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(** * Imp: Simple Imperative Programs *)
(** In this chapter, we take a more serious look at how to use Rocq as
a tool to study other things. Our case study is a _simple
imperative programming language_ called Imp, embodying a tiny core
fragment of conventional mainstream languages such as C and Java.
Here is a familiar mathematical function written in Imp.
Z := X;
Y := 1;
while Z <> 0 do
Y := Y * Z;
Z := Z - 1
end
*)
(** We concentrate here on defining the _syntax_ and _semantics_ of
Imp; later, in _Programming Language Foundations_ (_Software
Foundations_, volume 2), we develop a theory of _program
equivalence_ and introduce _Hoare Logic_, a popular logic for
reasoning about imperative programs. *)
Set Warnings "-notation-overridden".
From Stdlib Require Import Bool.
From Stdlib Require Import Init.Nat.
From Stdlib Require Import Arith.
From Stdlib Require Import EqNat. Import Nat.
From Stdlib Require Import Lia.
From Stdlib Require Import List. Import ListNotations.
From Stdlib Require Import Strings.String.
From LF Require Import Maps.
(* ################################################################# *)
(** * Arithmetic and Boolean Expressions *)
(** We'll present Imp in three parts: first a core language of
_arithmetic and boolean expressions_, then an extension of these
with _variables_, and finally a language of _commands_ including
assignment, conditionals, sequencing, and loops. *)
(* ================================================================= *)
(** ** Syntax *)
Module AExp.
(** These two definitions specify the _abstract syntax_ of
arithmetic and boolean expressions. *)
Inductive aexp : Type :=
| ANum (n : nat)
| APlus (a1 a2 : aexp)
| AMinus (a1 a2 : aexp)
| AMult (a1 a2 : aexp).
Inductive bexp : Type :=
| BTrue
| BFalse
| BEq (a1 a2 : aexp)
| BNeq (a1 a2 : aexp)
| BLe (a1 a2 : aexp)
| BGt (a1 a2 : aexp)
| BNot (b : bexp)
| BAnd (b1 b2 : bexp).
(** In this chapter, we'll mostly elide the translation from the
concrete syntax that a programmer would actually write to these
abstract syntax trees -- the process that, for example, would
translate the string ["1 + 2 * 3"] to the AST
APlus (ANum 1) (AMult (ANum 2) (ANum 3)).
The optional chapter [ImpParser] develops a simple lexical
analyzer and parser that can perform this translation. You do not
need to understand that chapter to understand this one, but if you
haven't already taken a course where these techniques are
covered (e.g., a course on compilers) you may want to skim it. *)
(** For comparison, here's a conventional BNF (Backus-Naur Form)
grammar defining the same abstract syntax:
a := nat
| a + a
| a - a
| a * a
b := true
| false
| a = a
| a <> a
| a <= a
| a > a
| ~ b
| b && b
*)
(** Compared to the Rocq version above...
- The BNF is more informal -- for example, it gives some
suggestions about the surface syntax of expressions (like the
fact that the addition operation is written with an infix
[+]) while leaving other aspects of lexical analysis and
parsing (like the relative precedence of [+], [-], and [*],
the use of parens to group subexpressions, etc.)
unspecified. Some additional information -- and human
intelligence -- would be required to turn this description
into a formal definition, e.g., for implementing a compiler.
The Rocq version consistently omits all this information and
concentrates on the abstract syntax only.
- Conversely, the BNF version is lighter and easier to read.
Its informality makes it flexible, a big advantage in
situations like discussions at the blackboard, where
conveying general ideas is more important than nailing down
every detail precisely.
Indeed, there are dozens of BNF-like notations and people
switch freely among them -- usually without bothering to say
which kind of BNF they're using, because there is no need to:
a rough-and-ready informal understanding is all that's
important.
It's good to be comfortable with both sorts of notations: informal
ones for communicating between humans and formal ones for carrying
out implementations and proofs. *)
(* ================================================================= *)
(** ** Evaluation *)
(** _Evaluating_ an arithmetic expression produces a number. *)
Fixpoint aeval (a : aexp) : nat :=
match a with
| ANum n => n
| APlus a1 a2 => (aeval a1) + (aeval a2)
| AMinus a1 a2 => (aeval a1) - (aeval a2)
| AMult a1 a2 => (aeval a1) * (aeval a2)
end.
Example test_aeval1:
aeval (APlus (ANum 2) (ANum 2)) = 4.
Proof. reflexivity. Qed.
(** Similarly, evaluating a boolean expression yields a boolean. *)
Fixpoint beval (b : bexp) : bool :=
match b with
| BTrue => true
| BFalse => false
| BEq a1 a2 => (aeval a1) =? (aeval a2)
| BNeq a1 a2 => negb ((aeval a1) =? (aeval a2))
| BLe a1 a2 => (aeval a1) <=? (aeval a2)
| BGt a1 a2 => negb ((aeval a1) <=? (aeval a2))
| BNot b1 => negb (beval b1)
| BAnd b1 b2 => andb (beval b1) (beval b2)
end.
(* ================================================================= *)
(** ** Optimization *)
(** We haven't defined very much yet, but we can already get
some mileage out of the definitions. Suppose we define a function
that takes an arithmetic expression and slightly simplifies it,
changing every occurrence of [0 + e] (i.e., [(APlus (ANum 0) e])
into just [e]. *)
Fixpoint optimize_0plus (a:aexp) : aexp :=
match a with
| ANum n => ANum n
| APlus (ANum 0) e2 => optimize_0plus e2
| APlus e1 e2 => APlus (optimize_0plus e1) (optimize_0plus e2)
| AMinus e1 e2 => AMinus (optimize_0plus e1) (optimize_0plus e2)
| AMult e1 e2 => AMult (optimize_0plus e1) (optimize_0plus e2)
end.
(** To gain confidence that our optimization is doing the right
thing we can test it on some examples and see if the output looks
OK. *)
Example test_optimize_0plus:
optimize_0plus (APlus (ANum 2)
(APlus (ANum 0)
(APlus (ANum 0) (ANum 1))))
= APlus (ANum 2) (ANum 1).
Proof. reflexivity. Qed.
(** But if we want to be certain the optimization is correct --
that evaluating an optimized expression _always_ gives the same
result as the original -- we should prove it! *)
Theorem optimize_0plus_sound: forall a,
aeval (optimize_0plus a) = aeval a.
Proof.
intros a. induction a.
- (* ANum *) reflexivity.
- (* APlus *) destruct a1 eqn:Ea1.
+ (* a1 = ANum n *) destruct n eqn:En.
* (* n = 0 *) simpl. apply IHa2.
* (* n <> 0 *) simpl. rewrite IHa2. reflexivity.
+ (* a1 = APlus a1_1 a1_2 *)
simpl. simpl in IHa1. rewrite IHa1.
rewrite IHa2. reflexivity.
+ (* a1 = AMinus a1_1 a1_2 *)
simpl. simpl in IHa1. rewrite IHa1.
rewrite IHa2. reflexivity.
+ (* a1 = AMult a1_1 a1_2 *)
simpl. simpl in IHa1. rewrite IHa1.
rewrite IHa2. reflexivity.
- (* AMinus *)
simpl. rewrite IHa1. rewrite IHa2. reflexivity.
- (* AMult *)
simpl. rewrite IHa1. rewrite IHa2. reflexivity. Qed.
(* ################################################################# *)
(** * Rocq Automation *)
(** The amount of repetition in this last proof is a little
annoying. And if either the language of arithmetic expressions or
the optimization being proved sound were significantly more
complex, it would start to be a real problem.
So far, we've been doing all our proofs using just a small handful
of Rocq's tactics and completely ignoring its powerful facilities
for constructing parts of proofs automatically. This section
introduces some of these facilities, and we will see more over the
next several chapters. Getting used to them will take some
energy -- Rocq's automation is a power tool -- but it will allow us
to scale up our efforts to more complex definitions and more
interesting properties without becoming overwhelmed by boring,
repetitive, low-level details. *)
(* ================================================================= *)
(** ** Tacticals *)
(** _Tacticals_ is Rocq's term for tactics that take other tactics as
arguments -- "higher-order tactics," if you will. *)
(* ----------------------------------------------------------------- *)
(** *** The [try] Tactical *)
(** If [T] is a tactic, then [try T] is a tactic that is just like [T]
except that, if [T] fails, [try T] _successfully_ does nothing at
all (rather than failing). *)
Theorem silly1 : forall (P : Prop), P -> P.
Proof.
intros P HP.
try reflexivity. (* Plain [reflexivity] would have failed. *)
apply HP. (* We can still finish the proof in some other way. *)
Qed.
Theorem silly2 : forall ae, aeval ae = aeval ae.
Proof.
try reflexivity. (* This just does [reflexivity]. *)
Qed.
