
diff -r 00ae320bb804 -r a8785fa80278 thys/turing_basic.thy
--- a/thys/turing_basic.thy	Mon Jan 14 21:40:38 2013 +0000
+++ b/thys/turing_basic.thy	Tue Jan 15 19:05:15 2013 +0000
@@ -1,400 +1,605 @@
-(* 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
-
+(* Title: Turing machines
+   Author: Xu Jian <xujian817@hotmail.com>
+   Maintainer: Xu Jian
+*)
+
+theory turing_basic
+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 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"
+
+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)
+
+fun 
+  tm_wf :: "tprog \<Rightarrow> bool"
+where
+  "tm_wf (p, off) = (length p \<ge> 2 \<and> length p mod 2 = 0 \<and> 
+                    (\<forall>(a, s) \<in> set p. s \<le> length p div 2
+                                             + off \<and> s \<ge> off))"
+
+
+(* 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 :: "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)"
+
+
+lemma length_shift [simp]: 
+  "length (shift p n) = length p"
+by (simp)
+
+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_adjust[simp]: 
+  shows "length (adjust p) = length p"
+by (induct p) (auto)
+
+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)))"
+
+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, off) 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, off) 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(case_tac p) (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 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)
+
+declare steps.simps[simp del]
+
+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')"
+proof -
+  from assms have "\<exists> n'. \<not> is_final (steps (1, tp) A n') \<and> 
+               steps (1, tp) A (Suc n') = (0, tp')"
+  proof(induct n arbitrary: tp', simp add: steps.simps)
+    fix n tp'
+    assume ind: 
+      "\<And>tp'. steps (1, tp) A n = (0, tp') \<Longrightarrow>
+      \<exists>n'. \<not> is_final (steps (1, tp) A n') \<and> steps (1, tp) A (Suc n') = (0, tp')"
+    and h: " steps (1, tp) A (Suc n) = (0, tp')"
+    from h show  "\<exists>n'. \<not> is_final (steps (1, tp) A n') \<and> steps (1, tp) A (Suc n') = (0, tp')"
+    proof(simp add: step_red del: steps.simps, 
+                     case_tac "(steps (Suc 0, tp) A n)", case_tac "a = 0", simp)
+      fix a b c
+      assume " steps (Suc 0, tp) A n = (0, tp')"
+      hence "\<exists>n'. \<not> is_final (steps (1, tp) A n') \<and> steps (1, tp) A (Suc n') = (0, tp')"
+        apply(rule_tac ind, simp)
+        done
+      thus "\<exists>n'. \<not> is_final (steps (Suc 0, tp) A n') \<and> step (steps (Suc 0, tp) A n') A = (0, tp')"
+        apply(simp)
+        done
+    next
+      fix a b c
+      assume "steps (Suc 0, tp) A n = (a, b, c)"
+             "step (steps (Suc 0, tp) A n) A = (0, tp')"
+        "a \<noteq> 0"
+      thus "\<exists>n'. \<not> is_final (steps (Suc 0, tp) A n') \<and> 
