(*  Title:      HOL/IMP/Compiler.thy

     ID:         $Id$

     Author:     Tobias Nipkow, TUM

     Copyright   1996 TUM

 *)

 theory Compiler imports Machines begin

 subsection "The compiler"

 primrec compile :: "com \<Rightarrow> instr list"

 where

   "compile \<SKIP> = []"

 | "compile (x:==a) = [SET x a]"

 | "compile (c1;c2) = compile c1 @ compile c2"

 | "compile (\<IF> b \<THEN> c1 \<ELSE> c2) =

     [JMPF b (length(compile c1) + 1)] @ compile c1 @

     [JMPF (\<lambda>x. False) (length(compile c2))] @ compile c2"

 | "compile (\<WHILE> b \<DO> c) = [JMPF b (length(compile c) + 1)] @ compile c @

     [JMPB (length(compile c)+1)]"

 subsection "Compiler correctness"

 theorem assumes A: "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t"

 shows "\<And>p q. \<langle>compile c @ p,q,s\<rangle> -*\<rightarrow> \<langle>p,rev(compile c)@q,t\<rangle>"

   (is "\<And>p q. ?P c s t p q")

 proof -

   from A show "\<And>p q. ?thesis p q"

   proof induct

     case Skip thus ?case by simp

   next

     case Assign thus ?case by force

   next

     case Semi thus ?case by simp (blast intro:rtrancl_trans)

   next

     fix b c0 c1 s0 s1 p q

     assume IH: "\<And>p q. ?P c0 s0 s1 p q"

     assume "b s0"

     thus "?P (\<IF> b \<THEN> c0 \<ELSE> c1) s0 s1 p q"

       by(simp add: IH[THEN rtrancl_trans])

   next

     case IfFalse thus ?case by(simp)

   next

     case WhileFalse thus ?case by simp

   next

     fix b c and s0::state and s1 s2 p q

     assume b: "b s0" and

       IHc: "\<And>p q. ?P c s0 s1 p q" and

       IHw: "\<And>p q. ?P (\<WHILE> b \<DO> c) s1 s2 p q"

     show "?P (\<WHILE> b \<DO> c) s0 s2 p q"

       using b  IHc[THEN rtrancl_trans] IHw by(simp)

   qed

 qed

 text {* The other direction! *}

 inductive_cases [elim!]: "(([],p,s),(is',p',s')) : stepa1"

 lemma [simp]: "(\<langle>[],q,s\<rangle> -n\<rightarrow> \<langle>p',q',t\<rangle>) = (n=0 \<and> p' = [] \<and> q' = q \<and> t = s)"

 apply(rule iffI)

  apply(erule rel_pow_E2, simp, fast)

 apply simp

 done

 lemma [simp]: "(\<langle>[],q,s\<rangle> -*\<rightarrow> \<langle>p',q',t\<rangle>) = (p' = [] \<and> q' = q \<and> t = s)"

 by(simp add: rtrancl_is_UN_rel_pow)

 definition

   forws :: "instr \<Rightarrow> nat set" where

   "forws instr = (case instr of

      SET x a \<Rightarrow> {0} |

      JMPF b n \<Rightarrow> {0,n} |

      JMPB n \<Rightarrow> {})"

 definition

   backws :: "instr \<Rightarrow> nat set" where

   "backws instr = (case instr of

      SET x a \<Rightarrow> {} |

      JMPF b n \<Rightarrow> {} |

      JMPB n \<Rightarrow> {n})"

 primrec closed :: "nat \<Rightarrow> nat \<Rightarrow> instr list \<Rightarrow> bool"

 where

   "closed m n [] = True"

 | "closed m n (instr#is) = ((\<forall>j \<in> forws instr. j \<le> size is+n) \<and>

                           (\<forall>j \<in> backws instr. j \<le> m) \<and> closed (Suc m) n is)"

 lemma [simp]:

  "\<And>m n. closed m n (C1@C2) =

          (closed m (n+size C2) C1 \<and> closed (m+size C1) n C2)"

 by(induct C1) (simp, simp add:add_ac)

 theorem [simp]: "\<And>m n. closed m n (compile c)"

 by(induct c) (simp_all add:backws_def forws_def)

 lemma drop_lem: "n \<le> size(p1@p2)

