header {* 
 {\em abacus} a kind of register machine
*}

theory abacus
imports Main turing_basic
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
  

text {*
  Abacus programs are defined as lists of Abacus instructions.
*}
type_synonym abc_prog = "abc_inst list"

section {*
  Sample Abacus programs
  *}

text {*
  Abacus for addition and clearance.
*}
fun plus_clear :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"  
  where
  "plus_clear m n e = [Dec m e, Inc n, Goto 0]"

text {*
  Abacus for clearing memory untis.
*}
fun clear :: "nat \<Rightarrow> nat \<Rightarrow> abc_prog"
  where
  "clear n e = [Dec n e, Goto 0]"

fun plus:: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"
  where
  "plus m n p e = [Dec m 4, Inc n, Inc p,
                   Goto 0, Dec p e, Inc m, Goto 4]"

fun mult :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"
  where
  "mult m1 m2 n p e = [Dec m1 e]@ plus m1 m2 p 1"

fun expo :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_prog"
  where
  "expo n m1 m2 p e = [Inc n, Dec m1 e] @ mult m2 n n p 2"


text {*
  The state of Abacus machine.
  *}
type_synonym abc_state = nat

(* text {*
  The memory of Abacus machine is defined as a function from address to contents.
*}
type_synonym abc_mem = "nat \<Rightarrow> nat" *)

text {*
  The memory of Abacus machine is defined as a list of contents, with 
  every units addressed by index into the list.
  *}
type_synonym abc_lm = "nat list"

text {*
  Fetching contents out of memory. Units not represented by list elements are considered
  as having content @{text "0"}.
*}
fun abc_lm_v :: "abc_lm \<Rightarrow> nat \<Rightarrow> nat"
  where 
    "abc_lm_v lm n = (if (n < length lm) then (lm!n) else 0)"         


text {*
  Set the content of memory unit @{text "n"} to value @{text "v"}.
  @{text "am"} is the Abacus memory before setting.
  If address @{text "n"} is outside to scope of @{text "am"}, @{text "am"} 
  is extended so that @{text "n"} becomes in scope.
*}

fun abc_lm_s :: "abc_lm \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> abc_lm"
  where
    "abc_lm_s am n v = (if (n < length am) then (am[n:=v]) else 
                           am@ (replicate (n - length am) 0) @ [v])"


text {*
  The configuration of Abaucs machines consists of its current state and its
  current memory:
*}
type_synonym abc_conf = "abc_state \<times> abc_lm"

text {*
  Fetch instruction out of Abacus program:
*}

fun abc_fetch :: "nat \<Rightarrow> abc_prog \<Rightarrow> abc_inst option" 
  where
  "abc_fetch s p = (if (s < length p) then Some (p ! s)
                    else None)"

text {*
  Single step execution of Abacus machine. If no instruction is feteched, 
  configuration does not change.
*}
fun abc_step_l :: "abc_conf \<Rightarrow> abc_inst option \<Rightarrow> abc_conf"
  where
  "abc_step_l (s, lm) a = (case a of 
               None \<Rightarrow> (s, lm) |
               Some (Inc n)  \<Rightarrow> (let nv = abc_lm_v lm n in
                       (s + 1, abc_lm_s lm n (nv + 1))) |
               Some (Dec n e) \<Rightarrow> (let nv = abc_lm_v lm n in
                       if (nv = 0) then (e, abc_lm_s lm n 0) 
                       else (s + 1,  abc_lm_s lm n (nv - 1))) |
               Some (Goto n) \<Rightarrow> (n, lm) 
               )"

text {*
  Multi-step execution of Abacus machine.
*}
fun abc_steps_l :: "abc_conf \<Rightarrow> abc_prog \<Rightarrow> nat \<Rightarrow> abc_conf"
  where
  "abc_steps_l (s, lm) p 0 = (s, lm)" |
  "abc_steps_l (s, lm) p (Suc n) = 
      abc_steps_l (abc_step_l (s, lm) (abc_fetch s p)) p n"

