Library Coq.Reals.Rsigma


Require Import Rbase.

Require Import Rfunctions.

Require Import Rseries.

Require Import PartSum.

Open Local Scope R_scope.




Set Implicit Arguments.




Section Sigma.




  Variable f : nat -> R.




  Definition sigma (low high:nat) : R :=

    sum_f_R0 (fun k:nat => f (low + k)) (high - low).




  Theorem sigma_split :

    forall low high k:nat,

      (low <= k)%nat ->

      (k < high)%nat -> sigma low high = sigma low k + sigma (S k) high.

  Proof.

    intros; induction  k as [| k Hreck].

    cut (low = 0%nat).

    intro; rewrite H1; unfold sigma in |- *; rewrite <- minus_n_n;

      rewrite <- minus_n_O; simpl in |- *; replace (high - 1)%nat with (pred high).

    apply (decomp_sum (fun k:nat => f k)).

    assumption.

    apply pred_of_minus.

    inversion H; reflexivity.

    cut ((low <= k)%nat \/ low = S k).

    intro; elim H1; intro.

    replace (sigma low (S k)) with (sigma low k + f (S k)).

    rewrite Rplus_assoc;

      replace (f (S k) + sigma (S (S k)) high) with (sigma (S k) high).

    apply Hreck.

    assumption.

    apply lt_trans with (S k); [ apply lt_n_Sn | assumption ].

    unfold sigma in |- *; replace (high - S (S k))%nat with (pred (high - S k)).

    pattern (S k) at 3 in |- *; replace (S k) with (S k + 0)%nat;

      [ idtac | ring ].

    replace (sum_f_R0 (fun k0:nat => f (S (S k) + k0)) (pred (high - S k))) with

    (sum_f_R0 (fun k0:nat => f (S k + S k0)) (pred (high - S k))).

    apply (decomp_sum (fun i:nat => f (S k + i))).

    apply lt_minus_O_lt; assumption.

    apply sum_eq; intros; replace (S k + S i)%nat with (S (S k) + i)%nat.

    reflexivity.

    ring.

    replace (high - S (S k))%nat with (high - S k - 1)%nat.

    apply pred_of_minus.

    omega.

    unfold sigma in |- *; replace (S k - low)%nat with (S (k - low)).

    pattern (S k) at 1 in |- *; replace (S k) with (low + S (k - low))%nat.

    symmetry  in |- *; apply (tech5 (fun i:nat => f (low + i))).

    omega.

    omega.

    rewrite <- H2; unfold sigma in |- *; rewrite <- minus_n_n; simpl in |- *;

      replace (high - S low)%nat with (pred (high - low)).

    replace (sum_f_R0 (fun k0:nat => f (S (low + k0))) (pred (high - low))) with

    (sum_f_R0 (fun k0:nat => f (low + S k0)) (pred (high - low))).

    apply (decomp_sum (fun k0:nat => f (low + k0))).

    apply lt_minus_O_lt.

    apply le_lt_trans with (S k); [ rewrite H2; apply le_n | assumption ].

    apply sum_eq; intros; replace (S (low + i)) with (low + S i)%nat.

    reflexivity.

    ring.

    omega.

    inversion H; [ right; reflexivity | left; assumption ].

  Qed.




  Theorem sigma_diff :

    forall low high k:nat,

      (low <= k)%nat ->

      (k < high)%nat -> sigma low high - sigma low k = sigma (S k) high.

  Proof.

    intros low high k H1 H2; symmetry  in |- *; rewrite (sigma_split H1 H2); ring.

  Qed.




  Theorem sigma_diff_neg :

    forall low high k:nat,

      (low <= k)%nat ->

      (k < high)%nat -> sigma low k - sigma low high = - sigma (S k) high.

  Proof.

    intros low high k H1 H2; rewrite (sigma_split H1 H2); ring.

  Qed.




  Theorem sigma_first :

    forall low high:nat,

      (low < high)%nat -> sigma low high = f low + sigma (S low) high.

  Proof.

    intros low high H1; generalize (lt_le_S low high H1); intro H2;

      generalize (lt_le_weak low high H1); intro H3;

        replace (f low) with (sigma low low).

    apply sigma_split.

    apply le_n.

    assumption.

    unfold sigma in |- *; rewrite <- minus_n_n.

    simpl in |- *.

    replace (low + 0)%nat with low; [ reflexivity | ring ].

  Qed.




  Theorem sigma_last :

    forall low high:nat,

      (low < high)%nat -> sigma low high = f high + sigma low (pred high).

  Proof.

    intros low high H1; generalize (lt_le_S low high H1); intro H2;

      generalize (lt_le_weak low high H1); intro H3;

        replace (f high) with (sigma high high).

    rewrite Rplus_comm; cut (high = S (pred high)).

    intro; pattern high at 3 in |- *; rewrite H.

    apply sigma_split.

    apply le_S_n; rewrite <- H; apply lt_le_S; assumption.

    apply lt_pred_n_n; apply le_lt_trans with low; [ apply le_O_n | assumption ].

    apply S_pred with 0%nat; apply le_lt_trans with low;

      [ apply le_O_n | assumption ].

    unfold sigma in |- *; rewrite <- minus_n_n; simpl in |- *;

      replace (high + 0)%nat with high; [ reflexivity | ring ].

  Qed.




  Theorem sigma_eq_arg : forall low:nat, sigma low low = f low.

  Proof.

    intro; unfold sigma in |- *; rewrite <- minus_n_n.

    simpl in |- *; replace (low + 0)%nat with low; [ reflexivity | ring ].

  Qed.




End Sigma.