theory turing_basic

imports Main

begin

section {* Basic definitions of Turing machine *}

(* Title: Turing machine's definition and its charater

   Author: Xu Jian <xujian817@hotmail.com>

   Maintainer: Xu Jian

*)

text {*

\label{description of turing machine}

*}

section {* Basic definitions of Turing machine *}

(* Title: Turing machine's definition and its charater

   Author: Xu Jian <xujian817@hotmail.com>

   Maintainer: Xu Jian

*)

text {*

  Actions of Turing machine (Abbreviated TM in the following* ).

*}

datatype taction = 

  -- {* Write zero *}

  W0 | 

  -- {* Write one *}

  W1 | 

  -- {* Move left *}

  L | 

  -- {* Move right *}

  R | 

  -- {* Do nothing *}

  Nop

text {*

  Tape contents in every block.

*}

datatype block = 

  -- {* Blank *}

  Bk | 

  -- {* Occupied *}

  Oc

text {*

  Tape is represented as a pair of lists $(L_{left}, L_{right})$,

  where $L_left$, named {\em left list}, is used to represent

  the tape to the left of RW-head and

  $L_{right}$, named {\em right list}, is used to represent the tape

  under and to the right of RW-head.

*}

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

text {* The state of turing machine.*}

type_synonym tstate = nat

text {*

  Turing machine instruction is represented as a 

  pair @{text "(action, next_state)"},

  where @{text "action"} is the action to take at the current state 

  and @{text "next_state"} is the next state the machine is getting into

  after the action.

*}

type_synonym tinst = "taction \<times> tstate"

text {*

  Program of Turing machine is represented as a list of Turing instructions

  and the execution of the program starts from the head of the list.

  *}

type_synonym tprog = "tinst list"

text {*

  Turing machine configuration, which consists of the current state 

  and the tape.

*}

type_synonym t_conf = "tstate \<times> tape"

fun nth_of ::  "'a list \<Rightarrow> nat \<Rightarrow> 'a option"

  where

  "nth_of xs n = (if n < length xs then Some (xs!n)

                  else None)"

text {*

  The function used to fetech instruction out of Turing program.

  *}

fun fetch :: "tprog \<Rightarrow> tstate \<Rightarrow> block \<Rightarrow> tinst"

  where

  "fetch p s b = (if s = 0 then (Nop, 0) else

                  case b of 

                     Bk \<Rightarrow> case nth_of p (2 * (s - 1)) of

                          Some i \<Rightarrow> i

                        | None \<Rightarrow> (Nop, 0) 

                   | Oc \<Rightarrow> case nth_of p (2 * (s - 1) +1) of

                          Some i \<Rightarrow> i

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

fun new_tape :: "taction \<Rightarrow> tape \<Rightarrow> tape"

where 

   "new_tape action (leftn, rightn) = (case action of

                                         W0 \<Rightarrow> (leftn, Bk#(tl rightn)) |

                                         W1 \<Rightarrow> (leftn, Oc#(tl rightn)) |

                                         L  \<Rightarrow>  (if leftn = [] then (tl leftn, Bk#rightn)

                                               else (tl leftn, (hd leftn) # rightn)) |

                                         R  \<Rightarrow> if rightn = [] then (Bk#leftn,tl rightn) 

                                               else ((hd rightn)#leftn, tl rightn) |

                                         Nop \<Rightarrow> (leftn, rightn)

                                       )"

text {*

  The one step function used to transfer Turing machine configuration.

*}

fun tstep :: "t_conf \<Rightarrow> tprog \<Rightarrow> t_conf"

  where

  "tstep c p = (let (s, l, r) = c in 

                     let (ac, ns) = (fetch p s (case r of [] \<Rightarrow> Bk |     

                                                               x # xs \<Rightarrow> x)) in

                       (ns, new_tape ac (l, r)))"

text {*

  The many-step function.

