (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
-by (auto split:prod.splits sum.splits)
- (erule notE, rule accpI, elim mrec_rel.cases, simp)+
-
+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 Heap_Monad.execute (f x) h of
- (Inl (Inl r), h') \<Rightarrow> (Inl r, h')
- | (Inl (Inr s), h') \<Rightarrow>
+ (case execute (f x) h of
+ Some (Inl r, h') \<Rightarrow> Some (r, h')
+ | Some (Inr s, h') \<Rightarrow>
(case mrec s h' of
- (Inl z, h'') \<Rightarrow> Heap_Monad.execute (g x s z) h''
- | (Inr e, h'') \<Rightarrow> (Inr e, h''))
- | (Inr e, h') \<Rightarrow> (Inr e, h')
+ 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 prod.split simp: mrec_default)
-
+by (auto split: sum.split option.split simp: mrec_default)
definition
"MREC x = Heap (mrec x)"
@@ -340,32 +280,31 @@
apply simp
apply (rule ext)
apply (unfold mrec_rule[of x])
- by (auto split:prod.splits sum.splits)
-
+ by (auto split: option.splits prod.splits sum.splits)
lemma MREC_pinduct:
- assumes "Heap_Monad.execute (MREC x) h = (Inl r, h')"
- assumes non_rec_case: "\<And> x h h' r. Heap_Monad.execute (f x) h = (Inl (Inl r), h') \<Longrightarrow> P x h h' r"
- assumes rec_case: "\<And> x h h1 h2 h' s z r. Heap_Monad.execute (f x) h = (Inl (Inr s), h1) \<Longrightarrow> Heap_Monad.execute (MREC s) h1 = (Inl z, h2) \<Longrightarrow> P s h1 h2 z
- \<Longrightarrow> Heap_Monad.execute (g x s z) h2 = (Inl r, h') \<Longrightarrow> P x h h' r"
+ 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 = (Inl r, h')"
+ 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 (mrec x h))" "r = (Sum_Type.Projl (fst (mrec x h)))"
+ 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 (mrec x h)) (Sum_Type.Projl (fst (mrec x h)))"
+ 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 "mrec x h = (rr, h')" by fastsimp
- obtain fret h1 where exec_f: "Heap_Monad.execute (f x) h = (fret, h1)" by fastsimp
+ obtain rr h' where "the (mrec x h) = (rr, h')" by fastsimp
show ?case
- proof (cases fret)
- case (Inl a)
+ 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)
@@ -375,23 +314,28 @@
next
case (Inr b)
note Inr' = this
- obtain ret_mrec h2 where mrec_rec: "mrec b h1 = (ret_mrec, h2)" by fastsimp
- from this Inl 1(1) exec_f mrec show ?thesis
- proof (cases "ret_mrec")
- case (Inl aaa)
- from this mrec exec_f Inl' Inr' 1(1) mrec_rec 1(2) [OF exec_f [symmetric] Inl' Inr', of "aaa" "h2"] 1(3)
- show ?thesis
- apply auto
- apply (rule rec_case)
- unfolding MREC_def by auto
+ 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 (Inr b)
- from this Inl 1(1) exec_f mrec Inr' mrec_rec 1(3) show ?thesis by auto
+ case None
+ from this 1(1) exec_f mrec Inr' 1(3) show ?thesis by auto
qed
qed
next
- case (Inr b)
- from this 1(1) mrec exec_f 1(3) show ?thesis by simp
+ case None
+ from this 1(1) mrec 1(3) show ?thesis by simp
qed
qed
from this h'_r show ?thesis by simp
@@ -412,38 +356,29 @@
subsubsection {* Logical intermediate layer *}
-definition
- Fail :: "String.literal \<Rightarrow> exception"
-where
- [code del]: "Fail s = Exn"
+primrec raise' :: "String.literal \<Rightarrow> 'a Heap" where
+ [code del, code_post]: "raise' (STR s) = raise s"
-definition
- raise_exc :: "exception \<Rightarrow> 'a Heap"
-where
- [code del]: "raise_exc e = raise []"
+lemma raise_raise' [code_inline]:
+ "raise s = raise' (STR s)"
+ by simp
-lemma raise_raise_exc [code, code_unfold]:
- "raise s = raise_exc (Fail (STR s))"
- unfolding Fail_def raise_exc_def raise_def ..
