Theory Machines

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theory Machines
imports Natural
begin

(* $Id: Machines.thy,v 1.11 2007/07/11 09:18:52 berghofe Exp $ *)

theory Machines imports Natural begin

lemma rtrancl_eq: "R^* = Id ∪ (R O R^*)"
  by (fast intro: rtrancl_into_rtrancl elim: rtranclE)

lemma converse_rtrancl_eq: "R^* = Id ∪ (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 n) (simp, simp add: O_assoc [symmetric])

lemma converse_in_rel_pow_eq:
  "((x,z) ∈ R^n) = (n=0 ∧ z=x ∨ (∃m y. n = Suc m ∧ (x,y) ∈ R ∧ (y,z) ∈ 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: "[| (x,y) ∈ R^m; (y,z) ∈ R^n |] ==> (x,z) ∈ 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"

inductive_set
  exec01 :: "instr list => ((nat×state) × (nat×state))set"
  and exec01' :: "[instrs, nat,state, nat,state] => bool"
    ("(_/ \<turnstile> (1⟨_,/_⟩)/ -1-> (1⟨_,/_⟩))" [50,0,0,0,0] 50)
  for P :: "instr list"
where
  "p \<turnstile> ⟨i,s⟩ -1-> ⟨j,t⟩ == ((i,s),j,t) : (exec01 p)"
| SET: "[| n<size P; P!n = SET x a |] ==> P \<turnstile> ⟨n,s⟩ -1-> ⟨Suc n,s[x\<mapsto> a s]⟩"
| JMPFT: "[| n<size P; P!n = JMPF b i;  b s |] ==> P \<turnstile> ⟨n,s⟩ -1-> ⟨Suc n,s⟩"
| JMPFF: "[| n<size P; P!n = JMPF b i; ¬b s; m=n+i+1; m ≤ size P |]
        ==> P \<turnstile> ⟨n,s⟩ -1-> ⟨m,s⟩"
| JMPB:  "[| n<size P; P!n = JMPB i; i ≤ n; j = n-i |] ==> P \<turnstile> ⟨n,s⟩ -1-> ⟨j,s⟩"

abbreviation
  exec0s :: "[instrs, nat,state, nat,state] => bool"
    ("(_/ \<turnstile> (1⟨_,/_⟩)/ -*-> (1⟨_,/_⟩))" [50,0,0,0,0] 50)  where
  "p \<turnstile> ⟨i,s⟩ -*-> ⟨j,t⟩ == ((i,s),j,t) : (exec01 p)^*"

abbreviation
  exec0n :: "[instrs, nat,state, nat, nat,state] => bool"
    ("(_/ \<turnstile> (1⟨_,/_⟩)/ -_-> (1⟨_,/_⟩))" [50,0,0,0,0] 50)  where
  "p \<turnstile> ⟨i,s⟩ -n-> ⟨j,t⟩ == ((i,s),j,t) : (exec01 p)^n"

subsection "M0 with lists"

text {* We describe execution of programs in the machine by
  an operational (small step) semantics:
*}

types config = "instrs × instrs × state"


inductive_set
  stepa1 :: "(config × config)set"
  and stepa1' :: "[instrs,instrs,state, instrs,instrs,state] => bool"
    ("((1⟨_,/_,/_⟩)/ -1-> (1⟨_,/_,/_⟩))" 50)
where
  "⟨p,q,s⟩ -1-> ⟨p',q',t⟩ == ((p,q,s),p',q',t) : stepa1"
| "⟨SET x a#p,q,s⟩ -1-> ⟨p,SET x a#q,s[x\<mapsto> a s]⟩"
| "b s ==> ⟨JMPF b i#p,q,s⟩ -1-> ⟨p,JMPF b i#q,s⟩"
| "[| ¬ b s; i ≤ size p |]
   ==> ⟨JMPF b i # p, q, s⟩ -1-> ⟨drop i p, rev(take i p) @ JMPF b i # q, s⟩"
| "i ≤ size q
   ==> ⟨JMPB i # p, q, s⟩ -1-> ⟨rev(take i q) @ JMPB i # p, drop i q, s⟩"

abbreviation
  stepa :: "[instrs,instrs,state, instrs,instrs,state] => bool"
    ("((1⟨_,/_,/_⟩)/ -*-> (1⟨_,/_,/_⟩))" 50)  where
  "⟨p,q,s⟩ -*-> ⟨p',q',t⟩ == ((p,q,s),p',q',t) : (stepa1^*)"

abbreviation
  stepan :: "[instrs,instrs,state, nat, instrs,instrs,state] => bool"
    ("((1⟨_,/_,/_⟩)/ -_-> (1⟨_,/_,/_⟩))" 50) where
  "⟨p,q,s⟩ -i-> ⟨p',q',t⟩ == ((p,q,s),p',q',t) : (stepa1^i)"

