# SmallstepSmall-step Operational Semantics

(* $Date: 2011-05-07 16:41:12 -0400 (Sat, 07 May 2011) $ *)

Require Export Imp.

Require Import Relations.

The evaluators we have seen so far (e.g., the ones for aexps,
bexps, and commands) have been formulated in a "big-step"
style — they specify how a given expression can be evaluated to
its final value (or a command plus a store to a final store) "all
in one big step."
This style is simple and natural for many purposes (indeed, Gilles
Kahn, who popularized its use, called it
Another shortcoming of the big-step style is more technical, but
critical for some applications. Consider the variant of Imp with
lists that we introduced in ImpList.v. We chose to define the
meaning of programs like 0 + nil by specifying that a list
should be interpreted as 0 when it occurs in a context expecting
a number, but this was a bit of a hack. It would be better simply
to say that the behavior of such a program is
However, this way of defining Imp has a serious deficiency. In
this language, a command might
These two outcomes — nontermination vs. getting stuck in an
erroneous configuration — are quite different. In particular, we
want to allow the first (permitting the possibility of infinite
loops is the price we pay for the convenience of programming with
general looping constructs like while) but prevent the
second (which is just wrong), for example by adding some form of
So, for lots of reasons, we'd like to have a finer-grained way of
defining and reasoning about program behaviors. This is the topic
of the present chapter. We replace the "big-step" eval relation
with a "small-step" relation that specifies, for a given program,
how the "atomic steps" of computation are performed.
To save space in the discussion, let's go back to an
incredibly simple language containing just constants and addition.
At the end of the chapter, we'll see how to apply the same
techniques to the full Imp language.

*natural semantics*), but there are some things it does not do well. In particular, it does not give us a natural way of talking about*concurrent*programming languages, where the "semantics" of a program — i.e., the essence of how it behaves — is not just which input states get mapped to which output states, but also includes the intermediate states that it passes through along the way, since these states can also be observed by concurrently executing code.*undefined*— it doesn't evaluate to any result. We could easily do this: we'd just have to use the formulations of aeval and beval as inductive propositions (rather than Fixpoints), so that we can make them partial functions instead of total ones.*fail*to map a given starting state to any ending state for two quite different reasons: either because the execution gets into an infinite loop or because, at some point, the program tries to do an operation that makes no sense, such as taking the successor of a boolean variable, and none of the evaluation rules can be applied.*typechecking*to the language. Indeed, this will be a major topic for the rest of the course. As a first step, we need a different way of presenting the semantics that allows us to distinguish nontermination from erroneous "stuck states."# A Toy Language

Inductive tm : Type :=

| tm_const : nat → tm

| tm_plus : tm → tm → tm.

Tactic Notation "tm_cases" tactic(first) ident(c) :=

first;

[ Case_aux c "tm_const" | Case_aux c "tm_plus" ].

Module SimpleArith0.

Here is a standard evaluator for this language, written in the
same (big-step) style as we've been using up to this point.

Fixpoint eval (t : tm) : nat :=

match t with

| tm_const n => n

| tm_plus a1 a2 => eval a1 + eval a2

end.

End SimpleArith0.

Module SimpleArith1.

Now, here is the same evaluator, written in exactly the same
style, but formulated as an inductively defined relation. Again,
we use the notation t ⇓ n for "t evaluates to n."

We write plus n1 n2 in the second rule, rather than n1 + n2,
to emphasize the fact that the + on the left of the ⇓ is an
abstract-syntax-tree node while the addition on the right of the
⇓ is the mathematical sum of the numbers n1 and n2. The
formal Coq version of the definition is more pedantic about this
distinction:

(E_Const) | |

n ⇓ n |

t1 ⇓ n1 | |

t2 ⇓ n2 | (E_Plus) |

t1 + t2 ⇓ plus n1 n2 |

Reserved Notation " t '⇓' n " (at level 50, left associativity).

Inductive eval : tm → nat → Prop :=

| E_Const : ∀ n,

tm_const n ⇓ n

| E_Plus : ∀ t1 t2 n1 n2,

t1 ⇓ n1 →

t2 ⇓ n2 →

tm_plus t1 t2 ⇓ plus n1 n2

where " t '⇓' n " := (eval t n).

End SimpleArith1.

Here is a slight variation, still in big-step style, where
the final result of evaluating a term is also a term, rather than
a bare number.

Reserved Notation " t '⇓' t' " (at level 50, left associativity).

Inductive eval : tm → tm → Prop :=

| E_Const : ∀ n1,

tm_const n1 ⇓ tm_const n1

| E_Plus : ∀ t1 n1 t2 n2,

t1 ⇓ tm_const n1 →

t2 ⇓ tm_const n2 →

tm_plus t1 t2 ⇓ tm_const (plus n1 n2)

where " t '⇓' t' " := (eval t t').

(Note that this doesn't change anything in the informal rules,
where we are eliding the distinction between the constant

*term*n and the bare*number*n.)Tactic Notation "eval_cases" tactic(first) ident(c) :=

first;

[ Case_aux c "E_Const" | Case_aux c "E_Plus" ].

Module SimpleArith2.

Now, here is a small-step version.

Note that we're using variable names here to lighten the notation:
by convention, n1 and n2 refer only to constants (constructed
with tm_const), while t1 and t2 refer to arbitrary terms.
In the formal rules, we use explicit constructors to make the same
distinction.

(ST_PlusConstConst) | |

n1 + n2 ⇒ plus n1 n2 |

t1 ⇓ t1' | (ST_Plus1) |

t1 + t2 ⇓ t1' + t2 |

t2 ⇓ t2' | (ST_Plus2) |

n1 + t2 ⇓ n1 + t2' |

Reserved Notation " t '⇒' t' " (at level 40).

Inductive step : tm → tm → Prop :=

| ST_PlusConstConst : ∀ n1 n2,

tm_plus (tm_const n1) (tm_const n2) ⇒ tm_const (plus n1 n2)

| ST_Plus1 : ∀ t1 t1' t2,

t1 ⇒ t1' →

tm_plus t1 t2 ⇒ tm_plus t1' t2

| ST_Plus2 : ∀ n1 t2 t2',

t2 ⇒ t2' →

tm_plus (tm_const n1) t2 ⇒ tm_plus (tm_const n1) t2'

where " t '⇒' t' " := (step t t').