  \<Longrightarrow> (p1' @ p2 = drop n p1 @ drop (n - size p1) p2) =

     (n \<le> size p1 & p1' = drop n p1)"

 apply(rule iffI)

  defer apply simp

 apply(subgoal_tac "n \<le> size p1")

  apply simp

 apply(rule ccontr)

 apply(drule_tac f = length in arg_cong)

 apply simp

 done

 lemma reduce_exec1:

  "\<langle>i # p1 @ p2,q1 @ q2,s\<rangle> -1\<rightarrow> \<langle>p1' @ p2,q1' @ q2,s'\<rangle> \<Longrightarrow>

   \<langle>i # p1,q1,s\<rangle> -1\<rightarrow> \<langle>p1',q1',s'\<rangle>"

 by(clarsimp simp add: drop_lem split:instr.split_asm split_if_asm)

 lemma closed_exec1:

  "\<lbrakk> closed 0 0 (rev q1 @ instr # p1);

     \<langle>instr # p1 @ p2, q1 @ q2,r\<rangle> -1\<rightarrow> \<langle>p',q',r'\<rangle> \<rbrakk> \<Longrightarrow>

   \<exists>p1' q1'. p' = p1'@p2 \<and> q' = q1'@q2 \<and> rev q1' @ p1' = rev q1 @ instr # p1"

 apply(clarsimp simp add:forws_def backws_def

                split:instr.split_asm split_if_asm)

 done

 theorem closed_execn_decomp: "\<And>C1 C2 r.

  \<lbrakk> closed 0 0 (rev C1 @ C2);

    \<langle>C2 @ p1 @ p2, C1 @ q,r\<rangle> -n\<rightarrow> \<langle>p2,rev p1 @ rev C2 @ C1 @ q,t\<rangle> \<rbrakk>

  \<Longrightarrow> \<exists>s n1 n2. \<langle>C2,C1,r\<rangle> -n1\<rightarrow> \<langle>[],rev C2 @ C1,s\<rangle> \<and>

      \<langle>p1@p2,rev C2 @ C1 @ q,s\<rangle> -n2\<rightarrow> \<langle>p2, rev p1 @ rev C2 @ C1 @ q,t\<rangle> \<and>

          n = n1+n2"

 (is "\<And>C1 C2 r. \<lbrakk>?CL C1 C2; ?H C1 C2 r n\<rbrakk> \<Longrightarrow> ?P C1 C2 r n")

 proof(induct n)

   fix C1 C2 r

   assume "?H C1 C2 r 0"

   thus "?P C1 C2 r 0" by simp

 next

   fix C1 C2 r n

   assume IH: "\<And>C1 C2 r. ?CL C1 C2 \<Longrightarrow> ?H C1 C2 r n \<Longrightarrow> ?P C1 C2 r n"

   assume CL: "?CL C1 C2" and H: "?H C1 C2 r (Suc n)"

   show "?P C1 C2 r (Suc n)"

   proof (cases C2)

     assume "C2 = []" with H show ?thesis by simp

   next

     fix instr tlC2

     assume C2: "C2 = instr # tlC2"

     from H C2 obtain p' q' r'

       where 1: "\<langle>instr # tlC2 @ p1 @ p2, C1 @ q,r\<rangle> -1\<rightarrow> \<langle>p',q',r'\<rangle>"

       and n: "\<langle>p',q',r'\<rangle> -n\<rightarrow> \<langle>p2,rev p1 @ rev C2 @ C1 @ q,t\<rangle>"

       by(fastsimp simp add:rel_pow_commute)

     from CL closed_exec1[OF _ 1] C2

     obtain C2' C1' where pq': "p' = C2' @ p1 @ p2 \<and> q' = C1' @ q"

       and same: "rev C1' @ C2' = rev C1 @ C2"

       by fastsimp

     have rev_same: "rev C2' @ C1' = rev C2 @ C1"

     proof -

       have "rev C2' @ C1' = rev(rev C1' @ C2')" by simp

       also have "\<dots> = rev(rev C1 @ C2)" by(simp only:same)

       also have "\<dots> =  rev C2 @ C1" by simp

       finally show ?thesis .

