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 ≝ ratio × ℚ. (* base (Qpos) , height *) -record q_f : Type ≝ { start : ℚ; bars: list bar }. +definition bar ≝ 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 ≝ 〈one,OQ〉. +definition empty_bar : bar ≝ 〈Qpos one,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). - -let rec sum_bases (l:list bar) (i:nat) on i ≝ - match i with - [ O ⇒ OQ - | S m ⇒ - match l with - [ nil ⇒ sum_bases l m + Qpos one - | cons x tl ⇒ sum_bases tl m + Qpos (\fst x)]]. - -axiom sum_bases_empty_nat_of_q_ge_OQ: - ∀q:ℚ.OQ ≤ sum_bases [] (nat_of_q q). -axiom sum_bases_empty_nat_of_q_le_q: - ∀q:ℚ.sum_bases [] (nat_of_q q) ≤ q. -axiom sum_bases_empty_nat_of_q_le_q_one: - ∀q:ℚ.q < sum_bases [] (nat_of_q q) + Qpos one. - -definition eject1 ≝ +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 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 bars; + bars_begin_OQ : nth_base bars O = OQ; + bars_tail_OQ : nth_height bars (pred (len bars)) = OQ +}. + +lemma nth_nil: ∀T,i.∀def:T. nth [] def i = def. +intros; elim i; simplify; [reflexivity;] assumption; qed. + +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. + +inductive non_empty_list (A:Type) : list A → Type := +| show_head: ∀x,l. non_empty_list A (x::l). + +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. + +lemma sorted_tail: ∀x,l.sorted (x::l) → sorted l. +intros; inversion H; intros; [destruct H1;|destruct H1;constructor 1;] +destruct H4; assumption; +qed. + +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. + +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;] +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]] +qed. + +definition eject_NxQ ≝ λP.λp:∃x:nat × ℚ.P x.match p with [ex_introT p _ ⇒ p]. -coercion eject1. -definition inject1 ≝ λP.λp:nat × ℚ.λh:P p. ex_introT ? P p h. -coercion inject1 with 0 1 nocomposites. - -definition value : - ∀f:q_f.∀i:ℚ.∃p:nat × ℚ. - match q_cmp i (start f) with - [ q_lt _ ⇒ \snd p = OQ - | _ ⇒ - And3 - (sum_bases (bars f) (\fst p) ≤ ⅆ[i,start f]) - (ⅆ[i, start f] < sum_bases (bars f) (S (\fst p))) - (\snd p = \snd (nth (bars f) ▭ (\fst 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 "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 (p: ℚ) (l : list bar) on l ≝ + let rec value (acc: nat × ℚ) (l : list bar) on l : nat × ℚ ≝ match l with - [ nil ⇒ 〈nat_of_q p,OQ〉 + [ nil ⇒ acc | cons x tl ⇒ - match q_cmp p (Qpos (\fst x)) with - [ q_lt _ ⇒ 〈O, \snd x〉 - | _ ⇒ - let rc ≝ value (p - Qpos (\fst x)) tl in - 〈S (\fst rc),\snd rc〉]] + 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. OQ ≤ acc → - And3 - (sum_bases l (\fst p) ≤ acc) - (acc < sum_bases l (S (\fst p))) - (\snd p = \snd (nth l ▭ (\fst p)))); + ∀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; reflexivity; - |*: cases (value ⅆ[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 〈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))); - [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption] - simplify; apply q_le_minus; assumption; + [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))); - [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption] - clear H3 H2 value; - change with (q < sum_bases l1 (S (\fst w)) + Qpos (\fst b)); - apply q_lt_plus; assumption; + [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))); - [1,3: intros; [apply q_lt_to_le|apply q_eq_to_le;symmetry] assumption] - assumption;] -|2: clear value H2; simplify; intros; split; [assumption|3:reflexivity] + [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: simplify; intros; split; - [1: apply sum_bases_empty_nat_of_q_le_q; - |2: apply sum_bases_empty_nat_of_q_le_q_one; - |3: elim (nat_of_q q); [reflexivity] simplify; 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:q_f. - (∀i.\fst (nth (bars l1) ▭ i) = \fst (nth (bars l2) ▭ i)). + 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. @@ -123,64 +373,6 @@ cases (?:False); [ apply (q_lt_corefl ? H)|apply (q_neg_gt ? H)] qed. -notation < "\blacksquare" non associative with precedence 90 for @{'hide}. -definition hide ≝ λT:Type.λx:T.x. -interpretation "hide" 'hide = (hide _ _). - -lemma sum_bases_ge_OQ: - ∀l,n. OQ ≤ sum_bases l n. -intro; elim l; simplify; intros; -[1: elim n; [apply q_eq_to_le;reflexivity] simplify; - apply q_le_plus_trans; try assumption; apply q_lt_to_le; apply q_pos_lt_OQ; -|2: cases n; [apply q_eq_to_le;reflexivity] simplify; - apply q_le_plus_trans; [apply H| apply q_lt_to_le; apply q_pos_lt_OQ;]] -qed. - -lemma sum_bases_O: - ∀l:q_f.∀x.sum_bases (bars l) x ≤ OQ → x = O. -intros; cases x in H; [intros; reflexivity] intro; cases (?:False); -cases (q_le_cases ?? H); -[1: apply (q_lt_corefl OQ); rewrite < H1 in ⊢ (?? %); -|2: apply (q_lt_antisym ??? H1);] clear H H1; cases (bars l); -simplify; apply q_lt_plus_trans; -try apply q_pos_lt_OQ; -try apply (sum_bases_ge_OQ []); -apply (sum_bases_ge_OQ l1); -qed. - -lemma sum_bases_increasing: - ∀l,x.sum_bases l x < sum_bases l (S x). -intro; elim l; -[1: elim x; - [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ; - apply q_pos_lt_OQ; - |2: simplify in H ⊢ %; - apply q_lt_plus; rewrite > q_elim_minus; - rewrite < q_plus_assoc; rewrite < q_elim_minus; - rewrite > q_plus_minus; rewrite > q_plus_OQ; - assumption;] -|2: elim x; - [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ; - apply q_pos_lt_OQ; - |2: simplify; change in ⊢ (? ? (? % ?)) with (sum_bases l1 (S n)) ; - apply q_lt_plus; rewrite > q_elim_minus; - rewrite < q_plus_assoc; rewrite < q_elim_minus; - rewrite > q_plus_minus; rewrite > q_plus_OQ; apply H]] -qed. - -lemma sum_bases_lt_canc: - ∀l,x,y.sum_bases l (S x) < sum_bases l (S y) → sum_bases l x < sum_bases l y. -intro; elim l; [apply (q_lt_canc_plus_r ?? (Qpos one));apply H] -generalize in match H1;apply (nat_elim2 (?:? → ? → CProp) ??? x y); -intros 2; -[3: intros 2; simplify; apply q_lt_inj_plus_r; apply H; - apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H3; -|2: cases (?:False); simplify in H2; - apply (q_lt_le_incompat (sum_bases l1 (S n)) OQ);[2: apply sum_bases_ge_OQ;] - apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H2; -|1: cases n in H2; intro; - [1: cases (?:False); apply (q_lt_corefl ? H2); - |2: simplify; apply q_lt_plus_trans; [apply sum_bases_ge_OQ] - apply q_pos_lt_OQ;]] -qed. +notation < "x \blacksquare" non associative with precedence 50 for @{'unpos $x}. +interpretation "hide unpos proof" 'unpos x = (unpos x _).