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(**  * Connecting nominal and LN semantics *)
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(** Our final goal is to show that the abstract nominal machine implements the same semantics as the LN substitution-based small step relation.  We'll do this by proving that any time the abstract machine takes a step, we can decode the machine configuration before and after the step to LN expressions, and those expressions are either identical or related by the LN step relation.  << machine_step (h,t,s)  ------>  (h',t',s')            nominal terms |                  | | decode           | decode v                  v e       - - ->     e'           locally nameless terms step e e' or e = e' >>  This result is the [simulate_step] lemma near the end of this file.  *)
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(** ** Translating nominal terms to LN terms *)
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(** We decode named terms to LN terms through the use of the [close_exp_wrt_exp] function. This function replaces all occurrences of a given atom by a new bound variable.  This function is defined in the [Stlc.Lemmas] module and can be automatically generated by LNgen.  *)
Fixpoint nom_to_exp (ne : n_exp) : exp :=
match ne with
| n_var x => var_f x
| n_app e1 e2 => app (nom_to_exp e1) (nom_to_exp e2)
| n_abs x e1 => abs (close_exp_wrt_exp x (nom_to_exp e1))
end.
(** We also define a translation from machine configurations to LN terms. In this case we must substitute all definitions in the heap through the terms and create all of the applications in the stack. *)
Fixpoint apply_heap (h : heap) (e : exp) : exp  :=
match h with
| nil => e
| (x , e') :: h' => apply_heap h' ([x ~> nom_to_exp e'] e)
end.
(* Note that the stack could have some heap definitions in it, so we apply the substitutions in the heap there too. *)
Fixpoint apply_stack h (s : list frame) (e :exp) : exp :=
match s with
| nil => e
| n_app2 e' :: s' => apply_stack h s' (app e (apply_heap h (nom_to_exp e')))
end.
(** The full decode function puts all of these parts together. *)
Definition decode (c:configuration) : exp  :=
match c with
| (h,e,s) => apply_stack h s (apply_heap h (nom_to_exp e))
end.
(** Here's an example translation from the machine step demo. *)
Definition conf1 := ([(Y,n_var Z)], n_abs X (n_var X), [n_app2 (n_var Y)]).
Example decode1 : decode conf1 = (app (abs (var_b 0)) (var_f Z)).
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(** ** Connecting free variable functions.                             *)
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(** Here is the first result about our decoding: that the two free variable functions agree.  In this part of the file, we will take advantage of automation provided by the Lemmas module. In particular, this module defines a database of rewriting hints (called [lngen]) that can be used to automatically rewrite LN terms into simpler form. *)
(** As we prove new lemmas, we can also extend the hint database with new rewritings. *)
Hint Rewrite fv_nom_fv_exp_eq : lngen.
Hint Resolve fv_nom_fv_exp_eq : lngen.
(** The Metatheory library contains two powerful tactics for simplifying goals:  - [default_steps]: repeat a bunch of simplifying steps, such as simplifying the goal, inverting simple hypotheses, etc.  - [default_simp]: above plus case analysis for booleans and other sums  Below, we modify the behavior of these tactics by updating the following two definitions, so that the [lngen] hint databases will be available.  *)
Ltac default_auto        ::= auto with lngen.
Ltac default_autorewrite ::= autorewrite with lngen.
(** We also add a few more rewriting lemmas to the hint database to automate our proofs. *)
Hint Rewrite subst_exp_open_exp_wrt_exp : lngen.
Hint Rewrite swap_size_eq : lngen.
Hint Resolve le_S_n : lngen.
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(** ** Decoded terms are locally closed *)
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(** Next, we show that our decoding of nominal terms and configurations produces locally closed LN terms.  Again, these proofs are highly automatable. *)
Hint Resolve nom_to_exp_lc : lngen.
Hint Resolve apply_heap_lc : lngen.
(** *** Exercise: [apply_stack_lc]  State and prove a lemma called [apply_stack_lc] and add it to the [lngen] hint database so that the proof for [decode_lc] goes through. *)
Hint Resolve apply_stack_lc : lngen.
(* /SOLUTION *)
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(** ** Properties of apply_heap *)
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(** Since the heap is just an iterated substitution, it inherits properties from [subst_exp].  Below, [alist induction] is a tactic from the metatheory library for induction over association lists (such as the heap).  *)
Hint Rewrite apply_heap_abs : lngen.
Hint Rewrite apply_heap_app : lngen.
(** *** Exercise: [apply_heap_open]  This function is the [apply_heap] analogue to [subst_exp_open_exp_wrt_exp], commuting the heap-based substitution with the [open_exp_wrt_exp] operation. *)
Hint Rewrite apply_heap_open : lngen.
(** This last lemma "unsimpl"s the [apply_heap] function. *)
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(** ** Stacks as evaluation contexts                                    *)
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(** Here is a quick lemma that uses the properties defined above. It shows that the stack behaves like an *evaluation context*, lifting a small-step reduction to a larger term. *)
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(** * Connecting "freshening" *)
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(** The abstract machine uses the nominal [swap] operation to make sure that the variables in abstractions are "fresh".  To be able prove that this machine implements the step relation for LN terms, we need to connect this operation to a LN version of "freshening".  The [swap_spec] lemma below states that connection. If a variable [y] is "fresh", then substituting with it (in the LN representation) is the same as swapping with it (in the nominal representation).  << Lemma swap_spec : forall  n w y, y `notin` fv_exp (nom_to_exp n) -> w <> y -> [w ~> var_f y] (nom_to_exp n) = nom_to_exp (swap w y n). >>  *)
(** *** Exercise [close_exp_wrt_exp_freshen] *)

(** One difficulty of [swap_spec] is that we need to use the induction not on direct subterms, but on those that have had a swapping applied to them. Swapping preserves the size of a nominal term, so we can prove this result by induction on [m], a bound on the size of the term.  The difficulty in this proof comes from the [n_abs] case. In this case, the term is is of the form [n_abs x t] for some binding variable [x]. We don't know much about [x]; it could be [w], or [y] or some other variable. The first and last case are straightforward, but the middle case causes difficulty: even though [x] is not free in [n_abs x t], it _is_ free in [t]. Therefore, our induction hypothesis doesn't apply.  To solve this problem, we need to generate a completely fresh variable [z] for the binder, use the lemma above to replace [y] and [w] with it, and then use the induction hypothesis.  *)
Lemma swap_spec_aux : forall m t w y,
size t <= m ->
y `notin` fv_exp (nom_to_exp t) ->
w <> y ->
[w ~> var_f y] (nom_to_exp t) =
nom_to_exp (swap w y t).