(* 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" 

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 :: "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))"

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 = (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 length_shift [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 length_adjust[simp]: 
  shows "length (adjust p) = length p"
by (induct p) (auto)

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)"

lemma is_final_steps:
  assumes "is_final (s, l, r)"
  shows "is_final (steps (s, l, r) p n)"
using assms by (induct n) (auto)

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)

lemma is_final_holds[simp]:
  assumes "is_final c"
  shows "Q holds_for (steps c p n) = Q holds_for c"
using assms 
apply(induct n)
apply(case_tac [!] c)
apply(auto)
done

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))"

lemma holds_for_imp:
  assumes "P holds_for c"
  and "P \<mapsto> Q"
  shows "Q holds_for c"
using assms unfolding assert_imp_def by (case_tac c, auto)

lemma test:
  assumes "is_final (steps (1, (l, r)) p n1)"
  and     "is_final (steps (1, (l, r)) p n2)"
  shows "Q holds_for (steps (1, (l, r)) p n1) \<longleftrightarrow> Q holds_for (steps (1, (l, r)) p n2)"
proof -
  obtain n3 where "n1 = n2 + n3 \<or> n2 = n1 + n3"
    by (metis le_iff_add nat_le_linear)
  with assms show "Q holds_for (steps (1, (l, r)) p n1) \<longleftrightarrow> Q holds_for (steps (1, (l, r)) p n2)"  
    by auto
qed

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

text {*
{P1} A {Q1}   {P2} B {Q2}  Q1 \<mapsto> P2
-----------------------------------
    {P1} A |+| B {Q2}
*}

lemma before_final: 
  assumes "steps (1, tp) A n = (0, tp')"
  obtains n' where "\<not> is_final (steps (1, tp) A n')" and "steps (1, tp) A (Suc n') = (0, tp')"
using assms 
apply(induct n)
apply(auto)
by (metis is_final.simps step_red steps.simps steps_0 surj_pair)

lemma t_merge_fetch_pre:
  assumes "fetch A s b = (ac, ns)" "s \<le> length A div 2" "tm_wf A" "s \<noteq> 0" 
  shows "fetch (adjustA |+| B) s b = (ac, if ns = 0 then Suc (length A div 2) else ns)"
using assms
unfolding tm_comp_def
apply(induct A)
apply(auto)
apply(subgoal_tac "2 * (s - Suc 0) < length A \<and> Suc (2 * (s - Suc 0)) < length A")
apply(auto simp: tm_wf_def iseven_def tm_comp_def split: if_splits cell.splits)
oops

lemma t_merge_pre_eq_step: 
  "\<lbrakk>step (a, b, c) A = cf; tm_wf A; \<not> is_final cf\<rbrakk> 
        \<Longrightarrow> step (a, b, c) (A |+| B) = cf"
apply(subgoal_tac "a \<le> length A div 2 \<and> a \<noteq> 0")
apply(simp)
apply(case_tac "fetch A a (read c)", simp)
apply(auto)
oops

lemma t_merge_pre_eq:  
  "\<lbrakk>steps (Suc 0, tp) A stp = cf; \<not> is_final cf; tm_wf A\<rbrakk>
  \<Longrightarrow> steps (Suc 0, tp) (A |+| B) stp = cf"
apply(induct stp arbitrary: cf)
apply(auto)[1]
apply(auto)
oops

lemma t_merge_pre_halt_same: 
  assumes a_ht: "steps (1, tp) A n = (0, tp')"
  and a_wf: "t_correct A"
  obtains n' where "steps (1, tp) (A |+| B) n' = (Suc (length A div 2), tp')"
proof -
  assume a: "\<And>n. steps (1, tp) (A |+| B) n = (Suc (length A div 2), tp') \<Longrightarrow> thesis"
  
  obtain stp' where "\<not> is_final (steps (1, tp) A stp')" and "steps (1, tp) A (Suc stp') = (0, tp')"
  using a_ht before_final by blast
  then have "steps (1, tp) (A |+| B) (Suc stp') = (Suc (length A div 2), tp')"
    sorry (*using a_wf t_merge_pre_halt_same' by blast*)
  with a show thesis by blast
qed



lemma steps_comp:
  assumes a1: "steps (1, l, r) A n1 = (s1, l1, r1)"
  and a2: "steps (1, l1, r1) B n2 = (s2, l2, r2)"
  shows "steps (1, l, r) (A |+| B) (n1 + n2) = (s2, l2, r2)"
using assms
apply(induct n2)
apply(simp)
apply(simp add: tm_comp_def)
oops

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 l r
  assume h: "P1 (l, r)"
  then obtain n1 where a: "is_final (steps (1, l, r) A n1)" and b: "Q1 holds_for (steps (1, l, r) A n1)"
    using A_halt unfolding Hoare_def by auto
  from b aimpb have "P2 holds_for (steps (1, l, r) A n1)"
    by (rule holds_for_imp)
  then obtain l' r' where "P2 (l', r')"
    apply(auto)
    apply(case_tac "steps (Suc 0, l, r) A n1")
    apply(simp)
    done
  then obtain n2 where a: "is_final (steps (1, l', r') B n2)" and b: "Q2 holds_for (steps (1, l', r') B n2)"
    using B_halt unfolding Hoare_def by auto
  show "\<exists>n. is_final (steps (1, l, r) (A |+| B) n) \<and> Q2 holds_for (steps (1, l, r) (A |+| B) n)"
    apply(rule_tac x="n1 + n2" in exI)
    apply(rule conjI)
    apply(simp)
    apply(simp only: steps_add[symmetric])
    sorry
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