X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_bars.ma;h=65066590f4baef3f754305ba709df1353d149bae;hb=02b4aca8654dd4b0c16cab14bf145bbc1ae963f8;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..65066590f 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_bars.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_bars.ma @@ -14,7 +14,7 @@ include "nat_ordered_set.ma". include "models/q_support.ma". -include "models/list_support.ma". +include "models/list_support.ma". include "cprop_connectives.ma". definition bar ≝ ℚ × ℚ. @@ -29,93 +29,58 @@ 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 → Prop ≝ -| sorted_nil : sorted [] -| sorted_one : ∀x. sorted [x] -| sorted_cons : ∀x,y,tl. \fst x < \fst y → sorted (y::tl) → sorted (x::y::tl). +definition q2_lt := mk_rel bar (λx,y:bar.\fst x < \fst y). -definition nth_base ≝ λf,n. \fst (nth f ▭ n). -definition nth_height ≝ λf,n. \snd (nth f ▭ n). +interpretation "bar lt" 'lt x y = (rel_op _ q2_lt x y). -record q_f : Type ≝ { - 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 : ∀a,b,c:bar. a < b → b < c → 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: ∀T,i.∀def: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: ∀T:Type.∀l1,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 → Type := -| show_head: ∀x,l. non_empty_list A (x::l). +definition canonical_q_lt : rel bar → trans_rel ≝ λx:rel bar.q2_trel. -lemma bars_not_nil: ∀f: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: ∀x,l.sorted (x::l) → 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: ∀x,y,l. sorted (x::y::l) → 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 ≝ λf,n. \fst (\nth f ▭ n). +definition nth_height ≝ λf,n. \snd (\nth f ▭ n). -lemma sorted_tail_bigger : ∀x,l.sorted (x::l) → ∀i. i < len l → \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 ≝ { + 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 +}. + +lemma len_bases_gt_O: ∀f.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 : ∀f:q_f.∀i. 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 ?? ▭ 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 : ∀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 nth_concat_lt_len: - ∀T:Type.∀l1,l2:list T.∀def.∀i.i < len l1 → 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: - ∀T:Type.∀l1,l2:list T.∀def.∀i. - len l1 ≤ i → 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: - ∀T:Type.∀l1,l2:list T.∀def,x. - nth (l1@x::l2) def (len l1) = x. -intros 2; elim l1;[reflexivity] simplify; apply H; qed. - +(* lemma all_bigger_can_concat_bigger: ∀l1,l2,start,b,x,n. (∀i.i< len l1 → nth_base l1 i < \fst b) → @@ -134,193 +99,81 @@ intros; cases (cmp_nat n (len l1)); [intros; assumption] intros; apply (q_le_trans ??? H5); apply (H1 n1); assumption;] qed. - -lemma sorted_head_smaller: - ∀l,p. sorted (p::l) → ∀i.i < len l → \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: - ∀l1,l2,p. sorted (l1@p::l2) → - (∀i. i < len l1 → nth_base l1 i < \fst p) ∧ - (∀i. i < len l2 → \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]] +*) + + +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 eject_NxQ ≝ - λP.λp:∃x:nat × ℚ.P x.match p with [ex_introT p _ ⇒ p]. -coercion eject_NxQ. -definition inject_NxQ ≝ λP.λp:nat × ℚ.λh:P p. ex_introT ? P p h. -coercion inject_NxQ with 0 1 nocomposites. - -definition value_spec : q_f → ℚ → nat × ℚ → Prop ≝ - λf,i,q. nth_height (bars f) (\fst q) = \snd q ∧ - (nth_base (bars f) (\fst q) < i ∧ - ∀n.\fst q < n → n < len (bars f) → i ≤ nth_base (bars f) n). - -definition value : ∀f:q_f.∀i:ratio.∃p:ℚ.∃j.value_spec f (Qpos i) 〈j,p〉. +definition value : ∀f:q_f.∀i:ratio.∃p:ℚ.∃j.value_spec f (Qpos i) p j. 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 ≝ ( - 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. +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; +[1: reflexivity +|2: cases (cases_find bar P (bars f) ▭); + [1: cases i1 in H H1 H2 H3; simplify; intros; + [1: generalize in match (bars_begin_OQ f); + cases (len_gt_non_empty ?? (len_bases_gt_O f)); simplify; intros; + rewrite > 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; + 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; + 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; + simplify; intros; [destruct H2] assumption;] +|3: intro; cases (cases_find bar P (bars 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; + 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_OQ_l: ∀l,i.i < start l → \snd (\fst (value l i)) = OQ.