Things to notice:

- We are defining just a single reduction step, in which
one tm_plus node is replaced by its value.
- Each step finds the
*leftmost*tm_plus node that is ready to go (both of its operands are constants) and rewrites it in place. The first rule tells how to rewrite this tm_plus node itself; the other two rules tell how to find it. - A term that is just a constant cannot take a step.

Tactic Notation "step_cases" tactic(first) ident(c) :=

first;

[ Case_aux c "ST_PlusConstConst"

| Case_aux c "ST_Plus1" | Case_aux c "ST_Plus2" ].

A couple of examples of reasoning with the step
relation...
If t1 can take a step to t1', then tm_plus t1 t2 steps
to plus t1' t2:

Example test_step_1 :

tm_plus

(tm_plus (tm_const 0) (tm_const 3))

(tm_plus (tm_const 2) (tm_const 4))

⇒

tm_plus

(tm_const (plus 0 3))

(tm_plus (tm_const 2) (tm_const 4)).

Proof.

apply ST_Plus1. apply ST_PlusConstConst. Qed.

#### Exercise: 2 stars (test_step_2)

Right-hand sides of sums can take a step only when the left-hand side is finished: if t2 can take a step to t2', then tm_plus (tm_const n) t2 steps to tm_plus (tm_const n) t2':Example test_step_2 :

tm_plus

(tm_const 0)

(tm_plus

(tm_const 2)

(tm_plus (tm_const 0) (tm_const 3)))

⇒

tm_plus

(tm_const 0)

(tm_plus

(tm_const 2)

(tm_const (plus 0 3))).

Proof.

(* FILL IN HERE *) Admitted.

☐
One interesting property of the ⇒ relation is that, like the
evaluation relation for our language of Imp programs, it is

*deterministic*.*Theorem*: For each t, there is at most one t' such that t steps to t' (t ⇒ t' is provable). Formally, this is the same as saying that ⇒ is a partial function.*Proof sketch*: We show that if x steps to both y1 and y2 then y1 and y2 are equal, by induction on a derivation of step x y1. There are several cases to consider, depending on the last rule used in this derivation and in the given derivation of step x y2.- If both are ST_PlusConstConst, the result is immediate.
- The cases when both derivations end with ST_Plus1 or
ST_Plus2 follow by the induction hypothesis.
- It cannot happen that one is ST_PlusConstConst and the other
is ST_Plus1 or ST_Plus2, since this would imply that x has
the form tm_plus t1 t2 where both t1 and t2 are
constants (by ST_PlusConstConst)
*and*one of t1 or t2 has the form tm_plus .... - Similarly, it cannot happen that one is ST_Plus1 and the other is ST_Plus2, since this would imply that x has the form tm_plus t1 t2 where t1 has both the form tm_plus t1 t2 and the form tm_const n. ☐

Theorem step_deterministic:

partial_function step.

Proof.

unfold partial_function. intros x y1 y2 Hy1 Hy2.

generalize dependent y2.

step_cases (induction Hy1) Case; intros y2 Hy2.

Case "ST_PlusConstConst". step_cases (inversion Hy2) SCase.

SCase "ST_PlusConstConst". reflexivity.

SCase "ST_Plus1". inversion H2.

SCase "ST_Plus2". inversion H2.

Case "ST_Plus1". step_cases (inversion Hy2) SCase.

SCase "ST_PlusConstConst". rewrite ← H0 in Hy1. inversion Hy1.

SCase "ST_Plus1".

rewrite ← (IHHy1 t1'0).

reflexivity. assumption.

SCase "ST_Plus2". rewrite ← H in Hy1. inversion Hy1.

Case "ST_Plus2". step_cases (inversion Hy2) SCase.

SCase "ST_PlusConstConst". rewrite ← H1 in Hy1. inversion Hy1.

SCase "ST_Plus1". inversion H2.

SCase "ST_Plus2".

rewrite ← (IHHy1 t2'0).

reflexivity. assumption. Qed.

End SimpleArith2.

## Values

*abstract machine*:

- At any moment, the
*state*of the machine is a term. - A
*step*of the machine is an atomic unit of computation — here, a single "add" operation. - The
*halting states*of the machine are ones where there is no more computation to be done.

- Take t as the starting state of the machine.
- Repeatedly use the ⇒ relation to find a sequence of
machine states, starting with t, where each state steps to
the next.
- When no more reduction is possible, "read out" the final state of the machine as the result of execution.

*values*.

Having introduced the idea of values, we can use it in the
definition of the ⇒ relation to write ST_Plus2 rule in a
slightly more intuitive way:

Again, the variable names here carry important information:
by convention, v1 ranges only over values, while t1 and t2
range over arbitrary terms. In the Coq version of the rules, we
use an explicit value hypothesis for the same purpose.

(ST_PlusConstConst) | |

n1 + n2 ⇒ plus n1 n2 |

t1 ⇓ t1' | (ST_Plus1) |

t1 + t2 ⇓ t1' + t2 |

t2 ⇓ t2' | (ST_Plus2) |

v1 + t2 ⇓ v1 + t2' |

Reserved Notation " t '⇒' t' " (at level 40).

Inductive step : tm → tm → Prop :=

| ST_PlusConstConst : ∀ n1 n2,

tm_plus (tm_const n1) (tm_const n2)

⇒ tm_const (plus n1 n2)

| ST_Plus1 : ∀ t1 t1' t2,

t1 ⇒ t1' →

tm_plus t1 t2 ⇒ tm_plus t1' t2

| ST_Plus2 : ∀ v1 t2 t2',

value v1 → (* <----- n.b. *)

t2 ⇒ t2' →

tm_plus v1 t2 ⇒ tm_plus v1 t2'

where " t '⇒' t' " := (step t t').

Tactic Notation "step_cases" tactic(first) ident(c) :=

first;

[ Case_aux c "ST_PlusConstConst"

| Case_aux c "ST_Plus1" | Case_aux c "ST_Plus2" ].