     qed

     hence rev_same': "\<And>p. rev C2' @ C1' @ p = rev C2 @ C1 @ p" by simp

     from n have n': "\<langle>C2' @ p1 @ p2,C1' @ q,r'\<rangle> -n\<rightarrow>

                      \<langle>p2,rev p1 @ rev C2' @ C1' @ q,t\<rangle>"

       by(simp add:pq' rev_same')

     from IH[OF _ n'] CL

     obtain s n1 n2 where n1: "\<langle>C2',C1',r'\<rangle> -n1\<rightarrow> \<langle>[],rev C2 @ C1,s\<rangle>" and

       "\<langle>p1 @ p2,rev C2 @ C1 @ q,s\<rangle> -n2\<rightarrow> \<langle>p2,rev p1 @ rev C2 @ C1 @ q,t\<rangle> \<and>

        n = n1 + n2"

       by(fastsimp simp add: same rev_same rev_same')

     moreover

     from 1 n1 pq' C2 have "\<langle>C2,C1,r\<rangle> -Suc n1\<rightarrow> \<langle>[],rev C2 @ C1,s\<rangle>"

       by (simp del:relpow.simps exec_simp) (fast dest:reduce_exec1)

     ultimately show ?thesis by (fastsimp simp del:relpow.simps)

   qed

 qed

 lemma execn_decomp:

 "\<langle>compile c @ p1 @ p2,q,r\<rangle> -n\<rightarrow> \<langle>p2,rev p1 @ rev(compile c) @ q,t\<rangle>

  \<Longrightarrow> \<exists>s n1 n2. \<langle>compile c,[],r\<rangle> -n1\<rightarrow> \<langle>[],rev(compile c),s\<rangle> \<and>

      \<langle>p1@p2,rev(compile c) @ q,s\<rangle> -n2\<rightarrow> \<langle>p2, rev p1 @ rev(compile c) @ q,t\<rangle> \<and>

          n = n1+n2"

 using closed_execn_decomp[of "[]",simplified] by simp

 lemma exec_star_decomp:

 "\<langle>compile c @ p1 @ p2,q,r\<rangle> -*\<rightarrow> \<langle>p2,rev p1 @ rev(compile c) @ q,t\<rangle>

  \<Longrightarrow> \<exists>s. \<langle>compile c,[],r\<rangle> -*\<rightarrow> \<langle>[],rev(compile c),s\<rangle> \<and>

      \<langle>p1@p2,rev(compile c) @ q,s\<rangle> -*\<rightarrow> \<langle>p2, rev p1 @ rev(compile c) @ q,t\<rangle>"

 by(simp add:rtrancl_is_UN_rel_pow)(fast dest: execn_decomp)

 (* Alternative:

 lemma exec_comp_n:

 "\<And>p1 p2 q r t n.

  \<langle>compile c @ p1 @ p2,q,r\<rangle> -n\<rightarrow> \<langle>p2,rev p1 @ rev(compile c) @ q,t\<rangle>

  \<Longrightarrow> \<exists>s n1 n2. \<langle>compile c,[],r\<rangle> -n1\<rightarrow> \<langle>[],rev(compile c),s\<rangle> \<and>

      \<langle>p1@p2,rev(compile c) @ q,s\<rangle> -n2\<rightarrow> \<langle>p2, rev p1 @ rev(compile c) @ q,t\<rangle> \<and>

          n = n1+n2"

  (is "\<And>p1 p2 q r t n. ?H c p1 p2 q r t n \<Longrightarrow> ?P c p1 p2 q r t n")

 proof (induct c)

 *)

 text{*Warning: 

 @{prop"\<langle>compile c @ p,q,s\<rangle> -*\<rightarrow> \<langle>p,rev(compile c)@q,t\<rangle> \<Longrightarrow> \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t"}

 is not true! *}

 theorem "\<And>s t.

