--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys/StateMonad.thy Fri Mar 21 15:07:59 2014 +0000
@@ -0,0 +1,481 @@
+theory StateMonad
+imports
+ "~~/src/HOL/Library/Monad_Syntax"
+begin
+
+datatype ('result, 'state) SM = SM "'state => ('result \<times> 'state) option"
+
+fun execute :: "('result, 'state) SM \<Rightarrow> 'state \<Rightarrow> ('result \<times> 'state) option" where
+ "execute (SM f) = f"
+
+lemma SM_cases [case_names succeed fail]:
+ fixes f and s
+ assumes succeed: "\<And>x s'. execute f h = Some (x, s') \<Longrightarrow> P"
+ assumes fail: "execute f h = None \<Longrightarrow> P"
+ shows P
+ using assms by (cases "execute f h") auto
+
+lemma SM_execute [simp]:
+ "SM (execute f) = f" by (cases f) simp_all
+
+lemma SM_eqI:
+ "(\<And>h. execute f h = execute g h) \<Longrightarrow> f = g"
+ by (cases f, cases g) (auto simp: fun_eq_iff)
+
+ML {* structure Execute_Simps = Named_Thms
+(
+ val name = @{binding execute_simps}
+ val description = "simplification rules for execute"
+) *}
+
+setup Execute_Simps.setup
+
+lemma execute_Let [execute_simps]:
+ "execute (let x = t in f x) = (let x = t in execute (f x))"
+ by (simp add: Let_def)
+
+
+subsubsection {* Specialised lifters *}
+
+definition sm :: "('state \<Rightarrow> 'a \<times> 'state) \<Rightarrow> ('a, 'state) SM" where
+ "sm f = SM (Some \<circ> f)"
+
+definition tap :: "('state \<Rightarrow> 'a) \<Rightarrow> ('a, 'state) SM" where
+ "tap f = SM (\<lambda>s. Some (f s, s))"
+
+definition "sm_get = tap id"
+
+definition "sm_map f = sm (\<lambda> s.((), f s))"
+
+definition "sm_set s' = sm_map (\<lambda> s. s)"
+
+lemma execute_tap [execute_simps]:
+ "execute (tap f) h = Some (f h, h)"
+ by (simp add: tap_def)
+
+
+lemma execute_heap [execute_simps]:
+ "execute (sm f) = Some \<circ> f"
+ by (simp add: sm_def)
+
+definition guard :: "('state \<Rightarrow> bool) \<Rightarrow> ('state \<Rightarrow> 'a \<times> 'state)
+ \<Rightarrow> ('a, 'state) SM" where
+ "guard P f = SM (\<lambda>h. if P h then Some (f h) else None)"
+
+lemma execute_guard [execute_simps]:
+ "\<not> P h \<Longrightarrow> execute (guard P f) h = None"
+ "P h \<Longrightarrow> execute (guard P f) h = Some (f h)"
+ by (simp_all add: guard_def)
+
+
+subsubsection {* Predicate classifying successful computations *}
+
+definition success :: "('a, 'state) SM \<Rightarrow> 'state \<Rightarrow> bool" where
+ "success f h = (execute f h \<noteq> None)"
+
+lemma successI:
+ "execute f h \<noteq> None \<Longrightarrow> success f h"
+ by (simp add: success_def)
+
+lemma successE:
+ assumes "success f h"
+ obtains r h' where "r = fst (the (execute c h))"
+ and "h' = snd (the (execute c h))"
+ and "execute f h \<noteq> None"
+ using assms by (simp add: success_def)
+
+ML {* structure Success_Intros = Named_Thms
+(
+ val name = @{binding success_intros}
+ val description = "introduction rules for success"
+) *}
+
+setup Success_Intros.setup
+
+lemma success_tapI [success_intros]:
+ "success (tap f) h"
+ by (rule successI) (simp add: execute_simps)
+
+lemma success_heapI [success_intros]:
+ "success (sm f) h"
+ by (rule successI) (simp add: execute_simps)
+
+lemma success_guardI [success_intros]:
+ "P h \<Longrightarrow> success (guard P f) h"
+ by (rule successI) (simp add: execute_guard)
+
+lemma success_LetI [success_intros]:
+ "x = t \<Longrightarrow> success (f x) h \<Longrightarrow> success (let x = t in f x) h"
+ by (simp add: Let_def)
+
+lemma success_ifI:
+ "(c \<Longrightarrow> success t h) \<Longrightarrow> (\<not> c \<Longrightarrow> success e h) \<Longrightarrow>
+ success (if c then t else e) h"
+ by (simp add: success_def)
+
+subsubsection {* Predicate for a simple relational calculus *}
+
+text {*
+ The @{text effect} predicate states that when a computation @{text c}
+ runs with the state @{text h} will result in return value @{text r}
+ and a state @{text "h'"}, i.e.~no exception occurs.
