(* This module defines an equivalence relation
 * on type-terms.
 *
 * In general, we want distributivity of ladders
 * over all other type constructors, e.g. type-
 * specializations or function-types.
 *
 * Formally, we define two rewrite rules that
 * shall define how we are able to rewrite type-terms
 * such that they remain equivalent.
 *
 * (1) `-->distribute-ladder` which distributes
 *    ladder types over other type constructors, e.g.
 *    <S T~U>  -->distribute-ladder   <S T> ~ <S U>.
 *
 * (2) `-->condense-ladder` which does the
 *    inverse operation, e.g.
 *    <S T> ~ <S U>  -->condense-ladder  <S T~U>
 *
 * When applied exhaustively, each rewrite rule yields
 * a normal form, namely *Ladder Normal Form (LNF)*, which
 * is reached by exhaustive application of `-->distribute-ladder`,
 * and *Parameter Normal Form (PNF)*, which is reached by exhaustive
 * application of `-->condense-ladder`.
 *
 * The equivalence relation `===` over type-terms is then defined
 * as the reflexive and transitive hull over `-->distribute-ladder`
 * and `-->condense-ladder`. Since the latter two are the inverse
 * rewrite-step of each other, `===` is symmetric and thus `===`
 * satisfies all properties required of an equivalence relation.
 *)

Require Import terms.
Require Import subst.
From Coq Require Import Strings.String.

Include Terms.
Include Subst.

Module Equiv.


(** Alpha conversion in types *)

Reserved Notation "S '--->α' T" (at level 40).
Inductive type_conv_alpha : type_term -> type_term -> Prop :=
  | TAlpha_Rename : forall x y t t',

  (type_subst1 x (type_var y) t t') ->
  (type_univ x t) --->α (type_univ y t')

  | TAlpha_SubUniv : forall x τ τ',
    (τ --->α τ') ->
    (type_univ x τ) --->α (type_univ x τ')

  | TAlpha_SubSpec1 : forall τ1 τ1' τ2,
    (τ1 --->α τ1') ->
    (type_spec τ1 τ2) --->α (type_spec τ1' τ2)

  | TAlpha_SubSpec2 : forall τ1 τ2 τ2',
    (τ2 --->α τ2') ->
    (type_spec τ1 τ2) --->α (type_spec τ1 τ2')

  | TAlpha_SubFun1 : forall τ1 τ1' τ2,
    (τ1 --->α τ1') ->
    (type_fun τ1 τ2) --->α (type_fun τ1' τ2)

  | TAlpha_SubFun2 : forall τ1 τ2 τ2',
    (τ2 --->α τ2') ->
    (type_fun τ1 τ2) --->α (type_fun τ1 τ2')

  | TAlpha_SubMorph1 : forall τ1 τ1' τ2,
    (τ1 --->α τ1') ->
    (type_morph τ1 τ2) --->α (type_morph τ1' τ2)

  | TAlpha_SubMorph2 : forall τ1 τ2 τ2',
    (τ2 --->α τ2') ->
    (type_morph τ1 τ2) --->α (type_morph τ1 τ2')

  | TAlpha_SubLadder1 : forall τ1 τ1' τ2,
    (τ1 --->α τ1') ->
    (type_ladder τ1 τ2) --->α (type_ladder τ1' τ2)

  | TAlpha_SubLadder2 : forall τ1 τ2 τ2',
    (τ2 --->α τ2') ->
    (type_ladder τ1 τ2) --->α (type_ladder τ1 τ2')

where "S '--->α' T" := (type_conv_alpha S T).

(** Alpha conversion is symmetric *)
Lemma type_alpha_symm :
  forall σ τ,
  (σ --->α τ) -> (τ --->α σ).
Proof.
 (* TODO *)
Admitted.


