(**************************************************************)
(*   Copyright Dominique Larchey-Wendling *                 *)
(*                                                            *)
(*                             * Affiliation LORIA -- CNRS  *)
(**************************************************************)
(*      This file is distributed under the terms of the       *)
(*         CeCILL v2 FREE SOFTWARE LICENSE AGREEMENT          *)
(**************************************************************)

Require Import List Arith Lia Max Wellfounded Setoid Eqdep_dec.

From Undecidability.Shared.Libs.DLW.Utils
  Require Import utils_tac utils_list finite.

From Undecidability.Shared.Libs.DLW.Vec
  Require Import pos.

From Undecidability.Shared.Libs.DLW.Wf
  Require Import wf_finite wf_chains.

From Undecidability.TRAKHTENBROT
  Require Import notations fol_ops.

Set Implicit Arguments.

Inductive bt : Set := bt_leaf | bt_node : bt -> bt -> bt.

Local Notation "∅" := bt_leaf. (* Empty set *)
Local Infix "⪧" := bt_node. (* x⪧s is {x} ∪ s *)

Section bt_rect'.

  Variables (P : bt -> Type)
            (HP0 : P ∅)
            (HP1 : forall x t, P t -> P (x⪧t)).

  Theorem bt_rect' t : P t.
  Proof. induction t; auto. Qed.

End bt_rect'.

Fact bt_eq_dec (s t : bt) : { s = t } + { s <> t }.
Proof. decide equality. Qed.

Fact bt_eq_pirr (s t : bt) (H1 H2 : s = t) : H1 = H2.
Proof. apply UIP_dec, bt_eq_dec. Qed.

Fixpoint bt_depth t :=
  match t with
    | ∅ => 0
    | r⪧s => max (S ⌞r⌟) ⌞s⌟
  end
where "⌞ t ⌟" := (bt_depth t).

Inductive bt_equiv : bt -> bt -> Prop :=
    | in_bte_refl : forall s, s ≈ s
    | in_bte_sym : forall s t, s ≈ t
                                       -> t ≈ s
    | in_bte_tran : forall r s t, r ≈ s
                                       -> s ≈ t
                                       -> r ≈ t
    | in_bte_cntr : forall s t, s⪧s⪧t ≈ s⪧t
    | in_bte_comm : forall s t u, s⪧t⪧u ≈ t⪧s⪧u
    | in_bte_cngr : forall s s' t t', s ≈ s'
                                       -> t ≈ t'
                                   -> s⪧t ≈ s'⪧t'
where "s ≈ t" := (bt_equiv s t).

Hint Constructors bt_equiv : core.

Local Notation "s ≉ t" := (~ s ≈ t).
Notation bte_refl := in_bte_refl.
Notation bte_trans := in_bte_tran.

Fact bte_sym x y : x ≈ y <-> y ≈ x.
Proof. split; auto. Qed.

Add Parametric Relation: (bt) (bt_equiv)
      reflexivity proved by bte_refl
      symmetry proved by in_bte_sym
      transitivity proved by bte_trans
as bte_rst.

Add Parametric Morphism: (bt_node) with signature
       (bt_equiv) ==> (bt_equiv) ==> (bt_equiv) as bt_node_congr.
Proof. intros; auto. Qed.

Fact bte_leaf_eq s t : s ≈ t -> (s = ∅ <-> t = ∅).
Proof. induction 1; try tauto; split; discriminate. Qed.

Fact bte_discr s t : s⪧t ≉ ∅.
Proof.
 intros H; apply bte_leaf_eq in H.
 generalize (proj2 H eq_refl); discriminate.
Qed.

Fact bte_inv_0 s : s ≈ ∅ <-> s = ∅.
Proof.
  split.
  + intros H; destruct s; auto.
    apply bte_discr in H; tauto.
  + intros ->; auto.
Qed.

(* s belongs to t if adding s to t does not change t *)

Definition bt_mem s t := s⪧t ≈ t.

Local Infix "∈" := bt_mem.
Local Notation "s ∉ t" := (~ s ∈ t).

Section establishing_membership_inversion.

  (* A restricted definition of membership, not up to equivalence *)

  Reserved Notation "x ⋷ y" (at level 70, no associativity).

  Inductive bt_restr_mem : bt -> bt -> Prop :=
    | in_btrm_0 : forall s t, s ⋷ s⪧t
    | in_btrm_1 : forall s t u, s ⋷ u -> s ⋷ t⪧u
  where "s ⋷ t" := (bt_restr_mem s t).

  Hint Constructors bt_restr_mem : core.

  Fact btrm_inv u s t : u ⋷ s⪧t <-> u = s \/ u ⋷ t.
  Proof.
    split.
    + inversion 1; auto.
    + intros [ <- | ]; constructor; auto.
  Qed.

  Notation "s ≾ t" := (forall u, u ⋷ s -> exists v, v ⋷ t /\ u ≈ v).

  Fact bt_rincl_refl s : s ≾ s.
  Proof. intros u; exists u; auto. Qed.

  Fact bt_rincl_trans r s t : r ≾ s -> s ≾ t -> r ≾ t.
  Proof.
    intros H1 H2 u Hu.
    destruct H1 with (1 := Hu) as (v & Hv & H3).
    destruct H2 with (1 := Hv) as (w & ? & ?).
    exists w; split; auto.
    apply bte_trans with v; auto.
  Qed.

  Hint Resolve bt_rincl_refl bt_rincl_trans : core.

  Lemma bte_rincl s t : s ≈ t -> s ≾ t.
  Proof.
    intros H.
    assert (s ≾ t /\ t ≾ s) as K; [ | apply K ].
    induction H as [ s | s t H IH | r s t H1 [] H2 []
                   | s t | s t u | s s' t t' H1 [H2 H3] H4 [H5 H6] ];
        auto; try tauto.
    + split; apply bt_rincl_trans with s; auto.
    + split; intros u; rewrite btrm_inv; intros [ <- | ]; exists u; auto.
    + split; intros v; rewrite btrm_inv; intros [ <- | ]; exists v; auto;
        revert H; rewrite btrm_inv; intros [ <- | ]; auto.
    + split.
      * intros u; rewrite btrm_inv.
        intros [ <- | ].
        - exists s'; auto.
        - destruct (H5 _ H) as (v & ? & ?); exists v; auto.
      * intros u; rewrite btrm_inv.
        intros [ <- | ].
        - exists s; auto.
        - destruct (H6 _ H) as (v & ? & ?); exists v; auto.
  Qed.

