header {* {\em abacus} a kind of register machine*}theory abacusimports Main (*StateMonad*)begintext {* {\em Abacus} instructions:*}datatype abc_inst = -- {* @{text "Inc n"} increments the memory cell (or register) with address @{text "n"} by one. *} Inc nat -- {* @{text "Dec n label"} decrements the memory cell with address @{text "n"} by one. If cell @{text "n"} is already zero, no decrements happens and the executio jumps to the instruction labeled by @{text "label"}. *} | Dec nat nat -- {* @{text "Goto label"} unconditionally jumps to the instruction labeled by @{text "label"}. *} | Goto natfun splits :: "'a set \<Rightarrow> ('a set \<times> 'a set) \<Rightarrow> bool"where "splits s (u, v) = (u \<union> v = s \<and> u \<inter> v = {})"declare splits.simps [simp del]definition "stimes p q = {s . \<exists> u v. u \<in> p \<and> v \<in> q \<and> splits s (u, v)}"lemmas st_def = stimes_def[unfolded splits.simps]notation stimes (infixr "*" 70)lemma stimes_comm: "(p::('a set set)) * q = q * p" by (unfold st_def, auto)lemma splits_simp: "splits s (u, v) = (v = (s - u) \<and> v \<subseteq> s \<and> u \<subseteq> s)" by (unfold splits.simps, auto)lemma stimes_assoc: "p * q * r = (p * q) * (r::'a set set)" by (unfold st_def, blast)definition "emp = {{}}"lemma emp_unit_r [simp]: "p * emp = p" by (unfold st_def emp_def, auto)lemma emp_unit_l [simp]: "emp * p = p" by (metis emp_unit_r stimes_comm)lemma stimes_mono: "p \<subseteq> q \<Longrightarrow> p * r \<subseteq> q * r" by (unfold st_def, auto)lemma stimes_left_commute: "(q * (p * r)) = ((p::'a set set) * (q * r))"by (metis stimes_assoc stimes_comm)lemmas stimes_ac = stimes_comm stimes_assoc stimes_left_commutelemma "x * y * z = z * y * (x::'a set set)"by (metis stimes_ac)definition pasrt :: "bool \<Rightarrow> ('a set set)" ("<_>" [71] 71)where "pasrt b = {s . s = {} \<and> b}"datatype apg = Instr abc_inst | Label nat | Seq apg apg | Local "(nat \<Rightarrow> apg)"notation Local (binder "LOCAL " 10)term "LOCAL a b. Seq (Label a) (Label b)"abbreviation prog_instr :: "abc_inst \<Rightarrow> apg" ("\<guillemotright>_" [61] 61)where "\<guillemotright>i \<equiv> Instr i"abbreviation prog_seq :: "apg \<Rightarrow> apg \<Rightarrow> apg" (infixl ";" 52)where "c1 ; c2 \<equiv> Seq c1 c2"type_synonym aconf = "((nat \<rightharpoonup> abc_inst) \<times> nat \<times> (nat \<rightharpoonup> nat) \<times> nat)"fun astep :: "aconf \<Rightarrow> aconf" where "astep (prog, pc, m, faults) = (case (prog pc) of Some (Inc i) \<Rightarrow> case m(i) of Some n \<Rightarrow> (prog, pc + 1, m(i:= Some (n + 1)), faults) | None \<Rightarrow> (prog, pc, m, faults + 1) | Some (Dec i e) \<Rightarrow> case m(i) of Some n \<Rightarrow> if (n = 0) then (prog, e, m, faults) else (prog, pc + 1, m(i:= Some (n - 1)), faults) | None \<Rightarrow> (prog, pc, m, faults + 1) | Some (Goto pc') \<Rightarrow> (prog, pc', m, faults) | None \<Rightarrow> (prog, pc, m, faults + 1))"definition "run n = astep ^^ n"datatype aresource = M nat nat | C nat abc_inst | At nat | Faults natdefinition "prog_set prog = {C i inst | i inst. prog i = Some inst}"definition "pc_set pc = {At pc}"definition "mem_set m = {M i n | i n. m (i) = Some n} "definition "faults_set faults = {Faults faults}"lemmas cpn_set_def = prog_set_def pc_set_def mem_set_def faults_set_deffun