X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=inline;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_bars.ma;h=d5a7806e799bdc99aaedfa33b0048937920487a2;hb=910c252965fe17d6b5af92e4658e7d02bac82d58;hp=d0d043f47e053ae50df4e8aaba1ac9f510a83d72;hpb=88b32d4e8fe371d59e41cd272064c9d486ae7ec5;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 d0d043f47..d5a7806e7 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_bars.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_bars.ma @@ -12,109 +12,359 @@ (* *) (**************************************************************************) +include "nat_ordered_set.ma". include "models/q_support.ma". include "models/list_support.ma". include "cprop_connectives.ma". -definition bar â ratio à â. (* base (Qpos) , height *) -record q_f : Type â { start : â; bars: list bar }. +definition bar â â à â. notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}. interpretation "Q x Q" 'q2 = (Prod Q Q). -definition empty_bar : bar â â©one,OQâª. +definition empty_bar : bar â â©Qpos one,OQâª. notation "\rect" with precedence 90 for @{'empty_bar}. interpretation "q0" 'empty_bar = empty_bar. notation < "\ldots\rect\square\EmptySmallSquare\ldots" with precedence 90 for @{'lq2}. -interpretation "lq2" 'lq2 = (list bar). - -let rec sum_bases (l:list bar) (i:nat) on i â - match i with - [ O â OQ - | S m â - match l with - [ nil â sum_bases l m + Qpos one - | cons x tl â sum_bases tl m + Qpos (\fst x)]]. - -axiom sum_bases_empty_nat_of_q_ge_OQ: - âq:â.OQ ⤠sum_bases [] (nat_of_q q). -axiom sum_bases_empty_nat_of_q_le_q: - âq:â.sum_bases [] (nat_of_q q) ⤠q. -axiom sum_bases_empty_nat_of_q_le_q_one: - âq:â.q < sum_bases [] (nat_of_q q) + Qpos one. - -definition eject1 â +interpretation "lq2" 'lq2 = (list bar). + +inductive sorted : list bar â Prop â +| sorted_nil : sorted [] +| sorted_one : âx. sorted [x] +| sorted_cons : âx,y,tl. \fst x < \fst y â sorted (y::tl) â sorted (x::y::tl). + +definition nth_base â λf,n. \fst (nth f â n). +definition nth_height â λf,n. \snd (nth f â n). + +record q_f : Type â { + bars: list bar; + bars_sorted : sorted bars; + bars_begin_OQ : nth_base bars O = OQ; + bars_tail_OQ : nth_height bars (pred (len bars)) = OQ +}. + +lemma nth_nil: âT,i.âdef:T. nth [] def i = def. +intros; elim i; simplify; [reflexivity;] assumption; qed. + +lemma len_concat: âT:Type.âl1,l2:list T. len (l1@l2) = len l1 + len l2. +intros; elim l1; [reflexivity] simplify; rewrite < H; reflexivity; +qed. + +inductive non_empty_list (A:Type) : list A â Type := +| show_head: âx,l. non_empty_list A (x::l). + +lemma bars_not_nil: âf:q_f.non_empty_list ? (bars f). +intro f; generalize in match (bars_begin_OQ f); cases (bars f); +[1: intro X; normalize in X; destruct X; +|2: intros; constructor 1;] +qed. + +lemma sorted_tail: âx,l.sorted (x::l) â sorted l. +intros; inversion H; intros; [destruct H1;|destruct H1;constructor 1;] +destruct H4; assumption; +qed. + +lemma sorted_skip: âx,y,l. sorted (x::y::l) â sorted (x::l). +intros; inversion H; intros; [1,2: destruct H1] +destruct H4; inversion H2; intros; [destruct H4] +[1: destruct H4; constructor 2; +|2: destruct H7; constructor 3; [apply (q_lt_trans ??? H1 H4);] + apply (sorted_tail ?? H2);] +qed. + +lemma sorted_tail_bigger : âx,l.sorted (x::l) â âi. i < len l â \fst x < nth_base l i. +intros 2; elim l; [ cases (not_le_Sn_O i H1);] +cases i in H2; +[2: intros; apply (H ? n);[apply (sorted_skip ??? H1)|apply le_S_S_to_le; apply H2] +|1: intros; inversion H1; intros; [1,2: destruct H3] + destruct H6; simplify; assumption;] +qed. + +lemma all_bases_positive : âf:q_f.