X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_bars.ma;h=1d2107b7c934722dc281345fb033a0d226f0059e;hb=e2a6d130d5274760f4591197bbbc66c3191e8a6a;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..1d2107b7c 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 +14,186 @@ 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 ≝ 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. 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). + +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,OQ〉 +}. -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;] +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);] -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) → - (∀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. - -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]] +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. -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〉. -intros; -alias symbol "pi2" = "pair pi2". -alias symbol "pi1" = "pair pi1". -alias symbol "lt" (instance 7) = "Q less than". +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". -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)); +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 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: ]]] - - - - - - - - - +alias symbol "lt" (instance 6) = "natural 'less than'". +definition value_lemma : + ∀f:list bar.sorted q2_lt f → O < length bar f → + ∀i:ratio.nth_base f O < Qpos i → ∃p:ℚ×ℚ.value_spec f (Qpos i) p. +intros (f bars_sorted_f len_bases_gt_O_f i bars_begin_OQ_f); +exists [apply (value_simple f i);] +apply (value_of ?? (match_domain f i)); +[1: reflexivity +|2: unfold match_domain; cases (cases_find bar (match_pred i) 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; + 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; + simplify; intros; lapply (H (\len l) (le_n ?)) as K; clear H; + unfold match_pred in K; cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ (\len l)))) in K; + simplify; intros; [destruct H2] assumption;] +|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 match_pred; + 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 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; + 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;] + 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]] +|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. -[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;] +lemma bars_begin_lt_Qpos : ∀q,r. nth_base (bars q) O bars_begin_OQ; apply q_pos_OQ; 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); +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}.