+lemma sorted_near:
+ ∀r,l. sorted r l → ∀i,d. S i < \len l → r (\nth l d i) (\nth l d (S i)).
+ intros 3; elim H;
+ [1: cases (not_le_Sn_O ? H1);
+ |2: simplify in H1; cases (not_le_Sn_O ? (le_S_S_to_le ?? H1));
+ |3: simplify; cases i in H4; intros; [apply H1]
+ apply H3; apply le_S_S_to_le; apply H4]
+ qed.
+
+lemma sorted_copy:
+ ∀l:list bar.sorted q2_lt l → sorted q2_lt (copy l).
+intros 2; unfold copy; generalize in match (le_n (\len l));
+elim (\len l) in ⊢ (?%?→? ? (? ? ? %));
+simplify; [apply (sorted_nil q2_lt);] cases n in H1 H2;
+simplify; intros; [apply (sorted_one q2_lt);]
+apply (sorted_cons q2_lt);
+[2: apply H1; apply lt_to_le; apply H2;
+|1: elim l in H2 H; simplify; [simplify in H2; cases (not_le_Sn_O ? H2)]
+ simplify in H3; unfold nth_base;
+ unfold canonical_q_lt; unfold q2_trel; unfold q2_lt; simplify;
+ change with (q2_lt (\nth (a::l1) ▭ (\len l1-S n1)) (\nth (a::l1) ▭ (\len l1-n1)));
+ cut (∃w.w = \len l1 - S n1); [2: exists[apply (\len l1 - S n1)] reflexivity]
+ cases Hcut; rewrite < H4; rewrite < (?:S w = \len l1 - n1);
+ [1: apply (sorted_near q2_lt (a::l1) H2); rewrite > H4;
+ simplify; apply le_S_S; elim (\len l1) in H3; simplify;
+ [ cases (not_le_Sn_O ? (le_S_S_to_le ?? H3));
+ | lapply le_S_S_to_le to H5 as H6;
+ lapply le_S_S_to_le to H6 as H7; clear H5 H6;
+ cases H7 in H3; intros; [rewrite < minus_n_n; apply le_S_S; apply le_O_n]
+ simplify in H5; apply le_S_S; apply (trans_le ???? (H5 ?));
+ [2: apply le_S_S; apply le_S_S; assumption;
+ |1: apply (lt_minus_S_n_to_le_minus_n n1 (S m) (S (minus m n1)) ?).
+ apply (not_le_to_lt (S (minus m n1)) (minus (S m) (S n1)) ?).
+ apply (not_le_Sn_n (minus (S m) (S n1))).]]
+ |2: rewrite > H4; lapply le_S_S_to_le to H3 as K;
+ clear H4 Hcut H3 H H1 H2; generalize in match K; clear K;
+ apply (nat_elim2 ???? n1 (\len l1)); simplify; intros;
+ [1: rewrite < minus_n_O; cases n2 in H; [intro; cases (not_le_Sn_O ? H)]
+ intros; cases n3; simplify; reflexivity;
+ |2: cases (not_le_Sn_O ? H);
+ |3: apply H; apply le_S_S_to_le; apply H1;]]]
+qed.
+
+lemma make_list_ext: ∀T,f1,f2,n. (∀x.x<n → f1 x = f2 x) → make_list T f1 n = make_list T f2 n.
+intros 4;elim n; [reflexivity] simplify; rewrite > H1; [2: apply le_n]
+apply eq_f; apply H; intros; apply H1; apply (trans_le ??? H2); apply le_S; apply le_n;
+qed.
+
+lemma len_copy: ∀l. \len l = \len (copy l).
+intro; elim l; [reflexivity] simplify; rewrite > H; clear H;
+apply eq_f; elim (\len (copy l1)) in ⊢ (??%(??(???%))); [reflexivity] simplify;
+rewrite > H in ⊢ (??%?); reflexivity;
+qed.
+
+lemma same_bases_cons: ∀a,b,l1,l2.
+ same_bases l1 l2 → \fst a = \fst b → same_bases (a::l1) (b::l2).
+intros; intro; cases i; simplify; [assumption;] apply (H n);
+qed.
+
+lemma minus_lt : ∀i,j. i < j → j - i = S (j - S i).
+intros 2;
+apply (nat_elim2 ???? i j); simplify; intros;
+[1: cases n in H; intros; rewrite < minus_n_O; [cases (not_le_Sn_O ? H);]
+ simplify; rewrite < minus_n_O; reflexivity;
+|2: cases (not_le_Sn_O ? H);
+|3: apply H; apply le_S_S_to_le; assumption;]
+qed.
+
+lemma copy_same_bases: ∀l. same_bases l (copy l).
+intro; unfold copy; elim l using list_elim_with_len; [1: intro;reflexivity]
+simplify; rewrite < minus_n_n;
+simplify in ⊢ (? ? (? ? (? ? ? % ?) ?));
+apply same_bases_cons; [2: reflexivity]
+cases l1 in H; [intros 2; reflexivity]
+simplify in ⊢ (? ? (? ? (λ_:?.? ? ? (? ? %) ?) ?)→?);
+simplify in ⊢ (?→? ? (? ? (λ_:?.? ? ? (? ? (? % ?)) ?) ?));
+intro; rewrite > (make_list_ext ?? (λn:nat.〈nth_base (b::l2) (\len l2-n),〈OQ,OQ〉〉));[assumption]
+intro; elim x; [simplify; rewrite < minus_n_O; reflexivity]
+simplify in ⊢ (? ? (? ? ? (? ? %) ?) ?);
+simplify in H2:(? ? %); rewrite > minus_lt; [reflexivity] apply le_S_S_to_le;
+assumption;
+qed.
+