(**
This file is part of the Flocq formalization of floating-point
arithmetic in Coq: http://flocq.gforge.inria.fr/

Copyright (C) 2010-2013 Sylvie Boldo
#<br />#
Copyright (C) 2010-2013 Guillaume Melquiond

This library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 3 of the License, or (at your option) any later version.

This library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
COPYING file for more details.
*)

(** * Error of the rounded-to-nearest addition is representable. *)

Require Import Psatz.
Require Import Fcore_Raux.
Require Import Fcore_defs.
Require Import Fcore_float_prop.
Require Import Fcore_generic_fmt.
Require Import Fcore_FIX.
Require Import Fcore_FLX.
Require Import Fcore_FLT.
Require Import Fcore_ulp.
Require Import Fcalc_ops.


Section Fprop_plus_error.

Variable beta : radix.
Notation bpow e := (bpow beta e).

Variable fexp : Z -> Z.
Context { valid_exp : Valid_exp fexp }.

Section round_repr_same_exp.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Theorem round_repr_same_exp :
  forall m e,
  exists m',
  round beta fexp rnd (F2R (Float beta m e)) = F2R (Float beta m' e).
Proof with auto with typeclass_instances.
intros m e.
set (e' := canonic_exp beta fexp (F2R (Float beta m e))).
unfold round, scaled_mantissa. fold e'.
destruct (Zle_or_lt e' e) as [He|He].
exists m.
unfold F2R at 2. simpl.
rewrite Rmult_assoc, <- bpow_plus.
rewrite <- Z2R_Zpower. 2: omega.
rewrite <- Z2R_mult, Zrnd_Z2R...
(* Goal: @eq R (@F2R beta (let 'pair (pair m1 m2 as p) e := Falign beta (Float beta mx ex) (Float beta my ey) in Float beta (Z.add m1 m2) e)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
unfold F2R. simpl.
rewrite Z2R_mult.
rewrite Rmult_assoc.
rewrite Z2R_Zpower. 2: omega.
rewrite <- bpow_plus.
apply (f_equal (fun v => Z2R m * bpow v)%R).
ring.
exists ((rnd (Z2R m * bpow (e - e'))) * Zpower beta (e' - e))%Z.
(* Goal: @eq R (@F2R beta (let 'pair (pair m1 m2 as p) e := Falign beta (Float beta mx ex) (Float beta my ey) in Float beta (Z.add m1 m2) e)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
unfold F2R. simpl.
rewrite Z2R_mult.
rewrite Z2R_Zpower. 2: omega.
rewrite 2!Rmult_assoc.
rewrite <- 2!bpow_plus.
apply (f_equal (fun v => _ * bpow v)%R).
ring.
Qed.

End round_repr_same_exp.

Context { monotone_exp : Monotone_exp fexp }.
Notation format := (generic_format beta fexp).

Variable choice : Z -> bool.

