--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys/Turing.thy	Sun Feb 10 19:49:07 2013 +0000
@@ -0,0 +1,428 @@
+(* Title: Turing machines
+   Author: Xu Jian <xujian817@hotmail.com>
+   Maintainer: Xu Jian
+*)
+
+theory Turing
+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 after_is_final:
+  assumes "is_final c"
+  shows "is_final (steps c p n)"
+using assms 
+apply(induct n) 
+apply(case_tac [!] c)
+apply(auto)
+done
+
+lemma not_is_final:
+  assumes a: "\<not> is_final (steps c p n1)"
+  and b: "n2 \<le> n1"
+  shows "\<not> is_final (steps c p n2)"
+proof (rule notI)
+  obtain n3 where eq: "n1 = n2 + n3" using b by (metis le_iff_add)
+  assume "is_final (steps c p n2)"
+  then have "is_final (steps c p n1)" unfolding eq
+    by (simp add: after_is_final)
+  with a show "False" by simp
+qed
+
+(* 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 *)
+abbreviation "is_even n \<equiv> (n::nat) mod 2 = 0"
+
+fun 
+  tm_wf :: "tprog \<Rightarrow> bool"
+where
+  "tm_wf (p, off) = (length p \<ge> 2 \<and> is_even (length p) \<and> 
+                    (\<forall>(a, s) \<in> set p. s \<le> length p div 2 + off \<and> s \<ge> off))"
+
+abbreviation
+  "tm_wf0 p \<equiv> tm_wf (p, 0)"
+
+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)
+
+defs (overloaded)
+  tape_of_nat_abv: "<(n::nat)> \<equiv> Oc \<up> (Suc n)"
+
+fun tape_of_nat_list :: "'a list \<Rightarrow> cell list" 
+  where 
+  "tape_of_nat_list [] = []" |
+  "tape_of_nat_list [n] = <n>" |
+  "tape_of_nat_list (n#ns) = <n> @ Bk # (tape_of_nat_list ns)"
+
+fun tape_of_nat_pair :: "'a \<times> 'b \<Rightarrow> cell list" 
+  where 
+  "tape_of_nat_pair (n, m) = <n> @ [Bk] @ <m>" 
+
+
+defs (overloaded)
+  tape_of_nl_abv: "<(ns::'a list)> \<equiv> tape_of_nat_list ns"
+  tape_of_nat_pair: "<(np::'a\<times>'b)> \<equiv> tape_of_nat_pair np"
+
+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
+
+lemma tm_comp_wf[intro]: 
+  "\<lbrakk>tm_wf (A, 0); tm_wf (B, 0)\<rbrakk> \<Longrightarrow> tm_wf (A |+| B, 0)"
+by (auto simp: tm_wf.simps shift.simps adjust.simps tm_comp_length tm_comp.simps)
+
+
+lemma tm_comp_step: 
+  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
+
+lemma tm_comp_steps:  
+  assumes "\<not> is_final (steps0 c A n)" 
+  shows "steps0 c (A |+| B) n = steps0 c A n"
+using assms
+proof(induct n)
+  case 0
+  then show "steps0 c (A |+| B) 0 = steps0 c A 0" by auto
+next 
+  case (Suc n)
+  have ih: "\<not> is_final (steps0 c A n) \<Longrightarrow> steps0 c (A |+| B) n = steps0 c A n" by fact
+  have fin: "\<not> is_final (steps0 c A (Suc n))" by fact
+  then have fin1: "\<not> is_final (step0 (steps0 c A n) A)" 
+    by (auto simp only: step_red)
+  then have fin2: "\<not> is_final (steps0 c A n)"
+    by (metis is_final_eq step_0 surj_pair) 
+ 
+  have "steps0 c (A |+| B) (Suc n) = step0 (steps0 c (A |+| B) n) (A |+| B)" 
