ImpSimple Imperative Programs

In this chapter, we take a more serious look at how to use Coq 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 chapters in Programming Language Foundations (Software Foundations, volume 2) develop a theory of program equivalence and introduce Hoare Logic, a widely used logic for reasoning about imperative programs.

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 expressions with variables, and finally a language of commands including assignment, conditions, sequencing, and loops.

Syntax

Abstract syntax trees for 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)
  | BLe (a1 a2 : aexp)
  | BNot (b : bexp)
  | BAnd (b1 b2 : bexp).

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)
  | BLe a1 a2 ⇒ (aeval a1) <=? (aeval a2)
  | BNot b1 ⇒ negb (beval b1)
  | BAnd b1 b2 ⇒ andb (beval b1) (beval b2)
  end.
What does the following expression evaluate to?
  aeval (APlus (ANum 3) (AMinus (ANum 4) (ANum 1)))
(1) true
(2) false
(3) 0
(4) 3
(5) 6

Optimization


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.
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.


Theorem optimize_0plus_sound: ∀ 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.

Coq Automation

That last proof was getting a little repetitive. Time to learn a few more Coq tricks...

Tacticals

Tacticals is Coq'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 : ∀ ae, aeval ae = aeval ae.
Proof. try reflexivity. (* This just does reflexivity. *) Qed.
Theorem silly2 : ∀ (P : Prop), P → P.
Proof.
  intros P HP.
  try reflexivity. (* Just reflexivity would have failed. *)
  apply HP. (* We can still finish the proof in some other way. *)
Qed.

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:

Lemma foo : ∀ 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' : ∀ 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': ∀ 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.

The repeat Tactical

The repeat tactical takes another tactic and keeps applying this tactic until it fails to make progress. Here is an example showing 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.
repeat can loop forever.

Theorem repeat_loop : ∀ (m n : nat),
    m + n = n + m.
Proof.
  intros m n.
  (* Uncomment the next line to see the infinite loop occur.
     In Proof General, C-c C-c will break out of the loop. *)
  (* repeat rewrite Nat.add_comm. *)
Admitted.

Defining New Tactic Notations

Coq also provides several ways of "programming" tactic scripts:
  • Tactic Notation: syntax extension for tactics (good for simple macros)
  • Ltac: scripting language for tactics (good for more sophisticated proof engineering)
  • OCaml tactic scripting API (for wizards)
An example Tactic Notation:

Tactic Notation "simpl_and_try" tactic(c) :=
  simpl;
  try c.

The omega Tactic

The omega tactic implements a decision procedure for a subset of first-order logic called Presburger arithmetic. It is based on the Omega algorithm invented by William Pugh [Pugh 1991].
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
  • the logical connectives ∧, ∨, ¬, and →,
then invoking omega will either solve the goal or fail, meaning that the goal is actually false. (If the goal is not of this form, omega will also fail.)

Example silly_presburger_example : ∀ m n o p,
  m + n ≤ n + o ∧ o + 3 = p + 3 →
  m ≤ p.
Proof.
  intros. omega.
Qed.
Example plus_comm__omega : ∀ m n,
    m + n = n + m.
Proof.
  intros. omega.
Qed.
Example plus_assoc__omega : ∀ m n p,
    m + (n + p) = m + n + p.
Proof.
  intros. omega.
Qed.

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: For 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 current 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 Fixpoints. Another way to think about evaluation -- one that we will see is often more flexible -- is as a relation between expressions and their values. This leads naturally to Inductive definitions like the following one for arithmetic expressions...

Inductive aevalR : aexp → nat → Prop :=
  | E_ANum n :
      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).

