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Short_Theory_8_1.thy
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Short_Theory_8_1.thy
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theory Short_Theory_8_1
imports "HOL-IMP.Big_Step" "HOL-IMP.Star"
begin
declare [[coercion_enabled]]
declare [[coercion "int :: nat \<Rightarrow> int"]]
primrec (nonexhaustive) 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 (induct xs arbitrary: i) (auto simp: algebra_simps)
lemma nth_inth: "i < length (x # xs) \<Longrightarrow> (x # xs) !! i = (x # xs) ! i"
proof (induct xs arbitrary: i x)
case (Cons a xs)
then show ?case
proof (cases "i = 0")
case False
from Cons False have Hl: "i - 1 < length (a # xs)" by auto
from Cons(2) False have "(x # a # xs) !! int i = (a # xs) !! (int (i - 1))" using int_ops(6) by auto
also from Cons(1) Hl have "\<dots> = (a # xs) ! (i - 1)" by blast
also from Cons(2) False have "\<dots> = (x # a # xs) ! i" by simp
finally show ?thesis .
qed simp
qed simp
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)"
(* note: by using hd / tl functions rather than pattern matching, we limit
reliance on the structure of stk in the behavior of iexec, allowing us
to simplify preconditions on lemmas that don't rely on the structure of
the stack
*)
fun iexec :: "instr \<Rightarrow> config \<Rightarrow> config" where
"iexec (LOADI n) (i, s, stk) = (i + 1, s, n # stk)" |
"iexec (LOAD x) (i, s, stk) = (i + 1, s, s x # stk)" |
"iexec ADD (i, s, stk) = (i + 1, s, (hd2 stk + hd stk) # tl2 stk)" |
"iexec (STORE x) (i, s, stk) = (i + 1, s(x := hd stk), tl stk)" |
"iexec (JMP n) (i, s, stk) = (i + 1 + n, s, stk)" |
"iexec (JMPLESS n) (i, s, stk) = (if hd2 stk < hd stk then i + 1 + n else i + 1, s, tl2 stk)" |
"iexec (JMPGE n) (i, s, stk) = (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' \<longleftrightarrow>
(\<exists>i s stk. c = (i, s, stk) \<and> c' = iexec (P !! i) (i, s, stk) \<and> 0 \<le> i \<and> i < size P)"
(* an introduction rule that expects the LHS config parameters to already be known *)
lemma exec1I [intro, code_pred_intro]:
"\<lbrakk>c' = iexec (P !! i) (i, s, stk);
0 \<le> i;
i < size P
\<rbrakk> \<Longrightarrow> P \<turnstile> (i, s, stk) \<rightarrow> c'"
by (simp add: exec1_def)
code_pred exec1 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)]
(* proof by case analysis on instructions, and that each case changes the PC relative to its
initial value
*)
lemma iexec_shift [simp]:
"(n + i', s') = iexec x (n + i, s) \<longleftrightarrow>
(i', s') = iexec x (i, s)"
proof -
{
fix fs' ss' fs ss
have "(n + i', fs', ss') = iexec x (n + i, fs, ss) \<longleftrightarrow>
(i', fs', ss') = iexec x (i, fs, ss)"
by (cases x, auto)
}
then have "(n + i', fst s', snd s') = iexec x (n + i, fst s, snd s) \<longleftrightarrow>
(i', fst s', snd s') = iexec x (i, fst s, snd s)" .
then show "(n + i', s') = iexec x (n + i, s) \<longleftrightarrow>
(i', s') = iexec x (i, s)" by simp
qed
(* trivial: iexec (P !! i) depends only on first i elements of P, and 0 \<le> i < size P *)
lemma exec1_appendR: "P \<turnstile> c \<rightarrow> c' \<Longrightarrow> P @ P' \<turnstile> c \<rightarrow> c'"
by (auto simp add: exec1_def)
lemma exec_appendR: "P \<turnstile> c \<rightarrow>* c' \<Longrightarrow> P @ P' \<turnstile> c \<rightarrow>* c'"
by (induct rule: star.induct) (blast intro: star.step exec1_appendR)+
lemma exec1_appendL:"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 (auto simp add: exec1_def)
lemma exec_appendL: "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 (induct rule: exec_induct) (blast intro: star.step exec1_appendL)+
(* specialize append lemmas to discuss execution through concrete instructions
while assuming the execution of preceding and following code.
