X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_bars.ma;h=1d2107b7c934722dc281345fb033a0d226f0059e;hb=e2a6d130d5274760f4591197bbbc66c3191e8a6a;hp=65066590f4baef3f754305ba709df1353d149bae;hpb=02b4aca8654dd4b0c16cab14bf145bbc1ae963f8;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 65066590f..1d2107b7c 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_bars.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_bars.ma @@ -15,14 +15,14 @@ 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. +definition bar ≝ ℚ × (ℚ × ℚ). notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}. interpretation "Q x Q" 'q2 = (Prod Q Q). -definition empty_bar : bar ≝ 〈Qpos one,OQ〉. +definition empty_bar : bar ≝ 〈Qpos one,〈OQ,OQ〉〉. notation "\rect" with precedence 90 for @{'empty_bar}. interpretation "q0" 'empty_bar = empty_bar. @@ -55,7 +55,7 @@ record q_f : Type ≝ { bars: list bar; bars_sorted : sorted q2_lt bars; bars_begin_OQ : nth_base bars O = OQ; - bars_end_OQ : nth_height bars (pred (\len bars)) = OQ + bars_end_OQ : nth_height bars (pred (\len bars)) = 〈OQ,OQ〉 }. lemma len_bases_gt_O: ∀f.O < \len (bars f). @@ -75,80 +75,54 @@ cases (cmp_nat (\len l) i); apply (H2 n1); simplify in H3; apply (le_S_S_to_le ?? H3);] qed. -(* -lemma lt_n_plus_n_Sm : ∀n,m:nat.n < n + S m. -intros; rewrite > sym_plus; apply (le_S_S n (m+n)); apply (le_plus_n m n); qed. -*) - -(* -lemma all_bigger_can_concat_bigger: - ∀l1,l2,start,b,x,n. - (∀i.i< len l1 → nth_base l1 i < \fst b) → - (∀i.i< len l2 → \fst b ≤ nth_base l2 i) → - (∀i.i< len l1 → start ≤ i → x ≤ nth_base l1 i) → - start ≤ n → n < len (l1@b::l2) → x ≤ \fst b → x ≤ nth_base (l1@b::l2) n. -intros; cases (cmp_nat n (len l1)); -[1: unfold nth_base; rewrite > (nth_concat_lt_len ????? H6); - apply (H2 n); assumption; -|2: rewrite > H6; unfold nth_base; rewrite > nth_len; assumption; -|3: unfold nth_base; rewrite > nth_concat_ge_len; [2: apply lt_to_le; assumption] - rewrite > len_concat in H4; simplify in H4; rewrite < plus_n_Sm in H4; - lapply linear le_S_S_to_le to H4 as K; rewrite > sym_plus in K; - lapply linear le_plus_to_minus to K as X; - generalize in match X; generalize in match (n - len l1); intro W; cases W; clear W X; - [intros; assumption] intros; - apply (q_le_trans ??? H5); apply (H1 n1); assumption;] -qed. -*) - - -inductive value_spec (f : q_f) (i : ℚ) : ℚ → nat → CProp ≝ - value_of : ∀q,j. - 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 j. - - -inductive break_spec (T : Type) (n : nat) (l : list T) : list T → CProp ≝ -| break_to: ∀l1,x,l2. \len l1 = n → l = l1 @ [x] @ l2 → break_spec T n l l. - -lemma list_break: ∀T,n,l. n < \len l → break_spec T n l l. -intros 2; elim n; -[1: elim l in H; [cases (not_le_Sn_O ? H)] - apply (break_to ?? ? [] a l1); reflexivity; -|2: cases (H l); [2: apply lt_S_to_lt; assumption;] cases l2 in H3; intros; - [1: rewrite < H2 in H1; rewrite > H3 in H1; rewrite > append_nil in H1; - rewrite > len_append in H1; rewrite > plus_n_SO in H1; - cases (not_le_Sn_n ? H1); - |2: apply (break_to ?? ? (l1@[x]) t l3); - [2: simplify; rewrite > associative_append; assumption; - |1: rewrite < H2; rewrite > len_append; rewrite > plus_n_SO; reflexivity]]] -qed. - -definition value : ∀f:q_f.∀i:ratio.∃p:ℚ.∃j.value_spec f (Qpos i) p j. -intros; -letin P ≝ - (λx:bar.match q_cmp (Qpos i) (\fst x) with[ q_leq _ ⇒ true| q_gt _ ⇒ false]); -exists [apply (nth_height (bars f) (pred (find ? P (bars f) ▭)));] -exists [apply (pred (find ? P (bars f) ▭))] apply value_of; +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 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); @@ -157,13 +131,13 @@ exists [apply (pred (find ? P (bars f) ▭))] apply value_of; 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;] @@ -172,58 +146,54 @@ exists [apply (pred (find ? P (bars f) ▭))] apply value_of; 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 value_OQ_l: - ∀l,i.i < start l → \snd (\fst (value l i)) = OQ. -intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1; -try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H H6); -qed. - -lemma value_OQ_r: - ∀l,i.start l + sum_bases (bars l) (len (bars l)) ≤ i → \snd (\fst (value l i)) = OQ. -intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1; -try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H7 H); +lemma bars_begin_lt_Qpos : ∀q,r. nth_base (bars q) O bars_begin_OQ; apply q_pos_OQ; qed. - -lemma value_OQ_e: - ∀l,i.bars l = [] → \snd (\fst (value l i)) = OQ. -intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1; -try assumption; cases H2; cases (?:False); apply (H1 H); + +lemma value : q_f → ratio → ℚ × ℚ. +intros; cases (value_lemma (bars q) ?? r); +[ apply bars_sorted. +| apply len_bases_gt_O; +| apply w; +| apply bars_begin_lt_Qpos;] qed. -inductive value_ok_spec (f : q_f) (i : ℚ) : nat × ℚ → Type ≝ - | value_ok : ∀n,q. n ≤ (len (bars f)) → - q = \snd (nth (bars f) ▭ n) → - sum_bases (bars f) n ≤ ⅆ[i,start f] → - ⅆ[i, start f] < sum_bases (bars f) (S n) → value_ok_spec f i 〈n,q〉. - -lemma value_ok: - ∀f,i.bars f ≠ [] → start f ≤ i → i < start f + sum_bases (bars f) (len (bars f)) → - value_ok_spec f i (\fst (value f i)). -intros; cases (value f i); simplify; -cases H3; simplify; clear H3; cases H4; clear H4; -[1,2,3: cases (?:False); - [1: apply (q_lt_le_incompat ?? H3 H1); - |2: apply (q_lt_le_incompat ?? H2 H3); - |3: apply (H H3);] -|4: cases H7; clear H7; cases w in H3 H4 H5 H6 H8; simplify; intros; - constructor 1; assumption;] +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.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)). +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). -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] cases (?:False); -[ apply (q_lt_corefl ? H)|apply (q_neg_gt ? H)] +[ apply (q_lt_corefl ? H)| cases (q_lt_le_incompat ?? (q_neg_gt ?) (q_lt_to_le ?? H))] qed. notation < "x \blacksquare" non associative with precedence 50 for @{'unpos $x}.