X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fbasics%2Flists%2Flist.ma;h=a330f6224ef1fdb6c94e356b71955f9bad0129e5;hb=85a42e4a2a4c62818b6a98eff545e58ceb8770a4;hp=2ed7fcc868a9f1434e888f2f2400e365958ea130;hpb=65a3b93b01f2d00960c56df3563b879f36f3cbfd;p=helm.git

diff --git a/matita/matita/lib/basics/lists/list.ma b/matita/matita/lib/basics/lists/list.ma
index 2ed7fcc86..a330f6224 100644
--- a/matita/matita/lib/basics/lists/list.ma
+++ b/matita/matita/lib/basics/lists/list.ma
@@ -55,6 +55,11 @@ definition hd ≝ λA.λl: list A.λd:A.
 
 definition tail ≝  λA.λl: list A.
   match l with [ nil ⇒  [] | cons hd tl ⇒  tl].
+  
+definition option_hd ≝ 
+  λA.λl:list A. match l with
+  [ nil ⇒ None ?
+  | cons a _ ⇒ Some ? a ].
 
 interpretation "append" 'append l1 l2 = (append ? l1 l2).
 
@@ -79,6 +84,14 @@ theorem nil_to_nil:  ∀A.∀l1,l2:list A.
 #A #l1 #l2 #isnil @(nil_append_elim A l1 l2) /2/
 qed.
 
+lemma cons_injective_l : ∀A.∀a1,a2:A.∀l1,l2.a1::l1 = a2::l2 → a1 = a2.
+#A #a1 #a2 #l1 #l2 #Heq destruct //
+qed.
+
+lemma cons_injective_r : ∀A.∀a1,a2:A.∀l1,l2.a1::l1 = a2::l2 → l1 = l2.
+#A #a1 #a2 #l1 #l2 #Heq destruct //
+qed.
+
 (**************************** iterators ******************************)
 
 let rec map (A,B:Type[0]) (f: A → B) (l:list A) on l: list B ≝
@@ -162,7 +175,7 @@ let rec length (A:Type[0]) (l:list A) on l ≝
     [ nil ⇒ 0
     | cons a tl ⇒ S (length A tl)].
 
-notation "|M|" non associative with precedence 60 for @{'norm $M}.
+notation "|M|" non associative with precedence 65 for @{'norm $M}.
 interpretation "norm" 'norm l = (length ? l).
 
 lemma length_tail: ∀A,l. length ? (tail A l) = pred (length ? l).
@@ -178,6 +191,190 @@ lemma length_map: ∀A,B,l.∀f:A→B. length ? (map ?? f l) = length ? l.
 #A #B #l #f elim l // #a #tl #Hind normalize //
 qed.
 
