X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fng_assembly2%2Fcommon%2Flist.ma;fp=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fng_assembly2%2Fcommon%2Flist.ma;h=a1077f697d02b4b2858b57e4daf9152364bf504d;hb=bd112857523fc543c78cd29b74417585033ec464;hp=0000000000000000000000000000000000000000;hpb=c49f650fe86d8ff5d388418e9f603549d765a2ea;p=helm.git

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+(**************************************************************************)
+(*       ___	                                                          *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* ********************************************************************** *)
+(*                          Progetto FreeScale                            *)
+(*                                                                        *)
+(*   Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it              *)
+(*   Sviluppo: 2008-2010                                                  *)
+(*                                                                        *)
+(* ********************************************************************** *)
+
+include "common/comp.ma".
+include "common/option.ma".
+include "common/nat.ma".
+
+(* *************** *)
+(* LISTE GENERICHE *)
+(* *************** *)
+
+ninductive list (A:Type) : Type ≝
+  nil: list A
+| cons: A → list A → list A.
+
+nlet rec append A (l1: list A) l2 on l1 ≝
+ match l1 with
+  [ nil ⇒ l2
+  | (cons hd tl) ⇒ cons A hd (append A tl l2) ].
+
+notation "hvbox(hd break :: tl)"
+  right associative with precedence 47
+  for @{'cons $hd $tl}.
+
+notation "[ list0 x sep ; ]"
+  non associative with precedence 90
+  for ${fold right @'nil rec acc @{'cons $x $acc}}.
+
+notation "hvbox(l1 break @ l2)"
+  right associative with precedence 47
+  for @{'append $l1 $l2 }.
+
+interpretation "nil" 'nil = (nil ?).
+interpretation "cons" 'cons hd tl = (cons ? hd tl).
+interpretation "append" 'append l1 l2 = (append ? l1 l2).
+
+nlemma list_destruct_1 : ∀T.∀x1,x2:T.∀y1,y2:list T.cons T x1 y1 = cons T x2 y2 → x1 = x2.
+ #T; #x1; #x2; #y1; #y2; #H;
+ nchange with (match cons T x2 y2 with [ nil ⇒ False | cons a _ ⇒ x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma list_destruct_2 : ∀T.∀x1,x2:T.∀y1,y2:list T.cons T x1 y1 = cons T x2 y2 → y1 = y2.
+ #T; #x1; #x2; #y1; #y2; #H;
+ nchange with (match cons T x2 y2 with [ nil ⇒ False | cons _ b ⇒ y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma list_destruct_cons_nil : ∀T.∀x:T.∀y:list T.cons T x y = nil T → False.
+ #T; #x; #y; #H;
+ nchange with (match cons T x y with [ nil ⇒ True | cons a b ⇒ False ]);
+ nrewrite > H;
+ nnormalize;
+ napply I.
+nqed.
+
+nlemma list_destruct_nil_cons : ∀T.∀x:T.∀y:list T.nil T = cons T x y → False.
+ #T; #x; #y; #H;
+ nchange with (match cons T x y with [ nil ⇒ True | cons a b ⇒ False ]);
+ nrewrite < H;
+ nnormalize;
+ napply I.
+nqed.
+
+nlemma append_nil : ∀T:Type.∀l:list T.(l@[]) = l.
+ #T; #l;
+ nelim l;
+ nnormalize;
+ ##[ ##1: napply refl_eq
+ ##| ##2: #x; #y; #H;
+          nrewrite > H;
+          napply refl_eq
+ ##]
+nqed.
+
+nlemma associative_list : ∀T.associative (list T) (append T).
+ #T; #x; #y; #z;
+ nelim x;
+ nnormalize;
+ ##[ ##1: napply refl_eq
+ ##| ##2: #a; #b; #H;
+          nrewrite > H;
+          napply refl_eq
+ ##]
+nqed.
+
+nlemma cons_append_commute : ∀T:Type.∀l1,l2:list T.∀a:T.a :: (l1 @ l2) = (a :: l1) @ l2.
