src/HOL/Isar_examples/ExprCompiler.thy

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(*  Title:      HOL/Isar_examples/ExprCompiler.thy
    ID:         $Id$
    Author:     Markus Wenzel, TU Muenchen

Correctness of a simple expression/stack-machine compiler.
*)

header {* Correctness of a simple expression compiler *}

theory ExprCompiler imports Main begin

text {*
 This is a (rather trivial) example of program verification.  We model
 a compiler for translating expressions to stack machine instructions,
 and prove its correctness wrt.\ some evaluation semantics.
*}


subsection {* Binary operations *}

text {*
 Binary operations are just functions over some type of values.  This
 is both for abstract syntax and semantics, i.e.\ we use a ``shallow
 embedding'' here.
*}

types
  'val binop = "'val => 'val => 'val"


subsection {* Expressions *}

text {*
 The language of expressions is defined as an inductive type,
 consisting of variables, constants, and binary operations on
 expressions.
*}

datatype ('adr, 'val) expr =
  Variable 'adr |
  Constant 'val |
  Binop "'val binop" "('adr, 'val) expr" "('adr, 'val) expr"

text {*
 Evaluation (wrt.\ some environment of variable assignments) is
 defined by primitive recursion over the structure of expressions.
*}

consts
  eval :: "('adr, 'val) expr => ('adr => 'val) => 'val"

primrec
  "eval (Variable x) env = env x"
  "eval (Constant c) env = c"
  "eval (Binop f e1 e2) env = f (eval e1 env) (eval e2 env)"


subsection {* Machine *}

text {*
 Next we model a simple stack machine, with three instructions.
*}

datatype ('adr, 'val) instr =
  Const 'val |
  Load 'adr |
  Apply "'val binop"

text {*
 Execution of a list of stack machine instructions is easily defined
 as follows.
*}

consts
  exec :: "(('adr, 'val) instr) list
    => 'val list => ('adr => 'val) => 'val list"

primrec
  "exec [] stack env = stack"
  "exec (instr # instrs) stack env =
    (case instr of
      Const c => exec instrs (c # stack) env
    | Load x => exec instrs (env x # stack) env
    | Apply f => exec instrs (f (hd stack) (hd (tl stack))
                   # (tl (tl stack))) env)"

constdefs
  execute :: "(('adr, 'val) instr) list => ('adr => 'val) => 'val"
  "execute instrs env == hd (exec instrs [] env)"


subsection {* Compiler *}

text {*
 We are ready to define the compilation function of expressions to
 lists of stack machine instructions.
*}

consts
  compile :: "('adr, 'val) expr => (('adr, 'val) instr) list"

primrec
  "compile (Variable x) = [Load x]"
  "compile (Constant c) = [Const c]"
  "compile (Binop f e1 e2) = compile e2 @ compile e1 @ [Apply f]"


text {*
 The main result of this development is the correctness theorem for
 $\idt{compile}$.  We first establish a lemma about $\idt{exec}$ and
 list append.
*}

lemma exec_append:
  "exec (xs @ ys) stack env =
    exec ys (exec xs stack env) env"
proof (induct xs fixing: stack)
  case Nil
  show ?case by simp
next
  case (Cons x xs)
  show ?case
  proof (induct x)
    case Const show ?case by simp
  next
    case Load show ?case by simp
  next
    case Apply show ?case by simp
  qed
qed

theorem correctness: "execute (compile e) env = eval e env"
proof -
  have "\<And>stack. exec (compile e) stack env = eval e env # stack"
  proof (induct e)
    case Variable show ?case by simp
  next
    case Constant show ?case by simp
  next
    case Binop then show ?case by (simp add: exec_append)
  qed
  thus ?thesis by (simp add: execute_def)
qed


text {*
 \bigskip In the proofs above, the \name{simp} method does quite a lot
 of work behind the scenes (mostly ``functional program execution'').
 Subsequently, the same reasoning is elaborated in detail --- at most
 one recursive function definition is used at a time.  Thus we get a
 better idea of what is actually going on.
*}

lemma exec_append':
  "exec (xs @ ys) stack env = exec ys (exec xs stack env) env"
proof (induct xs fixing: stack)
  case (Nil s)
  have "exec ([] @ ys) s env = exec ys s env" by simp
  also have "... = exec ys (exec [] s env) env" by simp
  finally show ?case .
next
  case (Cons x xs s)
  show ?case
  proof (induct x)
    case (Const val)
    have "exec ((Const val # xs) @ ys) s env = exec (Const val # xs @ ys) s env"
      by simp
    also have "... = exec (xs @ ys) (val # s) env" by simp
    also from Cons have "... = exec ys (exec xs (val # s) env) env" .
    also have "... = exec ys (exec (Const val # xs) s env) env" by simp
    finally show ?case .
  next
    case (Load adr)
    from Cons show ?case by simp -- {* same as above *}
  next
    case (Apply fun)
    have "exec ((Apply fun # xs) @ ys) s env =
        exec (Apply fun # xs @ ys) s env" by simp
    also have "... =
        exec (xs @ ys) (fun (hd s) (hd (tl s)) # (tl (tl s))) env" by simp
    also from Cons have "... =
        exec ys (exec xs (fun (hd s) (hd (tl s)) # tl (tl s)) env) env" .
    also have "... = exec ys (exec (Apply fun # xs) s env) env" by simp
    finally show ?case .
  qed
qed

theorem correctness': "execute (compile e) env = eval e env"
proof -
  have exec_compile: "\<And>stack. exec (compile e) stack env = eval e env # stack"
  proof (induct e)
    case (Variable adr s)
    have "exec (compile (Variable adr)) s env = exec [Load adr] s env"
      by simp
    also have "... = env adr # s" by simp
    also have "env adr = eval (Variable adr) env" by simp
    finally show ?case .
  next
    case (Constant val s)
    show ?case by simp -- {* same as above *}
  next
    case (Binop fun e1 e2 s)
    have "exec (compile (Binop fun e1 e2)) s env =
        exec (compile e2 @ compile e1 @ [Apply fun]) s env" by simp
    also have "... = exec [Apply fun]
        (exec (compile e1) (exec (compile e2) s env) env) env"
      by (simp only: exec_append)
    also have "exec (compile e2) s env = eval e2 env # s" by fact
    also have "exec (compile e1) ... env = eval e1 env # ..." by fact
    also have "exec [Apply fun] ... env =
        fun (hd ...) (hd (tl ...)) # (tl (tl ...))" by simp
    also have "... = fun (eval e1 env) (eval e2 env) # s" by simp
    also have "fun (eval e1 env) (eval e2 env) =
        eval (Binop fun e1 e2) env"
      by simp
    finally show ?case .
  qed

  have "execute (compile e) env = hd (exec (compile e) [] env)"
    by (simp add: execute_def)
  also from exec_compile
    have "exec (compile e) [] env = [eval e env]" .
  also have "hd ... = eval e env" by simp
  finally show ?thesis .
qed

end