# Definition of STLC

This file containes all of the definitions for a locally-nameless representation of a Curry-Style simply-typed lambda calculus.
This file was generated via Ott from `stlc.ott` and then edited to include explanation about the definitions. As a result, it is gathers all of the STLC definitions in one place, but the associated exercises are found elsewhere in the tutorial. You'll want to refer back to this file as you progress through the rest of the material.

Require Import Metalib.Metatheory.

# Syntax of STLC

We use a locally nameless representation for the simply-typed lambda calculus, where bound variables are represented as natural numbers (de Bruijn indices) and free variables are represented as atoms.
The type atom, defined in the Metatheory library, represents names. Equality on names is decidable, and it is possible to generate an atom fresh for any given finite set of atoms (atom_fresh).
Note: the type var is notation for atom.

Inductive typ : Set :=
| typ_base : typ
| typ_arrow (T1:typ) (T2:typ).

Inductive exp : Set :=
| var_b (_:nat)
| var_f (x:var)
| abs (e:exp)
| app (e1:exp) (e2:exp).

# Substitution

Substitution replaces a free variable with a term. The definition below is simple for two reasons:
• Because bound variables are represented using indices, there is no need to worry about variable capture.
• We assume that the term being substituted in is locally closed. Thus, there is no need to shift indices when passing under a binder.
The Fixpoint keyword defines a Coq function. As all functions in Coq must be total. The annotation {struct e} indicates the termination metric---all recursive calls in this definition are made to arguments that are structurally smaller than e.
Note also that subst_exp uses x == y for decidable equality. This operation is defined in the Metatheory library.

Fixpoint subst_exp (u:exp) (y:var) (e:exp) {struct e} : exp :=
match e with
| (var_b n) ⇒ var_b n
| (var_f x) ⇒ (if x == y then u else (var_f x))
| (abs e1) ⇒ abs (subst_exp u y e1)
| (app e1 e2) ⇒ app (subst_exp u y e1) (subst_exp u y e2)
end.

# Free variables

The function fv_exp, defined below, calculates the set of free variables in an expression. Because we are using a locally nameless representation, where bound variables are represented as indices, any name we see is a free variable of a term. In particular, this makes the abs case simple.

Fixpoint fv_exp (e_5:exp) : vars :=
match e_5 with
| (var_b nat) ⇒ {}
| (var_f x) ⇒ {{x}}
| (abs e) ⇒ fv_exp e
| (app e1 e2) ⇒ fv_exp e1 \u fv_exp e2
end.

The type vars represents a finite set of elements of type atom. The notations for the finite set definitions (empty set `{}`, singleton `x` and union `\u`) is also defined in the Metatheory library.

# Opening

Opening replaces an index with a term. It corresponds to informal substitution for a bound variable, such as in the rule for beta reduction. Note that only "dangling" indices (those that do not refer to any abstraction) can be opened. Opening has no effect for terms that are locally closed.
Natural numbers are just an inductive datatype with two constructors: O (as in the letter 'oh', not 'zero') and S, defined in Coq.Init.Datatypes. Coq allows literal natural numbers to be written using standard decimal notation, e.g., 0, 1, 2, etc. The function lt_eq_lt_dec compares its two arguments for ordering.
We do not assume that zero is the only unbound index in the term. Consequently, we must substract one when we encounter other unbound indices (i.e. the inright case).
However, we do assume that the argument u is locally closed. This assumption simplifies the implementation since we do not need to shift indices in u when passing under a binder.

Fixpoint open_exp_wrt_exp_rec (k:nat) (u:exp) (e:exp) {struct e}: exp :=
match e with
| (var_b n) ⇒
match lt_eq_lt_dec n k with
| inleft (left _) ⇒ var_b n
| inleft (right _) ⇒ u
| inright _var_b (n - 1)
end
| (var_f x) ⇒ var_f x
| (abs e) ⇒ abs (open_exp_wrt_exp_rec (S k) u e)
| (app e1 e2) ⇒ app (open_exp_wrt_exp_rec k u e1)
(open_exp_wrt_exp_rec k u e2)
end.