section {*
  Compiling Abacus machines into Truing machines
*}

subsection {*
  Compiling functions
*}

text {*
  @{text "findnth n"} returns the TM which locates the represention of
  memory cell @{text "n"} on the tape and changes representation of zero
  on the way.
*}

fun findnth :: "nat \<Rightarrow> instr list"
  where
  "findnth 0 = []" |
  "findnth (Suc n) = (findnth n @ [(W1, 2 * n + 1), 
           (R, 2 * n + 2), (R, 2 * n + 3), (R, 2 * n + 2)])"

text {*
  @{text "tinc_b"} returns the TM which increments the representation 
  of the memory cell under rw-head by one and move the representation 
  of cells afterwards to the right accordingly.
  *}

definition tinc_b :: "instr list"
  where
  "tinc_b \<equiv> [(W1, 1), (R, 2), (W1, 3), (R, 2), (W1, 3), (R, 4), 
             (L, 7), (W0, 5), (R, 6), (W0, 5), (W1, 3), (R, 6),
             (L, 8), (L, 7), (R, 9), (L, 7), (R, 10), (W0, 9)]" 

text {*
  @{text "tinc ss n"} returns the TM which simulates the execution of 
  Abacus instruction @{text "Inc n"}, assuming that TM is located at
  location @{text "ss"} in the final TM complied from the whole
  Abacus program.
*}

fun tinc :: "nat \<Rightarrow> nat \<Rightarrow> instr list"
  where
  "tinc ss n = shift (findnth n @ shift tinc_b (2 * n)) (ss - 1)"

text {*
  @{text "tinc_b"} returns the TM which decrements the representation 
  of the memory cell under rw-head by one and move the representation 
  of cells afterwards to the left accordingly.
  *}

definition tdec_b :: "instr list"
  where
  "tdec_b \<equiv>  [(W1, 1), (R, 2), (L, 14), (R, 3), (L, 4), (R, 3),
              (R, 5), (W0, 4), (R, 6), (W0, 5), (L, 7), (L, 8),
              (L, 11), (W0, 7), (W1, 8), (R, 9), (L, 10), (R, 9),
              (R, 5), (W0, 10), (L, 12), (L, 11), (R, 13), (L, 11),
              (R, 17), (W0, 13), (L, 15), (L, 14), (R, 16), (L, 14),
              (R, 0), (W0, 16)]"

text {*
  @{text "sete tp e"} attaches the termination edges (edges leading to state @{text "0"}) 
  of TM @{text "tp"} to the intruction labelled by @{text "e"}.
  *}

fun sete :: "instr list \<Rightarrow> nat \<Rightarrow> instr list"
  where
  "sete tp e = map (\<lambda> (action, state). (action, if state = 0 then e else state)) tp"

text {*
  @{text "tdec ss n label"} returns the TM which simulates the execution of 
  Abacus instruction @{text "Dec n label"}, assuming that TM is located at
  location @{text "ss"} in the final TM complied from the whole
  Abacus program.
*}

fun tdec :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> instr list"
  where
  "tdec ss n e = sete (shift (findnth n @ shift tdec_b (2 * n)) 
                 (ss - 1)) e"
 
text {*
  @{text "tgoto f(label)"} returns the TM simulating the execution of Abacus instruction
  @{text "Goto label"}, where @{text "f(label)"} is the corresponding location of
  @{text "label"} in the final TM compiled from the overall Abacus program.
*}

fun tgoto :: "nat \<Rightarrow> instr list"
  where
  "tgoto n = [(Nop, n), (Nop, n)]"

text {*
  The layout of the final TM compiled from an Abacus program is represented
  as a list of natural numbers, where the list element at index @{text "n"} represents the 
  starting state of the TM simulating the execution of @{text "n"}-th instruction
  in the Abacus program.
*}

type_synonym layout = "nat list"

text {*
  @{text "length_of i"} is the length of the 
  TM simulating the Abacus instruction @{text "i"}.
*}
fun length_of :: "abc_inst \<Rightarrow> nat"
  where
  "length_of i = (case i of 
                    Inc n   \<Rightarrow> 2 * n + 9 |
                    Dec n e \<Rightarrow> 2 * n + 16 |
                    Goto n  \<Rightarrow> 1)"