*)
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
(* as exec_appendL, with (i := i - size P'), precondition necessary to satisfy exec1 precondition *)
lemma exec_appendL_if [intro]:
"\<lbrakk>size P' \<le> i;
P \<turnstile> (i - size P', s, stk) \<rightarrow>* (j, s', stk');
i' = size P' + j
\<rbrakk> \<Longrightarrow> P' @ P \<turnstile> (i, s, stk) \<rightarrow>* (i', s', stk')"
by (drule exec_appendL [where P'=P']) simp
lemma exec_append_trans[intro]:
"\<lbrakk>P \<turnstile> (0, s, stk) \<rightarrow>* (i', s', stk');
size P \<le> i';
P' \<turnstile> (i' - size P, s', stk') \<rightarrow>* (i'', s'', stk'');
j'' = size P + i''
\<rbrakk> \<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+
(* f = True means that we intend to jump n spaces upon the expression evaluating to True, and
step to next instruction upon the expression evaluating to False
f = False means vice versa
Suppose f = True in bcomp (And b1 b2). Then we want b1 to jump to just past b2 on False,
(we know early that the And expression evaluates to False),
and to continue with b2 on True. Thus we let cb1 = bcomp b1 False (size cb2)
suppose f = False in bcomp (And b1 b2). Then we want b1 to jump to (size cb2 + n) on False
(we know early that the And expression evaluates to False),
and to continue with b2 on True.
*)
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 + 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]: "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 (induct b arbitrary: f n)
case Not
from Not(1) [where f="\<not>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+
fun ccomp :: "com \<Rightarrow> instr list" where
"ccomp SKIP = []" |
"ccomp (x ::= a) = acomp a @ [STORE x]" |
"ccomp (c\<^sub>1;; c\<^sub>2) = ccomp c\<^sub>1 @ ccomp c\<^sub>2" |
"ccomp (IF b THEN c\<^sub>1 ELSE c\<^sub>2) = (let
cc\<^sub>1 = ccomp c\<^sub>1;
cc\<^sub>2 = ccomp c\<^sub>2;
cb = bcomp b False (size cc\<^sub>1 + 1) in
cb @ cc\<^sub>1 @ JMP (size cc\<^sub>2) # cc\<^sub>2)" |
"ccomp (WHILE b DO c) = (let
cc = ccomp c;
cb = bcomp b False (size cc + 1) in
cb @ cc @ [JMP (-(size cb + size cc + 1))])"
lemma ccomp_bigstep:
"(c, s) \<Rightarrow> t \<Longrightarrow> ccomp c \<turnstile> (0, s, stk) \<rightarrow>* (size (ccomp c), t, stk)"
proof (induct arbitrary: stk rule: big_step_induct)
case (Assign x a s)
show ?case by (fastforce simp:fun_upd_def cong: if_cong)
next
case (Seq c1 s1 s2 c2 s3)
let ?cc1 = "ccomp c1" and ?cc2 = "ccomp c2"
have "?cc1 @ ?cc2 \<turnstile> (0, s1 ,stk) \<rightarrow>* (size ?cc1, s2, stk)"
using Seq(2) by fastforce
moreover
have "?cc1 @ ?cc2 \<turnstile> (size ?cc1, s2, stk) \<rightarrow>* (size (?cc1 @ ?cc2), s3, stk)"
using Seq(4) by fastforce
ultimately show ?case by simp (blast intro: star_trans)
next
case (WhileTrue b s1 c s2 s3)
let ?cc = "ccomp c"
let ?cb = "bcomp b False (size ?cc + 1)"
and ?cw = "ccomp (WHILE b DO c)"
have "?cw \<turnstile> (0, s1, stk) \<rightarrow>* (size ?cb, s1, stk)"
using \<open>bval b s1\<close> by fastforce
moreover have "?cw \<turnstile> (size ?cb, s1, stk) \<rightarrow>* (size ?cb + size ?cc, s2, stk)"
using WhileTrue(3) by fastforce
moreover have "?cw \<turnstile> (size ?cb + size ?cc, s2, stk) \<rightarrow>* (0, s2, stk)"
by fastforce
moreover have "?cw \<turnstile> (0, s2, stk) \<rightarrow>* (size ?cw, s3, stk)" by (rule WhileTrue(5))
ultimately show ?case by(blast intro: star_trans)
qed fastforce+
text \<open>
The preservation of the source code semantics is already shown in the
parent theory \<open>Compiler\<close>. This here shows the second direction.
