Library Lambda_Infrastructure
Require Import Metatheory Lambda_Definitions.
Computing free variables of a term
Fixpoint fv (t : trm) {struct t} : vars :=
match t with
| trm_bvar i => {}
| trm_fvar x => {{x}}
| trm_app t1 t2 => (fv t1) \u (fv t2)
| trm_abs t1 => (fv t1)
end.
Abstracting a name out of a term
Fixpoint close_var_rec (k : nat) (z : var) (t : trm) {struct t} : trm :=
match t with
| trm_bvar i => trm_bvar i
| trm_fvar x => if x == z then (trm_bvar k) else (trm_fvar x)
| trm_app t1 t2 => trm_app (close_var_rec k z t1) (close_var_rec k z t2)
| trm_abs t1 => trm_abs (close_var_rec (S k) z t1)
end.
Definition close_var z t := close_var_rec 0 z t.
Substitution for a name
Fixpoint subst (z : var) (u : trm) (t : trm) {struct t} : trm :=
match t with
| trm_bvar i => trm_bvar i
| trm_fvar x => if x == z then u else (trm_fvar x)
| trm_app t1 t2 => trm_app (subst z u t1) (subst z u t2)
| trm_abs t1 => trm_abs (subst z u t1)
end.
Notation "[ z ~> u ] t" := (subst z u t) (at level 68).
Definition of parallel relation
Inductive para : relation :=
| para_red : forall L t1 t1' t2 t2',
(forall x, x \notin L -> para (t1 ^ x) (t1' ^ x) ) ->
para t2 t2' ->
para (trm_app (trm_abs t1) t2) (t1' ^^ t2')
| para_var : forall x,
para (trm_fvar x) (trm_fvar x)
| para_app : forall t1 t1' t2 t2',
para t1 t1' -> para t2 t2' ->
para (trm_app t1 t2) (trm_app t1' t2')
| para_abs : forall L t1 t1',
(forall x, x \notin L -> para (t1 ^ x) (t1' ^ x) ) ->
para (trm_abs t1) (trm_abs t1').
Definition of the transitive closure of a relation
Inductive iter_ (R : relation) : relation :=
| iter_trans : forall t2 t1 t3,
iter_ R t1 t2 -> iter_ R t2 t3 -> iter_ R t1 t3
| iter_step : forall t t',
R t t' -> iter_ R t t'.
Notation "R 'iter'" := (iter_ R) (at level 69).
Inclusion between relations (simulation or a relation by another)
Definition simulated (R1 R2 : relation) :=
forall (t t' : trm), R1 t t' -> R2 t t'.
Infix "simulated_by" := simulated (at level 69).
Properties of relations
Definition red_regular (R : relation) :=
forall t t', R t t' -> term t /\ term t'.
Definition red_refl (R : relation) :=
forall t, term t -> R t t.
Definition red_in (R : relation) :=
forall t x u u', term t -> R u u' ->
R ([x ~> u]t) ([x ~> u']t).
Definition red_all (R : relation) :=
forall x t t', R t t' ->
forall u u', R u u' ->
R ([x~>u]t) ([x~>u']t').
Definition red_out (R : relation) :=
forall x u t t', term u -> R t t' ->
R ([x~>u]t) ([x~>u]t').
Definition red_rename (R : relation) :=
forall x t t' y,
x \notin (fv t) -> x \notin (fv t') ->
R (t ^ x) (t' ^ x) ->
R (t ^ y) (t' ^ y).
Definition red_through (R : relation) :=
forall x t1 t2 u1 u2,
x \notin (fv t1) -> x \notin (fv u1) ->
R (t1 ^ x) (u1 ^ x) -> R t2 u2 ->
R (t1 ^^ t2) (u1 ^^ u2).
Ltac gather_vars :=
let A := gather_vars_with (fun x : vars => x) in
let B := gather_vars_with (fun x : var => {{ x }}) in
let C := gather_vars_with (fun x : trm => fv x) in
constr:(A \u B \u C).
Ltac pick_fresh X :=
let L := gather_vars in (pick_fresh_gen L X).
Tactic Notation "apply_fresh" constr(T) "as" ident(x) :=
apply_fresh_base T gather_vars x.