rset_of :: "aconf \<Rightarrow> aresource set" where "rset_of (prog, pc, m, faults) = prog_set prog \<union> pc_set pc \<union> mem_set m \<union> faults_set faults"definition "pc l = {pc_set l}"definition "m a v = {{M a v}}"declare rset_of.simps[simp del]type_synonym assert = "aresource set set"primrec assemble_to :: "apg \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> assert" where "assemble_to (Instr ai) i j = ({{C i ai}} * <(j = i + 1)>)" | "assemble_to (Seq p1 p2) i j = (\<Union> j'. (assemble_to p1 i j') * (assemble_to p2 j' j))" | "assemble_to (Local fp) i j = (\<Union> l. (assemble_to (fp l) i j))" | "assemble_to (Label l) i j = <(i = j \<and> j = l)>"abbreviation asmb_to :: "nat \<Rightarrow> apg \<Rightarrow> nat \<Rightarrow> assert" ("_ :[ _ ]: _" [60, 60, 60] 60) where "i :[ apg ]: j \<equiv> assemble_to apg i j"lemma stimes_sgD: "s \<in> {x} * q \<Longrightarrow> (s - x) \<in> q \<and> x \<subseteq> s" apply (unfold st_def, auto)by (smt Diff_disjoint Un_Diff_cancel2 Un_Int_distrib Un_commute Un_empty_right Un_left_absorb)lemma pcD: "rset_of (prog, i', mem, fault) \<in> pc i * r \<Longrightarrow> i' = i"proof - assume "rset_of (prog, i', mem, fault) \<in> pc i * r" from stimes_sgD [OF this[unfolded pc_def], unfolded rset_of.simps] have "pc_set i \<subseteq> prog_set prog \<union> pc_set i' \<union> mem_set mem \<union> faults_set fault" by auto thus ?thesis by (unfold cpn_set_def, auto)qedlemma codeD: "rset_of (prog, pos, mem, fault) \<in> pc i * {{C i inst}} * r \<Longrightarrow> prog pos = Some inst"proof - assume h: "rset_of (prog, pos, mem, fault) \<in> pc i * {{C i inst}} * r" (is "?c \<in> ?X") from pcD[OF this] have "i = pos" by simp with h show ?thesis by (unfold rset_of.simps st_def pc_def prog_set_def pc_set_def mem_set_def faults_set_def, auto)qedlemma memD: "rset_of (prog, pos, mem, fault) \<in> (m a v) * r \<Longrightarrow> mem a = Some v"proof - assume "rset_of (prog, pos, mem, fault) \<in> (m a v) * r" from stimes_sgD[OF this[unfolded rset_of.simps cpn_set_def m_def]] have "{M a v} \<subseteq> {C i inst |i inst. prog i = Some inst} \<union> {At pos} \<union> {M i n |i n. mem i = Some n} \<union> {Faults fault}" by auto thus ?thesis by autoqeddefinition Hoare_abc :: "assert \<Rightarrow> assert \<Rightarrow> assert \<Rightarrow> bool" ("({(1_)}/ (_)/ {(1_)})" 50)where "{p} c {q} \<equiv> (\<forall> s r. (rset_of s) \<in> (p*c*r) \<longrightarrow> (\<exists> k. ((rset_of (run k s)) \<in> (q*c*r))))" definition "dec_fun v j e = (if (v = 0) then (e, v) else (j, v - 1))"lemma disj_Diff: "a \<inter> b = {} \<Longrightarrow> a \<union> b - b = a"by (metis (lifting) Diff_cancel Un_Diff Un_Diff_Int)lemma diff_pc_set: "prog_set aa \<union> pc_set i \<union> mem_set ab \<union> faults_set b - pc_set i = prog_set aa \<union> mem_set ab \<union> faults_set b" (is "?L = ?R")proof - have "?L = (prog_set aa \<union> mem_set ab \<union> faults_set b \<union> pc_set i) - pc_set i" by auto also have "\<dots> = ?R" proof(rule disj_Diff) show " (prog_set aa \<union> mem_set ab \<union> faults_set b) \<inter> pc_set i = {}" by (unfold cpn_set_def, auto) qed finally show ?thesis .qedlemma M_in_simp: "({M a v} \<subseteq> prog_set x \<union> pc_set y \<union> mem_set mem \<union> faults_set f) = ({M a v} \<subseteq> mem_set mem)" by (unfold