(** Define all rewrite steps $\label{coq:type-dist}$ *)

Reserved Notation "S '-->distribute-ladder' T" (at level 40).
Inductive type_distribute_ladder : type_term -> type_term -> Prop :=
  | L_DistributeOverSpec1 : forall x x' y,
    (type_spec (type_ladder x x') y)
    -->distribute-ladder
    (type_ladder (type_spec x y) (type_spec x' y))

  | L_DistributeOverSpec2 : forall x y y',
    (type_spec x (type_ladder y y'))
    -->distribute-ladder
    (type_ladder (type_spec x y) (type_spec x y'))

  | L_DistributeOverFun1 : forall x x' y,
    (type_fun (type_ladder x x') y)
    -->distribute-ladder
    (type_ladder (type_fun x y) (type_fun x' y))

  | L_DistributeOverFun2 : forall x y y',
    (type_fun x (type_ladder y y'))
    -->distribute-ladder
    (type_ladder (type_fun x y) (type_fun x y'))

  | L_DistributeOverMorph1 : forall x x' y,
    (type_morph (type_ladder x x') y)
    -->distribute-ladder
    (type_ladder (type_morph x y) (type_morph x' y))

  | L_DistributeOverMorph2 : forall x y y',
    (type_morph x (type_ladder y y'))
    -->distribute-ladder
    (type_ladder (type_morph x y) (type_morph x y'))

where "S '-->distribute-ladder' T" := (type_distribute_ladder S T).



Reserved Notation "S '-->condense-ladder' T" (at level 40).
Inductive type_condense_ladder : type_term -> type_term -> Prop :=
  | L_CondenseOverSpec1 : forall x x' y,
    (type_ladder (type_spec x y) (type_spec x' y))
    -->condense-ladder
    (type_spec (type_ladder x x') y)

  | L_CondenseOverSpec2 : forall x y y',
    (type_ladder (type_spec x y) (type_spec x y'))
    -->condense-ladder
    (type_spec x (type_ladder y y'))

  | L_CondenseOverFun1 : forall x x' y,
    (type_ladder (type_fun x y) (type_fun x' y))
    -->condense-ladder
    (type_fun (type_ladder x x') y)

  | L_CondenseOverFun2 : forall x y y',
    (type_ladder (type_fun x y) (type_fun x y'))
    -->condense-ladder
    (type_fun x (type_ladder y y'))

  | L_CondenseOverMorph1 : forall x x' y,
    (type_ladder (type_morph x y) (type_morph x' y))
    -->condense-ladder
    (type_morph (type_ladder x x') y)

  | L_CondenseOverMorph2 : forall x y y',
    (type_ladder (type_morph x y) (type_morph x y'))
    -->condense-ladder
    (type_morph x (type_ladder y y'))

where "S '-->condense-ladder' T" := (type_condense_ladder S T).


(** Inversion Lemma:
   `-->distribute-ladder` is the inverse of `-->condense-ladder
*)
Lemma distribute_inverse :
  forall x y,
  x -->distribute-ladder y ->
  y -->condense-ladder x.
Proof.
  intros.
  destruct H.
  apply L_CondenseOverSpec1.
  apply L_CondenseOverSpec2.
  apply L_CondenseOverFun1.
  apply L_CondenseOverFun2.
  apply L_CondenseOverMorph1.
  apply L_CondenseOverMorph2.
Qed.

(** Inversion Lemma:
    `-->condense-ladder`  is the inverse of  `-->distribute-ladder`
*)
Lemma condense_inverse :
  forall x y,
  x -->condense-ladder y ->
  y -->distribute-ladder x.
Proof.
  intros.
  destruct H.
  apply L_DistributeOverSpec1.
  apply L_DistributeOverSpec2.
  apply L_DistributeOverFun1.
  apply L_DistributeOverFun2.
  apply L_DistributeOverMorph1.
  apply L_DistributeOverMorph2.
Qed.


(** Define the equivalence relation as reflexive, transitive hull. $\label{coq:type-equiv}$ *)

Reserved Notation " S '===' T " (at level 40).
Inductive type_eq : type_term -> type_term -> Prop :=
  | TEq_Refl : forall x,
    x === x

  | TEq_Trans : forall x y z,
    x === y ->
    y === z ->
    x === z

  | TEq_Alpha : forall x y,
    x --->α y ->
    x === y

  | TEq_Distribute : forall x y,
    x -->distribute-ladder y ->
    x === y

  | TEq_Condense : forall x y,
    x -->condense-ladder y ->
    x === y

where "S '===' T" := (type_eq S T).



