(** A nominal representation of STLC terms.  This version looks a lot like we expect a representation of the lambda calculus to look like. Unlike the LN version, the syntax does not distinguish between bound and free variables and abstractions include the name of binding variables.  *)
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(** * Imports                                                *)
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(** Some of our proofs are by induction on the *size* of terms. We'll use Coq's [omega] tactic to easily dispatch reasoning about natural numbers. *)
Require Export Omega.
(** We will use the [atom] type from the metatheory library to represent variable names. *)
Require Export Metalib.Metatheory.
(** Although we are not using LNgen, some of the tactics from its library are useful for automating reasoning about names.  *)
Require Export Metalib.LibLNgen.
(** Some fresh variables *)
Notation X := (fresh nil).
Notation Y := (fresh (X :: nil)).
Notation Z := (fresh (X :: Y :: nil)).
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(** * A nominal representation of terms                      *)
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Inductive n_exp : Set :=
| n_var (x:atom)
| n_abs (x:atom) (t:n_exp)
| n_app (t1:n_exp) (t2:n_exp).
(** For example, we can encode the expression [(\X.Y X)] as below.  *)
Definition demo_rep1 := n_abs X (n_app (n_var Y) (n_var X)).
(** For example, we can encode the expression [(\Z.Y Z)] as below.  *)
Definition demo_rep2 := n_abs Z (n_app (n_var Y) (n_var Z)).
(** As usual, the free variable function needs to remove the bound variable in the [n_abs] case. *)
Fixpoint fv_nom (n : n_exp) : atoms :=
match n with
| n_var x => {{x}}
| n_abs x n => remove x (fv_nom n)
| n_app t1 t2 => fv_nom t1 `union` fv_nom t2
end.
(** The tactics for reasoning about lists and sets of atoms are useful here too. *)
Example fv_nom_rep1 : fv_nom demo_rep1 [=] {{ Y }}.
(** What makes this a *nominal* representation is that our operations are based on the following swapping function for names.  Note that this operation is symmetric: [x] becomes [y] and [y] becomes [x]. *)
Definition swap_var (x:atom) (y:atom) (z:atom) :=
if (z == x) then y else if (z == y) then x else z.
(** The main insight of nominal representations is that we can rename variables, without capture, using a simple structural induction. Note how in the [n_abs] case we swap all variables, both bound and free.  For example:  (swap x y) (\z. (x y)) = \z. (y x))  (swap x y) (\x. x) = \y.y  (swap x y) (\y. x) = \x.y  *)
Fixpoint swap (x:atom) (y:atom) (t:n_exp) : n_exp :=
match t with
| n_var z     => n_var (swap_var x y z)
| n_abs z t1  => n_abs (swap_var x y z) (swap x y t1)
| n_app t1 t2 => n_app (swap x y t1) (swap x y t2)
end.
(** Because swapping is a simple, structurally recursive function, it is highly automatable using the [default_simp] tactic from LNgen library.  This tactic "simplifies" goals using a combination of common proof steps, including case analysis of all disjoint sums in the goal. Because atom equality returns a sumbool, this makes this tactic useful for reasoning by cases about atoms.  For more information about the [default_simp] tactic, see [metalib/LibDefaultSimp.v].  WARNING: this tactic is not always safe. It's a power tool and can put your proof in an irrecoverable state. *)
Example swap1 : forall x y z, x <> z -> y <> z ->
swap x y (n_abs z (n_app (n_var x)(n_var y))) = n_abs z (n_app (n_var y) (n_var x)).
Example swap2 : forall x y, x <> y ->
swap x y (n_abs x (n_var x)) = n_abs y (n_var y).
Example swap3 : forall x y, x <> y ->
swap x y (n_abs y (n_var x)) = n_abs x (n_var y).
(** We define the "alpha-equivalence" relation that declares when two nominal terms are equivalent (up to renaming of bound variables).  Note the two different rules for [n_abs]: either the binders are the same and we can directly compare the bodies, or the binders differ, but we can swap one side to make it look like the other.  *)
Inductive aeq : n_exp -> n_exp -> Prop :=
| aeq_var : forall x,
aeq (n_var x) (n_var x)
| aeq_abs_same : forall x t1 t2,
aeq t1 t2 -> aeq (n_abs x t1) (n_abs x t2)
| aeq_abs_diff : forall x y t1 t2,
x <> y ->
x `notin` fv_nom t2 ->
aeq t1 (swap y x t2) ->
aeq (n_abs x t1) (n_abs y t2)
| aeq_app : forall t1 t2 t1' t2',
aeq t1 t1' -> aeq t2 t2' ->
aeq (n_app t1 t2) (n_app t1' t2').
Hint Constructors aeq : core.
Example aeq1 : forall x y, x <> y -> aeq (n_abs x (n_var x)) (n_abs y (n_var y)).