cpn_set_def, auto)lemma mem_set_upd: "{M a v} \<subseteq> mem_set mem \<Longrightarrow> mem_set (mem(a:=Some v')) = ((mem_set mem) - {M a v}) \<union> {M a v'}" by (unfold cpn_set_def, auto)lemma mem_set_disj: "{M a v} \<subseteq> mem_set mem \<Longrightarrow> {M a v'} \<inter> (mem_set mem - {M a v}) = {}" by (unfold cpn_set_def, auto)lemma stimesE: assumes h: "s \<in> x * y" obtains s1 s2 where "s = s1 \<union> s2" and "s1 \<inter> s2 = {}" and "s1 \<in> x" and "s2 \<in> y" by (insert h, auto simp:st_def)lemma stimesI: "\<lbrakk>s = s1 \<union> s2; s1 \<inter> s2 = {}; s1 \<in> x; s2 \<in> y\<rbrakk> \<Longrightarrow> s \<in> x * y" by (auto simp:st_def)lemma smem_upd: "(rset_of (x, y, mem, f)) \<in> (m a v)*r \<Longrightarrow> (rset_of (x, y, mem(a := Some v'), f) \<in> (m a v')*r)"proof - assume h: " rset_of (x, y, mem, f) \<in> m a v * r" from h[unfolded rset_of.simps m_def] have "prog_set x \<union> pc_set y \<union> mem_set mem \<union> faults_set f \<in> {{M a v}} * r" . from stimes_sgD [OF this] have h1: "prog_set x \<union> pc_set y \<union> mem_set mem \<union> faults_set f - {M a v} \<in> r" "{M a v} \<subseteq> prog_set x \<union> pc_set y \<union> mem_set mem \<union> faults_set f" by auto moreover have "prog_set x \<union> pc_set y \<union> mem_set mem \<union> faults_set f - {M a v} = prog_set x \<union> pc_set y \<union> (mem_set mem - {M a v}) \<union> faults_set f" by (unfold cpn_set_def, auto) ultimately have h0: "prog_set x \<union> pc_set y \<union> (mem_set mem - {M a v}) \<union> faults_set f \<in> r" by simp from h1(2) and M_in_simp have "{M a v} \<subseteq> mem_set mem" by simp from mem_set_upd [OF this] mem_set_disj[OF this] have h2: "mem_set (mem(a \<mapsto> v')) = {M a v'} \<union> (mem_set mem - {M a v})" "{M a v'} \<inter> (mem_set mem - {M a v}) = {}" by auto show ?thesis proof - have "mem_set (mem(a \<mapsto> v')) \<union> prog_set x \<union> pc_set y \<union> faults_set f \<in> m a v' * r" proof(rule stimesI[OF _ _ _ h0]) show "{M a v'} \<in> m a v'" by (unfold m_def, auto) next show "mem_set (mem(a \<mapsto> v')) \<union> prog_set x \<union> pc_set y \<union> faults_set f = {M a v'} \<union> (prog_set x \<union> pc_set y \<union> (mem_set mem - {M a v}) \<union> faults_set f)" apply (unfold h2(1)) by (smt Un_commute Un_insert_left Un_insert_right Un_left_commute `prog_set x \<union> pc_set y \<union> mem_set mem \<union> faults_set f - {M a v} =prog_set x \<union> pc_set y \<union> (mem_set mem - {M a v}) \<union> faults_set f`) next from h2(2) show "{M a v'} \<inter> (prog_set x \<union> pc_set y \<union> (mem_set mem - {M a v}) \<union> faults_set f) = {}" by (unfold cpn_set_def, auto) qed thus ?thesis apply (unfold rset_of.simps) by (metis `mem_set (mem(a \<mapsto> v')) \<union> prog_set x \<union> pc_set y \<union> faults_set f \<in> m a v' * r` stimes_comm sup_commute sup_left_commute) qedqedlemma spc_upd: "rset_of (x, i, y, z) \<in> pc i' * r \<Longrightarrow> rset_of (x, i'', y, z) \<in> pc i'' * r"proof - assume h: "rset_of (x, i, y, z) \<in> pc i' * r" from stimes_sgD [OF h[unfolded rset_of.simps pc_set_def pc_def]] have h1: "prog_set x \<union> {At i} \<union> mem_set y \<union> faults_set z - {At i'} \<in> r" "{At i'} \<subseteq> prog_set x \<union> {At i} \<union> mem_set y \<union> faults_set z" by auto from