Replaced force by fast because force may now take forever to fail
(due to a recend change of David's)
(* 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 = Main:;
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:
"ALL stack. exec (xs @ ys) stack env =
exec ys (exec xs stack env) env" (is "?P xs");
proof (induct ?P xs type: list);
show "?P []"; by simp;
fix x xs; assume "?P xs";
show "?P (x # xs)" (is "?Q x");
proof (induct ?Q x type: instr);
show "!!val. ?Q (Const val)"; by (simp!);
show "!!adr. ?Q (Load adr)"; by (simp!);
show "!!fun. ?Q (Apply fun)"; by (simp!);
qed;
qed;
theorem correctness: "execute (compile e) env = eval e env";
proof -;
have "ALL stack. exec (compile e) stack env =
eval e env # stack" (is "?P e");
proof (induct ?P e type: expr);
show "!!adr. ?P (Variable adr)"; by simp;
show "!!val. ?P (Constant val)"; by simp;
show "!!fun e1 e2. (?P e1 ==> ?P e2 ==> ?P (Binop fun e1 e2))";
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:
"ALL stack. exec (xs @ ys) stack env
= exec ys (exec xs stack env) env" (is "?P xs");
proof (induct ?P xs);
show "?P []" (is "ALL s. ?Q s");
proof;
fix s; have "exec ([] @ ys) s env = exec ys s env"; by simp;
also; have "... = exec ys (exec [] s env) env"; by simp;
finally; show "?Q s"; .;
qed;
fix x xs; assume hyp: "?P xs";
show "?P (x # xs)";
proof (induct x);
fix val;
show "?P (Const val # xs)" (is "ALL s. ?Q s");
proof;
fix s;
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 hyp; have "... = exec ys (exec xs (val # s) env) env"; ..;
also; have "... = exec ys (exec (Const val # xs) s env) env";
by simp;
finally; show "?Q s"; .;
qed;
next;
fix adr; from hyp; show "?P (Load adr # xs)"; by simp -- {* same as above *};
next;
fix fun;
show "?P (Apply fun # xs)" (is "ALL s. ?Q s");
proof;
fix s;
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 hyp; 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 "?Q s"; .;
qed;
qed;
qed;
theorem correctness: "execute (compile e) env = eval e env";
proof -;
have exec_compile:
"ALL stack. exec (compile e) stack env = eval e env # stack" (is "?P e");
proof (induct e);
fix adr; show "?P (Variable adr)" (is "ALL s. ?Q s");
proof;
fix 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 "?Q s"; .;
qed;
next;
fix val; show "?P (Constant val)"; by simp -- {* same as above *};
next;
fix fun e1 e2; assume hyp1: "?P e1" and hyp2: "?P e2";
show "?P (Binop fun e1 e2)" (is "ALL s. ?Q s");
proof;
fix 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; from hyp2; have "exec (compile e2) s env = eval e2 env # s"; ..;
also; from hyp1; have "exec (compile e1) ... env = eval e1 env # ..."; ..;
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 "?Q s"; .;
qed;
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;