frap/Interpreters_template.v
2016-02-09 09:07:37 -05:00

324 lines
8.1 KiB
Coq

Require Import Frap.
(* We begin with a return to our arithmetic language from the last chapter,
* adding subtraction*, which will come in handy later.
* *: good pun, right? *)
Inductive arith : Set :=
| Const (n : nat)
| Var (x : var)
| Plus (e1 e2 : arith)
| Minus (e1 e2 : arith)
| Times (e1 e2 : arith).
Example ex1 := Const 42.
Example ex2 := Plus (Var "y") (Times (Var "x") (Const 3)).
Definition valuation := fmap var nat.
(* A valuation is a finite map from [var] to [nat]. *)
(* The interpreter is a fairly innocuous-looking recursive function. *)
Fixpoint interp (e : arith) (v : valuation) : nat :=
match e with
| Const n => n
| Var x =>
(* Note use of infix operator to look up a key in a finite map. *)
match v $? x with
| None => 0 (* goofy default value! *)
| Some n => n
end
| Plus e1 e2 => interp e1 v + interp e2 v
| Minus e1 e2 => interp e1 v - interp e2 v
(* For anyone who's wondering: this [-] sticks at 0,
* if we would otherwise underflow. *)
| Times e1 e2 => interp e1 v * interp e2 v
end.
(* Here's an example valuation, using an infix operator for map extension. *)
Definition valuation0 : valuation :=
$0 $+ ("x", 17) $+ ("y", 3).
Theorem interp_ex1 : interp ex1 valuation0 = 42.
Proof.
simplify.
equality.
Qed.
Theorem interp_ex2 : interp ex2 valuation0 = 54.
Proof.
unfold valuation0.
simplify.
equality.
Qed.
(* Here's the silly transformation we defined last time. *)
Fixpoint commuter (e : arith) : arith :=
match e with
| Const _ => e
| Var _ => e
| Plus e1 e2 => Plus (commuter e2) (commuter e1)
| Minus e1 e2 => Minus (commuter e1) (commuter e2)
(* ^-- NB: didn't change the operand order here! *)
| Times e1 e2 => Times (commuter e2) (commuter e1)
end.
(* Instead of proving various odds-and-ends properties about it,
* let's show what we *really* care about: it preserves the
* *meanings* of expressions! *)
Theorem commuter_ok : forall v e, interp (commuter e) v = interp e v.
Proof.
admit.
Qed.
(* Let's also revisit substitution. *)
Fixpoint substitute (inThis : arith) (replaceThis : var) (withThis : arith) : arith :=
match inThis with
| Const _ => inThis
| Var x => if x ==v replaceThis then withThis else inThis
| Plus e1 e2 => Plus (substitute e1 replaceThis withThis) (substitute e2 replaceThis withThis)
| Minus e1 e2 => Minus (substitute e1 replaceThis withThis) (substitute e2 replaceThis withThis)
| Times e1 e2 => Times (substitute e1 replaceThis withThis) (substitute e2 replaceThis withThis)
end.
(* How should we state a correctness property for [substitute]?
Theorem substitute_ok : forall v replaceThis withThis inThis,
...
Proof.
Qed.*)
(* Let's also defined a pared-down version of the expression-simplificaton
* functions from last chapter. *)
Fixpoint doSomeArithmetic (e : arith) : arith :=
match e with
| Const _ => e
| Var _ => e
| Plus (Const n1) (Const n2) => Const (n1 + n2)
| Plus e1 e2 => Plus (doSomeArithmetic e1) (doSomeArithmetic e2)
| Minus e1 e2 => Minus (doSomeArithmetic e1) (doSomeArithmetic e2)
| Times (Const n1) (Const n2) => Const (n1 * n2)
| Times e1 e2 => Times (doSomeArithmetic e1) (doSomeArithmetic e2)
end.
Theorem doSomeArithmetic_ok : forall e v, interp (doSomeArithmetic e) v = interp e v.
Proof.
admit.
Qed.
(* Of course, we're going to get bored if we confine ourselves to arithmetic
* expressions for the rest of our journey. Let's get a bit fancier and define
* a *stack machine*, related to postfix calculators that some of you may have
* experienced. *)
Inductive instruction :=
| PushConst (n : nat)
| PushVar (x : var)
| Add
| Subtract
| Multiply.
