From 2ddda3a0f1e22c9b5c9572896cdaf69b3c4d19d2 Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Thu, 3 Jul 2008 14:20:35 +0000 Subject: [PATCH] ...snapshot --- .../contribs/dama/dama/models/list_support.ma | 121 +++++++++++++-- .../contribs/dama/dama/models/q_bars.ma | 138 ++++++++++++++++-- 2 files changed, 234 insertions(+), 25 deletions(-) diff --git a/helm/software/matita/contribs/dama/dama/models/list_support.ma b/helm/software/matita/contribs/dama/dama/models/list_support.ma index d9188b0a6..3aec1a024 100644 --- a/helm/software/matita/contribs/dama/dama/models/list_support.ma +++ b/helm/software/matita/contribs/dama/dama/models/list_support.ma @@ -16,8 +16,8 @@ include "list/list.ma". interpretation "list nth" 'nth = (nth _). interpretation "list nth" 'nth_appl l d i = (nth _ l d i). -notation "'nth'" with precedence 90 for @{'nth}. -notation < "'nth' \nbsp term 90 l \nbsp term 90 d \nbsp term 90 i" +notation "\nth" with precedence 90 for @{'nth}. +notation < "\nth \nbsp term 90 l \nbsp term 90 d \nbsp term 90 i" with precedence 69 for @{'nth_appl $l $d $i}. definition make_list ≝ @@ -26,18 +26,121 @@ definition make_list ≝ match n with [ O ⇒ nil ? | S m ⇒ def m :: make_list m] in make_list. -interpretation "'mk_list' appl" 'mk_list_appl f n = (make_list _ f n). -interpretation "'mk_list'" 'mk_list = (make_list _). -notation "'mk_list'" with precedence 90 for @{'mk_list}. -notation < "'mk_list' \nbsp term 90 f \nbsp term 90 n" +interpretation "\mk_list appl" 'mk_list_appl f n = (make_list _ f n). +interpretation "\mk_list" 'mk_list = (make_list _). +notation "\mk_list" with precedence 90 for @{'mk_list}. +notation < "\mk_list \nbsp term 90 f \nbsp term 90 n" with precedence 69 for @{'mk_list_appl $f $n}. -notation "'len'" with precedence 90 for @{'len}. +notation "\len" with precedence 90 for @{'len}. interpretation "len" 'len = (length _). -notation < "'len' \nbsp term 90 l" with precedence 69 for @{'len_appl $l}. +notation < "\len \nbsp term 90 l" with precedence 69 for @{'len_appl $l}. interpretation "len appl" 'len_appl l = (length _ l). -lemma len_mk_list : ∀T:Type.∀f:nat→T.∀n.len (mk_list f n) = n. +lemma len_mk_list : ∀T:Type.∀f:nat→T.∀n.\len (\mk_list f n) = n. intros; elim n; [reflexivity] simplify; rewrite > H; reflexivity; qed. +inductive sorted (T:Type) (lt : T → T → Prop): list T → Prop ≝ +| sorted_nil : sorted T lt [] +| sorted_one : ∀x. sorted T lt [x] +| sorted_cons : ∀x,y,tl. lt x y → sorted T lt (y::tl) → sorted T lt (x::y::tl). + +lemma nth_nil: ∀T,i.∀def:T. \nth [] def i = def. +intros; elim i; simplify; [reflexivity;] assumption; qed. + +lemma len_append: ∀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 len_gt_non_empty : + ∀T.∀l:list T.O < \len l → non_empty_list T l. +intros; cases l in H; [intros; cases (not_le_Sn_O ? H);] intros; constructor 1; +qed. + +lemma sorted_tail: ∀T,r,x,l.sorted T r (x::l) → sorted T r l. +intros; inversion H; intros; [destruct H1;|destruct H1;constructor 1;] +destruct H4; assumption; +qed. + +lemma sorted_skip: + ∀T,r,x,y,l. + transitive T r → sorted T r (x::y::l) → sorted T r (x::l). +intros; inversion H1; intros; [1,2: destruct H2] +destruct H5; inversion H3; intros; [destruct H5] +[1: destruct H5; constructor 2; +|2: destruct H8; constructor 3; [apply (H ??? H2 H5);] + apply (sorted_tail ???? H3);] +qed. + +lemma sorted_tail_bigger : ∀T,r,x,l,d.sorted T r (x::l) → ∀i. i < \len l → r x (\nth l d i). +intros 4; elim l; [ cases (not_le_Sn_O i H1);] +cases i in H2; +[2: intros; apply (H d ? 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. 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 c1a017dcc..d5a7806e7 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_bars.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_bars.ma @@ -47,6 +47,10 @@ record q_f : Type ≝ { 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). @@ -90,6 +94,77 @@ cases (cmp_nat i (len l)); 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 eject_NxQ. @@ -105,12 +180,14 @@ definition value : ∀f:q_f.∀i:ratio.∃p:ℚ.∃j.value_spec f (Qpos i) 〈j intros; alias symbol "pi2" = "pair pi2". alias symbol "pi1" = "pair pi1". -alias symbol "lt" (instance 6) = "Q less than". +alias symbol "lt" (instance 7) = "Q less than". alias symbol "leq" = "Q less or equal than". letin value_spec_aux ≝ ( - λf,i,q. - \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)); + 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 ≝ @@ -122,25 +199,54 @@ letin value ≝ ( | q_gt _ ⇒ acc]] in value : ∀acc,l.∃p:nat × ℚ. - ∀story. story @ l = bars f → + ∀story. story @ l = bars f → S (\fst acc) = len story → value_spec_aux story (Qpos i) acc → - value_spec_aux l (Qpos i) p); + 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 (S (\fst p))] simplify; - cases (Hp [x] (refl_eq ??) ?) (Hg HV); - [unfold; split[reflexivity]simplify;split; - [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 ?? H3))]] - split;[rewrite > Hg; reflexivity]split; cases HV; [assumption;] + 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 n1);apply le_S_S_to_le; assumption; -|1: unfold value_spec_aux; clear value value_spec_aux H2;intros; split[2:split] - [1: apply (q_lt_le_trans ??? (H4 (\fst p))); clear H4 H5; + 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)); -- 2.39.2