I agree Our site saves small pieces of text information (cookies) on your device in order to deliver better content and for statistical purposes. You can disable the usage of cookies by changing the settings of your browser. By browsing our website without changing the browser settings you grant us permission to store that information on your device.
theory Defs
imports "HOL-IMP.BExp" "HOL-IMP.Star"
begin
section "Source language"
datatype
com = SKIP
| Assign vname aexp ("_ ::= _" [1000, 61] 61)
| Seq com com ("_;;/ _" [60, 61] 60)
| If bexp com com ("(IF _/ THEN _/ ELSE _)" [0, 0, 61] 61)
| While bexp com ("(WHILE _/ DO _)" [0, 61] 61)
| THROW
| Attempt com com ("(ATTEMPT _/ CLEANUP _)" [0, 61] 61)
inductive big_step :: "com \<times> state \<Rightarrow> com \<times> state \<Rightarrow> bool" (infix "\<Rightarrow>" 55)
where
Skip: "(SKIP,s) \<Rightarrow> (SKIP, s)" |
Assign: "(x ::= a,s) \<Rightarrow> (SKIP, s(x := aval a s))" |
Seq: "\<lbrakk> (c\<^sub>1,s\<^sub>1) \<Rightarrow> (SKIP, s\<^sub>2); (c\<^sub>2,s\<^sub>2) \<Rightarrow> r \<rbrakk> \<Longrightarrow>
(c\<^sub>1;;c\<^sub>2, s\<^sub>1) \<Rightarrow> r" |
SeqThrow: "\<lbrakk> (c\<^sub>1,s\<^sub>1) \<Rightarrow> (THROW, s\<^sub>2) \<rbrakk> \<Longrightarrow>
(c\<^sub>1;;c\<^sub>2, s\<^sub>1) \<Rightarrow> (THROW, s\<^sub>2)" |
IfTrue: "\<lbrakk> bval b s; (c\<^sub>1,s) \<Rightarrow> r \<rbrakk> \<Longrightarrow>
(IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> r" |
IfFalse: "\<lbrakk> \<not>bval b s; (c\<^sub>2,s) \<Rightarrow> r \<rbrakk> \<Longrightarrow>
(IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> r" |
WhileFalse: "\<not>bval b s \<Longrightarrow> (WHILE b DO c,s) \<Rightarrow> (SKIP, s)" |
WhileTrueSkip: "\<lbrakk> bval b s\<^sub>1; (c,s\<^sub>1) \<Rightarrow> (SKIP,s\<^sub>2); (WHILE b DO c, s\<^sub>2) \<Rightarrow> r \<rbrakk>
\<Longrightarrow> (WHILE b DO c, s\<^sub>1) \<Rightarrow> r" |
WhileTrueThrow: "\<lbrakk> bval b s\<^sub>1; (c,s\<^sub>1) \<Rightarrow> (THROW,s\<^sub>2) \<rbrakk>
\<Longrightarrow> (WHILE b DO c, s\<^sub>1) \<Rightarrow> (THROW,s\<^sub>2)" |
Throw: "(THROW,s) \<Rightarrow> (THROW,s)" |
Attempt: "\<lbrakk> (c\<^sub>1,s\<^sub>1) \<Rightarrow> (_, s\<^sub>2); (c\<^sub>2, s\<^sub>2) \<Rightarrow> r \<rbrakk> \<Longrightarrow> (ATTEMPT c\<^sub>1 CLEANUP c\<^sub>2,s\<^sub>1) \<Rightarrow> r"
inductive_cases SkipE[elim!]: "(SKIP,s) \<Rightarrow> t"
inductive_cases AssignE[elim!]: "(x ::= a,s) \<Rightarrow> t"
inductive_cases SeqE[elim!]: "(c1;;c2,s1) \<Rightarrow> s3"
inductive_cases IfE[elim!]: "(IF b THEN c1 ELSE c2,s) \<Rightarrow> t"
inductive_cases WhileE[elim]: "(WHILE b DO c,s) \<Rightarrow> t"
inductive_cases ThrowE[elim!]: "(THROW,s) \<Rightarrow> t"
thm ThrowE
inductive_cases AttemptE[elim!]: "(ATTEMPT c1 CLEANUP c2,s) \<Rightarrow> t"
thm AttemptE
lemmas big_step_induct = big_step.induct[split_format(complete)]
declare big_step.intros[intro]