*}

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

  where

  "steps c p 0 = c" |

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

lemma tstep_red: "steps c p (Suc n) = tstep (steps c p n) p"

proof(induct n arbitrary: c)

  fix c

  show "steps c p (Suc 0) = tstep (steps c p 0) p" by(simp add: steps.simps)

next

  fix n c

  assume ind: "\<And> c. steps c p (Suc n) = tstep (steps c p n) p"

  have "steps (tstep c p) p (Suc n) = tstep (steps (tstep c p) p n) p"

    by(rule ind)

  thus "steps c p (Suc (Suc n)) = tstep (steps c p (Suc n)) p" by(simp add: steps.simps)

qed

declare Let_def[simp] option.split[split]

definition 

  "iseven n \<equiv> \<exists> x. n = 2 * x"

text {*

  The following @{text "t_correct"} function is used to specify the wellformedness of Turing

  machine.

*}

fun t_correct :: "tprog \<Rightarrow> bool"

  where

  "t_correct p = (length p \<ge> 2 \<and> iseven (length p) \<and> 

                   list_all (\<lambda> (acn, s). s \<le> length p div 2) p)"

declare t_correct.simps[simp del]

lemma allimp: "\<lbrakk>\<forall>x. P x \<longrightarrow> Q x; \<forall>x. P x\<rbrakk> \<Longrightarrow> \<forall>x. Q x"

by(auto elim: allE)

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)"

apply(rule exCI, drule allimp, auto)

apply(drule_tac f = f  in wf_inv_image, simp add: inv_image_def)

apply(erule wf_induct, auto)

done

lemma steps_add: "steps c t (x + y) = steps (steps c t x) t y"

by(induct x arbitrary: c, auto simp: steps.simps tstep_red)

lemma listall_set: "list_all p t \<Longrightarrow> \<forall> a \<in> set t. p a"

by(induct t, auto)

lemma fetch_ex: "\<exists>b a. fetch T aa ab = (b, a)"

by(simp add: fetch.simps)

definition exponent :: "'a \<Rightarrow> nat \<Rightarrow> 'a list" ("_\<^bsup>_\<^esup>" [0, 0]100)

  where "exponent x n = replicate n x"

text {* 

  @{text "tinres l1 l2"} means left list @{text "l1"} is congruent with

  @{text "l2"} with respect to the execution of Turing machine. 

  Appending Blank to the right of eigther one does not affect the 

  outcome of excution. 

*}

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

  where

  "tinres bx by = (\<exists> n. bx = by@Bk\<^bsup>n\<^esup> \<or> by = bx @ Bk\<^bsup>n\<^esup>)"

lemma exp_zero: "a\<^bsup>0\<^esup> = []"

by(simp add: exponent_def)

lemma exp_ind_def: "a\<^bsup>Suc x \<^esup> = a # a\<^bsup>x\<^esup>"

by(simp add: exponent_def)

text {*

  The following lemma shows the meaning of @{text "tinres"} with respect to 

  one step execution.

  *}

lemma tinres_step: 

  "\<lbrakk>tinres l l'; tstep (ss, l, r) t = (sa, la, ra); tstep (ss, l', r) t = (sb, lb, rb)\<rbrakk>

    \<Longrightarrow> tinres la lb \<and> ra = rb \<and> sa = sb"

apply(auto simp: tstep.simps fetch.simps new_tape.simps 

        split: if_splits taction.splits list.splits

                 block.splits)

apply(case_tac [!] "t ! (2 * (ss - Suc 0))", 

     auto simp: exponent_def tinres_def split: if_splits taction.splits list.splits

                 block.splits)

apply(case_tac [!] "t ! (2 * (ss - Suc 0) + Suc 0)", 

     auto simp: exponent_def tinres_def split: if_splits taction.splits list.splits

                 block.splits)

done

declare tstep.simps[simp del] steps.simps[simp del]

text {*

  The following lemma shows the meaning of @{text "tinres"} with respect to 

  many step execution.

