(* Title: Turing machines

   Author: Xu Jian <xujian817@hotmail.com>

   Maintainer: Xu Jian

*)

theory turing_basic

imports Main

begin

section {* Basic definitions of Turing machine *}

datatype action = W0 | W1 | L | R | Nop

datatype cell = Bk | Oc

type_synonym tape = "cell list \<times> cell list"

type_synonym state = nat

type_synonym instr = "action \<times> state"

type_synonym tprog = "instr list \<times> nat"

type_synonym tprog0 = "instr list"

type_synonym config = "state \<times> tape"

fun nth_of where

  "nth_of xs i = (if i \<ge> length xs then None

                  else Some (xs ! i))"

lemma nth_of_map [simp]:

  shows "nth_of (map f p) n = (case (nth_of p n) of None \<Rightarrow> None | Some x \<Rightarrow> Some (f x))"

apply(induct p arbitrary: n)

apply(auto)

apply(case_tac n)

apply(auto)

done

fun 

  fetch :: "instr list \<Rightarrow> state \<Rightarrow> cell \<Rightarrow> instr"

where

  "fetch p 0 b = (Nop, 0)"

| "fetch p (Suc s) Bk = 

     (case nth_of p (2 * s) of

        Some i \<Rightarrow> i

      | None \<Rightarrow> (Nop, 0))"

|"fetch p (Suc s) Oc = 

     (case nth_of p ((2 * s) + 1) of

         Some i \<Rightarrow> i

       | None \<Rightarrow> (Nop, 0))"

lemma fetch_Nil [simp]:

  shows "fetch [] s b = (Nop, 0)"

apply(case_tac s)

apply(auto)

apply(case_tac b)

apply(auto)

done

fun 

  update :: "action \<Rightarrow> tape \<Rightarrow> tape"

where 

  "update W0 (l, r) = (l, Bk # (tl r))" 

| "update W1 (l, r) = (l, Oc # (tl r))"

| "update L (l, r) = (if l = [] then ([], Bk # r) else (tl l, (hd l) # r))" 

| "update R (l, r) = (if r = [] then (Bk # l, []) else ((hd r) # l, tl r))" 

| "update Nop (l, r) = (l, r)"

abbreviation 

  "read r == if (r = []) then Bk else hd r"

fun step :: "config \<Rightarrow> tprog \<Rightarrow> config"

  where 

  "step (s, l, r) (p, off) = 

     (let (a, s') = fetch p (s - off) (read r) in (s', update a (l, r)))"

fun steps :: "config \<Rightarrow> tprog \<Rightarrow> nat \<Rightarrow> config"

  where

  "steps c p 0 = c" |

  "steps c p (Suc n) = steps (step c p) p n"

abbreviation

  "step0 c p \<equiv> step c (p, 0)"

abbreviation

  "steps0 c p n \<equiv> steps c (p, 0) n"

lemma step_red [simp]: 

  shows "steps c p (Suc n) = step (steps c p n) p"

by (induct n arbitrary: c) (auto)

lemma steps_add [simp]: 

  shows "steps c p (m + n) = steps (steps c p m) p n"

by (induct m arbitrary: c) (auto)

lemma step_0 [simp]: 

  shows "step (0, (l, r)) p = (0, (l, r))"

by (case_tac p, simp)

lemma steps_0 [simp]: 

  shows "steps (0, (l, r)) p n = (0, (l, r))"

by (induct n) (simp_all)

fun

  is_final :: "config \<Rightarrow> bool"

where

  "is_final (s, l, r) = (s = 0)"

lemma is_final_eq: 

  shows "is_final (s, tp) = (s = 0)"

by (case_tac tp) (auto)

lemma is_final_steps:

  assumes "is_final (s, l, r)"

  shows "is_final (steps (s, l, r) (p, off) n)"

using assms 

by (induct n) (auto)

(* if the machine is in the halting state, there must have 

   been a state just before the halting state *)

lemma before_final: 

  assumes "steps0 (1, tp) A n = (0, tp')"

  shows "\<exists> n'. \<not> is_final (steps0 (1, tp) A n') \<and> steps0 (1, tp) A (Suc n') = (0, tp')"

using assms

proof(induct n arbitrary: tp')

  case (0 tp')

  have asm: "steps0 (1, tp) A 0 = (0, tp')" by fact

  then show "\<exists>n'. \<not> is_final (steps0 (1, tp) A n') \<and> steps0 (1, tp) A (Suc n') = (0, tp')"

    by simp

next

  case (Suc n tp')

  have ih: "\<And>tp'. steps0 (1, tp) A n = (0, tp') \<Longrightarrow>

    \<exists>n'. \<not> is_final (steps0 (1, tp) A n') \<and> steps0 (1, tp) A (Suc n') = (0, tp')" by fact

  have asm: "steps0 (1, tp) A (Suc n) = (0, tp')" by fact

  obtain s l r where cases: "steps0 (1, tp) A n = (s, l, r)"

