src/HOL/IMP/Compiler.thy

tuned proofs -- eliminated vacuous "induct arbitrary: ..." situations;

(* Author: Tobias Nipkow *)

header "A Compiler for IMP"

theory Compiler imports Big_Step 
begin

subsection "List setup"

text {*
  We are going to define a small machine language where programs are
  lists of instructions. For nicer algebraic properties in our lemmas
  later, we prefer @{typ int} to @{term nat} as program counter.
  
  Therefore, we define notation for size and indexing for lists 
  on @{typ int}:
*}
abbreviation "isize xs == int (length xs)" 

primrec
  inth :: "'a list => int => 'a" (infixl "!!" 100) where
  inth_Cons: "(x # xs) !! n = (if n = 0 then x else xs !! (n - 1))"

text {*
  The only additional lemma we need is indexing over append:
*}
lemma inth_append [simp]:
  "0 \<le> n \<Longrightarrow>
  (xs @ ys) !! n = (if n < isize xs then xs !! n else ys !! (n - isize xs))"
  by (induct xs arbitrary: n) (auto simp: algebra_simps)

subsection "Instructions and Stack Machine"

datatype instr = 
  LOADI int | 
  LOAD string | 
  ADD |
  STORE string |
  JMP int |
  JMPFLESS int |
  JMPFGE 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)"

inductive iexec1 :: "instr \<Rightarrow> config \<Rightarrow> config \<Rightarrow> bool"
    ("(_/ \<turnstile>i (_ \<rightarrow>/ _))" [59,0,59] 60)
where
"LOADI n \<turnstile>i (i,s,stk) \<rightarrow> (i+1,s, n#stk)" |
"LOAD x  \<turnstile>i (i,s,stk) \<rightarrow> (i+1,s, s x # stk)" |
"ADD     \<turnstile>i (i,s,stk) \<rightarrow> (i+1,s, (hd2 stk + hd stk) # tl2 stk)" |
"STORE n \<turnstile>i (i,s,stk) \<rightarrow> (i+1,s(n := hd stk),tl stk)" |
"JMP n   \<turnstile>i (i,s,stk) \<rightarrow> (i+1+n,s,stk)" |
"JMPFLESS n \<turnstile>i (i,s,stk) \<rightarrow> (if hd2 stk < hd stk then i+1+n else i+1,s,tl2 stk)" |
"JMPFGE n \<turnstile>i (i,s,stk) \<rightarrow> (if hd2 stk >= hd stk then i+1+n else i+1,s,tl2 stk)"

code_pred iexec1 .

declare iexec1.intros

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> P!!i \<turnstile>i (i,s,stk) \<rightarrow> c' \<and> 0 \<le> i \<and> i < isize P)"

declare exec1_def [simp] 

lemma exec1I [intro, code_pred_intro]:
  assumes "P!!i \<turnstile>i (i,s,stk) \<rightarrow> c'" "0 \<le> i" "i < isize P" 
  shows "P \<turnstile> (i,s,stk) \<rightarrow> c'"
  using assms by simp

inductive exec :: "instr list \<Rightarrow> config \<Rightarrow> config \<Rightarrow> bool" ("_/ \<turnstile> (_ \<rightarrow>*/ _)" 50)
where
refl: "P \<turnstile> c \<rightarrow>* c" |
step: "P \<turnstile> c \<rightarrow> c' \<Longrightarrow> P \<turnstile> c' \<rightarrow>* c'' \<Longrightarrow> P \<turnstile> c \<rightarrow>* c''"

declare refl[intro] step[intro]

lemmas exec_induct = exec.induct[split_format(complete)]

code_pred exec by force

values
  "{(i,map t [''x'',''y''],stk) | i t stk.
    [LOAD ''y'', STORE ''x''] \<turnstile>
    (0, <''x'' := 3, ''y'' := 4>, []) \<rightarrow>* (i,t,stk)}"


subsection{* Verification infrastructure *}

lemma exec_trans: "P \<turnstile> c \<rightarrow>* c' \<Longrightarrow> P \<turnstile> c' \<rightarrow>* c'' \<Longrightarrow> P \<turnstile> c \<rightarrow>* c''"
  by (induct rule: exec.induct) fastforce+

inductive_cases iexec1_cases [elim!]:
  "LOADI n \<turnstile>i c \<rightarrow> c'" 
  "LOAD x  \<turnstile>i c \<rightarrow> c'"
  "ADD     \<turnstile>i c \<rightarrow> c'"
  "STORE n \<turnstile>i c \<rightarrow> c'" 
  "JMP n   \<turnstile>i c \<rightarrow> c'"
  "JMPFLESS n \<turnstile>i c \<rightarrow> c'"
  "JMPFGE n \<turnstile>i c \<rightarrow> c'"

