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(* 02110-1301 USA                                                     *)
(****************************************************************************  IEEE754  :  Rpower  Laurent Thery  ***************************************************************************** Definition of an exponential function over relative numbers *)
(* We have already an exponential over natural number, we prove some basic properties for this function *)
simpl in |- *; auto with real.
Qed.
simpl in |- *; auto with real.
Qed.
intros e n; elim n; simpl in |- *; auto with real.
Qed.
intros e n; elim n; simpl in |- *; auto with real.
intros n0 H' m; rewrite H'; auto with real.
Qed.
Hint Resolve pow_O pow_1 pow_NR0 pow_add: real.
intros e n; elim n; simpl in |- *; auto with real.
intros n0 H' m H'0.
rewrite Rmult_assoc; rewrite <- H'; auto.
Qed.
intros e n; elim n; simpl in |- *; auto with real.
intros n0 H' H'0; replace 0%R with (e * 0)%R; auto with real.
Qed.
Hint Resolve pow_lt: real.
intros e n; elim n; simpl in |- *; auto with real.
intros H' H'0; Contradict H'0; auto with arith.
intros n0; case n0.
simpl in |- *; rewrite Rmult_1_r; auto.
intros n1 H' H'0 H'1.
replace 1%R with (1 * 1)%R; auto with real.
apply Rlt_trans with (r2 := (e * 1)%R); auto with real.
apply Rmult_lt_compat_l; auto with real.
apply Rlt_trans with (r2 := 1%R); auto with real.
apply H'; auto with arith.
Qed.
Hint Resolve Rlt_pow_R1: real.
intros e n m H' H'0; replace m with (m - n + n).
rewrite pow_add.
pattern (e ^ n)%R at 1 in |- *; replace (e ^ n)%R with (1 * e ^ n)%R;
auto with real.
apply Rminus_lt.
repeat rewrite (fun x : R => Rmult_comm x (e ^ n));
rewrite <- Rmult_minus_distr_l.
replace 0%R with (e ^ n * 0)%R; auto with real.
apply Rmult_lt_compat_l; auto with real.
apply pow_lt; auto with real.
apply Rlt_trans with (r2 := 1%R); auto with real.
apply Rlt_minus; auto with real.
apply Rlt_pow_R1; auto with arith.
apply plus_lt_reg_l with (p := n); auto with arith.
rewrite le_plus_minus_r; auto with arith; rewrite <- plus_n_O; auto.
rewrite plus_comm; auto with arith.
Qed.
Hint Resolve Rlt_pow: real.
intros r n H'.
case (Req_dec (Rabs r) 1); auto; intros H'1.
case (Rdichotomy _ _ H'1); intros H'2.
generalize H'; case n; auto.
intros n0 H'0.
cut (r <> 0%R); [ intros Eq1 | idtac ].
2: Contradict H'0; auto with arith.
2: simpl in |- *; rewrite H'0; rewrite Rmult_0_l; auto with real.
cut (Rabs r <> 0%R); [ intros Eq2 | apply Rabs_no_R0 ]; auto.
absurd (Rabs (/ r) ^ 0 < Rabs (/ r) ^ S n0)%R; auto.
replace (Rabs (/ r) ^ S n0)%R with 1%R.
simpl in |- *; apply Rlt_irrefl; auto.
rewrite Rabs_Rinv; auto.
rewrite <- Rinv_pow; auto.
rewrite RPow_abs; auto.
rewrite H'0; rewrite Rabs_right; auto with real.
apply Rlt_pow; auto with arith.
rewrite Rabs_Rinv; auto.
apply Rmult_lt_reg_l with (r := Rabs r).
case (Rabs_pos r); auto.
intros H'3; case Eq2; auto.
rewrite Rmult_1_r; rewrite Rinv_r; auto with real.
generalize H'; case n; auto.
intros n0 H'0.
cut (r <> 0%R); [ intros Eq1 | auto with real ].
2: Contradict H'0; simpl in |- *; rewrite H'0; rewrite Rmult_0_l;
auto with real.
cut (Rabs r <> 0%R); [ intros Eq2 | apply Rabs_no_R0 ]; auto.
absurd (Rabs r ^ 0 < Rabs r ^ S n0)%R; auto with real arith.
repeat rewrite RPow_abs; rewrite H'0; simpl in |- *; auto with real.
Qed.
intros e n; elim n; unfold Zpower_nat in |- *; simpl in |- *;
auto with zarith.
Qed.
intros e n; elim n; unfold Zpower_nat in |- *; simpl in |- *;
auto with zarith.
Qed.
Hint Resolve Zpower_NR0 Zpower_NR1: zarith.
(* To define exponential over relative number, we simply do a case analysis on the sign of the number *)
Definition powerRZ (e : R) (n : BinInt.Z) :=
match n with
| BinInt.Z0 => 1%R
| BinInt.Zpos p => (e ^ BinPos.nat_of_P p)%R
| BinInt.Zneg p => (/ e ^ BinPos.nat_of_P p)%R
end.
