(*  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:

       "\<lbrakk> n<size P; P!n = ASIN x a \<rbrakk> \<Longrightarrow> P |- <s,n> -1-> <s[x::= a s],Suc n>"

JMPFT:

       "\<lbrakk> n<size P; P!n = JMPF b i;  b s \<rbrakk> \<Longrightarrow> P |- <s,n> -1-> <s,Suc n>"

JMPFF:

       "\<lbrakk> n<size P; P!n = JMPF b i; ~b s; m=n+i \<rbrakk> \<Longrightarrow> P |- <s,n> -1-> <s,m>"

JMPB:

      "\<lbrakk> n<size P; P!n = JMPB i; i <= n \<rbrakk> \<Longrightarrow> 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];

(* Lemmas for lifting an execution into a prefix and suffix

   of instructions; only needed for the first proof *)

lemma app_right_1:

  "is1 |- <s1,i1> -1-> <s2,i2> \<Longrightarrow> is1 @ is2 |- <s1,i1> -1-> <s2,i2>"

apply(erule stepa1.induct);

   apply (simp add:ASIN)

  apply (force intro!:JMPFT)

 apply (force intro!:JMPFF)

apply (simp add: JMPB)

done      

lemma app_left_1:

  "is2 |- <s1,i1> -1-> <s2,i2> \<Longrightarrow>

   is1 @ is2 |- <s1,size is1+i1> -1-> <s2,size is1+i2>"

apply(erule stepa1.induct);

   apply (simp add:ASIN)

  apply (fastsimp intro!:JMPFT)

 apply (fastsimp intro!:JMPFF)

apply (simp add: JMPB)

done

lemma app_right:

  "is1 |- <s1,i1> -*-> <s2,i2> \<Longrightarrow> is1 @ is2 |- <s1,i1> -*-> <s2,i2>"

apply(erule rtrancl_induct2);

 apply simp

apply(blast intro:app_right_1 rtrancl_trans)

done

lemma app_left:

  "is2 |- <s1,i1> -*-> <s2,i2> \<Longrightarrow>

   is1 @ is2 |- <s1,size is1+i1> -*-> <s2,size is1+i2>"

apply(erule rtrancl_induct2);

 apply simp

apply(blast intro:app_left_1 rtrancl_trans)

done

lemma app_left2:

  "\<lbrakk> is2 |- <s1,i1> -*-> <s2,i2>; j1 = size is1+i1; j2 = size is1+i2 \<rbrakk> \<Longrightarrow>

   is1 @ is2 |- <s1,j1> -*-> <s2,j2>"

by (simp add:app_left)

lemma app1_left:

  "is |- <s1,i1> -*-> <s2,i2> \<Longrightarrow>

   instr # is |- <s1,Suc i1> -*-> <s2,Suc i2>"

by(erule app_left[of _ _ _ _ _ "[instr]",simplified])

(* The first proof; statement very intuitive,

   but application of induction hypothesis requires the above lifting lemmas

*)

theorem "<c,s> -c-> t ==> compile c |- <s,0> -*-> <t,length(compile c)>"

apply(erule evalc.induct);

      apply simp;

     apply(force intro!: ASIN);

    apply simp

    apply(rule rtrancl_trans)

    apply(erule app_right)

    apply(erule app_left[of _ 0,simplified])

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

   apply(simp);

   (* execute JMPF: *)

   apply (rule rtrancl_into_rtrancl2)

    apply(force intro!: JMPFT);

   (* execute compile c0: *)

   apply(rule app1_left)

   apply(rule rtrancl_into_rtrancl);

    apply(erule app_right)

   (* execute JMPF: *)

   apply(force intro!: JMPFF);

(* end of case b is true *)

  apply simp

  apply (rule rtrancl_into_rtrancl2)

   apply(force intro!: JMPFF)

  apply(force intro!: app_left2 app1_left)

(* WHILE False *)

 apply(force intro: JMPFF);

(* WHILE True *)

apply(simp)

apply(rule rtrancl_into_rtrancl2);

 apply(force intro!: JMPFT);

apply(rule rtrancl_trans);

 apply(rule app1_left)

 apply(erule app_right)

apply(rule rtrancl_into_rtrancl2);

 apply(force intro!: JMPB)

apply(simp)

done

(* Second proof; statement is generalized to cater for prefixes and suffixes;

   needs none of the lifting lemmas, but instantiations of pre/suffix.

*)

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(force intro!: JMPFT);

   (* execute compile c0: *)

   apply(rule rtrancl_trans);

    apply(erule allE);

    apply assumption;

   (* execute JMPF: *)

   apply(rule r_into_rtrancl);

   apply(force intro!: JMPFF);

(* 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(force intro!: JMPFF);

  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(force intro!: JMPFT);

apply(rule rtrancl_trans);

 apply(erule allE);

 apply assumption;

apply(rule rtrancl_into_rtrancl2);

 apply(force intro!: JMPB);

apply(simp);

done

(* Missing: the other direction! *)

end