path: root/theories/IntMap/Lsort.v

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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
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(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
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(*i 	$Id: Lsort.v 5920 2004-07-16 20:01:26Z herbelin $	 i*)

Require Import Bool.
Require Import Sumbool.
Require Import Arith.
Require Import ZArith.
Require Import Addr.
Require Import Adist.
Require Import Addec.
Require Import Map.
Require Import List.
Require Import Mapiter.

Section LSort.

  Variable A : Set.

  Fixpoint ad_less_1 (a a':ad) (p:positive) {struct p} : bool :=
    match p with
    | xO p' => ad_less_1 (ad_div_2 a) (ad_div_2 a') p'
    | _ => andb (negb (ad_bit_0 a)) (ad_bit_0 a')
    end.

  Definition ad_less (a a':ad) :=
    match ad_xor a a' with
    | ad_z => false
    | ad_x p => ad_less_1 a a' p
    end.

  Lemma ad_bit_0_less :
   forall a a':ad,
     ad_bit_0 a = false -> ad_bit_0 a' = true -> ad_less a a' = true.
  Proof.
    intros. elim (ad_sum (ad_xor a a')). intro H1. elim H1. intros p H2. unfold ad_less in |- *.
    rewrite H2. generalize H2. elim p. intros. simpl in |- *. rewrite H. rewrite H0. reflexivity.
    intros. cut (ad_bit_0 (ad_xor a a') = false). intro. rewrite (ad_xor_bit_0 a a') in H5.
    rewrite H in H5. rewrite H0 in H5. discriminate H5.
    rewrite H4. reflexivity.
    intro. simpl in |- *. rewrite H. rewrite H0. reflexivity.
    intro H1. cut (ad_bit_0 (ad_xor a a') = false). intro. rewrite (ad_xor_bit_0 a a') in H2.
    rewrite H in H2. rewrite H0 in H2. discriminate H2.
    rewrite H1. reflexivity.
  Qed.

  Lemma ad_bit_0_gt :
   forall a a':ad,
     ad_bit_0 a = true -> ad_bit_0 a' = false -> ad_less a a' = false.
  Proof.
    intros. elim (ad_sum (ad_xor a a')). intro H1. elim H1. intros p H2. unfold ad_less in |- *.
    rewrite H2. generalize H2. elim p. intros. simpl in |- *. rewrite H. rewrite H0. reflexivity.
    intros. cut (ad_bit_0 (ad_xor a a') = false). intro. rewrite (ad_xor_bit_0 a a') in H5.
    rewrite H in H5. rewrite H0 in H5. discriminate H5.
    rewrite H4. reflexivity.
    intro. simpl in |- *. rewrite H. rewrite H0. reflexivity.
    intro H1. unfold ad_less in |- *. rewrite H1. reflexivity.
  Qed.

  Lemma ad_less_not_refl : forall a:ad, ad_less a a = false.
  Proof.
    intro. unfold ad_less in |- *. rewrite (ad_xor_nilpotent a). reflexivity.
  Qed.

  Lemma ad_ind_double :
   forall (a:ad) (P:ad -> Prop),
     P ad_z ->
     (forall a:ad, P a -> P (ad_double a)) ->
     (forall a:ad, P a -> P (ad_double_plus_un a)) -> P a.
  Proof.
    intros; elim a. trivial.
    simple induction p. intros. 
    apply (H1 (ad_x p0)); trivial.
    intros; apply (H0 (ad_x p0)); trivial.
    intros; apply (H1 ad_z); assumption.
  Qed.

  Lemma ad_rec_double :
   forall (a:ad) (P:ad -> Set),
     P ad_z ->
     (forall a:ad, P a -> P (ad_double a)) ->
     (forall a:ad, P a -> P (ad_double_plus_un a)) -> P a.
  Proof.
    intros; elim a. trivial.
    simple induction p. intros. 
    apply (H1 (ad_x p0)); trivial.
    intros; apply (H0 (ad_x p0)); trivial.
    intros; apply (H1 ad_z); assumption.
  Qed.