(** There is not much reason to use [try] in completely manual
proofs like these, but it is very useful for doing automated
proofs in conjunction with the [;] tactical, which we show
next. *)
(* ----------------------------------------------------------------- *)
(** *** The [;] Tactical (Simple Form) *)
(** In its most common form, the [;] tactical takes two tactics as
arguments. The compound tactic [T;T'] first performs [T] and then
performs [T'] on _each subgoal_ generated by [T]. *)
(** For example, consider the following trivial lemma: *)
Lemma foo : forall n, 0 <=? n = true.
Proof.
intros.
destruct n.
(* Leaves two subgoals, which are discharged identically... *)
- (* n=0 *) simpl. reflexivity.
- (* n=Sn' *) simpl. reflexivity.
Qed.
(** We can simplify this proof using the [;] tactical: *)
Lemma foo' : forall n, 0 <=? n = true.
Proof.
intros.
(* [destruct] the current goal *)
destruct n;
(* then [simpl] each resulting subgoal *)
simpl;
(* and do [reflexivity] on each resulting subgoal *)
reflexivity.
Qed.
(** Using [try] and [;] together, we can get rid of the repetition in
the proof that was bothering us a little while ago. *)
Theorem optimize_0plus_sound': forall a,
aeval (optimize_0plus a) = aeval a.
Proof.
intros a.
induction a;
(* Most cases follow directly by the IH... *)
try (simpl; rewrite IHa1; rewrite IHa2; reflexivity).
(* ... but the remaining cases -- ANum and APlus --
are different: *)
- (* ANum *) reflexivity.
- (* APlus *)
destruct a1 eqn:Ea1;
(* Again, most cases follow directly by the IH: *)
try (simpl; simpl in IHa1; rewrite IHa1;
rewrite IHa2; reflexivity).
(* The interesting case, on which the [try...]
does nothing, is when [e1 = ANum n]. In this
case, we have to destruct [n] (to see whether
the optimization applies) and rewrite with the
induction hypothesis. *)
+ (* a1 = ANum n *) destruct n eqn:En;
simpl; rewrite IHa2; reflexivity. Qed.
(** Rocq experts often use this "[...; try... ]" idiom after a tactic
like [induction] to take care of many similar cases all at once.
Indeed, this practice has an analog in informal proofs. For
example, here is an informal proof of the optimization theorem
that matches the structure of the formal one:
_Theorem_: For all arithmetic expressions [a],
aeval (optimize_0plus a) = aeval a.
_Proof_: By induction on [a]. Most cases follow directly from the
IH. The remaining cases are as follows:
- Suppose [a = ANum n] for some [n]. We must show
aeval (optimize_0plus (ANum n)) = aeval (ANum n).
This is immediate from the definition of [optimize_0plus].
- Suppose [a = APlus a1 a2] for some [a1] and [a2]. We must
show
aeval (optimize_0plus (APlus a1 a2)) = aeval (APlus a1 a2).
Consider the possible forms of [a1]. For most of them,
[optimize_0plus] simply calls itself recursively for the
subexpressions and rebuilds a new expression of the same form
as [a1]; in these cases, the result follows directly from the
IH.
The interesting case is when [a1 = ANum n] for some [n]. If
[n = 0], then
optimize_0plus (APlus a1 a2) = optimize_0plus a2
and the IH for [a2] is exactly what we need. On the other
hand, if [n = S n'] for some [n'], then again [optimize_0plus]
simply calls itself recursively, and the result follows from
the IH. [] *)
(** However, this proof can still be improved: the first case (for
[a = ANum n]) is very trivial -- even more trivial than the cases
that we said simply followed from the IH -- yet we have chosen to
write it out in full. It would be better and clearer to drop it
and just say, at the top, "Most cases are either immediate or
direct from the IH. The only interesting case is the one for
[APlus]..." We can make the same improvement in our formal proof
too. Here's how it looks: *)
Theorem optimize_0plus_sound'': forall a,
aeval (optimize_0plus a) = aeval a.
Proof.
intros a.
induction a;
(* Most cases follow directly by the IH *)
try (simpl; rewrite IHa1; rewrite IHa2; reflexivity);
(* ... or are immediate by definition *)
try reflexivity.
(* The interesting case is when a = APlus a1 a2. *)
- (* APlus *)
destruct a1; try (simpl; simpl in IHa1; rewrite IHa1;
rewrite IHa2; reflexivity).
+ (* a1 = ANum n *) destruct n;
simpl; rewrite IHa2; reflexivity. Qed.
(* ----------------------------------------------------------------- *)
(** *** The [;] Tactical (General Form) *)
(** The [;] tactical also has a more general form than the simple
[T;T'] we've seen above. If [T], [T1], ..., [Tn] are tactics,
then
T; [T1 | T2 | ... | Tn]
is a tactic that first performs [T] and then performs [T1] on the
first subgoal generated by [T], performs [T2] on the second
subgoal, etc.
So [T;T'] is just special notation for the case when all of the
[Ti]'s are the same tactic; i.e., [T;T'] is shorthand for:
T; [T' | T' | ... | T']
*)
(* ----------------------------------------------------------------- *)
(** *** The [repeat] Tactical *)
(** The [repeat] tactical takes another tactic and keeps applying this
tactic until it fails or until it succeeds but doesn't make any
progress.
Here is an example proving that [10] is in a long list using
[repeat]. *)
Theorem In10 : In 10 [1;2;3;4;5;6;7;8;9;10].
Proof.
repeat (try (left; reflexivity); right).
Qed.
(** The tactic [repeat T] never fails: if the tactic [T] doesn't apply
to the original goal, then repeat _succeeds_ without changing the
goal at all (i.e., it repeats zero times). *)
Theorem In10' : In 10 [1;2;3;4;5;6;7;8;9;10].
Proof.
repeat simpl.
repeat (left; reflexivity).
repeat (right; try (left; reflexivity)).
Qed.
(** The tactic [repeat T] does not have any upper bound on the
number of times it applies [T]. If [T] is a tactic that _always_
succeeds (and makes progress), then repeat [T] will loop
forever. *)
Theorem repeat_loop : forall (m n : nat),
m + n = n + m.
Proof.
intros m n.
(* Uncomment the next line to see the infinite loop occur. You will
then need to interrupt Rocq to make it listen to you again. (In
Proof General, [C-c C-c] does this.) *)
(* repeat rewrite Nat.add_comm. *)
Admitted.
(** Wait -- did we just write an infinite loop in Rocq?!?!
Sort of.
While evaluation in Rocq's term language, Gallina, is guaranteed to
terminate, _tactic_ evaluation is not. This does not affect Rocq's
logical consistency, however, since the job of [repeat] and other
tactics is to guide Rocq in constructing proofs; if the
construction process diverges (i.e., it does not terminate), this
simply means that we have failed to construct a proof at all, not
that we have constructed a bad proof. *)
(** **** Exercise: 3 stars, standard (optimize_0plus_b_sound)
Since the [optimize_0plus] transformation doesn't change the value
of [aexp]s, we should be able to apply it to all the [aexp]s that
appear in a [bexp] without changing the [bexp]'s value. Write a
function that performs this transformation on [bexp]s and prove
it is sound. Use the tacticals we've just seen to make the proof
as short and elegant as possible. *)
Fixpoint optimize_0plus_b (b : bexp) : bexp
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Example optimize_0plus_b_test1:
optimize_0plus_b (BNot (BGt (APlus (ANum 0) (ANum 4)) (ANum 8))) =
(BNot (BGt (ANum 4) (ANum 8))).
Proof. (* FILL IN HERE *) Admitted.
Example optimize_0plus_b_test2:
optimize_0plus_b (BAnd (BLe (APlus (ANum 0) (ANum 4)) (ANum 5)) BTrue) =
(BAnd (BLe (ANum 4) (ANum 5)) BTrue).
Proof. (* FILL IN HERE *) Admitted.
Theorem optimize_0plus_b_sound : forall b,
beval (optimize_0plus_b b) = beval b.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 4 stars, standard, optional (optimize)
_Design exercise_: The optimization implemented by our
[optimize_0plus] function is only one of many possible
optimizations on arithmetic and boolean expressions. Write a more
sophisticated optimizer and prove it correct. (You will probably
find it easiest to start small -- add just a single, simple
optimization and its correctness proof -- and build up
incrementally to something more interesting.) *)
(* FILL IN HERE
[] *)
(* ================================================================= *)
(** ** Defining New Tactics *)
(** Rocq also provides facilities for "programming" in tactic
scripts.
The [Ltac] idiom illustrated below gives a handy way to define
"shorthand tactics" that bundle several tactics into a single
command.
Ltac also includes syntactic pattern-matching on the goal and
context, as well as general programming facilities.
