theory turing_basicimports Mainbeginsection {* Basic definitions of Turing machine *}(* Title: Turing machine's definition and its charater Author: Xu Jian <xujian817@hotmail.com> Maintainer: Xu Jian*)text {* Actions of Turing machine (Abbreviated TM in the following* ).*}datatype taction = -- {* Write zero *} W0 | -- {* Write one *} W1 | -- {* Move left *} L | -- {* Move right *} R | -- {* Do nothing *} Noptext {* Tape contents in every block.*}datatype block = -- {* Blank *} Bk | -- {* Occupied *} Octext {* Tape is represented as a pair of lists $(L_{left}, L_{right})$, where $L_left$, named {\em left list}, is used to represent the tape to the left of RW-head and $L_{right}$, named {\em right list}, is used to represent the tape under and to the right of RW-head.*}type_synonym tape = "block list \<times> block list"text {* The state of turing machine.*}type_synonym tstate = nattext {* Turing machine instruction is represented as a pair @{text "(action, next_state)"}, where @{text "action"} is the action to take at the current state and @{text "next_state"} is the next state the machine is getting into after the action.*}type_synonym tinst = "taction \<times> tstate"text {* Program of Turing machine is represented as a list of Turing instructions and the execution of the program starts from the head of the list. *}type_synonym tprog = "tinst list"text {* Turing machine configuration, which consists of the current state and the tape.*}type_synonym t_conf = "tstate \<times> tape"fun nth_of :: "'a list \<Rightarrow> nat \<Rightarrow> 'a option" where "nth_of xs n = (if n < length xs then Some (xs!n) else None)"text {* The function used to fetech instruction out of Turing program. *}fun fetch :: "tprog \<Rightarrow> tstate \<Rightarrow> block \<Rightarrow> tinst" where "fetch p s b = (if s = 0 then (Nop, 0) else case b of Bk \<Rightarrow> case nth_of p (2 * (s - 1)) of Some i \<Rightarrow> i | None \<Rightarrow> (Nop, 0) | Oc \<Rightarrow> case nth_of p (2 * (s - 1) +1) of Some i \<Rightarrow> i | None \<Rightarrow> (Nop, 0))"fun new_tape :: "taction \<Rightarrow> tape \<Rightarrow> tape"where "new_tape action (leftn, rightn) = (case action of W0 \<Rightarrow> (leftn, Bk#(tl rightn)) | W1 \<Rightarrow> (leftn, Oc#(tl rightn)) | L \<Rightarrow> (if leftn = [] then (tl leftn, Bk#rightn) else (tl leftn, (hd leftn) # rightn)) | R \<Rightarrow> if rightn = [] then (Bk#leftn,tl rightn) else ((hd rightn)#leftn, tl rightn) | Nop \<Rightarrow> (leftn, rightn) )"text {* The one step function used to transfer Turing machine configuration.*}fun tstep :: "t_conf \<Rightarrow> tprog \<Rightarrow> t_conf" where "tstep c p = (let (s, l, r) = c in let (ac, ns) = (fetch p s (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)) in (ns, new_tape ac (l, r)))"text {* The many-step function.*}fun steps :: "t_conf \<Rightarrow> tprog \<Rightarrow> nat \<Rightarrow> t_conf" where "steps c p 0 = c" | "steps c p (Suc n) = steps (tstep c p) p n"lemma tstep_red: "steps c p (Suc n) = tstep (steps c p n) p"proof(induct n arbitrary: c) fix c show "steps c p (Suc 0) = tstep (steps c p 0) p" by(simp add: steps.simps)next fix n c assume ind: "\<And> c. steps c p (Suc n) = tstep (steps c p n) p" have "steps (tstep c p) p (Suc n) = tstep (steps (tstep c p) p n) p" by(rule ind) thus "steps c p (Suc (Suc n)) = tstep (steps c p (Suc n)) p" by(simp add: steps.simps)qeddeclare Let_def[simp] option.split[split]definition "iseven n \<equiv> \<exists> x. n = 2 * x"text {* The following @{text "t_correct"} function is used to specify the wellformedness of Turing machine.