+*}
+
+definition effect :: "('a, 'state) SM \<Rightarrow> 'state \<Rightarrow> 'state \<Rightarrow> 'a \<Rightarrow> bool" where
+ effect_def: "effect c h h' r = (execute c h = Some (r, h'))"
+
+lemma effectI:
+ "execute c h = Some (r, h') \<Longrightarrow> effect c h h' r"
+ by (simp add: effect_def)
+
+lemma effectE:
+ assumes "effect c h h' r"
+ obtains "r = fst (the (execute c h))"
+ and "h' = snd (the (execute c h))"
+ and "success c h"
+proof (rule that)
+ from assms have *: "execute c h = Some (r, h')" by (simp add: effect_def)
+ then show "success c h" by (simp add: success_def)
+ from * have "fst (the (execute c h)) = r" and "snd (the (execute c h)) = h'"
+ by simp_all
+ then show "r = fst (the (execute c h))"
+ and "h' = snd (the (execute c h))" by simp_all
+qed
+
+lemma effect_success:
+ "effect c h h' r \<Longrightarrow> success c h"
+ by (simp add: effect_def success_def)
+
+lemma success_effectE:
+ assumes "success c h"
+ obtains r h' where "effect c h h' r"
+ using assms by (auto simp add: effect_def success_def)
+
+lemma effect_deterministic:
+ assumes "effect f h h' a"
+ and "effect f h h'' b"
+ shows "a = b" and "h' = h''"
+ using assms unfolding effect_def by auto
+
+ML {* structure Effect_Intros = Named_Thms
+(
+ val name = @{binding effect_intros}
+ val description = "introduction rules for effect"
+) *}
+
+ML {* structure Effect_Elims = Named_Thms
+(
+ val name = @{binding effect_elims}
+ val description = "elimination rules for effect"
+) *}
+
+setup "Effect_Intros.setup #> Effect_Elims.setup"
+
+lemma effect_LetI [effect_intros]:
+ assumes "x = t" "effect (f x) h h' r"
+ shows "effect (let x = t in f x) h h' r"
+ using assms by simp
+
+lemma effect_LetE [effect_elims]:
+ assumes "effect (let x = t in f x) h h' r"
+ obtains "effect (f t) h h' r"
+ using assms by simp
+
+lemma effect_ifI:
+ assumes "c \<Longrightarrow> effect t h h' r"
+ and "\<not> c \<Longrightarrow> effect e h h' r"
+ shows "effect (if c then t else e) h h' r"
+ by (cases c) (simp_all add: assms)
+
+lemma effect_ifE:
+ assumes "effect (if c then t else e) h h' r"
+ obtains "c" "effect t h h' r"
+ | "\<not> c" "effect e h h' r"
+ using assms by (cases c) simp_all
+
+lemma effect_tapI [effect_intros]:
+ assumes "h' = h" "r = f h"
+ shows "effect (tap f) h h' r"
+ by (rule effectI) (simp add: assms execute_simps)
+
+lemma effect_tapE [effect_elims]:
+ assumes "effect (tap f) h h' r"
+ obtains "h' = h" and "r = f h"
+ using assms by (rule effectE) (auto simp add: execute_simps)
+
+lemma effect_heapI [effect_intros]:
+ assumes "h' = snd (f h)" "r = fst (f h)"
+ shows "effect (sm f) h h' r"
+ by (rule effectI) (simp add: assms execute_simps)
+
+lemma effect_heapE [effect_elims]:
+ assumes "effect (sm f) h h' r"
+ obtains "h' = snd (f h)" and "r = fst (f h)"
+ using assms by (rule effectE) (simp add: execute_simps)
+
+lemma effect_guardI [effect_intros]:
+ assumes "P h" "h' = snd (f h)" "r = fst (f h)"
+ shows "effect (guard P f) h h' r"
+ by (rule effectI) (simp add: assms execute_simps)
+
+lemma effect_guardE [effect_elims]:
+ assumes "effect (guard P f) h h' r"
+ obtains "h' = snd (f h)" "r = fst (f h)" "P h"
+ using assms by (rule effectE)
