X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_bars.ma;h=1d2107b7c934722dc281345fb033a0d226f0059e;hb=fe7b82f0aaed4ecbf84f70ec6fb7dce3c7da04e9;hp=7279fe8c04bad616587f9e57e970d551e8248a6d;hpb=8d367045e504f594c280d2c87f906695ef9671ee;p=helm.git diff --git a/helm/software/matita/contribs/dama/dama/models/q_bars.ma b/helm/software/matita/contribs/dama/dama/models/q_bars.ma index 7279fe8c0..1d2107b7c 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_bars.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_bars.ma @@ -15,7 +15,7 @@ include "nat_ordered_set.ma". include "models/q_support.ma". include "models/list_support.ma". -include "cprop_connectives.ma". +include "logic/cprop_connectives.ma". definition bar ≝ ℚ × (ℚ × ℚ). @@ -75,35 +75,54 @@ cases (cmp_nat (\len l) i); apply (H2 n1); simplify in H3; apply (le_S_S_to_le ?? H3);] qed. -coinductive value_spec (f : q_f) (i : ℚ) : ℚ × ℚ → CProp ≝ +alias symbol "lt" (instance 9) = "Q less than". +alias symbol "lt" (instance 7) = "natural 'less than'". +alias symbol "lt" (instance 6) = "natural 'less than'". +alias symbol "lt" (instance 5) = "Q less than". +alias symbol "lt" (instance 4) = "natural 'less than'". +alias symbol "lt" (instance 2) = "natural 'less than'". +alias symbol "leq" = "Q less or equal than". +coinductive value_spec (f : list bar) (i : ℚ) : ℚ × ℚ → CProp ≝ | value_of : ∀j,q. - nth_height (bars f) j = q → nth_base (bars f) j < i → - (∀n.j < n → n < \len (bars f) → i ≤ nth_base (bars f) n) → value_spec f i q. + nth_height f j = q → nth_base f j < i → j < \len f → + (∀n.n H4; apply q_pos_OQ; - |2: cases (len_gt_non_empty ?? (len_bases_gt_O f)) in H3; - intros; lapply (H3 n (le_n ?)) as K; unfold P in K; + [1: generalize in match (bars_begin_OQ_f); + cases (len_gt_non_empty ?? (len_bases_gt_O_f)); simplify; intros; + assumption; + |2: cases (len_gt_non_empty ?? (len_bases_gt_O_f)) in H3; + intros; lapply (H3 n (le_n ?)) as K; unfold match_pred in K; cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ n))) in K; simplify; intros; [destruct H5] assumption] - |2: destruct H; cases (len_gt_non_empty ?? (len_bases_gt_O f)) in H2; + |2: destruct H; cases (len_gt_non_empty ?? (len_bases_gt_O_f)) in H2; simplify; intros; lapply (H (\len l) (le_n ?)) as K; clear H; - unfold P in K; cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ (\len l)))) in K; + unfold match_pred in K; cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ (\len l)))) in K; simplify; intros; [destruct H2] assumption;] -|3: intro; cases (cases_find bar P (bars f) ▭); intros; - [1: generalize in match (bars_sorted f); +|5: intro; unfold match_domain; cases (cases_find bar (match_pred i) f ▭); intros; + [1: generalize in match (bars_sorted_f); cases (list_break ??? H) in H1; rewrite > H6; - rewrite < H1; simplify; rewrite > nth_len; unfold P; + rewrite < H1; simplify; rewrite > nth_len; unfold match_pred; cases (q_cmp (Qpos i) (\fst x)); simplify; intros (X Hs); [2: destruct X] clear X; cases (sorted_pivot q2_lt ??? ▭ Hs); @@ -112,13 +131,13 @@ apply (value_of ?? (pred (find ? P (bars f) ▭))); apply lt_to_le; assumption;] unfold