*}fun t_correct :: "tprog \<Rightarrow> bool" where "t_correct p = (length p \<ge> 2 \<and> iseven (length p) \<and> list_all (\<lambda> (acn, s). s \<le> length p div 2) p)"declare t_correct.simps[simp del]lemma allimp: "\<lbrakk>\<forall>x. P x \<longrightarrow> Q x; \<forall>x. P x\<rbrakk> \<Longrightarrow> \<forall>x. Q x"by(auto elim: allE)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)"apply(rule exCI, drule allimp, auto)apply(drule_tac f = f in wf_inv_image, simp add: inv_image_def)apply(erule wf_induct, auto)donelemma steps_add: "steps c t (x + y) = steps (steps c t x) t y"by(induct x arbitrary: c, auto simp: steps.simps tstep_red)lemma listall_set: "list_all p t \<Longrightarrow> \<forall> a \<in> set t. p a"by(induct t, auto)lemma fetch_ex: "\<exists>b a. fetch T aa ab = (b, a)"by(simp add: fetch.simps)definition exponent :: "'a \<Rightarrow> nat \<Rightarrow> 'a list" ("_\<^bsup>_\<^esup>" [0, 0]100) where "exponent x n = replicate n x"text {* @{text "tinres l1 l2"} means left list @{text "l1"} is congruent with @{text "l2"} with respect to the execution of Turing machine. Appending Blank to the right of eigther one does not affect the outcome of excution. *}definition tinres :: "block list \<Rightarrow> block list \<Rightarrow> bool" where "tinres bx by = (\<exists> n. bx = by@Bk\<^bsup>n\<^esup> \<or> by = bx @ Bk\<^bsup>n\<^esup>)"lemma exp_zero: "a\<^bsup>0\<^esup> = []"by(simp add: exponent_def)lemma exp_ind_def: "a\<^bsup>Suc x \<^esup> = a # a\<^bsup>x\<^esup>"by(simp add: exponent_def)text {* The following lemma shows the meaning of @{text "tinres"} with respect to one step execution. *}lemma tinres_step: "\<lbrakk>tinres l l'; tstep (ss, l, r) t = (sa, la, ra); tstep (ss, l', r) t = (sb, lb, rb)\<rbrakk> \<Longrightarrow> tinres la lb \<and> ra = rb \<and> sa = sb"apply(auto simp: tstep.simps fetch.simps new_tape.simps split: if_splits taction.splits list.splits block.splits)apply(case_tac [!] "t ! (2 * (ss - Suc 0))", auto simp: exponent_def tinres_def split: if_splits taction.splits list.splits block.splits)apply(case_tac [!] "t ! (2 * (ss - Suc 0) + Suc 0)", auto simp: exponent_def tinres_def split: if_splits taction.splits list.splits block.splits)donedeclare tstep.simps[simp del] steps.simps[simp del]text {* The following lemma shows the meaning of @{text "tinres"} with respect to many step execution. *}lemma tinres_steps: "\<lbrakk>tinres l l'; steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l', r) t stp = (sb, lb, rb)\<rbrakk> \<Longrightarrow> tinres la lb \<and> ra = rb \<and> sa = sb"apply(induct stp arbitrary: sa la ra sb lb rb, simp add: steps.simps)apply(simp add: tstep_red)apply(case_tac "(steps (ss, l, r) t stp)")apply(case_tac "(steps (ss, l', r) t stp)")proof - fix stp sa la ra sb lb rb a b c aa ba ca assume ind: "\<And>sa la ra sb lb rb. \<lbrakk>steps (ss, l, r) t stp = (sa, la, ra); steps (ss, l', r) t stp = (sb, lb, rb)\<rbrakk> \<Longrightarrow> tinres la lb \<and> ra = rb \<and> sa = sb" and h: " tinres l l'" "tstep (steps (ss, l, r) t stp) t = (sa, la, ra)" "tstep (steps (ss, l', r) t stp) t = (sb, lb, rb)" "steps (ss, l, r) t stp = (a, b, c)" "steps (ss, l', r) t stp = (aa, ba, ca)" have "tinres b ba \<and> c = ca \<and> a = aa" apply(rule_tac ind, simp_all add: h) done thus "tinres la lb \<and> ra = rb \<and> sa = sb" apply(rule_tac l = b and l' = ba and r = c and ss = a and t = t in tinres_step) using h apply(simp, simp, simp) doneqedtext {* The following function @{text "tshift tp n"} is used to shift Turing programs @{text "tp"} by @{text "n"} when it is going to be combined with others. *}fun tshift :: "tprog \<Rightarrow> nat \<Rightarrow> tprog" where "tshift tp off = (map (\<lambda> (action, state). (action, (if state = 0 then 0 else state + off))) tp)"text {* When two Turing programs are combined, the end state (state @{text "0"}) of the one at the prefix position needs to be connected to the start state of the one at postfix position. If @{text "tp"} is the Turing program to be at the prefix, @{text "change_termi_state tp"} is the transformed Turing program. *}fun change_termi_state :: "tprog \<Rightarrow> tprog" where "change_termi_state t = (map (\<lambda> (acn, ns). if ns = 0 then (acn, Suc ((length t) div 2)) else (acn, ns)) t)"text {* @{text "t_add tp1 tp2"} is the combined Truing program.*}fun t_add :: "tprog \<Rightarrow> tprog \<Rightarrow> tprog" ("_ |+| _" [0, 0] 100) where "t_add t1 t2 = ((change_termi_state t1) @ (tshift t2 ((length t1) div 2)))"text {* Tests whether the current configuration is at state @{text "0"}.*}definition isS0 :: "t_conf \<Rightarrow> bool" where "isS0 c = (let (s, l, r) = c in s = 0)"declare tstep.simps[simp del] steps.simps[simp del] t_add.simps[simp del] fetch.simps[simp del] new_tape.simps[simp del]text {* Single step execution starting from state @{text "0"} will not make any progress.*}lemma tstep_0: "tstep (0, tp) p = (0, tp)"apply(simp add: tstep.simps fetch.simps new_tape.simps)donetext {* Many step executions starting from state @{text "0"} will not make any progress.*}lemma steps_0: "steps (0, tp) p stp = (0, tp)"apply(induct stp)apply(simp add: steps.simps)apply(simp add: tstep_red tstep_0)donelemma s_keep_step: "\<lbrakk>a \<le> length A div 2; tstep (a, b, c) A = (s, l, r); t_correct A\<rbrakk> \<Longrightarrow> s \<le> length A div 2"apply(simp add: tstep.simps fetch.simps t_correct.simps iseven_def split: if_splits block.splits list.splits)apply(case_tac [!] a, auto simp: list_all_length)apply(erule_tac x = "2 * nat" in allE, auto)apply(erule_tac x = "2 * nat" in allE, auto)apply(erule_tac x = "Suc (2 * nat)" in allE, auto)donelemma s_keep: "\<lbrakk>steps (Suc 0, tp) A stp = (s, l, r); t_correct A\<rbrakk> \<Longrightarrow> s \<le> length A div 2"proof(induct stp arbitrary: s l r) case 0 thus "?case" by(auto simp: t_correct.simps steps.simps)next fix stp s l r assume ind: "\<And>s l r. \<lbrakk>steps (Suc 0, tp) A stp = (s, l, r); t_correct A\<rbrakk> \<Longrightarrow> s \<le> length A div 2" and h1: "steps (Suc 0, tp) A (Suc stp) = (s, l, r)" and h2: "t_correct A" from h1 h2 show "s \<le> length A div 2" proof(simp add: tstep_red, cases "(steps (Suc 0, tp) A stp)", simp) fix a b c assume h3: "tstep (a, b, c) A = (s, l, r)" and h4: "steps (Suc 0, tp) A stp = (a, b, c)" have "a \<le> length A div 2" using h2 h4 by(rule_tac l = b and r = c in ind, auto) thus "?thesis" using h3 h2 by(simp add: s_keep_step) qedqedlemma t_merge_fetch_pre: "\<lbrakk>fetch A s b = (ac, ns); s \<le> length A div 2; t_correct A; s \<noteq> 0\<rbrakk> \<Longrightarrow> fetch (A |+| B) s b = (ac, if ns = 0 then Suc (length A div 2) else ns)"apply(subgoal_tac "2 * (s - Suc 0) < length A \<and> Suc (2 * (s - Suc 0)) < length A")apply(auto simp: fetch.simps t_add.simps split: if_splits block.splits)apply(simp_all add: nth_append change_termi_state.simps)donelemma [simp]: "\<lbrakk>\<not> a \<le> length A div 2; t_correct A\<rbrakk> \<Longrightarrow> fetch A a b = (Nop, 0)"apply(auto simp: fetch.simps del: nth_of.simps split: block.splits)apply(case_tac [!] a, auto simp: t_correct.simps iseven_def)donelemma [elim]: "\<lbrakk>t_correct A; \<not> isS0 (tstep (a, b, c) A)\<rbrakk> \<Longrightarrow> a \<le> length A div 2"apply(rule_tac classical, auto simp: tstep.simps new_tape.simps isS0_def)donelemma [elim]: "\<lbrakk>t_correct A; \<not> isS0 (tstep (a, b, c) A)\<rbrakk> \<Longrightarrow> 0 < a"apply(rule_tac classical, simp add: tstep_0 isS0_def)donelemma t_merge_pre_eq_step: "\<lbrakk>tstep (a, b, c) A = cf; t_correct A; \<not> isS0 cf\<rbrakk> \<Longrightarrow> tstep (a, b, c) (A |+| B) = cf"apply(subgoal_tac "a \<le> length A div 2 \<and> a \<noteq> 0")apply(simp add: tstep.simps)apply(case_tac "fetch A a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp)apply(drule_tac B = B in t_merge_fetch_pre, simp, simp, simp, simp add: isS0_def, auto)donelemma t_merge_pre_eq: "\<lbrakk>steps (Suc 0, tp) A stp = cf; \<not> isS0 cf; t_correct A\<rbrakk> \<Longrightarrow> steps (Suc 0, tp) (A |+| B) stp = cf"proof(induct stp arbitrary: cf) case 0 thus "?case" by(simp add: steps.simps)next fix stp cf assume ind: "\<And>cf. \<lbrakk>steps (Suc 0, tp) A stp = cf; \<not> isS0 cf; t_correct A\<rbrakk> \<Longrightarrow> steps (Suc 0, tp) (A |+| B) stp = cf" and h1: "steps (Suc 0, tp) A (Suc stp) = cf" and h2: "\<not> isS0 cf" and h3: "t_correct A" from h1 h2 h3 show "steps (Suc 0, tp) (A |+| B) (Suc stp) = cf" proof(simp add: tstep_red, cases "steps (Suc 0, tp) (A) stp", simp) fix a b c assume h4: "tstep (a, b, c) A = cf" and h5: "steps (Suc 0, tp) A stp = (a, b, c)" have "steps (Suc 0, tp) (A |+| B) stp = (a, b, c)" proof(cases a) case 0 thus "?thesis" using h4 h2 apply(simp add: tstep_0, cases cf, simp add: isS0_def) done next case (Suc n) thus "?thesis" using h5 h3 apply(rule_tac ind, auto simp: isS0_def) done qed thus "tstep (steps (Suc 0, tp) (A |+| B) stp) (A |+| B) = cf" using h4 h5 h3 h2 apply(simp) apply(rule t_merge_pre_eq_step, auto) done qedqeddeclare nth.simps[simp del] tshift.simps[simp del] change_termi_state.simps[simp del]lemma [simp]: "length (change_termi_state A) = length A"by(simp add: change_termi_state.simps)lemma first_halt_point: "steps (Suc 0, tp) A stp = (0, tp') \<Longrightarrow> \<exists>stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> steps (Suc 0, tp) A (Suc stp) = (0, tp')"proof(induct stp) case 0 thus "?case" by(simp add: steps.simps)next case (Suc n) fix stp assume ind: "steps (Suc 0, tp) A stp = (0, tp') \<Longrightarrow> \<exists>stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> steps (Suc 0, tp) A (Suc stp) = (0, tp')" and h: "steps (Suc 0, tp) A (Suc stp) = (0, tp')" from h show "\<exists>stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> steps (Suc 0, tp) A (Suc stp) = (0, tp')" proof(simp add: tstep_red, cases "steps (Suc 0, tp) A stp", simp, case_tac a) fix a b c assume g1: "a = (0::nat)" and g2: "tstep (a, b, c) A = (0, tp')" and g3: "steps (Suc 0, tp) A stp = (a, b, c)" have "steps (Suc 0, tp) A stp = (0, tp')" using g2 g1 g3 by(simp add: tstep_0) hence "\<exists> stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> steps (Suc 0, tp) A (Suc stp) = (0, tp')" by(rule ind) thus "\<exists>stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> tstep (steps (Suc 0, tp) A stp) A = (0, tp')" apply(simp add: tstep_red) done next fix a b c nat assume g1: "steps (Suc 0, tp) A stp = (a, b, c)" and g2: "steps (Suc 0, tp) A (Suc stp) = (0, tp')" "a= Suc nat" thus "\<exists>stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> tstep (steps (Suc 0, tp) A stp) A = (0, tp')" apply(rule_tac x = stp in exI) apply(simp add: isS0_def tstep_red) done qedqed lemma t_merge_pre_halt_same': "\<lbrakk>\<not> isS0 (steps (Suc 0, tp) A stp) ; steps (Suc 0, tp) A (Suc stp) = (0, tp'); t_correct A\<rbrakk> \<Longrightarrow> steps (Suc 0, tp) (A |+| B) (Suc stp) = (Suc (length A div 2), tp')" proof(simp add: tstep_red, cases "steps (Suc 0, tp) A stp", simp) fix a b c assume h1: "\<not> isS0 (a, b, c)" and h2: "tstep (a, b, c) A = (0, tp')" and h3: "t_correct A" and h4: "steps (Suc 0, tp) A stp = (a, b, c)" have "steps (Suc 0, tp) (A |+| B) stp = (a, b, c)" using h1 h4 h3 apply(rule_tac t_merge_pre_eq, auto) done moreover have "tstep (a, b, c) (A |+| B) = (Suc (length A div 2), tp')" using h2 h3 h1 h4 apply(simp add: tstep.simps) apply(case_tac " fetch A a (case c of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)", simp) apply(drule_tac B = B in t_merge_fetch_pre, auto simp: isS0_def intro: s_keep) done ultimately show "tstep (steps (Suc 0, tp) (A |+| B) stp) (A |+| B) = (Suc (length A div 2), tp')" by(simp)qedtext {* When Turing machine @{text "A"} and @{text "B"} are combined and the execution of @{text "A"} can termination within @{text "stp"} steps, the combined machine @{text "A |+| B"} will eventually get into the starting state of machine @{text "B"}.*}lemma t_merge_pre_halt_same: " \<lbrakk>steps (Suc 0, tp) A stp = (0, tp'); t_correct A; t_correct B\<rbrakk> \<Longrightarrow> \<exists> stp. steps (Suc 0, tp) (A |+| B) stp = (Suc (length A div 2), tp')"proof - assume a_wf: "t_correct A" and b_wf: "t_correct B" and a_ht: "steps (Suc 0, tp) A stp = (0, tp')" have halt_point: "\<exists> stp. \<not> isS0 (steps (Suc 0, tp) A stp) \<and> steps (Suc 0, tp) A (Suc stp) = (0, tp')" using a_ht by(erule_tac first_halt_point) then obtain stp' where "\<not> isS0 (steps (Suc 0, tp) A stp') \<and> steps (Suc 0, tp) A (Suc stp') = (0, tp')".. hence "steps (Suc 0, tp) (A |+| B) (Suc stp') = (Suc (length A div 2), tp')" using a_wf apply(rule_tac t_merge_pre_halt_same', auto) done thus "?thesis" ..qedlemma fetch_0: "fetch p 0 b = (Nop, 0)"by(simp add: fetch.simps)lemma [simp]: "length (tshift B x) = length B"by(simp add: tshift.simps)lemma [simp]: "t_correct A \<Longrightarrow> 2 * (length A div 2) = length A"apply(simp add: t_correct.simps iseven_def, auto)donelemma t_merge_fetch_snd: "\<lbrakk>fetch B a b = (ac, ns); t_correct A; t_correct B; a > 0 \<rbrakk> \<Longrightarrow> fetch (A |+| B) (a + length A div 2) b = (ac, if ns = 0 then 0 else ns + length A div 2)"apply(auto simp: fetch.simps t_add.simps split: if_splits block.splits)apply(case_tac [!] a, simp_all)apply(simp_all add: nth_append change_termi_state.simps tshift.simps)donelemma