A standard notation for "evaluates to":

Notation "e '==>' n"
         := (aevalR e n)
            (at level 90, left associativity)
         : type_scope.
With infix notation:

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

For example, the constructor E_APlus...
      | E_APlus : ∀ (e1 e2: aexp) (n1 n2: nat),
          aevalR e1 n1 →
          aevalR e2 n2 →
          aevalR (APlus e1 e2) (n1 + n2)
...would be written like this as an inference rule:
e1 ==> n1
e2 ==> n2 (E_APlus)  
APlus e1 e2 ==> n1+n2

There is nothing very deep going on here:
  • the rule name corresponds to a constructor name
  • above the line are premises
  • below the line is conclusion
  • metavariables like e1 and n1 are implicitly universally quantified
  • the whole collection of rules is implicitly wrapped in an Inductive (sometimes we write this slightly more explicitly, as "...closed under these rules...")

Equivalence of the Definitions

It is straightforward to prove that the relational and functional definitions of evaluation agree:

Theorem aeval_iff_aevalR : ∀ 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.
We can make the proof quite a bit shorter by making more use of tacticals.

Theorem aeval_iff_aevalR' : ∀ a n,
  (a ==> n) ↔ aeval a = n.
Proof.
  (* WORK IN CLASS *) Admitted.

Computational vs. Relational Definitions

Sometimes relational (not functional) definitions are the only reasonable option...

Adding division


Inductive aexp : Type :=
  | ANum (n : nat)
  | APlus (a1 a2 : aexp)
  | AMinus (a1 a2 : aexp)
  | AMult (a1 a2 : aexp)
  | ADiv (a1 a2 : aexp). (* <--- NEW *)
What could aeval do with division by zero?

Adding division, relationally


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 the evaluation relation has now become partial: There are some inputs for which it simply does not specify an output.

Adding Nondeterminism

A nondeterministic number generator:

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).
What could aeval do with nondeterminism?

Adding nondeterminism, relationally


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.

Tradeoffs

Which to prefer: functional or relational?
  • Functional: take advantage of computation.
  • Relational: more (easily) expressive.
  • Best of both worlds: define both, and prove equivalence.

Expressions With Variables

Now we return to defining Imp. 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 reuse maps from the Maps chapter, and strings will be used to represent variables in Imp.
A machine state (or just state) represents the current values of all variables at some point in the execution of a program.

Definition state := total_map nat.

Syntax

We can add variables to the arithmetic expressions we had before by simply adding 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".

Notations

To make Imp programs easier to read and write, we introduce some notations and implicit coercions.

Coercion AId : string >-> aexp.
Coercion ANum : nat >-> aexp.
Definition bool_to_bexp (b : bool) : bexp :=
  if b then BTrue else BFalse.
Coercion bool_to_bexp : bool >-> bexp.
Bind Scope imp_scope with aexp.
Bind Scope imp_scope with bexp.
Delimit Scope imp_scope with imp.
Notation "x + y" := (APlus x y) (at level 50, left associativity) : imp_scope.
Notation "x - y" := (AMinus x y) (at level 50, left associativity) : imp_scope.
Notation "x * y" := (AMult x y) (at level 40, left associativity) : imp_scope.
Notation "x <= y" := (BLe x y) (at level 70, no associativity) : imp_scope.
Notation "x = y" := (BEq x y) (at level 70, no associativity) : imp_scope.
Notation "x && y" := (BAnd x y) (at level 40, left associativity) : imp_scope.
Notation "'~' b" := (BNot b) (at level 75, right associativity) : imp_scope.

Now we can write Imp expressions directly in Coq.

Definition example_aexp : aexp := 3 + (X × 2).
Definition example_bexp : bexp := true && ~(X ≤ 4).

One downside of these coercions is that they can make it a little harder for humans to calculate the types of expressions. If you get confused, try doing Set Printing Coercions to see exactly what is going on.

Set Printing Coercions.
Print example_bexp.
(* ===> (example_bexp = bool_to_bexp true && ~ (AId X <= ANum 4))*)
Unset Printing Coercions.

