include "russell_support.ma".
include "models/q_bars.ma".
-definition rebase_spec ≝
- 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 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).
-
-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].
-coercion eject.
-definition inject ≝ λP.λp:(list bar) × (list bar).λh:P p. ex_introT ? P p h.
-coercion inject with 0 1 nocomposites.
+(* move in nat/minus *)
+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.
-axiom devil : False.
-
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));
|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;
+lemma len_copy: ∀l. \len (copy l) = \len l.
+intro; unfold copy; apply len_mk_list;
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;
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; rewrite > (mk_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 prepend_sorted_with_same_head:
+ ∀r,x,l1,l2,d1,d2.
+ sorted r (x::l1) → sorted r l2 →
+ (r x (\nth l1 d1 O) → r x (\nth l2 d2 O)) → (l1 = [] → r x d1) →
+ sorted r (x::l2).
+intros 8 (R x l1 l2 d1 d2 Sl1 Sl2); inversion Sl1; inversion Sl2;
+intros; destruct; try assumption; [3: apply (sorted_one R);]
+[1: apply sorted_cons;[2:assumption] apply H2; apply H3; reflexivity;
+|2: apply sorted_cons;[2: assumption] apply H5; apply H6; reflexivity;
+|3: apply sorted_cons;[2: assumption] apply H5; assumption;
+|4: apply sorted_cons;[2: assumption] apply H8; apply H4;]
+qed.
+
+lemma move_head_sorted: ∀x,l1,l2.
+ sorted q2_lt (x::l1) → sorted q2_lt l2 → nth_base l2 O = nth_base l1 O →
+ l1 ≠ [] → sorted q2_lt (x::l2).
+intros; apply (prepend_sorted_with_same_head q2_lt x l1 l2 ▭ ▭);
+try assumption; intros; unfold nth_base in H2; whd in H4;
+[1: rewrite < H2 in H4; assumption;
+|2: cases (H3 H4);]
+qed.
+
+definition rebase_spec ≝
+ 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 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).
+
+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))).
+
lemma copy_rebases:
∀l1.rebase_spec_aux l1 [] 〈l1, copy l1〉.
intros; elim l1; intros 4;
[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: apply (sorted_copy ? H2);
|2: intro; symmetry; apply (copy_same_bases (a::l));]]
qed.
-
+
+definition eject ≝
+ λP.λp:∃x:(list bar) × (list bar).P x.match p with [ex_introT p _ ⇒ p].
+coercion eject.
+definition inject ≝ λP.λp:(list bar) × (list bar).λh:P p. ex_introT ? P p h.
+coercion inject with 0 1 nocomposites.
+
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 "leq" = "natural 'less or equal to'".
alias symbol "minus" = "Q minus".
letin aux ≝ (
let rec aux (l1,l2:list bar) (n : nat) on n : (list bar) × (list bar) ≝
|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≫);
+ intros; cases (aux l2 l3 n1); cases w in H4 (w1 w2); clear w;
+ intros 5;
simplify in match (\fst 〈?,?〉); simplify in match (\snd 〈?,?〉);
- cases H4;
+ cases H5;
[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]
+ |3,4: apply (sorted_tail q2_lt); [2: apply H4|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;
+ clear aux H5;
+ simplify in match (\fst 〈?,?〉) in H9 H10 H11 H12;
+ simplify in match (\snd 〈?,?〉) in H9 H10 H11 H12;
+ split;
[1: left; reflexivity;
- |2: cases H10;
+ |2: cases H10; cases H12; clear H15 H16 H12 H7 H8 H11 H10;
+ cases H9; clear H9;
+ [1: lapply (H14 O) as K1; clear H14; change in K1 with (nth_base w1 O = nth_base w2 O);
+ split;
+ [1: apply (move_head_sorted ??? H4 H5 H7); STOP
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