X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_bars.ma;h=5098fa724d380b3239c247eb94b02a16fb944c64;hb=a660b97f5a882da420809831581a7c3202fdaf35;hp=d5a7806e799bdc99aaedfa33b0048937920487a2;hpb=2ddda3a0f1e22c9b5c9572896cdaf69b3c4d19d2;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 d5a7806e7..5098fa724 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_bars.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_bars.ma @@ -14,363 +14,142 @@ include "nat_ordered_set.ma". include "models/q_support.ma". -include "models/list_support.ma". -include "cprop_connectives.ma". +include "models/list_support.ma". +include "logic/cprop_connectives.ma". -definition bar â â à â. +definition bar â â à (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©Qpos one,OQâª. +definition empty_bar : bar â â©Qpos one,â©OQ,OQâªâª. 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). -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 q2_lt := mk_rel bar (λx,y:bar.\fst x < \fst y). -definition nth_base â λf,n. \fst (nth f â n). -definition nth_height â λf,n. \snd (nth f â n). +interpretation "bar lt" 'lt x y = (rel_op _ q2_lt x y). -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 q2_trans : âa,b,c:bar. a < b â b < c â a < c. +intros 3; cases a; cases b; cases c; unfold q2_lt; simplify; intros; +apply (q_lt_trans ??? H H1); +qed. -lemma nth_nil: âT,i.âdef:T. nth [] def i = def. -intros; elim i; simplify; [reflexivity;] assumption; qed. +definition q2_trel := mk_trans_rel bar q2_lt q2_trans. -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. +interpretation "bar lt" 'lt x y = (FunClass_2_OF_trans_rel q2_trel x y). -inductive non_empty_list (A:Type) : list A â Type := -| show_head: âx,l. non_empty_list A (x::l). +definition canonical_q_lt : rel bar â trans_rel â λx:rel bar.q2_trel. -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. +coercion canonical_q_lt with nocomposites. -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. +interpretation "bar lt" 'lt x y = (FunClass_2_OF_trans_rel (canonical_q_lt _) x y). -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. +definition nth_base â λf,n. \fst (\nth f â n). +definition nth_height â λf,n. \snd (\nth f â n). -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;] +record q_f : Type â { + 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)) = â©OQ,OQ⪠+}. + +lemma len_bases_gt_O: âf.O < \len (bars f). +intros; generalize in match (bars_begin_OQ f); cases (bars f); intros; +[2: simplify; apply le_S_S; apply le_O_n; +|1: normalize in H; destruct H;] 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);] +cases (len_gt_non_empty ?? (len_bases_gt_O f)); intros; +cases (cmp_nat (\len l) i); +[2: lapply (sorted_tail_bigger q2_lt ?? â H ? H2) as K; + simplify in H1; rewrite < H1; apply K; +|1: simplify; elim l in i H2;[simplify; rewrite > nth_nil; apply (q_pos_OQ one)] + cases n in H3; intros; [simplify in H3; cases (not_le_Sn_O ? H3)] + apply (H2 n1); simplify in H3; 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 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 (acc: nat à â) (l : list bar) on l : nat à â â - match l with - [ nil â acc - | cons x tl â - 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. - 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; 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