[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,h1,h2,i,h3.
- value_simpl (copy l) h1 h2 i h3 = 〈OQ,OQ〉.
-intros; elim l in h1 h2 h3; [cases (not_le_Sn_O ? H1);]
-cases l1 in H H1 H2 H3;
-[1: intros; simplify in ⊢ (? ? (? % ? ? ? ?) ?);
+ ∀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));