  Fact btrm_btm s t : s ⋷ t -> s ∈ t.
  Proof.
    induction 1 as [ | s t u ]; try (constructor; auto; fail).
    constructor; apply bte_trans with (t⪧s⪧u); auto.
  Qed.

  Hint Resolve btrm_btm : core.

  Fact btm_congr_l s t u : s ≈ t -> s ∈ u -> t ∈ u.
  Proof. intros ? H; apply bte_trans with (2 := H); auto. Qed.

  Fact btm_congr_r s t u : s ≈ t -> u ∈ s -> u ∈ t.
  Proof.
    intros H1 H2.
    apply bte_trans with (2 := H1),
          bte_trans with (2 := H2); auto.
  Qed.

  Fact btm_inv_0 s : s ∈ ∅ <-> False.
  Proof. split; try tauto; apply bte_discr. Qed.

  Fact btm_inv u s t : u ∈ s⪧t <-> u ≈ s \/ u ∈ t.
  Proof.
    split.
    + intros H.
      destruct (@bte_rincl _ _ H u) as (v & H1 & ?); auto.
      revert H1; rewrite btrm_inv; intros [ <- | ]; auto.
      right; apply bte_trans with (v⪧t); auto.
      apply btrm_btm; auto.
    + intros [ H | H ].
      * apply btm_congr_l with s; auto.
      * apply bte_trans with (1 := in_bte_comm _ _ _); auto.
  Qed.

End establishing_membership_inversion.

Add Parametric Morphism: (bt_mem) with signature
       (bt_equiv) ==> (bt_equiv) ==> (iff) as bte_congr.
Proof.
  intros x y H1 a b H2; split; intros H.
  + apply btm_congr_l with (1 := H1),
          btm_congr_r with (1 := H2); auto.
  + rewrite bte_sym in H1.
    rewrite bte_sym in H2.
    apply btm_congr_l with (1 := H1),
          btm_congr_r with (1 := H2); auto.
Qed.

Tactic Notation "btm" "simpl" :=
  repeat match goal with
    | |- context[_ ∈ _ ⪧ _] => rewrite btm_inv; auto; try tauto
    | |- context[_ ∈ ∅] => rewrite btm_inv_0; auto; try tauto
  end.

Theorem btm_finite_t t : { l | forall s, s ∈ t <-> exists s', s ≈ s' /\ In s' l }.
Proof.
  induction t as [ | s (ls & Hs) t (lt & Ht) ].
  + exists nil; intros s; btm simpl; simpl; firstorder.
  + exists (s::lt); intros u; btm simpl; simpl.
    rewrite Ht; split.
    * intros [ H | (s' & H1 & H2) ].
      - exists s; auto.
      - exists s'; auto.
    * intros (s' & H1 & [ H2 | H2 ]); subst; auto.
      right; exists s'; auto.
Qed.

Section bte_depth.

  Opaque max.

  Fact bte_depth_eq s t : s ≈ t -> ⌞s⌟ = ⌞t⌟.
  Proof.
    induction 1; simpl; auto.
    + transitivity ⌞s⌟; auto.
    + rewrite max_assoc, max_idempotent; auto.
    + rewrite max_assoc, (max_comm (S _)), max_assoc; auto.
  Qed.

  Fact btm_depth s t : s ∈ t -> ⌞s⌟ < ⌞t⌟.
  Proof.
    intros H; apply bte_depth_eq in H.
    rewrite <- H; simpl.
    apply lt_le_trans with (1 := lt_n_Sn _).
    apply le_max_l.
  Qed.

  Theorem btm_wf : well_founded (fun s t => s ∈ t).
  Proof.
    apply wf_incl with (1 := btm_depth),
          wf_inverse_image, lt_wf.
  Qed.

End bte_depth.

Definition btree_wf_rect := well_founded_induction_type btm_wf.

Section establishing_decidability.

  Reserved Notation "x ≺ y" (at level 70, no associativity).

  Inductive bt_lt : bt -> bt -> Prop :=
    | in_btlt_0 : forall s t, ∅ ≺ s⪧t
    | in_btlt_1 : forall s s' t t', s ≺ s' -> s⪧t ≺ s'⪧t'
    | in_btlt_2 : forall s t t', t ≺ t' -> s⪧t ≺ s⪧t'
  where "s ≺ t" := (bt_lt s t).

  Hint Constructors bt_lt : core.

  Fact bt_lt_irrefl s : ~ s ≺ s.
  Proof.
    intros H.
    assert (forall t, s ≺ t -> s <> t) as D;
      [ | apply D in H; destruct H; auto ].
    clear H; induction 1 as [ s t | s s' t t' H IH | s t t' H IH ];
      try discriminate; contradict IH; inversion IH; auto.
  Qed.

  Fact bt_lt_trans r s t : r ≺ s -> s ≺ t -> r ≺ t.
  Proof.
    intros H1; revert H1 t.
    induction 1; inversion 1; auto.
  Qed.

  Fact bt_lt_eq_lt_dec s t : { s ≺ t } + { s = t } + { t ≺ s }.
  Proof.
    revert t; induction s as [ | s1 H1 s2 H2 ] using bt_rect; intros [ | t1 t2 ]; auto.
    destruct (H1 t1) as [ [] | ]; auto;
      destruct (H2 t2) as [ [] | ]; subst; auto.
  Qed.

  Fixpoint bt_insert s t : bt :=
    match t with
      | ∅ => s⪧∅
      | t⪧u => match bt_lt_eq_lt_dec s t with
        | inleft (left _) => s⪧t⪧u
        | inleft (right _) => t⪧u
        | inright _ => t⪧(s→u)
      end
    end
  where "s → t" := (bt_insert s t).

