header {* 

 {\em abacus} a kind of register machine

*}

theory abacus

begin

text {*

  {\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 nat

definition "stimes p q = {s . \<exists> u v. u \<in> p \<and> v \<in> q \<and> (u \<union> v = s) \<and> (u \<inter> v = {})}"

no_notation times (infixl "*" 70)

notation stimes (infixl "*" 70)

lemma stimes_comm: "p * q = q * p"

  by (unfold stimes_def, auto)

lemma stimes_assoc: "(p * q) * r = p * (q * r)"

  by (unfold stimes_def, blast)

definition

  "emp = {{}}"

lemma emp_unit_r [simp]: "p * emp = p"

  by (unfold stimes_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 stimes_def, auto)

thm mult_cancel_left

lemma stimes_left_commute:

  "(p * (q * r)) = (q * (p * r))"

by (metis stimes_assoc stimes_comm)

lemmas stimes_ac = stimes_comm stimes_assoc stimes_left_commute

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)"

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 nat

fun rset_of :: "aconf \<Rightarrow> aresource set"

  where "rset_of (prog, pc, m, faults) = 

               {M i n | i n. m (i) = Some n} \<union> {At pc} \<union>

               {C i inst | i inst. prog i = Some inst} \<union> {Faults faults}"

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"

definition

  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 "pc l = {{At l}}"

definition "m a v = {{M a v}}"

lemma hoare_dec_suc: "{pc i * m a v * <(v > 0)>} 

                          i:[ \<guillemotright>(Dec a e) ]:j  

                      {pc (i+1) * m a (v - 1)}"

  sorry

lemma hoare_dec_fail: "{pc i * m a 0} 

                          i:[ \<guillemotright>(Dec a e) ]:j   

                       {pc e * m a 0}"

  sorry

lemma hoare_inc: "{pc i * m a v} 

                      i:[ \<guillemotright>(Inc a) ]:j   

                  {pc (i+1) * m a (v + 1)}"

  sorry

interpretation foo: comm_monoid_mult "op * :: 'a set set => 'a set set => 'a set set" "{{}}::'a set set"

apply(default)

apply(simp add: stimes_assoc)

apply(simp add: stimes_comm)

apply(simp add: emp_def[symmetric])

done

(*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(simp add: stimes_ac)

done

lemma 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(simp add: stimes_ac)

  done

lemma 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

    qed

qed

lemma asm_end_unique: "\<lbrakk>s \<in> (i:[c]:j1); s' \<in> (i:[c]:j2)\<rbrakk> \<Longrightarrow> j1 = j2"

(* proof(induct c arbitrary:i j1 j2 s s') *) sorry

lemma union_unique: "(\<forall> j. j \<noteq> i \<longrightarrow> c(j) = {}) \<Longrightarrow> (\<Union> j. c(j)) = (c i)"

  by auto

lemma asm_consist: "i:[c1]:j \<noteq> {}"

  sorry

lemma seq_comp: "\<lbrakk>{p} i:[c1]:j {q}; 

                  {q} j:[c2]:k {r}\<rbrakk> \<Longrightarrow> {p} i:[(c1 ; c2)]:k {r}"

apply (unfold assemble_to.simps)

proof -

  assume h: "{p} i :[ c1 ]: j {q}" "{q} j :[ c2 ]: k {r}"

  have " (\<Union>j'. (i :[ c1 ]: j') * (j' :[ c2 ]: k)) = 

             (i :[ c1 ]: j) * (j :[ c2 ]: k)"

  proof -

    { fix j' 

      assume "j' \<noteq> j"

      have "(i :[ c1 ]: j') * (j' :[ c2 ]: k) = {}" (is "?X * ?Y = {}")

      proof -

        { fix s 

          assume "s \<in> ?X*?Y"

          then obtain s1 s2 where h1: "s1 \<in> ?X" by (unfold stimes_def, auto)

        }

      qed

    } thus ?thesis by (auto intro!:union_unique)

  qed

  moreover have "{p} \<dots> {r}" by (rule composition [OF h])

  ultimately show "{p} \<Union>j'. (i :[ c1 ]: j') * (j' :[ c2 ]: k) {r}" by metis

qed

end