Section Z_compl.
Local Open Scope Z_scope.
(* Provides a way to reason directly on Z in terms of nats instead of positive *)
Inductive Z_spec (x : Z) : Z -> Type :=
| ZintNull : x = 0 -> Z_spec x 0
| ZintPos (n : nat) : x = n -> Z_spec x n
| ZintNeg (n : nat) : x = - n -> Z_spec x (- n).
End Z_compl.
Definition powerRZ (x:R) (n:Z) :=
match n with
| Z0 => 1
| Zpos p => x ^ Pos.to_nat p
| Zneg p => / x ^ Pos.to_nat p
end.
Local Infix "^Z" := powerRZ (at level 30, right associativity) : R_scope.
Hint Resolve powerRZ_O powerRZ_1 powerRZ_NOR powerRZ_add: real.
Hint Resolve powerRZ_lt: real.
Hint Resolve powerRZ_le: real.
Local Open Scope Z_scope.
End PowerRZ.
Local Infix "^Z" := powerRZ (at level 30, right associativity) : R_scope.
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(* For easy interface          *)
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(* decimal_exp r z is defined as r 10^z *)
Definition decimal_exp (r:R) (z:Z) : R := (r * 10 ^Z z).
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(** * Sum of n first naturals  *)
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Fixpoint sum_nat_f_O (f:nat -> nat) (n:nat) : nat :=
match n with
| O => f 0%nat
| S n' => (sum_nat_f_O f n' + f (S n'))%nat
end.
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Definition sum_nat_f (s n:nat) (f:nat -> nat) : nat :=
sum_nat_f_O (fun x:nat => f (x + s)%nat) (n - s).
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Definition sum_nat_O (n:nat) : nat := sum_nat_f_O (fun x:nat => x) n.
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Definition sum_nat (s n:nat) : nat := sum_nat_f s n (fun x:nat => x).
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(** *          Sum             *)
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Fixpoint sum_f_R0 (f:nat -> R) (N:nat) : R :=
match N with
| O => f 0%nat
| S i => sum_f_R0 f i + f (S i)
end.
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Definition sum_f (s n:nat) (f:nat -> R) : R :=
sum_f_R0 (fun x:nat => f (x + s)%nat) (n - s).
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(** *     Distance  in R       *)
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Definition R_dist (x y:R) : R := Rabs (x - y).

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Lemma R_dist_plus :
forall a b c d:R, R_dist (a + c) (b + d) <= R_dist a b + R_dist c d.