It is useful for proof automation and there are several idioms for
programming with Ltac. Because it is a language style you might
not have seen before, a good reference is the textbook "Certified
Programming with dependent types" [CPDT], which is more advanced
that what we will need in this course, but is considered by many
the best reference for Ltac programming.
Just for future reference: Rocq provides two other ways of defining
new tactics. There is a [Tactic Notation] command that allows
defining new tactics with custom control over their concrete
syntax. And there is also a low-level API that can be used to
build tactics that directly manipulate Rocq's internal structures.
We will not need either of these for present purposes.
Here's an example [Ltac] script called [invert]. *)
Ltac invert H :=
inversion H; subst; clear H.
(** This defines a new tactic called [invert] that takes a hypothesis
[H] as an argument and performs the sequence of commands
[inversion H; subst; clear H]. This gives us quick way to do
inversion on evidence and constructors, rewrite with the generated
equations, and remove the redundant hypothesis at the end. *)
Lemma invert_example1: forall {a b c: nat}, [a ;b] = [a;c] -> b = c.
intros.
invert H.
reflexivity.
Qed.
(* ================================================================= *)
(** ** The [lia] Tactic *)
(** The [lia] tactic implements a decision procedure for integer linear
arithmetic, a subset of propositional logic and arithmetic.
If the goal is a universally quantified formula made out of
- numeric constants, addition ([+] and [S]), subtraction ([-]
and [pred]), and multiplication by constants (this is what
makes it Presburger arithmetic),
- equality ([=] and [<>]) and ordering ([<=] and [>]), and
- the logical connectives [/\], [\/], [~], and [->],
then invoking [lia] will either solve the goal or fail, meaning
that the goal is actually false. (If the goal is _not_ of this
form, [lia] will fail.) *)
Example silly_presburger_example : forall m n o p,
m + n <= n + o /\ o + 3 = p + 3 ->
m <= p.
Proof.
intros. lia.
Qed.
Example add_comm__lia : forall m n,
m + n = n + m.
Proof.
intros. lia.
Qed.
Example add_assoc__lia : forall m n p,
m + (n + p) = m + n + p.
Proof.
intros. lia.
Qed.
(** (Note the [From Stdlib Require Import Lia.] at the top of
this file, which makes [lia] available.) *)
(* ================================================================= *)
(** ** A Few More Handy Tactics *)
(** Finally, here are some miscellaneous tactics that you may find
convenient.
- [clear H]: Delete hypothesis [H] from the context.
- [subst x]: Given a variable [x], find an assumption [x = e] or
[e = x] in the context, replace [x] with [e] throughout the
context and current goal, and clear the assumption.
- [subst]: Substitute away _all_ assumptions of the form [x = e]
or [e = x] (where [x] is a variable).
- [rename... into...]: Change the name of a hypothesis in the
proof context. For example, if the context includes a variable
named [x], then [rename x into y] will change all occurrences
of [x] to [y].
- [assumption]: Try to find a hypothesis [H] in the context that
exactly matches the goal; if one is found, solve the goal.
- [contradiction]: Try to find a hypothesis [H] in the context
that is logically equivalent to [False]. If one is found,
solve the goal.
- [constructor]: Try to find a constructor [c] (from some
[Inductive] definition in the current environment) that can be
applied to solve the current goal. If one is found, behave
like [apply c].
We'll see examples of all of these as we go along. *)
(* ################################################################# *)
(** * Evaluation as a Relation *)
(** We have presented [aeval] and [beval] as functions defined by
[Fixpoint]s. Another way to think about evaluation -- one that is
often more flexible -- is as a _relation_ between expressions and
their values. This perspective leads to [Inductive] definitions
like the following... *)
Module aevalR_first_try.
Inductive aevalR : aexp -> nat -> Prop :=
| E_ANum (n : nat) :
aevalR (ANum n) n
| E_APlus (e1 e2 : aexp) (n1 n2 : nat) :
aevalR e1 n1 ->
aevalR e2 n2 ->
aevalR (APlus e1 e2) (n1 + n2)
| E_AMinus (e1 e2 : aexp) (n1 n2 : nat) :
aevalR e1 n1 ->
aevalR e2 n2 ->
aevalR (AMinus e1 e2) (n1 - n2)
| E_AMult (e1 e2 : aexp) (n1 n2 : nat) :
aevalR e1 n1 ->
aevalR e2 n2 ->
aevalR (AMult e1 e2) (n1 * n2).
Module HypothesisNames.
(** A small notational aside. We could also write the definition of
[aevalR] as follow, with explicit names for the hypotheses in each
case: *)
Inductive aevalR : aexp -> nat -> Prop :=
| E_ANum (n : nat) :
aevalR (ANum n) n
| E_APlus (e1 e2 : aexp) (n1 n2 : nat)
(H1 : aevalR e1 n1)
(H2 : aevalR e2 n2) :
aevalR (APlus e1 e2) (n1 + n2)
| E_AMinus (e1 e2 : aexp) (n1 n2 : nat)
(H1 : aevalR e1 n1)
(H2 : aevalR e2 n2) :
aevalR (AMinus e1 e2) (n1 - n2)
| E_AMult (e1 e2 : aexp) (n1 n2 : nat)
(H1 : aevalR e1 n1)
(H2 : aevalR e2 n2) :
aevalR (AMult e1 e2) (n1 * n2).
(** This style gives us more control over the names that Rocq chooses
during proofs involving [aevalR], at the cost of making the
definition a little more verbose. *)
End HypothesisNames.
(** It will be convenient to have an infix notation for
[aevalR]. We'll write [e ==> n] to mean that arithmetic expression
[e] evaluates to value [n]. *)
Notation "e '==>' n"
:= (aevalR e n)
(at level 90, left associativity)
: type_scope.
End aevalR_first_try.
(** As we saw in our case study of regular expressions in
chapter [IndProp], Rocq provides a way to use this notation in
the definition of [aevalR] itself. *)
Reserved Notation "e '==>' n" (at level 90, left associativity).
Inductive aevalR : aexp -> nat -> Prop :=
| E_ANum (n : nat) :
(ANum n) ==> n
| E_APlus (e1 e2 : aexp) (n1 n2 : nat) :
(e1 ==> n1) ->
(e2 ==> n2) ->
(APlus e1 e2) ==> (n1 + n2)
| E_AMinus (e1 e2 : aexp) (n1 n2 : nat) :
(e1 ==> n1) ->
(e2 ==> n2) ->
(AMinus e1 e2) ==> (n1 - n2)
| E_AMult (e1 e2 : aexp) (n1 n2 : nat) :
(e1 ==> n1) ->
(e2 ==> n2) ->
(AMult e1 e2) ==> (n1 * n2)
where "e '==>' n" := (aevalR e n) : type_scope.
(* ================================================================= *)
(** ** Inference Rule Notation *)
(** In informal discussions, it is convenient to write the rules
for [aevalR] and similar relations in the more readable graphical
form of _inference rules_, where the premises above the line
justify the conclusion below the line.
For example, the constructor [E_APlus]...
| E_APlus : forall (e1 e2 : aexp) (n1 n2 : nat),
aevalR e1 n1 ->
aevalR e2 n2 ->
aevalR (APlus e1 e2) (n1 + n2)
...can be written like this as an inference rule:
e1 ==> n1
e2 ==> n2
-------------------- (E_APlus)
APlus e1 e2 ==> n1+n2
*)
(** Formally, there is nothing deep about inference rules: they
are just implications.
You can read the rule name on the right as the name of the
constructor and read each of the linebreaks between the premises
above the line (as well as the line itself) as [->].
All the variables mentioned in the rule ([e1], [n1], etc.) are
implicitly bound by universal quantifiers at the beginning. (Such
variables are often called _metavariables_ to distinguish them
from the variables of the language we are defining. At the
moment, our arithmetic expressions don't include variables, but
we'll soon be adding them.)
The whole collection of rules is understood as being wrapped in an
[Inductive] declaration. In informal prose, this is sometimes
indicated by saying something like "Let [aevalR] be the smallest
relation closed under the following rules...". *)
(** For example, we could define [==>] as the smallest relation
closed under these rules:
----------- (E_ANum)
ANum n ==> n
e1 ==> n1
e2 ==> n2
-------------------- (E_APlus)
APlus e1 e2 ==> n1+n2
e1 ==> n1
e2 ==> n2
--------------------- (E_AMinus)
AMinus e1 e2 ==> n1-n2
e1 ==> n1
e2 ==> n2
-------------------- (E_AMult)
AMult e1 e2 ==> n1*n2
*)
(** **** Exercise: 1 star, standard, optional (beval_rules)
Here, again, is the Rocq definition of the [beval] function:
Fixpoint beval (e : bexp) : bool :=
match e with
| BTrue => true
| BFalse => false
| BEq a1 a2 => (aeval a1) =? (aeval a2)
| BNeq a1 a2 => negb ((aeval a1) =? (aeval a2))
| BLe a1 a2 => (aeval a1) <=? (aeval a2)
| BGt a1 a2 => ~((aeval a1) <=? (aeval a2))
| BNot b => negb (beval b)
| BAnd b1 b2 => andb (beval b1) (beval b2)
end.