+    by (simp only: step_red)
+  also have "... = step0 (steps0 c A n) (A |+| B)" by (simp only: ih[OF fin2])
+  also have "... = step0 (steps0 c A n) A" by (simp only: tm_comp_step[OF fin1])
+  finally show "steps0 c (A |+| B) (Suc n) = steps0 c A (Suc n)"
+    by (simp only: step_red)
+qed
+
+lemma tm_comp_fetch_in_A:
+  assumes h1: "fetch A s x = (a, 0)"
+  and h2: "s \<le> length A div 2" 
+  and h3: "s \<noteq> 0"
+  shows "fetch (A |+| B) s x = (a, Suc (length A div 2))"
+using h1 h2 h3
+apply(case_tac s)
+apply(case_tac [!] x)
+apply(auto simp: tm_comp_length nth_append)
+done
+
+lemma tm_comp_exec_after_first:
+  assumes h1: "\<not> is_final c" 
+  and h2: "step0 c A = (0, tp)"
+  and h3: "fst c \<le> length A div 2"
+  shows "step0 c (A |+| B) = (Suc (length A div 2), tp)"
+using h1 h2 h3
+apply(case_tac c)
+apply(auto simp del: tm_comp.simps)
+apply(case_tac "fetch A a Bk")
+apply(simp del: tm_comp.simps)
+apply(subst tm_comp_fetch_in_A)
+apply(auto)[4]
+apply(case_tac "fetch A a (hd c)")
+apply(simp del: tm_comp.simps)
+apply(subst tm_comp_fetch_in_A)
+apply(auto)[4]
+done
+
+lemma step_in_range: 
+  assumes h1: "\<not> is_final (step0 c A)"
+  and h2: "tm_wf (A, 0)"
+  shows "fst (step0 c A) \<le> length A div 2"
+using h1 h2
+apply(case_tac c)
+apply(case_tac a)
+apply(auto simp add: prod_case_unfold Let_def)
+apply(case_tac "hd c")
+apply(auto simp add: prod_case_unfold)
+done
+
+lemma steps_in_range: 
+  assumes h1: "\<not> is_final (steps0 (1, tp) A stp)"
+  and h2: "tm_wf (A, 0)"
+  shows "fst (steps0 (1, tp) A stp) \<le> length A div 2"
+using h1
+proof(induct stp)
+  case 0
+  then show "fst (steps0 (1, tp) A 0) \<le> length A div 2" using h2
+    by (auto simp add: steps.simps tm_wf.simps)
+next
+  case (Suc stp)
+  have ih: "\<not> is_final (steps0 (1, tp) A stp) \<Longrightarrow> fst (steps0 (1, tp) A stp) \<le> length A div 2" by fact
+  have h: "\<not> is_final (steps0 (1, tp) A (Suc stp))" by fact
+  from ih h h2 show "fst (steps0 (1, tp) A (Suc stp)) \<le> length A div 2"
+    by (metis step_in_range step_red)
+qed
+
+lemma tm_comp_pre_halt_same: 
+  assumes a_ht: "steps0 (1, tp) A n = (0, tp')"
+  and a_wf: "tm_wf (A, 0)"
+  obtains n' where "steps0 (1, tp) (A |+| B) n' = (Suc (length A div 2), tp')"
+proof -
+  assume a: "\<And>n. steps (1, tp) (A |+| B, 0) n = (Suc (length A div 2), tp') \<Longrightarrow> thesis"
+  obtain stp' where fin: "\<not> is_final (steps0 (1, tp) A stp')" and h: "steps0 (1, tp) A (Suc stp') = (0, tp')"
+  using before_final[OF a_ht] by blast
+  from fin have h1:"steps0 (1, tp) (A |+| B) stp' = steps0 (1, tp) A stp'"
+    by (rule tm_comp_steps)
+  from h have h2: "step0 (steps0 (1, tp) A stp') A = (0, tp')"
+    by (simp only: step_red)
+
+  have "steps0 (1, tp) (A |+| B) (Suc stp') = step0 (steps0 (1, tp) (A |+| B) stp') (A |+| B)" 
