update in lib

[helm.git] / matita / matita / lib / basics / lists / list.ma

diff --git a/matita/matita/lib/basics/lists/list.ma b/matita/matita/lib/basics/lists/list.ma
index 725f57135f42428760b67c42cdfa1b9e88d4b531..ed988b760d283732144e949e2b8c189b4df46283 100644 (file)
--- a/matita/matita/lib/basics/lists/list.ma
+++ b/matita/matita/lib/basics/lists/list.ma
@@ -11,6 +11,7 @@
 
 include "basics/types.ma".
 include "arithmetics/nat.ma".
+include "basics/core_notation/card_1.ma".
 
 inductive list (A:Type[0]) : Type[0] :=
   | nil: list A
@@ -92,6 +93,15 @@ lemma cons_injective_r : ∀A.∀a1,a2:A.∀l1,l2.a1::l1 = a2::l2 → l1 = l2.
 #A #a1 #a2 #l1 #l2 #Heq destruct //
 qed.
 
+(* option cons *)
+
+definition option_cons ≝ λsig.λc:option sig.λl.
+  match c with [ None ⇒ l | Some c0 ⇒ c0::l ].
+
+lemma opt_cons_tail_expand : ∀A,l.l = option_cons A (option_hd ? l) (tail ? l). 
+#A * //
+qed.
+
 (* comparing lists *)
 
 lemma compare_append : ∀A,l1,l2,l3,l4. l1@l2 = l3@l4 → 
@@ -110,6 +120,7 @@ lemma compare_append : ∀A,l1,l2,l3,l4. l1@l2 = l3@l4 →
     ]
   ]
 qed.
+
 (**************************** iterators ******************************)
 
 let rec map (A,B:Type[0]) (f: A → B) (l:list A) on l: list B ≝
@@ -199,6 +210,10 @@ lemma length_tail: ∀A,l. length ? (tail A l) = pred (length ? l).
 #A #l elim l // 
 qed.
 
+lemma length_tail1 : ∀A,l.0 < |l| → |tail A l| < |l|.
+#A * normalize //
+qed.
+
 lemma length_append: ∀A.∀l1,l2:list A. 
   |l1@l2| = |l1|+|l2|.
 #A #l1 elim l1 // normalize /2/
@@ -218,6 +233,19 @@ lemma lenght_to_nil: ∀A.∀l:list A.
 #A * // #a #tl normalize #H destruct
 qed.
  
+lemma lists_length_split : 
+ ∀A.∀l1,l2:list A.(∃la,lb.(|la| = |l1| ∧ l2 = la@lb) ∨ (|la| = |l2| ∧ l1 = la@lb)).
+#A #l1 elim l1
+[ #l2 %{[ ]} %{l2} % % %
+| #hd1 #tl1 #IH *
+  [ %{[ ]} %{(hd1::tl1)} %2 % %
+  | #hd2 #tl2 cases (IH tl2) #x * #y *
+    [ * #IH1 #IH2 %{(hd2::x)} %{y} % normalize % //
+    | * #IH1 #IH2 %{(hd1::x)} %{y} %2 normalize % // ]
+  ]
+]
+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.
@@ -334,6 +362,109 @@ lemma mem_map_forward: ∀A,B.∀f:A→B.∀a,l.
   ]
 qed.
 