+ #T; #l1; #l2; #a;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma append_cons_commute : ∀T:Type.∀a:T.∀l,l1:list T.l @ (a::l1) = (l@[a]) @ l1.
+ #T; #a; #l; #l1;
+ nrewrite > (associative_list T l [a] l1);
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+(* listlen *)
+nlet rec len_list (T:Type) (l:list T) on l ≝
+ match l with [ nil ⇒ O | cons _ t ⇒ S (len_list T t) ].
+
+(* vuota? *)
+ndefinition is_empty_list ≝
+λT:Type.λl:list T.match l with [ nil ⇒ True | cons _ _ ⇒ False ].
+
+ndefinition isb_empty_list ≝
+λT:Type.λl:list T.match l with [ nil ⇒ true | cons _ _ ⇒ false ].
+
+ndefinition isnot_empty_list ≝
+λT:Type.λl:list T.match l with [ nil ⇒ False | cons _ _ ⇒ True ].
+
+ndefinition isnotb_empty_list ≝
+λT:Type.λl:list T.match l with [ nil ⇒ false | cons _ _ ⇒ true ].
+
+(* reverse *)
+nlet rec reverse_list (T:Type) (l:list T) on l ≝
+ match l with
+  [ nil ⇒ nil T
+  | cons h t ⇒ (reverse_list T t)@[h]
+  ].
+
+(* getFirst *)
+ndefinition get_first_list ≝
+λT:Type.λl:list T.match l with
+ [ nil ⇒ None ?
+ | cons h _ ⇒ Some ? h ].
+
+(* getLast *)
+ndefinition get_last_list ≝
+λT:Type.λl:list T.match reverse_list T l with
+ [ nil ⇒ None ?
+ | cons h _ ⇒ Some ? h ].
+
+(* cutFirst *)
+ndefinition cut_first_list ≝
+λT:Type.λl:list T.match l with
+ [ nil ⇒ nil T
+ | cons _ t ⇒ t ].
+
+(* cutLast *)
+ndefinition cut_last_list ≝
+λT:Type.λl:list T.match reverse_list T l with
+ [ nil ⇒ nil T
+ | cons _ t ⇒ reverse_list T t ].
+
+(* apply f *)
+nlet rec apply_f_list (T1,T2:Type) (l:list T1) (f:T1 → T2) on l ≝
+match l with
+ [ nil ⇒ nil T2
+ | cons h t ⇒ cons T2 (f h) (apply_f_list T1 T2 t f) ].
+
+(* fold right *)
+nlet rec fold_right_list (T1,T2:Type) (f:T1 → T2 → T2) (acc:T2) (l:list T1) on l ≝
+  match l with
+   [ nil ⇒ acc
+   | cons h t ⇒ f h (fold_right_list T1 T2 f acc t)
+   ].
+
+(* double fold right *)
+nlemma fold_right_list2_aux1 :
+∀T.∀h,t.len_list T [] = len_list T (h::t) → False.
+ #T; #h; #t;
+ nnormalize;
+ #H;
+ ndestruct (*napply (nat_destruct_0_S ? H)*).
+nqed.
+
+nlemma fold_right_list2_aux2 :
+∀T.∀h,t.len_list T (h::t) = len_list T [] → False.
+ #T; #h; #t;
+ nnormalize;
+ #H;
+ ndestruct (*napply (nat_destruct_S_0 ? H)*).
+nqed.
+
+nlemma fold_right_list2_aux3 :
+∀T.∀h,h',t,t'.len_list T (h::t) = len_list T (h'::t') → len_list T t = len_list T t'.
+ #T; #h; #h'; #t; #t';
+ nelim t;
+ nelim t';
+ ##[ ##1: nnormalize; #H; napply refl_eq
+ ##| ##2: #a; #l'; #H; #H1;
+          nchange in H1:(%) with ((S O) = (S (S (len_list T l'))));
+          ndestruct (*nelim (nat_destruct_0_S ? (nat_destruct_S_S … H1))*)
+ ##| ##3: #a; #l'; #H; #H1;
+          nchange in H1:(%) with ((S (S (len_list T l'))) = (S O));
+          ndestruct (*nelim (nat_destruct_S_0 ? (nat_destruct_S_S … H1))*)
+ ##| ##4: #a; #l; #H; #a1; #l1; #H1; #H2;
+          nchange in H2:(%) with ((S (S (len_list T l1))) = (S (S (len_list T l))));
+          nchange with ((S (len_list T l1)) = (S (len_list T l)));
+          nrewrite > (nat_destruct_S_S … H2);
+          napply refl_eq
+ ##]
+nqed.
