(* Title: Turing machines

   Author: Xu Jian <xujian817@hotmail.com>

   Maintainer: Xu Jian

*)

theory turing_basic

imports Main

begin

section {* Basic definitions of Turing machine *}

definition 

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

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"

type_synonym config = "state \<times> tape"

fun nth_of where

  "nth_of [] _ = None"

| "nth_of (x # xs) 0 = Some x"

| "nth_of (x # xs) (Suc n) = nth_of xs n" 

fun 

  fetch :: "tprog \<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))"

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 = (let (a, s') = fetch p s (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"

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)

definition 

  tm_wf :: "tprog \<Rightarrow> bool"

where

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

(* FIXME: needed? *)

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"

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

where

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

lemma [simp]: 

  "length (shift p n) = length p"

by (simp)

fun 

  adjust :: "tprog \<Rightarrow> tprog"

where

  "adjust p = (map (\<lambda> (a, s). (a, if s = 0 then ((length p) div 2) + 1 else s)) p)"

lemma [simp]: 

  shows "length (adjust p) = length p"

by (simp)

definition

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

where

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

fun

  is_final :: "config \<Rightarrow> bool"

where

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

fun 

  holds_for :: "(tape \<Rightarrow> bool) \<Rightarrow> config \<Rightarrow> bool" ("_ holds'_for _" [100, 99] 100)

where

  "P holds_for (s, l, r) = P (l, r)"  

lemma step_0 [simp]: 

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

by simp

lemma steps_0 [simp]: 

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

by (induct n) (simp_all)

type_synonym assert = "tape \<Rightarrow> bool"

definition assert_imp :: "assert \<Rightarrow> assert \<Rightarrow> bool" ("_ \<mapsto> _" [0, 0] 100)

  where

  "assert_imp P Q = (\<forall>l r. P (l, r) \<longrightarrow> Q (l, r))"

definition

  Hoare :: "assert \<Rightarrow> tprog \<Rightarrow> assert \<Rightarrow> bool" ("({(1_)}/ (_)/ {(1_)})" 50)

where

  "{P} p {Q} \<equiv> 

     (\<forall>l r. P (l, r) \<longrightarrow> (\<exists>n. is_final (steps (1, (l, r)) p n) \<and> Q holds_for (steps (1, (l, r)) p n)))"

lemma HoareI:

  assumes "\<And>l r. P (l, r) \<Longrightarrow> \<exists>n. is_final (steps (1, (l, r)) p n) \<and> Q holds_for (steps (1, (l, r)) p n)"

  shows "{P} p {Q}"

unfolding Hoare_def using assms by auto

lemma HoareI2:

  assumes "\<And>l r n. P (l, r) \<and> is_final (steps (1, (l, r)) p n) \<and> Q holds_for (steps (1, (l, r)) p n)"

  shows "{P} p {Q}"

unfolding Hoare_def using assms by auto

text {*

{P1} A {Q1}   {P2} B {Q2}  Q1 \<mapsto> P2

-----------------------------------

    {P1} A |+| B {Q2}

*}

lemma Hoare_plus: 

  assumes aimpb: "Q1 \<mapsto> P2"

  and A_wf : "tm_wf A"

  and B_wf : "tm_wf B"

  and A_halt : "{P1} A {Q1}"

  and B_halt : "{P2} B {Q2}"

  shows "{P1} A |+| B {Q2}"

proof(rule HoareI)

  fix a b

  assume h: "P1 (a, b)"

  hence "\<exists>n. let (s, tp') = steps (Suc 0, a, b) A n in s = 0 \<and> Q1 tp'"

    using A_halt unfolding hoare_def by simp

  from this obtain stp1 where "let (s, tp') = steps (Suc 0, a, b) A stp1 in s = 0 \<and> Q1 tp'" ..

  then show "\<exists>n. case steps (Suc 0, a, b) (A |+| B) n of (s, tp') \<Rightarrow> s = 0 \<and> Q2 tp'"

  proof(case_tac "steps (Suc 0, a, b) A stp1", simp, erule_tac conjE)

    fix aa ba c

    assume g1: "Q1 (ba, c)" 

      and g2: "steps (Suc 0, a, b) A stp1 = (0, ba, c)"

    hence "P2 (ba, c)"

      using aimpb apply(simp add: assert_imp_def)

      done

    hence "\<exists> stp. let (s, tp') = steps (Suc 0, ba, c) B stp in s = 0 \<and> Q2 tp'"

      using B_halt unfolding hoare_def by simp

    from this obtain stp2 where "let (s, tp') = steps (Suc 0, ba, c) B stp2 in s = 0 \<and> Q2 tp'" ..