    by (auto intro: is_final.cases)

  then show "\<exists>n'. \<not> is_final (steps0 (1, tp) A n') \<and> steps0 (1, tp) A (Suc n') = (0, tp')"

  proof (cases "s = 0")

    case True (* in halting state *)

    then have "steps0 (1, tp) A n = (0, tp')"

      using asm cases by (simp del: steps.simps)

    then show ?thesis using ih by simp

  next

    case False (* not in halting state *)

    then have "\<not> is_final (steps0 (1, tp) A n) \<and> steps0 (1, tp) A (Suc n) = (0, tp')"

      using asm cases by simp

    then show ?thesis by auto

  qed

qed

(* well-formedness of Turing machine programs *)

fun 

  tm_wf :: "tprog \<Rightarrow> bool"

where

  "tm_wf (p, off) = (length p \<ge> 2 \<and> length p mod 2 = 0 \<and> 

                    (\<forall>(a, s) \<in> set p. s \<le> length p div 2 + off \<and> s \<ge> off))"

lemma halt_lemma: 

  "\<lbrakk>wf LE; \<forall>n. (\<not> P (f n) \<longrightarrow> (f (Suc n), (f n)) \<in> LE)\<rbrakk> \<Longrightarrow> \<exists>n. P (f n)"

by (metis wf_iff_no_infinite_down_chain)

abbreviation exponent :: "'a \<Rightarrow> nat \<Rightarrow> 'a list" ("_ \<up> _" [100, 99] 100)

  where "x \<up> n == replicate n x"

consts tape_of :: "'a \<Rightarrow> cell list" ("<_>" 100)

fun tape_of_nat_list :: "nat list \<Rightarrow> cell list" 

  where 

  "tape_of_nat_list [] = []" |

  "tape_of_nat_list [n] = Oc\<up>(Suc n)" |

  "tape_of_nat_list (n#ns) = Oc\<up>(Suc n) @ Bk # (tape_of_nat_list ns)"

defs (overloaded)

  tape_of_nl_abv: "<am> \<equiv> tape_of_nat_list am"

  tape_of_nat_abv : "<(n::nat)> \<equiv> Oc\<up>(Suc n)"

definition tinres :: "cell list \<Rightarrow> cell list \<Rightarrow> bool"

  where

  "tinres xs ys = (\<exists>n. xs = ys @ Bk \<up> n \<or> ys = xs @ Bk \<up> n)"

fun 

  shift :: "instr list \<Rightarrow> nat \<Rightarrow> instr list"

where

  "shift p n = (map (\<lambda> (a, s). (a, (if s = 0 then 0 else s + n))) p)"

fun 

  adjust :: "instr list \<Rightarrow> instr list"

where

  "adjust p = map (\<lambda> (a, s). (a, if s = 0 then (Suc (length p div 2)) else s)) p"

lemma length_shift [simp]: 

  shows "length (shift p n) = length p"

by simp

lemma length_adjust [simp]: 

  shows "length (adjust p) = length p"

by (induct p) (auto)

(* composition of two Turing machines *)

fun

  tm_comp :: "instr list \<Rightarrow> instr list \<Rightarrow> instr list" ("_ |+| _" [0, 0] 100)

where

  "tm_comp p1 p2 = ((adjust p1) @ (shift p2 (length p1 div 2)))"

lemma tm_comp_length:

  shows "length (A |+| B) = length A + length B"

by auto

  assumes unfinal: "\<not> is_final (step0 c A)"

  shows "step0 c (A |+| B) = step0 c A"

proof -

  obtain s l r where eq: "c = (s, l, r)" by (metis is_final.cases) 

  have "\<not> is_final (step0 (s, l, r) A)" using unfinal eq by simp

  then have "case (fetch A s (read r)) of (a, s) \<Rightarrow> s \<noteq> 0"

    by (auto simp add: is_final_eq)

  then  have "fetch (A |+| B) s (read r) = fetch A s (read r)"

    apply(case_tac [!] "read r")

    apply(case_tac [!] s)

    apply(auto simp: tm_comp_length nth_append)

    done

  then show "step0 c (A |+| B) = step0 c A" by (simp add: eq) 

qed

using assms

proof(induct n)

  case 0

next 

  case (Suc n)

    by (auto simp only: step_red)

    by (metis is_final_eq step_0 surj_pair) 

    by (simp only: step_red)

    by (simp only: step_red)

qed

done

done

done

lemma steps_in_range: 

next

qed

  and a_wf: "tm_wf (A, 0)"

proof -

  assume a: "\<And>n. steps (1, tp) (A |+| B, 0) n = (Suc (length A div 2), tp') \<Longrightarrow> thesis"

qed

lemma tm_comp_fetch_second_zero:

apply(case_tac x)

done 

lemma tm_comp_fetch_second_inst:

apply(case_tac x)

done 

lemma t_merge_second_same:

  assumes a_wf: "tm_wf (A, 0)"

    done

qed

lemma t_merge_second_halt_same:

end