+        step (steps (Suc 0, tp) A n') A = (0, tp')"
+        apply(rule_tac x = n in exI, simp)
+        done
+    qed
+  qed
+  thus "(\<And>n'. \<lbrakk>\<not> is_final (steps (1, tp) A n'); 
+    steps (1, tp) A (Suc n') = (0, tp')\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis"
+    by auto
+qed
+
+declare tm_comp.simps [simp del] adjust.simps[simp del] shift.simps[simp del]
+
+lemma length_comp:
+"length (A |+| B) = length A + length B"
+apply(auto simp: tm_comp.simps)
+done
+
+lemma tmcomp_fetch_in_first:
+  assumes "case (fetch A a x) of (ac, ns) \<Rightarrow> ns \<noteq> 0"
+  shows "fetch (A |+| B) a x = fetch A a x"
+using assms
+apply(case_tac a, case_tac [!] x, 
+auto simp: length_comp tm_comp.simps length_adjust nth_append)
+apply(simp_all add: adjust.simps)
+done
+
+
+lemma is_final_eq: "is_final (ba, tp) = (ba = 0)"
+apply(case_tac tp, simp add: is_final.simps)
+done
+
+lemma t_merge_pre_eq_step: 
+  assumes step: "step (a, b, c) (A, 0) = cf"
+  and     tm_wf: "tm_wf (A, 0)" 
+  and     unfinal: "\<not> is_final cf"
+  shows "step (a, b, c) (A |+| B, 0) = cf"
+proof -
+  have "fetch (A |+| B) a (read c) = fetch A a (read c)"
+  proof(rule_tac tmcomp_fetch_in_first)
+    from step and unfinal show "case fetch A a (read c) of (ac, ns) \<Rightarrow> ns \<noteq> 0"
+      apply(auto simp: is_final.simps)
+      apply(case_tac "fetch A a (read c)", simp_all add: is_final_eq)
+      done
+  qed      
+  thus "?thesis"
+    using step
+    apply(auto simp: step.simps is_final.simps)
+    done
+qed
+
+declare tm_wf.simps[simp del] step.simps[simp del]
+
+lemma t_merge_pre_eq:  
+  "\<lbrakk>steps (Suc 0, tp) (A, 0) stp = cf; \<not> is_final cf; tm_wf (A, 0)\<rbrakk>
+  \<Longrightarrow> steps (Suc 0, tp) (A |+| B, 0) stp = cf"
+proof(induct stp arbitrary: cf, simp add: steps.simps)
+  fix stp cf
+  assume ind: "\<And>cf. \<lbrakk>steps (Suc 0, tp) (A, 0) stp = cf; \<not> is_final cf; tm_wf (A, 0)\<rbrakk> 
+    \<Longrightarrow> steps (Suc 0, tp) (A |+| B, 0) stp = cf"
+  and h: "steps (Suc 0, tp) (A, 0) (Suc stp) = cf"
+      "\<not> is_final cf" "tm_wf (A, 0)"
+  from h show "steps (Suc 0, tp) (A |+| B, 0) (Suc stp) = cf"
+  proof(simp add: step_red, case_tac "(steps (Suc 0, tp) (A, 0) stp)", simp)
+    fix a b c
+    assume g: "steps (Suc 0, tp) (A, 0) stp = (a, b, c)"
+      "step (a, b, c) (A, 0) = cf"
+    have "(steps (Suc 0, tp) (A |+| B, 0) stp) = (a, b, c)"
+    proof(rule ind, simp_all add: h g)
+      show "0 < a"
+        using g h
+        apply(simp add: step_red)
+        apply(case_tac a, auto simp: step_0)
+        apply(case_tac "steps (Suc 0, tp) (A, 0) stp", simp)
+        done
+    qed
+    thus "step (steps (Suc 0, tp) (A |+| B, 0) stp) (A |+| B, 0) = cf"
+      apply(simp)
+      apply(rule_tac t_merge_pre_eq_step, simp_all add: g h)
+      done
+  qed
+qed
+
+lemma tmcomp_fetch_in_first2:
+  assumes "fetch A a x = (ac, 0)"
+          "tm_wf (A, 0)"
+          "a \<le> length A div 2" "a > 0"
+  shows "fetch (A |+| B) a x = (ac, Suc (length A div 2))"
+using assms
+apply(case_tac a, case_tac [!] x, 
+auto simp: length_comp tm_comp.simps length_adjust nth_append)
+apply(simp_all add: adjust.simps)
+done
+
+lemma tmcomp_exec_after_first:
+  "\<lbrakk>0 < a; step (a, b, c) (A, 0) = (0, tp'); tm_wf (A, 0); 
+       a \<le> length A div 2\<rbrakk>
+       \<Longrightarrow> step (a, b, c) (A |+| B, 0) = (Suc (length A div 2), tp')"
+apply(simp add: step.simps, auto)
+apply(case_tac "fetch A a Bk", simp add: tmcomp_fetch_in_first2)
+apply(case_tac "fetch A a (hd c)", simp add: tmcomp_fetch_in_first2)
+done
+
+lemma step_nothalt_pre: "\<lbrakk>step (aa, ba, ca) (A, 0) = (a, b, c);  0 < a\<rbrakk> \<Longrightarrow> 0 < aa"
+apply(case_tac "aa = 0", simp add: step_0, simp)
+done
+
+lemma nth_in_set: 
+  "\<lbrakk> A ! i = x; i <  length A\<rbrakk> \<Longrightarrow> x \<in> set A"
+by auto
+
+lemma step_nothalt: 
+  "\<lbrakk>step (aa, ba, ca) (A, 0) = (a, b, c); 0 < a; tm_wf (A, 0)\<rbrakk> \<Longrightarrow> 
+  a \<le> length A div 2"
+apply(simp add: step.simps)
+apply(case_tac aa, case_tac [!] aa, auto split: if_splits simp: tm_wf.simps)
+apply(case_tac "A ! (2 * nat)", simp)
+apply(erule_tac x = "(aa, a)" in ballE, simp_all add: nth_in_set)
+apply(case_tac "hd ca", auto split: if_splits simp: tm_wf.simps)
+apply(case_tac "A ! (2 * nat)", simp)
+apply(erule_tac x = "(aa, a)" in ballE, simp_all add: nth_in_set)
+apply(case_tac "A ! (Suc (2 * nat))")
+apply(erule_tac x = "(aa,bb)" in ballE, simp_all add: nth_in_set)
+done
+
+lemma steps_in_range: 
+  " \<lbrakk>0 < a; steps (Suc 0, tp) (A, 0) stp = (a, b, c); tm_wf (A, 0)\<rbrakk>
+  \<Longrightarrow> a \<le> length A div 2"
+proof(induct stp arbitrary: a b c)
+  fix a b c
+  assume h: "0 < a" "steps (Suc 0, tp) (A, 0) 0 = (a, b, c)" 
+            "tm_wf (A, 0)"
+  thus "a \<le> length A div 2"
+    apply(simp add: steps.simps tm_wf.simps, auto)
+    done
+next
+  fix stp a b c
+  assume ind: "\<And>a b c. \<lbrakk>0 < a; steps (Suc 0, tp) (A, 0) stp = (a, b, c); 
+    tm_wf (A, 0)\<rbrakk> \<Longrightarrow> a \<le> length A div 2"
+  and h: "0 < a" "steps (Suc 0, tp) (A, 0) (Suc stp) = (a, b, c)" "tm_wf (A, 0)"
+  from h show "a \<le> length A div 2"
+  proof(simp add: step_red, case_tac "(steps (Suc 0, tp) (A, 0) stp)", simp)
+    fix aa ba ca
+    assume g: "step (aa, ba, ca) (A, 0) = (a, b, c)" 
+           "steps (Suc 0, tp) (A, 0) stp = (aa, ba, ca)"
+    hence "aa \<le> length A div 2"
+      apply(rule_tac ind, auto simp: h step_nothalt_pre)
+      done
+    thus "?thesis"
+      using g h
+      apply(rule_tac step_nothalt, auto)
+      done
+  qed
+qed
+
+lemma t_merge_pre_halt_same: 
+  assumes a_ht: "steps (1, tp) (A, 0) n = (0, tp')"
+  and a_wf: "tm_wf (A, 0)"