#### Exercise: 3 stars, recommended (redo_determinacy)

As a sanity check on this change, let's re-verify determinacy- If both are ST_PlusConstConst, the result is immediate.
- It cannot happen that one is ST_PlusConstConst and the other
is ST_Plus1 or ST_Plus2, since this would imply that x has
the form tm_plus t1 t2 where both t1 and t2 are
constants (by ST_PlusConstConst) AND one of t1 or t2 has
the form tm_plus ....
- Similarly, it cannot happen that one is ST_Plus1 and the other
is ST_Plus2, since this would imply that x has the form
tm_plus t1 t2 where t1 both has the form tm_plus t1 t2 and
is a value (hence has the form tm_const n).
- The cases when both derivations end with ST_Plus1 or ST_Plus2 follow by the induction hypothesis. ☐

☐
The definition of single-step reduction for our toy language is
fairly simple, but for a larger language it would be pretty easy
to forget one of the rules and create a situation where some term
cannot take a step even though it has not been completely reduced
to a value. The following theorem shows that we did not, in fact,
make such a mistake here.
This important property is called

## Strong Progress and Normal Forms

*Theorem*(*Strong Progress*): For all t:tm, either t is a value, or there exists a term t' such that t ⇒ t'.*Proof*: By induction on t.- Suppose t = tm_const n. Then t is a value.
- Suppose t = tm_plus t1 t2, where (by the IH) t1 is either a
value or can step to some t1', and where t2 is either a
value or can step to some t2'. We must show tm_plus t1 t2 is
either a value or steps to some t'.
- If t1 and t2 are both values, then t can take a step, by
ST_PlusConstConst.
- If t1 is a value and t2 can take a step, then so can t,
by ST_Plus2.
- If t1 can take a step, then so can t, by ST_Plus1. ☐

- If t1 and t2 are both values, then t can take a step, by
ST_PlusConstConst.

*strong progress*, because every term either is a value or can "make progress" by stepping to some other term. (The qualifier "strong" distinguishes it from a more refined version that we'll see in later chapters, called simply "progress.")Theorem strong_progress : ∀ t,

value t ∨ (∃ t', t ⇒ t').

Proof.

tm_cases (induction t) Case.

Case "tm_const". left. apply v_const.

Case "tm_plus". right. inversion IHt1.

SCase "l". inversion IHt2.

SSCase "l". inversion H. inversion H0.

∃ (tm_const (plus n n0)).

apply ST_PlusConstConst.

SSCase "r". inversion H0 as [t' H1].

∃ (tm_plus t1 t').

apply ST_Plus2. apply H. apply H1.

SCase "r". inversion H as [t' H0].

∃ (tm_plus t' t2).

apply ST_Plus1. apply H0. Qed.

The idea of "making progress" can be extended to tell us something
interesting about values: they are exactly the terms that

*cannot*make progress in this sense. To state this fact, let's begin by giving a name to terms that cannot make progress: We'll call them*normal forms*.
This definition actually specifies what it is to be a normal form
for an
We can use this terminology to generalize the observation we made
in the strong progress theorem: in this language, normal forms and
values are actually the same thing.

*arbitrary*relation R over an arbitrary set X, not just for the particular single-step reduction relation over terms that we are interested in at the moment. We'll re-use the same terminology for talking about other relations later in the course.Lemma value_is_nf : ∀ t,

value t → normal_form step t.

Proof.

unfold normal_form. intros t H. inversion H.

intros contra. inversion contra. inversion H1.

Qed.

Lemma nf_is_value : ∀ t,

normal_form step t → value t.

Proof. (* a corollary of strong_progress... *)

unfold normal_form. intros t H.

assert (G : value t ∨ ∃ t', t ⇒ t').

SCase "Proof of assertion". apply strong_progress.

inversion G.

SCase "l". apply H0.

SCase "r". apply ex_falso_quodlibet. apply H. assumption. Qed.

Corollary nf_same_as_value : ∀ t,

normal_form step t ↔ value t.

Proof.

split. apply nf_is_value. apply value_is_nf.

Qed.

Why is this interesting? For two reasons:
Let's examine how this can happen...
We might, for example, mistakenly define value so that it
includes some terms that are not finished reducing.

- Because value is a syntactic concept — it is a defined by
looking at the form of a term — while normal_form is a
semantic one — it is defined by looking at how the term steps.
It is not obvious that these concepts should coincide!
- Indeed, there are lots of languages in which the concepts of
normal form and value do
*not*coincide.

Module Temp1.

(* Open an inner module so we can redefine value and step. *)

Inductive value : tm → Prop :=

| v_const : ∀ n, value (tm_const n)

| v_funny : ∀ t1 n2, (* <---- *)

value (tm_plus t1 (tm_const n2)).

Reserved Notation " t '⇒' t' " (at level 40).

Inductive step : tm → tm → Prop :=

| ST_PlusConstConst : ∀ n1 n2,

tm_plus (tm_const n1) (tm_const n2) ⇒ tm_const (plus n1 n2)

| ST_Plus1 : ∀ t1 t1' t2,

t1 ⇒ t1' →

tm_plus t1 t2 ⇒ tm_plus t1' t2

| ST_Plus2 : ∀ v1 t2 t2',

value v1 →

t2 ⇒ t2' →

tm_plus v1 t2 ⇒ tm_plus v1 t2'

where " t '⇒' t' " := (step t t').

Lemma value_not_same_as_normal_form :

∃ t, value t ∧ ~ normal_form step t.

Proof.

(* FILL IN HERE *) Admitted.

∃ t, value t ∧ ~ normal_form step t.

Proof.

(* FILL IN HERE *) Admitted.

☐

Alternatively, we might mistakenly define step so that it
permits something designated as a value to reduce further.

Module Temp2.

Inductive value : tm → Prop :=

| v_const : ∀ n, value (tm_const n).

Reserved Notation " t '⇒' t' " (at level 40).

Inductive step : tm → tm → Prop :=

| ST_Funny : ∀ n, (* <---- *)

tm_const n ⇒ tm_plus (tm_const n) (tm_const 0)

| ST_PlusConstConst : ∀ n1 n2,

tm_plus (tm_const n1) (tm_const n2) ⇒ tm_const (plus n1 n2)

| ST_Plus1 : ∀ t1 t1' t2,

t1 ⇒ t1' →

tm_plus t1 t2 ⇒ tm_plus t1' t2

| ST_Plus2 : ∀ v1 t2 t2',

value v1 →

t2 ⇒ t2' →

tm_plus v1 t2 ⇒ tm_plus v1 t2'

where " t '⇒' t' " := (step t t').

Lemma value_not_same_as_normal_form :

∃ t, value t ∧ ~ normal_form step t.

Proof.

(* FILL IN HERE *) Admitted.

∃ t, value t ∧ ~ normal_form step t.

