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thys/turing_basic.thy
changeset 41 6d89ed67ba27
parent 40 a37a21f7ccf4
child 43 a8785fa80278
equal deleted inserted replaced
40:a37a21f7ccf4 41:6d89ed67ba27 
    28 
    28 
    29 fun nth_of where
 
    29 fun nth_of where
 
    30   "nth_of [] _ = None"
 
    30   "nth_of [] _ = None"
 
    31 | "nth_of (x # xs) 0 = Some x"
 
    31 | "nth_of (x # xs) 0 = Some x"
 
    32 | "nth_of (x # xs) (Suc n) = nth_of xs n" 
    32 | "nth_of (x # xs) (Suc n) = nth_of xs n" 
       
    33 
       
    34 lemma nth_of_map [simp]:
 
       
    35   shows "nth_of (map f p) n = (case (nth_of p n) of None \<Rightarrow> None | Some x \<Rightarrow> Some (f x))"
 
       
    36 apply(induct p arbitrary: n)
 
       
    37 apply(auto)
 
       
    38 apply(case_tac n)
 
       
    39 apply(auto)
 
       
    40 done
 
    33 
    41 
    34 fun 
    42 fun 
    35   fetch :: "tprog \<Rightarrow> state \<Rightarrow> cell \<Rightarrow> instr"
 
    43   fetch :: "tprog \<Rightarrow> state \<Rightarrow> cell \<Rightarrow> instr"
 
    36 where
 
    44 where
 
    37   "fetch p 0 b = (Nop, 0)"
 
    45   "fetch p 0 b = (Nop, 0)"
 
    42 |"fetch p (Suc s) Oc = 
    50 |"fetch p (Suc s) Oc = 
    43      (case nth_of p ((2 * s) + 1) of
 
    51      (case nth_of p ((2 * s) + 1) of
 
    44          Some i \<Rightarrow> i
 
    52          Some i \<Rightarrow> i
 
    45        | None \<Rightarrow> (Nop, 0))"
 
    53        | None \<Rightarrow> (Nop, 0))"
 
    46 
    54 
       
    55 lemma fetch_Nil [simp]:
 
       
    56   shows "fetch [] s b = (Nop, 0)"
 
       
    57 apply(case_tac s)
 
       
    58 apply(auto)
 
       
    59 apply(case_tac b)
 
       
    60 apply(auto)
 
       
    61 done
 
       
    62 
    47 fun 
    63 fun 
    48   update :: "action \<Rightarrow> tape \<Rightarrow> tape"
 
    64   update :: "action \<Rightarrow> tape \<Rightarrow> tape"
 
    49 where 
    65 where 
    50   "update W0 (l, r) = (l, Bk # (tl r))" 
    66   "update W0 (l, r) = (l, Bk # (tl r))" 
    51 | "update W1 (l, r) = (l, Oc # (tl r))"
 
    67 | "update W1 (l, r) = (l, Oc # (tl r))"
 
    78 definition 
    94 definition 
    79   tm_wf :: "tprog \<Rightarrow> bool"
 
    95   tm_wf :: "tprog \<Rightarrow> bool"
 
    80 where
 
    96 where
 
    81   "tm_wf p = (length p \<ge> 2 \<and> iseven (length p) \<and> (\<forall>(a, s) \<in> set p. s \<le> length p div 2))"
 
    97   "tm_wf p = (length p \<ge> 2 \<and> iseven (length p) \<and> (\<forall>(a, s) \<in> set p. s \<le> length p div 2))"
 
    82 
    98 
       
    99 
    83 (* FIXME: needed? *)
 
   100 (* FIXME: needed? *)
 
    84 lemma halt_lemma: 
   101 lemma halt_lemma: 
    85   "\<lbrakk>wf LE; \<forall>n. (\<not> P (f n) \<longrightarrow> (f (Suc n), (f n)) \<in> LE)\<rbrakk> \<Longrightarrow> \<exists>n. P (f n)"
 
   102   "\<lbrakk>wf LE; \<forall>n. (\<not> P (f n) \<longrightarrow> (f (Suc n), (f n)) \<in> LE)\<rbrakk> \<Longrightarrow> \<exists>n. P (f n)"
 