âi. OQ < nth_base (bars f) (S i). +intro f; generalize in match (bars_begin_OQ f); generalize in match (bars_sorted f); +cases (bars_not_nil f); intros; +cases (cmp_nat i (len l)); +[1: lapply (sorted_tail_bigger ?? H ? H2) as K; simplify in H1; + rewrite > H1 in K; apply K; +|2: rewrite > H2; simplify; elim l; simplify; [apply (q_pos_OQ one)] + assumption; +|3: simplify; elim l in i H2;[simplify; rewrite > nth_nil; apply (q_pos_OQ one)] + cases n in H3; intros; [cases (not_le_Sn_O ? H3)] apply (H2 n1); + apply (le_S_S_to_le ?? H3);] +qed. + +lemma lt_n_plus_n_Sm : ân,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 nth_concat_lt_len: + âT:Type.âl1,l2:list T.âdef.âi.i < len l1 â nth (l1@l2) def i = nth l1 def i. +intros 4; elim l1; [cases (not_le_Sn_O ? H)] cases i in H H1; simplify; intros; +[reflexivity| rewrite < H;[reflexivity] apply le_S_S_to_le; apply H1] +qed. + +lemma nth_concat_ge_len: + âT:Type.âl1,l2:list T.âdef.âi. + len l1 ⤠i â nth (l1@l2) def i = nth l2 def (i - len l1). +intros 4; elim l1; [ rewrite < minus_n_O; reflexivity] +cases i in H1; simplify; intros; [cases (not_le_Sn_O ? H1)] +apply H; apply le_S_S_to_le; apply H1; +qed. + +lemma nth_len: + âT:Type.âl1,l2:list T.âdef,x. + nth (l1@x::l2) def (len l1) = x. +intros 2; elim l1;[reflexivity] simplify; apply H; qed. + +lemma all_bigger_can_concat_bigger: + âl1,l2,start,b,x,n. + (âi.i< len l1 â nth_base l1 i < \fst b) â + (âi.i< len l2 â \fst b ⤠nth_base l2 i) â + (âi.i< len l1 â start ⤠i â x ⤠nth_base l1 i) â + start ⤠n â n < len (l1@b::l2) â x ⤠\fst b â 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. + +lemma sorted_head_smaller: + âl,p. sorted (p::l) â âi.i < len l â \fst p < nth_base l i. +intro l; elim l; intros; [cases (not_le_Sn_O ? H1)] cases i in H2; simplify; intros; +[1: inversion H1; [1,2: simplify; intros; destruct H3] intros; destruct H6; assumption; +|2: apply (H p ? n ?); [apply (sorted_skip ??? H1)] apply le_S_S_to_le; apply H2] +qed. + + +alias symbol "pi1" = "pair pi1". +alias symbol "lt" (instance 6) = "Q less than". +alias symbol "lt" (instance 2) = "Q less than". +alias symbol "and" = "logical and". +lemma sorted_pivot: + âl1,l2,p. sorted (l1@p::l2) â + (âi. i < len l1 â nth_base l1 i < \fst p) ⧠+ (âi. i < len l2 â \fst p < nth_base l2 i). +intro l; elim l; +[1: split; [intros; cases (not_le_Sn_O ? H1);] intros; + apply sorted_head_smaller; assumption; +|2: cases (H ?? (sorted_tail a (l1@p::l2) H1)); + lapply depth = 0 (sorted_head_smaller (l1@p::l2) a H1) as Hs; + split; simplify; intros; + [1: cases i in H4; simplify; intros; + [1: lapply depth = 0 (Hs (len l1)) as HS; + unfold nth_base in HS; rewrite > nth_len in HS; apply HS; + rewrite > len_concat; simplify; apply lt_n_plus_n_Sm; + |2: apply (H2 n); apply le_S_S_to_le; apply H4] + |2: apply H3; assumption]] +qed. + +definition eject_NxQ â λP.