  *}

lemma tinres_steps: 

  "\<lbrakk>tinres l l'; steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l', r) t stp = (sb, lb, rb)\<rbrakk>

    \<Longrightarrow> tinres la lb \<and> ra = rb \<and> sa = sb"

apply(induct stp arbitrary: sa la ra sb lb rb, simp add: steps.simps)

apply(simp add: tstep_red)

apply(case_tac "(steps (ss, l, r) t stp)")

apply(case_tac "(steps (ss, l', r) t stp)")

proof -

  fix stp sa la ra sb lb rb a b c aa ba ca

  assume ind: "\<And>sa la ra sb lb rb. \<lbrakk>steps (ss, l, r) t stp = (sa, la, ra); 

          steps (ss, l', r) t stp = (sb, lb, rb)\<rbrakk> \<Longrightarrow> tinres la lb \<and> ra = rb \<and> sa = sb"

  and h: " tinres l l'" "tstep (steps (ss, l, r) t stp) t = (sa, la, ra)"

         "tstep (steps (ss, l', r) t stp) t = (sb, lb, rb)" "steps (ss, l, r) t stp = (a, b, c)" 

         "steps (ss, l', r) t stp = (aa, ba, ca)"

  have "tinres b ba \<and> c = ca \<and> a = aa"

    apply(rule_tac ind, simp_all add: h)

    done

  thus "tinres la lb \<and> ra = rb \<and> sa = sb"

    apply(rule_tac l = b and l' = ba and r = c  and ss = a   

            and t = t in tinres_step)

    using h

    apply(simp, simp, simp)

    done

qed

text {*

  The following function @{text "tshift tp n"} is used to shift Turing programs 

  @{text "tp"} by @{text "n"} when it is going to be combined with others.

  *}

fun tshift :: "tprog \<Rightarrow> nat \<Rightarrow> tprog"

  where

  "tshift tp off = (map (\<lambda> (action, state). (action, (if state = 0 then 0

                                                      else state + off))) tp)"

text {*

  When two Turing programs are combined, the end state (state @{text "0"}) of the one 

  at the prefix position needs to be connected to the start state 

  of the one at postfix position. If @{text "tp"} is the Turing program

  to be at the prefix, @{text "change_termi_state tp"} is the transformed Turing program.

  *}

fun change_termi_state :: "tprog \<Rightarrow> tprog"

  where

  "change_termi_state t = 

       (map (\<lambda> (acn, ns). if ns = 0 then (acn, Suc ((length t) div 2)) else (acn, ns)) t)"

text {*

  @{text "t_add tp1 tp2"} is the combined Truing program.

*}

fun t_add :: "tprog \<Rightarrow> tprog \<Rightarrow> tprog" ("_ |+| _" [0, 0] 100)

  where

  "t_add t1 t2 = ((change_termi_state t1) @ (tshift t2 ((length t1) div 2)))"

text {*

  Tests whether the current configuration is at state @{text "0"}.

*}

definition isS0 :: "t_conf \<Rightarrow> bool"

  where

  "isS0 c = (let (s, l, r) = c in s = 0)"

declare tstep.simps[simp del] steps.simps[simp del] 

        t_add.simps[simp del] fetch.simps[simp del]

        new_tape.simps[simp del]

text {*

  Single step execution starting from state @{text "0"} will not make any progress.

*}

lemma tstep_0: "tstep (0, tp) p = (0, tp)"

apply(simp add: tstep.simps fetch.simps new_tape.simps)

done

text {*

  Many step executions starting from state @{text "0"} will not make any progress.

*}

lemma steps_0: "steps (0, tp) p stp = (0, tp)"

apply(induct stp)

apply(simp add: steps.simps)

apply(simp add: tstep_red tstep_0)

done

lemma s_keep_step: "\<lbrakk>a \<le> length A div 2; tstep (a, b, c) A = (s, l, r); t_correct A\<rbrakk>