    86 by (metis wf_iff_no_infinite_down_chain)
 
   103 by (metis wf_iff_no_infinite_down_chain)
 
    87 
   104 
    96   shift :: "tprog \<Rightarrow> nat \<Rightarrow> tprog"
 
   113   shift :: "tprog \<Rightarrow> nat \<Rightarrow> tprog"
 
    97 where
 
   114 where
 
    98   "shift p n = (map (\<lambda> (a, s). (a, (if s = 0 then 0 else s + n))) p)"
 
   115   "shift p n = (map (\<lambda> (a, s). (a, (if s = 0 then 0 else s + n))) p)"
 
    99 
   116 
   100 
   117 
   101 lemma [simp]: 
   118 lemma length_shift [simp]: 
   102   "length (shift p n) = length p"
 
   119   "length (shift p n) = length p"
 
   103 by (simp)
 
   120 by (simp)
 
   104 
   121 
   105 fun 
   122 fun 
   106   adjust :: "tprog \<Rightarrow> tprog"
 
   123   adjust :: "tprog \<Rightarrow> tprog"
 
   107 where
 
   124 where
 
   108   "adjust p = (map (\<lambda> (a, s). (a, if s = 0 then ((length p) div 2) + 1 else s)) p)"
 
   125   "adjust p = map (\<lambda> (a, s). (a, if s = 0 then ((length p) div 2) + 1 else s)) p"
 
   109 
   126 
   110 lemma [simp]: 
   127 lemma length_adjust[simp]: 
   111   shows "length (adjust p) = length p"
 
   128   shows "length (adjust p) = length p"
 
   112 by (simp)
 
   129 by (induct p) (auto)
 
   113 
   130 
   114 definition
 
   131 definition
 
   115   tm_comp :: "tprog \<Rightarrow> tprog \<Rightarrow> tprog" ("_ |+| _" [0, 0] 100)
 
   132   tm_comp :: "tprog \<Rightarrow> tprog \<Rightarrow> tprog" ("_ |+| _" [0, 0] 100)
 
   116 where
 
   133 where
 
   117   "tm_comp p1 p2 = ((adjust p1) @ (shift p2 ((length p1) div 2)))"
 
   134   "tm_comp p1 p2 = ((adjust p1) @ (shift p2 ((length p1) div 2)))"
 
   151 type_synonym assert = "tape \<Rightarrow> bool"
 
   168 type_synonym assert = "tape \<Rightarrow> bool"
 
   152 
   169 
   153 definition assert_imp :: "assert \<Rightarrow> assert \<Rightarrow> bool" ("_ \<mapsto> _" [0, 0] 100)
 
   170 definition assert_imp :: "assert \<Rightarrow> assert \<Rightarrow> bool" ("_ \<mapsto> _" [0, 0] 100)
 
   154   where
 
   171   where
 
   155   "assert_imp P Q = (\<forall>l r. P (l, r) \<longrightarrow> Q (l, r))"
 
   172   "assert_imp P Q = (\<forall>l r. P (l, r) \<longrightarrow> Q (l, r))"
 
       
   173 
       
   174 lemma holds_for_imp:
 
       
   175   assumes "P holds_for c"
 
       
   176   and "P \<mapsto> Q"
 
       
   177   shows "Q holds_for c"
 
       
   178 using assms unfolding assert_imp_def by (case_tac c, auto)
 
   156 
   179 
   157 lemma test:
 
   180 lemma test:
 
   158   assumes "is_final (steps (1, (l, r)) p n1)"
 
   181   assumes "is_final (steps (1, (l, r)) p n1)"
 
   159   and     "is_final (steps (1, (l, r)) p n2)"
 
   182   and     "is_final (steps (1, (l, r)) p n2)"
 
   160   shows "Q holds_for (steps (1, (l, r)) p n1) \<longleftrightarrow> Q holds_for (steps (1, (l, r)) p n2)"
 
   183   shows "Q holds_for (steps (1, (l, r)) p n1) \<longleftrightarrow> Q holds_for (steps (1, (l, r)) p n2)"
 