  Lemma ad_less_def_1 :
   forall a a':ad, ad_less (ad_double a) (ad_double a') = ad_less a a'.
  Proof.
    simple induction a. simple induction a'. reflexivity.
    trivial.
    simple induction a'. unfold ad_less in |- *. simpl in |- *. elim p; trivial.
    unfold ad_less in |- *. simpl in |- *. intro. case (p_xor p p0). reflexivity.
    trivial.
  Qed.

  Lemma ad_less_def_2 :
   forall a a':ad,
     ad_less (ad_double_plus_un a) (ad_double_plus_un a') = ad_less a a'.
  Proof.
    simple induction a. simple induction a'. reflexivity.
    trivial.
    simple induction a'. unfold ad_less in |- *. simpl in |- *. elim p; trivial.
    unfold ad_less in |- *. simpl in |- *. intro. case (p_xor p p0). reflexivity.
    trivial.
  Qed.

  Lemma ad_less_def_3 :
   forall a a':ad, ad_less (ad_double a) (ad_double_plus_un a') = true.
  Proof.
    intros. apply ad_bit_0_less. apply ad_double_bit_0.
    apply ad_double_plus_un_bit_0.
  Qed.

  Lemma ad_less_def_4 :
   forall a a':ad, ad_less (ad_double_plus_un a) (ad_double a') = false.
  Proof.
    intros. apply ad_bit_0_gt. apply ad_double_plus_un_bit_0.
    apply ad_double_bit_0.
  Qed.

  Lemma ad_less_z : forall a:ad, ad_less a ad_z = false.
  Proof.
    simple induction a. reflexivity.
    unfold ad_less in |- *. intro. rewrite (ad_xor_neutral_right (ad_x p)). elim p; trivial.
  Qed.

  Lemma ad_z_less_1 :
   forall a:ad, ad_less ad_z a = true -> {p : positive | a = ad_x p}.
  Proof.
    simple induction a. intro. discriminate H.
    intros. split with p. reflexivity.
  Qed.

  Lemma ad_z_less_2 : forall a:ad, ad_less ad_z a = false -> a = ad_z.
  Proof.
    simple induction a. trivial.
    unfold ad_less in |- *. simpl in |- *. cut (forall p:positive, ad_less_1 ad_z (ad_x p) p = false -> False).
    intros. elim (H p H0).
    simple induction p. intros. discriminate H0.
    intros. exact (H H0).
    intro. discriminate H.
  Qed.