    thus "?thesis"

    proof(case_tac "steps (Suc 0, ba, c) B stp2", simp, erule_tac conjE)

      fix aa bb ca

      assume g3: " Q2 (bb, ca)" "steps (Suc 0, ba, c) B stp2 = (0, bb, ca)"

      have "\<exists> stp. steps (Suc 0, a, b) (A |+| B) stp = (Suc (length A div 2), ba , c)"

        using g2 A_wf B_wf

        sorry

      moreover have "\<exists> stp. steps (Suc (length A div 2), ba, c) (A |+| B) stp = (0, bb, ca)"

        using g3 A_wf B_wf

        sorry

      ultimately show "\<exists>n. case steps (Suc 0, a, b) (A |+| B) n of (s, tp') \<Rightarrow> s = 0 \<and> Q2 tp'"

        apply(erule_tac exE, erule_tac exE)

        apply(rule_tac x = "stp + stpa" in exI, simp)

        using g3 by simp

    qed

  qed

qed

lemma hoare_plus: 

  assumes A_wf : "tm_wf A"

  and B_wf : "tm_wf B"

  and A_halt : "{P1} A {Q1}"

  and B_halt : "{P2} B {Q2}"

  and aimpb: "Q1 \<mapsto> P2"

  shows "{P1} A |+| B {Q2}"

unfolding hoare_def

proof(auto split: )

  fix a b

  assume h: "P1 (a, b)"

  hence "\<exists>n. let (s, tp') = steps (Suc 0, a, b) A n in s = 0 \<and> Q1 tp'"

    using A_halt unfolding hoare_def by simp

  from this obtain stp1 where "let (s, tp') = steps (Suc 0, a, b) A stp1 in s = 0 \<and> Q1 tp'" ..

  then show "\<exists>n. case steps (Suc 0, a, b) (A |+| B) n of (s, tp') \<Rightarrow> s = 0 \<and> Q2 tp'"

  proof(case_tac "steps (Suc 0, a, b) A stp1", simp, erule_tac conjE)

    fix aa ba c

    assume g1: "Q1 (ba, c)" 

      and g2: "steps (Suc 0, a, b) A stp1 = (0, ba, c)"

    hence "P2 (ba, c)"

      using aimpb apply(simp add: assert_imp_def)

      done

    hence "\<exists> stp. let (s, tp') = steps (Suc 0, ba, c) B stp in s = 0 \<and> Q2 tp'"

      using B_halt unfolding hoare_def by simp

    from this obtain stp2 where "let (s, tp') = steps (Suc 0, ba, c) B stp2 in s = 0 \<and> Q2 tp'" ..

    thus "?thesis"

    proof(case_tac "steps (Suc 0, ba, c) B stp2", simp, erule_tac conjE)

      fix aa bb ca

      assume g3: " Q2 (bb, ca)" "steps (Suc 0, ba, c) B stp2 = (0, bb, ca)"

      have "\<exists> stp. steps (Suc 0, a, b) (A |+| B) stp = (Suc (length A div 2), ba , c)"

        using g2 A_wf B_wf

        sorry

      moreover have "\<exists> stp. steps (Suc (length A div 2), ba, c) (A |+| B) stp = (0, bb, ca)"

        using g3 A_wf B_wf

        sorry

      ultimately show "\<exists>n. case steps (Suc 0, a, b) (A |+| B) n of (s, tp') \<Rightarrow> s = 0 \<and> Q2 tp'"

        apply(erule_tac exE, erule_tac exE)

        apply(rule_tac x = "stp + stpa" in exI, simp)

        using g3 by simp

    qed

  qed

qed

locale turing_merge =

  fixes A :: "tprog" and B :: "tprog" and P1 :: "assert"

  and P2 :: "assert"

  and P3 :: "assert"

  and P4 :: "assert"

  and Q1:: "assert"

  and Q2 :: "assert"

  assumes 

  A_wf : "tm_wf A"

  and B_wf : "tm_wf B"

  and A_halt : "P1 tp \<Longrightarrow> \<exists> stp. let (s, tp') = steps (Suc 0, tp) A stp in s = 0 \<and> Q1 tp'"

  and B_halt : "P2 tp \<Longrightarrow> \<exists> stp. let (s, tp') = steps (Suc 0, tp) B stp in s = 0 \<and> Q2 tp'"

  and A_uhalt : "P3 tp \<Longrightarrow> \<not> (\<exists> stp. is_final (steps (Suc 0, tp) A stp))"

  and B_uhalt: "P4 tp \<Longrightarrow> \<not> (\<exists> stp. is_final (steps (Suc 0, tp) B stp))"

begin

text {*

  The following lemma tries to derive the Hoare logic rule for sequentially combined TMs.

