From fe7b82f0aaed4ecbf84f70ec6fb7dce3c7da04e9 Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Thu, 20 Nov 2008 16:02:44 +0000 Subject: [PATCH] ... --- .../library/dama/models/list_support.ma | 282 ------------------ helm/software/matita/library/depends | 1 - 2 files changed, 283 deletions(-) delete mode 100644 helm/software/matita/library/dama/models/list_support.ma diff --git a/helm/software/matita/library/dama/models/list_support.ma b/helm/software/matita/library/dama/models/list_support.ma deleted file mode 100644 index eb70322a9..000000000 --- a/helm/software/matita/library/dama/models/list_support.ma +++ /dev/null @@ -1,282 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "nat/minus.ma". -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" -with precedence 69 for @{'nth_appl $l $d $i}. - -definition make_list ≝ - λA:Type.λdef:nat→A. - let rec make_list (n:nat) on n ≝ - 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" -with precedence 69 for @{'mk_list_appl $f $n}. - -notation "\len" with precedence 90 for @{'len}. -interpretation "len" 'len = (length _). -notation < "\len \nbsp term 90 l" with precedence 69 for @{'len_appl $l}. -interpretation "len appl" 'len_appl l = (length _ l). - -lemma mk_list_ext: ∀T:Type.∀f1,f2:nat→T.∀n. (∀x.x H1; [2: apply le_n] -apply eq_f; apply H; intros; apply H1; apply (trans_le ??? H2); apply le_S; apply le_n; -qed. - -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. - -record rel (rel_T : Type) : Type ≝ { - rel_op :2> rel_T → rel_T → Prop -}. - -record trans_rel : Type ≝ { - o_T :> Type; - o_rel :> rel o_T; - o_tra : ∀x,y,z: o_T.o_rel x y → o_rel y z → o_rel x z -}. - -lemma trans: ∀r:trans_rel.∀x,y,z:r.r x y → r y z → r x z. -apply o_tra; -qed. - -inductive sorted (lt : trans_rel): list (o_T lt) → Prop ≝ -| sorted_nil : sorted lt [] -| sorted_one : ∀x. sorted lt [x] -| sorted_cons : ∀x,y,tl. lt x y → sorted lt (y::tl) → sorted 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. - -coinductive 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: ∀r,x,l.sorted r (x::l) → sorted r l. -intros; inversion H; intros; [destruct H1;|destruct H1;constructor 1;] -destruct H4; assumption; -qed. - -lemma sorted_skip: ∀r,x,y,l. sorted r (x::y::l) → sorted r (x::l). -intros (r x y l H1); inversion H1; intros; [1,2: destruct H] -destruct H4; inversion H2; intros; [destruct H4] -[1: destruct H4; constructor 2; -|2: destruct H7; constructor 3; - [ apply (o_tra ? ??? H H4); | apply (sorted_tail ??? H2);]] -qed. - -lemma sorted_tail_bigger : ∀r,x,l,d.sorted 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 ? 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. - -(* move in nat/ *) -lemma lt_n_plus_n_Sm : ∀n,m:nat.n < n + S m. -intros; rewrite > sym_plus; apply (le_S_S n (m+n)); alias id "le_plus_n" = "cic:/matita/nat/le_arith/le_plus_n.con". -apply (le_plus_n m n); qed. - -lemma nth_append_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_append_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 sorted_head_smaller: - ∀r,l,p,d. sorted r (p::l) → ∀i.i < \len l → r p (\nth l d i). -intros 2 (r l); elim l; [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 "lt" = "natural 'less than'". -lemma sorted_pivot: - ∀r,l1,l2,p,d. sorted r (l1@p::l2) → - (∀i. i < \len l1 → r (\nth l1 d i) p) ∧ - (∀i. i < \len l2 → r p (\nth l2 d i)). -intros 2 (r l); elim l; -[1: split; [intros; cases (not_le_Sn_O ? H1);] intros; - apply sorted_head_smaller; assumption; -|2: simplify in H1; cases (H ?? d (sorted_tail ??? H1)); - lapply depth = 0 (sorted_head_smaller ??? d H1) as Hs; - split; simplify; intros; - [1: cases i in H4; simplify; intros; - [1: lapply depth = 0 (Hs (\len l1)) as HS; - rewrite > nth_len in HS; apply HS; - rewrite > len_append; simplify; apply lt_n_plus_n_Sm; - |2: apply (H2 n); apply le_S_S_to_le; apply H4] - |2: apply H3; assumption]] -qed. - -coinductive cases_bool (p:bool) : bool → CProp ≝ -| case_true : p = true → cases_bool p true -| cases_false : p = false → cases_bool p false. - -lemma case_b : ∀A:Type.∀f:A → bool. ∀x.cases_bool (f x) (f x). -intros; cases (f x);[left;|right] reflexivity; -qed. - -coinductive 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. - -include "logic/cprop_connectives.ma". - -definition eject_N ≝ - λP.λp:∃x:nat.P x.match p with [ex_introT p _ ⇒ p]. -coercion eject_N. -definition inject_N ≝ λP.λp:nat.λh:P p. ex_introT ? P p h. -coercion inject_N with 0 1 nocomposites. - -coinductive find_spec (T:Type) (P:T→bool) (l:list T) (d:T) (res : nat) : nat → CProp ≝ -| found: - ∀i. i < \len l → P (\nth l d i) = true → res = i → - (∀j. j < i → P (\nth l d j) = false) → find_spec T P l d res i -| not_found: ∀i. i = \len l → res = i → - (∀j.j < \len l → P (\nth l d j) = false) → find_spec T P l d res i. - -lemma find_lemma : ∀T.