λp:âx:nat à â.P x.match p with [ex_introT p _ â p]. -coercion eject1. -definition inject1 â λP.λp:nat à â.λh:P p. ex_introT ? P p h. -coercion inject1 with 0 1 nocomposites. - -definition value : - âf:q_f.âi:â.âp:nat à â. - match q_cmp i (start f) with - [ q_lt _ â \snd p = OQ - | _ â - And3 - (sum_bases (bars f) (\fst p) ⤠â [i,start f]) - (â [i, start f] < sum_bases (bars f) (S (\fst p))) - (\snd p = \snd (nth (bars f) â (\fst p)))]. +coercion eject_NxQ. +definition inject_NxQ â λP.λp:nat à â.λh:P p. ex_introT ? P p h. +coercion inject_NxQ with 0 1 nocomposites. + +definition value_spec : q_f â â â nat à â â Prop â + λf,i,q. nth_height (bars f) (\fst q) = \snd q ⧠+ (nth_base (bars f) (\fst q) < i ⧠+ ân.\fst q < n â n < len (bars f) â i ⤠nth_base (bars f) n). + +definition value : âf:q_f.âi:ratio.âp:â.âj.value_spec f (Qpos i) â©j,pâª. intros; 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 â ( + λf,i,q. And4 + (\fst q < len f) + (\snd q = nth_height f (\fst q)) + (nth_base f (\fst q) < i) + (ân.(\fst q) < n â n < len f â i ⤠nth_base f n)); +alias symbol "lt" (instance 5) = "Q less than". letin value â ( - let rec value (p: â) (l : list bar) on l â + let rec value (acc: nat à â) (l : list bar) on l : nat à â â match l with - [ nil â â©nat_of_q p,OQ⪠+ [ nil â acc | cons x tl â - match q_cmp p (Qpos (\fst x)) with - [ q_lt _ â â©O, \snd x⪠- | _ â - let rc â value (p - Qpos (\fst x)) tl in - â©S (\fst rc),\snd rcâª]] + match q_cmp (\fst x) (Qpos i) with + [ q_leq _ â value â©S (\fst acc), \snd x⪠tl + | q_gt _ â acc]] in value : - âacc,l.âp:nat à â. OQ ⤠acc â - And3 - (sum_bases l (\fst p) ⤠acc) - (acc < sum_bases l (S (\fst p))) - (\snd p = \snd (nth l â (\fst p)))); + âacc,l.âp:nat à â. + âstory. story @ l = bars f â S (\fst acc) = len story â + value_spec_aux story (Qpos i) acc â + 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 â©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 â©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 â©O,OQâª] simplify; reflexivity; - |*: cases (value â [i,start f] (bars f)) (p Hp); - cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value; - exists[1,3:apply p]; simplify; split; assumption;] -|1,3: intros; split; - [1,4: clear H2; cases (value (q-Qpos (\fst b)) l1); + [2: exists [apply â©O,OQâª] simplify; constructor 1; split; try assumption; + try reflexivity; apply q_lt_to_le; assumption; + |1: cases (bars f); [exists [apply â©O,OQâª] simplify; constructor 3; split;try assumption;reflexivity;] + cases (value â [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 â©O,OQâª] simplify; constructor 2; split; try assumption; + try reflexivity; rewrite > H1; apply q_eq_to_le; reflexivity; + |3: exists [apply â©O,OQâª] 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 ⢠(??? % â %); intro X; cases X; clear X; + intros; + [1: exists [apply â©O,OQâª] simplify; constructor 3; split; reflexivity; + |2: cases (value â [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