t_merge_snd_eq_step: "\<lbrakk>tstep (s, l, r) B = (s', l', r'); t_correct A; t_correct B; s > 0\<rbrakk> \<Longrightarrow> tstep (s + length A div 2, l, r) (A |+| B) = (if s' = 0 then 0 else s' + length A div 2, l' ,r') "apply(simp add: tstep.simps)apply(cases "fetch B s (case r of [] \<Rightarrow> Bk | x # xs \<Rightarrow> x)")apply(auto simp: t_merge_fetch_snd)apply(frule_tac [!] t_merge_fetch_snd, auto)done text {* Relates the executions of TM @{text "B"}, one is when @{text "B"} is executed alone, the other is the execution when @{text "B"} is in the combined TM.*}lemma t_merge_snd_eq_steps: "\<lbrakk>t_correct A; t_correct B; steps (s, l, r) B stp = (s', l', r'); s > 0\<rbrakk> \<Longrightarrow> steps (s + length A div 2, l, r) (A |+| B) stp = (if s' = 0 then 0 else s' + length A div 2, l', r')"proof(induct stp arbitrary: s' l' r') case 0 thus "?case" by(simp add: steps.simps)next fix stp s' l' r' assume ind: "\<And>s' l' r'. \<lbrakk>t_correct A; t_correct B; steps (s, l, r) B stp = (s', l', r'); 0 < s\<rbrakk> \<Longrightarrow> steps (s + length A div 2, l, r) (A |+| B) stp = (if s' = 0 then 0 else s' + length A div 2, l', r')" and h1: "steps (s, l, r) B (Suc stp) = (s', l', r')" and h2: "t_correct A" and h3: "t_correct B" and h4: "0 < s" from h1 show "steps (s + length A div 2, l, r) (A |+| B) (Suc stp) = (if s' = 0 then 0 else s' + length A div 2, l', r')" proof(simp only: tstep_red, cases "steps (s, l, r) B stp") fix a b c assume h5: "steps (s, l, r) B stp = (a, b, c)" "tstep (steps (s, l, r) B stp) B = (s', l', r')" hence h6: "(steps (s + length A div 2, l, r) (A |+| B) stp) = ((if a = 0 then 0 else a + length A div 2, b, c))" using h2 h3 h4 by(rule_tac ind, auto) thus "tstep (steps (s + length A div 2, l, r) (A |+| B) stp) (A |+| B) = (if s' = 0 then 0 else s'+ length A div 2, l', r')" using h5 proof(auto) assume "tstep (0, b, c) B = (0, l', r')" thus "tstep (0, b, c) (A |+| B) = (0, l', r')" by(simp add: tstep_0) next assume "tstep (0, b, c) B = (s', l', r')" "0 < s'" thus "tstep (0, b, c) (A |+| B) = (s' + length A div 2, l', r')" by(simp add: tstep_0) next assume "tstep (a, b, c) B = (0, l', r')" "0 < a" thus "tstep (a + length A div 2, b, c) (A |+| B) = (0, l', r')" using h2 h3 by(drule_tac t_merge_snd_eq_step, auto) next assume "tstep (a, b, c) B = (s', l', r')" "0 < a" "0 < s'" thus "tstep (a + length A div 2, b, c) (A |+| B) = (s' + length A div 2, l', r')" using h2 h3 by(drule_tac t_merge_snd_eq_step, auto) qed qedqedlemma t_merge_snd_halt_eq: "\<lbrakk>steps (Suc 0, tp) B stp = (0, tp'); t_correct A; t_correct B\<rbrakk> \<Longrightarrow> \<exists>stp. steps (Suc (length A div 2), tp) (A |+| B) stp = (0, tp')"apply(case_tac tp, cases tp', simp)apply(drule_tac s = "Suc 0" in t_merge_snd_eq_steps, auto)donelemma t_inj: "\<lbrakk>steps (Suc 0, tp) A stpa = (0, tp1); steps (Suc 0, tp) A stpb = (0, tp2)\<rbrakk> \<Longrightarrow> tp1 = tp2"proof - assume h1: "steps (Suc 0, tp) A stpa = (0, tp1)" and h2: "steps (Suc 0, tp) A stpb = (0, tp2)" thus "?thesis" proof(cases "stpa < stpb") case True thus "?thesis" using h1 h2 apply(drule_tac less_imp_Suc_add, auto) apply(simp del: add_Suc_right add_Suc add: add_Suc_right[THEN sym] steps_add steps_0) done next case False thus "?thesis" using h1 h2 apply(drule_tac leI) apply(case_tac "stpb = stpa", auto) apply(subgoal_tac "stpb < stpa") apply(drule_tac