Require Import monoid.
Require Import Arith.
Require Import ZArith.
Require Import Compare_dec.
Require Import fmpc.

Record Mat2 : Set := mat2 {M11 : Z; M12 : Z; M21 : Z; M22 : Z}.

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 :
M M' M'' : Mat2,
Mat_mult M (Mat_mult M' M'') = Mat_mult (Mat_mult M M') M''.

Lemma matrix : monoid Mat2.
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 elim Zplus_0_r_reverse.
case M13; case M14; case M23; case M24; auto.
simple induction a.
intros.
unfold Id2, Mat_mult in |- *; simpl in |- ×.
repeat elim Zmult_0_r_reverse.
repeat rewrite Zmult_1_r.
simpl in |- *; repeat elim Zplus_0_r_reverse.
auto with arith.
Defined.

Fixpoint Fib (n : nat) : nat :=
match n return nat with
| O ⇒ 1

| S p
match p return nat with
| O ⇒ 1

| S qFib q + Fib p
end
end.

Lemma Unfold_FibO : Fib 0 = 1.
Proof.
unfold Fib in |- *; simpl in |- *; auto with arith.
Qed.

Lemma Unfold_Fib1 : Fib 1 = 1.
Proof.
unfold Fib in |- *; simpl in |- *; auto with arith.
Qed.

Lemma Unfold_FibSSn : n : nat, Fib (S (S n)) = Fib (S n) + Fib n.
Proof.
intro n; unfold Fib at 1 in |- ×.
simpl in |- *; auto with arith.
Qed.

Definition shift_Fib (n : nat) :=
match n return nat with
| O ⇒ 0
| S pFib p
end.

Lemma Unfold_shift_Fib : n : nat, shift_Fib (S n) = Fib n.
Proof.
intro n; unfold shift_Fib in |- *; auto with arith.
Qed.

Lemma Simpl_shift_Fib :
n : nat, shift_Fib (S (S n)) = shift_Fib (S n) + shift_Fib n.
Proof.
simple induction n.
unfold shift_Fib, Fib in |- *; simpl in |- *; auto with arith.
intros.
unfold shift_Fib in |- ×.
rewrite Unfold_FibSSn; auto with arith.
Qed.

Definition fib_mat := mat2 1 1 1 0.

Lemma fib_mat_n :
(n : nat) (a b d : Z),
power Mat2 matrix fib_mat n = mat2 a b b d
power Mat2 matrix fib_mat (S n) = mat2 (a + b) (b + d) a b.
Proof.
intros; simpl in |- ×.
rewrite H.
unfold Mat_mult in |- ×.
simpl in |- ×.
repeat elim Zplus_0_r_reverse.
case a; case b; case d; auto.
Qed.

Lemma fib_n :
n : nat,
power Mat2 matrix fib_mat (S n) =
mat2 (Z_of_nat (shift_Fib (S (S n)))) (Z_of_nat (shift_Fib (S n)))
(Z_of_nat (shift_Fib (S n))) (Z_of_nat (shift_Fib n)).
Proof.
simple induction n.
unfold power, shift_Fib, o, u in |- *; simpl in |- ×.
unfold fib_mat in |- *; simpl in |- ×.
unfold Mat_mult, Id2 in |- *; simpl in |- *; auto with arith.
intros.
rewrite (fib_mat_n (S n0) _ _ _ H).
repeat rewrite <- Znat.inj_plus.
rewrite (Simpl_shift_Fib (S n0)).
pattern (shift_Fib (S (S n0))) at 4 in |- ×.
rewrite (Simpl_shift_Fib n0).
auto with arith.
Qed.

Lemma fib_computation :
n : nat,
0 < n Z_of_nat (Fib n) = M11 (power Mat2 matrix fib_mat n).
Proof.
simple induction n.
intro; absurd (0 < 0); auto with arith.
intros.
rewrite fib_n; unfold M11 in |- *; auto with arith.
Qed.