text {* Simplification rules for @{const iexec1}. *}
lemma iexec1_simps [simp]:
  "LOADI n \<turnstile>i c \<rightarrow> c' = (\<exists>i s stk. c = (i, s, stk) \<and> c' = (i + 1, s, n # stk))"
  "LOAD x \<turnstile>i c \<rightarrow> c' = (\<exists>i s stk. c = (i, s, stk) \<and> c' = (i + 1, s, s x # stk))"
  "ADD \<turnstile>i c \<rightarrow> c' = 
  (\<exists>i s stk. c = (i, s, stk) \<and> c' = (i + 1, s, (hd2 stk + hd stk) # tl2 stk))"
  "STORE x \<turnstile>i c \<rightarrow> c' = 
  (\<exists>i s stk. c = (i, s, stk) \<and> c' = (i + 1, s(x := hd stk), tl stk))"
  "JMP n \<turnstile>i c \<rightarrow> c' = (\<exists>i s stk. c = (i, s, stk) \<and> c' = (i + 1 + n, s, stk))"
   "JMPFLESS n \<turnstile>i c \<rightarrow> c' = 
  (\<exists>i s stk. c = (i, s, stk) \<and> 
             c' = (if hd2 stk < hd stk then i + 1 + n else i + 1, s, tl2 stk))"  
  "JMPFGE n \<turnstile>i c \<rightarrow> c' = 
  (\<exists>i s stk. c = (i, s, stk) \<and> 
             c' = (if hd stk \<le> hd2 stk then i + 1 + n else i + 1, s, tl2 stk))"
  by (auto split del: split_if intro!: iexec1.intros)


text{* Below we need to argue about the execution of code that is embedded in
larger programs. For this purpose we show that execution is preserved by
appending code to the left or right of a program. *}

lemma exec1_appendR: "P \<turnstile> c \<rightarrow> c' \<Longrightarrow> P@P' \<turnstile> c \<rightarrow> c'"
  by auto

lemma exec_appendR: "P \<turnstile> c \<rightarrow>* c' \<Longrightarrow> P@P' \<turnstile> c \<rightarrow>* c'"
  by (induct rule: exec.induct) (fastforce intro: exec1_appendR)+

lemma iexec1_shiftI:
  assumes "X \<turnstile>i (i,s,stk) \<rightarrow> (i',s',stk')"
  shows "X \<turnstile>i (n+i,s,stk) \<rightarrow> (n+i',s',stk')"
  using assms by cases auto

lemma iexec1_shiftD:
  assumes "X \<turnstile>i (n+i,s,stk) \<rightarrow> (n+i',s',stk')"
  shows "X \<turnstile>i (i,s,stk) \<rightarrow> (i',s',stk')"
  using assms by cases auto

lemma iexec_shift [simp]: 
  "(X \<turnstile>i (n+i,s,stk) \<rightarrow> (n+i',s',stk')) = (X \<turnstile>i (i,s,stk) \<rightarrow> (i',s',stk'))"
  by (blast intro: iexec1_shiftI dest: iexec1_shiftD)
  
lemma exec1_appendL:
  "P \<turnstile> (i,s,stk) \<rightarrow> (i',s',stk') \<Longrightarrow>
   P' @ P \<turnstile> (isize(P')+i,s,stk) \<rightarrow> (isize(P')+i',s',stk')"
  by simp

lemma exec_appendL:
 "P \<turnstile> (i,s,stk) \<rightarrow>* (i',s',stk')  \<Longrightarrow>
  P' @ P \<turnstile> (isize(P')+i,s,stk) \<rightarrow>* (isize(P')+i',s',stk')"
  by (induct rule: exec_induct) (blast intro!: exec1_appendL)+

text{* Now we specialise the above lemmas to enable automatic proofs of
@{prop "P \<turnstile> c \<rightarrow>* c'"} where @{text P} is a mixture of concrete instructions and
pieces of code that we already know how they execute (by induction), combined
by @{text "@"} and @{text "#"}. Backward jumps are not supported.
The details should be skipped on a first reading.

If we have just executed the first instruction of the program, drop it: *}

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]:
 "isize P' <= i
  \<Longrightarrow> P \<turnstile> (i - isize P',s,stk) \<rightarrow>* (i',s',stk')
  \<Longrightarrow> P' @ P \<turnstile> (i,s,stk) \<rightarrow>* (isize P' + i',s',stk')"
  by (drule exec_appendL[where P'=P']) simp

text{* Split the execution of a compound program up into the excution of its
parts: *}