text {*
  @{text "layout_of ap"} returns the layout of Abacus program @{text "ap"}.
*}
fun layout_of :: "abc_prog \<Rightarrow> layout"
  where "layout_of ap = map length_of ap"


text {*
  @{text "start_of layout n"} looks out the starting state of @{text "n"}-th
  TM in the finall TM.
*}
thm listsum_def

fun start_of :: "nat list \<Rightarrow> nat \<Rightarrow> nat"
  where
  "start_of ly x = (Suc (listsum (take x ly))) "

text {*
  @{text "ci lo ss i"} complies Abacus instruction @{text "i"}
  assuming the TM of @{text "i"} starts from state @{text "ss"} 
  within the overal layout @{text "lo"}.
*}

fun ci :: "layout \<Rightarrow> nat \<Rightarrow> abc_inst \<Rightarrow> instr list"
  where
  "ci ly ss (Inc n) = tinc ss n"
| "ci ly ss (Dec n e) = tdec ss n (start_of ly e)"
| "ci ly ss (Goto n) = tgoto (start_of ly n)"

text {*
  @{text "tpairs_of ap"} transfroms Abacus program @{text "ap"} pairing
  every instruction with its starting state.
*}

fun tpairs_of :: "abc_prog \<Rightarrow> (nat \<times> abc_inst) list"
  where "tpairs_of ap = (zip (map (start_of (layout_of ap)) 
                         [0..<(length ap)]) ap)"

text {*
  @{text "tms_of ap"} returns the list of TMs, where every one of them simulates
  the corresponding Abacus intruction in @{text "ap"}.
*}

fun tms_of :: "abc_prog \<Rightarrow> (instr list) list"
  where "tms_of ap = map (\<lambda> (n, tm). ci (layout_of ap) n tm) 
                         (tpairs_of ap)"

text {*
  @{text "tm_of ap"} returns the final TM machine compiled from Abacus program @{text "ap"}.
*}
fun tm_of :: "abc_prog \<Rightarrow> instr list"
  where "tm_of ap = concat (tms_of ap)"

text {*
  The following two functions specify the well-formedness of complied TM.
*}
(*
fun t_ncorrect :: "tprog \<Rightarrow> bool"
  where
  "t_ncorrect tp = (length tp mod 2 = 0)"

fun abc2t_correct :: "abc_prog \<Rightarrow> bool"
  where 
  "abc2t_correct ap = list_all (\<lambda> (n, tm). 
             t_ncorrect (ci (layout_of ap) n tm)) (tpairs_of ap)"
*)

lemma length_findnth: 
  "length (findnth n) = 4 * n"
apply(induct n, auto)
done

lemma ci_length : "length (ci ns n ai) div 2 = length_of ai"
apply(auto simp: ci.simps tinc_b_def tdec_b_def length_findnth
                 split: abc_inst.splits)
done

subsection {*
  Representation of Abacus memory by TM tape
*}

text {*
  @{text "crsp acf tcf"} meams the abacus configuration @{text "acf"}
  is corretly represented by the TM configuration @{text "tcf"}.
*}

fun crsp :: "layout \<Rightarrow> abc_conf \<Rightarrow> config \<Rightarrow> cell list \<Rightarrow> bool"
  where 
  "crsp ly (as, lm) (s, l, r) inres = 
           (s = start_of ly as \<and> (\<exists> x. r = <lm> @ Bk\<up>x) \<and> 
            l = Bk # Bk # inres)"

declare crsp.simps[simp del]

subsection {*
  A more general definition of TM execution. 
*}

                    
text {*
  The type of invarints expressing correspondence between 
  Abacus configuration and TM configuration.
*}

type_synonym inc_inv_t = "abc_conf \<Rightarrow> config \<Rightarrow> cell list \<Rightarrow> bool"

declare tms_of.simps[simp del] tm_of.simps[simp del]
        layout_of.simps[simp del] abc_fetch.simps [simp del]  
        tpairs_of.simps[simp del] start_of.simps[simp del]
        ci.simps [simp del] length_of.simps[simp del] 
        layout_of.simps[simp del]