Definition open_exp_wrt_exp e u := open_exp_wrt_exp_rec 0 u e.

# Notations

Many common applications of opening replace index zero with an expression or variable. The following definition provides a convenient shorthand for such uses. Note that the order of arguments is switched relative to the definition above. For example, (open e x) can be read as "substitute the variable x for index 0 in e" and "open e with the variable x."

Module StlcNotations.
Notation "[ z ~> u ] e" := (subst_exp u z e) (at level 0) : exp_scope.
Notation open e1 e2 := (open_exp_wrt_exp e1 e2).
Notation "e ^ x" := (open_exp_wrt_exp e (var_f x)) : exp_scope.
End StlcNotations.
Import StlcNotations.
Open Scope exp_scope.

# Local closure

Recall that exp admits terms that contain unbound indices. We say that a term is locally closed when no indices appearing in it are unbound. The proposition lc_exp e holds when an expression e is locally closed.
The inductive definition below formalizes local closure such that the resulting induction principle serves as the structural induction principle over (locally closed) expressions. In particular, unlike induction for type exp, there are no cases for bound variables. Thus, the induction principle corresponds more closely to informal practice than the one arising from the definition of pre-terms.

Inductive lc_exp : exp Prop :=
| lc_var_f : (x:var),
lc_exp (var_f x)
| lc_abs : (e:exp),
( x , lc_exp (open e (var_f x)))
lc_exp (abs e)
| lc_app : (e1 e2:exp),
lc_exp e1
lc_exp e2
lc_exp (app e1 e2).

# Typing contexts

We represent typing contexts as association lists (lists of pairs of keys and values) whose keys are atoms.

Definition ctx : Set := list (atom × typ).

For STLC, contexts bind atoms to typs.
Lists are defined in Coq's standard library, with the constructors nil and cons. The list library includes the :: notation for cons as well as standard list operations such as append, map, and fold. The infix operation ++ is list append.
The Metatheory library extends this reasoning by instantiating the AssocList library to provide support for association lists whose keys are atoms. Everything in this library is polymorphic over the type of objects bound in the environment. Look in AssocList.v for additional details about the functions and predicates that we mention below.

# Typing relation

The definition of the typing relation is straightforward. In order to ensure that the relation holds for only well-formed environments, we check in the typing_var case that the environment is uniq. The structure of typing derivations implicitly ensures that the relation holds only for locally closed expressions.
Finally, note the use of cofinite quantification in the typing_abs case.

Inductive typing : ctx exp typ Prop :=
| typing_var : (G:ctx) (x:var) (T:typ),
uniq G
binds x T G
typing G (var_f x) T
| typing_abs : (L:vars) (G:ctx) (T1:typ) (e:exp) (T2:typ),
( x , x \notin L typing ([(x,T1)] ++ G) (e ^ x) T2)
typing G (abs e) (typ_arrow T1 T2)
| typing_app : (G:ctx) (e1 e2:exp) (T2 T1:typ),
typing G e1 (typ_arrow T1 T2)
typing G e2 T1
typing G (app e1 e2) T2 .

# Values and Small-step Evaluation

Finally, we define values and a call-by-name small-step evaluation relation. In STLC, abstractions are the only value.

Definition is_value (e : exp) : Prop :=
match e with
| abs _True
| _False
end.

For step_beta, note that we use open_exp_wrt_exp instead of substitution --- no variable names are involved.
Note also the hypotheses in step that ensure that the relation holds only for locally closed terms.

Inductive step : exp exp Prop :=
| step_beta : (e1 e2:exp),
lc_exp (abs e1)
lc_exp e2
step (app (abs e1) e2) (open e1 e2)
| step_app : (e1 e2 e1':exp),
lc_exp e2
step e1 e1'
step (app e1 e2) (app e1' e2).

Hint Constructors typing step lc_exp.