Lemma plus_error_aux :
  forall x y,
  (canonic_exp beta fexp x <= canonic_exp beta fexp y)%Z ->
  format x -> format y ->
  format (round beta fexp (Znearest choice) (x + y) - (x + y))%R.
Proof.
(* Goal: forall (x y : R) (_ : Z.le (canonic_exp beta fexp x) (canonic_exp beta fexp y)) (_ : generic_format beta fexp x) (_ : generic_format beta fexp y), generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
intros x y.
(* Goal: forall (_ : Z.le (canonic_exp beta fexp x) (canonic_exp beta fexp y)) (_ : generic_format beta fexp x) (_ : generic_format beta fexp y), generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
set (ex := canonic_exp beta fexp x).
(* Goal: forall (_ : Z.le ex (canonic_exp beta fexp y)) (_ : generic_format beta fexp x) (_ : generic_format beta fexp y), generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
set (ey := canonic_exp beta fexp y).
(* Goal: forall (_ : Z.le ex ey) (_ : generic_format beta fexp x) (_ : generic_format beta fexp y), generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
intros He Hx Hy.
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
destruct (Req_dec (round beta fexp (Znearest choice) (x + y) - (x + y)) R0) as [H0|H0].
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
rewrite H0.
(* Goal: generic_format beta fexp R0 *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
apply generic_format_0.
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
set (mx := Ztrunc (scaled_mantissa beta fexp x)).
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
set (my := Ztrunc (scaled_mantissa beta fexp y)).
(* *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
assert (Hxy: (x + y)%R = F2R (Float beta (mx + my * beta ^ (ey - ex)) ex)).
(* Goal: @eq R (Rplus x y) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
rewrite Hx, Hy.
(* Goal: @eq R (Rplus (@F2R beta (Float beta (Ztrunc (scaled_mantissa beta fexp x)) (canonic_exp beta fexp x))) (@F2R beta (Float beta (Ztrunc (scaled_mantissa beta fexp y)) (canonic_exp beta fexp y)))) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
fold mx my ex ey.
(* Goal: @eq R (Rplus (@F2R beta (Float beta mx ex)) (@F2R beta (Float beta my ey))) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
rewrite <- F2R_plus.
(* Goal: @eq R (@F2R beta (let 'pair (pair m1 m2 as p) e := Falign beta (Float beta mx ex) (Float beta my ey) in Float beta (Z.add m1 m2) e)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
unfold Fplus. simpl.
(* Goal: @eq R (@F2R beta (let 'pair (pair m1 m2 as p) e := if Z.leb ex ey then @pair (prod Z Z) Z (@pair Z Z mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex else @pair (prod Z Z) Z (@pair Z Z (Z.mul mx (Z.pow (radix_val beta) (Z.sub ex ey))) my) ey in Float beta (Z.add m1 m2) e)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
now rewrite Zle_imp_le_bool with (1 := He).
(* *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
rewrite Hxy.
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex))) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex))) *)
destruct (round_repr_same_exp (Znearest choice) (mx + my * beta ^ (ey - ex)) ex) as (mxy, Hxy').
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex))) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex))) *)
rewrite Hxy'.
assert (H: (F2R (Float beta mxy ex) - F2R (Float beta (mx + my * beta ^ (ey - ex)) ex))%R =
  F2R (Float beta (mxy - (mx + my * beta ^ (ey - ex))) ex)).
(* Goal: @eq R (Rminus (@F2R beta (Float beta mxy ex)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex))) (@F2R beta (Float beta (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (@F2R beta (Float beta mxy ex)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex))) *)
now rewrite <- F2R_minus, Fminus_same_exp.
(* Goal: generic_format beta fexp (Rminus (@F2R beta (Float beta mxy ex)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex))) *)
rewrite H.
(* Goal: generic_format beta fexp (@F2R beta (Float beta (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) ex)) *)
apply generic_format_F2R.
(* Goal: forall _ : not (@eq Z (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) Z0), Z.le (canonic_exp beta fexp (@F2R beta (Float beta (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) ex))) ex *)
intros _.
(* Goal: Z.le (canonic_exp beta fexp (@F2R beta (Float beta (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) ex))) ex *)
apply monotone_exp.
(* Goal: Z.le (ln_beta_val beta (@F2R beta (Float beta (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) ex)) (ln_beta beta (@F2R beta (Float beta (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) ex)))) (ln_beta_val beta x (ln_beta beta x)) *)
rewrite <- H, <- Hxy', <- Hxy.
(* Goal: Z.le (ln_beta_val beta (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) (ln_beta beta (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)))) (ln_beta_val beta x (ln_beta beta x)) *)
apply ln_beta_le_abs.
(* Goal: not (@eq R (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) (IZR Z0)) *)
(* Goal: Rle (Rabs (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y))) (Rabs x) *)
exact H0.
pattern x at 3 ; replace x with (-(y - (x + y)))%R by ring.
(* Goal: Rle (Rabs (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y))) (Rabs (Ropp (Rminus y (Rplus x y)))) *)
rewrite Rabs_Ropp.
(* Goal: Rle (Rabs (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y))) (Rabs (Rminus y (Rplus x y))) *)
now apply (round_N_pt beta _ choice (x + y)).
Qed.