\<close>
subsection \<open>Definitions\<close>
text \<open>Execution in \<^term>\<open>n\<close> steps for simpler induction\<close>
primrec
exec_n :: "instr list \<Rightarrow> config \<Rightarrow> nat \<Rightarrow> config \<Rightarrow> bool"
("_/ \<turnstile> (_ \<rightarrow>^_/ _)" [65,0,1000,55] 55)
where
"P \<turnstile> c \<rightarrow>^0 c' = (c'=c)" |
"P \<turnstile> c \<rightarrow>^(Suc n) c'' = (\<exists>c'. (P \<turnstile> c \<rightarrow> c') \<and> P \<turnstile> c' \<rightarrow>^n c'')"
(* Note: big-step notation causes parsing ambiguity that isn't well-typed *)
text \<open>The possible successor PCs of an instruction at position \<^term>\<open>n\<close>\<close>
definition isuccs :: "instr \<Rightarrow> int \<Rightarrow> int set" where
"isuccs i n = (case i of
JMP j \<Rightarrow> {n + 1 + j} |
JMPLESS j \<Rightarrow> {n + 1 + j, n + 1} |
JMPGE j \<Rightarrow> {n + 1 + j, n + 1} |
_ \<Rightarrow> {n + 1})"
lemma isuccs_LOADI_correct: "{fst c' | c' i s stk. c' = iexec (LOADI i) (n, s, stk)} = isuccs (LOADI i) n"
by (auto simp add: isuccs_def)
text \<open>The possible successors PCs of an instruction list starting from position n of P to its end\<close>
definition succs :: "instr list \<Rightarrow> int \<Rightarrow> int set" where
"succs P n = {s. \<exists>i\<ge>0. i < size P \<and> s \<in> isuccs (P !! i) (n + i)}"
text \<open>Possible exit PCs of a program\<close>
definition exits :: "instr list \<Rightarrow> int set" where
"exits P = succs P 0 - {0..<size P}"
subsection \<open>Basic properties of \<^term>\<open>exec_n\<close>\<close>
lemma exec_n_exec:
"P \<turnstile> c \<rightarrow>^n c' \<Longrightarrow> P \<turnstile> c \<rightarrow>* c'"
by (induct n arbitrary: c) (auto intro: star.step)
lemma exec_0 [intro!]: "P \<turnstile> c \<rightarrow>^0 c" by simp
lemma exec_Suc:
"\<lbrakk>P \<turnstile> c \<rightarrow> c'; P \<turnstile> c' \<rightarrow>^n c''\<rbrakk> \<Longrightarrow> P \<turnstile> c \<rightarrow>^(Suc n) c''"
by (fastforce simp del: split_paired_Ex)
lemma exec_exec_n:
"P \<turnstile> c \<rightarrow>* c' \<Longrightarrow> \<exists>n. P \<turnstile> c \<rightarrow>^n c'"
by (induct rule: star.induct) (auto intro: exec_Suc)
lemma exec_eq_exec_n:
"(P \<turnstile> c \<rightarrow>* c') = (\<exists>n. P \<turnstile> c \<rightarrow>^n c')"
by (blast intro: exec_exec_n exec_n_exec)
lemma exec_n_Nil [simp]:
"[] \<turnstile> c \<rightarrow>^k c' = (c' = c \<and> k = 0)"
by (induct k) (auto simp: exec1_def)
lemma exec1_exec_n [intro!]:
"P \<turnstile> c \<rightarrow> c' \<Longrightarrow> P \<turnstile> c \<rightarrow>^1 c'"
by (cases c') simp
subsection \<open>Concrete symbolic execution steps\<close>
lemma exec_n_step:
"n \<noteq> n' \<Longrightarrow>
P \<turnstile> (n,stk,s) \<rightarrow>^k (n',stk',s') =
(\<exists>c. P \<turnstile> (n,stk,s) \<rightarrow> c \<and> P \<turnstile> c \<rightarrow>^(k - 1) (n',stk',s') \<and> 0 < k)"
by (cases k) auto
text \<open>Note: fst c refers to the program counter\<close>
lemma exec1_end:
"size P \<le> fst c \<Longrightarrow> \<not> P \<turnstile> c \<rightarrow> c'"
by (auto simp: exec1_def)
lemma exec_n_end:
"size P \<le> (n::int) \<Longrightarrow>
P \<turnstile> (n,s,stk) \<rightarrow>^k (n',s',stk') = (n' = n \<and> stk'=stk \<and> s'=s \<and> k =0)"
by (cases k) (auto simp: exec1_end)
lemmas exec_n_simps = exec_n_step exec_n_end
subsection \<open>Basic properties of \<^term>\<open>succs\<close>\<close>
lemma succs_simps [simp]:
"succs [ADD] n = {n + 1}"
"succs [LOADI v] n = {n + 1}"
"succs [LOAD x] n = {n + 1}"
"succs [STORE x] n = {n + 1}"
"succs [JMP i] n = {n + 1 + i}"
"succs [JMPGE i] n = {n + 1 + i, n + 1}"
"succs [JMPLESS i] n = {n + 1 + i, n + 1}"
by (auto simp: succs_def isuccs_def)
lemma succs_empty [iff]: "succs [] n = {}"
by (simp add: succs_def)
lemma succs_Cons:
"succs (x # xs) n = isuccs x n \<union> succs xs (1 + n)" (is "_ = ?x \<union> ?xs")
proof
let ?isuccs = "\<lambda>p P n i::int. 0 \<le> i \<and> i < size P \<and> p \<in> isuccs (P!!i) (n+i)"
have "p \<in> ?x \<union> ?xs" if assm: "p \<in> succs (x#xs) n" for p
proof -
from assm obtain i::int where isuccs: "?isuccs p (x#xs) n i"
unfolding succs_def by auto
show ?thesis
proof cases
assume "i = 0" with isuccs show ?thesis by simp
next
assume "i \<noteq> 0"
with isuccs
have "?isuccs p xs (1+n) (i - 1)" by auto
hence "p \<in> ?xs" unfolding succs_def by blast
thus ?thesis ..