+lemma length_reverse: ∀A.∀l:list A. 
+  |reverse A l| = |l|.
+#A #l elim l // #a #l0 #IH >reverse_cons >length_append normalize //
+qed.
+
+lemma lenght_to_nil: ∀A.∀l:list A.
+  |l| = 0 → l = [ ].
+#A * // #a #tl normalize #H destruct
+qed.
+ 
+(****************** traversing two lists in parallel *****************)
+lemma list_ind2 : 
+  ∀T1,T2:Type[0].∀l1:list T1.∀l2:list T2.∀P:list T1 → list T2 → Prop.
+  length ? l1 = length ? l2 →
+  (P [] []) → 
+  (∀tl1,tl2,hd1,hd2. P tl1 tl2 → P (hd1::tl1) (hd2::tl2)) → 
+  P l1 l2.
+#T1 #T2 #l1 #l2 #P #Hl #Pnil #Pcons
+generalize in match Hl; generalize in match l2;
+elim l1
+[#l2 cases l2 // normalize #t2 #tl2 #H destruct
+|#t1 #tl1 #IH #l2 cases l2
+   [normalize #H destruct
+   |#t2 #tl2 #H @Pcons @IH normalize in H; destruct // ]
+]
+qed.
+
+lemma list_cases2 : 
+  ∀T1,T2:Type[0].∀l1:list T1.∀l2:list T2.∀P:Prop.
+  length ? l1 = length ? l2 →
+  (l1 = [] → l2 = [] → P) → 
+  (∀hd1,hd2,tl1,tl2.l1 = hd1::tl1 → l2 = hd2::tl2 → P) → P.
+#T1 #T2 #l1 #l2 #P #Hl @(list_ind2 … Hl)
+[ #Pnil #Pcons @Pnil //
+| #tl1 #tl2 #hd1 #hd2 #IH1 #IH2 #Hp @Hp // ]
+qed.
+
+(*********************** properties of append ***********************)
+lemma append_l1_injective : 
+  ∀A.∀l1,l2,l3,l4:list A. |l1| = |l2| → l1@l3 = l2@l4 → l1 = l2.
+#a #l1 #l2 #l3 #l4 #Hlen @(list_ind2 … Hlen) //
+#tl1 #tl2 #hd1 #hd2 #IH normalize #Heq destruct @eq_f /2/
+qed.
+  
+lemma append_l2_injective : 
+  ∀A.∀l1,l2,l3,l4:list A. |l1| = |l2| → l1@l3 = l2@l4 → l3 = l4.
+#a #l1 #l2 #l3 #l4 #Hlen @(list_ind2 … Hlen) normalize //
+#tl1 #tl2 #hd1 #hd2 #IH normalize #Heq destruct /2/
+qed.
+
+lemma append_l1_injective_r :
+  ∀A.∀l1,l2,l3,l4:list A. |l3| = |l4| → l1@l3 = l2@l4 → l1 = l2.
+#a #l1 #l2 #l3 #l4 #Hlen #Heq lapply (eq_f … (reverse ?) … Heq)
+>reverse_append >reverse_append #Heq1
+lapply (append_l2_injective … Heq1) [ // ] #Heq2
+lapply (eq_f … (reverse ?) … Heq2) //
+qed.
+  
+lemma append_l2_injective_r : 
+  ∀A.∀l1,l2,l3,l4:list A. |l3| = |l4| → l1@l3 = l2@l4 → l3 = l4.
+#a #l1 #l2 #l3 #l4 #Hlen #Heq lapply (eq_f … (reverse ?) … Heq)
+>reverse_append >reverse_append #Heq1
+lapply (append_l1_injective … Heq1) [ // ] #Heq2
+lapply (eq_f … (reverse ?) … Heq2) //
+qed.
+
+lemma length_rev_append: ∀A.∀l,acc:list A. 
+  |rev_append ? l acc| = |l|+|acc|.
+#A #l elim l // #a #tl #Hind normalize 
+#acc >Hind normalize // 
+qed.
+
+(****************************** mem ********************************)
+let rec mem A (a:A) (l:list A) on l ≝
+  match l with
+  [ nil ⇒ False
+  | cons hd tl ⇒ a=hd ∨ mem A a tl
+  ]. 
+
+lemma mem_map: ∀A,B.∀f:A→B.∀l,b. 
+  mem ? b (map … f l) → ∃a. mem ? a l ∧ f a = b.
+#A #B #f #l elim l 
+  [#b normalize @False_ind
+  |#a #tl #Hind #b normalize *
+    [#eqb @(ex_intro … a) /3/
+    |#memb cases (Hind … memb) #a * #mema #eqb
+     @(ex_intro … a) /3/
+    ]
+  ]
+qed.
+
+lemma mem_map_forward: ∀A,B.∀f:A→B.∀a,l. 
+  mem A a l → mem B (f a) (map ?? f l).
+ #A #B #f #a #l elim l
+  [normalize @False_ind
+  |#b #tl #Hind * 
+    [#eqab <eqab normalize %1 % |#memtl normalize %2 @Hind @memtl]
+  ]
+qed.
+