(** Error of the addition *)
Theorem plus_error :
  forall x y,
  format x -> format y ->
  format (round beta fexp (Znearest choice) (x + y) - (x + y))%R.
Proof.
(* Goal: forall (x y : R) (_ : generic_format beta fexp x) (_ : generic_format beta fexp y), generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
intros x y Hx Hy.
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
destruct (Zle_or_lt (canonic_exp beta fexp x) (canonic_exp beta fexp y)).
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
now apply plus_error_aux.
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
rewrite Rplus_comm.
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus y x)) (Rplus y x)) *)
apply plus_error_aux ; try easy.
(* Goal: Z.le (canonic_exp beta fexp y) (canonic_exp beta fexp x) *)
now apply Zlt_le_weak.
Qed.

End Fprop_plus_error.

Section Fprop_plus_zero.

Variable beta : radix.
Notation bpow e := (bpow beta e).

Variable fexp : Z -> Z.
Context { valid_exp : Valid_exp fexp }.
Context { exp_not_FTZ : Exp_not_FTZ fexp }.
Notation format := (generic_format beta fexp).

Section round_plus_eq_zero_aux.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Lemma round_plus_eq_zero_aux :
  forall x y,
  (canonic_exp beta fexp x <= canonic_exp beta fexp y)%Z ->
  format x -> format y ->
  (0 <= x + y)%R ->
  round beta fexp rnd (x + y) = 0%R ->
  (x + y = 0)%R.
Proof with auto with typeclass_instances.
intros x y He Hx Hy Hp Hxy.
destruct (Req_dec (x + y) 0) as [H0|H0].
(* Goal: not (@eq R (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) (IZR Z0)) *)
(* Goal: Rle (Rabs (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y))) (Rabs x) *)
exact H0.
destruct (ln_beta beta (x + y)) as (exy, Hexy).
(* Goal: @eq R (@F2R beta (let 'pair (pair m1 m2 as p) e := Falign beta (Float beta mx ex) (Float beta my ey) in Float beta (Z.add m1 m2) e)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
simpl.
specialize (Hexy H0).
destruct (Zle_or_lt exy (fexp exy)) as [He'|He'].
(* . *)
assert (H: (x + y)%R = F2R (Float beta (Ztrunc (x * bpow (- fexp exy)) +
  Ztrunc (y * bpow (- fexp exy))) (fexp exy))).
rewrite (subnormal_exponent beta fexp exy x He' Hx) at 1.
rewrite (subnormal_exponent beta fexp exy y He' Hy) at 1.
now rewrite <- F2R_plus, Fplus_same_exp.
rewrite H in Hxy.
rewrite round_generic in Hxy...
now rewrite <- H in Hxy.
(* Goal: generic_format beta fexp (@F2R beta (Float beta (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) ex)) *)
apply generic_format_F2R.
(* Goal: forall _ : not (@eq Z (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) Z0), Z.le (canonic_exp beta fexp (@F2R beta (Float beta (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) ex))) ex *)
intros _.
rewrite <- H.
unfold canonic_exp.
rewrite ln_beta_unique with (1 := Hexy).
apply Zle_refl.
(* . *)
elim Rle_not_lt with (1 := round_le beta _ rnd _ _ (proj1 Hexy)).
rewrite (Rabs_pos_eq _ Hp).
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
rewrite Hxy.
rewrite round_generic...
apply bpow_gt_0.
apply generic_format_bpow.
apply Zlt_succ_le.
now rewrite (Zsucc_pred exy) in He'.
Qed.

End round_plus_eq_zero_aux.