+  obtains n' where "steps (1, tp) (A |+| B, 0) 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 "\<not> is_final (steps (1, tp) (A, 0) stp')" and 
+                          "steps (1, tp) (A, 0) (Suc stp') = (0, tp')"
+  using a_ht before_final by blast
+  then have "steps (1, tp) (A |+| B, 0) (Suc stp') = (Suc (length A div 2), tp')"
+  proof(simp add: step_red)
+    assume "\<not> is_final (steps (Suc 0, tp) (A, 0) stp')"
+           " step (steps (Suc 0, tp) (A, 0) stp') (A, 0) = (0, tp')"
+    moreover hence "(steps (Suc 0, tp) (A |+| B, 0) stp') = (steps (Suc 0, tp) (A, 0) stp')"
+      apply(rule_tac t_merge_pre_eq)
+      apply(simp_all add: a_wf a_ht)
+      done
+    ultimately show "step (steps (Suc 0, tp) (A |+| B, 0) stp') (A |+| B, 0) = (Suc (length A div 2), tp')"
+      apply(case_tac " steps (Suc 0, tp) (A, 0) stp'", simp)
+      apply(rule tmcomp_exec_after_first, simp_all add: a_wf)
+      apply(erule_tac steps_in_range, auto simp: a_wf)
+      done
+  qed
+  with a show thesis by blast
+qed
+
+lemma tm_comp_fetch_second_zero:
+  "\<lbrakk>fetch B sa' x = (a, 0); tm_wf (A, 0); tm_wf (B, 0); sa' > 0\<rbrakk>
+     \<Longrightarrow> fetch (A |+| B) (sa' + (length A div 2)) x = (a, 0)"
+apply(case_tac x)
+apply(case_tac [!] sa',
+  auto simp: fetch.simps length_comp length_adjust nth_append tm_comp.simps
+             tm_wf.simps shift.simps split: if_splits)
+done 
+
+lemma tm_comp_fetch_second_inst:
+  "\<lbrakk>sa > 0; s > 0;  tm_wf (A, 0); tm_wf (B, 0); fetch B sa x = (a, s)\<rbrakk>
+     \<Longrightarrow> fetch (A |+| B) (sa + length A div 2) x = (a, s + length A div 2)"
+apply(case_tac x)
+apply(case_tac [!] sa,
+  auto simp: fetch.simps length_comp length_adjust nth_append tm_comp.simps
+             tm_wf.simps shift.simps split: if_splits)
+done 
+
+lemma t_merge_second_same:
+  assumes a_wf: "tm_wf (A, 0)"
+  and b_wf: "tm_wf (B, 0)"
+  and steps: "steps (Suc 0, l, r) (B, 0) stp = (s, l', r')"
+  shows "steps (Suc (length A div 2), l, r)  (A |+| B, 0) stp
+       = (if s = 0 then 0
+          else s + length A div 2, l', r')"
+using a_wf b_wf steps
+proof(induct stp arbitrary: s l' r', simp add: steps.simps, simp)
+  fix stpa sa l'a r'a
+  assume ind: "\<And>s l' r'. steps (Suc 0, l, r) (B, 0) stpa = (s, l', r') \<Longrightarrow>
+    steps (Suc (length A div 2), l, r) (A |+| B, 0) stpa = 
+                (if s = 0 then 0 else s + length A div 2, l', r')"
+  and h: "step (steps (Suc 0, l, r) (B, 0) stpa) (B, 0) = (sa, l'a, r'a)"
+  obtain sa' l'' r'' where a: "(steps (Suc 0, l, r) (B, 0) stpa) = (sa', l'', r'')"
+    apply(case_tac "steps (Suc 0, l, r) (B, 0) stpa", auto)
+    done
+  from this have b: "steps (Suc (length A div 2), l, r) (A |+| B, 0) stpa = 
+                (if sa' = 0 then 0 else sa' + length A div 2, l'', r'')"
+    apply(erule_tac ind)
+    done
+  from a b h show 
