(* 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)" 

fun inth :: "'a list \<Rightarrow> int \<Rightarrow> 'a" (infixl "!!" 100) where

"(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 (induction xs arbitrary: n) (auto simp: algebra_simps)

subsection "Instructions and Stack Machine"

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 < isize P)"

declare exec1_def [simp]

lemma exec1I [intro, code_pred_intro]:

  "c' = iexec (P!!i) (i,s,stk) \<Longrightarrow> 0 \<le> i \<Longrightarrow> i < isize P

  \<Longrightarrow> P \<turnstile> (i,s,stk) \<rightarrow> c'"

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 (induction rule: exec.induct) fastforce+

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 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

lemma exec_appendR: "P \<turnstile> c \<rightarrow>* c' \<Longrightarrow> P@P' \<turnstile> c \<rightarrow>* c'"

by (induction rule: exec.induct) (fastforce intro: exec1_appendR)+

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 (auto split: instr.split)

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 (induction 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 (induction a arbitrary: stk) fastforce+

fun bcomp :: "bexp \<Rightarrow> bool \<Rightarrow> int \<Rightarrow> instr list" where

"bcomp (Bc v) c n = (if v=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 [JMPLESS n] else [JMPGE 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 (Seq 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 Seq.IH(1) by fastforce

  moreover

  have "?cc1 @ ?cc2 \<turnstile> (isize ?cc1,s2,stk) \<rightarrow>* (isize(?cc1 @ ?cc2),s3,stk)"

    using Seq.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