(*  Title:      HOL/IMP/Compiler.thy

     ID:         $Id$

     Author:     Tobias Nipkow, TUM

     Copyright   1996 TUM

 A simple compiler for a simplistic machine.

 *)

 theory Compiler = Natural:

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

 consts  stepa1 :: "instr list => ((state*nat) * (state*nat))set"

 syntax

         "@stepa1" :: "[instr list,state,nat,state,nat] => bool"

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

         "@stepa" :: "[instr list,state,nat,state,nat] => bool"

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

 translations  "P |- <s,m> -1-> <t,n>" == "((s,m),t,n) : stepa1 P"

               "P |- <s,m> -*-> <t,n>" == "((s,m),t,n) : ((stepa1 P)^*)"

 inductive "stepa1 P"

 intros

 ASIN:  "P!n = ASIN x a ==> P |- <s,n> -1-> <s[x::= a s],Suc n>"

 JMPFT: "[| P!n = JMPF b i;  b s |] ==> P |- <s,n> -1-> <s,Suc n>"

 JMPFF: "[| P!n = JMPF b i; ~b s; m=n+i |] ==> P |- <s,n> -1-> <s,m>"

 JMPB:  "[| P!n = JMPB i |] ==> P |- <s,n> -1-> <s,n-i>"

 consts compile :: "com => instr list"

 primrec

 "compile SKIP = []"

 "compile (x:==a) = [ASIN x a]"

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

 "compile (IF b THEN c1 ELSE c2) =

  [JMPF b (length(compile c1)+2)] @ compile c1 @

  [JMPF (%x. False) (length(compile c2)+1)] @ compile c2"

 "compile (WHILE b DO c) = [JMPF b (length(compile c)+2)] @ compile c @

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

 declare nth_append[simp];

 lemma nth_tl[simp]: "tl(xs @ y # ys) ! (length xs + z) = ys!z";

 apply(induct_tac xs);

 by(auto);

 theorem "<c,s> -c-> t ==> 

  !a z. a@compile c@z |- <s,length a> -*-> <t,length a + length(compile c)>";

 apply(erule evalc.induct);

       apply simp;

      apply(force intro!: ASIN);

     apply(intro strip);

     apply(erule_tac x = a in allE);

     apply(erule_tac x = "a@compile c0" in allE);

     apply(erule_tac x = "compile c1@z" in allE);

     apply(erule_tac x = z in allE);

     apply(simp add:add_assoc[THEN sym]);

     apply(blast intro:rtrancl_trans);

 (* IF b THEN c0 ELSE c1; case b is true *)

    apply(intro strip);

    (* instantiate assumption sufficiently for later: *)

    apply(erule_tac x = "a@[?I]" in allE);

    apply(simp);

    (* execute JMPF: *)

    apply(rule rtrancl_into_rtrancl2);

     apply(rule JMPFT);

      apply(simp);

      apply(blast);

     apply assumption;

    (* execute compile c0: *)

    apply(rule rtrancl_trans);

     apply(erule allE);

     apply assumption;

    (* execute JMPF: *)

    apply(rule r_into_rtrancl);

    apply(rule JMPFF);

      apply(simp);

      apply(blast);

     apply(blast);

    apply(simp);

 (* end of case b is true *)

   apply(intro strip);

   apply(erule_tac x = "a@[?I]@compile c0@[?J]" in allE);

   apply(simp add:add_assoc);

   apply(rule rtrancl_into_rtrancl2);

    apply(rule JMPFF);

      apply(simp);

      apply(blast);

     apply assumption;

    apply(simp);

   apply(blast);

  apply(force intro: JMPFF);

 apply(intro strip);

 apply(erule_tac x = "a@[?I]" in allE);

 apply(erule_tac x = a in allE);

 apply(simp);

 apply(rule rtrancl_into_rtrancl2);

  apply(rule JMPFT);

   apply(simp);

   apply(blast);

  apply assumption;

 apply(rule rtrancl_trans);

  apply(erule allE);

  apply assumption;

 apply(rule rtrancl_into_rtrancl2);

  apply(rule JMPB);

  apply(simp);

 apply(simp);

 done

 (* Missing: the other direction! *)

 end