(*
subsection {* The compilation of @{text "Inc n"} *}
*)

text {*
  The lemmas in this section lead to the correctness of 
  the compilation of @{text "Inc n"} instruction.
*}


declare abc_step_l.simps[simp del] abc_steps_l.simps[simp del]
lemma [simp]: "start_of ly as > 0"
apply(simp add: start_of.simps)
done

lemma abc_step_red: 
 "abc_steps_l ac aprog stp = (as, am) \<Longrightarrow> 
   abc_steps_l ac aprog (Suc stp) = abc_step_l (as, am) (abc_fetch as aprog) "
oops

lemma tm_shift_fetch: 
  "\<lbrakk>fetch A s b = (ac, ns); ns \<noteq> 0 \<rbrakk>
  \<Longrightarrow> fetch (shift A off) s b = (ac, ns + off)"
apply(case_tac b)
apply(case_tac [!] s, auto simp: fetch.simps shift.simps)
done

lemma tm_shift_eq_step:
  assumes exec: "step (s, l, r) (A, 0) = (s', l', r')"
  and notfinal: "s' \<noteq> 0"
  shows "step (s + off, l, r) (shift A off, off) = (s' + off, l', r')"
using assms
apply(simp add: step.simps)
apply(case_tac "fetch A s (read r)", auto)
apply(drule_tac [!] off = off in tm_shift_fetch, simp_all)
done

lemma tm_shift_eq_steps: 
  assumes layout: "ly = layout_of ap"
  and compile: "tp = tm_of ap"
  and exec: "steps (s, l, r) (A, 0) stp = (s', l', r')"
  and notfinal: "s' \<noteq> 0"
  shows "steps (s + off, l, r) (shift A off, off) stp = (s' + off, l', r')"
  using exec notfinal
proof(induct stp arbitrary: s' l' r', simp add: steps.simps)
  fix stp s' l' r'
  assume ind: "\<And>s' l' r'. \<lbrakk>steps (s, l, r) (A, 0) stp = (s', l', r'); s' \<noteq> 0\<rbrakk> 
     \<Longrightarrow> steps (s + off, l, r) (shift A off, off) stp = (s' + off, l', r')"
  and h: " steps (s, l, r) (A, 0) (Suc stp) = (s', l', r')" "s' \<noteq> 0"
  obtain s1 l1 r1 where g: "steps (s, l, r) (A, 0) stp = (s1, l1, r1)" 
    apply(case_tac "steps (s, l, r) (A, 0) stp") by blast
  moreover then have "s1 \<noteq> 0"
    using h
    apply(simp add: step_red, case_tac "0 < s1", simp, simp)
    done
  ultimately have "steps (s + off, l, r) (shift A off, off) stp =
                   (s1 + off, l1, r1)"
    apply(rule_tac ind, simp_all)
    done
  thus "steps (s + off, l, r) (shift A off, off) (Suc stp) = (s' + off, l', r')"
    using h g assms
    apply(simp add: step_red)
    apply(rule_tac tm_shift_eq_step, auto)
    done
qed

lemma startof_not0[simp]: "0 < start_of ly as"
apply(simp add: start_of.simps)
done

lemma startof_ge1[simp]: "Suc 0 \<le> start_of ly as"
apply(simp add: start_of.simps)
done

lemma step_eq_fetch: 
  assumes layout: "ly = layout_of ap"
  and compile: "tp = tm_of ap"
  and abc_fetch: "abc_fetch as ap = Some ins" 
  and fetch: "fetch (ci ly (start_of ly as) ins)
       (Suc s - start_of ly as) b = (ac, ns)"
  and notfinal: "ns \<noteq> 0"
  shows "fetch tp s b = (ac, ns)"
oops
  