(* we now prove some basic properties of our exponential *)
simpl in |- *; auto.
Qed.
simpl in |- *; auto with real.
Qed.
intros e z; case z; simpl in |- *; auto with real.
Qed.
Hint Resolve powerRZ_O powerRZ_1 powerRZ_NOR: real.
intros e n m; case n; case m; simpl in |- *; auto with real.
intros n1 m1; rewrite Pnat.nat_of_P_plus_morphism; auto with real.
intros n1 m1. rewrite Z.pos_sub_spec; unfold Pos.compare.
CaseEq (Pcompare m1 n1 Datatypes.Eq); simpl in |- *;
auto with real.
intros H' H'0; rewrite BinPos.Pcompare_Eq_eq with (1 := H'); auto with real.
intros H' H'0; rewrite (Pnat.nat_of_P_minus_morphism n1 m1); auto with real.
rewrite
(pow_RN_plus e (BinPos.nat_of_P n1 - BinPos.nat_of_P m1)
(BinPos.nat_of_P m1)); auto with real.
rewrite plus_comm; rewrite le_plus_minus_r; auto with real.
rewrite Rinv_mult_distr; auto with real.
rewrite Rinv_involutive; auto with real.
apply lt_le_weak.
apply Pnat.nat_of_P_lt_Lt_compare_morphism; auto.
apply BinPos.ZC2; auto.
intros H' H'0; rewrite (Pnat.nat_of_P_minus_morphism m1 n1); auto with real.
rewrite
(pow_RN_plus e (BinPos.nat_of_P m1 - BinPos.nat_of_P n1)
(BinPos.nat_of_P n1)); auto with real.
rewrite plus_comm; rewrite le_plus_minus_r; auto with real.
apply lt_le_weak.
change (BinPos.nat_of_P m1 > BinPos.nat_of_P n1) in |- *.
apply Pnat.nat_of_P_gt_Gt_compare_morphism; auto.
intros n1 m1. rewrite Z.pos_sub_spec; unfold Pos.compare.
CaseEq (Pcompare n1 m1 Datatypes.Eq); simpl in |- *;
auto with real.
intros H' H'0; rewrite BinPos.Pcompare_Eq_eq with (1 := H'); auto with real.
intros H' H'0; rewrite (Pnat.nat_of_P_minus_morphism m1 n1); auto with real.
rewrite
(pow_RN_plus e (BinPos.nat_of_P m1 - BinPos.nat_of_P n1)
(BinPos.nat_of_P n1)); auto with real.
rewrite plus_comm; rewrite le_plus_minus_r; auto with real.
rewrite Rinv_mult_distr; auto with real.
apply lt_le_weak.
apply Pnat.nat_of_P_lt_Lt_compare_morphism; auto.
apply BinPos.ZC2; auto.
intros H' H'0; rewrite (Pnat.nat_of_P_minus_morphism n1 m1); auto with real.
rewrite
(pow_RN_plus e (BinPos.nat_of_P n1 - BinPos.nat_of_P m1)
(BinPos.nat_of_P m1)); auto with real.
rewrite plus_comm; rewrite le_plus_minus_r; auto with real.
apply lt_le_weak.
change (BinPos.nat_of_P n1 > BinPos.nat_of_P m1) in |- *.
apply Pnat.nat_of_P_gt_Gt_compare_morphism; auto.
intros n1 m1; rewrite Pnat.nat_of_P_plus_morphism; auto with real.
intros H'; rewrite pow_add; auto with real.
apply Rinv_mult_distr; auto.
apply pow_NR0; auto.
apply pow_NR0; auto.
Qed.
Hint Resolve powerRZ_O powerRZ_1 powerRZ_NOR powerRZ_add: real.
intros e z H; case z; simpl in |- *; auto with real.
intros p; apply sym_eq; apply Rinv_involutive.
apply pow_nonzero; auto.
Qed.
intros e n H'0.
replace (Z.succ n) with (n + Z.succ 0)%Z.
rewrite powerRZ_add; auto.
rewrite powerRZ_1.
rewrite Rmult_comm; auto.
auto with zarith.
Qed.
(* Conversion theorem between relative numbers and reals *)
intros n m; elim m; simpl in |- *; auto with real.
intros m1 H'; rewrite Pnat.nat_of_P_o_P_of_succ_nat_eq_succ; simpl in |- *.
replace (Zpower_nat n (S m1)) with (n * Zpower_nat n m1)%Z.
rewrite Rmult_IZR; auto with real.
rewrite H'; simpl in |- *.
case m1; simpl in |- *; auto with real.
intros m2; rewrite Pnat.nat_of_P_o_P_of_succ_nat_eq_succ; auto.
unfold Zpower_nat in |- *; auto.
Qed.

Theorem powerRZ_lt :
forall (e : R) (z : BinInt.Z), (0 < e)%R -> (0 < powerRZ e z)%R.