+    "(sa = 0 \<longrightarrow> step (steps (Suc (length A div 2), l, r) (A |+| B, 0) stpa) (A |+| B, 0) = (0, l'a, r'a)) \<and>
+    (0 < sa \<longrightarrow> step (steps (Suc (length A div 2), l, r) (A |+| B, 0) stpa) (A |+| B, 0) = (sa + length A div 2, l'a, r'a))"
+  proof(case_tac "sa' = 0", auto)
+    assume "step (sa', l'', r'') (B, 0) = (0, l'a, r'a)" "0 < sa'"
+    thus "step (sa' + length A div 2, l'', r'') (A |+| B, 0) = (0, l'a, r'a)"
+      using a_wf b_wf
+      apply(simp add:  step.simps)
+      apply(case_tac "fetch B sa' (read r'')", auto)
+      apply(simp_all add: step.simps tm_comp_fetch_second_zero)
+      done
+  next
+    assume "step (sa', l'', r'') (B, 0) = (sa, l'a, r'a)" "0 < sa'" "0 < sa"
+    thus "step (sa' + length A div 2, l'', r'') (A |+| B, 0) = (sa + length A div 2, l'a, r'a)"
+      using a_wf b_wf
+      apply(simp add: step.simps)
+      apply(case_tac "fetch B sa' (read r'')", auto)
+      apply(simp_all add: step.simps tm_comp_fetch_second_inst)
+      done
+  qed
+qed
+
+lemma t_merge_second_halt_same:
+  "\<lbrakk>tm_wf (A, 0); tm_wf (B, 0); 
+   steps (1, l, r) (B, 0) stp = (0, l', r')\<rbrakk>
+     \<Longrightarrow> steps (Suc (length A div 2), l, r)  (A |+| B, 0) stp
+       = (0, l', r')"
+using t_merge_second_same[where s = "0"]
+apply(auto)
+done
+
+lemma Hoare_plus_halt: 
+  assumes aimpb: "Q1 \<mapsto> P2"
+  and A_wf : "tm_wf (A, 0)"
+  and B_wf : "tm_wf (B, 0)"
+  and A_halt : "{P1} (A, 0) {Q1}"
+  and B_halt : "{P2} (B, 0) {Q2}"
+  shows "{P1} (A |+| B, 0) {Q2}"
+proof(rule HoareI)
+  fix l r
+  assume h: "P1 (l, r)"
+  then obtain n1 where a: "is_final (steps (1, l, r) (A, 0) n1)" and b: "Q1 holds_for (steps (1, l, r) (A, 0) n1)"
+    using A_halt unfolding Hoare_def by auto
+  then obtain l' r' where "steps (1, l, r) (A, 0) n1 = (0, l', r')" and c: "Q1 holds_for (0, l', r')"
+    by(case_tac "steps (1, l, r) (A, 0) n1", auto)
+  then obtain stpa where d: "steps (1, l, r) (A |+| B, 0) stpa = (Suc (length A div 2), l', r')"
+    using A_wf
+    by(rule_tac t_merge_pre_halt_same, auto)
+  from c aimpb have "P2 holds_for (0, l', r')"
+    by(rule holds_for_imp)
+  from this have "P2 (l', r')" by auto
+  from this obtain n2 where e: "is_final (steps (1, l', r') (B, 0) n2)" and f: "Q2 holds_for (steps (1, l', r') (B, 0) n2)"
+    using B_halt unfolding Hoare_def
+    by auto
+  then obtain l'' r'' where "steps (1, l', r') (B, 0) n2 = (0, l'', r'')" and g: "Q2 holds_for (0, l'', r'')"
+    by(case_tac "steps (1, l', r') (B, 0) n2", auto)
+  from this have "steps (Suc (length A div 2), l', r')  (A |+| B, 0) n2
+    = (0, l'', r'')"
+    apply(rule_tac t_merge_second_halt_same, auto simp: A_wf B_wf)
+    done
+  thus "\<exists>n. is_final (steps (1, l, r) (A |+| B, 0) n) \<and> Q2 holds_for (steps (1, l, r) (A |+| B, 0) n)"
+    using d g
+    apply(rule_tac x = "stpa + n2" in exI)
+    apply(simp add: steps_add)
+    done
+qed
