(* *)
(**************************************************************************)
+include "russell_support.ma".
include "models/q_bars.ma".
-alias symbol "pi2" = "pair pi2".
-alias symbol "pi1" = "pair pi1".
definition rebase_spec ≝
- ∀l1,l2:q_f.∃p:q_f × q_f.
- And4
- (start (\fst p) = start (\snd p))
+ λl1,l2:q_f.λp:q_f × q_f.
+ And3
(same_bases (bars (\fst p)) (bars (\snd p)))
(same_values l1 (\fst p))
(same_values l2 (\snd p)).
-definition rebase_spec_simpl ≝
- λstart.λl1,l2:list bar.λp:(list bar) × (list bar).
- And3
- (same_bases (\fst p) (\snd p))
- (same_values (mk_q_f start l1) (mk_q_f start (\fst p)))
- (same_values (mk_q_f start l2) (mk_q_f start (\snd p))).
+definition same_values_simpl ≝
+ λl1,l2.∀p1,p2,p3,p4,p5,p6.same_values (mk_q_f l1 p1 p2 p3) (mk_q_f l2 p4 p5 p6).
-(* a local letin makes russell fail *)
-definition cb0h : list bar → list bar ≝
- λl.mk_list (λi.〈\fst (nth l ▭ i),OQ〉) (len l).
+alias symbol "pi2" = "pair pi2".
+alias symbol "pi1" = "pair pi1".
+definition rebase_spec_aux ≝
+ λl1,l2:list bar.λp:(list bar) × (list bar).
+ sorted q2_lt l1 → sorted q2_lt l2 →
+ (l1 ≠ [] → \snd (\nth l1 ▭ (pred (\len l1))) = 〈OQ,OQ〉) →
+ (l2 ≠ [] → \snd (\nth l2 ▭ (pred (\len l2))) = 〈OQ,OQ〉) →
+ And4
+ (nth_base l1 O = nth_base (\fst p) O ∨
+ nth_base l2 O = nth_base (\fst p) O)
+ (sorted q2_lt (\fst p) ∧ sorted q2_lt (\snd p))
+ ((l1 ≠ [] → \snd (\nth (\fst p) ▭ (pred (\len (\fst p)))) = 〈OQ,OQ〉) ∧
+ (l2 ≠ [] → \snd (\nth (\snd p) ▭ (pred (\len (\snd p)))) = 〈OQ,OQ〉))
+ (And3
+ (same_bases (\fst p) (\snd p))
+ (same_values_simpl l1 (\fst p))
+ (same_values_simpl l2 (\snd p))).
definition eject ≝
λP.λp:∃x:(list bar) × (list bar).P x.match p with [ex_introT p _ ⇒ p].
axiom devil : False.
-definition rebase: rebase_spec.
-intros 2 (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
-letin spec ≝ (
- λs.λl1,l2.λm.λz.len l1 + len l2 < m → rebase_spec_simpl s l1 l2 z);
-alias symbol "pi1" (instance 34) = "exT \fst".
-alias symbol "pi1" (instance 21) = "exT \fst".
+definition copy ≝
+ 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; elim l1; intros 4;
+[1: split; [left; reflexivity]; split; try assumption; unfold; intros;
+ unfold same_values; intros; reflexivity;
+|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 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".
+alias symbol "pi2" = "pair pi2".
+alias symbol "pi1" = "pair pi1".
+alias symbol "minus" = "Q minus".