qed
qed
thus "succs (x#xs) n \<subseteq> ?x \<union> ?xs" ..
have "p \<in> succs (x#xs) n" if assm: "p \<in> ?x \<or> p \<in> ?xs" for p
proof -
from assm show ?thesis
proof
assume "p \<in> ?x" thus ?thesis by (fastforce simp: succs_def)
next
assume "p \<in> ?xs"
then obtain i where "?isuccs p xs (1+n) i"
unfolding succs_def by auto
hence "?isuccs p (x#xs) n (1+i)"
by (simp add: algebra_simps)
thus ?thesis unfolding succs_def by blast
qed
qed
thus "?x \<union> ?xs \<subseteq> succs (x#xs) n" by blast
qed
lemma succs_iexec1:
assumes "c' = iexec (P!!i) (i, s, stk)" "0 \<le> i" "i < size P"
shows "fst c' \<in> succs P 0"
using assms by (cases "P !! i", auto simp: succs_def isuccs_def)
lemma succs_shift:
"(p - n \<in> succs P 0) = (p \<in> succs P n)"
by (fastforce simp: succs_def isuccs_def split: instr.split)
lemma inj_op_plus [simp]:
"inj ((+) (i::int))"
by (metis add_minus_cancel inj_on_inverseI)
lemma succs_set_shift [simp]:
"(+) i ` succs xs 0 = succs xs i"
by (force simp: succs_shift [where n=i, symmetric] intro: set_eqI)
lemma succs_append [simp]:
"succs (xs @ ys) n = succs xs n \<union> succs ys (n + size xs)"
by (induct xs arbitrary: n) (auto simp: succs_Cons algebra_simps)
lemma exits_append [simp]:
"exits (xs @ ys) = exits xs \<union> ((+) (size xs)) ` exits ys -
{0..<size xs + size ys}"
by (auto simp: exits_def image_set_diff)
lemma exits_single:
"exits [x] = isuccs x 0 - {0}"
by (auto simp: exits_def succs_def)
lemma exits_Cons:
"exits (x # xs) = (isuccs x 0 - {0}) \<union> ((+) 1) ` exits xs -
{0..<1 + size xs}"
using exits_append [of "[x]" xs]
by (simp add: exits_single)
lemma exits_empty [iff]: "exits [] = {}" by (simp add: exits_def)
lemma exits_simps [simp]:
"exits [ADD] = {1}"
"exits [LOADI v] = {1}"
"exits [LOAD x] = {1}"
"exits [STORE x] = {1}"
"i \<noteq> -1 \<Longrightarrow> exits [JMP i] = {1 + i}"
"i \<noteq> -1 \<Longrightarrow> exits [JMPGE i] = {1 + i, 1}"
"i \<noteq> -1 \<Longrightarrow> exits [JMPLESS i] = {1 + i, 1}"
by (auto simp: exits_def)
lemma acomp_succs [simp]:
"succs (acomp a) n = {n + 1 .. n + size (acomp a)}"
by (induct a arbitrary: n) auto
lemma acomp_size:
"(1::int) \<le> size (acomp a)"
by (induct a) auto
lemma acomp_exits [simp]:
"exits (acomp a) = {size (acomp a)}"
by (auto simp: exits_def acomp_size)
lemma bcomp_succs:
"0 \<le> i \<Longrightarrow>
succs (bcomp b f i) n \<subseteq> {n .. n + size (bcomp b f i)}
\<union> {n + i + size (bcomp b f i)}"
proof (induction b arbitrary: f i n)
case (And b1 b2)
from And.prems
show ?case
by (cases f)
(auto dest: And.IH(1) [THEN subsetD, rotated]
And.IH(2) [THEN subsetD, rotated])
qed auto
lemmas bcomp_succsD [dest!] = bcomp_succs [THEN subsetD, rotated]
lemma bcomp_exits:
fixes i :: int
shows
"0 \<le> i \<Longrightarrow>
exits (bcomp b f i) \<subseteq> {size (bcomp b f i), i + size (bcomp b f i)}"
by (auto simp: exits_def)
lemma bcomp_exitsD [dest!]:
"p \<in> exits (bcomp b f i) \<Longrightarrow> 0 \<le> i \<Longrightarrow>
p = size (bcomp b f i) \<or> p = i + size (bcomp b f i)"
using bcomp_exits by auto
lemma ccomp_succs:
"succs (ccomp c) n \<subseteq> {n..n + size (ccomp c)}"
proof (induction c arbitrary: n)
case SKIP thus ?case by simp
next
case Assign thus ?case by simp
next
case (Seq c1 c2)
from Seq.prems
show ?case
by (fastforce dest: Seq.IH [THEN subsetD])
next