+ (auto simp add: execute_simps elim!: successE,
+ cases "P h", auto simp add: execute_simps)
+
+
+subsubsection {* Monad combinators *}
+
+definition return :: "'a \<Rightarrow> ('a, 'state) SM" where
+ "return x = sm (Pair x)"
+
+lemma execute_return [execute_simps]:
+ "execute (return x) = Some \<circ> Pair x"
+ by (simp add: return_def execute_simps)
+
+lemma success_returnI [success_intros]:
+ "success (return x) h"
+ by (rule successI) (simp add: execute_simps)
+
+lemma effect_returnI [effect_intros]:
+ "h = h' \<Longrightarrow> effect (return x) h h' x"
+ by (rule effectI) (simp add: execute_simps)
+
+lemma effect_returnE [effect_elims]:
+ assumes "effect (return x) h h' r"
+ obtains "r = x" "h' = h"
+ using assms by (rule effectE) (simp add: execute_simps)
+
+definition raise :: "string \<Rightarrow> ('a, 'state) SM" where
+ -- {* the string is just decoration *}
+ "raise s = SM (\<lambda>_. None)"
+
+lemma execute_raise [execute_simps]:
+ "execute (raise s) = (\<lambda>_. None)"
+ by (simp add: raise_def)
+
+lemma effect_raiseE [effect_elims]:
+ assumes "effect (raise x) h h' r"
+ obtains "False"
+ using assms by (rule effectE) (simp add: success_def execute_simps)
+
+definition bind :: "('a, 'state) SM \<Rightarrow> ('a \<Rightarrow> ('b, 'state) SM) \<Rightarrow> ('b, 'state) SM" where
+ "bind f g = SM (\<lambda>h. case execute f h of
+ Some (x, h') \<Rightarrow> execute (g x) h'
+ | None \<Rightarrow> None)"
+
+adhoc_overloading
+ Monad_Syntax.bind StateMonad.bind
+
+
+lemma execute_bind [execute_simps]:
+ "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: bind_def)
+
+lemma execute_bind_case:
+ "execute (f \<guillemotright>= g) h = (case (execute f h) of
+ Some (x, h') \<Rightarrow> execute (g x) h' | None \<Rightarrow> None)"
+ by (simp add: bind_def)
+
+lemma execute_bind_success:
+ "success f h \<Longrightarrow> execute (f \<guillemotright>= g) h = execute (g (fst (the (execute f h)))) (snd (the (execute f h)))"
+ by (cases f h rule: SM_cases) (auto elim!: successE simp add: bind_def)
+
+lemma success_bind_executeI:
+ "execute f h = Some (x, h') \<Longrightarrow> success (g x) h' \<Longrightarrow> success (f \<guillemotright>= g) h"
+ by (auto intro!: successI elim!: successE simp add: bind_def)
+
+lemma success_bind_effectI [success_intros]:
+ "effect f h h' x \<Longrightarrow> success (g x) h' \<Longrightarrow> success (f \<guillemotright>= g) h"
+ by (auto simp add: effect_def success_def bind_def)
+
+lemma effect_bindI [effect_intros]:
+ assumes "effect f h h' r" "effect (g r) h' h'' r'"
+ shows "effect (f \<guillemotright>= g) h h'' r'"
+ using assms
+ apply (auto intro!: effectI elim!: effectE successE)
+ apply (subst execute_bind, simp_all)
+ done
+
+lemma effect_bindE [effect_elims]:
+ assumes "effect (f \<guillemotright>= g) h h'' r'"
+ obtains h' r where "effect f h h' r" "effect (g r) h' h'' r'"
+ using assms by (auto simp add: effect_def bind_def split: option.split_asm)
+
+lemma execute_bind_eq_SomeI:
+ assumes "execute f h = Some (x, h')"
+ and "execute (g x) h' = Some (y, h'')"
+ shows "execute (f \<guillemotright>= g) h = Some (y, h'')"
+ using assms by (simp add: bind_def)
+
+lemma return_bind [simp]: "return x \<guillemotright>= f = f x"
