--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "models/q_bars.ma".
+
+(* 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.
+
+alias symbol "lt" = "bar lt".
+lemma inversion_sorted:
+ ∀a,l. sorted q2_lt (a::l) → Or (a < \hd ▭ l) (l = []).
+intros 2; elim l; [right;reflexivity] left; inversion H1; intros;
+[1,2:destruct H2| destruct H5; assumption]
+qed.
+
+lemma inversion_sorted2:
+ ∀a,b,l. sorted q2_lt (a::b::l) → a < b.
+intros; inversion H; intros; [1,2:destruct H1] destruct H4; assumption;
+qed.
+
+let rec copy (l : list bar) on l : list bar ≝
+ match l with
+ [ nil ⇒ []
+ | cons x tl ⇒ 〈\fst x, 〈OQ,OQ〉〉 :: copy tl].
+
+lemma sorted_copy:
+ ∀l:list bar.sorted q2_lt l → sorted q2_lt (copy l).
+intro l; elim l; [apply (sorted_nil q2_lt)] simplify;
+cases l1 in H H1; simplify; intros; [apply (sorted_one q2_lt)]
+apply (sorted_cons q2_lt); [2: apply H; apply (sorted_tail q2_lt ?? H1);]
+apply (inversion_sorted2 ??? H1);
+qed.
+
+lemma len_copy: ∀l. \len (copy l) = \len l.
+intro; elim l; [reflexivity] simplify; apply eq_f; assumption;
+qed.
+
+lemma copy_same_bases: ∀l. same_bases l (copy l).
+intros; elim l; [intro; reflexivity] intro; simplify; cases i; [reflexivity]
+simplify; apply (H n);
+qed.
+
+lemma copy_OQ : ∀l,n.nth_height (copy l) n = 〈OQ,OQ〉.
+intro; elim l; [elim n;[reflexivity] simplify; assumption]
+simplify; cases n; [reflexivity] simplify; apply (H n1);
+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.
+
+
+lemma sort_q2lt_same_base:
+ ∀b,h1,h2,l. sorted q2_lt (〈b,h1〉::l) → sorted q2_lt (〈b,h2〉::l).
+intros; cases (inversion_sorted ?? H); [2: rewrite > H1; apply (sorted_one q2_lt)]
+lapply (sorted_tail q2_lt ?? H) as K; clear H; cases l in H1 K; simplify; intros;
+[apply (sorted_one q2_lt);|apply (sorted_cons q2_lt);[2: assumption] apply H]
+qed.
+
+lemma value_head : ∀a,l,i.Qpos i ≤ \fst a → value_simple (a::l) i = \snd a.
+intros; unfold value_simple; unfold match_domain; cases (cases_find bar (match_pred i) (a::l) ▭);
+[1: cases i1 in H2 H3 H4; intros; [reflexivity] lapply (H4 O) as K; [2: apply le_S_S; apply le_O_n;]
+ simplify in K; unfold match_pred in K; cases (q_cmp (Qpos i) (\fst a)) in K;
+ simplify; intros; [destruct H6] lapply (q_le_lt_trans ??? H H5) as K; cases (q_lt_corefl ? K);
+|2: cases (?:False); lapply (H3 0); [2: simplify; apply le_S_S; apply le_O_n;]
+ unfold match_pred in Hletin; simplify in Hletin; cases (q_cmp (Qpos i) (\fst a)) in Hletin;
+ simplify; intros; [destruct H5] lapply (q_le_lt_trans ??? H H4); apply (q_lt_corefl ? Hletin);]
+qed.
+
+lemma value_unit : ∀x,i. value_simple [x] i = \snd x.
+intros; unfold value_simple; unfold match_domain;
+cases (cases_find bar (match_pred i) [x] ▭);
+[1: cases i1 in H; intros; [reflexivity] simplify in H;
+ cases (not_le_Sn_O ? (le_S_S_to_le ?? H));
+|2: simplify in H; destruct H; reflexivity;]
+qed.
+
+lemma value_tail :
+ ∀a,b,l,i.\fst a < Qpos i → \fst b ≤ Qpos i → value_simple (a::b::l) i = value_simple (b::l) i.
+intros; unfold value_simple; unfold match_domain;
+cases (cases_find bar (match_pred i) (a::b::l) ▭);
+[1: cases i1 in H3 H2 H4 H5; intros 1; simplify in ⊢ (? ? (? ? %) ?→?); unfold in ⊢ (? ? % ?→?); intros;
+ [1: cases (?:False); cases (q_cmp (Qpos i) (\fst a)) in H3; simplify; intros;[2: destruct H6]
+ apply (q_lt_corefl ? (q_lt_le_trans ??? H H3));
+ |2:
+
+normalize in ⊢ (? ? % ?→?); simplify;
+[1: rewrite > (value_head);
+
+lemma value_copy :
+ ∀l,i.rewrite > (value_u
+ value_simple (copy l) i = 〈OQ,OQ〉.
+intros; elim l;
+[1: reflexivity;
+|2: cases l1 in H; intros; simplify in ⊢ (? ? (? % ?) ?);
+ [1: rewrite > (value_unit); reflexivity;
+ |2: cases (q_cmp (\fst b) (Qpos i));
+
+ change with (\fst ▭ = \lamsimplify in ⊢ (? ? (? % ?) ?); unfold value_simple; unfold match_domain;
+ cases (cases_find bar (match_pred i) [〈\fst x,〈OQ,OQ〉〉] ▭);
+ [1: simplify in H1:(??%%);
+
+ unfold match_pred;
+ rewrite > (value_unit 〈\fst a,〈OQ,OQ〉〉); reflexivity;
+|2: intros; simplify in H2 H3 H4 ⊢ (? ? (? % ? ? ? ?) ?);
+ cases (q_cmp (Qpos i) (\fst b));
+ [2: rewrite > (value_tail ??? H2 H3 ? H4 H1); apply H;
+ |1: rewrite > (value_head ??? H2 H3 ? H4 H1); reflexivity]]
+qed.
+
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projects / helm.git / blobdiff