section "Compiler"
declare [[coercion_enabled]]
declare [[coercion "int :: nat \<Rightarrow> int"]]
fun inth :: "'a list \<Rightarrow> int \<Rightarrow> 'a" (infixl "!!" 100) where
"(x # xs) !! i = (if i = 0 then x else xs !! (i - 1))"
lemma inth_append [simp]:
"0 \<le> i \<Longrightarrow>
(xs @ ys) !! i = (if i < size xs then xs !! i else ys !! (i - size xs))"
by (induction xs arbitrary: i) (auto simp: algebra_simps)
abbreviation (output)
"isize xs == int (length xs)"
notation isize ("size")
datatype instr =
LOADI int | LOAD vname | ADD | STORE vname |
JMP int | JMPLESS int | JMPGE int
type_synonym stack = "val list"
type_synonym config = "int \<times> state \<times> stack"
abbreviation "hd2 xs == hd(tl xs)"
abbreviation "tl2 xs == tl(tl xs)"
fun iexec :: "instr \<Rightarrow> config \<Rightarrow> config" where
"iexec instr (i,s,stk) = (case instr of
LOADI n \<Rightarrow> (i+1,s, n#stk) |
LOAD x \<Rightarrow> (i+1,s, s x # stk) |
ADD \<Rightarrow> (i+1,s, (hd2 stk + hd stk) # tl2 stk) |
STORE x \<Rightarrow> (i+1,s(x := hd stk),tl stk) |
JMP n \<Rightarrow> (i+1+n,s,stk) |
JMPLESS n \<Rightarrow> (if hd2 stk < hd stk then i+1+n else i+1,s,tl2 stk) |
JMPGE n \<Rightarrow> (if hd2 stk >= hd stk then i+1+n else i+1,s,tl2 stk))"
definition
exec1 :: "instr list \<Rightarrow> config \<Rightarrow> config \<Rightarrow> bool"
("(_/ \<turnstile> (_ \<rightarrow>/ _))" [59,0,59] 60)
where
"P \<turnstile> c \<rightarrow> c' =
(\<exists>i s stk. c = (i,s,stk) \<and> c' = iexec(P!!i) (i,s,stk) \<and> 0 \<le> i \<and> i < size P)"
lemma exec1I [intro, code_pred_intro]:
"c' = iexec (P!!i) (i,s,stk) \<Longrightarrow> 0 \<le> i \<Longrightarrow> i < size P
\<Longrightarrow> P \<turnstile> (i,s,stk) \<rightarrow> c'"
by (simp add: exec1_def)
abbreviation
exec :: "instr list \<Rightarrow> config \<Rightarrow> config \<Rightarrow> bool" ("(_/ \<turnstile> (_ \<rightarrow>*/ _))" 50)
where
"exec P \<equiv> star (exec1 P)"
lemmas exec_induct = star.induct [of "exec1 P", split_format(complete)]
code_pred exec1 by (metis exec1_def)
subsection \<open>Verification infrastructure\<close>
lemma iexec_shift [simp]:
"((n+i',s',stk') = iexec x (n+i,s,stk)) = ((i',s',stk') = iexec x (i,s,stk))"
by(auto split:instr.split)
lemma exec1_appendR: "P \<turnstile> c \<rightarrow> c' \<Longrightarrow> P@P' \<turnstile> c \<rightarrow> c'"
by (auto simp: exec1_def)
lemma exec_appendR: "P \<turnstile> c \<rightarrow>* c' \<Longrightarrow> P@P' \<turnstile> c \<rightarrow>* c'"
by (induction rule: star.induct) (fastforce intro: star.step exec1_appendR)+
lemma exec1_appendL:
fixes i i' :: int
shows
"P \<turnstile> (i,s,stk) \<rightarrow> (i',s',stk') \<Longrightarrow>
P' @ P \<turnstile> (size(P')+i,s,stk) \<rightarrow> (size(P')+i',s',stk')"
unfolding exec1_def
by (auto simp del: iexec.simps)
lemma exec_appendL:
fixes i i' :: int
shows