+code_datatype raise' -- {* avoid @{const "Heap"} formally *}
-hide_const (open) Fail raise_exc
+hide_const (open) raise'
subsubsection {* SML and OCaml *}
code_type Heap (SML "unit/ ->/ _")
-code_const Heap (SML "raise/ (Fail/ \"bare Heap\")")
code_const "op \<guillemotright>=" (SML "!(fn/ f'_/ =>/ fn/ ()/ =>/ f'_/ (_/ ())/ ())")
code_const return (SML "!(fn/ ()/ =>/ _)")
-code_const "Heap_Monad.Fail" (SML "Fail")
-code_const "Heap_Monad.raise_exc" (SML "!(fn/ ()/ =>/ raise/ _)")
+code_const Heap_Monad.raise' (SML "!(raise/ Fail/ _)")
code_type Heap (OCaml "_")
-code_const Heap (OCaml "failwith/ \"bare Heap\"")
code_const "op \<guillemotright>=" (OCaml "!(fun/ f'_/ ()/ ->/ f'_/ (_/ ())/ ())")
code_const return (OCaml "!(fun/ ()/ ->/ _)")
-code_const "Heap_Monad.Fail" (OCaml "Failure")
-code_const "Heap_Monad.raise_exc" (OCaml "!(fun/ ()/ ->/ raise/ _)")
+code_const Heap_Monad.raise' (OCaml "failwith/ _")
setup {*
@@ -514,8 +449,6 @@
*}
-code_reserved OCaml Failure raise
-
subsubsection {* Haskell *}
@@ -556,10 +489,8 @@
text {* Monad *}
code_type Heap (Haskell "Heap.ST/ Heap.RealWorld/ _")
-code_const Heap (Haskell "error/ \"bare Heap\"")
code_monad "op \<guillemotright>=" Haskell
code_const return (Haskell "return")
-code_const "Heap_Monad.Fail" (Haskell "_")
-code_const "Heap_Monad.raise_exc" (Haskell "error")
+code_const Heap_Monad.raise' (Haskell "error/ _")
end
--- a/src/HOL/Imperative_HOL/Ref.thy Mon Jul 05 10:42:27 2010 +0200
+++ b/src/HOL/Imperative_HOL/Ref.thy Mon Jul 05 14:34:28 2010 +0200
@@ -1,5 +1,4 @@
(* Title: HOL/Library/Ref.thy
- ID: $Id$
Author: John Matthews, Galois Connections; Alexander Krauss, Lukas Bulwahn & Florian Haftmann, TU Muenchen
*)
@@ -53,7 +52,7 @@
lemma update_change [code]:
"r := e = Ref.change (\<lambda>_. e) r \<guillemotright> return ()"
- by (auto simp add: monad_simp change_def lookup_chain)
+ by (auto simp add: change_def lookup_chain)
subsection {* Code generator setup *}
--- a/src/HOL/Imperative_HOL/Relational.thy Mon Jul 05 10:42:27 2010 +0200
+++ b/src/HOL/Imperative_HOL/Relational.thy Mon Jul 05 14:34:28 2010 +0200
@@ -9,10 +9,10 @@
definition crel :: "'a Heap \<Rightarrow> heap \<Rightarrow> heap \<Rightarrow> 'a \<Rightarrow> bool"
where
- crel_def': "crel c h h' r \<longleftrightarrow> Heap_Monad.execute c h = (Inl r, h')"
+ crel_def': "crel c h h' r \<longleftrightarrow> Heap_Monad.execute c h = Some (r, h')"
lemma crel_def: -- FIXME
- "crel c h h' r \<longleftrightarrow> (Inl r, h') = Heap_Monad.execute c h"
+ "crel c h h' r \<longleftrightarrow> Some (r, h') = Heap_Monad.execute c h"
unfolding crel_def' by auto
lemma crel_deterministic: "\<lbrakk> crel f h h' a; crel f h h'' b \<rbrakk> \<Longrightarrow> (a = b) \<and> (h' = h'')"
@@ -28,8 +28,7 @@
lemma crelE[consumes 1]:
assumes "crel (f >>= g) h h'' r'"
obtains h' r where "crel f h h' r" "crel (g r) h' h'' r'"
- using assms
- by (auto simp add: crel_def bindM_def Let_def prod_case_beta split_def Pair_fst_snd_eq split add: sum.split_asm)
+ using assms by (auto simp add: crel_def bindM_def split: option.split_asm)
lemma crelE'[consumes 1]:
assumes "crel (f >> g) h h'' r'"
@@ -86,8 +85,8 @@
lemma crel_heap:
assumes "crel (Heap_Monad.heap f) h h' r"
obtains "h' = snd (f h)" "r = fst (f h)"
- using assms
- unfolding heap_def crel_def apfst_def split_def prod_fun_def by simp_all
+ using assms by (cases "f h") (simp add: crel_def)
+
subsection {* Elimination rules for array commands *}
@@ -369,11 +368,9 @@
apply (cases f)
apply simp
apply (simp add: expand_fun_eq split_def)
+apply (auto split: option.split)
+apply (erule_tac x="x" in meta_allE)
apply auto
-apply (case_tac "fst (fun x)")
-apply (simp_all add: Pair_fst_snd_eq)
-apply (erule_tac x="x" in meta_allE)
-apply fastsimp
done
section {* Introduction rules *}
@@ -502,10 +499,10 @@
shows "P x h h' r"
proof (rule MREC_pinduct[OF assms(1)[unfolded crel_def, symmetric]])
fix x h h1 h2 h' s z r
- assume "Heap_Monad.execute (f x) h = (Inl (Inr s), h1)"
- "Heap_Monad.execute (MREC f g s) h1 = (Inl z, h2)"
+ 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 = (Inl r, h')"
+ "Heap_Monad.execute (g x s z) h2 = Some (r, h')"
from assms(3)[unfolded crel_def, OF this(1)[symmetric] this(2)[symmetric] this(3) this(4)[symmetric]]
show "P x h h' r" .