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

(** rnd(x+y)=0 -> x+y = 0 provided this is not a FTZ format *)
Theorem round_plus_eq_zero :
  forall x y,
  format x -> format y ->
  round beta fexp rnd (x + y) = 0%R ->
  (x + y = 0)%R.
Proof with auto with typeclass_instances.
(* Goal: forall (x y : R) (_ : generic_format beta fexp x) (_ : generic_format beta fexp y), generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
intros x y Hx Hy.
destruct (Rle_or_lt 0 (x + y)) as [H1|H1].
(* . *)
revert H1.
destruct (Zle_or_lt (canonic_exp beta fexp x) (canonic_exp beta fexp y)) as [H2|H2].
now apply round_plus_eq_zero_aux.
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
rewrite Rplus_comm.
apply round_plus_eq_zero_aux ; try easy.
(* Goal: Z.le (canonic_exp beta fexp y) (canonic_exp beta fexp x) *)
now apply Zlt_le_weak.
(* . *)
revert H1.
rewrite <- (Ropp_involutive (x + y)), Ropp_plus_distr, <- Ropp_0.
intros H1.
rewrite round_opp.
intros Hxy.
apply f_equal.
cut (round beta fexp (Zrnd_opp rnd) (- x + - y) = 0)%R.
cut (0 <= -x + -y)%R.
destruct (Zle_or_lt (canonic_exp beta fexp (-x)) (canonic_exp beta fexp (-y))) as [H2|H2].
apply round_plus_eq_zero_aux ; try apply generic_format_opp...
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
rewrite Rplus_comm.
apply round_plus_eq_zero_aux ; try apply generic_format_opp...
(* Goal: Z.le (canonic_exp beta fexp y) (canonic_exp beta fexp x) *)
now apply Zlt_le_weak.
apply Rlt_le.
now apply Ropp_lt_cancel.
rewrite <- (Ropp_involutive (round _ _ _ _)).
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
rewrite Hxy.
apply Ropp_involutive.
Qed.

End Fprop_plus_zero.

Section Fprop_plus_FLT.
Variable beta : radix.

Notation bpow e := (bpow beta e).

Variable emin prec : Z.
Context { prec_gt_0_ : Prec_gt_0 prec }.

Theorem FLT_format_plus_small: forall x y,
  generic_format beta (FLT_exp emin prec) x ->
  generic_format beta (FLT_exp emin prec) y ->
   (Rabs (x+y) <= bpow (prec+emin))%R ->
    generic_format beta (FLT_exp emin prec) (x+y).
Proof with auto with typeclass_instances.
intros x y Fx Fy H.
apply generic_format_FLT_FIX...
rewrite Zplus_comm; assumption.
apply generic_format_FIX_FLT, FIX_format_generic in Fx.
apply generic_format_FIX_FLT, FIX_format_generic in Fy.
destruct Fx as (nx,(H1x,H2x)).
destruct Fy as (ny,(H1y,H2y)).
apply generic_format_FIX.
exists (Float beta (Fnum nx+Fnum ny)%Z emin).
split;[idtac|reflexivity].
(* Goal: @eq R (@F2R beta (let 'pair (pair m1 m2 as p) e := Falign beta (Float beta mx ex) (Float beta my ey) in Float beta (Z.add m1 m2) e)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
rewrite H1x,H1y; unfold F2R; simpl.
rewrite H2x, H2y.
rewrite Z2R_plus; ring.
Qed.

End Fprop_plus_FLT.

Section Fprop_plus_mult_ulp.

Variable beta : radix.
Notation bpow e := (bpow beta e).

Variable fexp : Z -> Z.
Context { valid_exp : Valid_exp fexp }.
Context { monotone_exp : Monotone_exp fexp }.
Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.

Notation format := (generic_format beta fexp).
Notation cexp := (canonic_exp beta fexp).

Lemma ex_shift :
  forall x e, format x -> (e <= cexp x)%Z ->
  exists m, (x = Z2R m * bpow e)%R.
Proof with auto with typeclass_instances.
intros x e Fx He.
exists (Ztrunc (scaled_mantissa beta fexp x)*Zpower beta (cexp x -e))%Z.
(* Goal: @eq R (@F2R beta (let 'pair (pair m1 m2 as p) e := Falign beta (Float beta mx ex) (Float beta my ey) in Float beta (Z.add m1 m2) e)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
rewrite Fx at 1; unfold F2R; simpl.
rewrite Z2R_mult, Rmult_assoc.
f_equal.
rewrite Z2R_Zpower.
2: omega.
rewrite <- bpow_plus; f_equal; ring.
Qed.