Proof.

(* FILL IN HERE *) Admitted.

☐

Finally, we might mistakenly define value and step so that
there is some term that is not a value but that cannot take a step
in the step relation. Such terms are said to be

*stuck*.Module Temp3.

Inductive value : tm → Prop :=

| v_const : ∀ n, value (tm_const n).

Reserved Notation " t '⇒' t' " (at level 40).

Inductive step : tm → tm → Prop :=

| ST_PlusConstConst : ∀ n1 n2,

tm_plus (tm_const n1) (tm_const n2) ⇒ tm_const (plus n1 n2)

| ST_Plus1 : ∀ t1 t1' t2,

t1 ⇒ t1' →

tm_plus t1 t2 ⇒ tm_plus t1' t2

where " t '⇒' t' " := (step t t').

Lemma value_not_same_as_normal_form :

∃ t, ~ value t ∧ normal_form step t.

Proof.

(* FILL IN HERE *) Admitted.

∃ t, ~ value t ∧ normal_form step t.

Proof.

(* FILL IN HERE *) Admitted.

☐

Here is another very simple language whose terms, instead of being
just plus and numbers, are just the booleans true and false and a
conditional expression...

Inductive tm : Type :=

| tm_true : tm

| tm_false : tm

| tm_if : tm → tm → tm → tm.

Inductive value : tm → Prop :=

| v_true : value tm_true

| v_false : value tm_false.

Reserved Notation " t '⇒' t' " (at level 40).

Inductive step : tm → tm → Prop :=

| ST_IfTrue : ∀ t1 t2,

tm_if tm_true t1 t2 ⇒ t1

| ST_IfFalse : ∀ t1 t2,

tm_if tm_false t1 t2 ⇒ t2

| ST_If : ∀ t1 t1' t2 t3,

t1 ⇒ t1' →

tm_if t1 t2 t3 ⇒ tm_if t1' t2 t3

where " t '⇒' t' " := (step t t').

#### Exercise: 1 star (smallstep_bools)

Which of the following propositions are provable? (This is just a thought exercise, but for an extra challenge feel free to prove your answers in Coq.)Definition bool_step_prop1 :=

tm_false ⇒ tm_false.

(* FILL IN HERE *)

Definition bool_step_prop2 :=

tm_if

tm_true

(tm_if tm_true tm_true tm_true)

(tm_if tm_false tm_false tm_false)

⇒

tm_true.

(* FILL IN HERE *)

Definition bool_step_prop3 :=

tm_if

(tm_if tm_true tm_true tm_true)

(tm_if tm_true tm_true tm_true)

tm_false

⇒

tm_if

tm_true

(tm_if tm_true tm_true tm_true)

tm_false.

(* FILL IN HERE *)

☐

#### Exercise: 3 stars, recommended (progress_bool)

Just as we proved a progress theorem for plus expressions, we can do so for boolean expressions, as well.
☐

#### Exercise: 2 stars (smallstep_bool_shortcut)

Suppose we want to add a "short circuit" to the step relation for boolean expressions, so that it can recognize when the then and else branches of a conditional are the same value (either tm_true or tm_false) and reduce the whole conditional to this value in a single step, even if the guard has not yet been reduced to a value. For example, we would like this proposition to be provable:
tm_if

(tm_if tm_true tm_true tm_true)

tm_false

tm_false

⇒

tm_false.

(tm_if tm_true tm_true tm_true)

tm_false

tm_false

⇒

tm_false.

Reserved Notation " t '⇒' t' " (at level 40).

Inductive step : tm → tm → Prop :=

| ST_IfTrue : ∀ t1 t2,

tm_if tm_true t1 t2 ⇒ t1

| ST_IfFalse : ∀ t1 t2,

tm_if tm_false t1 t2 ⇒ t2

| ST_If : ∀ t1 t1' t2 t3,

t1 ⇒ t1' →

tm_if t1 t2 t3 ⇒ tm_if t1' t2 t3

(* FILL IN HERE *)

where " t '⇒' t' " := (step t t').

☐

Definition bool_step_prop4 :=

tm_if

(tm_if tm_true tm_true tm_true)

tm_false

tm_false

⇒

tm_false.

Example bool_step_prop4_holds :

bool_step_prop4.

Proof.

(* FILL IN HERE *) Admitted.

☐

#### Exercise: 3 stars (properties_of_altered_step)

It can be shown that the determinism and strong progress theorems for the step relation in the lecture notes also hold for the definition of step given above. After we add the clause ST_ShortCircuit...- Is the step relation still deterministic? Write yes or no and
briefly (1 sentence) explain your answer.

(* FILL IN HERE *)

- Does a strong progress theorem hold? Write yes or no and
briefly (1 sentence) explain your answer.

(* FILL IN HERE *)

- In general, is there any way we could cause strong progress to fail if we took away one or more constructors from the original step relation? Write yes or no and briefly (1 sentence) explain your answer.

☐

# Multi-Step Reduction

*single-step reduction*relation ⇒, which formalizes the individual steps of an

*abstract machine*for executing programs.

- First, we define a
*multi-step reduction relation*⇒*, which relates terms t and t' if t can reach t' by any number of single reduction steps. - Then we define a "result" of a term t as a normal form that t can reach by multi-step reduction.

## Multi-Step Reduction

*multi-step reduction*relation ⇒* is the reflexive, transitive closure of the single-step relation ⇒.

Definition stepmany := refl_step_closure step.

Notation " t '⇒*' t' " := (stepmany t t') (at level 40).

For example...

Lemma test_stepmany_1:

tm_plus

(tm_plus (tm_const 0) (tm_const 3))

(tm_plus (tm_const 2) (tm_const 4))

⇒*

tm_const (plus (plus 0 3) (plus 2 4)).

Proof.

apply rsc_step with

(tm_plus

(tm_const (plus 0 3))

(tm_plus (tm_const 2) (tm_const 4))).

apply ST_Plus1. apply ST_PlusConstConst.

apply rsc_step with

(tm_plus

(tm_const (plus 0 3))

(tm_const (plus 2 4))).

apply ST_Plus2. apply v_const.

apply ST_PlusConstConst.

apply rsc_R.

apply ST_PlusConstConst. Qed.

Here's an alternate proof that uses eapply to avoid explicitly
constructing all the intermediate terms.

