axiom devil : False.
definition copy ≝
- λl:list bar.make_list ? (λn.〈nth_base l (n - \len l),〈OQ,OQ〉〉) (\len l).
+ λl:list bar.make_list ? (λn.〈nth_base l (\len l - S n),〈OQ,OQ〉〉) (\len l).
+
+lemma list_elim_with_len:
+ ∀T:Type.∀P: nat → list T → CProp.
+ P O [] → (∀l,a,n.P (\len l) l → P (S n) (a::l)) →
+ ∀l.P (\len l) l.
+intros;elim l; [assumption] simplify; apply H1; apply H2;
+qed.
+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.
+
lemma copy_rebases:
∀l1.rebase_spec_aux l1 [] 〈l1, copy l1〉.
-intros; cases l1; intros 4;
+intros; elim l1; intros 4;
[1: split; [left; reflexivity]; split; try assumption; unfold; intros;
unfold same_values; intros; reflexivity;
-|2: rewrite > H2; [2: intro X; destruct X] clear H2 H3;
+|2: rewrite > H3; [2: intro X; destruct X]
split; [left; reflexivity] split;
unfold same_values_simpl; unfold same_values; intros; try reflexivity;
- try assumption; [4: normalize in p2; destruct p2|2: cases H2; reflexivity;]
- simplify; clear H1;
- [1: elim (\len l) in H; simplify; [apply (sorted_one q2_lt);]
-
-
-
-
+ try assumption; [4: normalize in p2; destruct p2|2: cases H5; reflexivity;]
+ [1: apply (sorted_copy ? H1);
+ |2: apply (copy_same_bases (a::l));]]
+qed.
+
+lemma copy_rebases_r:
+ ∀l1.rebase_spec_aux [] l1 〈copy l1, l1〉.
+intros; elim l1; intros 4;
+[1: split; [left; reflexivity]; split; try assumption; unfold; intros;
+ unfold same_values; intros; reflexivity;
+|2: rewrite > H4; [2: intro X; destruct X]
+ split; [right; simplify; rewrite < minus_n_n; reflexivity] split;
+ unfold same_values_simpl; unfold same_values; intros; try reflexivity;
+ try assumption; [4: normalize in p2; destruct p2|2: cases H5; reflexivity;]
+ [1: apply (sorted_copy ? H2);
+ |2: intro; symmetry; apply (copy_same_bases (a::l));]]
+qed.
+
definition rebase: ∀l1,l2:q_f.∃p:q_f × q_f.rebase_spec l1 l2 p.
intros 2 (f1 f2); cases f1 (b1 Hs1 Hb1 He1); cases f2 (b2 Hs2 Hb2 He2); clear f1 f2;
alias symbol "plus" = "natural plus".
let rest ≝ base1 - base2 in
let rc ≝ aux (〈rest,height1〉 :: tl1) tl2 m in
〈〈base2,height1〉 :: \fst rc,〈base2,height2〉 :: \snd rc〉]]]]
-in aux : ∀l1,l2,m.∃z.m = \len l1 + \len l2 → rebase_spec_aux l1 l2 z);
+in aux : ∀l1,l2,m.∃z.\len l1 + \len l2 ≤ m → rebase_spec_aux l1 l2 z);
[7: clearbody aux; cases (aux b1 b2 (\len b1 + \len b2)) (w Hw); clear aux;
- cases (Hw (refl_eq ??) Hs1 Hs2 (λ_.He1) (λ_.He2)); clear Hw; cases H1; cases H2; cases H3; clear H3 H1 H2;
+ cases (Hw (le_n ?) Hs1 Hs2 (λ_.He1) (λ_.He2)); clear Hw; cases H1; cases H2; cases H3; clear H3 H1 H2;
exists [constructor 1;constructor 1;[apply (\fst w)|5:apply (\snd w)]] try assumption;
[1,3: apply hide; cases H (X X); try rewrite < (H8 O); try rewrite < X; assumption
|2,4: apply hide;[apply H6|apply H7]intro X;[rewrite > X in Hb1|rewrite > X in Hb2]
simplify in match (\snd 〈?,?〉); simplify in match (\fst 〈?,?〉);
split; [assumption; |apply H9;|apply H10]
|6: intro ABS; unfold; intros 4; clear H1 H2;
- cases l in ABS H3; intros 1; [2: simplify in H1; destruct H1]
- cases l1 in H4 H1; intros; [2: simplify in H2; destruct H2]
- split; [left;reflexivity|split; apply (sorted_nil q2_lt);|split; assumption;]
+ cases l in ABS H3; intros 1; [2: simplify in H1; cases (not_le_Sn_O ? H1)]
+ cases l1 in H4 H1; intros; [2: simplify in H2; cases (not_le_Sn_O ? H2)]
+ split; [ left; reflexivity|split; apply (sorted_nil q2_lt);|split; assumption;]
split; unfold; intros; unfold same_values; intros; reflexivity;
-|5: unfold rebase_spec_aux; intros; cases l1 in H2 H4 H6; intros; [ simplify in H2; destruct H2;]
+|5: intros; apply copy_rebases_r;
+|4: intros; rewrite < H1; apply copy_rebases;
+|3: cut (\fst b = \fst b3) as K; [2: apply q_le_to_le_to_eq; assumption] clear H6 H5 H4 H3;
+ intros; cases (aux l2 l3 n1); intros 4; simplify in match (\fst ≪w,H≫);
+ simplify in match (\fst 〈?,?〉); simplify in match (\snd 〈?,?〉);
+ cases H4;
+ [2: apply le_S_S_to_le; apply (trans_le ???? H3); simplify;
+ rewrite < plus_n_Sm; apply le_S; apply le_n;
+ |3,4: apply (sorted_tail q2_lt); [2: apply H5|4:apply H6]
+ |5: intro; cases l2 in H7 H9; intros; [cases H9; reflexivity]
+ simplify in H7 ⊢ %; apply H7; intro; destruct H10;
+ |6: intro; cases l3 in H8 H9; intros; [cases H9; reflexivity]
+ simplify in H8 ⊢ %; apply H8; intro; destruct H10;]
+ clear aux; split;
+ [1: left; reflexivity;
+ |2: cases H10;
+
+
+
+ unfold rebase_spec_aux; intros; cases l1 in H2 H4 H6; intros; [ simplify in H2; destruct H2;]
lapply H6 as H7; [2: intro X; destruct X] clear H6 H5;
rewrite > H7; split; [right; simplify;
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