X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_bars.ma;h=de39589073d51967d81528130c9afa59c4e858c4;hb=b5564e329d48efa6c2ca01da18203def26a70294;hp=b5c62219c616edbca96710eaaa4cf35b3e3126e0;hpb=8f4162a9db17a597d4fba49eb957009fc0268378;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 b5c62219c..de3958907 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). @@ -75,46 +75,17 @@ 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. - - -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]); +coinductive value_spec (f : q_f) (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. + +definition value_lemma : ∀f:q_f.∀i:ratio.∃p:ℚ×ℚ.value_spec f (Qpos i) p. +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; +apply (value_of ?? (pred (find ? P (bars f) ▭))); [1: reflexivity |2: cases (cases_find bar P (bars f) ▭); [1: cases i1 in H H1 H2 H3; simplify; intros; @@ -130,208 +101,50 @@ exists [apply (pred (find ? P (bars f) ▭))] apply value_of; unfold P 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_begin_OQ f); generalize in match (bars_sorted f); -generalize in match (bars_end_OQ f); -cases (len_gt_non_empty ?? (len_bases_gt_O f)); simplify; -intros; -[1: - - -alias symbol "pi2" = "pair pi2". -alias symbol "pi1" = "pair pi1". -alias symbol "lt" (instance 7) = "Q less than". -alias symbol "leq" = "Q less or equal than". -letin value_spec_aux ≝ ( - λf,i,q. And4 - (\fst q < len f) - (\snd q = nth_height f (\fst q)) - (nth_base f (\fst q) < i) - (∀n.(\fst q) < n → n < len f → i ≤ nth_base f n)); -alias symbol "lt" (instance 5) = "Q less than". -letin value ≝ ( - let rec value (acc: nat × ℚ) (l : list bar) on l : nat × ℚ ≝ - match l with - [ nil ⇒ acc - | cons x tl ⇒ - match q_cmp (\fst x) (Qpos i) with - [ q_leq _ ⇒ value 〈S (\fst acc), \snd x〉 tl - | q_gt _ ⇒ acc]] - in value : - ∀acc,l.∃p:nat × ℚ. - ∀story. story @ l = bars f → S (\fst acc) = len story → - value_spec_aux story (Qpos i) acc → - value_spec_aux (story @ l) (Qpos i) p); -[4: clearbody value; unfold value_spec; - generalize in match (bars_begin_OQ f); - generalize in match (bars_sorted f); - cases (bars_not_nil f) in value; intros (value S); generalize in match (sorted_tail_bigger ?? S); - clear S; cases (value 〈O,\snd x〉 l) (p Hp); intros; - exists[apply (\snd p)];exists [apply (\fst p)] simplify; - cases (Hp [x] (refl_eq ??) (refl_eq ??) ?) (Hg HV); - [unfold; split; [apply le_n|reflexivity|rewrite > H; apply q_pos_OQ;] - intros; cases n in H2 H3; [intro X; cases (not_le_Sn_O ? X)] - intros; cases (not_le_Sn_O ? (le_S_S_to_le (S n1) O H3))] - split;[rewrite > HV; reflexivity] split; [assumption;] - intros; cases n in H4 H5; intros [cases (not_le_Sn_O ? H4)] - apply (H3 (S n1)); assumption; -|1: unfold value_spec_aux; clear value value_spec_aux H2; intros; - cases H4; clear H4; split; - [1: apply (trans_lt ??? H5); rewrite > len_concat; simplify; apply lt_n_plus_n_Sm; - |2: unfold nth_height; rewrite > nth_concat_lt_len;[2:assumption]assumption; - |3: unfold nth_base; rewrite > nth_concat_lt_len;[2:assumption] - apply (q_le_lt_trans ???? H7); apply q_le_n; - |4: intros; (*clear H6 H5 H4 H l;*) lapply (bars_sorted f) as HS; - apply (all_bigger_can_concat_bigger story l1 (S (\fst p)));[6:apply q_lt_to_le]try assumption; - [1: rewrite < H2 in HS; cases (sorted_pivot ??? HS); assumption - |2: rewrite < H2 in HS; cases (sorted_pivot ??? HS); - intros; apply q_lt_to_le; apply H11; assumption; - |3: intros; apply H8; assumption;]] -|3: intro; rewrite > append_nil; intros; assumption; -|2: intros; cases (value 〈S (\fst p),\snd b〉 l1); unfold; simplify; - cases (H6 (story@[b]) ???); - [1: rewrite > associative_append; apply H3; - |2: simplify; rewrite > H4; rewrite > len_concat; rewrite > sym_plus; reflexivity; - |4: rewrite < (associative_append ? story [b] l1); split; assumption; - |3: cases H5; clear H5; split; simplify in match (\snd ?); simplify in match (\fst ?); - [1: rewrite > len_concat; simplify; rewrite < plus_n_SO; apply le_S_S; assumption; - |2: - |3: - |4: ]]] - - - - - - - - - - -[5: clearbody value; - cases (q_cmp i (start f)); - [2: exists [apply 〈O,OQ〉] simplify; constructor 1; split; try assumption; - try reflexivity; apply q_lt_to_le; assumption; - |1: cases (bars f); [exists [apply 〈O,OQ〉] simplify; constructor 3; split;try assumption;reflexivity;] - cases (value ⅆ[i,start f] (b::l)) (p Hp); - cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value; [cases H1; destruct H2] - cases H1; clear H1; lapply (sum_bases_O (b::l) (\fst p)) as H1; - [2: apply (q_le_trans ??? H2); rewrite > H; apply q_eq_to_le; - rewrite > q_d_x_x; reflexivity; - |1: exists [apply p] simplify; constructor 4; rewrite > H1; split; - try split; try