∀P:T→bool.∀l:list T.∀d.∃i.find_spec ? P l d i i. -intros; -letin find ≝ ( - let rec aux (acc: nat) (l : list T) on l : nat ≝ - match l with - [ nil ⇒ acc - | cons x tl ⇒ - match P x with - [ false ⇒ aux (S acc) tl - | true ⇒ acc]] - in aux : - ∀acc,l1.∃p:nat. - ∀story. story @ l1 = l → acc = \len story → - find_spec ? P story d acc acc → - find_spec ? P (story @ l1) d p p); -[4: clearbody find; cases (find 0 l); - lapply (H [] (refl_eq ??) (refl_eq ??)) as K; - [2: apply (not_found ?? [] d); intros; try reflexivity; cases (not_le_Sn_O ? H1); - |1: cases K; clear K; - [2: exists[apply (\len l)] - apply not_found; try reflexivity; apply H3; - |1: exists[apply i] apply found; try reflexivity; assumption;]] -|1: intros; cases (aux (S n) l2); simplify; clear aux; - lapply depth = 0 (H5 (story@[t])) as K; clear H5; - change with (find_spec ? P (story @ ([t] @ l2)) d w w); - rewrite < associative_append; apply K; clear K; - [1: rewrite > associative_append; apply H2; - |2: rewrite > H3; rewrite > len_append; rewrite > sym_plus; reflexivity; - |3: cases H4; clear H4; destruct H7; - [2: rewrite > H5; rewrite > (?:S (\len story) = \len (story @ [t])); [2: - rewrite > len_append; rewrite > sym_plus; reflexivity;] - apply not_found; try reflexivity; intros; cases (cmp_nat (S j) (\len story)); - [1: rewrite > (nth_append_lt_len ????? H8); apply H7; apply H8; - |2: rewrite > (nth_append_ge_len ????? (le_S_S_to_le ?? H8)); - rewrite > (?: j - \len story = 0);[assumption] - rewrite > (?:j = \len story);[rewrite > minus_n_n; reflexivity] - apply le_to_le_to_eq; [2: apply le_S_S_to_le; assumption;] - rewrite > len_append in H4;rewrite > sym_plus in H4; - apply le_S_S_to_le; apply H4;] - |1: rewrite < H3 in H5; cases (not_le_Sn_n ? H5);]] -|2: intros; cases H4; clear H4; - [1: destruct H7; rewrite > H3 in H5; cases (not_le_Sn_n ? H5); - |2: apply found; try reflexivity; - [1: rewrite > len_append; simplify; rewrite > H5; apply lt_n_plus_n_Sm; - |2: rewrite > H5; rewrite > nth_append_ge_len; [2: apply le_n] - rewrite < minus_n_n; assumption; - |3: intros; rewrite > H5 in H4; rewrite > nth_append_lt_len; [2:assumption] - apply H7; assumption]] -|3: intros; rewrite > append_nil; assumption;] -qed. - -lemma find : ∀T:Type.∀P:T→bool.∀l:list T.∀d:T.nat. -intros; cases (find_lemma ? f l t); apply w; qed. - -lemma cases_find: ∀T,P,l,d. find_spec T P l d (find T P l d) (find T P l d). -intros; unfold find; cases (find_lemma T P l d); simplify; assumption; qed. - -lemma list_elim_with_len: - ∀T:Type.∀P: nat → list T → CProp. - P O [] → (∀l,a,n.P (\len l) l → P (S n) (a::l)) → - ∀l.P (\len l) l. -intros;elim l; [assumption] simplify; apply H1; apply H2; -qed. - -lemma sorted_near: - ∀r,l. sorted r l → ∀i,d. S i < \len l → r (\nth l d i) (\nth l d (S i)). - intros 3; elim H; - [1: cases (not_le_Sn_O ? H1); - |2: simplify in H1; cases (not_le_Sn_O ? (le_S_S_to_le ?? H1)); - |3: simplify; cases i in H4; intros; [apply H1] - apply H3; apply le_S_S_to_le; apply H4] -qed. - -definition last ≝ - λT:Type.λd.λl:list T. \nth l d (pred (\len l)). - -notation > "\last" non associative with precedence 90 for @{'last}. -notation < "\last d l" non associative with precedence 70 for @{'last2 $d $l}. -interpretation "list last" 'last = (last _). -interpretation "list last applied" 'last2 d l = (last _ d l). - -definition head ≝ - λT:Type.λd.λl:list T.\nth l d O. - -notation > "\hd" non associative with precedence 90 for @{'hd}. -notation < "\hd d l" non associative with precedence 70 for @{'hd2 $d $l}. -interpretation "list head" 'hd = (head _). -interpretation "list head applied" 'hd2 d l = (head _ d l). - diff --git a/helm/software/matita/library/depends b/helm/software/matita/library/depends index d510318df..3e3d2d343 100644 --- a/helm/software/matita/library/depends +++ b/helm/software/matita/library/depends @@ -127,7 +127,6 @@ nat/primes.ma logic/connectives.ma nat/div_and_mod.ma nat/factorial.ma nat/minim nat/gcd_properties1.ma nat/gcd.ma list/sort.ma datatypes/bool.ma datatypes/constructors.ma list/in.ma didactic/exercises/natural_deduction.ma didactic/support/natural_deduction.ma -dama/models/list_support.ma nat/le_arith.ma list/list.ma logic/cprop_connectives.ma nat/minus.ma dama/bishop_set_rewrite.ma dama/bishop_set.ma Z/times.ma Z/plus.ma nat/lt_arith.ma Z/sigma_p.ma Z/plus.ma Z/times.ma nat/generic_iter_p.ma nat/ord.ma nat/primes.ma -- 2.39.2