Lemma test_stepmany_1':

tm_plus

(tm_plus (tm_const 0) (tm_const 3))

(tm_plus (tm_const 2) (tm_const 4))

⇒*

tm_const (plus (plus 0 3) (plus 2 4)).

Proof.

eapply rsc_step. apply ST_Plus1. apply ST_PlusConstConst.

eapply rsc_step. apply ST_Plus2. apply v_const.

apply ST_PlusConstConst.

eapply rsc_step. apply ST_PlusConstConst.

apply rsc_refl. Qed.

Lemma test_stepmany_3:

tm_plus (tm_const 0) (tm_const 3)

⇒*

tm_plus (tm_const 0) (tm_const 3).

Proof.

(* FILL IN HERE *) Admitted.

tm_plus (tm_const 0) (tm_const 3)

⇒*

tm_plus (tm_const 0) (tm_const 3).

Proof.

(* FILL IN HERE *) Admitted.

Lemma test_stepmany_4:

tm_plus

(tm_const 0)

(tm_plus

(tm_const 2)

(tm_plus (tm_const 0) (tm_const 3)))

⇒*

tm_plus

(tm_const 0)

(tm_const (plus 2 (plus 0 3))).

Proof.

(* FILL IN HERE *) Admitted.

tm_plus

(tm_const 0)

(tm_plus

(tm_const 2)

(tm_plus (tm_const 0) (tm_const 3)))

⇒*

tm_plus

(tm_const 0)

(tm_const (plus 2 (plus 0 3))).

Proof.

(* FILL IN HERE *) Admitted.

☐
If t reduces to t' in zero or more steps and t' is a
normal form, we say that "t' is a normal form of t."

## Normal Forms Again

Definition step_normal_form := normal_form step.

Definition normal_form_of (t t' : tm) :=

(t ⇒* t' ∧ step_normal_form t').

We have already seen that single-step reduction is
deterministic — i.e., a given term can take a single step in
at most one way. It follows from this that, if t can reach
a normal form, then this normal form is unique — i.e.,
normal_form_of is a partial function. In other words, we
can actually pronounce normal_form t t' as "t' is

*the*normal form of t."#### Exercise: 3 stars, optional (test_stepmany_3)

Theorem normal_forms_unique:

partial_function normal_form_of.

Proof.

unfold partial_function. unfold normal_form_of. intros x y1 y2 P1 P2.

destruct P1 as [P11 P12]. destruct P2 as [P21 P22].

generalize dependent y2.

(* We recommend using this initial setup as-is! *)

(* FILL IN HERE *) Admitted.

partial_function normal_form_of.

Proof.

unfold partial_function. unfold normal_form_of. intros x y1 y2 P1 P2.

destruct P1 as [P11 P12]. destruct P2 as [P21 P22].

generalize dependent y2.

(* We recommend using this initial setup as-is! *)

(* FILL IN HERE *) Admitted.

☐
Indeed, something stronger is true for this language (though not
for all languages): the reduction of

*any*term t will eventually reach a normal form — i.e., normal_form_of is a*total*function. Formally, we say the step relation is*normalizing*.Definition normalizing {X:Type} (R:relation X) :=

∀ t, ∃ t',

(refl_step_closure R) t t' ∧ normal_form R t'.

To prove that step is normalizing, we need a couple of lemmas.
First, we observe that, if t reduces to t' in many steps, then
the same sequence of reduction steps within t is also possible
when t appears as the left-hand child of a tm_plus node, and
similarly when t appears as the right-hand child of a tm_plus
node (whose left-hand child is a value).

Lemma stepmany_congr_1 : ∀ t1 t1' t2,

t1 ⇒* t1' →

tm_plus t1 t2 ⇒* tm_plus t1' t2.

Proof.

intros t1 t1' t2 H. rsc_cases (induction H) Case.

Case "rsc_refl". apply rsc_refl.

Case "rsc_step". apply rsc_step with (tm_plus y t2).

apply ST_Plus1. apply H.

apply IHrefl_step_closure. Qed.

Lemma stepmany_congr_2 : ∀ t1 t2 t2',

value t1 →

t2 ⇒* t2' →

tm_plus t1 t2 ⇒* tm_plus t1 t2'.

Proof.

(* FILL IN HERE *) Admitted.

value t1 →

t2 ⇒* t2' →

tm_plus t1 t2 ⇒* tm_plus t1 t2'.

Proof.

(* FILL IN HERE *) Admitted.

☐

*Theorem*: The step function is normalizing — i.e., for every t there exists some t' such that t steps to t' and t' is a normal form.*Proof sketch*: By induction on terms. There are two cases to consider:- t = tm_const n for some n. Here t doesn't take a step,
and we have t' = t. We can derive the left-hand side by
reflexivity and the right-hand side by observing (a) that values
are normal forms (by nf_same_as_value) and (b) that t is a
value (by v_const).
- t = tm_plus t1 t2 for some t1 and t2. By the IH, t1 and
t2 have normal forms t1' and t2'. Recall that normal
forms are values (by nf_same_as_value); we know that t1' =
tm_const n1 and t2' = tm_const n2, for some n1 and n2.
We can combine the ⇒* derivations for t1 and t2 to prove
that tm_plus t1 t2 reduces in many steps to tm_const (plus n1
n2).

Theorem step_normalizing :

normalizing step.

Proof.

unfold normalizing.

tm_cases (induction t) Case.

Case "tm_const".

∃ (tm_const n).

split.

SCase "l". apply rsc_refl.

SCase "r".

(* We can use rewrite with "iff" statements, not

just equalities: *)

rewrite nf_same_as_value. apply v_const.

Case "tm_plus".

destruct IHt1 as [t1' H1]. destruct IHt2 as [t2' H2].

destruct H1 as [H11 H12]. destruct H2 as [H21 H22].

rewrite nf_same_as_value in H12. rewrite nf_same_as_value in H22.

inversion H12 as [n1]. inversion H22 as [n2].

rewrite ← H in H11.

rewrite ← H0 in H21.

∃ (tm_const (plus n1 n2)).

split.

SCase "l".

apply rsc_trans with (tm_plus (tm_const n1) t2).

apply stepmany_congr_1. apply H11.

apply rsc_trans with

(tm_plus (tm_const n1) (tm_const n2)).

apply stepmany_congr_2. apply v_const. apply H21.

apply rsc_R. apply ST_PlusConstConst.

SCase "r".

rewrite nf_same_as_value. apply v_const. Qed.