   180 {P1} A {Q1}   {P2} B {Q2}  Q1 \<mapsto> P2
 
   203 {P1} A {Q1}   {P2} B {Q2}  Q1 \<mapsto> P2
 
   181 -----------------------------------
 
   204 -----------------------------------
 
   182     {P1} A |+| B {Q2}
 
   205     {P1} A |+| B {Q2}
 
   183 *}
 
   206 *}
 
   184 
   207 
       
   208 lemma before_final: 
       
   209   assumes "steps (1, tp) A n = (0, tp')"
 
       
   210   obtains n' where "\<not> is_final (steps (1, tp) A n')" and "steps (1, tp) A (Suc n') = (0, tp')"
 
       
   211 using assms 
       
   212 apply(induct n)
 
       
   213 apply(auto)
 
       
   214 by (metis is_final.simps step_red steps.simps steps_0 surj_pair)
 
       
   215 
       
   216 lemma t_merge_fetch_pre:
 
       
   217   assumes "fetch A s b = (ac, ns)" "s \<le> length A div 2" "tm_wf A" "s \<noteq> 0" 
       
   218   shows "fetch (adjustA |+| B) s b = (ac, if ns = 0 then Suc (length A div 2) else ns)"
 
       
   219 using assms
 
       
   220 unfolding tm_comp_def
 
       
   221 apply(induct A)
 
       
   222 apply(auto)
 
       
   223 apply(subgoal_tac "2 * (s - Suc 0) < length A \<and> Suc (2 * (s - Suc 0)) < length A")
 
       
   224 apply(auto simp: tm_wf_def iseven_def tm_comp_def split: if_splits cell.splits)
 
       
   225 oops
 
       
   226 
       
   227 lemma t_merge_pre_eq_step: 
       
   228   "\<lbrakk>step (a, b, c) A = cf; tm_wf A; \<not> is_final cf\<rbrakk> 
       
   229         \<Longrightarrow> step (a, b, c) (A |+| B) = cf"
 
       
   230 apply(subgoal_tac "a \<le> length A div 2 \<and> a \<noteq> 0")
 
       
   231 apply(simp)
 
       
   232 apply(case_tac "fetch A a (read c)", simp)
 
       
   233 apply(auto)
 
       
   234 oops
 
       
   235 
       
   236 lemma t_merge_pre_eq:  
       
   237   "\<lbrakk>steps (Suc 0, tp) A stp = cf; \<not> is_final cf; tm_wf A\<rbrakk>
 
       
   238   \<Longrightarrow> steps (Suc 0, tp) (A |+| B) stp = cf"
 
       
   239 apply(induct stp arbitrary: cf)
 
       
   240 apply(auto)[1]
 
       
   241 apply(auto)
 
       
   242 oops
 
       
   243 
       
   244 lemma t_merge_pre_halt_same: 
       
   245   assumes a_ht: "steps (1, tp) A n = (0, tp')"
 
       
   246   and a_wf: "t_correct A"
 
       
   247   obtains n' where "steps (1, tp) (A |+| B) n' = (Suc (length A div 2), tp')"
 
       
   248 proof -
 
       
   249   assume a: "\<And>n. steps (1, tp) (A |+| B) n = (Suc (length A div 2), tp') \<Longrightarrow> thesis"
 
       
   250   
       
   251   obtain stp' where "\<not> is_final (steps (1, tp) A stp')" and "steps (1, tp) A (Suc stp') = (0, tp')"
 
       
   252   using a_ht before_final by blast
 
       
   253   then have "steps (1, tp) (A |+| B) (Suc stp') = (Suc (length A div 2), tp')"
 
       
   254     sorry (*using a_wf t_merge_pre_halt_same' by blast*)
 
       
   255   with a show thesis by blast
 
       
   256 qed
 
       
   257 
       
   258 
       
   259 
       
   260 lemma steps_comp:
 
       
   261   assumes a1: "steps (1, l, r) A n1 = (s1, l1, r1)"
 
       
   262   and a2: "steps (1, l1, r1) B n2 = (s2, l2, r2)"
 
       
   263   shows "steps (1, l, r) (A |+| B) (n1 + n2) = (s2, l2, r2)"
 
       
   264 using assms
 
       
   265 apply(induct n2)
 
       
   266 apply(simp)
 
       
   267 apply(simp add: tm_comp_def)
 