+(***************************** split *******************************)
+let rec split_rev A (l:list A) acc n on n ≝ 
+  match n with 
+  [O ⇒ 〈acc,l〉
+  |S m ⇒ match l with 
+    [nil ⇒ 〈acc,[]〉
+    |cons a tl ⇒ split_rev A tl (a::acc) m
+    ]
+  ].
+  
+definition split ≝ λA,l,n.
+  let 〈l1,l2〉 ≝ split_rev A l [] n in 〈reverse ? l1,l2〉.
+
+lemma split_rev_len: ∀A,n,l,acc. n ≤ |l| →
+  |\fst (split_rev A l acc n)| = n+|acc|.
+#A #n elim n // #m #Hind *
+  [normalize #acc #Hfalse @False_ind /2/
+  |#a #tl #acc #Hlen normalize >Hind 
+    [normalize // |@le_S_S_to_le //]
+  ]
+qed.
+
+lemma split_len: ∀A,n,l. n ≤ |l| →
+  |\fst (split A l n)| = n.
+#A #n #l #Hlen normalize >(eq_pair_fst_snd ?? (split_rev …))
+normalize >length_reverse  >(split_rev_len … [ ] Hlen) normalize //
+qed.
+  
+lemma split_rev_eq: ∀A,n,l,acc. n ≤ |l| → 
+  reverse ? acc@ l = 
+    reverse ? (\fst (split_rev A l acc n))@(\snd (split_rev A l acc n)).
+ #A #n elim n //
+ #m #Hind * 
+   [#acc whd in ⊢ ((??%)→?); #False_ind /2/ 
+   |#a #tl #acc #Hlen >append_cons <reverse_single <reverse_append 
+    @(Hind tl) @le_S_S_to_le @Hlen
+   ]
+qed.
+ 
+lemma split_eq: ∀A,n,l. n ≤ |l| → 
+  l = (\fst (split A l n))@(\snd (split A l n)).
+#A #n #l #Hlen change with ((reverse ? [ ])@l) in ⊢ (??%?);
+>(split_rev_eq … Hlen) normalize 
+>(eq_pair_fst_snd ?? (split_rev A l [] n)) %
+qed.
+
+lemma split_exists: ∀A,n.∀l:list A. n ≤ |l| → 
+  ∃l1,l2. l = l1@l2 ∧ |l1| = n.
+#A #n #l #Hlen @(ex_intro … (\fst (split A l n)))
+@(ex_intro … (\snd (split A l n))) % /2/
+qed.
+  
+(****************************** flatten ******************************)
+definition flatten ≝ λA.foldr (list A) (list A) (append A) [].
+
+lemma flatten_to_mem: ∀A,n,l,l1,l2.∀a:list A. 0 < n →
+  (∀x. mem ? x l → |x| = n) → |a| = n → flatten ? l = l1@a@l2  →
+    (∃q.|l1| = n*q)  → mem ? a l.
+#A #n #l elim l
+  [normalize #l1 #l2 #a #posn #Hlen #Ha #Hnil @False_ind
+   cut (|a|=0) [@sym_eq @le_n_O_to_eq 
+   @(transitive_le ? (|nil A|)) // >Hnil >length_append >length_append //] /2/
+  |#hd #tl #Hind #l1 #l2 #a #posn #Hlen #Ha 
+   whd in match (flatten ??); #Hflat * #q cases q
+    [<times_n_O #Hl1 
+     cut (a = hd) [>(lenght_to_nil… Hl1) in Hflat; 
+     whd in ⊢ ((???%)→?); #Hflat @sym_eq @(append_l1_injective … Hflat)
+     >Ha >Hlen // %1 //   
+     ] /2/
+    |#q1 #Hl1 lapply (split_exists … n l1 ?) //
+     * #l11 * #l12 * #Heql1 #Hlenl11 %2
+     @(Hind l12 l2 … posn ? Ha) 
+      [#x #memx @Hlen %2 //
+      |@(append_l2_injective ? hd l11) 
+        [>Hlenl11 @Hlen %1 %
+        |>Hflat >Heql1 >associative_append %
+        ]
+      |@(ex_intro …q1) @(injective_plus_r n) 
+       <Hlenl11 in ⊢ (??%?); <length_append <Heql1 >Hl1 //
+      ]
+    ]
+  ]
+qed.
+
 (****************************** nth ********************************)
 let rec nth n (A:Type[0]) (l:list A) (d:A)  ≝  
   match n with
@@ -365,7 +562,7 @@ lemma lhd_cons_ltl: ∀A,n,l. lhd A l n @ ltl A l n = l.
 qed.
 
 lemma length_ltl: ∀A,n,l. |ltl A l n| = |l| - n.
-#A #n elim n -n /2/
+#A #n elim n -n // 
 #n #IHn *; normalize /2/
 qed.