## Equivalence of Big-Step and Small-Step Reduction

Lemma eval__value : ∀ t1 t2,

eval t1 t2 →

value t2.

Proof.

intros t1 t2 HE.

eval_cases (inversion HE) Case; apply v_const. Qed.

#### Exercise: 3 stars (eval__stepmany)

You'll want to use the congruences and some properties of rsc for this.
☐
(* FILL IN HERE *)

☐

#### Exercise: 3 stars (eval__stepmany_inf)

Write an informal version of the proof of eval__stepmany.☐

#### Exercise: 3 stars (step__eval)

☐

Theorem stepmany__eval : ∀ t v,

normal_form_of t v → t ⇓ v.

Proof.

intros t v Hnorm.

unfold normal_form_of in Hnorm.

inversion Hnorm as [Hs Hnf]; clear Hnorm.

(* v is a normal form -> v = tm_const n for some n *)

rewrite nf_same_as_value in Hnf. inversion Hnf. clear Hnf.

rsc_cases (induction Hs) Case; subst.

Case "rsc_refl".

apply E_Const.

Case "rsc_step".

eapply step__eval. eassumption. apply IHHs. reflexivity. Qed.

Bringing it all together, we can crisply say that the v is the
normal form of t iff t evaluates to v.

Corollary stepmany_iff_eval : ∀ t v,

normal_form_of t v ↔ t ⇓ v.

Proof.

split.

Case "→". apply stepmany__eval.

Case "←". unfold normal_form_of. intros E. split. apply eval__stepmany.

assumption. rewrite nf_same_as_value. eapply eval__value. eassumption. Qed.

## Additional Exercises

#### Exercise: 4 stars (combined_properties)

We've considered the arithmetic and conditional expressions separately. This exercise explores how the two interact.Module Combined.

Inductive tm : Type :=

| tm_const : nat → tm

| tm_plus : tm → tm → tm

| tm_true : tm

| tm_false : tm

| tm_if : tm → tm → tm → tm.

Tactic Notation "tm_cases" tactic(first) ident(c) :=

first;

[ Case_aux c "tm_const" | Case_aux c "tm_plus"

| Case_aux c "tm_true" | Case_aux c "tm_false" | Case_aux c "tm_if" ].

Inductive value : tm → Prop :=

| v_const : ∀ n, value (tm_const n)

| v_true : value tm_true

| v_false : value tm_false.

Reserved Notation " t '⇒' t' " (at level 40).

Inductive step : tm → tm → Prop :=

| ST_PlusConstConst : ∀ n1 n2,

tm_plus (tm_const n1) (tm_const n2) ⇒ tm_const (plus n1 n2)

| ST_Plus1 : ∀ t1 t1' t2,

t1 ⇒ t1' →

tm_plus t1 t2 ⇒ tm_plus t1' t2

| ST_Plus2 : ∀ v1 t2 t2',

value v1 →

t2 ⇒ t2' →

tm_plus v1 t2 ⇒ tm_plus v1 t2'

| ST_IfTrue : ∀ t1 t2,

tm_if tm_true t1 t2 ⇒ t1

| ST_IfFalse : ∀ t1 t2,

tm_if tm_false t1 t2 ⇒ t2

| ST_If : ∀ t1 t1' t2 t3,

t1 ⇒ t1' →

tm_if t1 t2 t3 ⇒ tm_if t1' t2 t3

where " t '⇒' t' " := (step t t').

Tactic Notation "step_cases" tactic(first) ident(c) :=

first;

[ Case_aux c "ST_PlusConstConst"

| Case_aux c "ST_Plus1" | Case_aux c "ST_Plus2"

| Case_aux c "ST_IfTrue" | Case_aux c "ST_IfFalse" | Case_aux c "ST_If" ].

Earlier, we separately proved for both plus- and if-expressions...

- that the step relation was a partial function (i.e., it was
deterministic), and
- a strong progress lemma, stating that every term is either a value or can take a step.

(* FILL IN HERE *)

☐

# Small-Step Imp

Inductive aval : aexp → Prop :=

av_num : ∀ n, aval (ANum n).

Reserved Notation " t '/' st '⇒a' t' " (at level 40, st at level 39).

Inductive astep : state → aexp → aexp → Prop :=

| AS_Id : ∀ st i,

AId i / st ⇒a ANum (st i)

| AS_Plus : ∀ st n1 n2,

APlus (ANum n1) (ANum n2) / st ⇒a ANum (plus n1 n2)

| AS_Plus1 : ∀ st a1 a1' a2,

a1 / st ⇒a a1' →

(APlus a1 a2) / st ⇒a (APlus a1' a2)

| AS_Plus2 : ∀ st v1 a2 a2',

aval v1 →

a2 / st ⇒a a2' →

(APlus v1 a2) / st ⇒a (APlus v1 a2')

| AS_Minus : ∀ st n1 n2,

(AMinus (ANum n1) (ANum n2)) / st ⇒a (ANum (minus n1 n2))

| AS_Minus1 : ∀ st a1 a1' a2,

a1 / st ⇒a a1' →

(AMinus a1 a2) / st ⇒a (AMinus a1' a2)

| AS_Minus2 : ∀ st v1 a2 a2',

aval v1 →

a2 / st ⇒a a2' →

(AMinus v1 a2) / st ⇒a (AMinus v1 a2')

| AS_Mult : ∀ st n1 n2,

(AMult (ANum n1) (ANum n2)) / st ⇒a (ANum (mult n1 n2))

| AS_Mult1 : ∀ st a1 a1' a2,

a1 / st ⇒a a1' →

(AMult (a1) (a2)) / st ⇒a (AMult (a1') (a2))

| AS_Mult2 : ∀ st v1 a2 a2',

aval v1 →

a2 / st ⇒a a2' →

(AMult v1 a2) / st ⇒a (AMult v1 a2')

where " t '/' st '⇒a' t' " := (astep st t t').

Reserved Notation " t '/' st '⇒b' t' " (at level 40, st at level 39).