Evaluation

Now we need to add an st parameter to both evaluation functions:

Fixpoint aeval (st : state) (a : aexp) : nat :=
  match a with
  | ANum n ⇒ n
  | AId x ⇒ st x (* <--- NEW *)
  | APlus a1 a2 ⇒ (aeval st a1) + (aeval st a2)
  | AMinus a1 a2 ⇒ (aeval st a1) - (aeval st a2)
  | AMult a1 a2 ⇒ (aeval st a1) × (aeval st a2)
  end.
Fixpoint beval (st : state) (b : bexp) : bool :=
  match b with
  | BTrue ⇒ true
  | BFalse ⇒ false
  | BEq a1 a2 ⇒ (aeval st a1) =? (aeval st a2)
  | BLe a1 a2 ⇒ (aeval st a1) <=? (aeval st a2)
  | BNot b1 ⇒ negb (beval st b1)
  | BAnd b1 b2 ⇒ andb (beval st b1) (beval st b2)
  end.

We specialize our notation for total maps to the specific case of states, i.e. using (_ !-> 0) as empty state.

Definition empty_st := (_ !-> 0).
Now we can add a notation for a "singleton state" with just one variable bound to a value.
Notation "x '!->' v" := (t_update empty_st x v) (at level 100).

Commands

Now we are ready define the syntax and behavior of Imp commands (sometimes called statements).

Syntax

Informally, commands c are described by the following BNF grammar.
     c ::= SKIP | x ::= a | c ;; c | TEST b THEN c ELSE c FI
         | WHILE b DO c END


Inductive com : Type :=
  | CSkip
  | CAss (x : string) (a : aexp)
  | CSeq (c1 c2 : com)
  | CIf (b : bexp) (c1 c2 : com)
  | CWhile (b : bexp) (c : com).

As for expressions, we can use a few Notation declarations to make reading and writing Imp programs more convenient.

Bind Scope imp_scope with com.
Notation "'SKIP'" :=
   CSkip : imp_scope.
Notation "x '::=' a" :=
  (CAss x a) (at level 60) : imp_scope.
Notation "c1 ;; c2" :=
  (CSeq c1 c2) (at level 80, right associativity) : imp_scope.
Notation "'WHILE' b 'DO' c 'END'" :=
  (CWhile b c) (at level 80, right associativity) : imp_scope.
Notation "'TEST' c1 'THEN' c2 'ELSE' c3 'FI'" :=
  (CIf c1 c2 c3) (at level 80, right associativity) : imp_scope.

Imp Factorial in Coq

For example, here is the factorial function again, written as a formal definition to Coq:

Definition fact_in_coq : com :=
  Z ::= X;;
  Y ::= 1;;
  WHILE ~(Z = 0) DO
    Y ::= Y × Z;;
    Z ::= Z - 1
  END.

More Examples

Assignment:

Definition plus2 : com :=
  X ::= X + 2.
Definition XtimesYinZ : com :=
  Z ::= X × Y.
Definition subtract_slowly_body : com :=
  Z ::= Z - 1 ;;
  X ::= X - 1.

Loops


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, omitting the WHILE case.

Open Scope imp_scope.
Fixpoint ceval_fun_no_while (st : state) (c : com)
                          : state :=
  match c with
    | SKIP ⇒
        st
    | x ::= a1 ⇒
        (x !-> (aeval st a1) ; st)
    | c1 ;; c2 ⇒
        let st' := ceval_fun_no_while st c1 in
        ceval_fun_no_while st' c2
    | TEST b THEN c1 ELSE c2 FI ⇒
        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.
Close Scope imp_scope.

Nontermination leads to Inconsistency

Consider the following "proof object": 
        Fixpoint loop_false (n : nat) : False :=
          loop_false n.
Accepting such a definition would be catastrophic, so Coq conservatively rejects all nonterminating programs.