(** Symmetry of === *)
Lemma TEq_Symm :
  forall x y,
  (x === y) -> (y === x).
Proof.
  intros.
  induction H.
  
  apply TEq_Refl.

  apply TEq_Trans with (y:=y).
  apply IHtype_eq2.
  apply IHtype_eq1.

  apply type_alpha_symm in H.
  apply TEq_Alpha.
  apply H.

  apply TEq_Condense.
  apply distribute_inverse.
  apply H.

  apply TEq_Distribute.
  apply condense_inverse.
  apply H.
Qed.

(** "flat" types do not contain ladders $\label{coq:type-flat}$ *)
Inductive type_is_flat : type_term -> Prop :=
  | FlatVar : forall x, (type_is_flat (type_var x))
  | FlatId : forall x, (type_is_flat (type_id x))
  | FlatApp : forall x y,
    (type_is_flat x) ->
    (type_is_flat y) ->
    (type_is_flat (type_spec x y))

  | FlatFun : forall x y,
    (type_is_flat x) ->
    (type_is_flat y) ->
    (type_is_flat (type_fun x y))

  | FlatSub : forall x v,
    (type_is_flat x) ->
    (type_is_flat (type_univ v x))
.

(** Ladder Normal Form:
    exhaustive application of -->distribute-ladder
 *)
Inductive type_is_lnf : type_term -> Prop :=
  | LNF : forall x : type_term,
  (not (exists y:type_term, x -->distribute-ladder y))
  -> (type_is_lnf x)
.

(** Parameter Normal Form:
    exhaustive application of -->condense-ladder
*)
Inductive type_is_pnf : type_term -> Prop :=
  | PNF : forall x : type_term,
  (not (exists y:type_term, x -->condense-ladder y))
  -> (type_is_pnf x)
.

(** Any term in LNF either is flat, or is a ladder T1~T2 where T1 is flat and T2 is in LNF *)
Lemma lnf_shape :
  forall τ:type_term,
  (type_is_lnf τ) -> ((type_is_flat τ) \/ (exists τ1 τ2, τ = (type_ladder τ1 τ2) /\ (type_is_flat τ1) /\ (type_is_lnf τ2)))
.
Proof.
  intros τ H.
  induction τ.

  left.
  apply FlatId.
  
  left.
  apply FlatVar.
(*
  left.
  apply IHτ1 in H.
  apply FlatFun.
  apply H.
  destruct H.
  destruct H.
  
  apply IHτ1.
*)
  admit.
  admit.
  admit.
  admit.
Admitted.

(** every type has an equivalent LNF *)
Lemma lnf_normalizing :
  forall t, exists t', (t === t') /\ (type_is_lnf t').
Proof.
  intros.
  destruct t.

  exists (type_id s).
  split. apply TEq_Refl.
  apply LNF.
  admit.
  admit.
  admit.
  
  exists (type_univ s t).
  split.
  apply TEq_Refl.
  apply LNF.
  
Admitted.

(** every type has an equivalent PNF *)
Lemma pnf_normalizing :
  forall t, exists t', (t === t') /\ (type_is_pnf t').
Proof.
Admitted.

Lemma lnf_uniqueness :
  forall t t' t'',
  (t === t') /\ (type_is_lnf t') /\ (t === t'') /\ (type_is_lnf t'')
  -> t = t'.
Proof.
Admitted.





(**
 *  Examples
 *)

Example example_flat_type :
    (type_is_flat (type_spec (type_id "PosInt") (type_id "10"))).
Proof.
    apply FlatApp.
    apply FlatId.
    apply FlatId.
Qed.

Example example_lnf_type :
    (type_is_lnf
    (type_ladder (type_spec (type_id "Seq") (type_id "Char"))
               (type_spec (type_id "Seq") (type_id "Byte")))).
Proof.
Admitted.

Example example_type_eq :
    (type_spec (type_id "Seq") (type_ladder (type_id "Char") (type_id "Byte")))
    ===
    (type_ladder (type_spec (type_id "Seq") (type_id "Char"))
               (type_spec (type_id "Seq") (type_id "Byte"))).
Proof.
  apply TEq_Distribute.
  apply L_DistributeOverSpec2.
Qed.


End Equiv.