+
+nlet rec fold_right_list2 (T1,T2:Type) (f:T1 → T1 → T2 → T2) (acc:T2) (l1:list T1) on l1 ≝
+ match l1
+  return λl1.Πl2.len_list T1 l1 = len_list T1 l2 → T2
+ with
+  [ nil ⇒ λl2.match l2 return λl2.len_list T1 [] = len_list T1 l2 → T2 with
+   [ nil ⇒ λp:len_list T1 [] = len_list T1 [].acc
+   | cons h t ⇒ λp:len_list T1 [] = len_list T1 (h::t).
+    False_rect_Type0 ? (fold_right_list2_aux1 T1 h t p)
+   ]
+  | cons h t ⇒ λl2.match l2 return λl2.len_list T1 (h::t) = len_list T1 l2 → T2 with
+   [ nil ⇒ λp:len_list T1 (h::t) = len_list T1 [].
+    False_rect_Type0 ? (fold_right_list2_aux2 T1 h t p)
+   | cons h' t' ⇒ λp:len_list T1 (h::t) = len_list T1 (h'::t').
+    f h h' (fold_right_list2 T1 T2 f acc t t' (fold_right_list2_aux3 T1 h h' t t' p))
+   ]
+  ].
+
+nlet rec bfold_right_list2 (T1:Type) (f:T1 → T1 → bool) (l1,l2:list T1) on l1 ≝
+ match l1 with
+  [ nil ⇒ match l2 with
+   [ nil ⇒ true | cons h t ⇒ false ]
+  | cons h t ⇒ match l2 with
+   [ nil ⇒ false | cons h' t' ⇒ (f h h') ⊗ (bfold_right_list2 T1 f t t')
+   ]
+  ].
+
+(* nth elem *)
+nlet rec nth_list (T:Type) (l:list T) (n:nat) on l ≝
+ match l with
+  [ nil ⇒ None ?
+  | cons h t ⇒ match n with
+   [ O ⇒ Some ? h | S n' ⇒ nth_list T t n' ]
+  ].
+
+(* abs elem *)
+ndefinition abs_list_aux1 : ∀T:Type.∀n.((len_list T []) > n) = true → False.
+ #T; nnormalize; #n; #H; ndestruct (*napply (bool_destruct … H)*). nqed.
+
+ndefinition abs_list_aux2 : ∀T:Type.∀h:T.∀t:list T.∀n.((len_list T (h::t)) > (S n) = true) → ((len_list T t) > n) = true.
+ #T; #h; #t; #n; nnormalize; #H; napply H. nqed.
+
+nlet rec abs_list (T:Type) (l:list T) on l ≝
+ match l
+  return λl.Πn.(((len_list T l) > n) = true) → T
+ with
+  [ nil ⇒ λn.λp:(((len_list T []) > n) = true).False_rect_Type0 ? (abs_list_aux1 T n p)
+  | cons h t ⇒ λn.
+   match n with
+    [ O ⇒ λp:(((len_list T (h::t)) > O) = true).h
+    | S n' ⇒ λp:(((len_list T (h::t)) > (S n')) = true).
+     abs_list T t n' (abs_list_aux2 T h t n' p)
+    ]
+  ].
+
+(* esempio: abs_list ? [ 1; 2; 3 ; 4 ] 0 (refl_eq …) = 1. *)
+
+nlemma symmetric_lenlist : ∀T.∀l1,l2:list T.len_list T l1 = len_list T l2 → len_list T l2 = len_list T l1.