  \<Longrightarrow> s \<le> length A div 2"

apply(simp add: tstep.simps fetch.simps t_correct.simps iseven_def 

  split: if_splits block.splits list.splits)

apply(case_tac [!] a, auto simp: list_all_length)

apply(erule_tac x = "2 * nat" in allE, auto)

apply(erule_tac x = "2 * nat" in allE, auto)

apply(erule_tac x = "Suc (2 * nat)" in allE, auto)

done

lemma s_keep: "\<lbrakk>steps (Suc 0, tp) A stp = (s, l, r);  t_correct A\<rbrakk> \<Longrightarrow> s \<le> length A div 2"

proof(induct stp arbitrary: s l r)

  case 0 thus "?case" by(auto simp: t_correct.simps steps.simps)

next

  fix stp s l r

  assume ind: "\<And>s l r. \<lbrakk>steps (Suc 0, tp) A stp = (s, l, r); t_correct A\<rbrakk> \<Longrightarrow> s \<le> length A div 2"

  and h1: "steps (Suc 0, tp) A (Suc stp) = (s, l, r)"

  and h2: "t_correct A"

  from h1 h2 show "s \<le> length A div 2"

  proof(simp add: tstep_red, cases "(steps (Suc 0, tp) A stp)", simp)

    fix a b c 

    assume h3: "tstep (a, b, c) A = (s, l, r)"

    and h4: "steps (Suc 0, tp) A stp = (a, b, c)"

    have "a \<le> length A div 2"

      using h2 h4

      by(rule_tac l = b and r = c in ind, auto)

    thus "?thesis"

      using h3 h2

      by(simp add: s_keep_step)

  qed

qed

lemma t_merge_fetch_pre:

  "\<lbrakk>fetch A s b = (ac, ns); s \<le> length A div 2; t_correct A; s \<noteq> 0\<rbrakk> \<Longrightarrow> 

  fetch (A |+| B) s b = (ac, if ns = 0 then Suc (length A div 2) else ns)"

apply(subgoal_tac "2 * (s - Suc 0) < length A \<and> Suc (2 * (s - Suc 0)) < length A")

apply(auto simp: fetch.simps t_add.simps split: if_splits block.splits)

apply(simp_all add: nth_append change_termi_state.simps)

done

lemma [simp]:  "\<lbrakk>\<not> a \<le> length A div 2; t_correct A\<rbrakk> \<Longrightarrow> fetch A a b = (Nop, 0)"

apply(auto simp: fetch.simps del: nth_of.simps split: block.splits)

apply(case_tac [!] a, auto simp: t_correct.simps iseven_def)

done

lemma  [elim]: "\<lbrakk>t_correct A; \<not> isS0 (tstep (a, b, c) A)\<rbrakk> \<Longrightarrow> a \<le> length A div 2"

apply(rule_tac classical, auto simp: tstep.simps new_tape.simps isS0_def)

done

lemma [elim]: "\<lbrakk>t_correct A; \<not> isS0 (tstep (a, b, c) A)\<rbrakk> \<Longrightarrow> 0 < a"

apply(rule_tac classical, simp add: tstep_0 isS0_def)

done

lemma t_merge_pre_eq_step: "\<lbrakk>tstep (a, b, c) A = cf; t_correct A; \<not> isS0 cf\<rbrakk> 

        \<Longrightarrow> tstep (a, b, c) (A |+| B) = cf"

apply(subgoal_tac "a \<le> length A div 2 \<and> a \<noteq> 0")

apply(simp add: tstep.simps)

apply(case_tac "fetch A a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)

apply(drule_tac B = B in t_merge_fetch_pre, simp, simp, simp, simp add: isS0_def, auto)

done

lemma t_merge_pre_eq:  "\<lbrakk>steps (Suc 0, tp) A stp = cf; \<not> isS0 cf; t_correct A\<rbrakk>

    \<Longrightarrow> steps (Suc 0, tp) (A |+| B) stp = cf"

proof(induct stp arbitrary: cf)

  case 0 thus "?case" by(simp add: steps.simps)

next

  fix stp cf

  assume ind: "\<And>cf. \<lbrakk>steps (Suc 0, tp) A stp = cf; \<not> isS0 cf; t_correct A\<rbrakk> 

