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(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(* ---                        matrix.v                                  --- *)
(*                                                                          *)
(* Author: Pierre Casteran.                                                 *)
(*    LABRI, URA CNRS 1304,                                                 *)
(*    Departement d'Informatique, Universite Bordeaux I,                    *)
(*    33405 Talence CEDEX,                                                  *)
(*    e-mail:  casteran@labri.u-bordeaux.fr                                 *)
(*          2-2 matrices with natural coefficients                          *)
(* original version using nat instead of Z                                  *)
Record Mat2 : Set := mat2 {M11 : nat; M12 : nat; M21 : nat; M22 : nat}.
Definition Id2 := mat2 1 0 0 1.
Definition Mat_mult (M M' : Mat2) :=
mat2 (M11 M * M11 M' + M12 M * M21 M') (M11 M * M12 M' + M12 M * M22 M')
(M21 M * M11 M' + M22 M * M21 M') (M21 M * M12 M' + M22 M * M22 M').
Axiom
Mat_assoc :
forall M M' M'' : Mat2,
Mat_mult M (Mat_mult M' M'') = Mat_mult (Mat_mult M M') M''.
refine (mkmonoid Mat2 Id2 Mat_mult _ _ _).
exact Mat_assoc.
simple induction a.
intros M13 M14 M23 M24.
unfold Id2, Mat_mult in |- *; simpl in |- *.
repeat rewrite plus_0_r.
case M13; case M14; case M23; case M24; auto.
simple induction a.
intros.
unfold Id2, Mat_mult in |- *; simpl in |- *.
repeat rewrite mult_0_r.
repeat rewrite mult_1_r.
simpl in |- *; repeat rewrite plus_0_r.
auto.
Defined.
(* Fibonacci numbers *)
(* definition *)
Fixpoint Fib (n : nat) : nat :=
match n return nat with
| O => (* O *)  1
(* (S p ) *)
| S p =>
match p return nat with
| O => (* O *)  1
(* S q *)
| S q => Fib q + Fib p
end
end.
(* A "Decaled" Fibonnacci function *)
Definition shift_Fib (n : nat) :=
match n return nat with
| O => 0
| S p => Fib p
end.

Lemma Simpl_shift_Fib :
forall n : nat, shift_Fib (S (S n)) = shift_Fib (S n) + shift_Fib n.