+ by (rule SM_eqI) (simp add: execute_bind execute_simps)
+
+lemma bind_return [simp]: "f \<guillemotright>= return = f"
+ by (rule SM_eqI) (simp add: bind_def execute_simps split: option.splits)
+
+lemma bind_bind [simp]: "(f \<guillemotright>= g) \<guillemotright>= k = (f :: ('a, 'state) SM) \<guillemotright>= (\<lambda>x. g x \<guillemotright>= k)"
+ by (rule SM_eqI) (simp add: bind_def execute_simps split: option.splits)
+
+lemma raise_bind [simp]: "raise e \<guillemotright>= f = raise e"
+ by (rule SM_eqI) (simp add: execute_simps)
+
+
+subsection {* Generic combinators *}
+
+subsubsection {* Assertions *}
+
+definition assert :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> ('a, 'state) SM" where
+ "assert P x = (if P x then return x else raise ''assert'')"
+
+lemma execute_assert [execute_simps]:
+ "P x \<Longrightarrow> execute (assert P x) h = Some (x, h)"
+ "\<not> P x \<Longrightarrow> execute (assert P x) h = None"
+ by (simp_all add: assert_def execute_simps)
+
+lemma success_assertI [success_intros]:
+ "P x \<Longrightarrow> success (assert P x) h"
+ by (rule successI) (simp add: execute_assert)
+
+lemma effect_assertI [effect_intros]:
+ "P x \<Longrightarrow> h' = h \<Longrightarrow> r = x \<Longrightarrow> effect (assert P x) h h' r"
+ by (rule effectI) (simp add: execute_assert)
+
+lemma effect_assertE [effect_elims]:
+ assumes "effect (assert P x) h h' r"
+ obtains "P x" "r = x" "h' = h"
+ using assms by (rule effectE) (cases "P x", simp_all add: execute_assert success_def)
+
+lemma assert_cong [fundef_cong]:
+ assumes "P = P'"
+ assumes "\<And>x. P' x \<Longrightarrow> f x = f' x"
+ shows "(assert P x >>= f) = (assert P' x >>= f')"
+ by (rule SM_eqI) (insert assms, simp add: assert_def)
+
+
+subsubsection {* Plain lifting *}
+
+definition lift :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> ('b, 'state) SM" where
+ "lift f = return o f"
+
+lemma lift_collapse [simp]:
+ "lift f x = return (f x)"
+ by (simp add: lift_def)
+
+lemma bind_lift:
+ "(f \<guillemotright>= lift g) = (f \<guillemotright>= (\<lambda>x. return (g x)))"
+ by (simp add: lift_def comp_def)
+
+
+subsubsection {* Iteration -- warning: this is rarely useful! *}
+
+primrec fold_map :: "('a \<Rightarrow> ('b, 'state) SM) \<Rightarrow> 'a list \<Rightarrow> ('b list, 'state) SM" where
+ "fold_map f [] = return []"
+| "fold_map f (x # xs) = do {
+ y \<leftarrow> f x;
+ ys \<leftarrow> fold_map f xs;
+ return (y # ys)
+ }"
+
+lemma fold_map_append:
+ "fold_map f (xs @ ys) = fold_map f xs \<guillemotright>= (\<lambda>xs. fold_map f ys \<guillemotright>= (\<lambda>ys. return (xs @ ys)))"
+ by (induct xs) simp_all
+
+lemma execute_fold_map_unchanged_heap [execute_simps]:
+ assumes "\<And>x. x \<in> set xs \<Longrightarrow> \<exists>y. execute (f x) h = Some (y, h)"
+ shows "execute (fold_map f xs) h =
+ Some (List.map (\<lambda>x. fst (the (execute (f x) h))) xs, h)"
+using assms proof (induct xs)
+ case Nil show ?case by (simp add: execute_simps)
+next
+ case (Cons x xs)
+ from Cons.prems obtain y
+ where y: "execute (f x) h = Some (y, h)" by auto
+ moreover from Cons.prems Cons.hyps have "execute (fold_map f xs) h =
