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(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
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(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(** list. Polymorphic lists with some operations.  @author Olga Caprotti and Martijn Oostdijk @version $Revision$ *)
(** * Polymorphic lists *)
Inductive list (A : Set) : Set :=
| Nil : list A
| Cons : A -> list A -> list A.
(** Length of a list *)
Fixpoint length (A : Set) (l : list A) {struct l} : nat :=
match l with
| Nil => 0
| Cons a r => S (length A r)
end.
(** Map a function over a list *)
Fixpoint map (A B : Set) (f : A -> B) (l : list A) {struct l} :
list B :=
match l with
| Nil => Nil B
| Cons a r => Cons B (f a) (map A B f r)
end.
Notation Map := (map _ _) (only parsing).
(** Checks that all members of a list are P *)
Fixpoint alllist (A : Set) (P : A -> Prop) (qlist : list A) {struct qlist} :
Prop :=
match qlist with
| Nil => True
| Cons m l => P m /\ alllist A P l
end.
(** Checks that some member of a list is P *)
Fixpoint exlist (A : Set) (P : A -> Prop) (qlist : list A) {struct qlist} :
Prop :=
match qlist with
| Nil => False
| Cons m l => P m \/ exlist A P l
end.
(** Membership *)
Definition inlist (A : Set) (a : A) := exlist A (fun b : A => a = b).
(** alllist and exlist behave nicely *)
(** * Lists of natural numbers. *)
Definition natlist := list nat.
(** Multiply all elements of a natlist *)
Fixpoint product (qlist : natlist) : nat :=
match qlist with
| Nil => 1
| Cons m l => m * product l
end.
(** Drop the first occurance of q from qlist *)
Fixpoint drop (q : nat) (qlist : natlist) {struct qlist} : natlist :=
match qlist with
| Nil => Nil nat
| Cons q' l => if beq_nat q q' then l else Cons nat q' (drop q l)
end.
(** Multiply all elements of qlist, except for (first occurance of) q *)
Definition multDrop (q : nat) (l : natlist) := product (drop q l).
(** * Lists of integers. *)
Definition Zlist := list Z.
(** allPos checks whether all members of a Zlist are non-negative. *)
Definition allPos : Zlist -> Prop := alllist Z (fun x : Z => (x >= 0)%Z).
(** Multiply all elements of a Zlist *)
Fixpoint zproduct (l : Zlist) : Z :=
match l with
| Nil => 1%Z
| Cons x t => (x * zproduct t)%Z
end.
(** Drop an element from a Zlist *)
Fixpoint zdrop (x : Z) (l : Zlist) {struct l} : Zlist :=
match l with
| Nil => Nil Z
| Cons h t => if Zeq_bool x h then t else Cons Z h (zdrop x t)
end.
(** Multiply all elements except first occurance of x *)
Definition zmultDrop (x : Z) (l : Zlist) := zproduct (zdrop x l).
(** Multiply all elements in list with a *)
Definition mapmult (a : Z) (l : Zlist) := map Z Z (fun x : Z => (a * x)%Z) l.

Lemma mapmult_image :
forall (a : Z) (l : Zlist) (x : Z),
inlist Z x l -> inlist Z (a * x)%Z (mapmult a l).