ladder-calculus/coq/subst.v

From Coq Require Import Strings.String.
From Coq Require Import Lists.List.
Import ListNotations.
Require Import terms.


Fixpoint type_fv (τ : type_term) {struct τ} : (list string) :=
  match τ with
  | type_id s => []
  | type_var α => [α]
  | type_univ α τ => (remove string_dec α (type_fv τ))
  | type_spec σ τ => (type_fv σ) ++ (type_fv τ)
  | type_fun σ τ => (type_fv σ) ++ (type_fv τ)
  | type_morph σ τ => (type_fv σ) ++ (type_fv τ)
  | type_ladder σ τ => (type_fv σ) ++ (type_fv τ)
  end.

Open Scope ladder_type_scope.
Example ex_type_fv1 :
  (In "T"%string (type_fv [< ∀"U",%"T"% >]))
.
Proof. simpl. left. auto. Qed.

Open Scope ladder_type_scope.
Example ex_type_fv2 :
  ~(In "T"%string (type_fv [< ∀"T",%"T"% >]))
.
Proof. simpl. auto. Qed.

(* scoped variable substitution in type terms $\label{coq:subst-type}$ *)
Fixpoint type_subst (v:string) (n:type_term) (t0:type_term) : type_term :=
  match t0 with
  | type_var name => if (eqb v name) then n else t0
  | type_fun t1 t2 => (type_fun (type_subst v n t1) (type_subst v n t2))
  | type_univ x t => if (eqb v x) then t0 else type_univ x (type_subst v n t)
  | type_spec t1 t2 => (type_spec (type_subst v n t1) (type_subst v n t2))
  | type_ladder t1 t2 => (type_ladder (type_subst v n t1) (type_subst v n t2))
  | t => t
  end.

(*

Inductive type_subst1 (x:string) (σ:type_term) : type_term -> type_term -> Prop :=
  | TSubst_VarReplace :
    (type_subst1 x σ (type_var x) σ)

  | TSubst_VarKeep : forall y,
    (x <> y) ->
    (type_subst1 x σ (type_var y) (type_var y))

  | TSubst_UnivReplace : forall y τ τ',
    ~(x = y) ->
    ~(type_var_free y σ) ->
    (type_subst1 x σ τ τ') ->
    (type_subst1 x σ (type_univ y τ) (type_univ y τ'))

  | TSubst_Id : forall n,
    (type_subst1 x σ (type_id n) (type_id n))

  | TSubst_Spec : forall τ1 τ2 τ1' τ2',
    (type_subst1 x σ τ1 τ1') ->
    (type_subst1 x σ τ2 τ2') ->
    (type_subst1 x σ (type_spec τ1 τ2) (type_spec τ1' τ2'))

  | TSubst_Fun : forall τ1 τ1' τ2 τ2',
    (type_subst1 x σ τ1 τ1') ->
    (type_subst1 x σ τ2 τ2') ->
    (type_subst1 x σ (type_fun τ1 τ2) (type_fun τ1' τ2'))

  | TSubst_Morph : forall τ1 τ1' τ2 τ2',
    (type_subst1 x σ τ1 τ1') ->
    (type_subst1 x σ τ2 τ2') ->
    (type_subst1 x σ (type_morph τ1 τ2) (type_morph τ1' τ2'))

  | TSubst_Ladder : forall τ1 τ1' τ2 τ2',
    (type_subst1 x σ τ1 τ1') ->
    (type_subst1 x σ τ2 τ2') ->
    (type_subst1 x σ (type_ladder τ1 τ2) (type_ladder τ1' τ2'))
.
*)

Lemma type_subst_symm :
  forall x y τ τ',
  ((type_subst x (type_var y) τ) = τ') ->
  ((type_subst y (type_var x) τ') = τ)
.
Proof.
  intros.
  induction H.

  unfold type_subst.
  induction τ.
  reflexivity.

  
Admitted.

Lemma type_subst_fresh :
  forall α τ u,
  ~ (In α (type_fv τ))
  -> (type_subst α u τ) = τ
.
Proof.
  intros.
  unfold type_subst.
  induction τ.
  reflexivity.

  
  unfold eqb.

  admit.
(*
  apply TSubst_Id.
  apply TSubst_VarKeep.
  contradict H.
  rewrite H.
  apply TFree_Var.

  apply TSubst_Fun.
  apply IHτ1.

  contradict H.
  apply TFree_Fun.
  apply H.
  apply 
  *)
Admitted.


(* scoped variable substitution, replaces free occurences of v with n in expression e *)
Fixpoint expr_subst (v:string) (n:expr_term) (e0:expr_term) :=
  match e0 with
  | expr_var name => if (eqb v name) then n else e0
  | expr_ty_abs x e => if (eqb v x) then e0 else expr_ty_abs x (expr_subst v n e)
  | expr_ty_app e t => expr_ty_app (expr_subst v n e) t
  | expr_abs x t e => if (eqb v x) then e0 else expr_abs x t (expr_subst v n e)
  | expr_morph x t e => if (eqb v x) then e0 else expr_morph x t (expr_subst v n e)
  | expr_app e a => expr_app (expr_subst v n e) (expr_subst v n a)
  | expr_let x a e => expr_let x (expr_subst v n a) (expr_subst v n e)
  | expr_ascend t e => expr_ascend t (expr_subst v n e)
  | expr_descend t e => expr_descend t (expr_subst v n e)
  end.

(* replace only type variables in expression *)
Fixpoint expr_specialize (v:string) (n:type_term) (e0:expr_term) :=
  match e0 with
  | expr_ty_app e t => expr_ty_app (expr_specialize v n e) (type_subst v n t)
  | expr_ascend t e => expr_ascend (type_subst v n t) (expr_specialize v n e)
  | expr_descend t e => expr_descend (type_subst v n t) (expr_specialize v n e)
  | e => e
  end.