  Fact bt_lt_eq_lt_dec_refl s : exists H, bt_lt_eq_lt_dec s s = inleft (right H).
  Proof.
    destruct (bt_lt_eq_lt_dec s s) as [ [ C | H ] | C ].
    1,3: exfalso; revert C; apply bt_lt_irrefl.
    exists H; auto.
  Qed.

  Ltac bt_lt_eq H :=
    match goal with
      |- context [bt_lt_eq_lt_dec ?x ?x]
           => destruct (bt_lt_eq_lt_dec_refl x) as (H & ->)
    end.

  Fact bt_insert_leaf s : s→∅ = s⪧∅.
  Proof. reflexivity. Qed.

  Fact bt_insert_eq s t : s→s⪧t = s⪧t.
  Proof. simpl; bt_lt_eq H; auto. Qed.

  Fact bt_insert_lt s t u : s ≺ t -> s→t⪧u = s⪧t⪧u.
  Proof.
    intros H; simpl.
    destruct (bt_lt_eq_lt_dec s t) as [[|]|]; auto.
    + contradict H; subst; apply bt_lt_irrefl.
    + destruct (@bt_lt_irrefl s); apply bt_lt_trans with t; auto.
  Qed.

  Fact bt_insert_gt s t u : t ≺ s -> s→t⪧u = t⪧(s→u).
  Proof.
    intros H; simpl.
    destruct (bt_lt_eq_lt_dec s t) as [[|]|]; auto.
    + destruct (@bt_lt_irrefl s); apply bt_lt_trans with t; auto.
    + contradict H; subst; apply bt_lt_irrefl.
  Qed.

  Fact bt_insert_equiv s t : s→t ≈ s⪧t.
  Proof.
    induction t as [ | t u Hu ] using bt_rect'; simpl; auto.
    destruct (bt_lt_eq_lt_dec s t) as [[]|]; subst; auto.
    apply bte_trans with (t⪧s⪧u); auto.
  Qed.

  Fixpoint bt_norm t :=
    match t with
      | ∅ => ∅
      | s⪧t => s† → t†
    end
  where "t †" := (bt_norm t).

  Hint Resolve bt_insert_equiv : core.

  Fact bt_norm_eq t : t† ≈ t.
  Proof.
    induction t as [ | s ? t ? ]; simpl; auto.
    apply bte_trans with (s†⪧t†); auto.
  Qed.

  Opaque bt_insert.

  Tactic Notation "rew" "bt" "insert" :=
      repeat match goal with
        | |- context[_→∅] => rewrite bt_insert_leaf
        | |- context[?x→?x⪧_] => rewrite bt_insert_eq
        | H : ?x ≺ ?y |- context[?x→?y⪧_] => rewrite bt_insert_lt with (1 := H)
        | H : ?x ≺ ?y |- context[?y→?x⪧_] => rewrite bt_insert_gt with (1 := H)
      end.

  Tactic Notation "bt" "insert" "case" hyp(x) hyp(y) :=
      destruct (bt_lt_eq_lt_dec x y) as [ [|] | ]; subst; rew bt insert; auto.

  Tactic Notation "bt" "lt" "trans" constr(t) := apply bt_lt_trans with t; auto.

  Tactic Notation "bt" "lt" "cycle" :=
      match goal with
        | H1 : ?x ≺ ?x |- _ => destruct (@bt_lt_irrefl x); auto
        | H1 : ?x ≺ ?y,
          H2 : ?y ≺ ?x |- _ => destruct (@bt_lt_irrefl x); bt lt trans y
        | H1 : ?x ≺ ?y,
          H2 : ?y ≺ ?z,
          H3 : ?z ≺ ?x |- _ => destruct (@bt_lt_irrefl x); bt lt trans y; bt lt trans z
      end.

  Fact bt_insert_cntr s t : s→s→t = s→t.
  Proof.
    induction t as [ | t1 t2 IH2 ] using bt_rect'; rew bt insert; auto.
    bt insert case s t1; f_equal; auto.
  Qed.

  Fact bt_insert_comm s t u : s→t→u = t→s→u.
  Proof.
    induction u as [ | u1 u2 IH ] using bt_rect'.
    + rew bt insert; bt insert case s t.
    + bt insert case t u1;
      bt insert case s u1;
      bt insert case s t;
      try bt lt cycle;
      f_equal; auto.
  Qed.

  Hint Resolve bt_insert_cntr bt_insert_comm bt_norm_eq : core.

  Theorem bte_norm_iff s t : s ≈ t <-> s† = t†.
  Proof.
    split.
    + induction 1; simpl; auto.
      * transitivity s†; auto.
      * f_equal; auto.
    + intros H.
      apply bte_trans with s †; auto.
      rewrite H; auto.
  Qed.

  Theorem bt_norm_idem s : s†† = s†.
  Proof. apply bte_norm_iff; auto. Qed.

  Theorem bte_dec s t : { s ≈ t } + { s ≉ t }.
  Proof.
    destruct (bt_eq_dec s† t†) as [ H | H ];
    rewrite <- bte_norm_iff in H; auto.
  Qed.

End establishing_decidability.

Corollary btm_dec s t : { s ∈ t } + { s ∉ t }.
Proof. apply bte_dec. Qed.

Section more_decidability.

  Section btm_ex_dec.

    Variable (P : bt -> Prop) (HP : forall s t, s ≈ t -> P s -> P t).

    Theorem btm_ex_dec t : (forall x, x ∈ t -> { P x } + { ~ P x })
                        -> { s | s ∈ t /\ P s } + { forall s, s ∈ t -> ~ P s }.
    Proof.
      induction t as [ | s t IHt ] using bt_rect'; intros Ht.
      + right; intros ?; btm simpl.
      + destruct (Ht s) as [ H | H ]; btm simpl; auto.
        * left; exists s; btm simpl; auto.
        * destruct IHt as [ (x & H1 & H2) | H1 ].
          - intros x Hx; apply Ht; btm simpl.
          - left; exists x; btm simpl.
          - right; intros x; btm simpl.
            intros [ | ]; auto.
            contradict H; revert H; apply HP; auto.
    Qed.

