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(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010     *)
(*   \VV/  **************************************************************)
(*    //   *      This file is distributed under the terms of the       *)
(*         *       GNU Lesser General Public License Version 2.1        *)
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(** * This module proves some logical properties of the axiomatics of Reals

1. Decidablity of arithmetical statements from
   the axiom that the order of the real numbers is decidable.

2. Derivability of the archimedean "axiom"
*)

(** 1- Proof of the decidablity of arithmetical statements from
excluded middle and the axiom that the order of the real numbers is
decidable. *)

(** Assuming a decidable predicate [P n], A series is constructed whose
[n]th term is 1/2^n if [P n] holds and 0 otherwise.  This sum reaches 2
only if [P n] holds for all [n], otherwise the sum is less than 2.
Comparing the sum to 2 decides if [forall n, P n] or [~forall n, P n] *)

(** One can iterate this lemma and use classical logic to decide any
statement in the arithmetical hierarchy. *)

(** Contributed by Cezary Kaliszyk and Russell O'Connor *)

Require Import ConstructiveEpsilon.
Require Import Rfunctions.
Require Import PartSum.
Require Import SeqSeries.
Require Import RiemannInt.
Require Import Fourier.

Section Arithmetical_dec.

Variable P : nat -> Prop.
Hypothesis HP : forall n, {P n} + {~P n}.

Let ge_fun_sums_ge_lemma : (forall (m n : nat) (f : nat -> R), (lt m n) -> (forall i : nat, 0 <= f i) -> sum_f_R0 f m <= sum_f_R0 f n).
intros m n f mn fpos.
replace (sum_f_R0 f m) with (sum_f_R0 f m + 0) by ring.
rewrite (tech2 f m n mn).
apply Rplus_le_compat_l.
 induction (n - S m)%nat; simpl in *.
 apply fpos.
replace 0 with (0 + 0) by ring.
apply (Rplus_le_compat _ _ _ _ IHn0 (fpos (S (m + S n0)%nat))).
Qed.

Let ge_fun_sums_ge : (forall (m n : nat) (f : nat -> R), (le m n) -> (forall i : nat, 0 <= f i) -> sum_f_R0 f m <= sum_f_R0 f n).
intros m n f mn pos.
 elim (le_lt_or_eq _ _ mn).
 intro; apply ge_fun_sums_ge_lemma; assumption.
intro H; rewrite H; auto with *.
Qed.

Let f:=fun n => (if HP n then (1/2)^n else 0)%R.

Lemma cauchy_crit_geometric_dec_fun : Cauchy_crit_series f.
intros e He.
assert (X:(Pser (fun n:nat => 1) (1/2) (/ (1 - (1/2))))%R).
 apply GP_infinite.
 apply Rabs_def1; fourier.
assert (He':e/2 > 0) by fourier.
destruct (X _ He') as [N HN].
clear X.
exists N.
intros n m Hn Hm.
replace e with (e/2 + e/2)%R by field.
set (g:=(fun n0 : nat => 1 * (1 / 2) ^ n0)) in *.
assert (R_dist (sum_f_R0 g n) (sum_f_R0 g m) < e / 2 + e / 2).
 apply Rle_lt_trans with (R_dist (sum_f_R0 g n) 2+R_dist 2 (sum_f_R0 g m))%R.
  apply R_dist_tri.
 replace (/(1 - 1/2)) with 2 in HN by field.
 cut (forall n, (n >= N)%nat -> R_dist (sum_f_R0 g n) 2 < e/2)%R.
  intros Z.
  apply Rplus_lt_compat.
   apply Z; assumption.
  rewrite R_dist_sym.
  apply Z; assumption.
 clear - HN He.
 intros n Hn.
 apply HN.
 auto.
eapply Rle_lt_trans;[|apply H].
clear -ge_fun_sums_ge n.
cut (forall n m, (m <= n)%nat -> R_dist (sum_f_R0 f n) (sum_f_R0 f m) <= R_dist (sum_f_R0 g n) (sum_f_R0 g m)).
 intros H.
 destruct (le_lt_dec m n).
  apply H; assumption.
 rewrite R_dist_sym.
 rewrite (R_dist_sym (sum_f_R0 g n)).
 apply H; auto with *.
clear n m.
intros n m Hnm.
unfold R_dist.
cut (forall i : nat, (1 / 2) ^ i >= 0). intro RPosPow.
rewrite Rabs_pos_eq.
 rewrite Rabs_pos_eq.
  cut (sum_f_R0 g m - sum_f_R0 f m <=  sum_f_R0 g n - sum_f_R0 f n).
   intros; fourier.
   do 2 rewrite <- minus_sum.
   apply (ge_fun_sums_ge m n (fun i : nat => g i - f i) Hnm).
   intro i.
   unfold f, g.
   elim (HP i); intro; ring_simplify; auto with *.
  cut (sum_f_R0 g m <= sum_f_R0 g n).
   intro; fourier.
  apply (ge_fun_sums_ge m n g Hnm).
  intro. unfold g.
  ring_simplify.
  apply Rge_le.
  apply RPosPow.
 cut (sum_f_R0 f m <= sum_f_R0 f n).
  intro; fourier.
 apply (ge_fun_sums_ge m n f Hnm).
 intro; unfold f.
 elim (HP i); intro; simpl.
  apply Rge_le.
  apply RPosPow.
 auto with *.
intro i.
apply Rle_ge.
apply pow_le.
fourier.
Qed.