Lemma ln_beta_minus1 :
  forall z, z <> 0%R ->
  (ln_beta beta z - 1)%Z = ln_beta beta (z / Z2R beta).
Proof.
(* Goal: forall (z : R) (_ : not (@eq R z (IZR Z0))), @eq Z (Z.sub (ln_beta_val beta z (ln_beta beta z)) (Zpos xH)) (ln_beta_val beta (Rdiv z (Z2R (radix_val beta))) (ln_beta beta (Rdiv z (Z2R (radix_val beta))))) *)
intros z Hz.
(* Goal: @eq Z (Z.sub (ln_beta_val beta z (ln_beta beta z)) (Zpos xH)) (ln_beta_val beta (Rdiv z (Z2R (radix_val beta))) (ln_beta beta (Rdiv z (Z2R (radix_val beta))))) *)
unfold Zminus.
(* Goal: @eq Z (Z.add (ln_beta_val beta z (ln_beta beta z)) (Z.opp (Zpos xH))) (ln_beta_val beta (Rdiv z (Z2R (radix_val beta))) (ln_beta beta (Rdiv z (Z2R (radix_val beta))))) *)
rewrite <- ln_beta_mult_bpow with (1 := Hz).
(* Goal: @eq Z (ln_beta_val beta (Rmult z (Fcore_Raux.bpow beta (Z.opp (Zpos xH)))) (ln_beta beta (Rmult z (Fcore_Raux.bpow beta (Z.opp (Zpos xH)))))) (ln_beta_val beta (Rdiv z (Z2R (radix_val beta))) (ln_beta beta (Rdiv z (Z2R (radix_val beta))))) *)
now rewrite bpow_opp, bpow_1.
Qed.

Theorem round_plus_mult_ulp :
  forall x y, format x -> format y -> (x <> 0)%R ->
  exists m, (round beta fexp rnd (x+y) = Z2R m * ulp beta fexp (x/Z2R beta))%R.
Proof with auto with typeclass_instances.
intros x y Fx Fy Zx.
case (Zle_or_lt (ln_beta beta (x/Z2R beta)) (ln_beta beta y)); intros H1.
pose (e:=cexp (x / Z2R beta)).
destruct (ex_shift x e) as (nx, Hnx); try exact Fx.
(* Goal: Z.le (canonic_exp beta fexp (@F2R beta (Float beta (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) ex))) ex *)
apply monotone_exp.
rewrite <- (ln_beta_minus1 x Zx); omega.
destruct (ex_shift y e) as (ny, Hny); try assumption.
(* Goal: Z.le (canonic_exp beta fexp (@F2R beta (Float beta (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) ex))) ex *)
apply monotone_exp...
destruct (round_repr_same_exp beta fexp rnd (nx+ny) e) as (n,Hn).
exists n.
apply trans_eq with (F2R (Float beta n e)).
rewrite <- Hn; f_equal.
rewrite Hnx, Hny; unfold F2R; simpl; rewrite Z2R_plus; ring.
(* Goal: @eq R (@F2R beta (let 'pair (pair m1 m2 as p) e := Falign beta (Float beta mx ex) (Float beta my ey) in Float beta (Z.add m1 m2) e)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
unfold F2R; simpl.
rewrite ulp_neq_0; try easy.
apply Rmult_integral_contrapositive_currified; try assumption.
apply Rinv_neq_0_compat.
apply Rgt_not_eq.
apply radix_pos.
(* *)
destruct (ex_shift (round beta fexp rnd (x + y)) (cexp (x/Z2R beta))) as (n,Hn).
apply generic_format_round...
apply Zle_trans with (cexp (x+y)).
(* Goal: Z.le (canonic_exp beta fexp (@F2R beta (Float beta (Z.sub mxy (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex))))) ex))) ex *)
apply monotone_exp.
rewrite <- ln_beta_minus1 by easy.
rewrite <- (ln_beta_abs beta (x+y)).
(* . *)
assert (U: (Rabs (x+y) = Rabs x + Rabs y)%R \/ (y <> 0 /\ Rabs (x+y) = Rabs x - Rabs y)%R).
assert (V: forall x y, (Rabs y <= Rabs x)%R ->
   (Rabs (x+y) = Rabs x + Rabs y)%R \/ (y <> 0 /\ Rabs (x+y) = Rabs x - Rabs y)%R).
(* Goal: forall (x y : R) (_ : Z.le (canonic_exp beta fexp x) (canonic_exp beta fexp y)) (_ : generic_format beta fexp x) (_ : generic_format beta fexp y), generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
clear; intros x y.
case (Rle_or_lt 0 y); intros Hy.
case Hy; intros Hy'.
case (Rle_or_lt 0 x); intros Hx.
intros _; rewrite (Rabs_pos_eq y) by easy.
rewrite (Rabs_pos_eq x) by easy.
left; apply Rabs_pos_eq.
now apply Rplus_le_le_0_compat.
rewrite (Rabs_pos_eq y) by easy.
rewrite (Rabs_left x) by easy.
intros H; right; split.
now apply Rgt_not_eq.
rewrite Rabs_left1.
ring.
apply Rplus_le_reg_l with (-x)%R; ring_simplify; assumption.
intros _; left.
now rewrite <- Hy', Rabs_R0, 2!Rplus_0_r.
case (Rle_or_lt 0 x); intros Hx.
rewrite (Rabs_left y) by easy.
rewrite (Rabs_pos_eq x) by easy.
intros H; right; split.
now apply Rlt_not_eq.
rewrite Rabs_pos_eq.
ring.
apply Rplus_le_reg_l with (-y)%R; ring_simplify; assumption.
intros _; left.
rewrite (Rabs_left y) by easy.
rewrite (Rabs_left x) by easy.
rewrite Rabs_left1.
ring.
lra.
apply V; left.
apply ln_beta_lt_pos with beta.
now apply Rabs_pos_lt.
rewrite <- ln_beta_minus1 in H1; try assumption.
rewrite 2!ln_beta_abs; omega.
(* . *)
destruct U as [U|U].
rewrite U; apply Zle_trans with (ln_beta beta x).
omega.
rewrite <- ln_beta_abs.
apply ln_beta_le.
now apply Rabs_pos_lt.
apply Rplus_le_reg_l with (-Rabs x)%R; ring_simplify.
apply Rabs_pos.
destruct U as (U',U); rewrite U.
rewrite <- ln_beta_abs.
apply ln_beta_minus_lb.
now apply Rabs_pos_lt.
now apply Rabs_pos_lt.
rewrite 2!ln_beta_abs.
assert (ln_beta beta y < ln_beta beta x - 1)%Z.
now rewrite (ln_beta_minus1 x Zx).
omega.
apply canonic_exp_round_ge...
intros K.
apply round_plus_eq_zero in K...
contradict H1; apply Zle_not_lt.
rewrite <- (ln_beta_minus1 x Zx).
replace y with (-x)%R.
rewrite ln_beta_opp; omega.
lra.
exists n.
rewrite ulp_neq_0.
assumption.
apply Rmult_integral_contrapositive_currified; try assumption.
apply Rinv_neq_0_compat.
apply Rgt_not_eq.
apply radix_pos.
Qed.