  Lemma ad_less_trans :
   forall a a' a'':ad,
     ad_less a a' = true -> ad_less a' a'' = true -> ad_less a a'' = true.
  Proof.
    intro a. apply ad_ind_double with
  (P := fun a:ad =>
          forall a' a'':ad,
            ad_less a a' = true ->
            ad_less a' a'' = true -> ad_less a a'' = true).
    intros. elim (sumbool_of_bool (ad_less ad_z a'')). trivial.
    intro H1. rewrite (ad_z_less_2 a'' H1) in H0. rewrite (ad_less_z a') in H0. discriminate H0.
    intros a0 H a'. apply ad_ind_double with
  (P := fun a':ad =>
          forall a'':ad,
            ad_less (ad_double a0) a' = true ->
            ad_less a' a'' = true -> ad_less (ad_double a0) a'' = true).
    intros. rewrite (ad_less_z (ad_double a0)) in H0. discriminate H0.
    intros a1 H0 a'' H1. rewrite (ad_less_def_1 a0 a1) in H1.
    apply ad_ind_double with
     (P := fun a'':ad =>
             ad_less (ad_double a1) a'' = true ->
             ad_less (ad_double a0) a'' = true).
    intro. rewrite (ad_less_z (ad_double a1)) in H2. discriminate H2.
    intros. rewrite (ad_less_def_1 a1 a2) in H3. rewrite (ad_less_def_1 a0 a2).
    exact (H a1 a2 H1 H3).
    intros. apply ad_less_def_3.
    intros a1 H0 a'' H1. apply ad_ind_double with
  (P := fun a'':ad =>
          ad_less (ad_double_plus_un a1) a'' = true ->
          ad_less (ad_double a0) a'' = true).
    intro. rewrite (ad_less_z (ad_double_plus_un a1)) in H2. discriminate H2.
    intros. rewrite (ad_less_def_4 a1 a2) in H3. discriminate H3.
    intros. apply ad_less_def_3.
    intros a0 H a'. apply ad_ind_double with
  (P := fun a':ad =>
          forall a'':ad,
            ad_less (ad_double_plus_un a0) a' = true ->
            ad_less a' a'' = true ->
            ad_less (ad_double_plus_un a0) a'' = true).
    intros. rewrite (ad_less_z (ad_double_plus_un a0)) in H0. discriminate H0.
    intros. rewrite (ad_less_def_4 a0 a1) in H1. discriminate H1.
    intros a1 H0 a'' H1. apply ad_ind_double with
  (P := fun a'':ad =>
          ad_less (ad_double_plus_un a1) a'' = true ->
          ad_less (ad_double_plus_un a0) a'' = true).
    intro. rewrite (ad_less_z (ad_double_plus_un a1)) in H2. discriminate H2.
    intros. rewrite (ad_less_def_4 a1 a2) in H3. discriminate H3.
    rewrite (ad_less_def_2 a0 a1) in H1. intros. rewrite (ad_less_def_2 a1 a2) in H3.
    rewrite (ad_less_def_2 a0 a2). exact (H a1 a2 H1 H3).
  Qed.

  Fixpoint alist_sorted (l:alist A) : bool :=
    match l with
    | nil => true
    | (a, _) :: l' =>
        match l' with
        | nil => true
        | (a', y') :: l'' => andb (ad_less a a') (alist_sorted l')
        end
    end.

  Fixpoint alist_nth_ad (n:nat) (l:alist A) {struct l} : ad :=
    match l with
    | nil => ad_z (* dummy *)
    | (a, y) :: l' => match n with
                      | O => a
                      | S n' => alist_nth_ad n' l'
                      end
    end.

  Definition alist_sorted_1 (l:alist A) :=
    forall n:nat,
      S (S n) <= length l ->
      ad_less (alist_nth_ad n l) (alist_nth_ad (S n) l) = true.

  Lemma alist_sorted_imp_1 :
   forall l:alist A, alist_sorted l = true -> alist_sorted_1 l.
  Proof.
    unfold alist_sorted_1 in |- *. simple induction l. intros. elim (le_Sn_O (S n) H0).
    intro r. elim r. intros a y. simple induction l0. intros. simpl in H1.
    elim (le_Sn_O n (le_S_n (S n) 0 H1)).
    intro r0. elim r0. intros a0 y0. simple induction n. intros. simpl in |- *. simpl in H1.
    exact (proj1 (andb_prop _ _ H1)).
    intros. change
   (ad_less (alist_nth_ad n0 ((a0, y0) :: l1))
      (alist_nth_ad (S n0) ((a0, y0) :: l1)) = true) 
  in |- *.
    apply H0. exact (proj2 (andb_prop _ _ H1)).
    apply le_S_n. exact H3.
  Qed.

  Definition alist_sorted_2 (l:alist A) :=
    forall m n:nat,
      m < n ->
      S n <= length l -> ad_less (alist_nth_ad m l) (alist_nth_ad n l) = true.

  Lemma alist_sorted_1_imp_2 :
   forall l:alist A, alist_sorted_1 l -> alist_sorted_2 l.
  Proof.
    unfold alist_sorted_1, alist_sorted_2, lt in |- *. intros l H m n H0. elim H0. exact (H m).
    intros. apply ad_less_trans with (a' := alist_nth_ad m0 l). apply H2. apply le_Sn_le.
    assumption.
    apply H. assumption.
  Qed.