rewrite > q_d_x_x; try autobatch depth=2; - [1: rewrite > H; rewrite > q_plus_sym; apply q_lt_plus; - rewrite > q_plus_minus; apply q_lt_plus_trans; [apply sum_bases_ge_OQ] - apply q_pos_lt_OQ; - |2: rewrite > H; rewrite > q_d_x_x; apply q_eq_to_le; reflexivity; - |3: rewrite > H; rewrite > q_d_x_x; apply q_lt_plus_trans; - try apply sum_bases_ge_OQ; apply q_pos_lt_OQ;]] - |3: cases (q_cmp i (start f+sum_bases (bars f) (len (bars f)))); - [1: exists [apply 〈O,OQ〉] simplify; constructor 2; split; try assumption; - try reflexivity; rewrite > H1; apply q_eq_to_le; reflexivity; - |3: exists [apply 〈O,OQ〉] simplify; constructor 2; split; try assumption; - try reflexivity; apply q_lt_to_le; assumption; - |2: generalize in match (refl_eq ? (bars f): bars f = bars f); - generalize in match (bars f) in ⊢ (??? % → %); intro X; cases X; clear X; - intros; - [1: exists [apply 〈O,OQ〉] simplify; constructor 3; split; reflexivity; - |2: cases (value ⅆ[i,start f] (b::l)) (p Hp); - cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value; [cases H3;destruct H4] - cases H3; clear H3; - exists [apply p]; constructor 4; split; try split; try assumption; - [1: intro X; destruct X; - |2: apply q_lt_to_le; assumption; - |3: rewrite < H2; assumption; - |4: cases (cmp_nat (\fst p) (len (bars f))); - [1:apply lt_to_le;rewrite H3;rewrite < H2;apply le_n] - cases (?:False); cases (\fst p) in H3 H4 H6; clear H5; - [1: intros; apply (not_le_Sn_O ? H5); - |2: rewrite > q_d_sym; rewrite > q_d_noabs; [2: apply q_lt_to_le; assumption] - intros; lapply (q_lt_inj_plus_r ?? (Qopp (start f)) H1); clear H1; - generalize in match Hletin; - rewrite > (q_plus_sym (start f)); rewrite < q_plus_assoc; - do 2 rewrite < q_elim_minus; rewrite > q_plus_minus; - rewrite > q_plus_OQ; intro K; apply (q_lt_corefl (i-start f)); - apply (q_lt_le_trans ???? H3); rewrite < H2; - apply (q_lt_trans ??? K); apply sum_bases_increasing; - assumption;]]]]] -|1,3: intros; right; split; - [1,4: clear H2; cases (value (q-Qpos (\fst b)) l1); - cases (H2 (q_le_to_diff_ge_OQ ?? (? H1))); - [1: intro; apply q_lt_to_le;assumption; - |3: simplify; cases H4; apply q_le_minus; assumption; - |2,5: simplify; cases H4; rewrite > H5; rewrite > H6; - apply q_le_minus; apply sum_bases_empty_nat_of_q_le_q; - |4: intro X; rewrite > X; apply q_eq_to_le; reflexivity; - |*: simplify; apply q_le_minus; cases H4; assumption;] - |2,5: cases (value (q-Qpos (\fst b)) l1); - cases (H4 (q_le_to_diff_ge_OQ ?? (? H1))); - [1,4: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption; - |3,6: cases H5; simplify; change with (q < sum_bases l1 (S (\fst w)) + Qpos (\fst b)); - apply q_lt_plus; assumption; - |2,5: simplify; cases H5; rewrite > H6; simplify; rewrite > H7; - apply q_lt_plus; apply sum_bases_empty_nat_of_q_le_q_one;] - |*: cases (value (q-Qpos (\fst b)) l1); simplify; - cases (H4 (q_le_to_diff_ge_OQ ?? (? H1))); - [1,4: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption; - |3,6: cases H5; assumption; - |*: cases H5; rewrite > H6; rewrite > H8; - elim (\fst w); [1,3:reflexivity;] simplify; assumption;]] -|2: clear value H2; simplify; intros; right; split; [assumption|3:reflexivity] - rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption; -|4: intros; left; split; reflexivity;] -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); -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); -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;] -qed. - -definition same_values ≝ - λl1,l2:q_f. - ∀input.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)). - -definition same_bases ≝ - λl1,l2:list bar. (∀i.\fst (nth l1 ▭ i) = \fst (nth l2 ▭ i)). + [1: generalize in match (bars_sorted f); + cases (list_break ??? H) in H1; rewrite > H6; + rewrite < H1; simplify; rewrite > nth_len; unfold P; + cases (q_cmp (Qpos i) (\fst x)); simplify; + intros (X Hs); [2: destruct X] clear X; + cases (sorted_pivot q2_lt ??? ▭ Hs); + cut (\len l1 ≤ n) as Hn; [2: + rewrite > H1; cases i1 in H4; simplify; intro X; [2: assumption] + 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: + 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);] + 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;] + cases (n - \len l1) in K; simplify; intros; [ assumption] + lapply (H9 ? (le_S_S_to_le ?? H10)) as W; apply (q_le_trans ??? H7); + 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]]] +qed. + +lemma value : q_f → ratio → ℚ × ℚ. +intros; cases (value_lemma q r); apply w; qed. + +lemma cases_value : ∀f,i. value_spec f (Qpos i) (value f i). +intros; unfold value; cases (value_lemma 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). 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}.