case (If b c1 c2)
from If.prems
show ?case
by (auto dest!: If.IH [THEN subsetD] simp: isuccs_def succs_Cons)
next
case (While b c)
from While.prems
show ?case by (auto dest!: While.IH [THEN subsetD])
qed
lemma ccomp_exits:
"exits (ccomp c) \<subseteq> {size (ccomp c)}"
using ccomp_succs [of c 0] by (auto simp: exits_def)
lemma ccomp_exitsD [dest!]:
"p \<in> exits (ccomp c) \<Longrightarrow> p = size (ccomp c)"
using ccomp_exits by auto
subsection \<open>Splitting up machine executions\<close>
lemma exec1_split:
fixes i j :: int
shows
"P @ c @ P' \<turnstile> (size P + i, s) \<rightarrow> (j,s') \<Longrightarrow> 0 \<le> i \<Longrightarrow> i < size c \<Longrightarrow>
c \<turnstile> (i,s) \<rightarrow> (j - size P, s')"
proof -
assume assm: "P @ c @ P' \<turnstile> (size P + i, s) \<rightarrow> (j, s')" "0 \<le> i" "i < size c"
from assm(1) have "(\<exists>ii ss stk. (size P + i, s) = (ii, ss, stk) \<and>
(j, s') = iexec ((P @ c @ P') !! ii) (ii, ss, stk) \<and>
0 \<le> ii \<and> ii < size (P @ c @ P'))"
using exec1_def by simp
then obtain ii ss stk where assm1: "(size P + i, s) = (ii, ss, stk)"
"(j, s') = iexec ((P @ c @ P') !! ii) (ii, ss, stk)"
"0 \<le> ii" "ii < size (P @ c @ P')" by auto
from assm1(1) assm(2, 3) have "(P @ c @ P') !! ii = c !! i" by auto
with assm1(2) have "(j, s') = iexec (c !! i) (ii, ss, stk)" by simp
with assm1(1) have "(j, s') = iexec (c !! i) (size P + i, ss, stk)" by simp
then have "((- size P) + j, s') = iexec (c !! i) ((- size P) + (size P + i), ss, stk)"
using iexec_shift by (fastforce intro: iexec_shift)
then have "(j - size P, s') = iexec (c !! i) (i, ss, stk)" by simp
with assm(2, 3) assm1(1) show "c \<turnstile> (i, s) \<rightarrow> (j - size P, s')" by auto
qed
lemma exec_n_split:
fixes i j :: int
assumes "P @ c @ P' \<turnstile> (size P + i, s) \<rightarrow>^n (j, s')"
"0 \<le> i" "i < size c"
"j \<notin> {size P ..< size P + size c}"
shows "\<exists>s'' (i'::int) k m.
c \<turnstile> (i, s) \<rightarrow>^k (i', s'') \<and>
i' \<in> exits c \<and>
P @ c @ P' \<turnstile> (size P + i', s'') \<rightarrow>^m (j, s') \<and>
n = k + m"
using assms proof (induction n arbitrary: i j s)
case 0
thus ?case by simp
next
case (Suc n)
have i: "0 \<le> i" "i < size c" by fact+
from Suc.prems
have j: "\<not> (size P \<le> j \<and> j < size P + size c)" by simp
from Suc.prems
obtain i0 s0 where
step: "P @ c @ P' \<turnstile> (size P + i, s) \<rightarrow> (i0,s0)" and
rest: "P @ c @ P' \<turnstile> (i0,s0) \<rightarrow>^n (j, s')"
by clarsimp
from step i
have c: "c \<turnstile> (i,s) \<rightarrow> (i0 - size P, s0)" by (rule exec1_split)
have "i0 = size P + (i0 - size P) " by simp
then obtain j0::int where j0: "i0 = size P + j0" ..
note split_paired_Ex [simp del]
have ?case if assm: "j0 \<in> {0 ..< size c}"
proof -
from assm j0 j rest c show ?case
by (fastforce dest!: Suc.IH intro!: exec_Suc)
qed
moreover
have ?case if assm: "j0 \<notin> {0 ..< size c}"
proof -
from c j0 have "j0 \<in> succs c 0"
by (auto dest: succs_iexec1 simp: exec1_def simp del: iexec.simps)
with assm have "j0 \<in> exits c" by (simp add: exits_def)
with c j0 rest show ?case by fastforce
qed
ultimately
show ?case by cases
qed
lemma exec_n_drop_right:
fixes j :: int
assumes "c @ P' \<turnstile> (0, s) \<rightarrow>^n (j, s')" "j \<notin> {0..<size c}"
shows "\<exists>s'' i' k m.