(* What does it all mean? An interpreter tells us unambiguously! *)
Definition run1 (i : instruction) (v : valuation) (stack : list nat) : list nat :=
match i with
| PushConst n => n :: stack
| PushVar x => (match v $? x with
| None => 0
| Some n => n
end) :: stack
| Add =>
match stack with
| arg2 :: arg1 :: stack' => arg1 + arg2 :: stack'
| _ => stack (* arbitrary behavior in erroneous case (stack underflow) *)
end
| Subtract =>
match stack with
| arg2 :: arg1 :: stack' => arg1 - arg2 :: stack'
| _ => stack (* arbitrary behavior in erroneous case *)
end
| Multiply =>
match stack with
| arg2 :: arg1 :: stack' => arg1 * arg2 :: stack'
| _ => stack (* arbitrary behavior in erroneous case *)
end
end.
(* That function explained how to run one instruction.
* Here's how to run several of them. *)
Fixpoint run (is : list instruction) (v : valuation) (stack : list nat) : list nat :=
match is with
| nil => stack
| i :: is' => run is' v (run1 i v stack)
end.
(* Instead of writing fiddly stack programs ourselves, let's *compile*
* arithmetic expressions into equivalent stack programs. *)
Fixpoint compile (e : arith) : list instruction :=
match e with
| Const n => PushConst n :: nil
| Var x => PushVar x :: nil
| Plus e1 e2 => compile e1 ++ compile e2 ++ Add :: nil
| Minus e1 e2 => compile e1 ++ compile e2 ++ Subtract :: nil
| Times e1 e2 => compile e1 ++ compile e2 ++ Multiply :: nil
end.
Theorem compile_ok : forall e v, run (compile e) v nil = interp e v :: nil.
Proof.
admit.
Qed.
(* Let's get a bit fancier, moving toward the level of general-purpose
* imperative languages. Here's a language of commands, building on the
* language of expressions we have defined. *)
Inductive cmd :=
| Skip
| Assign (x : var) (e : arith)
| Sequence (c1 c2 : cmd)
| Repeat (e : arith) (body : cmd).
Fixpoint selfCompose {A} (f : A -> A) (n : nat) : A -> A :=
match n with
| O => fun x => x
| S n' => fun x => selfCompose f n' (f x)
end.
Fixpoint exec (c : cmd) (v : valuation) : valuation :=
match c with
| Skip => v
| Assign x e => v $+ (x, interp e v)
| Sequence c1 c2 => exec c2 (exec c1 v)
| Repeat e body => selfCompose (exec body) (interp e v) v
end.
(* Let's define some programs and prove that they operate in certain ways. *)
Example factorial_ugly :=
Sequence
(Assign "output" (Const 1))
(Repeat (Var "input")
(Sequence
(Assign "output" (Times (Var "output") (Var "input")))
(Assign "input" (Minus (Var "input") (Const 1))))).
(* Ouch; that code is hard to read. Let's introduce some notations to make the
* concrete syntax more palatable. We won't explain the general mechanisms on
* display here, but see the Coq manual for details, or try to reverse-engineer
* them from our examples. *)
Coercion Const : nat >-> arith.
Coercion Var : var >-> arith.
Infix "+" := Plus : arith_scope.
Infix "-" := Minus : arith_scope.
Infix "*" := Times : arith_scope.
Delimit Scope arith_scope with arith.
Notation "x <- e" := (Assign x e%arith) (at level 75).
Infix ";" := Sequence (at level 76).
Notation "'repeat' e 'doing' body 'done'" := (Repeat e%arith body) (at level 75).
(* OK, let's try that program again. *)
Example factorial :=
"output" <- 1;
repeat "input" doing
"output" <- "output" * "input";
"input" <- "input" - 1
done.
(* Now we prove that it really computes factorial.
* First, a reference implementation as a functional program. *)
Fixpoint fact (n : nat) : nat :=
match n with
| O => 1
| S n' => n * fact n'
end.
Theorem factorial_ok : forall v input,
v $? "input" = Some input
-> exec factorial v $? "output" = Some (fact input).
Proof.
admit.
Qed.
(* One last example: let's try to do loop unrolling, for constant iteration
* counts. That is, we can duplicate the loop body instead of using an explicit
* loop. *)
(* This obvious-sounding fact will come in handy: self-composition gives the
* same result, when passed two functions that map equal inputs to equal
* outputs. *)
Lemma selfCompose_extensional : forall {A} (f g : A -> A) n x,
(forall y, f y = g y)
-> selfCompose f n x = selfCompose g n x.
Proof.
induct n; simplify; try equality.
rewrite H.
apply IHn.
trivial.
Qed.
(*Theorem unroll_ok : forall c v, exec (unroll c) v = exec c v.
Proof.
Qed.*)