  Lemma alist_sorted_2_imp :
   forall l:alist A, alist_sorted_2 l -> alist_sorted l = true.
  Proof.
    unfold alist_sorted_2, lt in |- *. simple induction l. trivial.
    intro r. elim r. intros a y. simple induction l0. trivial.
    intro r0. elim r0. intros a0 y0. intros.
    change (andb (ad_less a a0) (alist_sorted ((a0, y0) :: l1)) = true)
     in |- *.
    apply andb_true_intro. split. apply (H1 0 1). apply le_n.
    simpl in |- *. apply le_n_S. apply le_n_S. apply le_O_n.
    apply H0. intros. apply (H1 (S m) (S n)). apply le_n_S. assumption.
    exact (le_n_S _ _ H3).
  Qed.

  Lemma app_length :
   forall (C:Set) (l l':list C), length (l ++ l') = length l + length l'.
  Proof.
    simple induction l. trivial.
    intros. simpl in |- *. rewrite (H l'). reflexivity.
  Qed.

  Lemma aapp_length :
   forall l l':alist A, length (aapp A l l') = length l + length l'.
  Proof.
    exact (app_length (ad * A)).
  Qed.

  Lemma alist_nth_ad_aapp_1 :
   forall (l l':alist A) (n:nat),
     S n <= length l -> alist_nth_ad n (aapp A l l') = alist_nth_ad n l.
  Proof.
    simple induction l. intros. elim (le_Sn_O n H).
    intro r. elim r. intros a y l' H l''. simple induction n. trivial.
    intros. simpl in |- *. apply H. apply le_S_n. exact H1.
  Qed.

  Lemma alist_nth_ad_aapp_2 :
   forall (l l':alist A) (n:nat),
     S n <= length l' ->
     alist_nth_ad (length l + n) (aapp A l l') = alist_nth_ad n l'.
  Proof.
    simple induction l. trivial.
    intro r. elim r. intros a y l' H l'' n H0. simpl in |- *. apply H. exact H0.
  Qed.

  Lemma interval_split :
   forall p q n:nat,
     S n <= p + q -> {n' : nat | S n' <= q /\ n = p + n'} + {S n <= p}.
  Proof.
    simple induction p. simpl in |- *. intros. left. split with n. split; [ assumption | reflexivity ].
    intros p' H q. simple induction n. intros. right. apply le_n_S. apply le_O_n.
    intros. elim (H _ _ (le_S_n _ _ H1)). intro H2. left. elim H2. intros n' H3.
    elim H3. intros H4 H5. split with n'. split; [ assumption | rewrite H5; reflexivity ].
    intro H2. right. apply le_n_S. assumption.
  Qed.

  Lemma alist_conc_sorted :
   forall l l':alist A,
     alist_sorted_2 l ->
     alist_sorted_2 l' ->
     (forall n n':nat,
        S n <= length l ->
        S n' <= length l' ->
        ad_less (alist_nth_ad n l) (alist_nth_ad n' l') = true) ->
     alist_sorted_2 (aapp A l l').
  Proof.
    unfold alist_sorted_2, lt in |- *. intros. rewrite (aapp_length l l') in H3.
    elim
     (interval_split (length l) (length l') m
        (le_trans _ _ _ (le_n_S _ _ (lt_le_weak m n H2)) H3)).
    intro H4. elim H4. intros m' H5. elim H5. intros. rewrite H7.
    rewrite (alist_nth_ad_aapp_2 l l' m' H6). elim (interval_split (length l) (length l') n H3).
    intro H8. elim H8. intros n' H9. elim H9. intros. rewrite H11.
    rewrite (alist_nth_ad_aapp_2 l l' n' H10). apply H0. rewrite H7 in H2. rewrite H11 in H2.
    change (S (length l) + m' <= length l + n') in H2.
    rewrite (plus_Snm_nSm (length l) m') in H2. exact ((fun p n m:nat => plus_le_reg_l n m p) (length l) (S m') n' H2).
    exact H10.
    intro H8. rewrite H7 in H2. cut (S (length l) <= length l). intros. elim (le_Sn_n _ H9).
    apply le_trans with (m := S n). apply le_n_S. apply le_trans with (m := S (length l + m')).
    apply le_trans with (m := length l + m'). apply le_plus_l.
    apply le_n_Sn.
    exact H2.
    exact H8.
    intro H4. rewrite (alist_nth_ad_aapp_1 l l' m H4).
    elim (interval_split (length l) (length l') n H3). intro H5. elim H5. intros n' H6. elim H6.
    intros. rewrite H8. rewrite (alist_nth_ad_aapp_2 l l' n' H7). exact (H1 m n' H4 H7).
    intro H5. rewrite (alist_nth_ad_aapp_1 l l' n H5). exact (H m n H2 H5).
  Qed.