Context {exp_not_FTZ : Exp_not_FTZ fexp}.

Theorem round_plus_ge_ulp :
  forall x y, format x -> format y ->
  round beta fexp rnd (x+y) = 0%R \/
  (ulp beta fexp (x/Z2R beta) <= Rabs (round beta fexp rnd (x+y)))%R.
Proof with auto with typeclass_instances.
intros x y Fx Fy.
case (Req_dec x 0); intros Zx.
(* *)
rewrite Zx, Rplus_0_l.
rewrite round_generic...
unfold Rdiv; rewrite Rmult_0_l.
rewrite Fy at 2.
unfold F2R; simpl; rewrite Rabs_mult.
rewrite (Rabs_pos_eq (bpow _)) by apply bpow_ge_0.
case (Z.eq_dec (Ztrunc (scaled_mantissa beta fexp y)) 0); intros Hm.
left.
rewrite Fy, Hm; unfold F2R; simpl; ring.
right.
apply Rle_trans with (1*bpow (cexp y))%R.
rewrite Rmult_1_l.
rewrite <- ulp_neq_0.
apply ulp_ge_ulp_0...
intros K; apply Hm.
rewrite K, scaled_mantissa_0.
apply (Ztrunc_Z2R 0).
apply Rmult_le_compat_r.
apply bpow_ge_0.
rewrite <- Z2R_abs.
apply (Z2R_le 1).
apply (Zlt_le_succ 0).
now apply Z.abs_pos.
(* *)
destruct (round_plus_mult_ulp x y Fx Fy Zx) as (m,Hm).
case (Z.eq_dec m 0); intros Zm.
left.
rewrite Hm, Zm; simpl; ring.
right.
rewrite Hm, Rabs_mult.
rewrite (Rabs_pos_eq (ulp _ _ _)) by apply ulp_ge_0.
apply Rle_trans with (1*ulp beta fexp (x/Z2R beta))%R.
right; ring.
apply Rmult_le_compat_r.
apply ulp_ge_0.
rewrite <- Z2R_abs.
apply (Z2R_le 1).
apply (Zlt_le_succ 0).
now apply Z.abs_pos.
Qed.