+ Some (map (\<lambda>x. fst (the (execute (f x) h))) xs, h)" by auto
+ ultimately show ?case by (simp, simp only: execute_bind(1), simp add: execute_simps)
+qed
+
+
+subsection {* Partial function definition setup *}
+
+definition SM_ord :: "('a, 'state) SM \<Rightarrow> ('a, 'state) SM \<Rightarrow> bool" where
+ "SM_ord = img_ord execute (fun_ord option_ord)"
+
+definition SM_lub :: "('a , 'state) SM set \<Rightarrow> ('a, 'state) SM" where
+ "SM_lub = img_lub execute SM (fun_lub (flat_lub None))"
+
+interpretation sm!: partial_function_definitions SM_ord SM_lub
+proof -
+ have "partial_function_definitions (fun_ord option_ord) (fun_lub (flat_lub None))"
+ by (rule partial_function_lift) (rule flat_interpretation)
+ then have "partial_function_definitions (img_ord execute (fun_ord option_ord))
+ (img_lub execute SM (fun_lub (flat_lub None)))"
+ by (rule partial_function_image) (auto intro: SM_eqI)
+ then show "partial_function_definitions SM_ord SM_lub"
+ by (simp only: SM_ord_def SM_lub_def)
+qed
+
+declaration {* Partial_Function.init "sm" @{term sm.fixp_fun}
+ @{term sm.mono_body} @{thm sm.fixp_rule_uc} @{thm sm.fixp_induct_uc} NONE *}
+
+
+abbreviation "mono_SM \<equiv> monotone (fun_ord SM_ord) SM_ord"
+
+lemma SM_ordI:
+ assumes "\<And>h. execute x h = None \<or> execute x h = execute y h"
+ shows "SM_ord x y"
+ using assms unfolding SM_ord_def img_ord_def fun_ord_def flat_ord_def
+ by blast
+
+lemma SM_ordE:
+ assumes "SM_ord x y"
+ obtains "execute x h = None" | "execute x h = execute y h"
+ using assms unfolding SM_ord_def img_ord_def fun_ord_def flat_ord_def
+ by atomize_elim blast
+
+lemma bind_mono [partial_function_mono]:
+ assumes mf: "mono_SM B" and mg: "\<And>y. mono_SM (\<lambda>f. C y f)"
+ shows "mono_SM (\<lambda>f. B f \<guillemotright>= (\<lambda>y. C y f))"
+proof (rule monotoneI)
+ fix f g :: "'a \<Rightarrow> ('b, 'c) SM" assume fg: "fun_ord SM_ord f g"
+ from mf
+ have 1: "SM_ord (B f) (B g)" by (rule monotoneD) (rule fg)
+ from mg
+ have 2: "\<And>y'. SM_ord (C y' f) (C y' g)" by (rule monotoneD) (rule fg)
+
+ have "SM_ord (B f \<guillemotright>= (\<lambda>y. C y f)) (B g \<guillemotright>= (\<lambda>y. C y f))"
+ (is "SM_ord ?L ?R")
+ proof (rule SM_ordI)
+ fix h
+ from 1 show "execute ?L h = None \<or> execute ?L h = execute ?R h"
+ by (rule SM_ordE[where h = h]) (auto simp: execute_bind_case)
+ qed
+ also
+ have "SM_ord (B g \<guillemotright>= (\<lambda>y'. C y' f)) (B g \<guillemotright>= (\<lambda>y'. C y' g))"
+ (is "SM_ord ?L ?R")
+ proof (rule SM_ordI)
+ fix h
+ show "execute ?L h = None \<or> execute ?L h = execute ?R h"
+ proof (cases "execute (B g) h")
+ case None
+ then have "execute ?L h = None" by (auto simp: execute_bind_case)
+ thus ?thesis ..
+ next
+ case Some
+ then obtain r h' where "execute (B g) h = Some (r, h')"
+ by (metis surjective_pairing)
+ then have "execute ?L h = execute (C r f) h'"
+ "execute ?R h = execute (C r g) h'"
+ by (auto simp: execute_bind_case)
+ with 2[of r] show ?thesis by (auto elim: SM_ordE)
+ qed
+ qed
+ finally (sm.leq_trans)
+ show "SM_ord (B f \<guillemotright>= (\<lambda>y. C y f)) (B g \<guillemotright>= (\<lambda>y'. C y' g))" .
+qed
+
+end
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