lemma step_eq_in:
  assumes layout: "ly = layout_of ap"
  and compile: "tp = tm_of ap"
  and fetch: "abc_fetch as ap = Some ins"    
  and exec: "step (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) 
  = (s', l', r')"
  and notfinal: "s' \<noteq> 0"
  shows "step (s, l, r) (tp, 0) = (s', l', r')"
  using assms
  apply(simp add: step.simps)
  apply(case_tac "fetch (ci (layout_of ap) (start_of (layout_of ap) as) ins)
    (Suc s - start_of (layout_of ap) as) (read r)", simp)
  using layout
  apply(drule_tac s = s and b = "read r" and ac = a in step_eq_fetch, auto)
  done

lemma steps_eq_in:
  assumes layout: "ly = layout_of ap"
  and compile: "tp = tm_of ap"
  and crsp: "crsp ly (as, lm) (s, l, r) ires"
  and fetch: "abc_fetch as ap = Some ins"    
  and exec: "steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp 
  = (s', l', r')"
  and notfinal: "s' \<noteq> 0"
  shows "steps (s, l, r) (tp, 0) stp = (s', l', r')"
  using exec notfinal
proof(induct stp arbitrary: s' l' r', simp add: steps.simps)
  fix stp s' l' r'
  assume ind: 
    "\<And>s' l' r'. \<lbrakk>steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp = (s', l', r'); s' \<noteq> 0\<rbrakk>
              \<Longrightarrow> steps (s, l, r) (tp, 0) stp = (s', l', r')"
  and h: "steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) (Suc stp) = (s', l', r')" "s' \<noteq> 0"
  obtain s1 l1 r1 where g: "steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp = 
                        (s1, l1, r1)"
    apply(case_tac "steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp") by blast
  moreover hence "s1 \<noteq> 0"
    using h
    apply(simp add: step_red)
    apply(case_tac "0 < s1", simp_all)
    done
  ultimately have "steps (s, l, r) (tp, 0) stp = (s1, l1, r1)"
    apply(rule_tac ind, auto)
    done
  thus "steps (s, l, r) (tp, 0) (Suc stp) = (s', l', r')"
    using h g assms
    apply(simp add: step_red)
    apply(rule_tac step_eq_in, auto)
    done
qed

lemma tm_append_fetch_first: 
  "\<lbrakk>fetch A s b = (ac, ns); ns \<noteq> 0\<rbrakk> \<Longrightarrow> 
    fetch (A @ B) s b = (ac, ns)"
apply(case_tac b)
apply(case_tac [!] s, auto simp: fetch.simps nth_append split: if_splits)
done

lemma tm_append_first_step_eq: 
  assumes "step (s, l, r) (A, 0) = (s', l', r')"
  and "s' \<noteq> 0"
  shows "step (s, l, r) (A @ B, 0) = (s', l', r')"
using assms
apply(simp add: step.simps)
apply(case_tac "fetch A s (read r)")
apply(frule_tac  B = B and b = "read r" in tm_append_fetch_first, auto)
done

lemma tm_append_first_steps_eq: 
  assumes "steps (s, l, r) (A, 0) stp = (s', l', r')"
  and "s' \<noteq> 0"
  shows "steps (s, l, r) (A @ B, 0) stp = (s', l', r')"
using assms
sorry

lemma tm_append_second_steps_eq: 
  assumes 
  exec: "steps (s, l, r) (B, 0) stp = (s', l', r')"
  and notfinal: "s' \<noteq> 0"
  and off: "off = length A div 2"
  and even: "length A mod 2 = 0"
  shows "steps (s + off, l, r) (A @ shift B off, 0) stp = (s' + off, l', r')"
using assms
sorry

lemma tm_append_steps: 
  assumes 
  aexec: "steps (s, l, r) (A, 0) stpa = (Suc (length A div 2), la, ra)"
  and bexec: "steps (Suc 0, la, ra) (B, 0) stpb =  (sb, lb, rb)"
  and notfinal: "sb \<noteq> 0"
  and off: "off = length A div 2"
  and even: "length A mod 2 = 0"
  shows "steps (s, l, r) (A @ shift B off, 0) (stpa + stpb) = (sb + off, lb, rb)"
proof -
  have "steps (s, l, r) (A@shift B off, 0) stpa = (Suc (length A div 2), la, ra)"
    apply(rule_tac tm_append_first_steps_eq)
    apply(auto simp: assms)
    done
  moreover have "steps (1 + off, la, ra) (A @ shift B off, 0) stpb = (sb + off, lb, rb)"
    apply(rule_tac tm_append_second_steps_eq)
    apply(auto simp: assms bexec)
    done
  ultimately show "steps (s, l, r) (A @ shift B off, 0) (stpa + stpb) = (sb + off, lb, rb)"
    apply(simp add: steps_add off)
    done
qed
       