+ #T; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2; nnormalize;
+          ##[ ##1: #H; napply refl_eq
+          ##| ##2: #h; #t; #H; ndestruct (*nelim (nat_destruct_0_S ? H)*)
+          ##]
+ ##| ##2: #h; #l2; ncases l2; nnormalize;
+          ##[ ##1: #H; #l; #H1; nrewrite < H1; napply refl_eq
+          ##| ##2: #h; #l; #H; #l3; #H1; nrewrite < H1; napply refl_eq
+          ##]
+ ##]
+nqed.
+
+nlemma symmetric_foldrightlist2_aux :
+∀T1,T2:Type.∀f:T1 → T1 → T2 → T2.
+ (∀x,y,z.f x y z = f y x z) →
+ (∀acc:T2.∀l1,l2:list T1.
+  ∀H1:(len_list T1 l1 = len_list T1 l2).
+  ∀H2:(len_list T1 l2 = len_list T1 l1).
+   (fold_right_list2 T1 T2 f acc l1 l2 H1 = fold_right_list2 T1 T2 f acc l2 l1 H2)).
+ #T1; #T2; #f; #H; #acc; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+          ##[ ##1: nnormalize; #H1; #H2; napply refl_eq
+          ##| ##2: #h; #l; #H1; #H2;
+                   nchange in H1:(%) with (O = (S (len_list ? l)));
+                   ndestruct (*nelim (nat_destruct_0_S ? H1)*)
+          ##]
+ ##| ##2: #h3; #l3; #H1; #l2; ncases l2;
+          ##[ ##1: #H2; #H3; nchange in H2:(%) with ((S (len_list ? l3)) = O);
+                   ndestruct (*nelim (nat_destruct_S_0 ? H1)*)
+          ##| ##2: #h4; #l4; #H2; #H3;
+                   nchange in H2:(%) with ((S (len_list ? l3)) = (S (len_list ? l4)));
+                   nchange in H3:(%) with ((S (len_list ? l4)) = (S (len_list ? l3)));
+                   nchange with ((f h3 h4 (fold_right_list2 T1 T2 f acc l3 l4 (fold_right_list2_aux3 T1 h3 h4 l3 l4 ?))) =
+                                 (f h4 h3 (fold_right_list2 T1 T2 f acc l4 l3 (fold_right_list2_aux3 T1 h4 h3 l4 l3 ?))));
+                   nrewrite < (H1 l4 (fold_right_list2_aux3 T1 h3 h4 l3 l4 H2) (fold_right_list2_aux3 T1 h4 h3 l4 l3 H3));
+                   nrewrite > (H h3 h4 (fold_right_list2 T1 T2 f acc l3 l4 ?));
+                   napply refl_eq
+          ##]
+ ##]
+nqed.
+
+nlemma symmetric_foldrightlist2 :
+∀T1,T2:Type.∀f:T1 → T1 → T2 → T2.
+ (∀x,y,z.f x y z = f y x z) →
+ (∀acc:T2.∀l1,l2:list T1.∀H:len_list T1 l1 = len_list T1 l2.
+   fold_right_list2 T1 T2 f acc l1 l2 H = fold_right_list2 T1 T2 f acc l2 l1 (symmetric_lenlist T1 l1 l2 H)).
+ #T1; #T2; #f; #H; #acc; #l1; #l2; #H1;
+ nrewrite > (symmetric_foldrightlist2_aux T1 T2 f H acc l1 l2 H1 (symmetric_lenlist T1 l1 l2 H1));
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_bfoldrightlist2 :
+∀T1:Type.∀f:T1 → T1 → bool.
+ (∀x,y.f x y = f y x) →
+ (∀l1,l2:list T1.
+  bfold_right_list2 T1 f l1 l2 = bfold_right_list2 T1 f l2 l1).
+ #T; #f; #H; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+          ##[ ##1: nnormalize; napply refl_eq
+          ##| ##2: #hh2; #ll2; nnormalize; napply refl_eq
+          ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+          ##[ ##1: nnormalize; napply refl_eq
+          ##| ##2: #hh2; #ll2; nnormalize;
+                   nrewrite > (H1 ll2);
+                   nrewrite > (H hh1 hh2);
+                   napply refl_eq
+          ##]
+ ##]
+nqed.