       
   268 oops
 
   185 
   269 
   186 lemma Hoare_plus: 
   270 lemma Hoare_plus: 
   187   assumes aimpb: "Q1 \<mapsto> P2"
 
   271   assumes aimpb: "Q1 \<mapsto> P2"
 
   188   and A_wf : "tm_wf A"
 
   272   and A_wf : "tm_wf A"
 
   189   and B_wf : "tm_wf B"
 
   273   and B_wf : "tm_wf B"
 
   190   and A_halt : "{P1} A {Q1}"
 
   274   and A_halt : "{P1} A {Q1}"
 
   191   and B_halt : "{P2} B {Q2}"
 
   275   and B_halt : "{P2} B {Q2}"
 
   192   shows "{P1} A |+| B {Q2}"
 
   276   shows "{P1} A |+| B {Q2}"
 
   193 proof(rule HoareI)
 
   277 proof(rule HoareI)
 
   194   fix a b
 
   278   fix l r
 
   195   assume h: "P1 (a, b)"
 
   279   assume h: "P1 (l, r)"
 
   196   then have "\<exists>n. is_final (steps (1, a, b) A n) \<and> Q1 holds_for (steps (1, a, b) A n)"
 
   280   then obtain n1 where a: "is_final (steps (1, l, r) A n1)" and b: "Q1 holds_for (steps (1, l, r) A n1)"
 
   197     using A_halt unfolding Hoare_def by simp
 
   281     using A_halt unfolding Hoare_def by auto
 
   198   then obtain n1 where "is_final (steps (1, a, b) A n1) \<and> Q1 holds_for (steps (1, a, b) A stp1)" ..
 
   282   from b aimpb have "P2 holds_for (steps (1, l, r) A n1)"
 
   199   then show "\<exists>n. is_final (steps (1, a, b) (A |+| B) n) \<and> Q2 holds_for (steps (1, a, b) (A |+| B) n)"
 
   283     by (rule holds_for_imp)
 
   200   proof(case_tac "steps (Suc 0, a, b) A stp1", simp, erule_tac conjE)
 
   284   then obtain l' r' where "P2 (l', r')"
 
   201     fix aa ba c
 
   285     apply(auto)
 
   202     assume g1: "Q1 (ba, c)" 
   286     apply(case_tac "steps (Suc 0, l, r) A n1")
 
   203       and g2: "steps (Suc 0, a, b) A stp1 = (0, ba, c)"
 
   287     apply(simp)
 
   204     hence "P2 (ba, c)"
 
   288     done
 
   205       using aimpb apply(simp add: assert_imp_def)
 
   289   then obtain n2 where a: "is_final (steps (1, l', r') B n2)" and b: "Q2 holds_for (steps (1, l', r') B n2)"
 
   206       done
 
   290     using B_halt unfolding Hoare_def by auto
 
   207     hence "\<exists> stp. is_final (steps (Suc 0, ba, c) B stp) \<and> Q2 holds_for (steps (Suc 0, ba, c) B stp)"
 
   291   show "\<exists>n. is_final (steps (1, l, r) (A |+| B) n) \<and> Q2 holds_for (steps (1, l, r) (A |+| B) n)"
 
   208       using B_halt unfolding Hoare_def by simp
 
   292     apply(rule_tac x="n1 + n2" in exI)
 
   209     from this obtain stp2 where 
   293     apply(rule conjI)
 
   210       "is_final (steps (Suc 0, ba, c) B stp2) \<and> Q2 holds_for (steps (Suc 0, ba, c) B stp2)" ..
 
   294     apply(simp)
 
   211     thus "?thesis"
 
   295     apply(simp only: steps_add[symmetric])
 
   212     proof(case_tac "steps (Suc 0, ba, c) B stp2", simp, erule_tac conjE)
 
   296     sorry
 
   213       fix aa bb ca
 
       
   214       assume g3: " Q2 (bb, ca)" "steps (Suc 0, ba, c) B stp2 = (0, bb, ca)"
 
       
   215       have "\<exists> stp. steps (Suc 0, a, b) (A |+| B) stp = (Suc (length A div 2), ba , c)"
 
       
   216         using g2 A_wf B_wf
 
       
   217         sorry
 
       
   218       moreover have "\<exists> stp. steps (Suc (length A div 2), ba, c) (A |+| B) stp = (0, bb, ca)"
 
