Library CoC_Infrastructure

Require Import Metatheory CoC_Definitions.

Additional Definitions Used in the Proofs


Computing free variables of a term

Fixpoint fv (t : trm) {struct t} : vars :=
  match t with
  | trm_bvar i => {}
  | trm_fvar x => {{x}}
  | trm_type n => {}
  | trm_app t1 t2 => (fv t1) \u (fv t2)
  | trm_abs t1 t2 => (fv t1) \u (fv t2)
  | trm_prod t1 t2 => (fv t1) \u (fv t2)
  end.

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_type n => trm_type n
  | trm_app t1 t2 => trm_app (subst z u t1) (subst z u t2)
  | trm_abs t1 t2 => trm_abs (subst z u t1) (subst z u t2)
  | trm_prod t1 t2 => trm_prod (subst z u t1) (subst z u t2)
  end.

Notation "[ z ~> u ] t" := (subst z u t) (at level 68).

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_type n => trm_type n
  | trm_app t1 t2 => trm_app (close_var_rec k z t1) (close_var_rec k z t2)
  | trm_abs t1 t2 => trm_abs (close_var_rec k z t1) (close_var_rec (S k) z t2)
  | trm_prod t1 t2 => trm_prod (close_var_rec k z t1) (close_var_rec (S k) z t2)
  end.

Definition close_var z t := close_var_rec 0 z t.

An environment contains only terms

Definition contains_terms E :=
  forall x U, binds x U E -> term U.

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,
  R (t ^ x) (t' ^ x) ->
  x \notin (fv t) -> x \notin (fv t') ->
  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).

Instanciation of Tactics


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
  let D := gather_vars_with (fun x : env => dom x) in
  constr:(A \u B \u C \u D).

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*.

Ltac exists_fresh :=
  let L := gather_vars_with (fun x : vars => x) in exists L.

Scheme typing_induct := Induction for typing Sort Prop
  with wf_induct := Induction for wf Sort Prop.

Hint Constructors beta star_ equiv_ less typing wf.
Hint Unfold conv.

Lemmas


Properties of substitutions

Section SubstProperties.

Hint Constructors term.


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).

End SubstProperties.

Tactic to permute subst and open var

Ltac cross :=
  rewrite subst_open_var; try cross.

Tactic Notation "cross" "*" :=
  cross; auto*.

Lifting operations to terms

Hint Constructors term.

Terms are stable by substitution

Lemma subst_term : forall t z u,
  term u -> term t -> term ([z ~> u]t).

Terms are stable by open

Lemma open_term : forall t u,
  body t -> term u -> term (t ^^ u).

Hint Resolve subst_term open_term.

Properties of Body

Lemma term_abs1 : forall t2 t1,
  term (trm_abs t1 t2) -> term t1.

Lemma body_abs2 : forall t1 t2,
  term (trm_abs t1 t2) -> body t2.

Lemma body_prod2 : forall t1 t2,
  term (trm_prod t1 t2) -> body t2.

Hint Extern 1 (term ?t) =>
  match goal with H: term (trm_abs t ?t2) |- _ =>
    apply (@term_abs1 t2) end.

Hint Extern 1 (body ?t) =>
  match goal with
  | H: context [trm_abs ?t1 t] |- _ => apply (@body_abs2 t1)
  | H: context [trm_prod ?t1 t] |- _ => apply (@body_prod2 t1)
  | H: context [t ^ _] |- _ =>
    let x := fresh in let L := gather_vars in
    exists L; intros x Fr; destructi (H x); auto
  end.

Lemma term_abs_prove : forall t1 t2,
  term t1 -> body t2 -> term (trm_abs t1 t2).

Lemma term_prod_prove : forall t1 t2,
  term t1 -> body t2 -> term (trm_prod t1 t2).

Hint Resolve term_abs_prove term_prod_prove.

Lemma body_prove : forall L t,
  (forall x, x \notin L -> term (t ^ x)) -> body t.

Hint Extern 1 (body ?t) =>
  match goal with
  | H: forall _, _ \notin ?L -> term (t ^ _) |- _ =>
    apply (@body_prove L)
  end.

Lemma body_subst : forall x t u,
  term u -> body t -> body ([x ~> u]t).

Hint Resolve body_subst.

Additional results on primitive operations


Section PrimProperties.

Hint Constructors 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).

End PrimProperties.

Regularity: relations are restricted to terms


Lemma red_regular_beta : red_regular beta.

Lemma red_regular_beta_star : red_regular (beta star).

Lemma red_regular_conv : red_regular conv.

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: (beta star) t _ |- _ => apply (proj1 (red_regular_beta_star H))
  | H: (beta star) _ t |- _ => apply (proj2 (red_regular_beta_star H))
  | H: conv t _ |- _ => apply (proj1 (red_regular_conv H))
  | H: conv _ t |- _ => apply (proj2 (red_regular_conv H))
  end.

Hint Extern 1 (term (trm_abs ([?x ~> ?u]?t1) ([?x ~> ?u]?t2))) =>
  match goal with H: term (trm_abs t1 t2) |- _ =>
  unsimpl ([x ~> u](trm_abs t1 t2)) end.

Lemma red_regular_less : red_regular less.

Hint Extern 1 (term ?t) => match goal with
  | H: less t _ |- _ => apply (proj1 (red_regular_less H))
  | H: less _ t |- _ => apply (proj2 (red_regular_less H))
  end.

Lemma regular_typing : forall E t T, typing E t T ->
  (wf E /\ ok E /\ contains_terms E /\ term t /\ term T).

Hint Extern 1 (term ?t) => match goal with
  | H: typing _ t _ |- _ => apply (proj32 (proj33 (regular_typing H)))
  | H: typing _ _ t |- _ => apply (proj33 (proj33 (regular_typing H)))
  end.

Lemma ok_from_wf : forall E, wf E -> ok E.

Hint Extern 1 (ok ?E) => match goal with
  | H: wf E |- _ => apply (ok_from_wf H)
  end.

Hint Extern 1 (wf ?E) => match goal with
  | H: typing E _ _ |- _ => apply (proj1 (regular_typing H))
  end.

Lemma term_from_binds_in_wf : forall x E U,
  wf E -> binds x U E -> term U.

Hint Extern 1 (term ?t) => match goal with
  H: binds ?x t ?E |- _ => apply (@term_from_binds_in_wf x E)
  end.

Lemma wf_left : forall E F,
  wf (E & F) -> wf E.

Implicit Arguments wf_left [E F].

Some freshness results


Lemma fv_open_var : forall y x t,
  x <> y -> x \notin fv (t ^ y) -> x \notin fv t.

Lemma typing_fresh : forall E T i x,
  typing E T (trm_type i) -> x # E -> x \notin fv T.

Lemma notin_fv_from_wf : forall E F x V,
  wf (E & x ~ V & F) -> x \notin fv V.

Lemma notin_fv_from_binds : forall x y U E,
  wf E -> binds y U E -> x # E -> x \notin fv U.

Lemma notin_fv_from_binds' : forall E F x V y U,
  wf (E & x ~ V & F) -> binds y U E -> x \notin fv U.

Hint Extern 1 (?x \notin fv ?V) =>
  match goal with H: wf (?E & x ~ V & ?F) |- _ =>
    apply (@notin_fv_from_wf E F) end.

Hint Extern 1 (?x \notin fv ?U) =>
  match goal with H: wf (?E & x ~ ?V & ?F), B: binds ?y U ?E |- _ =>
    apply (@notin_fv_from_binds' E F x V y) end.