+
+nlemma bfoldrightlist2_to_eq :
+∀T1:Type.∀f:T1 → T1 → bool.
+ (∀x,y.(f x y = true → x = y)) →
+ (∀l1,l2:list T1.
+  (bfold_right_list2 T1 f l1 l2 = true → l1 = l2)).
+ #T; #f; #H; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+          ##[ ##1: #H1; napply refl_eq
+          ##| ##2: #hh2; #ll2; nnormalize; #H1;
+                   ndestruct (*napply (bool_destruct … H1)*)
+          ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+          ##[ ##1: nnormalize; #H2;
+                   ndestruct (*napply (bool_destruct … H2)*)
+          ##| ##2: #hh2; #ll2; #H2;
+                   nchange in H2:(%) with (((f hh1 hh2)⊗(bfold_right_list2 T f ll1 ll2)) = true);
+                   nrewrite > (H hh1 hh2 (andb_true_true_l … H2));
+                   nrewrite > (H1 ll2 (andb_true_true_r … H2));
+                   napply refl_eq
+          ##]
+ ##]
+nqed.
+
+nlemma eq_to_bfoldrightlist2 :
+∀T1:Type.∀f:T1 → T1 → bool.
+ (∀x,y.(x = y → f x y = true)) →
+ (∀l1,l2:list T1.
+  (l1 = l2 → bfold_right_list2 T1 f l1 l2 = true)).
+ #T; #f; #H; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+          ##[ ##1: #H1; nnormalize; napply refl_eq
+          ##| ##2: #hh2; #ll2; #H1;
+                   (* !!! ndestruct: assert false *)
+                   nelim (list_destruct_nil_cons ??? H1)
+          ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+          ##[ ##1: #H2;
+                   (* !!! ndestruct: assert false *)
+                   nelim (list_destruct_cons_nil ??? H2)
+          ##| ##2: #hh2; #ll2; #H2; nnormalize;
+                   nrewrite > (list_destruct_1 … H2);
+                   nrewrite > (H hh2 hh2 (refl_eq …));
+                   nnormalize;
+                   nrewrite > (H1 ll2 (list_destruct_2 … H2));
+                   napply refl_eq
+          ##]
+ ##]
+nqed.
+
+nlemma bfoldrightlist2_to_lenlist :
+∀T.∀f:T → T → bool.
+ (∀l1,l2:list T.bfold_right_list2 T f l1 l2 = true → len_list T l1 = len_list T l2).
+ #T; #f; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+          ##[ ##1: nnormalize; #H; napply refl_eq
+          ##| ##2: nnormalize; #hh; #tt; #H;
+                   ndestruct (*napply (bool_destruct … H)*)
+          ##]
+ ##| ##2: #hh; #tt; #H; #l2; ncases l2;
+          ##[ ##1: nnormalize; #H1;
+                   ndestruct (*napply (bool_destruct … H1)*)
+          ##| ##2: #hh1; #tt1; #H1; nnormalize;
+                   nrewrite > (H tt1 ?);
+                   ##[ ##1: napply refl_eq
+                   ##| ##2: nchange in H1:(%) with ((? ⊗ (bfold_right_list2 T f tt tt1)) = true);
+                            napply (andb_true_true_r … H1)
+                   ##]
+          ##]
+ ##]
+nqed.
+
+nlemma decidable_list :
+∀T.(∀x,y:T.decidable (x = y)) →
+   (∀x,y:list T.decidable (x = y)).