Evaluation as a Relation

Here's a better way: define ceval as a relation rather than a function -- i.e., define it in Prop instead of Type, as we did for aevalR above.
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 a1 = n (E_Ass)  
st =[ x := a1 ]=> (x !-> n ; st)
st =[ c1 ]=> st'
st' =[ c2 ]=> st'' (E_Seq)  
st =[ c1;;c2 ]=> st''
beval st b1 = true
st =[ c1 ]=> st' (E_IfTrue)  
st =[ TEST b1 THEN c1 ELSE c2 FI ]=> st'
beval st b1 = false
st =[ c2 ]=> st' (E_IfFalse)  
st =[ TEST b1 THEN c1 ELSE c2 FI ]=> 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''


Reserved Notation "st '=[' c ']=>' st'"
                  (at level 40).
Inductive ceval : com → state → state → Prop :=
  | E_Skip : ∀ st,
      st =[ SKIP ]=> st
  | E_Ass : ∀ st a1 n x,
      aeval st a1 = n →
      st =[ x ::= a1 ]=> (x !-> n ; st)
  | E_Seq : ∀ c1 c2 st st' st'',
      st =[ c1 ]=> st' →
      st' =[ c2 ]=> st'' →
      st =[ c1 ;; c2 ]=> st''
  | E_IfTrue : ∀ st st' b c1 c2,
      beval st b = true →
      st =[ c1 ]=> st' →
      st =[ TEST b THEN c1 ELSE c2 FI ]=> st'
  | E_IfFalse : ∀ st st' b c1 c2,
      beval st b = false →
      st =[ c2 ]=> st' →
      st =[ TEST b THEN c1 ELSE c2 FI ]=> st'
  | E_WhileFalse : ∀ b st c,
      beval st b = false →
      st =[ WHILE b DO c END ]=> st
  | E_WhileTrue : ∀ 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 "st =[ c ]=> st'" := (ceval c st st').

The cost of defining evaluation as a relation instead of a function is that we now need to construct proofs that some program evaluates to some result state, rather than just letting Coq's computation mechanism do it for us.

Example ceval_example1:
  empty_st =[
     X ::= 2;;
     TEST X ≤ 1
       THEN Y ::= 3
       ELSE Z ::= 4
     FI
  ]=> (Z !-> 4 ; X !-> 2).
Proof.
  (* We must supply the intermediate state *)
  apply E_Seq with (X !-> 2).
  - (* assignment command *)
    apply E_Ass. reflexivity.
  - (* if command *)
    apply E_IfFalse.
    reflexivity.
    apply E_Ass. reflexivity.
Qed.
Is the following proposition provable?
      ∀ (c : com) (st st' : state),
        st =[ SKIP ;; c ]=> st' →
        st =[ c ]=> st'
(1) Yes
(2) No
(3) Not sure
Is the following proposition provable?
      ∀ (c1 c2 : com) (st st' : state),
          st =[ c1;;c2 ]=> st' →
          st =[ c1 ]=> st →
          st =[ c2 ]=> st'
(1) Yes
(2) No
(3) Not sure
Is the following proposition provable?
      ∀ (b : bexp) (c : com) (st st' : state),
          st =[ TEST b THEN c ELSE c FI ]=> st' →
          st =[ c ]=> st'
(1) Yes
(2) No
(3) Not sure
Is the following proposition provable?
      ∀ b : bexp,
         (∀ st, beval st b = true) →
         ∀ (c : com) (st : state),
           ~(∃ st', st =[ WHILE b DO c END ]=> st')
(1) Yes
(2) No
(3) Not sure
Is the following proposition provable?
      ∀ (b : bexp) (c : com) (st : state),
         ~(∃ st', st =[ WHILE b DO c END ]=> st') →
         ∀ st'', beval st'' b = true
(1) Yes
(2) No
(3) Not sure

Determinism of Evaluation


Theorem ceval_deterministic: ∀ 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_Ass *) reflexivity.
  - (* E_Seq *)
    assert (st' = st'0) as EQ1.
    { (* Proof of assertion *) apply IHE1_1; assumption. }
    subst st'0.
    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 H5.
  - (* E_IfFalse, b evaluates to true (contradiction) *)
      rewrite H in H5. discriminate H5.
  - (* 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 H2.
  - (* E_WhileTrue, b evaluates to false (contradiction) *)
    rewrite H in H4. discriminate H4.
  - (* E_WhileTrue, b evaluates to true *)
      assert (st' = st'0) as EQ1.
      { (* Proof of assertion *) apply IHE1_1; assumption. }
      subst st'0.
      apply IHE1_2. assumption. Qed.