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

lemma exec_simp[simp]:
 "(⟨i#p,q,s⟩ -1-> ⟨p',q',t⟩) = (case i of
 SET x a => t = s[x\<mapsto> a s] ∧ p' = p ∧ q' = i#q |
 JMPF b n => t=s ∧ (if b s then p' = p ∧ q' = i#q
            else n ≤ size p ∧ p' = drop n p ∧ q' = rev(take n p) @ i # q) |
 JMPB n => n ≤ size q ∧ t=s ∧ p' = rev(take n q) @ i # p ∧ 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]:
"(⟨i#p,q,s⟩ -n-> ⟨p'',q'',u⟩) =
 (n=0 ∧ p'' = i#p ∧ q'' = q ∧ u = s ∨
  ((∃m p' q' t. n = Suc m ∧
                ⟨i#p,q,s⟩ -1-> ⟨p',q',t⟩ ∧ ⟨p',q',t⟩ -m-> ⟨p'',q'',u⟩)))"
by(subst converse_in_rel_pow_eq, simp)


lemma exec_star_simp[simp]: "(⟨i#p,q,s⟩ -*-> ⟨p'',q'',u⟩) =
 (p'' = i#p & q''=q & u=s |
 (∃p' q' t. ⟨i#p,q,s⟩ -1-> ⟨p',q',t⟩ ∧ ⟨p',q',t⟩ -*-> ⟨p'',q'',u⟩))"
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 ==> 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:
 "⟨q,p,s⟩ -1-> ⟨q',p',t⟩ ==>
  rev p' @ q' = rev p @ q ∧ rev p @ q \<turnstile> ⟨size p,s⟩ -1-> ⟨size p',t⟩"
apply(induct set: stepa1)
   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 simp
apply(fastsimp simp add:exec01.JMPB)
done

(*
lemma rev_take: "!!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: "!!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> ⟨sp,s⟩ -1-> ⟨sp',t⟩ ==>
  rpq = rev p @ q & sp = size p & sp' = size p' -->
          rev p' @ q' = rev p @ q --> ⟨q,p,s⟩ -1-> ⟨q',p',t⟩"
apply(induct arbitrary: p q p' q' set: exec01)
   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:
"(⟨q,p,s⟩ -1-> ⟨q',p',t⟩) =
 (rev p' @ q' = rev p @ q ∧ rev p @ q \<turnstile> ⟨size p,s⟩ -1-> ⟨size p',t⟩)"
  by (blast dest: direction1 direction2)

end

lemma rtrancl_eq:

  R* = IdR O R*

lemma converse_rtrancl_eq:

  R* = IdR* O R

lemma converse_rel_powE:

  [| (x, z) ∈ R ^ n; [| n = 0; x = z |] ==> P;
     !!y m. [| n = Suc m; (x, y) ∈ R; (y, z) ∈ R ^ m |] ==> P |]
  ==> P

lemma R_O_Rn_commute:

  R O R ^ n = R ^ n O R

lemma converse_in_rel_pow_eq:

  ((x, z) ∈ R ^ n) =
  (n = 0z = x ∨ (∃m y. n = Suc m ∧ (x, y) ∈ R ∧ (y, z) ∈ R ^ m))

lemma rel_pow_plus:

  R ^ (m + n) = R ^ n O R ^ m

lemma rel_pow_plusI:

  [| (x, y) ∈ R ^ m; (y, z) ∈ R ^ n |] ==> (x, z) ∈ R ^ (m + n)

Instructions

M0 with PC

M0 with lists

lemma exec_simp:

  (i # p,q,s -1-> p',q',t) =
  (case i of SET x a => t = s[x ::= a s] ∧ p' = pq' = i # q
   | JMPF b n =>
       t = s ∧
       (if b s then p' = pq' = i # q
        else n  length pp' = drop n pq' = rev (take n p) @ i # q)
   | JMPB n => n  length qt = sp' = rev (take n q) @ i # pq' = drop n q)

lemma execn_simp:

  (i # p,q,s -n-> p'',q'',u) =
  (n = 0p'' = i # pq'' = qu = s ∨
   (∃m p' q' t.
       n = Suc mi # p,q,s -1-> p',q',tp',q',t -m-> p'',q'',u))

lemma exec_star_simp:

  (i # p,q,s -*-> p'',q'',u) =
  (p'' = i # pq'' = qu = s ∨
   (∃p' q' t. i # p,q,s -1-> p',q',tp',q',t -*-> p'',q'',u))

lemma rev_revD:

  rev xs = rev ys ==> xs = ys

lemma

  (rev xs @ rev ys = rev zs) = (ys @ xs = zs)

lemma direction1:

  q,p,s -1-> q',p',t
  ==> rev p' @ q' = rev p @ q ∧
      rev p @ q \<turnstile> length p,s -1-> length p',t

lemma direction2:

  rpq \<turnstile> sp,s -1-> sp',t
  ==> rpq = rev p @ qsp = length psp' = length p' -->
      rev p' @ q' = rev p @ q --> q,p,s -1-> q',p',t

theorem M_eqiv:

  (q,p,s -1-> q',p',t) =
  (rev p' @ q' = rev p @ q ∧
   rev p @ q \<turnstile> length p,s -1-> length p',t)