nth_base; rewrite > (nth_append_ge_len ????? Hn); cut (n - \len l1 < \len (x::l2)) as K; [2: - simplify; rewrite > H1; rewrite > (?:\len l2 = \len (bars f) - \len (l1 @ [x]));[2: + simplify; rewrite > H1; rewrite > (?:\len l2 = \len f - \len (l1 @ [x]));[2: rewrite > H6; repeat rewrite > len_append; simplify; repeat rewrite < plus_n_Sm; rewrite < plus_n_O; simplify; rewrite > sym_plus; rewrite < minus_plus_m_m; reflexivity;] rewrite > len_append; rewrite > H1; simplify; rewrite < plus_n_SO; - apply le_S_S; clear H1 H6 H7 Hs H8 H9 Hn x l2 l1 H4 H3 H2 H P i; - elim (\len (bars f)) in i1 n H5; [cases (not_le_Sn_O ? H);] + apply le_S_S; clear H1 H6 H7 Hs H8 H9 Hn x l2 l1 H4 H3 H2 H; + elim (\len f) in i1 n H5; [cases (not_le_Sn_O ? H);] simplify; cases n2; [ repeat rewrite < minus_n_O; apply le_S_S_to_le; assumption] cases n1 in H1; [intros; rewrite > eq_minus_n_m_O; apply le_O_n] intros; simplify; apply H; apply le_S_S_to_le; assumption;] @@ -127,19 +146,49 @@ apply (value_of ?? (pred (find ? P (bars f) ▭))); apply q_lt_to_le; apply W; |2: cases (not_le_Sn_n i1); rewrite > H in ⊢ (??%); apply (trans_le ??? ? H4); cases i1 in H3; intros; apply le_S_S; - [ apply le_O_n; | assumption]]] + [ apply le_O_n; | assumption]] +|3: unfold match_domain; cases (cases_find bar (match_pred i) f ▭); [ + cases i1 in H; intros; simplify; [assumption] + apply lt_S_to_lt; assumption;] + rewrite > H; cases (\len f) in len_bases_gt_O_f; intros; [cases (not_le_Sn_O ? H3)] + simplify; apply le_n; +|4: intros; unfold match_domain in H; cases (cases_find bar (match_pred i) f ▭) in H; simplify; intros; + [1: lapply (H3 n); [2: cases i1 in H4; intros [assumption] apply le_S; assumption;] + unfold match_pred in Hletin; cases (q_cmp (Qpos i) (\fst (\nth f ▭ n))) in Hletin; + simplify; intros; [destruct H6] assumption; + |2: destruct H; cases f in len_bases_gt_O_f H2 H3; clear H1; simplify; intros; + [cases (not_le_Sn_O ? H)] lapply (H1 n); [2: apply le_S; assumption] + unfold match_pred in Hletin; cases (q_cmp (Qpos i) (\fst (\nth (b::l) ▭ n))) in Hletin; + simplify; intros; [destruct H4] assumption;]] qed. +lemma bars_begin_lt_Qpos : ∀q,r. nth_base (bars q) O bars_begin_OQ; apply q_pos_OQ; +qed. + lemma value : q_f → ratio → ℚ × ℚ. -intros; cases (value_lemma q r); apply w; qed. +intros; cases (value_lemma (bars q) ?? r); +[ apply bars_sorted. +| apply len_bases_gt_O; +| apply w; +| apply bars_begin_lt_Qpos;] +qed. -lemma cases_value : ∀f,i. value_spec f (Qpos i) (value f i). -intros; unfold value; cases (value_lemma f i); assumption; qed. +lemma cases_value : ∀f,i. value_spec (bars f) (Qpos i) (value f i). +intros; unfold value; +cases (value_lemma (bars f) (bars_sorted f) (len_bases_gt_O f) i (bars_begin_lt_Qpos f i)); +assumption; +qed. definition same_values ≝ λl1,l2:q_f.∀input. value l1 input = value l2 input. definition same_bases ≝ λl1,l2:list bar. ∀i.\fst (\nth l1 ▭ i) = \fst (\nth l2 ▭ i). +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. + alias symbol "lt" = "Q less than". lemma unpos: ∀x:ℚ.OQ < x → ∃r:ratio.Qpos r = x. intro; cases x; intros; [2:exists [apply r] reflexivity]