  Lemma alist_nth_ad_semantics :
   forall (l:alist A) (n:nat),
     S n <= length l ->
     {y : A | alist_semantics A l (alist_nth_ad n l) = SOME A y}.
  Proof.
    simple induction l. intros. elim (le_Sn_O _ H).
    intro r. elim r. intros a y l0 H. simple induction n. simpl in |- *. intro. split with y.
    rewrite (ad_eq_correct a). reflexivity.
    intros. elim (H _ (le_S_n _ _ H1)). intros y0 H2.
    elim (sumbool_of_bool (ad_eq a (alist_nth_ad n0 l0))). intro H3. split with y.
    rewrite (ad_eq_complete _ _ H3). simpl in |- *. rewrite (ad_eq_correct (alist_nth_ad n0 l0)).
    reflexivity.
    intro H3. split with y0. simpl in |- *. rewrite H3. assumption.
  Qed.

  Lemma alist_of_Map_nth_ad :
   forall (m:Map A) (pf:ad -> ad) (l:alist A),
     l =
     MapFold1 A (alist A) (anil A) (aapp A)
       (fun (a0:ad) (y:A) => acons A (a0, y) (anil A)) pf m ->
     forall n:nat, S n <= length l -> {a' : ad | alist_nth_ad n l = pf a'}.
  Proof.
    intros. elim (alist_nth_ad_semantics l n H0). intros y H1.
    apply (alist_of_Map_semantics_1_1 A m pf (alist_nth_ad n l) y).
    rewrite <- H. assumption.
  Qed.

  Definition ad_monotonic (pf:ad -> ad) :=
    forall a a':ad, ad_less a a' = true -> ad_less (pf a) (pf a') = true.

  Lemma ad_double_monotonic : ad_monotonic ad_double.
  Proof.
    unfold ad_monotonic in |- *. intros. rewrite ad_less_def_1. assumption.
  Qed.

  Lemma ad_double_plus_un_monotonic : ad_monotonic ad_double_plus_un.
  Proof.
    unfold ad_monotonic in |- *. intros. rewrite ad_less_def_2. assumption.
  Qed.

  Lemma ad_comp_monotonic :
   forall pf pf':ad -> ad,
     ad_monotonic pf ->
     ad_monotonic pf' -> ad_monotonic (fun a0:ad => pf (pf' a0)).
  Proof.
    unfold ad_monotonic in |- *. intros. apply H. apply H0. exact H1.
  Qed.

  Lemma ad_comp_double_monotonic :
   forall pf:ad -> ad,
     ad_monotonic pf -> ad_monotonic (fun a0:ad => pf (ad_double a0)).
  Proof.
    intros. apply ad_comp_monotonic. assumption.
    exact ad_double_monotonic.
  Qed.

  Lemma ad_comp_double_plus_un_monotonic :
   forall pf:ad -> ad,
     ad_monotonic pf -> ad_monotonic (fun a0:ad => pf (ad_double_plus_un a0)).
  Proof.
    intros. apply ad_comp_monotonic. assumption.
    exact ad_double_plus_un_monotonic.
  Qed.