                 \<Longrightarrow> steps (Suc 0, tp) (A |+| B) stp = cf"

  and h1: "steps (Suc 0, tp) A (Suc stp) = cf"

  and h2: "\<not> isS0 cf"

  and h3: "t_correct A"

  from h1 h2 h3 show "steps (Suc 0, tp) (A |+| B) (Suc stp) = cf"

  proof(simp add: tstep_red, cases "steps (Suc 0, tp) (A) stp", simp)

    fix a b c

    assume h4: "tstep (a, b, c) A = cf"

    and h5: "steps (Suc 0, tp) A stp = (a, b, c)"

    have "steps (Suc 0, tp) (A |+| B) stp = (a, b, c)"

    proof(cases a)

      case 0 thus "?thesis"

        using h4 h2

        apply(simp add: tstep_0, cases cf, simp add: isS0_def)

        done

    next

      case (Suc n) thus "?thesis"

        using h5 h3

        apply(rule_tac ind, auto simp: isS0_def)

        done

    qed

    thus "tstep (steps (Suc 0, tp) (A |+| B) stp) (A |+| B) = cf"

      using h4 h5 h3 h2

      apply(simp)

      apply(rule t_merge_pre_eq_step, auto)

      done

  qed

qed

declare nth.simps[simp del] tshift.simps[simp del] change_termi_state.simps[simp del]

lemma [simp]: "length (change_termi_state A) = length A"

by(simp add: change_termi_state.simps)

lemma first_halt_point: "steps (Suc 0, tp) A stp = (0, tp')

 \<Longrightarrow> \<exists>stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> steps (Suc 0, tp) A (Suc stp) = (0, tp')"

proof(induct stp)

  case 0  thus "?case" by(simp add: steps.simps)

next

  case (Suc n) 

  fix stp

  assume ind: "steps (Suc 0, tp) A stp = (0, tp') \<Longrightarrow> 

       \<exists>stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> steps (Suc 0, tp) A (Suc stp) = (0, tp')"

    and h: "steps (Suc 0, tp) A (Suc stp) = (0, tp')"

  from h show "\<exists>stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> steps (Suc 0, tp) A (Suc stp) = (0, tp')"

  proof(simp add: tstep_red, cases "steps (Suc 0, tp) A stp", simp, case_tac a)

    fix a b c

    assume g1: "a = (0::nat)"

    and g2: "tstep (a, b, c) A = (0, tp')"

    and g3: "steps (Suc 0, tp) A stp = (a, b, c)"

    have "steps (Suc 0, tp) A stp = (0, tp')"

      using g2 g1 g3

      by(simp add: tstep_0)

    hence "\<exists> stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> steps (Suc 0, tp) A (Suc stp) = (0, tp')"

      by(rule ind)

    thus "\<exists>stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> tstep (steps (Suc 0, tp) A stp) A = (0, tp')" 

      apply(simp add: tstep_red)

      done

  next

    fix a b c nat

    assume g1: "steps (Suc 0, tp) A stp = (a, b, c)"

    and g2: "steps (Suc 0, tp) A (Suc stp) = (0, tp')" "a= Suc nat"

    thus "\<exists>stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> tstep (steps (Suc 0, tp) A stp) A = (0, tp')"

      apply(rule_tac x = stp in exI)

      apply(simp add: isS0_def tstep_red)

      done

  qed

qed 

lemma t_merge_pre_halt_same': 

  "\<lbrakk>\<not> isS0 (steps (Suc 0, tp) A stp) ; steps (Suc 0, tp) A (Suc stp) = (0, tp'); t_correct A\<rbrakk>

