author | wenzelm |
Thu, 14 Oct 1999 16:02:39 +0200 | |
changeset 7869 | c007f801cd59 |
parent 7761 | 7fab9592384f |
child 7874 | 180364256231 |
permissions | -rw-r--r-- |
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(* Title: HOL/Isar_examples/ExprCompiler.thy |
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ID: $Id$ |
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Author: Markus Wenzel, TU Muenchen |
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Correctness of a simple expression/stack-machine compiler. |
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*) |
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header {* Correctness of a simple expression compiler *}; |
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theory ExprCompiler = Main:; |
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text {* |
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This is a (quite trivial) example of program verification. We model |
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a compiler translating expressions to stack machine instructions, and |
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prove its correctness wrt.\ evaluation semantics. |
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*}; |
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subsection {* Binary operations *}; |
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text {* |
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Binary operations are just functions over some type of values. This |
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is both for syntax and semantics, i.e.\ we use a ``shallow |
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embedding''. |
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*}; |
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types |
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'val binop = "'val => 'val => 'val"; |
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subsection {* Expressions *}; |
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text {* |
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The language of expressions is defined as an inductive type, |
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consisting of variables, constants, and binary operations on |
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expressions. |
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*}; |
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datatype ('adr, 'val) expr = |
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Variable 'adr | |
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Constant 'val | |
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Binop "'val binop" "('adr, 'val) expr" "('adr, 'val) expr"; |
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text {* |
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Evaluation (wrt.\ an environment) is defined by primitive recursion |
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over the structure of expressions. |
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*}; |
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consts |
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eval :: "('adr, 'val) expr => ('adr => 'val) => 'val"; |
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primrec |
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"eval (Variable x) env = env x" |
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"eval (Constant c) env = c" |
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"eval (Binop f e1 e2) env = f (eval e1 env) (eval e2 env)"; |
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subsection {* Machine *}; |
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text {* |
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Next we model a simple stack machine, with three instructions. |
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*}; |
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datatype ('adr, 'val) instr = |
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Const 'val | |
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Load 'adr | |
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Apply "'val binop"; |
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text {* |
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Execution of a list of stack machine instructions is easily defined |
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as follows. |
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*}; |
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consts |
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exec :: "(('adr, 'val) instr) list |
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=> 'val list => ('adr => 'val) => 'val list"; |
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primrec |
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"exec [] stack env = stack" |
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"exec (instr # instrs) stack env = |
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(case instr of |
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Const c => exec instrs (c # stack) env |
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| Load x => exec instrs (env x # stack) env |
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| Apply f => exec instrs (f (hd stack) (hd (tl stack)) |
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# (tl (tl stack))) env)"; |
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constdefs |
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execute :: "(('adr, 'val) instr) list => ('adr => 'val) => 'val" |
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"execute instrs env == hd (exec instrs [] env)"; |
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subsection {* Compiler *}; |
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text {* |
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We are ready to define the compilation function of expressions to |
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lists of stack machine instructions. |
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*}; |
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consts |
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comp :: "('adr, 'val) expr => (('adr, 'val) instr) list"; |
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primrec |
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"comp (Variable x) = [Load x]" |
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"comp (Constant c) = [Const c]" |
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"comp (Binop f e1 e2) = comp e2 @ comp e1 @ [Apply f]"; |
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text {* |
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The main result of this development is the correctness theorem for |
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$\idt{comp}$. We first establish some lemmas about $\idt{exec}$. |
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*}; |
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lemma exec_append: |
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"ALL stack. exec (xs @ ys) stack env = |
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exec ys (exec xs stack env) env" (is "?P xs"); |
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proof (induct ?P xs type: list); |
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show "?P []"; by simp; |
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fix x xs; |
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assume "?P xs"; |
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show "?P (x # xs)" (is "?Q x"); |
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proof (induct ?Q x type: instr); |
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fix val; show "?Q (Const val)"; by (simp!); |
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next; |
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fix adr; show "?Q (Load adr)"; by (simp!); |
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next; |
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fix fun; show "?Q (Apply fun)"; by (simp!); |
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qed; |
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qed; |
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lemma exec_comp: |
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"ALL stack. exec (comp e) stack env = |
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eval e env # stack" (is "?P e"); |
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proof (induct ?P e type: expr); |
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fix adr; show "?P (Variable adr)"; by (simp!); |
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next; |
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fix val; show "?P (Constant val)"; by (simp!); |
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next; |
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fix fun e1 e2; assume "?P e1" "?P e2"; show "?P (Binop fun e1 e2)"; |
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by (simp! add: exec_append); |
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qed; |
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text {* Main theorem ahead. *}; |
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theorem correctness: "execute (comp e) env = eval e env"; |
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by (simp add: execute_def exec_comp); |
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end; |