  End btm_ex_dec.

  Corollary btm_fa_dec (P : _ -> Prop) t :
                             (forall s t, s ≈ t -> P s -> P t)
                          -> (forall x, x ∈ t -> { P x } + { ~ P x })
                          -> { s | s ∈ t /\ ~ P s } + { forall s, s ∈ t -> P s }.
  Proof.
    intros H1 H2.
    destruct btm_ex_dec with (P := fun x => ~ P x) (t := t)
      as [ H | H ]; auto.
    + intros u v G1 G2; contradict G2; revert G2; apply H1; auto.
    + intros x Hx; specialize (H2 _ Hx); tauto.
    + right; intros s Hs; specialize (H _ Hs).
      destruct (H2 _ Hs); tauto.
  Qed.

  Definition btm_select (P : _ -> Prop) t :
                             (forall s t, s ≈ t -> P s -> P t)
                          -> (forall x, x ∈ t -> { P x } + { ~ P x })
                          -> { s | forall x, x ∈ s <-> x ∈ t /\ P x }.
  Proof.
    intros HP0.
    induction t as [ | s t Ht ] using bt_rect'; intros HP.
    + exists ∅; intros s; btm simpl.
    + destruct Ht as (t' & H').
      * intros x Hx; apply HP; btm simpl.
      * destruct (HP s) as [ H | H ]; btm simpl; auto.
        - exists (s⪧t'); intros x; btm simpl.
          rewrite H'.
          split; try tauto.
          intros [ H1 | (H1 & H2) ]; auto.
          split; auto.
          revert H; apply HP0; auto.
        - exists t'; intros x; rewrite H'; btm simpl.
          split; try tauto.
          intros ([H1|H1] & H2); split; auto.
          contradict H; revert H2; apply HP0; auto.
  Qed.

  Definition btm_partition x s : x ∈ s -> { t | s ≈ x⪧t /\ x ∉ t }.
  Proof.
    induction s as [ | y s IHs ] using bt_rect'.
    + intros H; exfalso; revert H; btm simpl.
    + intros H; rewrite btm_inv in H.
      destruct (bte_dec x y) as [ H1 | H1 ];
      destruct (btm_dec x s) as [ H2 | H2 ]; try (exfalso; tauto).
      * destruct (IHs H2) as (t & H3 & H4).
        exists t; split; auto.
        rewrite <- H1, H3; auto.
      * exists s; auto.
      * destruct (IHs H2) as (t & H3 & H4).
        exists (y⪧t); split; auto.
        - apply bte_trans with (y⪧x⪧t); auto.
        - contradict H4; revert H4; btm simpl.
  Qed.

End more_decidability.

Definition bt_incl x y := forall u, u ∈ x -> u ∈ y.
Local Infix "⊆" := bt_incl.

Add Parametric Morphism: (bt_incl) with signature
     (bt_equiv) ==> (bt_equiv) ==> (iff) as bti_congr.
Proof.
  intros x y H1 a b H2; split; intros H z.
  + rewrite <- H2, <- H1; apply H.
  + rewrite H1, H2; apply H.
Qed.

Fact bti_refl x : x ⊆ x.
Proof. red; auto. Qed.

Hint Resolve bti_refl : core.

Fact bti_trans x y z : x ⊆ y -> y ⊆ z -> x ⊆ z.
Proof. intros H1 H2 k Hx; apply H2, H1; auto. Qed.

Fact bti_comp s t : t ⊆ s⪧t.
Proof. intro; rewrite btm_inv; auto. Qed.

Fact bti_0 x : ∅ ⊆ x.
Proof. intro; btm simpl. Qed.

Fact bti_inv_0 x : x ⊆ ∅ <-> x = ∅.
Proof.
  split; intros H; subst; auto.
  destruct x as [ | s t ]; auto.
  specialize (H s); revert H; btm simpl.
  intros []; auto.
Qed.

Fact bti_mono_r x s t : s ⊆ t -> x⪧s ⊆ x⪧t.
Proof. intros ? ?; btm simpl; intros []; auto. Qed.

Fact bte_bti s t : s ≈ t -> s ⊆ t.
Proof. intros ? ?; apply btm_congr_r; auto. Qed.

Fact btm_bti x s : x ∈ s -> x⪧s ⊆ s.
Proof. intro; apply bte_bti; auto. Qed.

Fact bti_congr_l x y t : x ≈ y -> x ⊆ t-> y ⊆ t.
Proof.
  intros H1 H2 z Hz.
  apply H2; revert Hz; apply btm_congr_r; auto.
Qed.

Fact bti_congr_r x s t : s ≈ t -> x ⊆ s -> x ⊆ t.
Proof.
  intros H1 H2 z Hz.
  generalize (H2 _ Hz); apply btm_congr_r; auto.
Qed.

Hint Resolve bti_0 bti_refl bti_comp bti_mono_r : core.

Lemma bti_dec s t : { s ⊆ t } + { ~ s ⊆ t }.
Proof.
  destruct btm_fa_dec with (P := fun x => x ∈ t) (t := s)
    as [ (x & H1 & H2) | H1 ]; auto.
  + intros ? ?; apply btm_congr_l.
  + intros ? ?; apply btm_dec.
Qed.