subsection {* Crsp of Inc*}

lemma crsp_step_inc:
  assumes layout: "ly = layout_of ap"
  and crsp: "crsp ly (as, lm) (s, l, r) ires"
  and fetch: "abc_fetch as ap = Some (Inc n)"
  shows "\<exists>stp. crsp ly (abc_step_l (as, lm) (Some (Inc n)))
  (steps (s, l, r) (ci ly (start_of ly as) (Inc n), start_of ly as - Suc 0) stp) ires"
  sorry
    
subsection{* Crsp of Dec n e*}
lemma crsp_step_dec: 
  assumes layout: "ly = layout_of ap"
  and crsp: "crsp ly (as, lm) (s, l, r) ires"
  shows "\<exists>stp. crsp ly (abc_step_l (as, lm) (Some (Dec n e)))
  (steps (s, l, r) (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp) ires"
sorry

subsection{*Crsp of Goto*}
lemma crsp_step_goto:
  assumes layout: "ly = layout_of ap"
  and crsp: "crsp ly (as, lm) (s, l, r) ires"
  shows "\<exists>stp. crsp ly (abc_step_l (as, lm) (Some (Goto n)))
  (steps (s, l, r) (ci ly (start_of ly as) (Goto n), 
            start_of ly as - Suc 0) stp) ires"
sorry


lemma crsp_step_in:
  assumes layout: "ly = layout_of ap"
  and compile: "tp = tm_of ap"
  and crsp: "crsp ly (as, lm) (s, l, r) ires"
  and fetch: "abc_fetch as ap = Some ins"
  shows "\<exists> stp. crsp ly (abc_step_l (as, lm) (Some ins))
                      (steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp) ires"
  using assms
  apply(case_tac ins, simp_all)
  apply(rule crsp_step_inc, simp_all)
  apply(rule crsp_step_dec, simp_all)
  apply(rule_tac crsp_step_goto, simp_all)
  done

lemma crsp_step:
  assumes layout: "ly = layout_of ap"
  and compile: "tp = tm_of ap"
  and crsp: "crsp ly (as, lm) (s, l, r) ires"
  and fetch: "abc_fetch as ap = Some ins"
  shows "\<exists> stp. crsp ly (abc_step_l (as, lm) (Some ins))
                      (steps (s, l, r) (tp, 0) stp) ires"
proof -
  have "\<exists> stp. crsp ly (abc_step_l (as, lm) (Some ins))
                      (steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp) ires"
    using assms
    apply(erule_tac crsp_step_in, simp_all)
    done
  from this obtain stp where d: "crsp ly (abc_step_l (as, lm) (Some ins))
                      (steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp) ires" ..
  obtain s' l' r' where e:
    "(steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp) = (s', l', r')"
    apply(case_tac "(steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp)")
    by blast
  then have "steps (s, l, r) (tp, 0) stp = (s', l', r')"
    using assms d
    apply(rule_tac steps_eq_in)
    apply(simp_all)
    apply(case_tac "(abc_step_l (as, lm) (Some ins))", simp add: crsp.simps)
    done    
  thus " \<exists>stp. crsp ly (abc_step_l (as, lm) (Some ins)) (steps (s, l, r) (tp, 0) stp) ires"
    using d e
    apply(rule_tac x = stp in exI, simp)
    done
qed