lemma exec_append_trans[intro]:
"P \<turnstile> (0,s,stk) \<rightarrow>* (i',s',stk') \<Longrightarrow>
 isize P \<le> i' \<Longrightarrow>
 P' \<turnstile>  (i' - isize P,s',stk') \<rightarrow>* (i'',s'',stk'') \<Longrightarrow>
 j'' = isize P + i''
 \<Longrightarrow>
 P @ P' \<turnstile> (0,s,stk) \<rightarrow>* (j'',s'',stk'')"
  by(metis exec_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>* (isize(acomp a),s,aval a s#stk)"
  by (induct a arbitrary: stk) fastforce+

fun bcomp :: "bexp \<Rightarrow> bool \<Rightarrow> int \<Rightarrow> instr list" where
"bcomp (B bv) c n = (if bv=c then [JMP n] else [])" |
"bcomp (Not b) c n = bcomp b (\<not>c) n" |
"bcomp (And b1 b2) c n =
 (let cb2 = bcomp b2 c n;
        m = (if c then isize cb2 else isize cb2+n);
      cb1 = bcomp b1 False m
  in cb1 @ cb2)" |
"bcomp (Less a1 a2) c n =
 acomp a1 @ acomp a2 @ (if c then [JMPFLESS n] else [JMPFGE n])"

value
  "bcomp (And (Less (V ''x'') (V ''y'')) (Not(Less (V ''u'') (V ''v''))))
     False 3"

lemma bcomp_correct[intro]:
  "0 \<le> n \<Longrightarrow>
  bcomp b c n \<turnstile>
 (0,s,stk)  \<rightarrow>*  (isize(bcomp b c n) + (if c = bval b s then n else 0),s,stk)"
proof(induction b arbitrary: c n)
  case Not
  from Not(1)[where c="~c"] Not(2) show ?case by fastforce
next
  case (And b1 b2)
  from And(1)[of "if c then isize (bcomp b2 c n) else isize (bcomp b2 c n) + n" 
                 "False"] 
       And(2)[of n  "c"] 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\<^isub>1;c\<^isub>2) = ccomp c\<^isub>1 @ ccomp c\<^isub>2" |
"ccomp (IF b THEN c\<^isub>1 ELSE c\<^isub>2) =
  (let cc\<^isub>1 = ccomp c\<^isub>1; cc\<^isub>2 = ccomp c\<^isub>2; cb = bcomp b False (isize cc\<^isub>1 + 1)
   in cb @ cc\<^isub>1 @ JMP (isize cc\<^isub>2) # cc\<^isub>2)" |
"ccomp (WHILE b DO c) =
 (let cc = ccomp c; cb = bcomp b False (isize cc + 1)
  in cb @ cc @ [JMP (-(isize cb + isize cc + 1))])"


value "ccomp
 (IF Less (V ''u'') (N 1) THEN ''u'' ::= Plus (V ''u'') (N 1)
  ELSE ''v'' ::= V ''u'')"

value "ccomp (WHILE Less (V ''u'') (N 1) DO (''u'' ::= Plus (V ''u'') (N 1)))"


subsection "Preservation of semantics"

lemma ccomp_bigstep:
  "(c,s) \<Rightarrow> t \<Longrightarrow> ccomp c \<turnstile> (0,s,stk) \<rightarrow>* (isize(ccomp c),t,stk)"
proof(induction arbitrary: stk rule: big_step_induct)
  case (Assign x a s)
  show ?case by (fastforce simp:fun_upd_def cong: if_cong)
next
  case (Semi c1 s1 s2 c2 s3)
  let ?cc1 = "ccomp c1"  let ?cc2 = "ccomp c2"
  have "?cc1 @ ?cc2 \<turnstile> (0,s1,stk) \<rightarrow>* (isize ?cc1,s2,stk)"
    using Semi.IH(1) by fastforce
  moreover
  have "?cc1 @ ?cc2 \<turnstile> (isize ?cc1,s2,stk) \<rightarrow>* (isize(?cc1 @ ?cc2),s3,stk)"
    using Semi.IH(2) by fastforce
  ultimately show ?case by simp (blast intro: exec_trans)
next
  case (WhileTrue b s1 c s2 s3)
  let ?cc = "ccomp c"
  let ?cb = "bcomp b False (isize ?cc + 1)"
  let ?cw = "ccomp(WHILE b DO c)"
  have "?cw \<turnstile> (0,s1,stk) \<rightarrow>* (isize ?cb + isize ?cc,s2,stk)"
    using WhileTrue.IH(1) WhileTrue.hyps(1) by fastforce
  moreover
  have "?cw \<turnstile> (isize ?cb + isize ?cc,s2,stk) \<rightarrow>* (0,s2,stk)"
    by fastforce
  moreover
  have "?cw \<turnstile> (0,s2,stk) \<rightarrow>* (isize ?cw,s3,stk)" by(rule WhileTrue.IH(2))
  ultimately show ?case by(blast intro: exec_trans)
qed fastforce+

end