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
#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 →
+∃l:list A.(l1 = l3@l ∧ l4=l@l2) ∨ (l3 = l1@l ∧ l2=l@l4).
+#A #l1 elim l1
+ [#l2 #l3 #l4 #Heq %{l3} %2 % // @Heq
+ |#a1 #tl1 #Hind #l2 #l3 cases l3
+ [#l4 #Heq %{(a1::tl1)} %1 % // @sym_eq @Heq
+ |#a3 #tl3 #l4 normalize in ⊢ (%→?); #Heq cases (Hind l2 tl3 l4 ?)
+ [#l * * #Heq1 #Heq2 %{l}
+ [%1 % // >Heq1 >(cons_injective_l ????? Heq) //
+ |%2 % // >Heq1 >(cons_injective_l ????? Heq) //
+ ]
+ |@(cons_injective_r ????? Heq)
+ ]
+ ]
+ ]
+qed.
+
(**************************** iterators ******************************)
let rec map (A,B:Type[0]) (f: A → B) (l:list A) on l: list B ≝
#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/
#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.
]
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
]
] qed.
+lemma All_append: ∀A,P,l1,l2. All A P l1 → All A P l2 → All A P (l1@l2).
+#A #P #l1 elim l1 -l1 //
+#a #l1 #IHl1 #l2 * /3 width=1/
+qed.
+
+lemma All_inv_append: ∀A,P,l1,l2. All A P (l1@l2) → All A P l1 ∧ All A P l2.
+#A #P #l1 elim l1 -l1 /2 width=1/
+#a #l1 #IHl1 #l2 * #Ha #Hl12
+elim (IHl1 … Hl12) -IHl1 -Hl12 /3 width=1/
+qed-.
+
+(**************************** Allr ******************************)
+
let rec Allr (A:Type[0]) (R:relation A) (l:list A) on l : Prop ≝
match l with
[ nil ⇒ True
| cons a1 l ⇒ match l with [ nil ⇒ True | cons a2 _ ⇒ R a1 a2 ∧ Allr A R l ]
].
+lemma Allr_fwd_append_sn: ∀A,R,l1,l2. Allr A R (l1@l2) → Allr A R l1.
+#A #R #l1 elim l1 -l1 // #a1 * // #a2 #l1 #IHl1 #l2 * /3 width=2/
+qed-.
+
+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 ≝
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