(if c = [] then s'' = s \<and> i' = 0 \<and> k = 0
else c \<turnstile> (0, s) \<rightarrow>^k (i', s'') \<and>
i' \<in> exits c) \<and>
c @ P' \<turnstile> (i', s'') \<rightarrow>^m (j, s') \<and>
n = k + m"
using assms
by (cases "c = []")
(auto dest: exec_n_split [where P="[]", simplified])
text \<open>
Dropping the left context of a potentially incomplete execution of \<^term>\<open>c\<close>.
\<close>
lemma exec1_drop_left:
fixes i n :: int
assumes "P1 @ P2 \<turnstile> (i, s, stk) \<rightarrow> (n, s', stk')" and "size P1 \<le> i"
shows "P2 \<turnstile> (i - size P1, s, stk) \<rightarrow> (n - size P1, s', stk')"
proof -
have "i = size P1 + (i - size P1)" by simp
then obtain i' :: int where "i = size P1 + i'" ..
moreover
have "n = size P1 + (n - size P1)" by simp
then obtain n' :: int where "n = size P1 + n'" ..
ultimately
show ?thesis using assms
by (clarsimp simp: exec1_def simp del: iexec.simps)
qed
lemma exec_n_drop_left:
fixes i n :: int
assumes "P @ P' \<turnstile> (i, s, stk) \<rightarrow>^k (n, s', stk')"
"size P \<le> i" "exits P' \<subseteq> {0..}"
shows "P' \<turnstile> (i - size P, s, stk) \<rightarrow>^k (n - size P, s', stk')"
using assms proof (induction k arbitrary: i s stk)
case 0 thus ?case by simp
next
case (Suc k)
from Suc.prems
obtain i' s'' stk'' where
step: "P @ P' \<turnstile> (i, s, stk) \<rightarrow> (i', s'', stk'')" and
rest: "P @ P' \<turnstile> (i', s'', stk'') \<rightarrow>^k (n, s', stk')"
by auto
from step \<open>size P \<le> i\<close>
have *: "P' \<turnstile> (i - size P, s, stk) \<rightarrow> (i' - size P, s'', stk'')"
by (rule exec1_drop_left)
then have "i' - size P \<in> succs P' 0"
by (fastforce dest!: succs_iexec1 simp: exec1_def simp del: iexec.simps)
with \<open>exits P' \<subseteq> {0..}\<close>
have "size P \<le> i'" by (auto simp: exits_def)
from rest this \<open>exits P' \<subseteq> {0..}\<close>
have "P' \<turnstile> (i' - size P, s'', stk'') \<rightarrow>^k (n - size P, s', stk')"
by (rule Suc.IH)
with * show ?case by auto
qed
lemmas exec_n_drop_Cons =
exec_n_drop_left [where P="[instr]", simplified] for instr
definition
"closed P \<longleftrightarrow> exits P \<subseteq> {size P}"
lemma ccomp_closed [simp, intro!]: "closed (ccomp c)"
using ccomp_exits by (auto simp: closed_def)
lemma acomp_closed [simp, intro!]: "closed (acomp c)"
by (simp add: closed_def)
lemma exec_n_split_full:
fixes j :: int
assumes exec: "P @ P' \<turnstile> (0,s,stk) \<rightarrow>^k (j, s', stk')"
assumes P: "size P \<le> j"
assumes closed: "closed P"
assumes exits: "exits P' \<subseteq> {0..}"
shows "\<exists>k1 k2 s'' stk''. P \<turnstile> (0,s,stk) \<rightarrow>^k1 (size P, s'', stk'') \<and>
P' \<turnstile> (0,s'',stk'') \<rightarrow>^k2 (j - size P, s', stk')"
proof (cases "P")
case Nil with exec
show ?thesis by fastforce
next
case Cons
hence "0 < size P" by simp
with exec P closed
obtain k1 k2 s'' stk'' where
1: "P \<turnstile> (0,s,stk) \<rightarrow>^k1 (size P, s'', stk'')" and
2: "P @ P' \<turnstile> (size P,s'',stk'') \<rightarrow>^k2 (j, s', stk')"
by (auto dest!: exec_n_split [where P="[]" and i=0, simplified]
simp: closed_def)
moreover
have "j = size P + (j - size P)" by simp
then obtain j0 :: int where "j = size P + j0" ..