Inductive bstep : state → bexp → bexp → Prop :=

| BS_Eq : ∀ st n1 n2,

(BEq (ANum n1) (ANum n2)) / st ⇒b

(if (beq_nat n1 n2) then BTrue else BFalse)

| BS_Eq1 : ∀ st a1 a1' a2,

a1 / st ⇒a a1' →

(BEq a1 a2) / st ⇒b (BEq a1' a2)

| BS_Eq2 : ∀ st v1 a2 a2',

aval v1 →

a2 / st ⇒a a2' →

(BEq v1 a2) / st ⇒b (BEq v1 a2')

| BS_LtEq : ∀ st n1 n2,

(BLe (ANum n1) (ANum n2)) / st ⇒b

(if (ble_nat n1 n2) then BTrue else BFalse)

| BS_LtEq1 : ∀ st a1 a1' a2,

a1 / st ⇒a a1' →

(BLe a1 a2) / st ⇒b (BLe a1' a2)

| BS_LtEq2 : ∀ st v1 a2 a2',

aval v1 →

a2 / st ⇒a a2' →

(BLe v1 a2) / st ⇒b (BLe v1 (a2'))

| BS_NotTrue : ∀ st,

(BNot BTrue) / st ⇒b BFalse

| BS_NotFalse : ∀ st,

(BNot BFalse) / st ⇒b BTrue

| BS_NotStep : ∀ st b1 b1',

b1 / st ⇒b b1' →

(BNot b1) / st ⇒b (BNot b1')

| BS_AndTrueTrue : ∀ st,

(BAnd BTrue BTrue) / st ⇒b BTrue

| BS_AndTrueFalse : ∀ st,

(BAnd BTrue BFalse) / st ⇒b BFalse

| BS_AndFalse : ∀ st b2,

(BAnd BFalse b2) / st ⇒b BFalse

| BS_AndTrueStep : ∀ st b2 b2',

b2 / st ⇒b b2' →

(BAnd BTrue b2) / st ⇒b (BAnd BTrue b2')

| BS_AndStep : ∀ st b1 b1' b2,

b1 / st ⇒b b1' →

(BAnd b1 b2) / st ⇒b (BAnd b1' b2)

where " t '/' st '⇒b' t' " := (bstep st t t').

Notice that we didn't actually bother to define boolean values —
they aren't needed in the definition of ⇒b (why?), though
they might be if our language were a bit larger.
The semantics of commands is the interesting part. We need two
small tricks to make it work:

- We use SKIP as a "command value" — i.e., a command that
has reached a normal form.
- An assignment command reduces to SKIP (and an updated
state).
- The sequencing command waits until its left-hand
subcommand has reduced to SKIP, then throws it away so
that reduction can continue with the right-hand
subcommand.

- An assignment command reduces to SKIP (and an updated
state).
- We reduce a WHILE command by transforming it into a
conditional followed by the same WHILE.

Reserved Notation " t '/' st '⇒' t' '/' st' "

(at level 40, st at level 39, t' at level 39).

Inductive cstep : (com * state) → (com * state) → Prop :=

| CS_AssStep : ∀ st i a a',

a / st ⇒a a' →

(i ::= a) / st ⇒ (i ::= a') / st

| CS_Ass : ∀ st i n,

(i ::= (ANum n)) / st ⇒ SKIP / (update st i n)

| CS_SeqStep : ∀ st c1 c1' st' c2,

c1 / st ⇒ c1' / st' →

(c1 ; c2) / st ⇒ (c1' ; c2) / st'

| CS_SeqFinish : ∀ st c2,

(SKIP ; c2) / st ⇒ c2 / st

| CS_IfTrue : ∀ st c1 c2,

IFB BTrue THEN c1 ELSE c2 FI / st ⇒ c1 / st

| CS_IfFalse : ∀ st c1 c2,

IFB BFalse THEN c1 ELSE c2 FI / st ⇒ c2 / st

| CS_IfStep : ∀ st b b' c1 c2,

b / st ⇒b b' →

IFB b THEN c1 ELSE c2 FI / st ⇒ (IFB b' THEN c1 ELSE c2 FI) / st

| CS_While : ∀ st b c1,

(WHILE b DO c1 END) / st

⇒ (IFB b THEN (c1; (WHILE b DO c1 END)) ELSE SKIP FI) / st

where " t '/' st '⇒' t' '/' st' " := (cstep (t,st) (t',st')).

# Concurrent Imp

Module CImp.

Inductive com : Type :=

| CSkip : com

| CAss : id → aexp → com

| CSeq : com → com → com

| CIf : bexp → com → com → com

| CWhile : bexp → com → com

(* New: *)

| CPar : com → com → com.

Tactic Notation "com_cases" tactic(first) ident(c) :=

first;

[ Case_aux c "SKIP" | Case_aux c "::=" | Case_aux c ";"

| Case_aux c "IFB" | Case_aux c "WHILE" | Case_aux c "PAR" ].

Notation "'SKIP'" :=

CSkip.

Notation "l '::=' a" :=

(CAss l a) (at level 60).

Notation "c1 ; c2" :=

(CSeq c1 c2) (at level 80, right associativity).

Notation "'WHILE' b 'DO' c 'END'" :=

(CWhile b c) (at level 80, right associativity).

Notation "'IFB' e1 'THEN' e2 'ELSE' e3 'FI'" :=

(CIf e1 e2 e3) (at level 80, right associativity).

Notation "'PAR' c1 'WITH' c2 'END'" :=

(CPar c1 c2) (at level 80, right associativity).

Inductive cstep : (com * state) → (com * state) → Prop :=

| CS_AssStep : ∀ st i a a',

a / st ⇒a a' →

(i ::= a) / st ⇒ (i ::= a') / st

| CS_Ass : ∀ st i n,

(i ::= (ANum n)) / st ⇒ SKIP / (update st i n)

| CS_SeqStep : ∀ st c1 c1' st' c2,

c1 / st ⇒ c1' / st' →

(c1 ; c2) / st ⇒ (c1' ; c2) / st'

| CS_SeqFinish : ∀ st c2,

(SKIP ; c2) / st ⇒ c2 / st

| CS_IfTrue : ∀ st c1 c2,

(IFB BTrue THEN c1 ELSE c2 FI) / st ⇒ c1 / st

| CS_IfFalse : ∀ st c1 c2,

(IFB BFalse THEN c1 ELSE c2 FI) / st ⇒ c2 / st

| CS_IfStep : ∀ st b b' c1 c2,

b /st ⇒b b' →

(IFB b THEN c1 ELSE c2 FI) / st ⇒ (IFB b' THEN c1 ELSE c2 FI) / st

| CS_While : ∀ st b c1,

(WHILE b DO c1 END) / st ⇒

(IFB b THEN (c1; (WHILE b DO c1 END)) ELSE SKIP FI) / st

(* New: *)

| CS_Par1 : ∀ st c1 c1' c2 st',

c1 / st ⇒ c1' / st' →

(PAR c1 WITH c2 END) / st ⇒ (PAR c1' WITH c2 END) / st'

| CS_Par2 : ∀ st c1 c2 c2' st',

c2 / st ⇒ c2' / st' →

(PAR c1 WITH c2 END) / st ⇒ (PAR c1 WITH c2' END) / st'

| CS_ParDone : ∀ st,

(PAR SKIP WITH SKIP END) / st ⇒ SKIP / st

where " t '/' st '⇒' t' '/' st' " := (cstep (t,st) (t',st')).