less_imp_Suc_add, auto) apply(simp del: add_Suc_right add_Suc add: add_Suc_right[THEN sym] steps_add steps_0) done qedqedtype_synonym t_assert = "tape \<Rightarrow> bool"definition t_imply :: "t_assert \<Rightarrow> t_assert \<Rightarrow> bool" ("_ \<turnstile>-> _" [0, 0] 100) where "t_imply a1 a2 = (\<forall> tp. a1 tp \<longrightarrow> a2 tp)"locale turing_merge = fixes A :: "tprog" and B :: "tprog" and P1 :: "t_assert" and P2 :: "t_assert" and P3 :: "t_assert" and P4 :: "t_assert" and Q1:: "t_assert" and Q2 :: "t_assert" assumes A_wf : "t_correct A" and B_wf : "t_correct 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. isS0 (steps (Suc 0, tp) A stp))" and B_uhalt: "P4 tp \<Longrightarrow> \<not> (\<exists> stp. isS0 (steps (Suc 0, tp) B stp))"begintext {* 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_halt: assumes aimpb: "Q1 \<turnstile>-> P2" shows "P1 \<turnstile>-> \<lambda> tp. (\<exists> stp tp'. steps (Suc 0, tp) (A |+| B) stp = (0, tp') \<and> Q2 tp')"proof(simp add: t_imply_def, auto) fix a b assume h: "P1 (a, b)" hence "\<exists> stp. let (s, tp') = steps (Suc 0, a, b) A stp in s = 0 \<and> Q1 tp'" using A_halt by simp from this obtain stp1 where "let (s, tp') = steps (Suc 0, a, b) A stp1 in s = 0 \<and> Q1 tp'" .. thus "\<exists>stp aa ba. steps (Suc 0, a, b) (A |+| B) stp = (0, aa, ba) \<and> Q2 (aa, ba)" 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: t_imply_def) done hence "\<exists> stp. let (s, tp') = steps (Suc 0, ba, c) B stp in s = 0 \<and> Q2 tp'" using B_halt 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 by(rule_tac t_merge_pre_halt_same, auto) moreover have "\<exists> stp. steps (Suc (length A div 2), ba, c) (A |+| B) stp = (0, bb, ca)" using g3 A_wf B_wf apply(rule_tac t_merge_snd_halt_eq, auto) done ultimately show "\<exists>stp aa ba. steps (Suc 0, a, b) (A |+| B) stp = (0, aa, ba) \<and> Q2 (aa, ba)" apply(erule_tac exE, erule_tac exE) apply(rule_tac x = "stp + stpa" in exI, simp add: steps_add) using g3 by simp qed qedqedlemma t_merge_uhalt_tmp: assumes B_uh: "\<forall>stp. \<not> isS0 (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> isS0 (steps (Suc 0, tp) (A |+| B) stp)" using B_uh merge_ahapply(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> isS0 (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: isS0_def) moreover have "\<exists> stpb. stp = stpa + stpb" using h1 by(rule_tac x = "stp - stpa" in exI, simp) ultimately show "\<not> isS0 (steps (Suc 0, tp) (A |+| B) stp)" using merge_ah by(auto simp: steps_add isS0_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> isS0 (steps (Suc 0, tp) (A |+| B) stp)" using h apply(auto) apply(cases "steps (Suc 0, tp) (A |+| B) stp", simp add: steps_add isS0_def steps_0) doneqedtext {* The following lemma deals with the situation when TM @{text "B"} can not terminate. *}lemma t_merge_uhalt: assumes aimpb: "Q1 \<turnstile>-> P4" shows "P1 \<turnstile>-> \<lambda> tp. \<not> (\<exists> stp. isS0 (steps (Suc 0, tp) (A |+| B) stp))"proof(simp only: t_imply_def, rule_tac allI, rule_tac impI) fix tp assume init_asst: "P1 tp" show "\<not> (\<exists>stp. isS0 (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: t_imply_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. isS0 (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> isS0 (steps (Suc 0, tp) (A |+| B) stp)" by(rule_tac t_merge_uhalt_tmp, auto) qed qedqedendend