Write out a corresponding definition of boolean evaluation as a
relation (in inference rule notation). *)
(* FILL IN HERE *)
(* Do not modify the following line: *)
Definition manual_grade_for_beval_rules : option (nat*string) := None.
(** [] *)
(* ================================================================= *)
(** ** Equivalence of the Definitions *)
(** It is straightforward to prove that the relational and functional
definitions of evaluation agree: *)
Theorem aevalR_iff_aeval : forall a n,
(a ==> n) <-> aeval a = n.
Proof.
split.
- (* -> *)
intros H.
induction H; simpl.
+ (* E_ANum *)
reflexivity.
+ (* E_APlus *)
rewrite IHaevalR1. rewrite IHaevalR2. reflexivity.
+ (* E_AMinus *)
rewrite IHaevalR1. rewrite IHaevalR2. reflexivity.
+ (* E_AMult *)
rewrite IHaevalR1. rewrite IHaevalR2. reflexivity.
- (* <- *)
generalize dependent n.
induction a;
simpl; intros; subst.
+ (* ANum *)
apply E_ANum.
+ (* APlus *)
apply E_APlus.
* apply IHa1. reflexivity.
* apply IHa2. reflexivity.
+ (* AMinus *)
apply E_AMinus.
* apply IHa1. reflexivity.
* apply IHa2. reflexivity.
+ (* AMult *)
apply E_AMult.
* apply IHa1. reflexivity.
* apply IHa2. reflexivity.
Qed.
(** Again, we can make the proof quite a bit shorter using some
tacticals. *)
Theorem aevalR_iff_aeval' : forall a n,
(a ==> n) <-> aeval a = n.
Proof.
(* WORKED IN CLASS *)
split.
- (* -> *)
intros H; induction H; subst; reflexivity.
- (* <- *)
generalize dependent n.
induction a; simpl; intros; subst; constructor;
try apply IHa1; try apply IHa2; reflexivity.
Qed.
(** **** Exercise: 3 stars, standard (bevalR)
Write a relation [bevalR] in the same style as
[aevalR], and prove that it is equivalent to [beval]. *)
Reserved Notation "e '==>b' b" (at level 90, left associativity).
Inductive bevalR: bexp -> bool -> Prop :=
(* FILL IN HERE *)
where "e '==>b' b" := (bevalR e b) : type_scope
.
Lemma bevalR_iff_beval : forall b bv,
b ==>b bv <-> beval b = bv.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
End AExp.
(* ================================================================= *)
(** ** Computational vs. Relational Definitions *)
(** For the definitions of evaluation for arithmetic and boolean
expressions, the choice of whether to use functional or relational
definitions is mainly a matter of taste: either way works fine.
However, there are many situations where relational definitions of
evaluation work much better than functional ones. *)
Module aevalR_division.
(** For example, suppose that we wanted to extend the arithmetic
operations with division: *)
Inductive aexp : Type :=
| ANum (n : nat)
| APlus (a1 a2 : aexp)
| AMinus (a1 a2 : aexp)
| AMult (a1 a2 : aexp)
| ADiv (a1 a2 : aexp). (* <--- NEW *)
(** Extending the definition of [aeval] to handle this new
operation would not be straightforward (what should we return as
the result of [ADiv (ANum 5) (ANum 0)]?). But extending [aevalR]
is very easy. *)
Reserved Notation "e '==>' n"
(at level 90, left associativity).
Inductive aevalR : aexp -> nat -> Prop :=
| E_ANum (n : nat) :
(ANum n) ==> n
| E_APlus (a1 a2 : aexp) (n1 n2 : nat) :
(a1 ==> n1) -> (a2 ==> n2) -> (APlus a1 a2) ==> (n1 + n2)
| E_AMinus (a1 a2 : aexp) (n1 n2 : nat) :
(a1 ==> n1) -> (a2 ==> n2) -> (AMinus a1 a2) ==> (n1 - n2)
| E_AMult (a1 a2 : aexp) (n1 n2 : nat) :
(a1 ==> n1) -> (a2 ==> n2) -> (AMult a1 a2) ==> (n1 * n2)
| E_ADiv (a1 a2 : aexp) (n1 n2 n3 : nat) : (* <----- NEW *)
(a1 ==> n1) -> (a2 ==> n2) -> (n2 > 0) ->
(mult n2 n3 = n1) -> (ADiv a1 a2) ==> n3
where "a '==>' n" := (aevalR a n) : type_scope.
(** Notice that this evaluation relation corresponds to a _partial_
function: There are some inputs for which it does not specify an
output. *)
End aevalR_division.
Module aevalR_extended.
(** Or suppose that we want to extend the arithmetic operations
by a nondeterministic number generator [any] that, when evaluated,
may yield any number.
(Note that this is not the same as making a _probabilistic_ choice
among all possible numbers -- we're not specifying any particular
probability distribution for the results, just saying what results
are _possible_.) *)
Reserved Notation "e '==>' n" (at level 90, left associativity).
Inductive aexp : Type :=
| AAny (* <--- NEW *)
| ANum (n : nat)
| APlus (a1 a2 : aexp)
| AMinus (a1 a2 : aexp)
| AMult (a1 a2 : aexp).
(** Again, extending [aeval] would be tricky, since now
evaluation is _not_ a deterministic function from expressions to
numbers; but extending [aevalR] is no problem... *)
Inductive aevalR : aexp -> nat -> Prop :=
| E_Any (n : nat) :
AAny ==> n (* <--- NEW *)
| E_ANum (n : nat) :
(ANum n) ==> n
| E_APlus (a1 a2 : aexp) (n1 n2 : nat) :
(a1 ==> n1) -> (a2 ==> n2) -> (APlus a1 a2) ==> (n1 + n2)
| E_AMinus (a1 a2 : aexp) (n1 n2 : nat) :
(a1 ==> n1) -> (a2 ==> n2) -> (AMinus a1 a2) ==> (n1 - n2)
| E_AMult (a1 a2 : aexp) (n1 n2 : nat) :
(a1 ==> n1) -> (a2 ==> n2) -> (AMult a1 a2) ==> (n1 * n2)
where "a '==>' n" := (aevalR a n) : type_scope.
End aevalR_extended.
(** At this point you maybe wondering: Which of these styles
should I use by default?
In the examples we've just seen, relational definitions turned out
to be more useful than functional ones. For situations like
these, where the thing being defined is not easy to express as a
function, or indeed where it is _not_ a function, there is no real
choice. But what about when both styles are workable?
One point in favor of relational definitions is that they can be
more elegant and easier to understand.
Another is that Rocq automatically generates nice inversion and
induction principles from [Inductive] definitions.
On the other hand, functional definitions can often be more
convenient:
- Functions are automatically deterministic and total; for a
relational definition, we have to _prove_ these properties
explicitly if we need them.
- With functions we can also take advantage of Rocq's computation
mechanism to simplify expressions during proofs.
Furthermore, functions can be directly "extracted" from Gallina to
executable code in OCaml or Haskell.
Ultimately, the choice often comes down to either the specifics of
a particular situation or simply a question of taste. Indeed, in
large Rocq developments it is common to see a definition given in
_both_ functional and relational styles, plus a lemma stating that
the two coincide, allowing further proofs to switch from one point
of view to the other at will. *)
(* ################################################################# *)
(** * Expressions With Variables *)
(** Let's return to defining Imp, where the next thing we need to do
is to enrich our arithmetic and boolean expressions with
variables.
To keep things simple, we'll assume that all variables are global
and that they only hold numbers. *)
(* ================================================================= *)
(** ** States *)
(** Since we'll want to look variables up to find out their current
values, we'll use total maps from the [Maps] chapter.
A _machine state_ (or just _state_) represents the current values
of all variables at some point in the execution of a program. *)
(** For simplicity, we assume that the state is defined for
_all_ variables, even though any given program is only able to
mention a finite number of them. Because each variable stores a
natural number, we can represent the state as a total map from
strings (variable names) to [nat], and will use [0] as default
value in the store. *)
Definition state := total_map nat.
(* ================================================================= *)
(** ** Syntax *)
(** We can add variables to the arithmetic expressions we had before
simply by including one more constructor: *)
Inductive aexp : Type :=
| ANum (n : nat)
| AId (x : string) (* <--- NEW *)
| APlus (a1 a2 : aexp)
| AMinus (a1 a2 : aexp)
| AMult (a1 a2 : aexp).