Lemma forall_dec : {forall n, P n} + {~forall n, P n}.
Proof.
destruct (cv_cauchy_2 _ cauchy_crit_geometric_dec_fun).
 cut (2 <= x <-> forall n : nat, P n).
 intro H.
 elim (Rle_dec 2 x); intro X.
 left; tauto.
 right; tauto.
assert (A:Rabs(1/2) < 1) by (apply Rabs_def1; fourier).
assert (A0:=(GP_infinite (1/2) A)).
symmetry.
 split; intro.
 replace 2 with (/ (1 - (1 / 2))) by field.
 unfold Pser, infinite_sum in A0.
 eapply Rle_cv_lim;[|unfold Un_cv; apply A0 |apply u].
 intros n.
 clear -n H.
  induction n; unfold f;simpl.
  destruct (HP 0); auto with *.
  elim n; auto.
 apply Rplus_le_compat; auto.
 destruct (HP (S n)); auto with *.
 elim n0; auto.
intros n.
destruct (HP n); auto.
elim (RIneq.Rle_not_lt _ _ H).
assert (B:0< (1/2)^n).
 apply pow_lt.
 fourier.
apply Rle_lt_trans with (2-(1/2)^n);[|fourier].
replace (/(1-1/2))%R with 2 in A0 by field.
set (g:= fun m => if (eq_nat_dec m n) then (1/2)^n else 0).
assert (Z:  Un_cv (fun N : nat => sum_f_R0 g N) ((1/2)^n)).
 intros e He.
 exists n.
 intros a Ha.
 replace (sum_f_R0 g a) with ((1/2)^n).
  rewrite (R_dist_eq); assumption.
 symmetry.
 cut (forall a : nat, ((a >= n)%nat -> sum_f_R0 g a = (1 / 2) ^ n) /\ ((a < n)%nat -> sum_f_R0 g a = 0))%R.
  intros H0.
  destruct (H0 a).
  auto.
 clear - g.
 induction a.
  split;
   intros H;
   simpl; unfold g;
   destruct (eq_nat_dec 0 n) as [t|f]; try reflexivity.
   elim f; auto with *.
  exfalso; omega.
 destruct IHa as [IHa0 IHa1].
 split;
  intros H;
  simpl; unfold g at 2;
  destruct (eq_nat_dec (S a) n).
    rewrite IHa1.
     ring.
    omega.
   ring_simplify.
   apply IHa0.
   omega.
  exfalso; omega.
 ring_simplify.
 apply IHa1.
 omega.
assert (C:=CV_minus _ _ _ _ A0 Z).
eapply Rle_cv_lim;[|apply u |apply C].
clear - n0 B.
intros m.
simpl.
induction m.
 simpl.
 unfold f, g.
 destruct (eq_nat_dec 0 n).
  destruct (HP 0).
   elim n0.
   congruence.
  clear -n.
  induction n; simpl; fourier.
 destruct (HP); simpl; fourier.
cut (f (S m) <= 1 * ((1 / 2) ^ (S m)) - g (S m)).
 intros L.
 eapply Rle_trans.
  simpl.
  apply Rplus_le_compat.
   apply IHm.
  apply L.
 simpl; fourier.
unfold f, g.
destruct (eq_nat_dec (S m) n).
 destruct (HP (S m)).
  elim n0.
  congruence.
 rewrite e.
 fourier.
destruct (HP (S m)).
 fourier.
ring_simplify.
apply pow_le.
fourier.
Qed.

Lemma sig_forall_dec :  {n | ~P n}+{forall n, P n}.
destruct forall_dec.
 right; assumption.
left.
apply constructive_indefinite_description_nat; auto.
 clear - HP.
 firstorder.
apply Classical_Pred_Type.not_all_ex_not.
assumption.
Qed.

End Arithmetical_dec.

(** 2- Derivability of the Archimedean axiom *)

(* This is a standard proof (it has been taken from PlanetMath). It is
formulated negatively so as to avoid the need for classical
logic. Using a proof of {n | ~P n}+{forall n, P n} (the one above or a
variant of it that does not need classical axioms) , we can in
principle also derive [up] and its [specification] *)

Theorem not_not_archimedean :
  forall r : R, ~ (forall n : nat, (INR n <= r)%R).
intros r H.
set (E := fun r => exists n : nat, r = INR n).
assert (exists x : R, E x) by
  (exists 0%R; simpl; red; exists 0%nat; reflexivity).
assert (bound E) by (exists r; intros x (m,H2); rewrite H2; apply H).
destruct (completeness E) as (M,(H3,H4)); try assumption.
set (M' := (M + -1)%R).
assert (H2 : ~ is_upper_bound E M').
  intro H5.
  assert (M <= M')%R by (apply H4; exact H5).
  apply (Rlt_not_le M M').
    unfold M' in |- *.
    pattern M at 2 in |- *.
    rewrite <- Rplus_0_l.
    pattern (0 + M)%R in |- *.
    rewrite Rplus_comm.
    rewrite <- (Rplus_opp_r 1).
    apply Rplus_lt_compat_l.
    rewrite Rplus_comm.
    apply Rlt_plus_1.
  assumption.
apply H2.
intros N (n,H7).
rewrite H7.
unfold M' in |- *.
assert (H5 : (INR (S n) <= M)%R) by (apply H3; exists (S n); reflexivity).
rewrite S_INR in H5.
assert (H6 : (INR n + 1 + -1 <= M + -1)%R).
  apply Rplus_le_compat_r.
  assumption.
rewrite Rplus_assoc in H6.
rewrite Rplus_opp_r in H6.
rewrite (Rplus_comm (INR n) 0) in H6.
rewrite Rplus_0_l in H6.
assumption.
Qed.