theory Machines = Natural:

 lemma rtrancl_eq: "R^* = Id \<union> (R O R^*)"

 by(fast intro:rtrancl.intros elim:rtranclE)

 lemma converse_rtrancl_eq: "R^* = Id \<union> (R^* O R)"

 by(subst r_comp_rtrancl_eq[symmetric], rule rtrancl_eq)

 lemmas converse_rel_powE = rel_pow_E2

 lemma R_O_Rn_commute: "R O R^n = R^n O R"

 by(induct_tac n, simp, simp add: O_assoc[symmetric])

 lemma converse_in_rel_pow_eq:

 "((x,z) \<in> R^n) = (n=0 \<and> z=x \<or> (\<exists>m y. n = Suc m \<and> (x,y) \<in> R \<and> (y,z) \<in> R^m))"

 apply(rule iffI)

  apply(blast elim:converse_rel_powE)

 apply (fastsimp simp add:gr0_conv_Suc R_O_Rn_commute)

 done

 lemma rel_pow_plus: "R^(m+n) = R^n O R^m"

 by(induct n, simp, simp add:O_assoc)

 lemma rel_pow_plusI: "\<lbrakk> (x,y) \<in> R^m; (y,z) \<in> R^n \<rbrakk> \<Longrightarrow> (x,z) \<in> R^(m+n)"

 by(simp add:rel_pow_plus rel_compI)

 subsection "Instructions"

 text {* There are only three instructions: *}

 datatype instr = SET loc aexp | JMPF bexp nat | JMPB nat

 types instrs = "instr list"

 subsection "M0 with PC"

 consts  exec01 :: "instr list \<Rightarrow> ((nat\<times>state) \<times> (nat\<times>state))set" 

 syntax

   "_exec01" :: "[instrs, nat,state, nat,state] \<Rightarrow> bool"

                ("(_/ |- (1<_,/_>)/ -1-> (1<_,/_>))" [50,0,0,0,0] 50)

   "_exec0s" :: "[instrs, nat,state, nat,state] \<Rightarrow> bool"

                ("(_/ |- (1<_,/_>)/ -*-> (1<_,/_>))" [50,0,0,0,0] 50)

   "_exec0n" :: "[instrs, nat,state, nat, nat,state] \<Rightarrow> bool"

                ("(_/ |- (1<_,/_>)/ -_-> (1<_,/_>))" [50,0,0,0,0] 50)

 syntax (xsymbols)

   "_exec01" :: "[instrs, nat,state, nat,state] \<Rightarrow> bool"

                ("(_/ \<turnstile> (1\<langle>_,/_\<rangle>)/ -1\<rightarrow> (1\<langle>_,/_\<rangle>))" [50,0,0,0,0] 50)

   "_exec0s" :: "[instrs, nat,state, nat,state] \<Rightarrow> bool"

                ("(_/ \<turnstile> (1\<langle>_,/_\<rangle>)/ -*\<rightarrow> (1\<langle>_,/_\<rangle>))" [50,0,0,0,0] 50)

   "_exec0n" :: "[instrs, nat,state, nat, nat,state] \<Rightarrow> bool"

                ("(_/ \<turnstile> (1\<langle>_,/_\<rangle>)/ -_\<rightarrow> (1\<langle>_,/_\<rangle>))" [50,0,0,0,0] 50)

 syntax (HTML output)

   "_exec01" :: "[instrs, nat,state, nat,state] \<Rightarrow> bool"

                ("(_/ |- (1\<langle>_,/_\<rangle>)/ -1\<rightarrow> (1\<langle>_,/_\<rangle>))" [50,0,0,0,0] 50)

   "_exec0s" :: "[instrs, nat,state, nat,state] \<Rightarrow> bool"

                ("(_/ |- (1\<langle>_,/_\<rangle>)/ -*\<rightarrow> (1\<langle>_,/_\<rangle>))" [50,0,0,0,0] 50)

   "_exec0n" :: "[instrs, nat,state, nat, nat,state] \<Rightarrow> bool"

                ("(_/ |- (1\<langle>_,/_\<rangle>)/ -_\<rightarrow> (1\<langle>_,/_\<rangle>))" [50,0,0,0,0] 50)

 translations  

   "p \<turnstile> \<langle>i,s\<rangle> -1\<rightarrow> \<langle>j,t\<rangle>" == "((i,s),j,t) : (exec01 p)"

   "p \<turnstile> \<langle>i,s\<rangle> -*\<rightarrow> \<langle>j,t\<rangle>" == "((i,s),j,t) : (exec01 p)^*"