+    by (simp only: step_red)
+  also have "... = step0 (steps0 (1, tp) A stp') (A |+| B)" using h1 by simp
+  also have "... = (Suc (length A div 2), tp')" 
+    by (rule tm_comp_exec_after_first[OF fin h2 steps_in_range[OF fin a_wf]])
+  finally show thesis using a by blast
+qed
+
+lemma tm_comp_fetch_second_zero:
+  assumes h1: "fetch B s x = (a, 0)"
+  and hs: "tm_wf (A, 0)" "s \<noteq> 0"
+  shows "fetch (A |+| B) (s + (length A div 2)) x = (a, 0)"
+using h1 hs
+apply(case_tac x)
+apply(case_tac [!] s)
+apply(auto simp: tm_comp_length nth_append)
+done 
+
+lemma tm_comp_fetch_second_inst:
+  assumes h1: "fetch B sa x = (a, s)"
+  and hs: "tm_wf (A, 0)" "sa \<noteq> 0" "s \<noteq> 0"
+  shows "fetch (A |+| B) (sa + length A div 2) x = (a, s + length A div 2)"
+using h1 hs
+apply(case_tac x)
+apply(case_tac [!] sa)
+apply(auto simp: tm_comp_length nth_append)
+done 
+
+
+lemma tm_comp_second_same:
+  assumes a_wf: "tm_wf (A, 0)"
+  and steps: "steps0 (1, l, r) B stp = (s', l', r')"
+  shows "steps0 (Suc (length A div 2), l, r)  (A |+| B) stp 
+    = (if s' = 0 then 0 else s' + length A div 2, l', r')"
+using steps
+proof(induct stp arbitrary: s' l' r')
+  case 0
+  then show ?case by (simp add: steps.simps)
+next
+  case (Suc stp s' l' r')
+  obtain s'' l'' r'' where a: "steps0 (1, l, r) B stp = (s'', l'', r'')"
+    by (metis is_final.cases)
+  then have ih1: "s'' = 0 \<Longrightarrow> steps0 (Suc (length A div 2), l, r) (A |+| B) stp = (0, l'', r'')"
+  and ih2: "s'' \<noteq> 0 \<Longrightarrow> steps0 (Suc (length A div 2), l, r) (A |+| B) stp = (s'' + length A div 2, l'', r'')"
+    using Suc by (auto)
+  have h: "steps0 (1, l, r) B (Suc stp) = (s', l', r')" by fact
+
+  { assume "s'' = 0"
+    then have ?case using a h ih1 by (simp del: steps.simps) 
+  } moreover
+  { assume as: "s'' \<noteq> 0" "s' = 0"
+    from as a h 
+    have "step0 (s'', l'', r'') B = (0, l', r')" by (simp del: steps.simps)
+    with as have ?case
+    apply(simp add: ih2[OF as(1)] step.simps del: tm_comp.simps steps.simps)
+    apply(case_tac "fetch B s'' (read r'')")
+    apply(auto simp add: tm_comp_fetch_second_zero[OF _ a_wf] simp del: tm_comp.simps)
+    done
+  } moreover
+  { assume as: "s'' \<noteq> 0" "s' \<noteq> 0"
+    from as a h
+    have "step0 (s'', l'', r'') B = (s', l', r')" by (simp del: steps.simps)
+    with as have ?case
+    apply(simp add: ih2[OF as(1)] step.simps del: tm_comp.simps steps.simps)
+    apply(case_tac "fetch B s'' (read r'')")
+    apply(auto simp add: tm_comp_fetch_second_inst[OF _ a_wf as] simp del: tm_comp.simps)
+    done
+  }
+  ultimately show ?case by blast
+qed
+
+lemma tm_comp_second_halt_same:
+  assumes "tm_wf (A, 0)"  
+  and "steps0 (1, l, r) B stp = (0, l', r')"
+  shows "steps0 (Suc (length A div 2), l, r)  (A |+| B) stp = (0, l', r')"
+using tm_comp_second_same[OF assms] by (simp)
+
+end
+