theory StateMonadimports "~~/src/HOL/Library/Monad_Syntax" begindatatype ('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") autolemma SM_execute [simp]: "SM (execute f) = f" by (cases f) simp_alllemma 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.setuplemma 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.setuplemma 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_allqedlemma 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 autoML {* 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 simplemma 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 simplemma 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_alllemma 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.bindlemma 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) donelemma 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_alllemma 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)qedsubsection {* 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_lubproof - 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)qeddeclaration {* 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 blastlemma 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 blastlemma 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))" .qedend