(** Defining a few variable names as notational shorthands will make
examples easier to read: *)
Definition W : string := "W".
Definition X : string := "X".
Definition Y : string := "Y".
Definition Z : string := "Z".
(** (This convention for naming program variables ([X], [Y],
[Z]) clashes a bit with our earlier use of uppercase letters for
types. Since we're not using polymorphism heavily in the chapters
developed to Imp, this overloading should not cause confusion.) *)
(** The definition of [bexp]s is unchanged (except that it now refers
to the new [aexp]s): *)
Inductive bexp : Type :=
| BTrue
| BFalse
| BEq (a1 a2 : aexp)
| BNeq (a1 a2 : aexp)
| BLe (a1 a2 : aexp)
| BGt (a1 a2 : aexp)
| BNot (b : bexp)
| BAnd (b1 b2 : bexp).
(* ================================================================= *)
(** ** Notations *)
(** To make Imp programs easier to read and write, we introduce some
notations and implicit coercions. *)
(** You do not need to understand exactly what these declarations do.
Briefly, though:
- The [Coercion] declaration stipulates that a function (or
constructor) can be implicitly used by the type system to
coerce a value of the input type to a value of the output
type. For instance, the coercion declaration for [AId]
allows us to use plain strings when an [aexp] is expected;
the string will implicitly be wrapped with [AId].
- [Declare Custom Entry com] tells Rocq to create a new "custom
grammar" for parsing Imp expressions and programs. The first
notation declaration after this tells Rocq that anything
between [<{] and [}>] should be parsed using the Imp
grammar. Again, it is not necessary to understand the
details, but it is important to recognize that we are
defining _new_ interpretations for some familiar operators
like [+], [-], [*], [=], [<=], etc., when they occur between
[<{] and [}>]. *)
Coercion AId : string >-> aexp.
Coercion ANum : nat >-> aexp.
Declare Custom Entry com.
Declare Scope com_scope.
Notation "<{ e }>" := e
(e custom com, format "'[hv' <{ '/ ' '[v' e ']' '/' }> ']'") : com_scope.
Notation "( x )" := x (in custom com, x at level 99).
Notation "x" := x (in custom com at level 0, x constr at level 0).
Notation "f x .. y" := (.. (f x) .. y)
(in custom com at level 0, only parsing,
f constr at level 0, x constr at level 1,
y constr at level 1).
Notation "x + y" := (APlus x y) (in custom com at level 50, left associativity).
Notation "x - y" := (AMinus x y) (in custom com at level 50, left associativity).
Notation "x * y" := (AMult x y) (in custom com at level 40, left associativity).
Notation "'true'" := true (at level 1).
Notation "'true'" := BTrue (in custom com at level 0).
Notation "'false'" := false (at level 1).
Notation "'false'" := BFalse (in custom com at level 0).
Notation "x <= y" := (BLe x y) (in custom com at level 70, no associativity).
Notation "x > y" := (BGt x y) (in custom com at level 70, no associativity).
Notation "x = y" := (BEq x y) (in custom com at level 70, no associativity).
Notation "x <> y" := (BNeq x y) (in custom com at level 70, no associativity).
Notation "x && y" := (BAnd x y) (in custom com at level 80, left associativity).
Notation "'~' b" := (BNot b) (in custom com at level 75, right associativity).
Open Scope com_scope.
(** We can now write [3 + (X * 2)] instead of [APlus 3 (AMult X 2)],
and [true && ~(X <= 4)] instead of [BAnd true (BNot (BLe X 4))]. *)
Definition example_aexp : aexp := <{ 3 + (X * 2) }>.
Definition example_bexp : bexp := <{ true && ~(X <= 4) }>.
(* ================================================================= *)
(** ** Evaluation *)
(** The arith and boolean evaluators must now be extended to
handle variables in the obvious way, taking a state [st] as an
extra argument: *)
Fixpoint aeval (st : state) (* <--- NEW *)
(a : aexp) : nat :=
match a with
| ANum n => n
| AId x => st x (* <--- NEW *)
| <{a1 + a2}> => (aeval st a1) + (aeval st a2)
| <{a1 - a2}> => (aeval st a1) - (aeval st a2)
| <{a1 * a2}> => (aeval st a1) * (aeval st a2)
end.
Fixpoint beval (st : state) (* <--- NEW *)
(b : bexp) : bool :=
match b with
| <{true}> => true
| <{false}> => false
| <{a1 = a2}> => (aeval st a1) =? (aeval st a2)
| <{a1 <> a2}> => negb ((aeval st a1) =? (aeval st a2))
| <{a1 <= a2}> => (aeval st a1) <=? (aeval st a2)
| <{a1 > a2}> => negb ((aeval st a1) <=? (aeval st a2))
| <{~ b1}> => negb (beval st b1)
| <{b1 && b2}> => andb (beval st b1) (beval st b2)
end.
(** We can use our notation for total maps in the specific case of
states -- i.e., we write the empty state as [(__ !-> 0)]. *)
Definition empty_st := (__ !-> 0).
(** Also, we can add a notation for a "singleton state" with just one
variable bound to a value. *)
Notation "x '!->' v" := (x !-> v ; empty_st) (at level 100, right associativity).
Example aexp1 :
aeval (X !-> 5) <{ 3 + (X * 2) }>
= 13.
Proof. reflexivity. Qed.
Example aexp2 :
aeval (X !-> 5 ; Y !-> 4) <{ Z + (X * Y) }>
= 20.
Proof. reflexivity. Qed.
Example bexp1 :
beval (X !-> 5) <{ true && ~(X <= 4) }>
= true.
Proof. reflexivity. Qed.
(* ################################################################# *)
(** * Commands *)
(** Now we are ready to define the syntax and behavior of Imp
_commands_ (or _statements_). *)
(* ================================================================= *)
(** ** Syntax *)
(** Informally, commands [c] are described by the following BNF
grammar.
c := skip
| x := a
| c ; c
| if b then c else c end
| while b do c end
*)
(** Here is the formal definition of the abstract syntax of
commands: *)
Inductive com : Type :=
| CSkip
| CAsgn (x : string) (a : aexp)
| CSeq (c1 c2 : com)
| CIf (b : bexp) (c1 c2 : com)
| CWhile (b : bexp) (c : com).
(** As we did for expressions, we can use a few [Notation]
declarations to make reading and writing Imp programs more
convenient. *)
(* SOON: (NOTATION NDS'25)
I considered changing maps to also span multiple lines, but I
have not attempted this yet, as this would have required changes
in earlier chapters. *)
Notation "'skip'" := CSkip
(in custom com at level 0) : com_scope.
Notation "x := y" := (CAsgn x y)
(in custom com at level 0, x constr at level 0, y at level 85, no associativity,
format "x := y") : com_scope.
Notation "x ; y" := (CSeq x y)
(in custom com at level 90,
right associativity,
format "'[v' x ; '/' y ']'") : com_scope.
Notation "'if' x 'then' y 'else' z 'end'" := (CIf x y z)
(in custom com at level 89, x at level 99, y at level 99, z at level 99,
format "'[v' 'if' x 'then' '/ ' y '/' 'else' '/ ' z '/' 'end' ']'") : com_scope.
Notation "'while' x 'do' y 'end'" := (CWhile x y)
(in custom com at level 89, x at level 99, y at level 99,
format "'[v' 'while' x 'do' '/ ' y '/' 'end' ']'") : com_scope.
(** For example, here is the factorial function again, written as a
formal Rocq definition. When this command terminates, the variable
[Y] will contain the factorial of the initial value of [X]. *)
Definition fact_in_coq : com :=
<{ Z := X;
Y := 1;
while Z <> 0 do
Y := Y * Z;
Z := Z - 1
end }>.
Print fact_in_coq.
(* ================================================================= *)
(** ** Desugaring Notations *)
(** Rocq offers a rich set of features to manage the increasing
complexity of the objects we work with, such as coercions and
notations. However, their heavy usage can make it hard to
understand what the expressions we enter actually mean. In such
situations it is often instructive to "turn off" those features to
get a more elementary picture of things, using the following
commands:
- [Unset Printing Notations] (undo with [Set Printing Notations])
- [Set Printing Coercions] (undo with [Unset Printing Coercions])
- [Set Printing All] (undo with [Unset Printing All])
These commands can also be used in the middle of a proof, to
elaborate the current goal and context. *)
Unset Printing Notations.
Print fact_in_coq.
(* ===>
fact_in_coq =
CSeq (CAsgn Z X)
(CSeq (CAsgn Y (S O))
(CWhile (BNot (BEq Z O))
(CSeq (CAsgn Y (AMult Y Z))
(CAsgn Z (AMinus Z (S O))))))
: com *)
Set Printing Notations.
