(*  Title:      HOL/IMP/Compiler.thy

    ID:         $Id$

    Author:     Tobias Nipkow, TUM

    Copyright   1996 TUM

*)

header "A Simple Compiler"

theory Compiler = Natural:

subsection "An abstract, simplistic machine"

text {* There are only three instructions: *}

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

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

  an operational (small step) semantics:

*}

consts  stepa1 :: "instr list \<Rightarrow> ((state\<times>nat) \<times> (state\<times>nat))set"

syntax

  "_stepa1" :: "[instr list,state,nat,state,nat] \<Rightarrow> bool"

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

  "_stepa" :: "[instr list,state,nat,state,nat] \<Rightarrow> bool"

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

syntax (xsymbols)

  "_stepa1" :: "[instr list,state,nat,state,nat] \<Rightarrow> bool"

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

  "_stepa" :: "[instr list,state,nat,state,nat] \<Rightarrow> bool"

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

translations  

  "P \<turnstile> \<langle>s,m\<rangle> -1\<rightarrow> \<langle>t,n\<rangle>" == "((s,m),t,n) : stepa1 P"

  "P \<turnstile> \<langle>s,m\<rangle> -*\<rightarrow> \<langle>t,n\<rangle>" == "((s,m),t,n) : ((stepa1 P)^*)"

inductive "stepa1 P"

intros

ASIN[simp]:

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

JMPFT[simp,intro]:

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

JMPFF[simp,intro]:

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

JMPB[simp]:

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

subsection "The compiler"

consts compile :: "com \<Rightarrow> 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]

subsection "Context lifting lemmas"

text {* 

  Some lemmas for lifting an execution into a prefix and suffix

  of instructions; only needed for the first proof.

*}

lemma app_right_1:

  "is1 \<turnstile> \<langle>s1,i1\<rangle> -1\<rightarrow> \<langle>s2,i2\<rangle> \<Longrightarrow> is1 @ is2 \<turnstile> \<langle>s1,i1\<rangle> -1\<rightarrow> \<langle>s2,i2\<rangle>"

  (is "?P \<Longrightarrow> _")

proof -

 assume ?P

 then show ?thesis

 by induct force+

qed

lemma app_left_1:

  "is2 \<turnstile> \<langle>s1,i1\<rangle> -1\<rightarrow> \<langle>s2,i2\<rangle> \<Longrightarrow>

   is1 @ is2 \<turnstile> \<langle>s1,size is1+i1\<rangle> -1\<rightarrow> \<langle>s2,size is1+i2\<rangle>"

  (is "?P \<Longrightarrow> _")

proof -

 assume ?P

 then show ?thesis

 by induct force+

qed

declare rtrancl_induct2 [induct set: rtrancl]

lemma app_right:

  "is1 \<turnstile> \<langle>s1,i1\<rangle> -*\<rightarrow> \<langle>s2,i2\<rangle> \<Longrightarrow> is1 @ is2 \<turnstile> \<langle>s1,i1\<rangle> -*\<rightarrow> \<langle>s2,i2\<rangle>"

  (is "?P \<Longrightarrow> _")

proof -

 assume ?P

 then show ?thesis

 proof induct

   show "is1 @ is2 \<turnstile> \<langle>s1,i1\<rangle> -*\<rightarrow> \<langle>s1,i1\<rangle>" by simp

 next

   fix s1' i1' s2 i2

   assume "is1 @ is2 \<turnstile> \<langle>s1,i1\<rangle> -*\<rightarrow> \<langle>s1',i1'\<rangle>"

          "is1 \<turnstile> \<langle>s1',i1'\<rangle> -1\<rightarrow> \<langle>s2,i2\<rangle>"

   thus "is1 @ is2 \<turnstile> \<langle>s1,i1\<rangle> -*\<rightarrow> \<langle>s2,i2\<rangle>"

     by(blast intro:app_right_1 rtrancl_trans)

 qed

qed

lemma app_left:

  "is2 \<turnstile> \<langle>s1,i1\<rangle> -*\<rightarrow> \<langle>s2,i2\<rangle> \<Longrightarrow>

   is1 @ is2 \<turnstile> \<langle>s1,size is1+i1\<rangle> -*\<rightarrow> \<langle>s2,size is1+i2\<rangle>"

  (is "?P \<Longrightarrow> _")

proof -

 assume ?P

 then show ?thesis

 proof induct

   show "is1 @ is2 \<turnstile> \<langle>s1,length is1 + i1\<rangle> -*\<rightarrow> \<langle>s1,length is1 + i1\<rangle>" by simp

 next

   fix s1' i1' s2 i2

   assume "is1 @ is2 \<turnstile> \<langle>s1,length is1 + i1\<rangle> -*\<rightarrow> \<langle>s1',length is1 + i1'\<rangle>"

          "is2 \<turnstile> \<langle>s1',i1'\<rangle> -1\<rightarrow> \<langle>s2,i2\<rangle>"

   thus "is1 @ is2 \<turnstile> \<langle>s1,length is1 + i1\<rangle> -*\<rightarrow> \<langle>s2,length is1 + i2\<rangle>"

     by(blast intro:app_left_1 rtrancl_trans)

 qed

qed

lemma app_left2:

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

   is1 @ is2 \<turnstile> \<langle>s1,j1\<rangle> -*\<rightarrow> \<langle>s2,j2\<rangle>"

  by (simp add:app_left)

lemma app1_left:

  "is \<turnstile> \<langle>s1,i1\<rangle> -*\<rightarrow> \<langle>s2,i2\<rangle> \<Longrightarrow>

   instr # is \<turnstile> \<langle>s1,Suc i1\<rangle> -*\<rightarrow> \<langle>s2,Suc i2\<rangle>"

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

subsection "Compiler correctness"

declare rtrancl_into_rtrancl[trans]

        rtrancl_into_rtrancl2[trans]

        rtrancl_trans[trans]

text {*

  The first proof; The statement is very intuitive,

  but application of induction hypothesis requires the above lifting lemmas

*}

theorem "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t \<Longrightarrow> compile c \<turnstile> \<langle>s,0\<rangle> -*\<rightarrow> \<langle>t,length(compile c)\<rangle>"

        (is "?P \<Longrightarrow> ?Q c s t")

proof -

  assume ?P

  then show ?thesis

  proof induct

    show "\<And>s. ?Q \<SKIP> s s" by simp

  next

    show "\<And>a s x. ?Q (x :== a) s (s[x\<mapsto> a s])" by force

  next

    fix c0 c1 s0 s1 s2

    assume "?Q c0 s0 s1"

    hence "compile c0 @ compile c1 \<turnstile> \<langle>s0,0\<rangle> -*\<rightarrow> \<langle>s1,length(compile c0)\<rangle>"

      by(rule app_right)

    moreover assume "?Q c1 s1 s2"

    hence "compile c0 @ compile c1 \<turnstile> \<langle>s1,length(compile c0)\<rangle> -*\<rightarrow>

           \<langle>s2,length(compile c0)+length(compile c1)\<rangle>"

    proof -

      note app_left[of _ 0]

      thus

	      "\<And>is1 is2 s1 s2 i2.

	      is2 \<turnstile> \<langle>s1,0\<rangle> -*\<rightarrow> \<langle>s2,i2\<rangle> \<Longrightarrow>

	      is1 @ is2 \<turnstile> \<langle>s1,size is1\<rangle> -*\<rightarrow> \<langle>s2,size is1+i2\<rangle>"

	      by simp

    qed

    ultimately have "compile c0 @ compile c1 \<turnstile> \<langle>s0,0\<rangle> -*\<rightarrow>

                       \<langle>s2,length(compile c0)+length(compile c1)\<rangle>"

      by (rule rtrancl_trans)

    thus "?Q (c0; c1) s0 s2" by simp

  next

    fix b c0 c1 s0 s1

    let ?comp = "compile(\<IF> b \<THEN> c0 \<ELSE> c1)"

    assume "b s0" and IH: "?Q c0 s0 s1"

    hence "?comp \<turnstile> \<langle>s0,0\<rangle> -1\<rightarrow> \<langle>s0,1\<rangle>" by auto

    also from IH

    have "?comp \<turnstile> \<langle>s0,1\<rangle> -*\<rightarrow> \<langle>s1,length(compile c0)+1\<rangle>"

      by(auto intro:app1_left app_right)

    also have "?comp \<turnstile> \<langle>s1,length(compile c0)+1\<rangle> -1\<rightarrow> \<langle>s1,length ?comp\<rangle>"

      by(auto)

    finally show "?Q (\<IF> b \<THEN> c0 \<ELSE> c1) s0 s1" .

  next

    fix b c0 c1 s0 s1

    let ?comp = "compile(\<IF> b \<THEN> c0 \<ELSE> c1)"

    assume "\<not>b s0" and IH: "?Q c1 s0 s1"

    hence "?comp \<turnstile> \<langle>s0,0\<rangle> -1\<rightarrow> \<langle>s0,length(compile c0) + 2\<rangle>" by auto

    also from IH

    have "?comp \<turnstile> \<langle>s0,length(compile c0)+2\<rangle> -*\<rightarrow> \<langle>s1,length ?comp\<rangle>"

      by(force intro!:app_left2 app1_left)

    finally show "?Q (\<IF> b \<THEN> c0 \<ELSE> c1) s0 s1" .

  next

    fix b c and s::state

    assume "\<not>b s"

    thus "?Q (\<WHILE> b \<DO> c) s s" by force

  next

    fix b c and s0::state and s1 s2

    let ?comp = "compile(\<WHILE> b \<DO> c)"

    assume "b s0" and

      IHc: "?Q c s0 s1" and IHw: "?Q (\<WHILE> b \<DO> c) s1 s2"

    hence "?comp \<turnstile> \<langle>s0,0\<rangle> -1\<rightarrow> \<langle>s0,1\<rangle>" by auto

    also from IHc

    have "?comp \<turnstile> \<langle>s0,1\<rangle> -*\<rightarrow> \<langle>s1,length(compile c)+1\<rangle>"

      by(auto intro:app1_left app_right)

    also have "?comp \<turnstile> \<langle>s1,length(compile c)+1\<rangle> -1\<rightarrow> \<langle>s1,0\<rangle>" by simp

    also note IHw

    finally show "?Q (\<WHILE> b \<DO> c) s0 s2".

  qed

qed

text {*

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

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

  *}

theorem "\<langle>c,s\<rangle> \<longrightarrow>\<^sub>c t \<Longrightarrow> 

  !a z. a@compile c@z \<turnstile> \<langle>s,length a\<rangle> -*\<rightarrow> \<langle>t,length a + length(compile c)\<rangle>";

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

text {* Missing: the other direction! *}

end