End Fprop_plus_mult_ulp.

Section Fprop_plus_ge_ulp.

Variable beta : radix.
Notation bpow e := (bpow beta e).

Variable rnd : R -> Z.
Context { valid_rnd : Valid_rnd rnd }.
Variable emin prec : Z.
Context { prec_gt_0_ : Prec_gt_0 prec }.

Theorem round_plus_ge_ulp_FLT : forall x y e,
  generic_format beta (FLT_exp emin prec) x -> generic_format beta (FLT_exp emin prec) y ->
  (bpow e <= Rabs x)%R ->
  round beta (FLT_exp emin prec) rnd (x+y) = 0%R \/
  (bpow (e - prec) <= Rabs (round beta (FLT_exp emin prec) rnd (x+y)))%R.
Proof with auto with typeclass_instances.
intros x y e Fx Fy He.
assert (Zx: x <> 0%R).
  contradict He.
  apply Rlt_not_le; rewrite He, Rabs_R0.
  apply bpow_gt_0.
case round_plus_ge_ulp with beta (FLT_exp emin prec) rnd x y...
intros H; right.
apply Rle_trans with (2:=H).
rewrite ulp_neq_0.
unfold canonic_exp.
rewrite <- ln_beta_minus1 by easy.
unfold FLT_exp; apply bpow_le.
apply Zle_trans with (2:=Z.le_max_l _ _).
(* Goal: @eq R (@F2R beta (let 'pair (pair m1 m2 as p) e := Falign beta (Float beta mx ex) (Float beta my ey) in Float beta (Z.add m1 m2) e)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
destruct (ln_beta beta x) as (n,Hn); simpl.
assert (e < n)%Z; try omega.
apply lt_bpow with beta.
apply Rle_lt_trans with (1:=He).
now apply Hn.
apply Rmult_integral_contrapositive_currified; try assumption.
apply Rinv_neq_0_compat.
apply Rgt_not_eq.
apply radix_pos.
Qed.

Theorem round_plus_ge_ulp_FLX : forall x y e,
  generic_format beta (FLX_exp prec) x -> generic_format beta (FLX_exp prec) y ->
  (bpow e <= Rabs x)%R ->
  round beta (FLX_exp prec) rnd (x+y) = 0%R \/
  (bpow (e - prec) <= Rabs (round beta (FLX_exp prec) rnd (x+y)))%R.
Proof with auto with typeclass_instances.
intros x y e Fx Fy He.
assert (Zx: x <> 0%R).
  contradict He.
  apply Rlt_not_le; rewrite He, Rabs_R0.
  apply bpow_gt_0.
case round_plus_ge_ulp with beta (FLX_exp prec) rnd x y...
intros H; right.
apply Rle_trans with (2:=H).
rewrite ulp_neq_0.
unfold canonic_exp.
rewrite <- ln_beta_minus1 by easy.
unfold FLX_exp; apply bpow_le.
(* Goal: @eq R (@F2R beta (let 'pair (pair m1 m2 as p) e := Falign beta (Float beta mx ex) (Float beta my ey) in Float beta (Z.add m1 m2) e)) (@F2R beta (Float beta (Z.add mx (Z.mul my (Z.pow (radix_val beta) (Z.sub ey ex)))) ex)) *)
(* Goal: generic_format beta fexp (Rminus (round beta fexp (Znearest choice) (Rplus x y)) (Rplus x y)) *)
destruct (ln_beta beta x) as (n,Hn); simpl.
assert (e < n)%Z; try omega.
apply lt_bpow with beta.
apply Rle_lt_trans with (1:=He).
now apply Hn.
apply Rmult_integral_contrapositive_currified; try assumption.
apply Rinv_neq_0_compat.
apply Rgt_not_eq.
apply radix_pos.
Qed.

End Fprop_plus_ge_ulp.