  Lemma alist_of_Map_sorts_1 :
   forall (m:Map A) (pf:ad -> ad),
     ad_monotonic pf ->
     alist_sorted_2
       (MapFold1 A (alist A) (anil A) (aapp A)
          (fun (a:ad) (y:A) => acons A (a, y) (anil A)) pf m).
  Proof.
    simple induction m. simpl in |- *. intros. apply alist_sorted_1_imp_2. apply alist_sorted_imp_1. reflexivity.
    intros. simpl in |- *. apply alist_sorted_1_imp_2. apply alist_sorted_imp_1. reflexivity.
    intros. simpl in |- *. apply alist_conc_sorted.
    exact
     (H (fun a0:ad => pf (ad_double a0)) (ad_comp_double_monotonic pf H1)).
    exact
     (H0 (fun a0:ad => pf (ad_double_plus_un a0))
        (ad_comp_double_plus_un_monotonic pf H1)).
    intros. elim
  (alist_of_Map_nth_ad m0 (fun a0:ad => pf (ad_double a0))
     (MapFold1 A (alist A) (anil A) (aapp A)
        (fun (a0:ad) (y:A) => acons A (a0, y) (anil A))
        (fun a0:ad => pf (ad_double a0)) m0) (refl_equal _) n H2).
    intros a H4. rewrite H4. elim
  (alist_of_Map_nth_ad m1 (fun a0:ad => pf (ad_double_plus_un a0))
     (MapFold1 A (alist A) (anil A) (aapp A)
        (fun (a0:ad) (y:A) => acons A (a0, y) (anil A))
        (fun a0:ad => pf (ad_double_plus_un a0)) m1) (
     refl_equal _) n' H3).
    intros a' H5. rewrite H5. unfold ad_monotonic in H1. apply H1. apply ad_less_def_3.
  Qed.

  Lemma alist_of_Map_sorts :
   forall m:Map A, alist_sorted (alist_of_Map A m) = true.
  Proof.
    intro. apply alist_sorted_2_imp.
    exact
     (alist_of_Map_sorts_1 m (fun a0:ad => a0)
        (fun (a a':ad) (p:ad_less a a' = true) => p)).
  Qed.

  Lemma alist_of_Map_sorts1 :
   forall m:Map A, alist_sorted_1 (alist_of_Map A m).
  Proof.
    intro. apply alist_sorted_imp_1. apply alist_of_Map_sorts.
  Qed.
 
  Lemma alist_of_Map_sorts2 :
   forall m:Map A, alist_sorted_2 (alist_of_Map A m).
  Proof.
    intro. apply alist_sorted_1_imp_2. apply alist_of_Map_sorts1.
  Qed.
 
  Lemma ad_less_total :
   forall a a':ad, {ad_less a a' = true} + {ad_less a' a = true} + {a = a'}.
  Proof.
    intro a. refine
  (ad_rec_double a
     (fun a:ad =>
        forall a':ad,
          {ad_less a a' = true} + {ad_less a' a = true} + {a = a'}) _ _ _).
    intro. elim (sumbool_of_bool (ad_less ad_z a')). intro H. left. left. assumption.
    intro H. right. rewrite (ad_z_less_2 a' H). reflexivity.
    intros a0 H a'. refine
  (ad_rec_double a'
     (fun a':ad =>
        {ad_less (ad_double a0) a' = true} +
        {ad_less a' (ad_double a0) = true} + {ad_double a0 = a'}) _ _ _).
    elim (sumbool_of_bool (ad_less ad_z (ad_double a0))). intro H0. left. right. assumption.
    intro H0. right. exact (ad_z_less_2 _ H0).
    intros a1 H0. rewrite ad_less_def_1. rewrite ad_less_def_1. elim (H a1). intro H1.
    left. assumption.
    intro H1. right. rewrite H1. reflexivity.
    intros a1 H0. left. left. apply ad_less_def_3.
    intros a0 H a'. refine
  (ad_rec_double a'
     (fun a':ad =>
        {ad_less (ad_double_plus_un a0) a' = true} +
        {ad_less a' (ad_double_plus_un a0) = true} +
        {ad_double_plus_un a0 = a'}) _ _ _).
    left. right. case a0; reflexivity.
    intros a1 H0. left. right. apply ad_less_def_3.
    intros a1 H0. rewrite ad_less_def_2. rewrite ad_less_def_2. elim (H a1). intro H1.
    left. assumption.
    intro H1. right. rewrite H1. reflexivity.
  Qed.