(* I wonder whether the proof of this important result could be split 

   The proof can be done by generalising seteq.v
   to contraction under an equivalence (instead of @eq)
   This would avoid using decidable equality and would
   allow for the development of HF-Sets with UR-elements
   over a non-decidable type *)

Lemma bti_equiv s t : s ⊆ t -> t ⊆ s -> s ≈ t.
Proof.
  revert t; induction s as [ | x s Hs ] using bt_rect'.
  + intros t _ Ht.
    destruct t as [ | y t ]; auto.
    generalize (Ht y); btm simpl.
    intros []; auto.
  + induction t as [ | y _ t Ht ].
    * intros H _; generalize (H x).
      btm simpl; intros []; auto.
    * intros H1 H2.
      destruct (btm_dec x s) as [ H3 | H3 ].
      - apply bte_trans with s; auto.
        apply Hs.
        ++ apply bti_trans with (2 := H1); auto.
        ++ apply bti_trans with (1 := H2), btm_bti; auto.
      - assert (x ∈ y⪧t) as H4 by (apply H1; auto; btm simpl).
        destruct btm_partition with (1 := H4)
          as (u & H5 & H6).
        assert (s ≈ u) as H7.
        { apply Hs.
          + intros z Hz.
            assert (z ∈ x⪧u) as H7.
            { apply btm_congr_r with (1 := H5), H1, btm_inv; auto. }
            apply btm_inv in H7; destruct H7 as [ H7 | ]; auto.
            contradict H3; revert Hz; apply btm_congr_l; auto.
          + intros z Hz.
            assert (z ∈ x⪧s) as H7.
            { apply H2, btm_congr_r with (1 := in_bte_sym H5), btm_inv; auto. }
            apply btm_inv in H7; destruct H7 as [ H7 | ]; auto.
            contradict H6; revert Hz; apply btm_congr_l; auto. }
        apply bte_trans with (x⪧u); auto.
Qed.

Fact bte_incl_equiv s t : s ≈ t <-> s ⊆ t /\ t ⊆ s.
Proof.
  split.
  + intro; split; apply bte_bti; auto.
  + intros []; apply bti_equiv; auto.
Qed.

Fact bte_ext s t : s ≈ t <-> forall x, x ∈ s <-> x ∈ t.
Proof. rewrite bte_incl_equiv; firstorder. Qed.

Fact bti_inv x s t : x ⊆ s⪧t <-> x ⊆ t \/ exists y, y ⊆ t /\ x ≈ s⪧y.
Proof.
  split.
  + intros H.
    destruct (btm_dec s x) as [ H1 | H1 ].
    * destruct btm_partition with (1 := H1)
        as (y & H2 & H3).
      right; exists y; split; auto.
      intros z Hz.
      assert (z ∈ s⪧t) as C.
      { apply H.
        apply btm_congr_r with (1 := in_bte_sym H2), btm_inv; auto. }
      apply btm_inv in C; destruct C as [C|]; auto.
      contradict H3; revert Hz; apply btm_congr_l; auto.
    * left.
      intros y Hy.
      specialize (H _ Hy).
      apply btm_inv in H.
      destruct H as [ H | ]; auto.
      contradict H1.
      revert Hy; apply btm_congr_l; auto.
  + intros [ H | (y & H1 & H2) ].
    * apply bti_trans with (1 := H); auto.
    * intros z Hz.
      apply bti_congr_r with (1 := H2) in Hz; auto.
      revert Hz; btm simpl; intros []; auto.
Qed.

Fixpoint bt_cup s t :=
  match s with
    | ∅ => t
    | x⪧s => x⪧(s ∪ t)
  end
where "s ∪ t" := (bt_cup s t).

Theorem bt_cup_spec x s t : x ∈ s ∪ t <-> x ∈ s \/ x ∈ t.
Proof.
  revert x; induction s as [ | y s Hs ] using bt_rect';
    intros x; simpl; btm simpl.
  rewrite Hs; tauto.
Qed.

Fact bt_cup_left s t : s ⊆ s ∪ t.
Proof. intro; rewrite bt_cup_spec; auto. Qed.

Fact bt_cup_right s t : t ⊆ s ∪ t.
Proof. intro; rewrite bt_cup_spec; auto. Qed.

Hint Resolve bt_cup_left bt_cup_right : core.

Add Parametric Morphism: (bt_cup) with signature
       (bt_equiv) ==> (bt_equiv) ==> (bt_equiv) as bt_cup_congr.
Proof.
  intros s s' H t t'; revert H.
  do 3 rewrite bte_ext; intros H1 H2 x.
  do 2 rewrite bt_cup_spec.
  rewrite H1, H2; tauto.
Qed.

Fact bt_cup_incl s t x : s ∪ t ⊆ x <-> s ⊆ x /\ t ⊆ x.
Proof.
  split.
  + intros H; split; apply bti_trans with (2 := H); auto.
  + intros [] z; rewrite bt_cup_spec; intros []; auto.
Qed.

Fact bt_cup_mono s s' t t' : s ⊆ s' -> t ⊆ t' -> s ∪ t ⊆ s' ∪ t'.
Proof.
  intros H1 H2 x; do 2 rewrite bt_cup_spec.
  intros [ H | H ]; [ left | right ]; revert H; auto.
Qed.

Hint Resolve bt_cup_mono : core.

Definition bt_transitive t := forall u v, u ∈ v -> v ∈ t -> u ∈ t.

Fact bt_transitive_incl u t : bt_transitive t -> u ∈ t -> u ⊆ t.
Proof.
  intros H H1 x H2.
  revert H2 H1; apply H.
Qed.

Fixpoint bt_tc t :=
  match t with
    | ∅ => ∅
    | s⪧t => s⪧(↓s ∪ ↓t)
  end
where "↓ t" := (bt_tc t).

Fact bt_tc_incr t : t ⊆ ↓t.
Proof.
  intro; induction t; simpl; auto; btm simpl.
  rewrite bt_cup_spec; tauto.
Qed.

Hint Resolve bt_tc_incr : core.

Theorem bt_tc_trans t : bt_transitive ↓t.
Proof.
  induction t as [ | s Hs t Ht ]; simpl; intros u v H1; btm simpl; intros H2.
  rewrite bt_cup_spec in H2.
  rewrite bt_cup_spec.
  destruct H2 as [ H2 | [ H2 | H2 ] ].
  + rewrite H2 in H1; right; left; auto; apply bt_tc_incr; auto.
  + right; left; revert H2; apply Hs; auto.
  + right; right; revert H2; apply Ht; auto.
Qed.

Hint Resolve bt_tc_trans : core.