ultimately
show ?thesis using exits
by (fastforce dest: exec_n_drop_left)
qed
subsection \<open>Correctness theorem\<close>
lemma acomp_neq_Nil [simp]:
"acomp a \<noteq> []"
by (induct a) auto
lemma acomp_exec_n [dest!]:
"acomp a \<turnstile> (0,s,stk) \<rightarrow>^n (size (acomp a),s',stk') \<Longrightarrow>
s' = s \<and> stk' = aval a s#stk"
proof (induction a arbitrary: n s' stk stk')
case (Plus a1 a2)
let ?sz = "size (acomp a1) + (size (acomp a2) + 1)"
from Plus.prems
have "acomp a1 @ acomp a2 @ [ADD] \<turnstile> (0,s,stk) \<rightarrow>^n (?sz, s', stk')"
by (simp add: algebra_simps)
then obtain n1 s1 stk1 n2 s2 stk2 n3 where
"acomp a1 \<turnstile> (0,s,stk) \<rightarrow>^n1 (size (acomp a1), s1, stk1)"
"acomp a2 \<turnstile> (0,s1,stk1) \<rightarrow>^n2 (size (acomp a2), s2, stk2)"
"[ADD] \<turnstile> (0,s2,stk2) \<rightarrow>^n3 (1, s', stk')"
by (auto dest!: exec_n_split_full)
thus ?case by (fastforce dest: Plus.IH simp: exec_n_simps exec1_def)
qed (auto simp: exec_n_simps exec1_def)
lemma bcomp_split:
fixes i j :: int
assumes "bcomp b f i @ P' \<turnstile> (0, s, stk) \<rightarrow>^n (j, s', stk')"
"j \<notin> {0..<size (bcomp b f i)}" "0 \<le> i"
shows "\<exists>s'' stk'' (i'::int) k m.
bcomp b f i \<turnstile> (0, s, stk) \<rightarrow>^k (i', s'', stk'') \<and>
(i' = size (bcomp b f i) \<or> i' = i + size (bcomp b f i)) \<and>
bcomp b f i @ P' \<turnstile> (i', s'', stk'') \<rightarrow>^m (j, s', stk') \<and>
n = k + m"
using assms by (cases "bcomp b f i = []") (fastforce dest!: exec_n_drop_right)+
lemma bcomp_exec_n [dest]:
fixes i j :: int
assumes "bcomp b f j \<turnstile> (0, s, stk) \<rightarrow>^n (i, s', stk')"
"size (bcomp b f j) \<le> i" "0 \<le> j"
shows "i = size(bcomp b f j) + (if f = bval b s then j else 0) \<and>
s' = s \<and> stk' = stk"
using assms proof (induction b arbitrary: f j i n s' stk')
case Bc thus ?case
by (simp split: if_split_asm add: exec_n_simps exec1_def)
next
case (Not b)
from Not.prems show ?case
by (fastforce dest!: Not.IH)
next
case (And b1 b2)
let ?b2 = "bcomp b2 f j"
let ?m = "if f then size ?b2 else size ?b2 + j"
let ?b1 = "bcomp b1 False ?m"
have j: "size (bcomp (And b1 b2) f j) \<le> i" "0 \<le> j" by fact+
from And(3-5)
obtain s'' stk'' and i'::int and k m where
b1: "?b1 \<turnstile> (0, s, stk) \<rightarrow>^k (i', s'', stk'')"
"i' = size ?b1 \<or> i' = ?m + size ?b1" and
b2: "?b2 \<turnstile> (i' - size ?b1, s'', stk'') \<rightarrow>^m (i - size ?b1, s', stk')"
by (auto dest!: bcomp_split dest: exec_n_drop_left)
from b1 j(2)
have "i' = size ?b1 + (if \<not>bval b1 s then ?m else 0) \<and> s'' = s \<and> stk'' = stk"
by (auto dest!: And.IH)
with b2 j
show ?case
by (fastforce dest!: And.IH simp: exec_n_end split: if_split_asm)
next
case Less
thus ?case by (auto dest!: exec_n_split_full simp: exec_n_simps exec1_def) (* takes time *)
qed
lemma ccomp_empty [elim!]:
"ccomp c = [] \<Longrightarrow> (c,s) \<Rightarrow> s"
by (induct c) auto
declare assign_simp [simp]
lemma ccomp_exec_n:
"ccomp c \<turnstile> (0,s,stk) \<rightarrow>^n (size(ccomp c),t,stk')
\<Longrightarrow> (c,s) \<Rightarrow> t \<and> stk'=stk"
proof (induction c arbitrary: s t stk stk' n)
case SKIP
thus ?case by auto
next
case (Assign x a)
thus ?case
by simp (fastforce dest!: exec_n_split_full simp: exec_n_simps exec1_def)
next
case (Seq c1 c2)
thus ?case by (fastforce dest!: exec_n_split_full)
next
case (If b c1 c2)
note If.IH [dest!]