h1(2) have eq_i: "i' = i" by (unfold cpn_set_def, auto) have "prog_set x \<union> {At i} \<union> mem_set y \<union> faults_set z - {At i'} = prog_set x \<union> mem_set y \<union> faults_set z " apply (unfold eq_i) by (metis (full_types) Un_insert_left Un_insert_right diff_pc_set faults_set_def insert_commute insert_is_Un pc_set_def sup_assoc sup_bot_left sup_commute) with h1(1) have in_r: "prog_set x \<union> mem_set y \<union> faults_set z \<in> r" by auto show ?thesis proof(unfold rset_of.simps, rule stimesI[OF _ _ _ in_r]) show "{At i''} \<in> pc i''" by (unfold pc_def pc_set_def, simp) next show "prog_set x \<union> pc_set i'' \<union> mem_set y \<union> faults_set z = {At i''} \<union> (prog_set x \<union> mem_set y \<union> faults_set z)" by (unfold pc_set_def, auto) next show "{At i''} \<inter> (prog_set x \<union> mem_set y \<union> faults_set z) = {}" by (auto simp:cpn_set_def) qedqedlemma condD: "s \<in> <b>*r \<Longrightarrow> b" by (unfold st_def pasrt_def, auto)lemma condD1: "s \<in> <b>*r \<Longrightarrow> s \<in> r" by (unfold st_def pasrt_def, auto)lemma hoare_dec_suc: "{(pc i * m a v) * <(v > 0)>} i:[\<guillemotright>(Dec a e) ]:j {pc j * m a (v - 1)}"proof(unfold Hoare_abc_def, clarify) fix prog i' ab b r assume h: "rset_of (prog, i', ab, b) \<in> ((pc i * m a v) * <(0 < v)>) * (i :[ \<guillemotright>Dec a e ]: j) * r" (is "?r \<in> ?S") show "\<exists>k. rset_of (run k (prog, i', ab, b)) \<in> (pc j * m a (v - 1)) * (i :[ \<guillemotright>Dec a e ]: j) * r" proof - from h [unfolded assemble_to.simps] have h1: "?r \<in> pc i * {{C i (Dec a e)}} * m a v * <(0 < v)> * <(j = i + 1)> * r" "?r \<in> m a v * pc i * {{C i (Dec a e)}} * <(0 < v)> * <(j = i + 1)> * r" "?r \<in> <(0 < v)> * <(j = i + 1)> * m a v * pc i * {{C i (Dec a e)}} * r" "?r \<in> <(j = i + 1)> * <(0 < v)> * m a v * pc i * {{C i (Dec a e)}} * r" by ((metis stimes_ac)+) note h2 = condD [OF h1(3)] condD[OF h1(4)] pcD[OF h1(1)] codeD[OF h1(1)] memD[OF h1(2)] hence stp: "run 1 (prog, i', ab, b) = (prog, Suc i, ab(a \<mapsto> v - Suc 0), b)" (is "?x = ?y") by (unfold run_def, auto) have "rset_of ?x \<in> (pc j * m a (v - 1)) * (i :[ \<guillemotright>Dec a e ]: j) * r" proof - have "rset_of ?y \<in> (pc j * m a (v - 1)) * (i :[ \<guillemotright>Dec a e ]: j) * r" proof - from spc_upd[OF h1(1), of "Suc i"] have "rset_of (prog, (Suc i), ab, b) \<in> m a v * pc (Suc i) * {{C i (Dec a e)}} * <(0 < v)> * <(j = i + 1)> * r" by (metis stimes_ac) from smem_upd[OF this, of "v - (Suc 0)"] have "rset_of ?y \<in> m a (v - Suc 0) * pc (Suc i) * {{C i (Dec a e)}} * <(0 < v)> * <(j = i + 1)> * r" . hence "rset_of ?y \<in> <(0 < v)> * (pc (Suc i) * m a (v - Suc 0)) * ({{C i (Dec a e)}} * <(j = i + 1)>) * r" by (metis stimes_ac) from condD1[OF this] have "rset_of ?y \<in> (pc (Suc i) * m a (v - Suc 0)) * ({{C i (Dec a e)}} * <(j = i + 1)>) * r" . thus ?thesis by (unfold h2(2) assemble_to.simps, simp) qed with stp show ?thesis by simp qed thus ?thesis by blast qedqedlemma hoare_dec_fail: "{pc i * m a 0} i:[ \<guillemotright>(Dec a e) ]:j {pc e * m a 0}"proof(unfold Hoare_abc_def, clarify) fix prog i' ab b r assume h: "rset_of (prog, i', ab, b) \<in> (pc i * m a 0) * (i :[ \<guillemotright>Dec a e ]: j) * r" (is "?r \<in> ?S") show "\<exists>k. rset_of (run k (prog, i', ab, b)) \<in> (pc e * m a 0) * (i :[ \<guillemotright>Dec a e ]: j) * r" proof - from h [unfolded assemble_to.simps] have h1: "?r \<in> pc i * {{C i (Dec a e)}} * m a 0 * <(j = i + 1)> * r" "?r \<in> m a 0 * pc i * {{C i (Dec a e)}} * <(j = i + 1)> * r" "?r \<in> <(j = i + 1)> * m a 0 * pc i * {{C i (Dec a e)}} * r" by ((metis stimes_ac)+) note h2 = condD [OF h1(3)] pcD[OF h1(1)] codeD[OF h1(1)] memD[OF h1(2)] hence stp: "run 1 (prog, i', ab, b) = (prog, e, ab, b)" (is "?x = ?y") by (unfold run_def, auto) have "rset_of ?x \<in> (pc e * m a 0) * (i :[ \<guillemotright>Dec a e ]: j) * r" proof - have "rset_of ?y \<in> (pc e * m a 0) * (i :[ \<guillemotright>Dec a e ]: j) * r" proof - from spc_upd[OF h1(1), of "e"] have "rset_of ?y \<in> pc e * {{C i (Dec a e)}} * m a 0 * <(j = i + 1)> * r" . thus ?thesis by (unfold assemble_to.simps, metis stimes_ac) qed with stp show ?thesis by simp qed thus ?thesis by blast qedqedlemma pasrtD_p: "\<lbrakk>{p*<b>} c {q}\<rbrakk> \<Longrightarrow> (b \<longrightarrow> {p} c {q})" apply (unfold Hoare_abc_def pasrt_def, auto) by (fold emp_def, simp add:emp_unit_r)lemma hoare_dec: "dec_fun v j e = (pc', v') \<Longrightarrow> {pc i * m a v} i:[ \<guillemotright>(Dec a e) ]:j {pc pc' * m a v'}"proof - assume "dec_fun v j e = (pc', v')" thus "{pc i * m a v} i:[ \<guillemotright>(Dec a e) ]:j {pc pc' * m a v'}" apply (auto split:if_splits simp:dec_fun_def) apply (insert hoare_dec_fail, auto)[1] apply (insert hoare_dec_suc, auto) apply (atomize) apply (erule_tac x = i in allE, erule_tac x = a in allE, erule_tac x = v in allE, erule_tac x = e in allE, erule_tac x = pc' in allE) by (drule_tac pasrtD_p, clarify)qedlemma hoare_inc: "{pc i * m a v} i:[ \<guillemotright>(Inc a) ]:j {pc j * m a (v + 1)}"proof(unfold Hoare_abc_def, clarify) fix prog i' ab b r assume h: "rset_of (prog, i', ab, b) \<in> (pc i * m a v) * (i :[ \<guillemotright>Inc a ]: j) * r" (is "?r \<in> ?S") show "\<exists>k. rset_of (run k (prog, i', ab, b)) \<in> (pc j * m a (v + 1)) * (i :[ \<guillemotright>Inc a ]: j) * r" proof - from h [unfolded assemble_to.simps] have h1: "?r \<in> pc i * {{C i (Inc a)}} * m a v * <(j = i + 1)> * r" "?r \<in> m a v * pc i * {{C i (Inc a)}} * <(j = i + 1)> * r" "?r \<in> <(j = i + 1)> * m a v * pc i * {{C i (Inc a)}} * r" by ((metis stimes_ac)+) note h2 = condD [OF h1(3)] pcD[OF h1(1)] codeD[OF h1(1)] memD[OF h1(2)] hence stp: "run 1 (prog, i', ab, b) = (prog, Suc i, ab(a \<mapsto> Suc v), b)" (is "?x = ?y") by (unfold run_def, auto) have "rset_of ?x \<in> (pc j * m a (v + 1)) * (i :[ \<guillemotright>Inc a]: j) * r" proof - have "rset_of ?y \<in> (pc j * m a (v + 1)) * (i :[ \<guillemotright>Inc a ]: j) * r" proof - from spc_upd[OF h1(1), of "Suc i"] have "rset_of (prog, (Suc i), ab, b) \<in> m a v * pc (Suc i) * {{C i (Inc a)}} * <(j = i + 1)> * r" by (metis stimes_ac) from smem_upd[OF this, of "Suc v"] have "rset_of ?y \<in> m a (v + 1) * pc (i + 1) * {{C i (Inc a)}} * <(j = i + 1)> * r" by simp thus ?thesis by (unfold h2(1) assemble_to.simps, metis stimes_ac) qed with stp show ?thesis by simp qed thus ?thesis by blast qedqedlemma hoare_goto: "{pc i} i:[ \<guillemotright>(Goto e) ]:j {pc e}"proof(unfold Hoare_abc_def, clarify) fix prog i' ab b r assume h: "rset_of (prog, i', ab, b) \<in> pc i * (i :[ \<guillemotright>Goto e ]: j) * r" (is "?r \<in> ?S") show "\<exists>k. rset_of (run k (prog, i', ab, b)) \<in> pc e * (i :[ \<guillemotright>Goto e ]: j) * r" proof - from h [unfolded assemble_to.simps] have h1: "?r \<in> pc i * {{C i (Goto e)}} * <(j = i + 1)> * r" by ((metis stimes_ac)+) note h2 = pcD[OF h1(1)] codeD[OF h1(1)] hence stp: "run 1 (prog, i', ab, b) = (prog, e, ab, b)" (is "?x = ?y") by (unfold run_def, auto) have "rset_of ?x \<in> pc e * (i :[ \<guillemotright>Goto e]: j) * r" proof - from spc_upd[OF h1(1), of "e"] show ?thesis by (unfold stp assemble_to.simps, metis stimes_ac) qed thus ?thesis by blast qedqedno_notation stimes (infixr "*" 70)interpretation foo: comm_monoid_mult "stimes :: 'a set set => 'a set set => 'a set set" "emp::'a set set"apply(default)apply(simp add: stimes_assoc)apply(simp add: stimes_comm)apply(simp add: emp_def[symmetric])donenotation stimes (infixr "*" 70)(*used by simplifier for numbers *)thm mult_cancel_left(*interpretation foo: comm_ring_1 "op * :: 'a set set => 'a set set => 'a set set" "{{}}::'a set set" apply(default)*)lemma frame: "{p} c {q} \<Longrightarrow> \<forall> r. {p * r} c {q * r}"apply (unfold Hoare_abc_def, clarify)apply (erule_tac x = "(a, aa, ab, b)" in allE)apply (erule_tac x = "r * ra" in allE) apply(metis stimes_ac)donelemma code_extension: "\<lbrakk>{p} c {q}\<rbrakk> \<Longrightarrow> (\<forall> e. {p} c * e {q})" apply (unfold Hoare_abc_def, clarify) apply (erule_tac x = "(a, aa, ab, b)" in allE) apply (erule_tac x = "e * r" in allE) apply(metis stimes_ac) donelemma run_add: "run (n1 + n2) s = run n1 (run n2 s)"apply (unfold run_def)by (metis funpow_add o_apply)lemma composition: "\<lbrakk>{p} c1 {q}; {q} c2 {r}\<rbrakk> \<Longrightarrow> {p} c1 * c2 {r}"proof - assume h: "{p} c1 {q}" "{q} c2 {r}" from code_extension [OF h(1), rule_format, of "c2"] have "{p} c1 * c2 {q}" . moreover from code_extension [OF h(2), rule_format, of "c1"] and stimes_comm have "{q} c1 * c2 {r}" by metis ultimately show "{p} c1 * c2 {r}" apply (unfold Hoare_abc_def, clarify) proof - fix a aa ab b ra assume h1: "\<forall>s r. rset_of s \<in> p * (c1 * c2) * r \<longrightarrow> (\<exists>k. rset_of (run k s) \<in> q * (c1 * c2) * r)" and h2: "\<forall>s ra. rset_of s \<in> q * (c1 * c2) * ra \<longrightarrow> (\<exists>k. rset_of (run k s) \<in> r * (c1 * c2) * ra)" and h3: "rset_of (a, aa, ab, b) \<in> p * (c1 * c2) * ra" show "\<exists>k. rset_of (run k (a, aa, ab, b)) \<in> r * (c1 * c2) * ra" proof - let ?s = "(a, aa, ab, b)" from h1 [rule_format, of ?s, OF h3] obtain n1 where "rset_of (run n1 ?s) \<in> q * (c1 * c2) * ra" by blast from h2 [rule_format, OF this] obtain n2 where "rset_of (run n2 (run n1 ?s)) \<in> r * (c1 * c2) * ra" by blast with run_add show ?thesis by metis qed qedqedlemma stimes_simp: "s \<in> x * y = (\<exists> s1 s2. (s = s1 \<union> s2 \<and> s1 \<inter> s2 = {} \<and> s1 \<in> x \<and> s2 \<in> y))"by (metis (lifting) stimesE stimesI)lemma hoare_seq: "\<lbrakk>\<forall> i j. {pc i * p} i:[c1]:j {pc j * q}; \<forall> j k. {pc j * q} j:[c2]:k {pc k * r}\<rbrakk> \<Longrightarrow> {pc i * p} i:[(c1 ; c2)]:k {pc k *r}"proof - assume h: "\<forall>i j. {pc i * p} i :[ c1 ]: j {pc j * q}" "\<forall> j k. {pc j * q} j:[c2]:k {pc k * r}" show "{pc i * p} i:[(c1 ; c2)]:k {pc k *r}" proof(subst Hoare_abc_def, clarify) fix a aa ab b ra assume "rset_of (a, aa, ab, b) \<in> (pc i * p) * (i :[ (c1 ; c2) ]: k) * ra" hence "rset_of (a, aa, ab, b) \<in> (i :[ (c1 ; c2) ]: k) * (pc i * p * ra)" (is "?s \<in> ?X * ?Y") by (metis stimes_ac) from stimesE[OF this] obtain s1 s2 where sp: "rset_of(a, aa, ab, b) = s1 \<union> s2" "s1 \<inter> s2 = {}" "s1 \<in> ?X" "s2 \<in> ?Y" by blast from sp (3) obtain j' where "s1 \<in> (i:[c1]:j') * (j':[c2]:k)" (is "s1 \<in> ?Z") by (auto simp:assemble_to.simps) from stimesI[OF sp(1, 2) this sp(4)] have "?s \<in> (pc i * p) * (i :[ c1 ]: j') * (j' :[ c2 ]: k) * ra" by (metis stimes_ac) from h(1)[unfolded Hoare_abc_def, rule_format, OF this] obtain ka where "rset_of (run ka (a, aa, ab, b)) \<in> (pc j' * q) * (j' :[ c2 ]: k) * ((i :[ c1 ]: j') * ra)" sorry from h(2)[unfolded Hoare_abc_def, rule_format, OF this] obtain kb where "rset_of (run kb (run ka (a, aa, ab, b))) \<in> (pc k * r) * (j' :[ c2 ]: k) * (i :[ c1 ]: j') * ra" by blast hence h3: "rset_of (run (kb + ka) (a, aa, ab, b)) \<in> pc k * r * (j' :[ c2 ]: k) * ((i :[ c1 ]: j') * ra)" sorry hence "rset_of (run (kb + ka) (a, aa, ab, b)) \<in> pc k * r * (i :[ (c1 ; c2) ]: k) * ra" proof - have "rset_of (run (kb + ka) (a, aa, ab, b)) \<in> (i :[ (c1 ; c2) ]: k) * (pc k * r * ra)" proof - from h3 have "rset_of (run (kb + ka) (a, aa, ab, b)) \<in> ((j' :[ c2 ]: k) * ((i :[ c1 ]: j'))) * (pc k * r * ra)" by (metis stimes_ac) then obtain s1 s2 where h4: "rset_of (run (kb + ka) (a, aa, ab, b)) = s1 \<union> s2" " s1 \<inter> s2 = {}" "s1 \<in> (j' :[ c2 ]: k) * (i :[ c1 ]: j')" "s2 \<in> pc k * r * ra" by (rule stimesE, blast) from h4(3) have "s1 \<in> (i :[ (c1 ; c2) ]: k)" sorry from stimesI [OF h4(1, 2) this h4(4)] show ?thesis . qed thus ?thesis by (metis stimes_ac) qed thus "\<exists>ka. rset_of (run ka (a, aa, ab, b)) \<in> (pc k * r) * (i :[ (c1 ; c2) ]: k) * ra" by (metis stimes_ac) qedqedlemma hoare_local: "\<forall> l i j. {pc i*p} i:[c(l)]:j {pc j * q} \<Longrightarrow> {pc i * p} i:[Local c]:j {pc j * q}"proof - assume h: "\<forall> l i j. {pc i*p} i:[c(l)]:j {pc j * q} " show "{pc i * p} i:[Local c]:j {pc j * q}" proof(unfold assemble_to.simps Hoare_abc_def, clarify) fix a aa ab b r assume h1: "rset_of (a, aa, ab, b) \<in> (pc i * p) * (\<Union>l. i :[ c l ]: j) * r" hence "rset_of (a, aa, ab, b) \<in> (\<Union>l. i :[ c l ]: j) * (pc i * p * r)" by (metis stimes_ac) then obtain s1 s2 l where "rset_of (a, aa, ab, b) = s1 \<union> s2" "s1 \<inter> s2 = {}" "s1 \<in> (i :[ c l ]: j)" "s2 \<in> pc i * p * r" by (rule stimesE, auto) from stimesI[OF this] have "rset_of (a, aa, ab, b) \<in> (pc i * p) * (i :[ c l ]: j) * r" by (metis stimes_ac) from h[unfolded Hoare_abc_def, rule_format, OF this] obtain k where "rset_of (run k (a, aa, ab, b)) \<in> (i :[ c l ]: j) * (pc j * q * r)" sorry then obtain s1 s2 where h3: "rset_of (run k (a, aa, ab, b)) = s1 \<union> s2" " s1 \<inter> s2 = {}" "s1 \<in> (\<Union> l. (i :[ c l ]: j))" "s2 \<in> pc j * q * r" by(rule stimesE, auto) from stimesI[OF this] show "\<exists>k. rset_of (run k (a, aa, ab, b)) \<in> (pc j * q) * (\<Union>l. i :[ c l ]: j) * r" by (metis stimes_ac) qedqedlemma move_pure: "{p*<b>} c {q} = (b \<longrightarrow> {p} c {q})"proof(unfold Hoare_abc_def, default, clarify) fix prog i' mem ft r assume h: "\<forall>s r. rset_of s \<in> (p * <b>) * c * r \<longrightarrow> (\<exists>k. rset_of (run k s) \<in> q * c * r)" "b" "rset_of (prog, i', mem, ft) \<in> p * c * r" show "\<exists>k. rset_of (run k (prog, i', mem, ft)) \<in> q * c * r" proof(rule h(1)[rule_format]) have "(p * <b>) * c * r = <b> * p * c * r" by (metis stimes_ac) moreover have "rset_of (prog, i', mem, ft) \<in> \<dots>" proof(rule stimesI[OF _ _ _ h(3)]) from h(2) show "{} \<in> <b>" by (auto simp:pasrt_def) qed auto ultimately show "rset_of (prog, i', mem, ft) \<in> (p * <b>) * c * r" by (simp) qednext assume h: "b \<longrightarrow> (\<forall>s r. rset_of s \<in> p * c * r \<longrightarrow> (\<exists>k. rset_of (run k s) \<in> q * c * r))" show "\<forall>s r. rset_of s \<in> (p * <b>) * c * r \<longrightarrow> (\<exists>k. rset_of (run k s) \<in> q * c * r)" proof - { fix s r assume "rset_of s \<in> (p * <b>) * c * r" hence h1: "rset_of s \<in> <b> * p * c * r" by (metis stimes_ac) have "(\<exists>k. rset_of (run k s) \<in> q * c * r)" proof(rule h[rule_format]) from condD[OF h1] show b . next from condD1[OF h1] show "rset_of s \<in> p * c * r" . qed } thus ?thesis by blast qedqedlemma precond_ex: "{\<Union> x. p x} c {q} = (\<forall> x. {p x} c {q})"proof(unfold Hoare_abc_def, default, clarify) fix x prog i' mem ft r assume h: "\<forall>s r. rset_of s \<in> UNION UNIV p * c * r \<longrightarrow> (\<exists>k. rset_of (run k s) \<in> q * c * r)" "rset_of (prog, i', mem, ft) \<in> p x * c * r" show "\<exists>k. rset_of (run k (prog, i', mem, ft)) \<in> q * c * r" proof(rule h[rule_format]) from h(2) show "rset_of (prog, i', mem, ft) \<in> UNION UNIV p * c * r" by (auto simp:stimes_def) qednext assume h: "\<forall>x s r. rset_of s \<in> p x * c * r \<longrightarrow> (\<exists>k. rset_of (run k s) \<in> q * c * r)" show "\<forall>s r. rset_of s \<in> UNION UNIV p * c * r \<longrightarrow> (\<exists>k. rset_of (run k s) \<in> q * c * r)" proof - { fix s r assume "rset_of s \<in> UNION UNIV p * c * r" then obtain x where "rset_of s \<in> p x * c * r" by (unfold st_def, auto, metis) hence "(\<exists>k. rset_of (run k s) \<in> q * c * r)" by(rule h[rule_format]) } thus ?thesis by blast qedqedlemma pre_stren: "\<lbrakk>{p} c {q}; r \<subseteq> p\<rbrakk> \<Longrightarrow> {r} c {q}"proof(unfold Hoare_abc_def, clarify) fix prog i' mem ft r' assume h: "\<forall>s r. rset_of s \<in> p * c * r \<longrightarrow> (\<exists>k. rset_of (run k s) \<in> q * c * r)" " r \<subseteq> p" " rset_of (prog, i', mem, ft) \<in> r * c * r'" show "\<exists>k. rset_of (run k (prog, i', mem, ft)) \<in> q * c * r'" proof(rule h(1)[rule_format]) from stimes_mono[OF h(2), of "c * r'"] h(3) show "rset_of (prog, i', mem, ft) \<in> p * c * r'" by auto qedqedlemma post_weaken: "\<lbrakk>{p} c {q}; q \<subseteq> r\<rbrakk> \<Longrightarrow> {p} c {r}"proof(unfold Hoare_abc_def, clarify) fix prog i' mem ft r' assume h: "\<forall>s r. rset_of s \<in> p * c * r \<longrightarrow> (\<exists>k. rset_of (run k s) \<in> q * c * r)" " q \<subseteq> r" "rset_of (prog, i', mem, ft) \<in> p * c * r'" show "\<exists>k. rset_of (run k (prog, i', mem, ft)) \<in> r * c * r'" proof - from h(1)[rule_format, OF h(3)] obtain k where "rset_of (run k (prog, i', mem, ft)) \<in> q * c * r'" by auto moreover from h(2) have "\<dots> \<subseteq> r * c * r'" by (metis stimes_mono) ultimately show ?thesis by auto qedqedend