Definition cstepmany := refl_step_closure cstep.

Notation " t '/' st '⇒*' t' '/' st' " :=

(refl_step_closure cstep (t,st) (t',st'))

(at level 40, st at level 39, t' at level 39).

Among the many interesting properties of this language is the fact
that the following program can terminate with the variable X set
to any value...

Definition par_loop : com :=

PAR

Y ::= ANum 1

WITH

WHILE BEq (AId Y) (ANum 0) DO

X ::= APlus (AId X) (ANum 1)

END

END.

In particular, it can terminate with X set to 0:

Example par_loop_example_0:

∃ st',

par_loop / empty_state ⇒* SKIP / st'

∧ st' X = 0.

Proof.

eapply ex_intro. split.

unfold par_loop.

eapply rsc_step. apply CS_Par1.

apply CS_Ass.

eapply rsc_step. apply CS_Par2. apply CS_While.

eapply rsc_step. apply CS_Par2. apply CS_IfStep.

apply BS_Eq1. apply AS_Id.

eapply rsc_step. apply CS_Par2. apply CS_IfStep.

apply BS_Eq. simpl.

eapply rsc_step. apply CS_Par2. apply CS_IfFalse.

eapply rsc_step. apply CS_ParDone.

eapply rsc_refl.

reflexivity. Qed.

It can also terminate with X set to 2:

Example par_loop_example_2:

∃ st',

par_loop / empty_state ⇒* SKIP / st'

∧ st' X = 2.

Proof.

eapply ex_intro. split.

eapply rsc_step. apply CS_Par2. apply CS_While.

eapply rsc_step. apply CS_Par2. apply CS_IfStep.

apply BS_Eq1. apply AS_Id.

eapply rsc_step. apply CS_Par2. apply CS_IfStep.

apply BS_Eq. simpl.

eapply rsc_step. apply CS_Par2. apply CS_IfTrue.

eapply rsc_step. apply CS_Par2. apply CS_SeqStep.

apply CS_AssStep. apply AS_Plus1. apply AS_Id.

eapply rsc_step. apply CS_Par2. apply CS_SeqStep.

apply CS_AssStep. apply AS_Plus.

eapply rsc_step. apply CS_Par2. apply CS_SeqStep.

apply CS_Ass.

eapply rsc_step. apply CS_Par2. apply CS_SeqFinish.

eapply rsc_step. apply CS_Par2. apply CS_While.

eapply rsc_step. apply CS_Par2. apply CS_IfStep.

apply BS_Eq1. apply AS_Id.

eapply rsc_step. apply CS_Par2. apply CS_IfStep.

apply BS_Eq. simpl.

eapply rsc_step. apply CS_Par2. apply CS_IfTrue.

eapply rsc_step. apply CS_Par2. apply CS_SeqStep.

apply CS_AssStep. apply AS_Plus1. apply AS_Id.

eapply rsc_step. apply CS_Par2. apply CS_SeqStep.

apply CS_AssStep. apply AS_Plus.

eapply rsc_step. apply CS_Par2. apply CS_SeqStep.

apply CS_Ass.

eapply rsc_step. apply CS_Par1. apply CS_Ass.

eapply rsc_step. apply CS_Par2. apply CS_SeqFinish.

eapply rsc_step. apply CS_Par2. apply CS_While.

eapply rsc_step. apply CS_Par2. apply CS_IfStep.

apply BS_Eq1. apply AS_Id.

eapply rsc_step. apply CS_Par2. apply CS_IfStep.

apply BS_Eq. simpl.

eapply rsc_step. apply CS_Par2. apply CS_IfFalse.

eapply rsc_step. apply CS_ParDone.

eapply rsc_refl.

reflexivity. Qed.

Lemma par_body_n__Sn : ∀ n st,

st X = n ∧ st Y = 0 →

par_loop / st ⇒* par_loop / (update st X (S n)).

Proof.

(* FILL IN HERE *) Admitted.

st X = n ∧ st Y = 0 →

par_loop / st ⇒* par_loop / (update st X (S n)).

Proof.

(* FILL IN HERE *) Admitted.

Lemma par_body_n : ∀ n st,

st X = 0 ∧ st Y = 0 →

∃ st',

par_loop / st ⇒* par_loop / st' ∧ st' X = n ∧ st' Y = 0.

Proof.

(* FILL IN HERE *) Admitted.

st X = 0 ∧ st Y = 0 →

∃ st',

par_loop / st ⇒* par_loop / st' ∧ st' X = n ∧ st' Y = 0.

Proof.

(* FILL IN HERE *) Admitted.

... the above loop can exit with X having any value
whatsoever.

Theorem par_loop_any_X:

∀ n, ∃ st',

par_loop / empty_state ⇒* SKIP / st'

∧ st' X = n.

Proof.

intros n.

destruct (par_body_n n empty_state).

split; unfold update; reflexivity.

rename x into st.

destruct H as [H' [HX HY]].

∃ (update st Y 1). split.

eapply rsc_trans with (par_loop,st). apply H'.

eapply rsc_step. apply CS_Par1. apply CS_Ass.

eapply rsc_step. apply CS_Par2. apply CS_While.

eapply rsc_step. apply CS_Par2. apply CS_IfStep.

apply BS_Eq1. apply AS_Id. rewrite update_eq.

eapply rsc_step. apply CS_Par2. apply CS_IfStep.

apply BS_Eq. simpl.

eapply rsc_step. apply CS_Par2. apply CS_IfFalse.

eapply rsc_step. apply CS_ParDone.

apply rsc_refl.

rewrite update_neq. assumption. reflexivity.

Qed.

End CImp.