       
   219         using g3 A_wf B_wf
 
       
   220         sorry
 
       
   221       ultimately show "\<exists>n. case steps (Suc 0, a, b) (A |+| B) n of (s, tp') \<Rightarrow> s = 0 \<and> Q2 tp'"
 
       
   222         apply(erule_tac exE, erule_tac exE)
 
       
   223         apply(rule_tac x = "stp + stpa" in exI, simp)
 
       
   224         using g3 by simp
 
       
   225     qed
 
       
   226   qed
 
       
   227 qed
 
   297 qed
 
   228 
   298 
   229 lemma Hoare_plus2: 
   299 
   230   assumes A_wf : "tm_wf A"
 
       
   231   and B_wf : "tm_wf B"
 
       
   232   and A_halt : "{P1} A {Q1}"
 
       
   233   and B_halt : "{P2} B {Q2}"
 
       
   234   and aimpb: "Q1 \<mapsto> P2"
 
       
   235   shows "{P1} A |+| B {Q2}"
 
       
   236 unfolding hoare_def
 
       
   237 proof(auto split: )
 
       
   238   fix a b
 
       
   239   assume h: "P1 (a, b)"
 
       
   240   hence "\<exists>n. let (s, tp') = steps (Suc 0, a, b) A n in s = 0 \<and> Q1 tp'"
 
       
   241     using A_halt unfolding hoare_def by simp
 
       
   242   from this obtain stp1 where "let (s, tp') = steps (Suc 0, a, b) A stp1 in s = 0 \<and> Q1 tp'" ..
 
       
   243   then show "\<exists>n. case steps (Suc 0, a, b) (A |+| B) n of (s, tp') \<Rightarrow> s = 0 \<and> Q2 tp'"
 
       
   244   proof(case_tac "steps (Suc 0, a, b) A stp1", simp, erule_tac conjE)
 
       
   245     fix aa ba c
 
       
   246     assume g1: "Q1 (ba, c)" 
       
   247       and g2: "steps (Suc 0, a, b) A stp1 = (0, ba, c)"
 
       
   248     hence "P2 (ba, c)"
 
       
   249       using aimpb apply(simp add: assert_imp_def)
 
       
   250       done
 
       
   251     hence "\<exists> stp. let (s, tp') = steps (Suc 0, ba, c) B stp in s = 0 \<and> Q2 tp'"
 
       
   252       using B_halt unfolding hoare_def by simp
 
       
   253     from this obtain stp2 where "let (s, tp') = steps (Suc 0, ba, c) B stp2 in s = 0 \<and> Q2 tp'" ..
 
       
   254     thus "?thesis"
 
       
   255     proof(case_tac "steps (Suc 0, ba, c) B stp2", simp, erule_tac conjE)
 
       
   256       fix aa bb ca
 
       
   257       assume g3: " Q2 (bb, ca)" "steps (Suc 0, ba, c) B stp2 = (0, bb, ca)"
 
       
   258       have "\<exists> stp. steps (Suc 0, a, b) (A |+| B) stp = (Suc (length A div 2), ba , c)"
 
       
   259         using g2 A_wf B_wf
 
       
   260         sorry
 
       
   261       moreover have "\<exists> stp. steps (Suc (length A div 2), ba, c) (A |+| B) stp = (0, bb, ca)"
 
       
   262         using g3 A_wf B_wf
 
       
   263         sorry
 
       
   264       ultimately show "\<exists>n. case steps (Suc 0, a, b) (A |+| B) n of (s, tp') \<Rightarrow> s = 0 \<and> Q2 tp'"
 
       
   265         apply(erule_tac exE, erule_tac exE)
 
       
   266         apply(rule_tac x = "stp + stpa" in exI, simp)
 
       
   267         using g3 by simp
 
       
   268     qed
 
       
   269   qed
 
       
   270 qed
 
       
   271 
   300 
   272 
   301 
   273 
   302 
   274 locale turing_merge =
 
   303 locale turing_merge =
 
   275   fixes A :: "tprog" and B :: "tprog" and P1 :: "assert"
 
   304   fixes A :: "tprog" and B :: "tprog" and P1 :: "assert"
tm