floating-levels/theories/properties.v

Require Import Autosubst2.syntax
               Autosubst2.core Autosubst2.unscoped typing ssreflect List ssrbool.

From Hammer Require Import Tactics.

Section ren.

  Variables (T : Set).

  Definition ren_ok (ξ : nat -> nat) (Γ Δ : list T) :=
    forall i A, lookup i Δ A -> lookup (ξ i) Γ A.

  Lemma ren_comp ξ ψ (Γ Δ : list T) Ξ :
    ren_ok ξ Γ Δ ->
    ren_ok ψ Δ Ξ ->
    ren_ok (funcomp ξ ψ) Γ Ξ.
  Proof using. sfirstorder unfold:ren_ok. Qed.

  Lemma ren_shift (A : T) Γ :
    ren_ok S (A :: Γ) Γ.
  Proof using. hauto lq:on ctrs:lookup inv:lookup unfold:ren_ok. Qed.

  Lemma ren_ext i ξ Γ Δ (A : T) :
    ren_ok ξ Γ Δ ->
    lookup i Γ A ->
    ren_ok (scons i ξ) Γ (A :: Δ).
  Proof using. qauto l:on unfold:ren_ok inv:lookup. Qed.

  Lemma ren_up ξ (A : T) Γ Δ :
    ren_ok ξ Γ Δ ->
    ren_ok (upRen_Tm_Tm ξ) (A :: Γ) (A :: Δ).
  Proof using.
    rewrite /upRen_Tm_Tm.
    move => h. apply ren_ext. apply : ren_comp; eauto. apply ren_shift.
    apply here.
  Qed.

End ren.

Arguments ren_ok {T}.

Definition bool_leb a b :=
  match a,b with
  | true, false => false
  | _,_ => true
  end.

Lemma bool_leb_refl a : bool_leb a a.
Proof. case : a => //=. Qed.

Lemma bool_le_spec a b : Bool.reflect (Bool.le a b) (bool_leb a b).
Proof. hauto l:on inv:bool ctrs:Bool.reflect. Qed.

Module Liftable.
  Lemma lookup_fixed i Γ ℓ A :
    lookup i Γ (ℓ, A) ->
    lookup i (fixed_ctx Γ) (squash ℓ).
  Proof.
    elim : Γ i ℓ A => //=.
    - hauto lq:on inv:lookup.
    - hauto lq:on inv:lookup ctrs:lookup.
  Qed.

  Lemma lookup_fixed' i Γ ℓ :
    lookup i (fixed_ctx Γ) ℓ ->
    exists A ℓ0, squash ℓ0 = ℓ /\ lookup i Γ (ℓ0, A).
  Proof.
    elim : Γ i ℓ => //=.
    - hauto lq:on inv:lookup.
    - hauto lq:on inv:lookup ctrs:lookup.
  Qed.

  Lemma fixed_ren_ok ξ Γ Δ :
    ren_ok ξ Γ Δ ->
    ren_ok ξ (fixed_ctx Γ) (fixed_ctx Δ).
  Proof.
    rewrite /ren_ok.
    move => h i ℓ / lookup_fixed' [A][ℓ0][?]hi. subst.
    apply : lookup_fixed; eauto.
  Qed.

  Lemma renaming ξ Φ Ψ a :
    liftable Ψ a ->
    ren_ok ξ Φ Ψ ->
    liftable Φ (ren_Tm ξ a).
  Proof.
    elim : a ξ Φ Ψ => //=; hauto l:on use:ren_up unfold:ren_ok.
  Qed.

  Fixpoint ctx_leq Φ Ψ :=
    match Φ, Ψ with
    | nil, nil => true
    | ℓ::Φ, ℓ'::Ψ => bool_leb ℓ ℓ' && ctx_leq Φ Ψ
    | _, _ => false
    end.

  Lemma ctx_leq_lookup Φ Ψ i ℓ1 :
    ctx_leq Φ Ψ ->
    lookup i Ψ ℓ1 ->
    exists ℓ0, lookup i Φ ℓ0 /\
    Bool.le ℓ0 ℓ1.
  Proof.
    move => + h. move : Φ.
    elim : i Ψ ℓ1 / h.
    - move => A Γ Ψ h.
      case : Ψ h => //=.
      move => B Ψ.
      case : bool_le_spec => //=.
      sfirstorder ctrs:lookup.
    - move => i Γ A B hi ihi Ψ.
      case : Ψ => //=.
      move => C Ψ.
      case : bool_le_spec => //=.
      hauto lq:on ctrs:lookup.
  Qed.

  Lemma narrowing Φ Ψ a :
    ctx_leq Φ Ψ ->
    liftable Ψ a ->
    liftable Φ a.
  Proof.
    elim : a Φ Ψ => //=.
    - move => n Φ Ψ.
      move : ctx_leq_lookup; repeat move/[apply].
      move => [ℓ1 [h0 h1]].
      suff : ℓ1 = false by congruence.
      hauto q:on inv:bool.
    - sfirstorder.
    - sfirstorder.
    - sfirstorder.
  Qed.