Fact bt_tc_incl_transitive s t : bt_transitive t -> s ⊆ t -> ↓s ⊆ t.
Proof.
  intros H.
  induction s as [ | u Hu v Hv ]; simpl; auto.
  intros H1 x; btm simpl; rewrite bt_cup_spec.
  intros [ H2 | [ H2 | H2 ] ].
  + rewrite H2; apply H1; btm simpl.
  + revert H2; apply Hu.
    intros a Ha; apply H with (1 := Ha), H1; btm simpl.
  + revert H2; apply Hv; intros ? ?; apply H1; btm simpl.
Qed.

Fact bt_tc_mono s t : s ⊆ t -> ↓s ⊆ ↓t.
Proof.
  intros H.
  apply bt_tc_incl_transitive; auto.
  apply bti_trans with (1 := H); auto.
Qed.

Fact bt_tc_idem t : (↓↓t) ⊆ ↓t.
Proof. apply bt_tc_incl_transitive; auto. Qed.

Hint Resolve bt_tc_mono bt_tc_idem : core.

Fact bt_tc_cup s t : ↓(s ∪ t) ⊆ ↓s ∪ ↓t.
Proof.
  apply bt_tc_incl_transitive; auto.
  intros x y; do 2 rewrite bt_cup_spec.
  intros H [ H1 | H1 ]; [ left | right ];
    revert H H1; apply bt_tc_trans.
Qed.

Fact bt_tc_congr_l u v t : u ≈ v -> u ∈ ↓t -> v ∈ ↓t.
Proof.
  revert u v; induction t using bt_rect'; simpl;
    intros ? ? ?; apply btm_congr_l; auto.
Qed.

Fact bt_tc_congr_r u s t : s ≈ t -> u ∈ ↓s <-> u ∈ ↓t.
Proof.
  intros H; revert u.
  rewrite bte_incl_equiv in H; destruct H.
  apply bte_ext, bti_equiv; auto.
Qed.

Add Parametric Morphism: (bt_tc) with signature
     (bt_equiv) ==> (bt_equiv) as bt_tc_congr.
Proof. intros; apply bte_ext; intro; apply bt_tc_congr_r; auto. Qed.

Section bt_pow.

  (* bt_mcomp x {t1,...,tk} => {{x} ∪ t1,...,{x} ∪ tk} **)

  Fixpoint bt_mcomp x t :=
    match t with
      | ∅ => ∅
      | s⪧t => (x⪧s) ⪧ (bt_mcomp x t)
    end.

  Fact bt_mcomp_spec x t u : u ∈ bt_mcomp x t <-> exists k, k ∈ t /\ u ≈ x⪧k.
  Proof.
    revert u; induction t as [ | s t Ht ] using bt_rect'; intros u; simpl.
    + rewrite btm_inv_0; split; try tauto.
      intros (k & Hk & _); revert Hk; rewrite btm_inv_0; auto.
    + rewrite btm_inv; split.
      * intros [ H | H ].
        - exists s; split; auto; btm simpl.
        - rewrite Ht in H.
          destruct H as (k & ? & ?).
          exists k; split; auto.
          btm simpl.
      * intros (k & H1 & H2); revert H1 H2.
        rewrite btm_inv, Ht.
        intros [H2|H2] H1; auto.
        - left; apply bte_trans with (1 := H1); auto.
        - right; exists k; auto.
  Qed.

  (* Powerset bti_inv x ⊆ s⪧t <-> x ⊆ t \/ exists y, y ⊆ t /\ x ≈ s⪧y *)

  Fixpoint bt_pow t :=
    match t with
      | ∅ => ∅⪧∅
      | x⪧t => bt_pow t ∪ bt_mcomp x (bt_pow t)
    end.

  Fact bt_pow_spec t x : x ∈ bt_pow t <-> x ⊆ t.
  Proof.
    revert x; induction t as [ | s t Ht ] using bt_rect'; intros x; simpl.
    + rewrite btm_inv, btm_inv_0, bti_inv_0, bte_inv_0; tauto.
    + rewrite bti_inv, bt_cup_spec, bt_mcomp_spec, Ht.
      split; intros [ | (y & H) ]; auto; right; exists y;
        revert H; rewrite Ht; auto.
  Qed.

  Fact bt_pow_transitive t : bt_transitive t -> bt_transitive (bt_pow t).
  Proof.
    intros H x y H1.
    do 2 rewrite bt_pow_spec.
    intros H2 z Hz.
    apply H2 in H1.
    revert Hz H1; apply H.
  Qed.

  Fact bt_pow_trans_incl t : bt_transitive t -> t ⊆ bt_pow t.
  Proof.
    intros Ht.
    intros x Hx.
    apply bt_pow_spec; intros y Hy.
    revert Hx; apply Ht; auto.
  Qed.

  Fact bt_pow_iter_trans t n : bt_transitive t -> bt_transitive (iter bt_pow t n)
                                               /\ t ⊆ iter bt_pow t n.
  Proof.
    revert t; induction n as [ | n IHn ]; simpl; intros t Ht; auto; split.
    + apply IHn, bt_pow_transitive; auto.
    + destruct (IHn _ (bt_pow_transitive Ht)) as [ _ H ].
      apply bti_trans with (2 := H), bt_pow_trans_incl; auto.
  Qed.

  Fact bt_pow_iter_mono t t' n : t ⊆ t' -> iter bt_pow t n ⊆ iter bt_pow t' n.
  Proof.
    revert t t'; induction n as [ | n IHn ]; simpl; auto; intros t t' H.
    apply IHn; intro; do 2 rewrite bt_pow_spec.
    intro; apply bti_trans with t; auto.
  Qed.

  Fact bt_pow_iter_le t n m : n <= m -> bt_transitive t -> iter bt_pow t n ⊆ iter bt_pow t m.
  Proof.
    intros H; revert H t.
    induction 1 as [ | m Hm IHm ]; auto; intros t Ht.
    apply bti_trans with (1 := IHm _ Ht); simpl.
    apply bt_pow_iter_mono, bt_pow_trans_incl; auto.
  Qed.

End bt_pow.