  \<langle>compile c,[],s\<rangle> -*\<rightarrow> \<langle>[],rev(compile c),t\<rangle> \<Longrightarrow> \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t"

 proof (induct c)

   fix s t

   assume "\<langle>compile SKIP,[],s\<rangle> -*\<rightarrow> \<langle>[],rev(compile SKIP),t\<rangle>"

   thus "\<langle>SKIP,s\<rangle> \<longrightarrow>\<^sub>c t" by simp

 next

   fix s t v f

   assume "\<langle>compile(v :== f),[],s\<rangle> -*\<rightarrow> \<langle>[],rev(compile(v :== f)),t\<rangle>"

   thus "\<langle>v :== f,s\<rangle> \<longrightarrow>\<^sub>c t" by simp

 next

   fix s1 s3 c1 c2

   let ?C1 = "compile c1" let ?C2 = "compile c2"

   assume IH1: "\<And>s t. \<langle>?C1,[],s\<rangle> -*\<rightarrow> \<langle>[],rev ?C1,t\<rangle> \<Longrightarrow> \<langle>c1,s\<rangle> \<longrightarrow>\<^sub>c t"

      and IH2: "\<And>s t. \<langle>?C2,[],s\<rangle> -*\<rightarrow> \<langle>[],rev ?C2,t\<rangle> \<Longrightarrow> \<langle>c2,s\<rangle> \<longrightarrow>\<^sub>c t"

   assume "\<langle>compile(c1;c2),[],s1\<rangle> -*\<rightarrow> \<langle>[],rev(compile(c1;c2)),s3\<rangle>"

   then obtain s2 where exec1: "\<langle>?C1,[],s1\<rangle> -*\<rightarrow> \<langle>[],rev ?C1,s2\<rangle>" and

              exec2: "\<langle>?C2,rev ?C1,s2\<rangle> -*\<rightarrow> \<langle>[],rev(compile(c1;c2)),s3\<rangle>"

     by(fastsimp dest:exec_star_decomp[of _ _ "[]" "[]",simplified])

   from exec2 have exec2': "\<langle>?C2,[],s2\<rangle> -*\<rightarrow> \<langle>[],rev ?C2,s3\<rangle>"

     using exec_star_decomp[of _ "[]" "[]"] by fastsimp

   have "\<langle>c1,s1\<rangle> \<longrightarrow>\<^sub>c s2" using IH1 exec1 by simp

   moreover have "\<langle>c2,s2\<rangle> \<longrightarrow>\<^sub>c s3" using IH2 exec2' by fastsimp

   ultimately show "\<langle>c1;c2,s1\<rangle> \<longrightarrow>\<^sub>c s3" ..

 next

   fix s t b c1 c2

   let ?if = "IF b THEN c1 ELSE c2" let ?C = "compile ?if"

   let ?C1 = "compile c1" let ?C2 = "compile c2"

   assume IH1: "\<And>s t. \<langle>?C1,[],s\<rangle> -*\<rightarrow> \<langle>[],rev ?C1,t\<rangle> \<Longrightarrow> \<langle>c1,s\<rangle> \<longrightarrow>\<^sub>c t"

      and IH2: "\<And>s t. \<langle>?C2,[],s\<rangle> -*\<rightarrow> \<langle>[],rev ?C2,t\<rangle> \<Longrightarrow> \<langle>c2,s\<rangle> \<longrightarrow>\<^sub>c t"

      and H: "\<langle>?C,[],s\<rangle> -*\<rightarrow> \<langle>[],rev ?C,t\<rangle>"

   show "\<langle>?if,s\<rangle> \<longrightarrow>\<^sub>c t"

   proof cases

     assume b: "b s"

     with H have "\<langle>?C1,[],s\<rangle> -*\<rightarrow> \<langle>[],rev ?C1,t\<rangle>"

       by (fastsimp dest:exec_star_decomp

             [of _ "[JMPF (\<lambda>x. False) (size ?C2)]@?C2" "[]",simplified])

     hence "\<langle>c1,s\<rangle> \<longrightarrow>\<^sub>c t" by(rule IH1)

     with b show ?thesis ..