End Liftable.


Definition lvl_leb a b :=
  match a, b with
  | Fixed ℓ0, Fixed ℓ1 => Bool.eqb ℓ0 ℓ1
  | Floating ℓ0, Floating ℓ1 => bool_leb ℓ0 ℓ1
  | _,_ => false
  end.

Lemma lvl_leb_refl a : lvl_leb a a.
Proof. destruct a,b; sfirstorder use:Bool.eqb_reflx, bool_leb_refl. Qed.

Module Typing.

  Inductive ctx_fleq : list bool -> ctx -> ctx -> Prop :=
  | FL_Nil : ctx_fleq nil nil nil
  | FL_ConsFixed Φ Γ Δ A :
    ctx_fleq Φ Γ Δ ->
    ctx_fleq (false :: Φ) (A :: Γ) (A :: Δ)
  | FL_ConsFloat Φ Γ Δ ℓ0 ℓ1 A :
    ctx_fleq Φ Γ Δ ->
    lvl_leb ℓ0 ℓ1 ->
    ctx_fleq (true :: Φ) ((ℓ0, A) :: Γ) ((ℓ1, A) :: Δ).

  Lemma ctx_fleq_fixed Φ Γ Δ :
    ctx_fleq Φ Γ Δ ->
    ctx_fleq (fixed_ctx Γ) Γ Δ.
  Proof.
    move => h. elim : Φ Γ Δ/ h => //=.
    - constructor.
    - move => Φ Γ Δ [ℓ B].
      move => h0 h1.
      destruct ℓ => //=; constructor; eauto using lvl_leb_refl.
    - move => Φ Γ Δ ℓ0 ℓ1 A hctx ihctx h0.
      destruct ℓ0, ℓ1 => //=.
      simpl in h0.
      move : h0. case : Bool.eqb_spec => //= ? _. subst.
      by constructor.
      by constructor.
  Qed.

  Lemma renaming ξ Γ Δ ℓ a A  :
    Δ ⊢ a ;; ℓ  ∈ A  ->
    ren_ok ξ Γ Δ ->
    Γ ⊢ ren_Tm ξ a ;; ℓ ∈  A.
  Proof.
    move => h. move : ξ Γ.
    elim : Δ a ℓ A / h.
    - hauto lq:on ctrs:Wt unfold:ren_ok.
    - hauto lq:on ctrs:Wt unfold:ren_ok.
    - hauto lq:on ctrs:Wt use:ren_up.
    - move => Γ ℓ ℓ0 a b A B hb ihb ha iha hl ξ Δ hξ.
      have ? : liftable (fixed_ctx Δ) (ren_Tm ξ a) by
        qauto l:on use:Liftable.renaming, Liftable.fixed_ren_ok.
      hauto lq:on ctrs:Wt.
    - hauto lq:on ctrs:Wt use:ren_up.
  Qed.

  Lemma ctx_fleq_lookup i Φ (Γ Δ  : ctx) A  :
    ctx_fleq Φ Γ Δ ->
    lookup i Φ false ->
    lookup i Γ A ->
    lookup i Δ A.
  Proof.
    move => h. move : A i.
    elim : Φ Γ Δ / h => //=; sauto.
  Qed.

  Lemma ctx_fleq_fixed_eq Φ Γ Δ :
    ctx_fleq Φ Γ Δ ->
    fixed_ctx Γ = fixed_ctx Δ.
  Proof.
    move => h. elim : Φ Γ Δ / h => //=.
    - hauto lq:on.
    - move => Φ Γ Δ ℓ0 ℓ1 h ih h0 h1.
      f_equal => //.
      move {h0 ih h}.
      hauto lq:on inv:Level.
  Qed.

  Lemma upgrading Φ Γ Δ a ℓ A :
    Γ ⊢ a ;; ℓ ∈ A ->
    liftable Φ a ->
    ctx_fleq Φ Γ Δ ->
    Δ ⊢ a ;; ℓ ∈ A.
  Proof.
    move => h. move : Φ Δ.
    elim : Γ a ℓ A /h.
    - move => Γ ℓ i ℓ0 A hℓ hi Φ Δ /= h0 h1.
      econstructor; eauto.
      eauto using ctx_fleq_lookup.
    - hauto lq:on ctrs:Wt.
    - move => Γ ℓ a A B ha iha Φ Δ hla hleq /=.
      constructor. apply : iha => /=; cycle 1; eauto.
      by constructor.
    - move => Γ ℓ ℓ0 a b A B hb ihb ha iha hla Φ Δ /= hlb hctx.
      apply : T_App; eauto.
      apply : iha; eauto.
      sfirstorder use:ctx_fleq_fixed.
      erewrite <- ctx_fleq_fixed_eq; eauto.
    - move => Γ ℓ ℓ0 b A B hb ihb Δ /=.
      move => h0 h1. constructor. apply : ihb; eauto.
      simpl. by constructor.
  Qed.

End Typing.