  It deals with the situtation when both @{text "A"} and @{text "B"} are terminated.

*}

lemma  t_merge_uhalt_tmp:

  assumes B_uh: "\<forall>stp. \<not> is_final (steps (Suc 0, b, c) B stp)"

  and merge_ah: "steps (Suc 0, tp) (A |+| B) stpa = (Suc (length A div 2), b, c)" 

  shows "\<forall> stp. \<not> is_final (steps (Suc 0, tp) (A |+| B) stp)"

  using B_uh merge_ah

apply(rule_tac allI)

apply(case_tac "stp > stpa")

apply(erule_tac x = "stp - stpa" in allE)

apply(case_tac "(steps (Suc 0, b, c) B (stp - stpa))", simp)

proof -

  fix stp a ba ca 

  assume h1: "\<not> is_final (a, ba, ca)" "stpa < stp"

  and h2: "steps (Suc 0, b, c) B (stp - stpa) = (a, ba, ca)"

  have "steps (Suc 0 + length A div 2, b, c) (A |+| B) (stp - stpa) = 

      (if a = 0 then 0 else a + length A div 2, ba, ca)"

    using A_wf B_wf h2

    by(rule_tac t_merge_snd_eq_steps, auto)

  moreover have "a > 0" using h1 by(simp add: is_final_def)

  moreover have "\<exists> stpb. stp = stpa + stpb" 

    using h1 by(rule_tac x = "stp - stpa" in exI, simp)

  ultimately show "\<not> is_final (steps (Suc 0, tp) (A |+| B) stp)"

    using merge_ah

    by(auto simp: steps_add is_final_def)

next

  fix stp

  assume h: "steps (Suc 0, tp) (A |+| B) stpa = (Suc (length A div 2), b, c)" "\<not> stpa < stp"

  hence "\<exists> stpb. stpa = stp + stpb" apply(rule_tac x = "stpa - stp" in exI, auto) done

  thus "\<not> is_final (steps (Suc 0, tp) (A |+| B) stp)"

    using h

    apply(auto)

    apply(cases "steps (Suc 0, tp) (A |+| B) stp", simp add: steps_add is_final_def steps_0)

    done

qed

text {*

  The following lemma deals with the situation when TM @{text "B"} can not terminate.

  *}

lemma t_merge_uhalt: 

  assumes aimpb: "Q1 \<mapsto> P4"

  shows "P1 \<mapsto> \<lambda> tp. \<not> (\<exists> stp. is_final (steps (Suc 0, tp) (A |+| B) stp))"

proof(simp only: assert_imp_def, rule_tac allI, rule_tac impI)

  fix tp 

  assume init_asst: "P1 tp"

  show "\<not> (\<exists>stp. is_final (steps (Suc 0, tp) (A |+| B) stp))"

  proof -

    have "\<exists> stp. let (s, tp') = steps (Suc 0, tp) A stp in s = 0 \<and> Q1 tp'"

      using A_halt[of tp] init_asst

      by(simp)

    from this obtain stpx where "let (s, tp') = steps (Suc 0, tp) A stpx in s = 0 \<and> Q1 tp'" ..

    thus "?thesis"

    proof(cases "steps (Suc 0, tp) A stpx", simp, erule_tac conjE)

      fix a b c

      assume "Q1 (b, c)"

        and h3: "steps (Suc 0, tp) A stpx = (0, b, c)"

      hence h2: "P4 (b, c)"  using aimpb

        by(simp add: assert_imp_def)

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

        using h3 A_wf B_wf

        apply(rule_tac stp = stpx in t_merge_pre_halt_same, auto)

        done

      from this obtain stpa where h4:"steps (Suc 0, tp) (A |+| B) stpa = (Suc (length A div 2), b, c)" ..

      have " \<not> (\<exists> stp. is_final (steps (Suc 0, b, c) B stp))"

        using B_uhalt [of "(b, c)"] h2 apply simp

        done

      from this and h4 show "\<forall>stp. \<not> is_final (steps (Suc 0, tp) (A |+| B) stp)"

        by(rule_tac t_merge_uhalt_tmp, auto)

    qed

  qed

qed

end

end