"P \<turnstile> (i,s,stk) \<rightarrow>* (i',s',stk') \<Longrightarrow>
P' @ P \<turnstile> (size(P')+i,s,stk) \<rightarrow>* (size(P')+i',s',stk')"
by (induction rule: exec_induct) (blast intro: star.step exec1_appendL)+
lemma exec_Cons_1 [intro]:
"P \<turnstile> (0,s,stk) \<rightarrow>* (j,t,stk') \<Longrightarrow>
instr#P \<turnstile> (1,s,stk) \<rightarrow>* (1+j,t,stk')"
by (drule exec_appendL[where P'="[instr]"]) simp
lemma exec_appendL_if[intro]:
fixes i i' j :: int
shows
"size P' <= i
\<Longrightarrow> P \<turnstile> (i - size P',s,stk) \<rightarrow>* (j,s',stk')
\<Longrightarrow> i' = size P' + j
\<Longrightarrow> P' @ P \<turnstile> (i,s,stk) \<rightarrow>* (i',s',stk')"
by (drule exec_appendL[where P'=P']) simp
lemma exec_append_trans[intro]:
fixes i' i'' j'' :: int
shows
"P \<turnstile> (0,s,stk) \<rightarrow>* (i',s',stk') \<Longrightarrow>
size P \<le> i' \<Longrightarrow>
P' \<turnstile> (i' - size P,s',stk') \<rightarrow>* (i'',s'',stk'') \<Longrightarrow>
j'' = size P + i''
\<Longrightarrow>
P @ P' \<turnstile> (0,s,stk) \<rightarrow>* (j'',s'',stk'')"
by(metis star_trans[OF exec_appendR exec_appendL_if])
declare Let_def[simp]
subsection "Compilation"
fun acomp :: "aexp \<Rightarrow> instr list" where
"acomp (N n) = [LOADI n]" |
"acomp (V x) = [LOAD x]" |
"acomp (Plus a1 a2) = acomp a1 @ acomp a2 @ [ADD]"
lemma acomp_correct[intro]:
"acomp a \<turnstile> (0,s,stk) \<rightarrow>* (size(acomp a),s,aval a s#stk)"
by (induction a arbitrary: stk) fastforce+
fun bcomp :: "bexp \<Rightarrow> bool \<Rightarrow> int \<Rightarrow> instr list" where
"bcomp (Bc v) f n = (if v=f then [JMP n] else [])" |
"bcomp (Not b) f n = bcomp b (\<not>f) n" |
"bcomp (And b1 b2) f n =
(let cb2 = bcomp b2 f n;
m = if f then size cb2 else (size cb2::int)+n;
cb1 = bcomp b1 False m
in cb1 @ cb2)" |
"bcomp (Less a1 a2) f n =
acomp a1 @ acomp a2 @ (if f then [JMPLESS n] else [JMPGE n])"
lemma bcomp_correct[intro]:
fixes n :: int
shows
"0 \<le> n \<Longrightarrow>
bcomp b f n \<turnstile>
(0,s,stk) \<rightarrow>* (size(bcomp b f n) + (if f = bval b s then n else 0),s,stk)"
proof(induction b arbitrary: f n)
case Not
from Not(1)[where f="~f"] Not(2) show ?case by fastforce
next
case (And b1 b2)
from And(1)[of "if f then size(bcomp b2 f n) else size(bcomp b2 f n) + n"
"False"]
And(2)[of n f] And(3)
show ?case by fastforce
qed fastforce+
consts ccomp :: "nat \<Rightarrow> com \<Rightarrow> instr list"
end
theory Submission imports Defs begin fun ccomp :: "nat \<Rightarrow> com \<Rightarrow> instr list" where "ccomp _ = undefined" lemma ccomp_bigstep: "(c,s) \<Rightarrow> (c',t) \<Longrightarrow> ccomp n c \<turnstile> (0,s,stk) \<rightarrow>* (size(ccomp n c) + a,t,stk)" sorry end
theory Check imports Submission begin lemma ccomp_bigstep: "(c,s) \<Rightarrow> (c',t) \<Longrightarrow> ccomp n c \<turnstile> (0,s,stk) \<rightarrow>* (size(ccomp n c) + a,t,stk)" by (rule Submission.ccomp_bigstep) end