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# Homework 6

This is the task corresponding to homework 6.

## Resources

### Definitions File

```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 =
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'"

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```

### Template File

```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```

### Check File

```theory Check
imports Submission
begin

lemma ccomp_bigstep: "\<exists>a. ((c,s) \<Rightarrow> (c',t) \<longrightarrow> ccomp n c \<turnstile> (0,s,stk) \<rightarrow>* (size(ccomp n c) + a,t,stk))"
by (rule exI, tactic \<open>Object_Logic.rulify_tac @{context} 1\<close>, erule Submission.ccomp_bigstep)

end```

Terms and Conditions