Print example_bexp.
(* ===> example_bexp = <{(true && ~ (X <= 4))}> *)
Set Printing Coercions.
Print example_bexp.
(* ===> example_bexp = <{(true && ~ (AId X <= ANum 4))}> *)
Print fact_in_coq.
(* ===>
fact_in_coq =
<{ Z := (AId X);
Y := (ANum 1);
while ~ (AId Z) = (ANum 0) do
Y := (AId Y) * (AId Z);
Z := (AId Z) - (ANum 1)
end }>
: com *)
Unset Printing Coercions.
(* ================================================================= *)
(** ** [Locate] Again *)
(* ----------------------------------------------------------------- *)
(** *** Finding identifiers *)
(** When used with an identifier, the [Locate] prints the full path to
every value in scope with the same name. This is useful to
troubleshoot problems due to variable shadowing. *)
Locate aexp.
(* ===>
Inductive LF.Imp.aexp
Inductive LF.Imp.AExp.aexp
(shorter name to refer to it in current context is AExp.aexp)
Inductive LF.Imp.aevalR_division.aexp
(shorter name to refer to it in current context is aevalR_division.aexp)
Inductive LF.Imp.aevalR_extended.aexp
(shorter name to refer to it in current context is aevalR_extended.aexp)
*)
(* ----------------------------------------------------------------- *)
(** *** Finding notations *)
(** When faced with an unknown notation, you can use [Locate] with a
string containing one of its symbols to see its possible
interpretations. *)
Locate "&&".
(* ===>
Notation
"x && y" := BAnd x y (default interpretation)
"x && y" := andb x y : bool_scope (default interpretation)
*)
Locate ";".
(* ===>
Notation
"x '|->' v ';' m" := (update m x v) (default interpretation)
"x ; y" := (CSeq x y) (default interpretation)
"x '!->' v ';' m" := (t_update m x v) (default interpretation)
"[ x ; y ; .. ; z ]" := cons x (cons y .. (cons z nil) ..) : list_scope
(default interpretation) *)
Locate "while".
(* ===>
Notation
"'while' x 'do' y 'end'" :=
(CWhile x y) (default interpretation)
*)
(* ================================================================= *)
(** ** More Examples *)
(* ----------------------------------------------------------------- *)
(** *** Assignment: *)
Definition plus2 : com :=
<{ X := X + 2 }>.
Definition XtimesYinZ : com :=
<{ Z := X * Y }>.
(* ----------------------------------------------------------------- *)
(** *** Loops *)
Definition subtract_slowly_body : com :=
<{ Z := Z - 1 ;
X := X - 1 }>.
Definition subtract_slowly : com :=
<{ while X <> 0 do
subtract_slowly_body
end }>.
Definition subtract_3_from_5_slowly : com :=
<{ X := 3 ;
Z := 5 ;
subtract_slowly }>.
(* ----------------------------------------------------------------- *)
(** *** An infinite loop: *)
Definition loop : com :=
<{ while true do
skip
end }>.
(* ################################################################# *)
(** * Evaluating Commands *)
(** Next we need to define what it means to evaluate an Imp command.
The fact that [while] loops don't necessarily terminate makes
defining an evaluation function tricky... *)
(* ================================================================= *)
(** ** Evaluation as a Function (Failed Attempt) *)
(** Here's an attempt at defining an evaluation function for commands
(with a bogus [while] case). *)
Fixpoint ceval_fun_no_while (st : state) (c : com) : state :=
match c with
| <{ skip }> =>
st
| <{ x := a }> =>
(x !-> aeval st a ; st)
| <{ c1 ; c2 }> =>
let st' := ceval_fun_no_while st c1 in
ceval_fun_no_while st' c2
| <{ if b then c1 else c2 end}> =>
if (beval st b)
then ceval_fun_no_while st c1
else ceval_fun_no_while st c2
| <{ while b do c end }> =>
st (* bogus *)
end.
(** In a more conventional functional programming language like OCaml or
Haskell we could add the [while] case as follows:
Fixpoint ceval_fun (st : state) (c : com) : state :=
match c with
...
| <{ while b do c end}> =>
if (beval st b)
then ceval_fun st <{c ; while b do c end}>
else st
end.
Rocq doesn't accept such a definition ("Error: Cannot guess
decreasing argument of fix") because the function we want to
define is not guaranteed to terminate. Indeed, it _doesn't_ always
terminate: for example, the full version of the [ceval_fun]
function applied to the [loop] program above would never
terminate. Since Rocq aims to be not just a functional programming
language but also a consistent logic, any potentially
non-terminating function needs to be rejected.
Here is an example showing what would go wrong if Rocq allowed
non-terminating recursive functions:
Fixpoint loop_false (n : nat) : False := loop_false n.
That is, propositions like [False] would become provable
([loop_false 0] would be a proof of [False]), which would be
a disaster for Rocq's logical consistency.
Thus, because it doesn't terminate on all inputs, [ceval_fun]
cannot be written in Rocq -- at least not without additional tricks
and workarounds (see chapter [ImpCEvalFun] if you're curious
about those). *)
(* ================================================================= *)
(** ** Evaluation as a Relation *)
(** Here's a better way: define [ceval] as a _relation_ rather than a
_function_ -- i.e., make its result a [Prop] rather than a
[state], similar to what we did for [aevalR] above. *)
(** This is an important change. Besides freeing us from awkward
workarounds, it gives us a ton more flexibility in the definition.
For example, if we add nondeterministic features like [any] to the
language, we want the definition of evaluation to be
nondeterministic -- i.e., not only will it not be total, it will
not even be a function! *)
(** We'll use the notation [st =[ c ]=> st'] for the [ceval] relation:
[st =[ c ]=> st'] means that executing program [c] in a starting
state [st] results in an ending state [st']. This can be
pronounced "[c] takes state [st] to [st']". *)
(* ----------------------------------------------------------------- *)
(** *** Operational Semantics *)
(** Here is an informal definition of evaluation, presented as inference
rules for readability:
----------------- (E_Skip)
st =[ skip ]=> st
aeval st a = n
------------------------------- (E_Asgn)
st =[ x := a ]=> (x !-> n ; st)
st =[ c1 ]=> st'
st' =[ c2 ]=> st''
--------------------- (E_Seq)
st =[ c1;c2 ]=> st''
beval st b = true
st =[ c1 ]=> st'
-------------------------------------- (E_IfTrue)
st =[ if b then c1 else c2 end ]=> st'
beval st b = false
st =[ c2 ]=> st'
-------------------------------------- (E_IfFalse)
st =[ if b then c1 else c2 end ]=> st'
beval st b = false
----------------------------- (E_WhileFalse)
st =[ while b do c end ]=> st
beval st b = true
st =[ c ]=> st'
st' =[ while b do c end ]=> st''
-------------------------------- (E_WhileTrue)
st =[ while b do c end ]=> st''
*)
(** Here is the formal definition. Make sure you understand
how it corresponds to the inference rules. *)
Reserved Notation
"st0 '=[' c ']=>' st1"
(at level 40, c custom com at level 99,
st0 constr, st1 constr at next level,
format "'[hv' st0 =[ '/ ' '[' c ']' '/' ]=> st1 ']'").
Inductive ceval : com -> state -> state -> Prop :=
| E_Skip : forall st,
st =[ skip ]=> st
| E_Asgn : forall st a n x,
aeval st a = n ->
st =[ x := a ]=> (x !-> n ; st)
| E_Seq : forall c1 c2 st st' st'',
st =[ c1 ]=> st' ->
st' =[ c2 ]=> st'' ->
st =[ c1 ; c2 ]=> st''
| E_IfTrue : forall st st' b c1 c2,
beval st b = true ->
st =[ c1 ]=> st' ->
st =[ if b then c1 else c2 end]=> st'
| E_IfFalse : forall st st' b c1 c2,
beval st b = false ->
st =[ c2 ]=> st' ->
st =[ if b then c1 else c2 end]=> st'
| E_WhileFalse : forall b st c,
beval st b = false ->
st =[ while b do c end ]=> st
| E_WhileTrue : forall st st' st'' b c,
beval st b = true ->
st =[ c ]=> st' ->
st' =[ while b do c end ]=> st'' ->
st =[ while b do c end ]=> st''
where "st0 =[ c ]=> st1" := (ceval c st0 st1).
(** The cost of defining evaluation as a relation instead of a
function is that we now need to construct a _proof_ that some
program evaluates to some result state, rather than just letting
Rocq's computation mechanism do it for us. *)
Example ceval_example1:
empty_st =[
X := 2;
if (X <= 1)
then Y := 3
else Z := 4
end
]=> (Z !-> 4 ; X !-> 2).
Proof.
(* We must supply the intermediate state *)
apply E_Seq with (X !-> 2).