let ?if = "IF b THEN c1 ELSE c2"
let ?cs = "ccomp ?if"
let ?bcomp = "bcomp b False (size (ccomp c1) + 1)"
from \<open>?cs \<turnstile> (0,s,stk) \<rightarrow>^n (size ?cs,t,stk')\<close>
obtain i' :: int and k m s'' stk'' where
cs: "?cs \<turnstile> (i',s'',stk'') \<rightarrow>^m (size ?cs,t,stk')" and
"?bcomp \<turnstile> (0,s,stk) \<rightarrow>^k (i', s'', stk'')"
"i' = size ?bcomp \<or> i' = size ?bcomp + size (ccomp c1) + 1"
by (auto dest!: bcomp_split)
hence i':
"s''=s" "stk'' = stk"
"i' = (if bval b s then size ?bcomp else size ?bcomp+size(ccomp c1)+1)"
by auto
with cs have cs':
"ccomp c1@JMP (size (ccomp c2))#ccomp c2 \<turnstile>
(if bval b s then 0 else size (ccomp c1)+1, s, stk) \<rightarrow>^m
(1 + size (ccomp c1) + size (ccomp c2), t, stk')"
by (fastforce dest: exec_n_drop_left simp: exits_Cons isuccs_def algebra_simps)
show ?case
proof (cases "bval b s")
case True with cs'
show ?thesis
by simp
(fastforce dest: exec_n_drop_right
split: if_split_asm
simp: exec_n_simps exec1_def)
next
case False with cs'
show ?thesis
by (auto dest!: exec_n_drop_Cons exec_n_drop_left
simp: exits_Cons isuccs_def)
qed
next
case (While b c)
from While.prems
show ?case
proof (induction n arbitrary: s rule: nat_less_induct)
case (1 n)
have ?case if assm: "\<not> bval b s"
proof -
from assm "1.prems"
show ?case
by simp (fastforce dest!: bcomp_split simp: exec_n_simps)
qed
moreover
have ?case if b: "bval b s"
proof -
let ?c0 = "WHILE b DO c"
let ?cs = "ccomp ?c0"
let ?bs = "bcomp b False (size (ccomp c) + 1)"
let ?jmp = "[JMP (-((size ?bs + size (ccomp c) + 1)))]"
from "1.prems" b
obtain k where
cs: "?cs \<turnstile> (size ?bs, s, stk) \<rightarrow>^k (size ?cs, t, stk')" and
k: "k \<le> n"
by (fastforce dest!: bcomp_split)
show ?case
proof cases
assume "ccomp c = []"
with cs k
obtain m where
"?cs \<turnstile> (0,s,stk) \<rightarrow>^m (size (ccomp ?c0), t, stk')"
"m < n"
by (auto simp: exec_n_step [where k=k] exec1_def)
with "1.IH"
show ?case by blast
next
assume "ccomp c \<noteq> []"
with cs
obtain m m' s'' stk'' where
c: "ccomp c \<turnstile> (0, s, stk) \<rightarrow>^m' (size (ccomp c), s'', stk'')" and
rest: "?cs \<turnstile> (size ?bs + size (ccomp c), s'', stk'') \<rightarrow>^m
(size ?cs, t, stk')" and
m: "k = m + m'"
by (auto dest: exec_n_split [where i=0, simplified])
from c
have "(c,s) \<Rightarrow> s''" and stk: "stk'' = stk"
by (auto dest!: While.IH)
moreover
from rest m k stk
obtain k' where
"?cs \<turnstile> (0, s'', stk) \<rightarrow>^k' (size ?cs, t, stk')"
"k' < n"
by (auto simp: exec_n_step [where k=m] exec1_def)
with "1.IH"
have "(?c0, s'') \<Rightarrow> t \<and> stk' = stk" by blast
ultimately
show ?case using b by blast
qed
qed
ultimately show ?case by cases
qed
qed
theorem ccomp_exec:
"ccomp c \<turnstile> (0,s,stk) \<rightarrow>* (size(ccomp c),t,stk') \<Longrightarrow> (c,s) \<Rightarrow> t"
by (auto dest: exec_exec_n ccomp_exec_n)
corollary ccomp_sound:
"ccomp c \<turnstile> (0,s,stk) \<rightarrow>* (size(ccomp c),t,stk) \<longleftrightarrow> (c,s) \<Rightarrow> t"
by (blast intro!: ccomp_exec ccomp_bigstep)
end