theory Abacus

imports Main

begin 

datatype abc_inst =

     Inc nat

   | Dec nat nat

   | Goto nat

type_synonym abc_prog = "abc_inst list"

type_synonym abc_state = 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"

fun abc_inst_shift :: "abc_inst \<Rightarrow> nat \<Rightarrow> abc_inst"

  where

  "abc_inst_shift (Inc m) n = Inc m" |

  "abc_inst_shift (Dec m e) n = Dec m (e + n)" |

  "abc_inst_shift (Goto m) n = Goto (m + n)"

fun abc_shift :: "abc_inst list \<Rightarrow> nat \<Rightarrow> abc_inst list" 

  where

  "abc_shift xs n = map (\<lambda> x. abc_inst_shift x n) xs" 

fun abc_comp :: "abc_inst list \<Rightarrow> abc_inst list \<Rightarrow> abc_inst list" (infixl ";" 99)

  where

  "al; bl = (let al_len = length al in al @ abc_shift bl al_len)"

end