- (* assignment command *)
apply E_Asgn. reflexivity.
- (* if command *)
apply E_IfFalse.
+ reflexivity.
+ apply E_Asgn. reflexivity.
Qed.
(** **** Exercise: 2 stars, standard (ceval_example2) *)
Example ceval_example2:
empty_st =[
X := 0;
Y := 1;
Z := 2
]=> (Z !-> 2 ; Y !-> 1 ; X !-> 0).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
Set Printing Implicit.
Check @ceval_example2.
(** **** Exercise: 3 stars, standard, optional (pup_to_n)
Write an Imp program that sums the numbers from [1] to [X]
(inclusive: [1 + 2 + ... + X]) in the variable [Y]. Your program
should update the state as shown in theorem [pup_to_2_ceval],
which you can reverse-engineer to discover the program you should
write. The proof of that theorem will be somewhat lengthy. *)
Definition pup_to_n : com
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Theorem pup_to_2_ceval :
(X !-> 2) =[
pup_to_n
]=> (X !-> 0 ; Y !-> 3 ; X !-> 1 ; Y !-> 2 ; Y !-> 0 ; X !-> 2).
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(* ================================================================= *)
(** ** Determinism of Evaluation *)
(** Changing from a computational to a relational definition of
evaluation is a good move because it frees us from the artificial
requirement that evaluation should be a total function. But it
also raises a question: Is the second definition of evaluation
really a partial _function_? Or is it possible that, beginning from
the same state [st], we could evaluate some command [c] in
different ways to reach two different output states [st'] and
[st'']?
In fact, this cannot happen: [ceval] _is_ a partial function: *)
Theorem ceval_deterministic: forall c st st1 st2,
st =[ c ]=> st1 ->
st =[ c ]=> st2 ->
st1 = st2.
Proof.
intros c st st1 st2 E1 E2.
generalize dependent st2.
induction E1; intros st2 E2; inversion E2; subst.
- (* E_Skip *) reflexivity.
- (* E_Asgn *) reflexivity.
- (* E_Seq *)
rewrite (IHE1_1 st'0 H1) in *.
apply IHE1_2. assumption.
- (* E_IfTrue, b evaluates to true *)
apply IHE1. assumption.
- (* E_IfTrue, b evaluates to false (contradiction) *)
rewrite H in H5. discriminate.
- (* E_IfFalse, b evaluates to true (contradiction) *)
rewrite H in H5. discriminate.
- (* E_IfFalse, b evaluates to false *)
apply IHE1. assumption.
- (* E_WhileFalse, b evaluates to false *)
reflexivity.
- (* E_WhileFalse, b evaluates to true (contradiction) *)
rewrite H in H2. discriminate.
- (* E_WhileTrue, b evaluates to false (contradiction) *)
rewrite H in H4. discriminate.
- (* E_WhileTrue, b evaluates to true *)
rewrite (IHE1_1 st'0 H3) in *.
apply IHE1_2. assumption. Qed.
(* ################################################################# *)
(** * Reasoning About Imp Programs *)
(** We'll get into more systematic and powerful techniques for
reasoning about Imp programs in _Programming Language
Foundations_, but we can already do a few things (albeit in a
somewhat low-level way) just by working with the bare definitions.
This section explores some examples. *)
Theorem plus2_spec : forall st n st',
st X = n ->
st =[ plus2 ]=> st' ->
st' X = n + 2.
Proof.
intros st n st' HX Heval.
(** Inverting [Heval] essentially forces Rocq to expand one step of
the [ceval] computation -- in this case revealing that [st']
must be [st] extended with the new value of [X], since [plus2]
is an assignment. *)
inversion Heval. subst. clear Heval. simpl.
apply t_update_eq. Qed.
(** **** Exercise: 3 stars, standard, optional (XtimesYinZ_spec)
State and prove a specification of [XtimesYinZ]. *)
(* FILL IN HERE *)
(* Do not modify the following line: *)
Definition manual_grade_for_XtimesYinZ_spec : option (nat*string) := None.
(** [] *)
(** **** Exercise: 3 stars, standard, especially useful (loop_never_stops) *)
Theorem loop_never_stops : forall st st',
~(st =[ loop ]=> st').
Proof.
intros st st' contra. unfold loop in contra.
remember <{ while true do skip end }> as loopdef
eqn:Heqloopdef.
(** Proceed by induction on the assumed derivation showing that
[loopdef] terminates. Most of the cases are immediately
contradictory and so can be solved in one step with
[discriminate]. *)
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, standard (no_whiles_eqv)
Consider the following function: *)
Fixpoint no_whiles (c : com) : bool :=
match c with
| <{ skip }> =>
true
| <{ _ := _ }> =>
true
| <{ c1 ; c2 }> =>
andb (no_whiles c1) (no_whiles c2)
| <{ if _ then ct else cf end }> =>
andb (no_whiles ct) (no_whiles cf)
| <{ while _ do _ end }> =>
false
end.
(** This predicate yields [true] just on programs that have no while
loops. Using [Inductive], write a property [no_whilesR] such that
[no_whilesR c] is provable exactly when [c] is a program with no
while loops. Then prove its equivalence with [no_whiles]. *)
Inductive no_whilesR: com -> Prop :=
(* FILL IN HERE *)
.
Theorem no_whiles_eqv:
forall c, no_whiles c = true <-> no_whilesR c.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 4 stars, standard (no_whiles_terminating)
Imp programs that don't involve while loops always terminate.
State and prove a theorem [no_whiles_terminating] that says this.
Use either [no_whiles] or [no_whilesR], as you prefer. *)
(* FILL IN HERE *)
(* Do not modify the following line: *)
Definition manual_grade_for_no_whiles_terminating : option (nat*string) := None.
(** [] *)
(* ################################################################# *)
(** * Additional Exercises *)
(** **** Exercise: 3 stars, standard (stack_compiler)
Old HP Calculators, programming languages like Forth and Postscript,
and abstract machines like the Java Virtual Machine all evaluate
arithmetic expressions using a _stack_. For instance, the expression
(2*3)+(3*(4-2))
would be written as
2 3 * 3 4 2 - * +
and evaluated like this (where we show the program being evaluated
on the right and the contents of the stack on the left):
[ ] | 2 3 * 3 4 2 - * +
[2] | 3 * 3 4 2 - * +
[3, 2] | * 3 4 2 - * +
[6] | 3 4 2 - * +
[3, 6] | 4 2 - * +
[4, 3, 6] | 2 - * +
[2, 4, 3, 6] | - * +
[2, 3, 6] | * +
[6, 6] | +
[12] |
The goal of this exercise is to write a small compiler that
translates [aexp]s into stack machine instructions.
The instruction set for our stack language will consist of the
following instructions:
- [SPush n]: Push the number [n] on the stack.
- [SLoad x]: Load the identifier [x] from the store and push it
on the stack
- [SPlus]: Pop the two top numbers from the stack, add them, and
push the result onto the stack.
- [SMinus]: Similar, but subtract the first number from the second.
- [SMult]: Similar, but multiply. *)
Inductive sinstr : Type :=
| SPush (n : nat)
| SLoad (x : string)
| SPlus
| SMinus
| SMult.
(** Write a function to evaluate programs in the stack language. It
should take as input a state, a stack represented as a list of
numbers (top stack item is the head of the list), and a program
represented as a list of instructions, and it should return the
stack after executing the program. Test your function on the
examples below.
Note that it is unspecified what to do when encountering an
[SPlus], [SMinus], or [SMult] instruction if the stack contains
fewer than two elements. In a sense, it is immaterial what we do,
since a correct compiler will never emit such a malformed program.
But for sake of later exercises, it would be best to skip the
offending instruction and continue with the next one. *)
Fixpoint s_execute (st : state) (stack : list nat)
(prog : list sinstr)
: list nat
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
Check s_execute.
Example s_execute1 :
s_execute empty_st []
[SPush 5; SPush 3; SPush 1; SMinus]
= [2; 5].
(* FILL IN HERE *) Admitted.
Example s_execute2 :
s_execute (X !-> 3) [3;4]
[SPush 4; SLoad X; SMult; SPlus]
= [15; 4].
(* FILL IN HERE *) Admitted.
(** Next, write a function that compiles an [aexp] into a stack
machine program. The effect of running the program should be the
same as pushing the value of the expression on the stack. *)
Fixpoint s_compile (e : aexp) : list sinstr
(* REPLACE THIS LINE WITH ":= _your_definition_ ." *). Admitted.
(** After you've defined [s_compile], prove the following to test
that it works. *)
Example s_compile1 :
s_compile <{ X - (2 * Y) }>
= [SLoad X; SPush 2; SLoad Y; SMult; SMinus].
