(* Title: HOL/IMP/Compiler.thy
ID: $Id$
Author: Tobias Nipkow, TUM
Copyright 1996 TUM
*)
theory Compiler imports Machines begin
subsection "The compiler"
consts compile :: "com \<Rightarrow> instr list"
primrec
"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),next) : 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 converse_rel_powE, 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)
constdefs
forws :: "instr \<Rightarrow> nat set"
"forws instr == case instr of
SET x a \<Rightarrow> {0} |
JMPF b n \<Rightarrow> {0,n} |
JMPB n \<Rightarrow> {}"
backws :: "instr \<Rightarrow> nat set"
"backws instr == case instr of
SET x a \<Rightarrow> {} |
JMPF b n \<Rightarrow> {} |
JMPB n \<Rightarrow> {n}"
consts closed :: "nat \<Rightarrow> nat \<Rightarrow> instr list \<Rightarrow> bool"
primrec
"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
apply arith
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:R_O_Rn_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
(* To Do: connect with Machine 0 using M_equiv *)
end