+ #T; #H; #x; nelim x;
+ ##[ ##1: #y; ncases y;
+          ##[ ##1: nnormalize; napply (or2_intro1 (? = ?) (? ≠ ?) (refl_eq …))
+          ##| ##2: #hh2; #tt2; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+                   nnormalize; #H1;
+                   (* !!! ndestruct: assert false *)
+                   napply (list_destruct_nil_cons T … H1)
+          ##]
+ ##| ##2: #hh1; #tt1; #H1; #y; ncases y;
+          ##[ ##1: nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+                   nnormalize; #H2;
+                   (* !!! ndestruct: assert false *)
+                   napply (list_destruct_cons_nil T … H2)
+          ##| ##2: #hh2; #tt2; nnormalize; napply (or2_elim (hh1 = hh2) (hh1 ≠ hh2) ? (H …));
+                   ##[ ##2: #H2; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+                            nnormalize; #H3; napply (H2 (list_destruct_1 T … H3))
+                   ##| ##1: #H2; napply (or2_elim (tt1 = tt2) (tt1 ≠ tt2) ? (H1 tt2));
+                            ##[ ##2: #H3; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+                                     nnormalize; #H4; napply (H3 (list_destruct_2 T … H4))
+                            ##| ##1: #H3; napply (or2_intro1 (? = ?) (? ≠ ?) ?);
+                                     nrewrite > H2; nrewrite > H3; napply refl_eq
+                            ##]
+                   ##]
+          ##]
+ ##]
+nqed.
+
+nlemma nbfoldrightlist2_to_neq :
+∀T1:Type.∀f:T1 → T1 → bool.
+ (∀x,y.(f x y = false → x ≠ y)) →
+ (∀l1,l2:list T1.
+  (bfold_right_list2 T1 f l1 l2 = false → l1 ≠ l2)).
+ #T; #f; #H; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+          ##[ ##1: nnormalize; #H1;
+                   ndestruct (*napply (bool_destruct … H1)*)
+          ##| ##2: #hh2; #ll2; #H1; nnormalize; #H2;
+                   (* !!! ndestruct: assert false *)
+                   napply (list_destruct_nil_cons T … H2)
+          ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+          ##[ ##1: #H2; nnormalize; #H3;
+                   (* !!! ndestruct: assert false *)
+                   napply (list_destruct_cons_nil T … H3)
+          ##| ##2: #hh2; #ll2; #H2; nnormalize; #H3;
+                   nchange in H2:(%) with (((f hh1 hh2)⊗(bfold_right_list2 T f ll1 ll2)) = false);
+                   napply (H1 ll2 ? (list_destruct_2 T … H3));
+                   napply (or2_elim ??? (andb_false2 … H2) );
+                   ##[ ##1: #H4; napply (absurd (hh1 = hh2) …);
+                            ##[ ##1: nrewrite > (list_destruct_1 T … H3); napply refl_eq
+                            ##| ##2: napply (H … H4)
+                            ##]
+                   ##| ##2: #H4; napply H4
+                   ##]
+          ##]
+ ##]
+nqed.
+
+nlemma list_destruct :
+∀T.(∀x,y:T.decidable (x = y)) →
+   (∀h1,h2:T.∀l1,l2:list T.
+    (h1::l1) ≠ (h2::l2) → h1 ≠ h2 ∨ l1 ≠ l2).
+ #T; #H; #h1; #h2; #l1; nelim l1;
+ ##[ ##1: #l2; ncases l2;
+          ##[ ##1: #H1; napply (or2_intro1 (h1 ≠ h2) ([] ≠ []) …);
+                   nnormalize; #H2; nrewrite > H2 in H1:(%);
+                   nnormalize; #H1; napply (H1 (refl_eq …))
+          ##| ##2: #hh2; #ll2; #H1; napply (or2_intro2 (h1 ≠ h2) ([] ≠ (hh2::ll2)) …);
+                   nnormalize; #H2;
+                   (* !!! ndestruct: assert false *)
+                   napply (list_destruct_nil_cons T … H2)
+          ##]
+ ##| ##2: #hh1; #ll1; #H1; #l2; ncases l2;
+          ##[ ##1: #H2; napply (or2_intro2 (h1 ≠ h2) ((hh1::ll1) ≠ []) …);
+                   nnormalize; #H3;
+                   (* !!! ndestruct: assert false *)
+                   napply (list_destruct_cons_nil T … H3)
+          ##| ##2: #hh2; #ll2; #H2;     
+                   napply (or2_elim (h1 = h2) (h1 ≠ h2) ? (H h1 h2) …);
+                   ##[ ##2: #H3; napply (or2_intro1 (h1 ≠ h2) ((hh1::ll1) ≠ (hh2::ll2)) H3)
+                   ##| ##1: #H3; napply (or2_intro2 (h1 ≠ h2) ((hh1::ll1) ≠ (hh2::ll2) …));
+                            nrewrite > H3 in H2:(%); #H2;
+                            nnormalize; #H4; nrewrite > (list_destruct_1 T … H4) in H2:(%); #H2;
+                            nrewrite > (list_destruct_2 T … H4) in H2:(%); #H2;
+                            napply (H2 (refl_eq …))
+                   ##]
+          ##]
+ ##]
+nqed.