   "p \<turnstile> \<langle>i,s\<rangle> -n\<rightarrow> \<langle>j,t\<rangle>" == "((i,s),j,t) : (exec01 p)^n"

 inductive "exec01 P"

 intros

 SET: "\<lbrakk> n<size P; P!n = SET x a \<rbrakk> \<Longrightarrow> P \<turnstile> \<langle>n,s\<rangle> -1\<rightarrow> \<langle>Suc n,s[x\<mapsto> a s]\<rangle>"

 JMPFT: "\<lbrakk> n<size P; P!n = JMPF b i;  b s \<rbrakk> \<Longrightarrow> P \<turnstile> \<langle>n,s\<rangle> -1\<rightarrow> \<langle>Suc n,s\<rangle>"

 JMPFF: "\<lbrakk> n<size P; P!n = JMPF b i; \<not>b s; m=n+i+1; m \<le> size P \<rbrakk>

         \<Longrightarrow> P \<turnstile> \<langle>n,s\<rangle> -1\<rightarrow> \<langle>m,s\<rangle>"

 JMPB:  "\<lbrakk> n<size P; P!n = JMPB i; i \<le> n; j = n-i \<rbrakk> \<Longrightarrow> P \<turnstile> \<langle>n,s\<rangle> -1\<rightarrow> \<langle>j,s\<rangle>"

 subsection "M0 with lists"

 text {* We describe execution of programs in the machine by

   an operational (small step) semantics:

 *}

 types config = "instrs \<times> instrs \<times> state"

 consts  stepa1 :: "(config \<times> config)set"

 syntax

   "_stepa1" :: "[instrs,instrs,state, instrs,instrs,state] \<Rightarrow> bool"

                ("((1<_,/_,/_>)/ -1-> (1<_,/_,/_>))" 50)

   "_stepa" :: "[instrs,instrs,state, instrs,instrs,state] \<Rightarrow> bool"

                ("((1<_,/_,/_>)/ -*-> (1<_,/_,/_>))" 50)

   "_stepan" :: "[state,instrs,instrs, nat, instrs,instrs,state] \<Rightarrow> bool"

                ("((1<_,/_,/_>)/ -_-> (1<_,/_,/_>))" 50)

 syntax (xsymbols)

   "_stepa1" :: "[instrs,instrs,state, instrs,instrs,state] \<Rightarrow> bool"

                ("((1\<langle>_,/_,/_\<rangle>)/ -1\<rightarrow> (1\<langle>_,/_,/_\<rangle>))" 50)

   "_stepa" :: "[instrs,instrs,state, instrs,instrs,state] \<Rightarrow> bool"

                ("((1\<langle>_,/_,/_\<rangle>)/ -*\<rightarrow> (1\<langle>_,/_,/_\<rangle>))" 50)

   "_stepan" :: "[instrs,instrs,state, nat, instrs,instrs,state] \<Rightarrow> bool"

                ("((1\<langle>_,/_,/_\<rangle>)/ -_\<rightarrow> (1\<langle>_,/_,/_\<rangle>))" 50)

 translations  

   "\<langle>p,q,s\<rangle> -1\<rightarrow> \<langle>p',q',t\<rangle>" == "((p,q,s),p',q',t) : stepa1"

   "\<langle>p,q,s\<rangle> -*\<rightarrow> \<langle>p',q',t\<rangle>" == "((p,q,s),p',q',t) : (stepa1^*)"

   "\<langle>p,q,s\<rangle> -i\<rightarrow> \<langle>p',q',t\<rangle>" == "((p,q,s),p',q',t) : (stepa1^i)"

 inductive "stepa1"

 intros

   "\<langle>SET x a#p,q,s\<rangle> -1\<rightarrow> \<langle>p,SET x a#q,s[x\<mapsto> a s]\<rangle>"

   "b s \<Longrightarrow> \<langle>JMPF b i#p,q,s\<rangle> -1\<rightarrow> \<langle>p,JMPF b i#q,s\<rangle>"

   "\<lbrakk> \<not> b s; i \<le> size p \<rbrakk>

    \<Longrightarrow> \<langle>JMPF b i # p, q, s\<rangle> -1\<rightarrow> \<langle>drop i p, rev(take i p) @ JMPF b i # q, s\<rangle>"

   "i \<le> size q

    \<Longrightarrow> \<langle>JMPB i # p, q, s\<rangle> -1\<rightarrow> \<langle>rev(take i q) @ JMPB i # p, drop i q, s\<rangle>"

 inductive_cases execE: "((i#is,p,s),next) : stepa1"

 lemma exec_simp[simp]:

  "(\<langle>i#p,q,s\<rangle> -1\<rightarrow> \<langle>p',q',t\<rangle>) = (case i of