   next

     assume b: "\<not> b s"

     with H have "\<langle>?C2,[],s\<rangle> -*\<rightarrow> \<langle>[],rev ?C2,t\<rangle>"

       using exec_star_decomp[of _ "[]" "[]"] by simp

     hence "\<langle>c2,s\<rangle> \<longrightarrow>\<^sub>c t" by(rule IH2)

     with b show ?thesis ..

   qed

 next

   fix b c s t

   let ?w = "WHILE b DO c" let ?W = "compile ?w" let ?C = "compile c"

   let ?j1 = "JMPF b (size ?C + 1)" let ?j2 = "JMPB (size ?C + 1)"

   assume IHc: "\<And>s t. \<langle>?C,[],s\<rangle> -*\<rightarrow> \<langle>[],rev ?C,t\<rangle> \<Longrightarrow> \<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t"

      and H: "\<langle>?W,[],s\<rangle> -*\<rightarrow> \<langle>[],rev ?W,t\<rangle>"

   from H obtain k where ob:"\<langle>?W,[],s\<rangle> -k\<rightarrow> \<langle>[],rev ?W,t\<rangle>"

     by(simp add:rtrancl_is_UN_rel_pow) blast

   { fix n have "\<And>s. \<langle>?W,[],s\<rangle> -n\<rightarrow> \<langle>[],rev ?W,t\<rangle> \<Longrightarrow> \<langle>?w,s\<rangle> \<longrightarrow>\<^sub>c t"

     proof (induct n rule: less_induct)

       fix n

       assume IHm: "\<And>m s. \<lbrakk>m < n; \<langle>?W,[],s\<rangle> -m\<rightarrow> \<langle>[],rev ?W,t\<rangle> \<rbrakk> \<Longrightarrow> \<langle>?w,s\<rangle> \<longrightarrow>\<^sub>c t"

       fix s

       assume H: "\<langle>?W,[],s\<rangle> -n\<rightarrow> \<langle>[],rev ?W,t\<rangle>"

       show "\<langle>?w,s\<rangle> \<longrightarrow>\<^sub>c t"

       proof cases

 	assume b: "b s"

 	then obtain m where m: "n = Suc m"

           and "\<langle>?C @ [?j2],[?j1],s\<rangle> -m\<rightarrow> \<langle>[],rev ?W,t\<rangle>"

 	  using H by fastsimp

 	then obtain r n1 n2 where n1: "\<langle>?C,[],s\<rangle> -n1\<rightarrow> \<langle>[],rev ?C,r\<rangle>"

           and n2: "\<langle>[?j2],rev ?C @ [?j1],r\<rangle> -n2\<rightarrow> \<langle>[],rev ?W,t\<rangle>"

           and n12: "m = n1+n2"

 	  using execn_decomp[of _ "[?j2]"]

 	  by(simp del: execn_simp) fast

 	have n2n: "n2 - 1 < n" using m n12 by arith

 	note b

 	moreover

 	{ from n1 have "\<langle>?C,[],s\<rangle> -*\<rightarrow> \<langle>[],rev ?C,r\<rangle>"

 	    by (simp add:rtrancl_is_UN_rel_pow) fast

 	  hence "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c r" by(rule IHc)

 	}

 	moreover

 	{ have "n2 - 1 < n" using m n12 by arith

 	  moreover from n2 have "\<langle>?W,[],r\<rangle> -n2- 1\<rightarrow> \<langle>[],rev ?W,t\<rangle>" by fastsimp

 	  ultimately have "\<langle>?w,r\<rangle> \<longrightarrow>\<^sub>c t" by(rule IHm)

 	}

 	ultimately show ?thesis ..

       next

 	assume b: "\<not> b s"

 	hence "t = s" using H by simp

 	with b show ?thesis by simp

       qed

     qed

   }

   with ob show "\<langle>?w,s\<rangle> \<longrightarrow>\<^sub>c t" by fast

 qed

 (* TODO: connect with Machine 0 using M_equiv *)

 end