Tactic Notation "apply_fresh" "*" constr(T) "as" ident(x) :=
apply_fresh T as x; auto*.
Hint Constructors term.
Properties of substitutions
Substitution for a fresh name is identity.
Lemma subst_fresh : forall x t u,
x \notin fv t -> [x ~> u] t = t.
Substitution distributes on the open operation.
Lemma subst_open : forall x u t1 t2, term u ->
[x ~> u] (t1 ^^ t2) = ([x ~> u]t1) ^^ ([x ~> u]t2).
Substitution and open_var for distinct names commute.
Lemma subst_open_var : forall x y u e, y <> x -> term u ->
([x ~> u]e) ^ y = [x ~> u] (e ^ y).
Opening up an abstraction of body t with a term u is the same as opening
up the abstraction with a fresh name x and then substituting u for x.
Lemma subst_intro : forall x t u,
x \notin (fv t) -> term u ->
t ^^ u = [x ~> u](t ^ x).
Tactic to permute subst and open var
Ltac cross :=
rewrite subst_open_var; try cross.
Tactic Notation "cross" "*" :=
cross; auto*.
Terms are stable by substitution
Lemma subst_term : forall t z u,
term u -> term t -> term ([z ~> u]t).
Lemma subst_body : forall z u t,
body t -> term u -> body ([z ~> u]t).
Hint Resolve subst_term subst_body.
Conversion from locally closed abstractions and bodies
Lemma term_abs_to_body : forall t1,
term (trm_abs t1) -> body t1.
Lemma body_to_term_abs : forall t1,
body t1 -> term (trm_abs t1).
Hint Resolve term_abs_to_body body_to_term_abs.
Lemma open_term : forall t u,
body t -> term u -> term (t ^^ u).
Hint Resolve open_term.
Open_var with fresh names is an injective operation
Lemma open_var_inj : forall x t1 t2,
x \notin (fv t1) -> x \notin (fv t2) ->
(t1 ^ x = t2 ^ x) -> (t1 = t2).
Close var is an operation returning a body of an abstraction
Lemma close_var_fresh : forall x t,
x \notin fv (close_var x t).
Close var is an operation returning a body of an abstraction
Lemma close_var_body : forall x t,
term t -> body (close_var x t).
Close var is the right inverse of open_var
Lemma close_var_open : forall x t,
term t -> t = (close_var x t) ^ x.
An abstract specification of close_var, which packages the
result of the operation and all the properties about it.
Lemma close_var_spec : forall t x, term t ->
exists u, t = u ^ x /\ body u /\ x \notin (fv u).
Regularity: relations only hold on well-formed terms
Section TermRelations.
Hint Extern 1 (term (trm_abs ?t)) =>
match goal with H: context [term (t ^ _) ] |- _ =>
let y := fresh in let K := fresh in
apply_fresh term_abs as y;
inst_notin H y as K; destruct K; auto end.
Lemma red_regular_beta : red_regular beta.
Lemma red_regular_beta_star : red_regular (beta star).
Lemma red_regular_para : red_regular para.
Lemma red_regular_para_iter : red_regular (para iter).
End TermRelations.
Hint Resolve red_regular_beta red_regular_beta_star red_regular_para red_regular_para_iter.
Hint Extern 1 (term ?t) => match goal with
| H: beta t _ |- _ => apply (proj1 (red_regular_beta H))
| H: beta _ t |- _ => apply (proj2 (red_regular_beta H))
| H: para t _ |- _ => apply (proj1 (red_regular_para H))
| H: para _ t |- _ => apply (proj2 (red_regular_para H))
| H: (beta star) t _ |- _ => apply (proj1 (red_regular_beta_star H))
| H: (beta star) _ t |- _ => apply (proj2 (red_regular_beta_star H))
| H: (para iter) t _ |- _ => apply (proj1 (red_regular_para_iter))
| H: (para iter) _ t |- _ => apply (proj2 (red_regular_para_iter))
end.
Hint Extern 1 (body ?t) =>
match goal with H: context [?R (t ^ _) _ ] |- _ =>
let y := fresh in apply term_abs_to_body;
apply_fresh term_abs as y;
inst_notin H y as K; clear H end.