  Lemma alist_too_low :
   forall (l:alist A) (a a':ad) (y:A),
     ad_less a a' = true ->
     alist_sorted_2 ((a', y) :: l) ->
     alist_semantics A ((a', y) :: l) a = NONE A.
  Proof.
    simple induction l. intros. simpl in |- *. elim (sumbool_of_bool (ad_eq a' a)). intro H1.
    rewrite (ad_eq_complete _ _ H1) in H. rewrite (ad_less_not_refl a) in H. discriminate H.
    intro H1. rewrite H1. reflexivity.
    intro r. elim r. intros a y l0 H a0 a1 y0 H0 H1.
    change
      (match ad_eq a1 a0 with
       | true => SOME A y0
       | false => alist_semantics A ((a, y) :: l0) a0
       end = NONE A) in |- *.
    elim (sumbool_of_bool (ad_eq a1 a0)). intro H2. rewrite (ad_eq_complete _ _ H2) in H0.
    rewrite (ad_less_not_refl a0) in H0. discriminate H0.
    intro H2. rewrite H2. apply H. apply ad_less_trans with (a' := a1). assumption.
    unfold alist_sorted_2 in H1. apply (H1 0 1). apply lt_n_Sn.
    simpl in |- *. apply le_n_S. apply le_n_S. apply le_O_n.
    apply alist_sorted_1_imp_2. apply alist_sorted_imp_1.
    cut (alist_sorted ((a1, y0) :: (a, y) :: l0) = true). intro H3.
    exact (proj2 (andb_prop _ _ H3)).
    apply alist_sorted_2_imp. assumption.
  Qed.

  Lemma alist_semantics_nth_ad :
   forall (l:alist A) (a:ad) (y:A),
     alist_semantics A l a = SOME A y ->
     {n : nat | S n <= length l /\ alist_nth_ad n l = a}.
  Proof.
    simple induction l. intros. discriminate H.
    intro r. elim r. intros a y l0 H a0 y0 H0. simpl in H0. elim (sumbool_of_bool (ad_eq a a0)).
    intro H1. rewrite H1 in H0. split with 0. split. simpl in |- *. apply le_n_S. apply le_O_n.
    simpl in |- *. exact (ad_eq_complete _ _ H1).
    intro H1. rewrite H1 in H0. elim (H a0 y0 H0). intros n' H2. split with (S n'). split.
    simpl in |- *. apply le_n_S. exact (proj1 H2).
    exact (proj2 H2).
  Qed.

  Lemma alist_semantics_tail :
   forall (l:alist A) (a:ad) (y:A),
     alist_sorted_2 ((a, y) :: l) ->
     eqm A (alist_semantics A l)
       (fun a0:ad =>
          if ad_eq a a0 then NONE A else alist_semantics A ((a, y) :: l) a0).
  Proof.
    unfold eqm in |- *. intros. elim (sumbool_of_bool (ad_eq a a0)). intro H0. rewrite H0.
    rewrite <- (ad_eq_complete _ _ H0). unfold alist_sorted_2 in H.
    elim (option_sum A (alist_semantics A l a)). intro H1. elim H1. intros y0 H2.
    elim (alist_semantics_nth_ad l a y0 H2). intros n H3. elim H3. intros.
    cut
     (ad_less (alist_nth_ad 0 ((a, y) :: l))
        (alist_nth_ad (S n) ((a, y) :: l)) = true).
    intro. simpl in H6. rewrite H5 in H6. rewrite (ad_less_not_refl a) in H6. discriminate H6.
    apply H. apply lt_O_Sn.
    simpl in |- *. apply le_n_S. assumption.
    trivial.
    intro H0. simpl in |- *. rewrite H0. reflexivity.
  Qed.