Add Parametric Morphism: (bt_pow) with signature
     (bt_equiv) ==> (bt_equiv) as bt_pow_congr.
Proof. intros ? ? H; apply bte_ext; intro; do 2 rewrite bt_pow_spec; rewrite H; tauto. Qed.

Definition bt_opair s t := (s⪧∅)⪧(s⪧t⪧∅)⪧∅.

Local Notation "⟬ s , t ⟭" := (bt_opair s t).

Section ordered_pairs.

  Fact bt_sg_inv x y : x⪧∅ ≈ y⪧∅ <-> x ≈ y.
  Proof.
    split; auto.
    rewrite bte_ext.
    intros H; specialize (H x).
    do 2 rewrite btm_inv, btm_inv_0 in H.
    apply proj1 in H; destruct H; try tauto; auto.
  Qed.

  Fact bt_sg_db_inv a x y : a⪧∅ ≈ x⪧y⪧∅ <-> a ≈ x /\ a ≈ y.
  Proof.
    split.
    + intros H.
      rewrite bte_ext in H.
      generalize (proj2 (H x)) (proj2 (H y)).
      btm simpl; intros [|[]] [|[]]; auto.
    + intros [].
      apply bte_trans with (a⪧a⪧∅); auto.
  Qed.

  Fact bt_db_inv a b x y : a⪧b⪧∅ ≈ x⪧y⪧∅ <-> a ≈ x /\ b ≈ x /\ x ≈ y
                                          \/ a ≈ x /\ b ≈ y
                                          \/ a ≈ y /\ b ≈ x.
  Proof.
    split.
    + intros H.
      rewrite bte_ext in H.
      generalize (proj1 (H a)) (proj1 (H b)) (proj2 (H x)) (proj2 (H y)).
      btm simpl; intros [|[|[]]] [|[|[]]] [|[|[]]] [|[|[]]]; auto.
    + intros [ (H1&H2&H3) | [ (H1&H2) | (H1&H2) ] ]; auto.
      * do 2 (apply in_bte_cngr; auto).
        apply bte_trans with x; auto.
      * apply bte_trans with (1 := in_bte_comm _ _ _).
        do 2 (apply in_bte_cngr; auto).
  Qed.

  Fact bt_opair_spec x s t : x ∈ ⟬s,t⟭ <-> x ≈ s⪧∅ \/ x ≈ s⪧t⪧∅.
  Proof. unfold bt_opair; btm simpl. Qed.

  Fact bt_opair_equiv s t s' t' : ⟬s,t⟭ ≈⟬s',t'⟭ <-> s≈s' /\ t≈t'.
  Proof.
    split.
    2: intros []; unfold bt_opair; auto.
    intros H; rewrite bte_ext in H.
    assert (forall x, (x ≈ s⪧∅ \/ x ≈ s⪧t⪧∅) <-> (x ≈ s'⪧∅ \/ x ≈ s'⪧t'⪧∅)) as H'.
    { intros x; generalize (H x); do 2 rewrite bt_opair_spec; tauto. }
    clear H; revert H'; intros H.
    generalize (proj1 (H _) (or_introl (in_bte_refl _)))
               (proj1 (H _) (or_intror (in_bte_refl _)))
               (proj2 (H _) (or_introl (in_bte_refl _)))
               (proj2 (H _) (or_intror (in_bte_refl _))).
    repeat rewrite bt_db_inv.
    do 2 rewrite (bte_sym (_⪧_⪧_)).
    repeat rewrite bt_sg_inv.
    repeat rewrite bt_sg_db_inv.
    intros.
    repeat match goal with
      | H : _ /\ _ |- _ => destruct H
      | H : _ \/ _ |- _ => destruct H
    end; split; auto;
    try (apply bte_trans with s); auto;
    try (apply bte_trans with s'); auto.
  Qed.

End ordered_pairs.

Add Parametric Morphism: (bt_opair) with signature
       (bt_equiv) ==> (bt_equiv) ==> (bt_equiv) as bt_pair_congr.
Proof. intros; apply bt_opair_equiv; auto. Qed.

Fact btm_opair_pow x y t : x ∈ t -> y ∈ t -> ⟬x,y⟭ ∈ bt_pow (bt_pow t).
Proof.
  rewrite bt_pow_spec.
  intros Hx Hy p.
  rewrite bt_opair_spec.
  intros [H|H]; apply bte_sym in H;
  apply btm_congr_l with (1 := H); apply bt_pow_spec.
  + intros z; btm simpl; intros [ Hz | [] ].
    revert Hx; apply btm_congr_l; auto.
  + intros z; btm simpl; intros [ Hz | [ Hz | [] ] ];
      [ revert Hx | revert Hy ]; apply btm_congr_l; auto.
Qed.

Section FOL_encoding.

  Fact bt_enc_equiv s t : s ≈ t <-> forall x, x ∈ s <-> x ∈ t.
  Proof. apply bte_ext. Qed.

  Fact bt_enc_empty s : s ≈ ∅ <-> forall x, ~ x ∈ s.
  Proof. rewrite bte_ext; apply forall_equiv; intro; btm simpl. Qed.

  Fact bt_enc_sg s t : s ≈ t⪧∅ <-> forall x, x ∈ s <-> x ≈ t.
  Proof. rewrite bte_ext; apply forall_equiv; intro; btm simpl. Qed.

  Fact bt_enc_db u s t : u ≈ s⪧t⪧∅ <-> forall x, x ∈ u <-> x ≈ s \/ x ≈ t.
  Proof. rewrite bte_ext; apply forall_equiv; intros; btm simpl. Qed.

  Fact bt_enc_opair p s t : p ≈ ⟬s,t⟭ <-> (forall x, x ∈ p <-> x ≈ s⪧∅ \/ x ≈ s⪧t⪧∅).
  Proof.
    rewrite bte_ext; apply forall_equiv; intro.
    rewrite bt_opair_spec; tauto.
  Qed.

  Fact bt_enc_otriple p r s t : p ≈ ⟬r,⟬s,t⟭⟭ <-> exists a, p ≈ ⟬r,a⟭ /\ a ≈ ⟬s,t⟭ .
  Proof.
    split.
    + exists ⟬s,t⟭; auto.
    + intros (a & H1 & H2).
      apply bte_trans with (1 := H1).
      apply bt_opair_equiv; auto.
  Qed.