+
+nlemma neq_to_nbfoldrightlist2 :
+∀T:Type.∀f:T → T → bool.
+ (∀x,y:T.decidable (x = y)) →
+ (∀x,y.(x ≠ y → f x y = false)) →
+ (∀l1,l2:list T.
+  (l1 ≠ l2 → bfold_right_list2 T f l1 l2 = false)).
+ #T; #f; #H; #H1; #l1;
+ nelim l1;
+ ##[ ##1: #l2; ncases l2;
+          ##[ ##1: nnormalize; #H2; nelim (H2 (refl_eq …))
+          ##| ##2: #hh2; #ll2; nnormalize; #H2; napply refl_eq
+          ##]
+ ##| ##2: #hh1; #ll1; #H2; #l2; ncases l2;
+          ##[ ##1: nnormalize; #H3; napply refl_eq
+          ##| ##2: #hh2; #ll2; #H3;
+                   nchange with (((f hh1 hh2)⊗(bfold_right_list2 T f ll1 ll2)) = false);
+                   napply (or2_elim (hh1 ≠ hh2) (ll1 ≠ ll2) ? (list_destruct T H … H3) …);
+                   ##[ ##1: #H4; nrewrite > (H1 hh1 hh2 H4); nnormalize; napply refl_eq
+                   ##| ##2: #H4; nrewrite > (H2 ll2 H4);
+                            nrewrite > (symmetric_andbool (f hh1 hh2) false);
+                            nnormalize; napply refl_eq
+                   ##]
+          ##]
+ ##]
+nqed.
+
+nlemma isbemptylist_to_isemptylist : ∀T,l.isb_empty_list T l = true → is_empty_list T l.
+ #T; #l;
+ ncases l;
+ nnormalize;
+ ##[ ##1: #H; napply I
+ ##| ##2: #x; #l; #H; ndestruct (*napply (bool_destruct … H)*)
+ ##]
+nqed.
+
+nlemma isnotbemptylist_to_isnotemptylist : ∀T,l.isnotb_empty_list T l = true → isnot_empty_list T l.
+ #T; #l;
+ ncases l;
+ nnormalize;
+ ##[ ##1: #H; ndestruct (*napply (bool_destruct … H)*)
+ ##| ##2: #x; #l; #H; napply I
+ ##]
+nqed.
+
+nlemma list_is_comparable : comparable → comparable.
+ #T; napply (mk_comparable (list T));
+ ##[ napply (nil ?)
+ ##| napply (λx.false)
+ ##| napply (bfold_right_list2 T (eqc T))
+ ##| napply (bfoldrightlist2_to_eq … (eqc T));
+     napply (eqc_to_eq T)
+ ##| napply (eq_to_bfoldrightlist2 … (eqc T));
+     napply (eq_to_eqc T)
+ ##| napply (nbfoldrightlist2_to_neq … (eqc T));
+     napply (neqc_to_neq T)
+ ##| napply (neq_to_nbfoldrightlist2 … (eqc T));
+     ##[ napply (decidable_c T)
+     ##| napply (neq_to_neqc T)
+     ##]
+ ##| napply decidable_list;
+     napply (decidable_c T)
+ ##| napply symmetric_bfoldrightlist2;
+     napply (symmetric_eqc T)
+ ##]
+nqed.
+
+unification hint 0 ≔ S: comparable;
+         T ≟ (carr S),
+         X ≟ (list_is_comparable S)
+ (*********************************************) ⊢
+         carr X ≡ list T.