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, standard (execute_app) *)
(** Execution can be decomposed in the following sense: executing
stack program [p1 ++ p2] is the same as executing [p1], taking
the resulting stack, and executing [p2] from that stack. Prove
that fact. *)
Theorem execute_app : forall st p1 p2 stack,
s_execute st stack (p1 ++ p2) = s_execute st (s_execute st stack p1) p2.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, standard (stack_compiler_correct) *)
(** Now we'll prove the correctness of the compiler implemented in the
previous exercise. Begin by proving the following lemma. If it
becomes difficult, consider whether your implementation of
[s_execute] or [s_compile] could be simplified. *)
Lemma s_compile_correct_aux : forall st e stack,
s_execute st stack (s_compile e) = aeval st e :: stack.
Proof.
(* FILL IN HERE *) Admitted.
(** The main theorem should be a very easy corollary of that lemma. *)
Theorem s_compile_correct : forall (st : state) (e : aexp),
s_execute st [] (s_compile e) = [ aeval st e ].
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, standard, optional (short_circuit)
Most modern programming languages use a "short-circuit" evaluation
rule for boolean [and]: to evaluate [BAnd b1 b2], first evaluate
[b1]. If it evaluates to [false], then the entire [BAnd]
expression evaluates to [false] immediately, without evaluating
[b2]. Otherwise, [b2] is evaluated to determine the result of the
[BAnd] expression.
Write an alternate version of [beval] that performs short-circuit
evaluation of [BAnd] in this manner, and prove that it is
equivalent to [beval]. (N.b. This is only true because expression
evaluation in Imp is rather simple. In a bigger language where
evaluating an expression might diverge, the short-circuiting [BAnd]
would _not_ be equivalent to the original, since it would make more
programs terminate.) *)
(* FILL IN HERE
[] *)
Module BreakImp.
(** **** Exercise: 4 stars, standard, optional (break_imp)
Imperative languages like C and Java often include a [break] or
similar statement for interrupting the execution of loops. In this
exercise we consider how to add [break] to Imp. First, we need to
enrich the language of commands with an additional case. *)
Inductive com : Type :=
| CSkip
| CBreak (* <--- NEW *)
| CAsgn (x : string) (a : aexp)
| CSeq (c1 c2 : com)
| CIf (b : bexp) (c1 c2 : com)
| CWhile (b : bexp) (c : com).
Notation "'break'" := CBreak (in custom com at level 0) : com_scope.
Notation "'skip'" := CSkip
(in custom com at level 0) : com_scope.
Notation "x := y" := (CAsgn x y)
(in custom com at level 0, x constr at level 0, y at level 85, no associativity,
format "x := y") : com_scope.
Notation "x ; y" := (CSeq x y)
(in custom com at level 90,
right associativity,
format "'[v' x ; '/' y ']'") : com_scope.
Notation "'if' x 'then' y 'else' z 'end'" := (CIf x y z)
(in custom com at level 89, x at level 99, y at level 99, z at level 99,
format "'[v' 'if' x 'then' '/ ' y '/' 'else' '/ ' z '/' 'end' ']'") : com_scope.
Notation "'while' x 'do' y 'end'" := (CWhile x y)
(in custom com at level 89, x at level 99, y at level 99,
format "'[v' 'while' x 'do' '/ ' y '/' 'end' ']'") : com_scope.
(** Next, we need to define the behavior of [break]. Informally,
whenever [break] is executed in a sequence of commands, it stops
the execution of that sequence and signals that the innermost
enclosing loop should terminate. (If there aren't any
enclosing loops, then the whole program simply terminates.) The
final state should be the same as the one in which the [break]
statement was executed.
One important point is what to do when there are multiple loops
enclosing a given [break]. In those cases, [break] should only
terminate the _innermost_ loop. Thus, after executing the
following...
X := 0;
Y := 1;
while 0 <> Y do
while true do
break
end;
X := 1;
Y := Y - 1
end
... the value of [X] should be [1], and not [0].
One way of expressing this behavior is to add another parameter to
the evaluation relation that specifies whether evaluation of a
command executes a [break] statement: *)
Inductive result : Type :=
| SContinue
| SBreak.
Reserved Notation
"st0 '=[' c ']=>' st1 '/' s"
(at level 40, c custom com at level 99,
st0 constr, st1 constr at next level,
format "'[hv' st0 =[ '/ ' '[' c ']' '/' ]=> st1 / s ']'").
(** Intuitively, [st =[ c ]=> st' / s] means that, if [c] is started in
state [st], then it terminates in state [st'] and either signals
that the innermost surrounding loop (or the whole program) should
exit immediately ([s = SBreak]) or that execution should continue
normally ([s = SContinue]).
The definition of the "[st =[ c ]=> st' / s]" relation is very
similar to the one we gave above for the regular evaluation
relation ([st =[ c ]=> st']) -- we just need to handle the
termination signals appropriately:
- If the command is [skip], then the state doesn't change and
execution of any enclosing loop can continue normally.
- If the command is [break], the state stays unchanged but we
signal a [SBreak].
- If the command is an assignment, then we update the binding for
that variable in the state accordingly and signal that execution
can continue normally.
- If the command is of the form [if b then c1 else c2 end], then
the state is updated as in the original semantics of Imp, except
that we also propagate the signal from the execution of
whichever branch was taken.
- If the command is a sequence [c1 ; c2], we first execute
[c1]. If this yields a [SBreak], we skip the execution of [c2]
and propagate the [SBreak] signal to the surrounding context;
the resulting state is the same as the one obtained by
executing [c1] alone. Otherwise, we execute [c2] on the state
obtained after executing [c1], and propagate the signal
generated there.
- Finally, for a loop of the form [while b do c end], the
semantics is almost the same as before. The only difference is
that, when [b] evaluates to true, we execute [c] and check the
signal that it raises. If that signal is [SContinue], then the
execution proceeds as in the original semantics. Otherwise, we
stop the execution of the loop, and the resulting state is the
same as the one resulting from the execution of the current
iteration. In either case, since [break] only terminates the
innermost loop, [while] signals [SContinue]. *)
(** Based on the above description, complete the definition of the
[ceval] relation. *)
Inductive ceval : com -> state -> result -> state -> Prop :=
| E_Skip : forall st,
st =[ CSkip ]=> st / SContinue
(* FILL IN HERE *)
where "st '=[' c ']=>' st' '/' s" := (ceval c st s st').
(** Now prove the following properties of your definition of [ceval]: *)
Theorem break_ignore : forall c st st' s,
st =[ break; c ]=> st' / s ->
st = st'.
Proof.
(* FILL IN HERE *) Admitted.
Theorem while_continue : forall b c st st' s,
st =[ while b do c end ]=> st' / s ->
s = SContinue.
Proof.
(* FILL IN HERE *) Admitted.
Theorem while_stops_on_break : forall b c st st',
beval st b = true ->
st =[ c ]=> st' / SBreak ->
st =[ while b do c end ]=> st' / SContinue.
Proof.
(* FILL IN HERE *) Admitted.
Theorem seq_continue : forall c1 c2 st st' st'',
st =[ c1 ]=> st' / SContinue ->
st' =[ c2 ]=> st'' / SContinue ->
st =[ c1 ; c2 ]=> st'' / SContinue.
Proof.
(* FILL IN HERE *) Admitted.
Theorem seq_stops_on_break : forall c1 c2 st st',
st =[ c1 ]=> st' / SBreak ->
st =[ c1 ; c2 ]=> st' / SBreak.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 3 stars, advanced, optional (while_break_true) *)
Theorem while_break_true : forall b c st st',
st =[ while b do c end ]=> st' / SContinue ->
beval st' b = true ->
exists st'', st'' =[ c ]=> st' / SBreak.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
(** **** Exercise: 4 stars, advanced, optional (ceval_deterministic) *)
Theorem ceval_deterministic: forall (c:com) st st1 st2 s1 s2,
st =[ c ]=> st1 / s1 ->
st =[ c ]=> st2 / s2 ->
st1 = st2 /\ s1 = s2.
Proof.
(* FILL IN HERE *) Admitted.
(** [] *)
End BreakImp.
(** **** Exercise: 4 stars, standard, optional (add_for_loop)
Add C-style [for] loops to the language of commands, update the
[ceval] definition to define the semantics of [for] loops, and add
cases for [for] loops as needed so that all the proofs in this
file are accepted by Rocq.
A [for] loop should be parameterized by (a) a statement executed
initially, (b) a test that is run on each iteration of the loop to
determine whether the loop should continue, (c) a statement
executed at the end of each loop iteration, and (d) a statement
that makes up the body of the loop. (You don't need to worry
about making up a concrete Notation for [for] loops, but feel free
to play with this too if you like.) *)
(* FILL IN HERE
[] *)
(* 2026-01-07 13:18 *)
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