next
@@ -519,15 +516,15 @@
definition noError :: "'a Heap \<Rightarrow> heap \<Rightarrow> bool"
where
- "noError c h \<longleftrightarrow> (\<exists>r h'. (Inl r, h') = Heap_Monad.execute c h)"
+ "noError c h \<longleftrightarrow> (\<exists>r h'. Some (r, h') = Heap_Monad.execute c h)"
lemma noError_def': -- FIXME
- "noError c h \<longleftrightarrow> (\<exists>r h'. Heap_Monad.execute c h = (Inl r, h'))"
+ "noError c h \<longleftrightarrow> (\<exists>r h'. Heap_Monad.execute c h = Some (r, h'))"
unfolding noError_def apply auto proof -
fix r h'
- assume "(Inl r, h') = Heap_Monad.execute c h"
- then have "Heap_Monad.execute c h = (Inl r, h')" ..
- then show "\<exists>r h'. Heap_Monad.execute c h = (Inl r, h')" by blast
+ assume "Some (r, h') = Heap_Monad.execute c h"
+ then have "Heap_Monad.execute c h = Some (r, h')" ..
+ then show "\<exists>r h'. Heap_Monad.execute c h = Some (r, h')" by blast
qed
subsection {* Introduction rules for basic monadic commands *}
@@ -640,7 +637,7 @@
(*TODO: move to HeapMonad *)
lemma mapM_append:
"mapM f (xs @ ys) = mapM f xs \<guillemotright>= (\<lambda>xs. mapM f ys \<guillemotright>= (\<lambda>ys. return (xs @ ys)))"
- by (induct xs) (simp_all add: monad_simp)
+ by (induct xs) simp_all
lemma noError_freeze:
shows "noError (freeze a) h"
--- a/src/HOL/Imperative_HOL/ex/Linked_Lists.thy Mon Jul 05 10:42:27 2010 +0200
+++ b/src/HOL/Imperative_HOL/ex/Linked_Lists.thy Mon Jul 05 14:34:28 2010 +0200
@@ -56,7 +56,7 @@
return (x#xs)
done"
unfolding traverse_def
-by (auto simp: traverse_def monad_simp MREC_rule)
+by (auto simp: traverse_def MREC_rule)
section {* Proving correctness with relational abstraction *}
@@ -531,7 +531,7 @@
done"
unfolding rev'_def MREC_rule[of _ _ "(q, p)"] unfolding rev'_def[symmetric]
thm arg_cong2
- by (auto simp add: monad_simp expand_fun_eq intro: arg_cong2[where f = "op \<guillemotright>="] split: node.split)
+ by (auto simp add: expand_fun_eq intro: arg_cong2[where f = "op \<guillemotright>="] split: node.split)
fun rev :: "('a:: heap) node \<Rightarrow> 'a node Heap"
where
--- a/src/HOL/Imperative_HOL/ex/SatChecker.thy Mon Jul 05 10:42:27 2010 +0200
+++ b/src/HOL/Imperative_HOL/ex/SatChecker.thy Mon Jul 05 14:34:28 2010 +0200
@@ -402,6 +402,12 @@
res_thm' l cli clj
done))"
+primrec
+ foldM :: "('a \<Rightarrow> 'b \<Rightarrow> 'b Heap) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b Heap"
+where
+ "foldM f [] s = return s"
+ | "foldM f (x#xs) s = f x s \<guillemotright>= foldM f xs"
+
fun doProofStep2 :: "Clause option array \<Rightarrow> ProofStep \<Rightarrow> Clause list \<Rightarrow> Clause list Heap"
where
"doProofStep2 a (Conflict saveTo (i, rs)) rcs =