lemma crsp_steps:
  assumes layout: "ly = layout_of ap"
  and compile: "tp = tm_of ap"
  and crsp: "crsp ly (as, lm) (s, l, r) ires"
  shows "\<exists> stp. crsp ly (abc_steps_l (as, lm) ap n)
                      (steps (s, l, r) (tp, 0) stp) ires"
  using crsp
  apply(induct n)
  apply(rule_tac x = 0 in exI) 
  apply(simp add: steps.simps abc_steps_l.simps, simp)
  apply(case_tac "(abc_steps_l (as, lm) ap n)", auto)
  apply(frule_tac abc_step_red, simp)
  apply(case_tac "abc_fetch a ap", simp add: abc_step_l.simps, auto)
  apply(case_tac "steps (s, l, r) (tp, 0) stp", simp)
  using assms
  apply(drule_tac s = ab and l = ba and r = c in crsp_step, auto)
  apply(rule_tac x = "stp + stpa" in exI, simp add: steps_add)
  done

lemma tp_correct': 
  assumes layout: "ly = layout_of ap"
  and compile: "tp = tm_of ap"
  and crsp: "crsp ly (0, lm) (Suc 0, l, r) ires"
  and abc_halt: "abc_steps_l (0, lm) ap stp = (length ap, am)"
  shows "\<exists> stp k. steps (Suc 0, l, r) (tp, 0) stp = (start_of ly (length ap), Bk # Bk # ires, <am> @ Bk\<up>k)"
  using assms
  apply(drule_tac n = stp in crsp_steps, auto)
  apply(rule_tac x = stpa in exI)
  apply(case_tac "steps (Suc 0, l, r) (tm_of ap, 0) stpa", simp add: crsp.simps)
  done

(***normalize the tp compiled from ap, and we can use Hoare_plus when composed with mopup machine*)
definition tp_norm :: "abc_prog \<Rightarrow> instr list"
  where
  "tp_norm ap = tm_of ap @ [(Nop, 0), (Nop, 0)]"

lemma tp_correct: 
  assumes layout: "ly = layout_of ap"
  and crsp: "crsp ly (0, lm) (Suc 0, l, r) ires"
  and abc_halt: "abc_steps_l (0, lm) ap stp = (length ap, am)"
  shows "\<exists> stp k. steps (Suc 0, l, r) (tp_norm ap, 0) stp = (0, Bk # Bk # ires, <am> @ Bk\<up>k)"
 sorry

(****mop_up***)
fun mopup_a :: "nat \<Rightarrow> instr list"
  where
  "mopup_a 0 = []" |
  "mopup_a (Suc n) = mopup_a n @ 
       [(R, 2*n + 3), (W0, 2*n + 2), (R, 2*n + 1), (W1, 2*n + 2)]"

definition mopup_b :: "instr list"
  where
  "mopup_b \<equiv> [(R, 2), (R, 1), (L, 5), (W0, 3), (R, 4), (W0, 3),
            (R, 2), (W0, 3), (L, 5), (L, 6), (R, 0), (L, 6)]"

fun mopup :: "nat \<Rightarrow> instr list"
  where 
  "mopup n = mopup_a n @ shift mopup_b (2*n)"
(****)

(*we can use Hoare_plus here*)
lemma compile_correct_halt: 
  assumes layout: "ly = layout_of ap"
  and crsp: "crsp ly (0, lm) (Suc 0, l, r) ires"
  and abc_halt: "abc_steps_l (0, lm) ap stp = (length ap, am)"
  and rs_loc: "n < length am"
  and rs: "rs = abc_lm_v am n"
  shows "\<exists> stp i j. steps (Suc 0, l, r) (tp_norm ap |+| mopup n, 0) stp = (0, Bk\<up>i @ Bk # Bk # ires, Oc\<up>Suc rs @ Bk\<up>j)"
sorry

lemma compile_correct_unhalt: 
  assumes layout: "ly = layout_of ap"
  and crsp: "crsp ly (0, lm) (Suc 0, l, r) ires"
  and abc_unhalt: "\<forall> stp. (\<lambda> (as, am). as < length ap) (abc_steps_l (0, lm) ap stp)"
  shows "\<forall> stp.\<not> is_final (steps (Suc 0, l, r) (tp_norm ap |+| mopup n, 0) stp)"
sorry

end