  \<Longrightarrow> steps (Suc 0, tp) (A |+| B) (Suc stp) = (Suc (length A div 2), tp')"    

proof(simp add: tstep_red, cases "steps (Suc 0, tp) A stp", simp)

  fix a b c 

  assume h1: "\<not> isS0 (a, b, c)"

  and h2: "tstep (a, b, c) A = (0, tp')"

  and h3: "t_correct A"

  and h4: "steps (Suc 0, tp) A stp = (a, b, c)"

  have "steps (Suc 0, tp) (A |+| B) stp = (a, b, c)"

    using h1 h4 h3

    apply(rule_tac  t_merge_pre_eq, auto)

    done

  moreover have "tstep (a, b, c) (A |+| B) = (Suc (length A div 2), tp')"

    using h2 h3 h1 h4 

    apply(simp add: tstep.simps)

    apply(case_tac " fetch A a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)

    apply(drule_tac B = B in t_merge_fetch_pre, auto simp: isS0_def intro: s_keep)

    done

  ultimately show "tstep (steps (Suc 0, tp) (A |+| B) stp) (A |+| B) = (Suc (length A div 2), tp')"

    by(simp)

qed

text {*

  When Turing machine @{text "A"} and @{text "B"} are combined and the execution

  of @{text "A"} can termination within @{text "stp"} steps, 

  the combined machine @{text "A |+| B"} will eventually get into the starting 

  state of machine @{text "B"}.

*}

lemma t_merge_pre_halt_same: "

  \<lbrakk>steps (Suc 0, tp) A stp = (0, tp'); t_correct A; t_correct B\<rbrakk>

     \<Longrightarrow> \<exists> stp. steps (Suc 0, tp) (A |+| B) stp = (Suc (length A div 2), tp')"

proof -

  assume a_wf: "t_correct A"

  and b_wf: "t_correct B"

  and a_ht: "steps (Suc 0, tp) A stp = (0, tp')"

  have halt_point: "\<exists> stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> steps (Suc 0, tp) A (Suc stp) = (0, tp')"

    using a_ht

    by(erule_tac first_halt_point)

  then obtain stp' where "\<not> isS0 (steps (Suc 0, tp) A stp') \<and> steps (Suc 0, tp) A (Suc stp') = (0, tp')"..

  hence "steps (Suc 0, tp) (A |+| B) (Suc stp') = (Suc (length A div 2), tp')"

    using a_wf

    apply(rule_tac t_merge_pre_halt_same', auto)

    done

  thus "?thesis" ..

qed

lemma fetch_0: "fetch p 0 b = (Nop, 0)"

by(simp add: fetch.simps)

lemma [simp]: "length (tshift B x) = length B"

by(simp add: tshift.simps)

lemma [simp]: "t_correct A \<Longrightarrow> 2 * (length A div 2) = length A"

apply(simp add: t_correct.simps iseven_def, auto)

done

lemma t_merge_fetch_snd: 

  "\<lbrakk>fetch B a b = (ac, ns); t_correct A; t_correct B; a > 0 \<rbrakk>

  \<Longrightarrow> fetch (A |+| B) (a + length A div 2) b

  = (ac, if ns = 0 then 0 else ns + length A div 2)"

apply(auto simp: fetch.simps t_add.simps split: if_splits block.splits)

apply(case_tac [!] a, simp_all)

apply(simp_all add: nth_append change_termi_state.simps tshift.simps)

done

lemma t_merge_snd_eq_step: 

  "\<lbrakk>tstep (s, l, r) B = (s', l', r'); t_correct A; t_correct B; s > 0\<rbrakk>

    \<Longrightarrow> tstep (s + length A div 2, l, r) (A |+| B) = 

       (if s' = 0 then 0 else s' + length A div 2, l' ,r') "

apply(simp add: tstep.simps)

apply(cases "fetch B s (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)")

apply(auto simp: t_merge_fetch_snd)

apply(frule_tac [!] t_merge_fetch_snd, auto)

done 

text {*

  Relates the executions of TM @{text "B"}, one is when @{text "B"} is executed alone,

  the other is the execution when @{text "B"} is in the combined TM.

*}

lemma t_merge_snd_eq_steps: 

  "\<lbrakk>t_correct A; t_correct B; steps (s, l, r) B stp = (s', l', r'); s > 0\<rbrakk>

  \<Longrightarrow> steps (s + length A div 2, l, r) (A |+| B) stp = 

      (if s' = 0 then 0 else s' + length A div 2, l', r')"