  SET x a \<Rightarrow> t = s[x\<mapsto> a s] \<and> p' = p \<and> q' = i#q |

  JMPF b n \<Rightarrow> t=s \<and> (if b s then p' = p \<and> q' = i#q

             else n \<le> size p \<and> p' = drop n p \<and> q' = rev(take n p) @ i # q) |

  JMPB n \<Rightarrow> n \<le> size q \<and> t=s \<and> p' = rev(take n q) @ i # p \<and> q' = drop n q)"

 apply(rule iffI)

 defer

 apply(clarsimp simp add: stepa1.intros split: instr.split_asm split_if_asm)

 apply(erule execE)

 apply(simp_all)

 done

 lemma execn_simp[simp]:

 "(\<langle>i#p,q,s\<rangle> -n\<rightarrow> \<langle>p'',q'',u\<rangle>) =

  (n=0 \<and> p'' = i#p \<and> q'' = q \<and> u = s \<or>

   ((\<exists>m p' q' t. n = Suc m \<and>

                 \<langle>i#p,q,s\<rangle> -1\<rightarrow> \<langle>p',q',t\<rangle> \<and> \<langle>p',q',t\<rangle> -m\<rightarrow> \<langle>p'',q'',u\<rangle>)))"

 by(subst converse_in_rel_pow_eq, simp)

 lemma exec_star_simp[simp]: "(\<langle>i#p,q,s\<rangle> -*\<rightarrow> \<langle>p'',q'',u\<rangle>) =

  (p'' = i#p & q''=q & u=s |

  (\<exists>p' q' t. \<langle>i#p,q,s\<rangle> -1\<rightarrow> \<langle>p',q',t\<rangle> \<and> \<langle>p',q',t\<rangle> -*\<rightarrow> \<langle>p'',q'',u\<rangle>))"

 apply(simp add: rtrancl_is_UN_rel_pow del:exec_simp)

 apply(blast)

 done

 declare nth_append[simp]

 lemma rev_revD: "rev xs = rev ys \<Longrightarrow> xs = ys"

 by simp

 lemma [simp]: "(rev xs @ rev ys = rev zs) = (ys @ xs = zs)"

 apply(rule iffI)

  apply(rule rev_revD, simp)

 apply fastsimp

 done

 lemma direction1:

  "\<langle>q,p,s\<rangle> -1\<rightarrow> \<langle>q',p',t\<rangle> \<Longrightarrow>

   rev p' @ q' = rev p @ q \<and> rev p @ q \<turnstile> \<langle>size p,s\<rangle> -1\<rightarrow> \<langle>size p',t\<rangle>"

 apply(erule stepa1.induct)

    apply(simp add:exec01.SET)

   apply(fastsimp intro:exec01.JMPFT)

  apply simp

  apply(rule exec01.JMPFF)

      apply simp

     apply fastsimp

    apply simp

   apply simp

   apply arith

  apply simp

  apply arith

 apply(fastsimp simp add:exec01.JMPB)

 done

 (*

 lemma rev_take: "\<And>i. rev (take i xs) = drop (length xs - i) (rev xs)"

 apply(induct xs)

  apply simp_all

 apply(case_tac i)

 apply simp_all

 done

 lemma rev_drop: "\<And>i. rev (drop i xs) = take (length xs - i) (rev xs)"

 apply(induct xs)

  apply simp_all

 apply(case_tac i)

 apply simp_all

 done

 *)

 lemma direction2:

  "rpq \<turnstile> \<langle>sp,s\<rangle> -1\<rightarrow> \<langle>sp',t\<rangle> \<Longrightarrow>

  \<forall>p q p' q'. rpq = rev p @ q & sp = size p & sp' = size p' \<longrightarrow>

           rev p' @ q' = rev p @ q \<longrightarrow> \<langle>q,p,s\<rangle> -1\<rightarrow> \<langle>q',p',t\<rangle>"

 apply(erule exec01.induct)

    apply(clarsimp simp add: neq_Nil_conv append_eq_conv_conj)

    apply(drule sym)

    apply simp

    apply(rule rev_revD)

    apply simp

   apply(clarsimp simp add: neq_Nil_conv append_eq_conv_conj)

   apply(drule sym)

   apply simp

   apply(rule rev_revD)

   apply simp

  apply(simp (no_asm_use) add: neq_Nil_conv append_eq_conv_conj, clarify)+

  apply(drule sym)

  apply simp

  apply(rule rev_revD)

  apply simp

 apply(clarsimp simp add: neq_Nil_conv append_eq_conv_conj)

 apply(drule sym)

 apply(simp add:rev_take)

 apply(rule rev_revD)

 apply(simp add:rev_drop)

 done

 theorem M_eqiv:

 "(\<langle>q,p,s\<rangle> -1\<rightarrow> \<langle>q',p',t\<rangle>) =

  (rev p' @ q' = rev p @ q \<and> rev p @ q \<turnstile> \<langle>size p,s\<rangle> -1\<rightarrow> \<langle>size p',t\<rangle>)"

 by(fast dest:direction1 direction2)

 end