+(****************************** mem filter ***************************)
+lemma mem_filter: ∀S,f,a,l. 
+  mem S a (filter S f l) → mem S a l.
+#S #f #a #l elim l [normalize //]
+#b #tl #Hind normalize (cases (f b)) normalize
+  [* [#eqab %1 @eqab | #H %2 @Hind @H]
+  |#H %2 @Hind @H]
+qed.
+
+lemma mem_filter_true: ∀S,f,a,l. 
+  mem S a (filter S f l)  → f a = true.
+#S #f #a #l elim l [normalize @False_ind ]
+#b #tl #Hind cases (true_or_false (f b)) #H
+normalize >H normalize [2:@Hind]
+* [#eqab // | @Hind]
+qed. 
+
+lemma mem_filter_l: ∀S,f,x,l. (f x = true) → mem S x l →
+mem S x (filter ? f l).
+#S #f #x #l #fx elim l [@False_ind]
+#b #tl #Hind * 
+  [#eqxb <eqxb >(filter_true ???? fx) %1 % 
+  |#Htl cases (true_or_false (f b)) #fb 
+    [>(filter_true ???? fb) %2 @Hind @Htl
+    |>(filter_false ???? fb) @Hind @Htl
+    ]
+  ]
+qed.
+
+lemma filter_case: ∀A,p,l,x. mem ? x l → 
+  mem ? x (filter A p l) ∨ mem ? x (filter A (λx.¬ p x) l).
+#A #p #l elim l 
+  [#x @False_ind 
+  |#a #tl #Hind #x * 
+    [#eqxa >eqxa cases (true_or_false (p a)) #Hcase
+      [%1 >(filter_true A tl a p Hcase) %1 % 
+      |%2 >(filter_true A tl a ??) [%1 % | >Hcase %]
+      ]
+    |#memx cases (Hind … memx) -memx #memx
+      [%1 cases (true_or_false (p a)) #Hpa 
+        [>(filter_true A tl a p Hpa) %2 @memx
+        |>(filter_false A tl a p Hpa) @memx
+        ]
+      |cases (true_or_false (p a)) #Hcase
+        [%2 >(filter_false A tl a) [@memx |>Hcase %]
+        |%2 >(filter_true A tl a) [%2 @memx|>Hcase %]
+        ]
+      ]
+    ]
+  ]
+qed.
+
+lemma filter_length2: ∀A,p,l. |filter A p l|+|filter A (λx.¬ p x) l| = |l|.
+#A #p #l elim l //
+#a #tl #Hind cases (true_or_false (p a)) #Hcase
+  [>(filter_true A tl a p Hcase) >(filter_false A tl a ??) 
+    [@(eq_f ?? S) @Hind | >Hcase %]
+  |>(filter_false A tl a p Hcase) >(filter_true A tl a ??) 
+    [<plus_n_Sm @(eq_f ?? S) @Hind | >Hcase %]
+  ]
+qed.
+
+(***************************** unique *******************************)
+let rec unique A (l:list A) on l ≝ 
+  match l with 
+  [nil ⇒ True
+  |cons a tl ⇒ ¬ mem A a tl ∧ unique A tl].
+
+lemma unique_filter : ∀S,l,f.
+ unique S l → unique S (filter S f l).
+#S #l #f elim l //
+#a #tl #Hind * 
+#memba #uniquetl cases (true_or_false … (f a)) #Hfa
+  [>(filter_true ???? Hfa) % 
+    [@(not_to_not … memba) @mem_filter |/2/ ]
+  |>filter_false /2/
+  ]
+qed.
+
+lemma filter_eqb : ∀m,l. unique ? l → 
+  (mem ? m l ∧ filter ? (eqb m) l = [m])∨(¬mem ? m l ∧ filter ? (eqb m) l = []).
+#m #l elim l
+  [#_ %2 % [% @False_ind | //]
+  |#a #tl #Hind * #Hmema #Hunique
+   cases (Hind Hunique)
+    [* #Hmemm #Hind %1 % [%2 //]
+     >filter_false // @not_eq_to_eqb_false % #eqma @(absurd ? Hmemm) //
+    |* #Hmemm #Hind cases (decidable_eq_nat m a) #eqma 
+      [%1 <eqma % [%1 //] >filter_true [2: @eq_to_eqb_true //] >Hind //
+      |%2 % 
+        [@(not_to_not … Hmemm) * // #H @False_ind  @(absurd … H) //
+        |>filter_false // @not_eq_to_eqb_false @eqma
+        ]
+      ]
+    ]
+  ]
+qed.
+
+lemma length_filter_eqb: ∀m,l. unique ? l → 
+  |filter ? (eqb m) l| ≤ 1.
+#m #l #Huni cases (filter_eqb m l Huni) * #_ #H >H // 
+qed. 
+
 (***************************** split *******************************)
 let rec split_rev A (l:list A) acc n on n ≝ 
   match n with 
@@ -490,6 +621,11 @@ lemma Allr_fwd_cons: ∀A,R,a,l. Allr A R (a::l) → Allr A R l.
 #A #R #a * // #a0 #l * //
 qed-.
 
+lemma Allr_fwd_append_dx: ∀A,R,l1,l2. Allr A R (l1@l2) → Allr A R l2.
+#A #R #l1 elim l1 -l1 // #a1 #l1 #IHl1 #l2 #H
+lapply (Allr_fwd_cons … (l1@l2) H) -H /2 width=1/
+qed-.  
+
 (**************************** Exists *******************************)
 
 let rec Exists (A:Type[0]) (P:A → Prop) (l:list A) on l : Prop ≝