  Fact bt_enc_rel3 a b c t : ⟬a,⟬b,c⟭⟭ ∈ t <-> exists p, p ≈ ⟬a,⟬b,c⟭⟭ /\ p ∈ t.
  Proof.
    split.
    + exists ⟬a,⟬b,c⟭⟭ ; auto.
    + intros (p & H1 & H2).
      revert H2; apply btm_congr_l; auto.
  Qed.

  Fixpoint tuple (l : list bt) :=
    match l with
      | nil => ∅
      | x::l => ⟬x,tuple l⟭
    end.

  Fact bt_enc_tuple_0 p : p ≈ tuple nil <-> p ≈ ∅.
  Proof. simpl; tauto. Qed.

  Fact bt_enc_tuple p x l : p ≈ tuple (x::l) <-> exists a, p ≈ ⟬x,a⟭ /\ a ≈ tuple l.
  Proof.
    simpl; split.
    + exists (tuple l); auto; split; auto.
    + intros (a & H1 & H2).
      apply bte_trans with (1 := H1).
      apply bt_opair_equiv; auto.
  Qed.

End FOL_encoding.

Section nat2bt.

  (* We build the usual embedding of nat into bt *)

  Fixpoint nat2bt n : bt :=
    match n with
      | 0 => ∅
      | S n => (nat2bt n)⪧(nat2bt n)
    end.

  Fact nat2bt_transitive n : bt_transitive (nat2bt n).
  Proof.
    induction n as [ | n IHn ]; simpl.
    + intros ? ?; btm simpl.
    + intros a b; btm simpl.
      intros H1 [ H2 | H3 ]; right.
      * rewrite <- H2; auto.
      * revert H1 H3; apply IHn.
  Qed.

  Hint Resolve nat2bt_transitive : core.

  Fact nat2bt_mono n m : n <= m -> nat2bt n ⊆ nat2bt m.
  Proof.
    induction 1 as [ | m Hm IHm ]; auto.
    apply bti_trans with (1 := IHm).
    apply bt_transitive_incl; auto.
    simpl; btm simpl.
  Qed.

  Let nat2bt_inv_full m x : x ∈ nat2bt m -> { n | n < m /\ x ≈ nat2bt n }.
  Proof.
    revert x; induction m as [ | m IHm ]; intros x; simpl; btm simpl.
    + intro E; exfalso; revert E; btm simpl.
    + intros H; rewrite btm_inv in H.
      destruct (bte_dec x (nat2bt m)) as [ H1 | H1 ].
      * exists m; split; auto.
      * destruct (IHm x) as (n & H2 & H3); try tauto.
        exists n; split; auto; lia.
  Qed.

  Definition bt2nat n x Hx := proj1_sig (@nat2bt_inv_full n x Hx).

  Fact bt2nat_lt n x Hx : @bt2nat n x Hx < n.
  Proof. apply (proj2_sig (@nat2bt_inv_full n x Hx)). Qed.

  Fact bt2nat_fix n x Hx : nat2bt (@bt2nat n x Hx) ≈ x.
  Proof. apply bte_sym, (proj2_sig (@nat2bt_inv_full n x Hx)). Qed.

  Fact nat2bt_inj n m : nat2bt n ≈ nat2bt m -> n = m.
  Proof.
    revert m; induction n as [ | n IHn ]; intros [ | m ]; simpl; auto.
    + intros H; symmetry in H; apply bte_discr in H; tauto.
    + intros H; apply bte_discr in H; tauto.
    + intros H; rewrite bte_ext in H.
      f_equal; apply IHn; clear IHn.
      generalize (proj1 (H (nat2bt n)))
                 (proj2 (H (nat2bt m))); btm simpl.
      intros [H0 | H0] [H1 | H1]; auto.
      apply btm_depth in H0.
      apply btm_depth in H1.
      lia.
  Qed.

  Hint Resolve nat2bt_mono : core.

  Fact nat2bt_order_iso n m : nat2bt n ⊆ nat2bt m <-> n <= m.
  Proof.
    split; eauto.
    intros H.
    destruct (le_lt_dec m n); try lia.
    cut (n = m); try lia.
    apply nat2bt_inj, bti_equiv; eauto.
  Qed.

  Fact nat2bt_strict n m : nat2bt n ∈ nat2bt m <-> n < m.
  Proof.
    split.
    + intros H.
      destruct (le_lt_dec m n) as [ H1 | ]; auto.
      generalize (nat2bt_mono H1 H); intros H2.
      apply btm_depth in H2; lia.
    + intros H; apply (nat2bt_mono H).
      simpl; btm simpl.
  Qed.

End nat2bt.

Section pos2bt.

  (* And now a bijection between pos n and nat2bt n *)

  Definition pos2bt n (p : pos n) := nat2bt (pos2nat p).

  Fact pos2bt_in n p : @pos2bt n p ∈ nat2bt n.
  Proof. apply nat2bt_strict, pos2nat_prop. Qed.

  Let bt2pos_full n x : x ∈ nat2bt n -> { p | x ≈ @pos2bt n p }.
  Proof.
    intros Hx.
    assert (bt2nat _ Hx < n).
    { apply nat2bt_strict; rewrite bt2nat_fix; auto. }
    exists (nat2pos H); unfold pos2bt; rewrite pos2nat_nat2pos.
    rewrite bt2nat_fix; auto.
  Qed.

  Definition bt2pos n x Hx := proj1_sig (@bt2pos_full n x Hx).

  Fact bt2pos_fix n x Hx : pos2bt (@bt2pos n x Hx) ≈ x.
  Proof. apply bte_sym, (proj2_sig (@bt2pos_full n x Hx)). Qed.

  Fact pos2bt_inj n p q : @pos2bt n p ≈ pos2bt q -> p = q.
  Proof. intro H; apply pos2nat_inj, nat2bt_inj, H. Qed.

End pos2bt.