+
+definition
+  Hoare_unhalt :: "assert \<Rightarrow> tprog \<Rightarrow> bool" ("({(1_)}/ (_))" 50)
+where
+  "{P} p \<equiv> 
+     (\<forall>l r. P (l, r) \<longrightarrow> (\<forall> n . \<not> (is_final (steps (1, (l, r)) p n))))"
+
+lemma Hoare_unhalt_I:
+  assumes "\<And>l r. P (l, r) \<Longrightarrow> \<forall> n. \<not> is_final (steps (1, (l, r)) p n)"
+  shows "{P} p"
+unfolding Hoare_unhalt_def using assms by auto
+
+lemma Hoare_plus_unhalt: 
+  assumes aimpb: "Q1 \<mapsto> P2"
+  and A_wf : "tm_wf (A, 0)"
+  and B_wf : "tm_wf (B, 0)"
+  and A_halt : "{P1} (A, 0) {Q1}"
+  and B_uhalt : "{P2} (B, 0)"
+  shows "{P1} (A |+| B, 0)"
+proof(rule_tac Hoare_unhalt_I)
+  fix l r
+  assume h: "P1 (l, r)"
+  then obtain n1 where a: "is_final (steps (1, l, r) (A, 0) n1)" and b: "Q1 holds_for (steps (1, l, r) (A, 0) n1)"
+    using A_halt unfolding Hoare_def by auto
+  then obtain l' r' where "steps (1, l, r) (A, 0) n1 = (0, l', r')" and c: "Q1 holds_for (0, l', r')"
+    by(case_tac "steps (1, l, r) (A, 0) n1", auto)
+  then obtain stpa where d: "steps (1, l, r) (A |+| B, 0) stpa = (Suc (length A div 2), l', r')"
+    using A_wf
+    by(rule_tac t_merge_pre_halt_same, auto)
+  from c aimpb have "P2 holds_for (0, l', r')"
+    by(rule holds_for_imp)
+  from this have "P2 (l', r')" by auto
+  from this have e: "\<forall> n. \<not> is_final (steps (Suc 0, l', r') (B, 0) n)  "
+    using B_uhalt unfolding Hoare_unhalt_def
+    by auto
+  from e show "\<forall>n. \<not> is_final (steps (1, l, r) (A |+| B, 0) n)"
+  proof(rule_tac allI, case_tac "n > stpa")
+    fix n
+    assume h2: "stpa < n"
+    hence "\<not> is_final (steps (Suc 0, l', r') (B, 0) (n - stpa))"
+      using e
+      apply(erule_tac x = "n - stpa" in allE) by simp
+    then obtain s'' l'' r'' where f: "steps (Suc 0, l', r') (B, 0) (n - stpa) = (s'', l'', r'')" and g: "s'' \<noteq> 0"
+      apply(case_tac "steps (Suc 0, l', r') (B, 0) (n - stpa)", auto)
+      done
+    have k: "steps (Suc (length A div 2), l', r') (A |+| B, 0) (n - stpa) = (s''+ length A div 2, l'', r'') "
+      using A_wf B_wf f g
+      apply(drule_tac t_merge_second_same, auto)
+      done
+    show "\<not> is_final (steps (1, l, r) (A |+| B, 0) n)"
+    proof -
+      have "\<not> is_final (steps (1, l, r) (A |+| B, 0) (stpa + (n  - stpa)))"
+        using d k A_wf
+        apply(simp only: steps_add d, simp add: tm_wf.simps)
+        done
+      thus "\<not> is_final (steps (1, l, r) (A |+| B, 0) n)"
+        using h2 by simp
+    qed
+  next
+    fix n
+    assume h2: "\<not> stpa < n"
+    with d show "\<not> is_final (steps (1, l, r) (A |+| B, 0) n)"
+      apply(auto)
+      apply(subgoal_tac "\<exists> d. stpa = n + d", auto)
+      apply(case_tac "(steps (Suc 0, l, r) (A |+| B, 0) n)", simp)
+      apply(rule_tac x = "stpa - n" in exI, simp)
+      done
+  qed
+qed
+    
+      
+       
+end
+