  Lemma alist_semantics_same_tail :
   forall (l l':alist A) (a:ad) (y:A),
     alist_sorted_2 ((a, y) :: l) ->
     alist_sorted_2 ((a, y) :: l') ->
     eqm A (alist_semantics A ((a, y) :: l))
       (alist_semantics A ((a, y) :: l')) ->
     eqm A (alist_semantics A l) (alist_semantics A l').
  Proof.
    unfold eqm in |- *. intros. rewrite (alist_semantics_tail _ _ _ H a0).
    rewrite (alist_semantics_tail _ _ _ H0 a0). case (ad_eq a a0). reflexivity.
    exact (H1 a0).
  Qed.

  Lemma alist_sorted_tail :
   forall (l:alist A) (a:ad) (y:A),
     alist_sorted_2 ((a, y) :: l) -> alist_sorted_2 l.
  Proof.
    unfold alist_sorted_2 in |- *. intros. apply (H (S m) (S n)). apply lt_n_S. assumption.
    simpl in |- *. apply le_n_S. assumption.
  Qed.

  Lemma alist_canonical :
   forall l l':alist A,
     eqm A (alist_semantics A l) (alist_semantics A l') ->
     alist_sorted_2 l -> alist_sorted_2 l' -> l = l'.
  Proof.
    unfold eqm in |- *. simple induction l. simple induction l'. trivial.
    intro r. elim r. intros a y l0 H H0 H1 H2. simpl in H0.
    cut
     (NONE A =
      match ad_eq a a with
      | true => SOME A y
      | false => alist_semantics A l0 a
      end).
    rewrite (ad_eq_correct a). intro. discriminate H3.
    exact (H0 a).
    intro r. elim r. intros a y l0 H. simple induction l'. intros. simpl in H0.
    cut
     (match ad_eq a a with
      | true => SOME A y
      | false => alist_semantics A l0 a
      end = NONE A).
    rewrite (ad_eq_correct a). intro. discriminate H3.
    exact (H0 a).
    intro r'. elim r'. intros a' y' l'0 H0 H1 H2 H3. elim (ad_less_total a a'). intro H4.
    elim H4. intro H5.
    cut
     (alist_semantics A ((a, y) :: l0) a =
      alist_semantics A ((a', y') :: l'0) a).
    intro. rewrite (alist_too_low l'0 a a' y' H5 H3) in H6. simpl in H6.
    rewrite (ad_eq_correct a) in H6. discriminate H6.
    exact (H1 a).
    intro H5. cut
  (alist_semantics A ((a, y) :: l0) a' =
   alist_semantics A ((a', y') :: l'0) a').
    intro. rewrite (alist_too_low l0 a' a y H5 H2) in H6. simpl in H6.
    rewrite (ad_eq_correct a') in H6. discriminate H6.
    exact (H1 a').
    intro H4. rewrite H4.
    cut
     (alist_semantics A ((a, y) :: l0) a =
      alist_semantics A ((a', y') :: l'0) a).
    intro. simpl in H5. rewrite H4 in H5. rewrite (ad_eq_correct a') in H5. inversion H5.
    rewrite H4 in H1. rewrite H7 in H1. cut (l0 = l'0). intro. rewrite H6. reflexivity.
    apply H. rewrite H4 in H2. rewrite H7 in H2.
    exact (alist_semantics_same_tail l0 l'0 a' y' H2 H3 H1).
    exact (alist_sorted_tail _ _ _ H2).
    exact (alist_sorted_tail _ _ _ H3).
    exact (H1 a).
  Qed.

End LSort.