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(** ** Properties about swapping                             *)
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(** Now let's look at some simple properties of swapping. *)
(** Demo: We will need the next two properties later in the tutorial, so we show that even though there are many cases to consider, [default_simp] can find these proofs. *)
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(** ** Exercises                                             *)
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(** *** Recommended Exercise: [swap] properties  Prove the following properties about swapping, either explicitly (by destructing [x == y]) or automatically (using [default_simp]).  *)
(* HINT: For a helpful fact about sets of atoms, check AtomSetImpl.union_1 *)
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(** * An abstract machine for cbn evaluation                 *)
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(** The advantage of named terms is an efficient operational semantics! Compared to LN or de Bruijn representation, we don't need always need to modify terms (such as via open or shifting) as we traverse binders. Instead, as long as the binder is "sufficiently fresh" we can use the name as is, and only rename (via swapping) when absolutely necessary. *)
(** Below, we define an operational semantics based on an abstract machine. As before, it will model execution as a sequence of small-step transitions. However, transition will be between _machine configurations_, not just expressions.  Our abstract machine configurations have three components:  - the current expression being evaluated the heap (aka - environment): a mapping between variables and expressions - the stack: the evaluation context of the current - expression  Because we have a heap, we don't need to use substitution to define our semantics! To implement [x ~> e] we add a definition that [x] maps to [e] in the heap and then look up the definition when we get to [x] during evaluation.  *)
Definition heap := list (atom * n_exp).
Inductive frame : Set := | n_app2 : n_exp -> frame.
Notation  stack := (list frame).
Definition configuration := (heap * n_exp * stack)%type.
(** The (small-step) semantics is a _function_ from configurations to configurations, until completion or error. *)
Inductive Step a := Error    : Step a
| Done     : Step a
| TakeStep : a -> Step a.
Definition isVal (t : n_exp) :=
match t with
| n_abs _ _ => true
| _         => false
end.
Definition machine_step (avoid : atoms) (c : configuration) : Step configuration :=
match c with
(h, t, s) =>
if isVal t then
match s with
| nil => Done _
| n_app2 t2 :: s' =>
match t with
| n_abs x t1 =>
(* if the bound name [x] is not fresh, we need to rename it *)
if AtomSetProperties.In_dec x (dom h `union` avoid)  then
let (y,_) := atom_fresh (dom h `union` avoid) in
TakeStep _ ((y,t2)::h, swap x y t1, s')
else
TakeStep _ ((x,t2)::h, t1, s')
| _ => Error _ (* non-function applied to argument *)
end
end
else match t with
| n_var x => match get x h with
| Some t1  => TakeStep _ (h, t1, s)
| None    => Error _ (* Unbound variable *)
end
| n_app t1 t2 => TakeStep _ (h, t1, n_app2 t2 :: s)
| _ => Error _ (* unreachable (value in nonValueStep) *)
end
end.
Definition initconf (t : n_exp) : configuration := (nil,t,nil).
(** Example: evaluation of  "(\y. (\x. x) y) Z"  << machine                                       corresponding term  {}, (\y. (\x. x) y) Z , nil                  (\y. (\x. x) y) Z ==> {}, (\y. (\x. x) y), n_app2 Z :: nil         (\y. (\x. x) y) Z ==> {y -> Z}, (\x.x) y, nil                      (\x. x) Z ==> {y -> Z}, (\x.x), n_app2 y :: nil            (\x. x) Z ==> {x -> y, y -> Z}, x, nil                     Z ==> {x -> y, y -> Z}, y, nil                     Z ==> {x -> y, y -> Z}, Z, nil                     Z ==> Done >>  (Note that the machine takes extra steps compared to the substitution semantics.)  We will prove that this machine implements the substitution semantics in the next section.  *)
(** ** Recommended Exercise [values_are_done]  Show that values don't step using this abstract machine. (This is a simple proof.) *)
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(** * Size based reasoning                                   *)
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(** Some properties about nominal terms require calling the induction hypothesis not on a direct subterm, but on one that has first had a swapping applied to it.  However, swapping names does not change the size of terms, so that means we can prove such properties by induction on that size.  *)
Fixpoint size (t : n_exp) : nat :=
match t with
| n_var x => 1
| n_abs x t => 1 + size t
| n_app t1 t2 => 1 + size t1 + size t2
end.
Hint Rewrite swap_size_eq : core.
(* HIDE *)
(** ** Nominal induction *)
(** ** Challenge Exercise [subst]  Use the definitions above to prove the following results about the nominal substitution function.  *)
(* HIDE *)
Fixpoint aeq_f  (n: nat) (L : atoms) (t1 : n_exp) (t2 : n_exp) : bool :=
match n with
| 0 => false
| S m =>
match t1 , t2 with
| n_var x , n_var y => if (x == y) then true else false
| n_abs x1 t1, n_abs x2 t2 =>
if (x1 == x2) then aeq_f m L t1 t2
else let (y,_) := atom_fresh L in
aeq_f m ({{y}} \u L) (swap y x1 t1) (swap y x2 t2)
| n_app t1 t2 , n_app t1' t2' =>
if aeq_f m L t1 t1' then aeq_f m L t2 t2' else false
| _ , _ => false
end
end.
Definition is_aeq t1 t2 := aeq_f (size t1) (fv_nom t1 \u fv_nom t2).
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Lemma subst_equivariance : forall u x y z t,
swap x y (subst u z t) = subst (swap x y u) (swap_var x y z) (swap x y t).