letin aux ≝ (
-let rec aux (l1,l2:list bar) (n:nat) on n : (list bar) × (list bar) ≝
+let rec aux (l1,l2:list bar) (n : nat) on n : (list bar) × (list bar) ≝
match n with
-[ O ⇒ 〈 nil ? , nil ? 〉
-| S m ⇒
+[ O ⇒ 〈[], []〉
+| S m ⇒
match l1 with
- [ nil ⇒ 〈cb0h l2, l2〉
+ [ nil ⇒ 〈copy l2, l2〉
| cons he1 tl1 ⇒
match l2 with
- [ nil ⇒ 〈l1, cb0h l1〉
+ [ nil ⇒ 〈l1, copy l1〉
| cons he2 tl2 ⇒
- let base1 ≝ Qpos (\fst he1) in
- let base2 ≝ Qpos (\fst he2) in
- let height1 ≝ (\snd he1) in
- let height2 ≝ (\snd he2) in
+ let base1 ≝ \fst he1 in
+ let base2 ≝ \fst he2 in
+ let height1 ≝ \snd he1 in
+ let height2 ≝ \snd he2 in
match q_cmp base1 base2 with
- [ q_eq _ ⇒
- let rc ≝ aux tl1 tl2 m in
- 〈he1 :: \fst rc,he2 :: \snd rc〉
- | q_lt Hp ⇒
- let rest ≝ base2 - base1 in
- let rc ≝ aux tl1 (〈\fst (unpos rest ?),height2〉 :: tl2) m in
- 〈〈\fst he1,height1〉 :: \fst rc,〈\fst he1,height2〉 :: \snd rc〉
+ [ q_leq Hp1 ⇒
+ match q_cmp base2 base1 with
+ [ q_leq Hp2 ⇒
+ let rc ≝ aux tl1 tl2 m in
+ 〈he1 :: \fst rc,he2 :: \snd rc〉
+ | q_gt Hp ⇒
+ let rest ≝ base2 - base1 in
+ let rc ≝ aux tl1 (〈rest,height2〉 :: tl2) m in
+ 〈〈base1,height1〉 :: \fst rc,〈base1,height2〉 :: \snd rc〉]
| q_gt Hp ⇒
let rest ≝ base1 - base2 in
- let rc ≝ aux (〈\fst (unpos rest ?),height1〉 :: tl1) tl2 m in
- 〈〈\fst he2,height1〉 :: \fst rc,〈\fst he2,height2〉 :: \snd rc〉
-]]]]
-in aux : ∀l1,l2,m.∃z.∀s.spec s l1 l2 m z); unfold spec;
-[9: clearbody aux; unfold spec in aux; clear spec;
- cases (q_cmp s1 s2);
- [1: cases (aux l1 l2 (S (len l1 + len l2)));
- cases (H1 s1 (le_n ?)); clear H1;
- exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s2 (\snd w)〉] split;
- [1,2: assumption;
- |3: intro; apply (H3 input);
- |4: intro; rewrite > H in H4;
- rewrite > (H4 input) in ⊢ (? ? % ?); reflexivity;]
- |2: letin l2' ≝ (〈\fst (unpos (s2-s1) ?),OQ〉::l2);[
- apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
- assumption]
- cases (aux l1 l2' (S (len l1 + len l2')));
- cases (H1 s1 (le_n ?)); clear H1 aux;
- exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s1 (\snd w)〉] split;
- [1: reflexivity
- |2: assumption;
- |3: assumption;
- |4: intro;
- rewrite > (initial_shift_same_values (mk_q_f s2 l2) s1 H input) in ⊢ (? ? % ?);
- rewrite < (H4 input)in ⊢ (? ? ? %); reflexivity;]
- |3: letin l1' ≝ (〈\fst (unpos (s1-s2) ?),OQ〉::l1);[
- apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
- assumption]
- cases (aux l1' l2 (S (len l1' + len l2)));
- cases (H1 s2 (le_n ?)); clear H1 aux;
- exists [apply 〈mk_q_f s2 (\fst w), mk_q_f s2 (\snd w)〉] split;
- [1: reflexivity
- |2: assumption;
- |4: assumption;
- |3: intro; simplify in ⊢ (? ? ? (? ? ? (? ? ? (? % ?))));
- rewrite > (initial_shift_same_values (mk_q_f s1 l1) s2 H input) in ⊢ (? ? % ?);
- rewrite < (H3 input) in ⊢ (? ? ? %); reflexivity;]]
-|1,2: unfold rest; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ let rc ≝ aux (〈rest,height1〉 :: tl1) tl2 m in
+ 〈〈base2,height1〉 :: \fst rc,〈base2,height2〉 :: \snd rc〉]]]]
+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 (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]
+ normalize in Hb1 Hb2; [destruct Hb1|destruct Hb2]]
+ unfold; unfold same_values; simplify in ⊢ (? (? % %) ? ?);
+ 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; 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: 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;
+
+ split; [left;reflexivity]
+ split;
+
+,2: unfold rest; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
assumption;
|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);
|3: intros; generalize in match (unpos ??); intro X; cases X; clear X;
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