Average.v 56.3 KB
 BOLDO Sylvie committed Nov 20, 2014 1 2 ``````Require Import Fcore. Require Import Fprop_plus_error. `````` BOLDO Sylvie committed Jun 05, 2015 3 ``````Require Import Fourier. `````` BOLDO Sylvie committed Nov 20, 2014 4 5 6 7 8 `````` Open Scope R_scope. Section av1. `````` BOLDO Sylvie committed Jun 05, 2015 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 `````` Lemma Rmin_Rmax_overflow: forall x y z M, Rabs x <= M -> Rabs y <= M -> Rmin x y <= z <= Rmax x y -> Rabs z <= M. Proof. intros x y z M Hx Hy H. case (Rle_or_lt 0 z); intros Hz. rewrite Rabs_right. apply Rle_trans with (1:=proj2 H). generalize (proj2 H). apply Rmax_case_strong. intros; apply Rle_trans with (2:=Hx). apply RRle_abs. intros; apply Rle_trans with (2:=Hy). apply RRle_abs. now apply Rle_ge. rewrite Rabs_left; try assumption. apply Rle_trans with (Rmax (-x) (-y)). rewrite Rmax_opp. apply Ropp_le_contravar, H. apply Rmax_case_strong. intros; apply Rle_trans with (2:=Hx). rewrite <- Rabs_Ropp. apply RRle_abs. intros; apply Rle_trans with (2:=Hy). rewrite <- Rabs_Ropp. apply RRle_abs. Qed. `````` BOLDO Sylvie committed Nov 20, 2014 37 38 39 40 41 42 43 44 45 46 47 `````` Definition radix2 := Build_radix 2 (refl_equal true). Notation bpow e := (bpow radix2 e). Variable emin prec : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Notation format := (generic_format radix2 (FLT_exp emin prec)). Notation round_flt :=(round radix2 (FLT_exp emin prec) ZnearestE). Notation ulp_flt :=(ulp radix2 (FLT_exp emin prec)). Notation cexp := (canonic_exp radix2 (FLT_exp emin prec)). `````` BOLDO Sylvie committed Jun 05, 2015 48 ``````Notation pred_flt := (pred radix2 (FLT_exp emin prec)). `````` BOLDO Sylvie committed Nov 20, 2014 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 `````` Lemma FLT_format_double: forall u, format u -> format (2*u). Proof with auto with typeclass_instances. intros u Fu. apply generic_format_FLT. apply FLT_format_generic in Fu... destruct Fu as (uf, (H1,(H2,H3))). exists (Float radix2 (Fnum uf) (Fexp uf+1)). split. rewrite H1; unfold F2R; simpl. rewrite bpow_plus, bpow_1. simpl;ring. split. now simpl. simpl; apply Zle_trans with (1:=H3). omega. Qed. Lemma FLT_format_half: forall u, format u -> bpow (prec+emin) <= Rabs u -> format (u/2). Proof with auto with typeclass_instances. intros u Fu H. apply FLT_format_generic in Fu... destruct Fu as ((n,e),(H1,(H2,H3))). simpl in H1, H2, H3. apply generic_format_FLT. exists (Float radix2 n (e-1)). split; simpl. rewrite H1; unfold F2R; simpl. unfold Zminus; rewrite bpow_plus. simpl; unfold Rdiv; ring. split;[assumption|idtac]. assert (prec + emin < prec +e)%Z;[idtac|omega]. apply lt_bpow with radix2. apply Rle_lt_trans with (1:=H). rewrite H1; unfold F2R; simpl. rewrite Rabs_mult; rewrite (Rabs_right (bpow e)). 2: apply Rle_ge, bpow_ge_0. rewrite bpow_plus. apply Rmult_lt_compat_r. apply bpow_gt_0. rewrite <- Z2R_abs. rewrite <- Z2R_Zpower. now apply Z2R_lt. unfold Prec_gt_0 in prec_gt_0_; omega. Qed. `````` BOLDO Sylvie committed Jun 05, 2015 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 ``````Lemma FLT_round_half: forall z, bpow (prec+emin) <= Rabs z -> round_flt (z/2)= round_flt z /2. Proof with auto with typeclass_instances. intros z Hz. apply Rmult_eq_reg_l with 2. 2: apply sym_not_eq; auto with real. apply trans_eq with (round_flt z). 2: field. assert (z <> 0)%R. intros K; contradict Hz. rewrite K, Rabs_R0; apply Rlt_not_le. apply bpow_gt_0. assert (cexp (z/2) = cexp z -1)%Z. assert (prec+emin < ln_beta radix2 z)%Z. apply lt_bpow with radix2. destruct ln_beta as (e,He); simpl. apply Rle_lt_trans with (1:=Hz). now apply He. unfold canonic_exp, FLT_exp. replace ((ln_beta radix2 (z/2))-prec)%Z with ((ln_beta radix2 z -1) -prec)%Z. rewrite Z.max_l; try omega. rewrite Z.max_l; try omega. apply Zplus_eq_compat; try reflexivity. apply sym_eq, ln_beta_unique. destruct (ln_beta radix2 z) as (e,He); simpl. unfold Rdiv; rewrite Rabs_mult. rewrite (Rabs_right (/2)). split. apply Rmult_le_reg_l with (bpow 1). apply bpow_gt_0. rewrite <- bpow_plus. replace (1+(e-1-1))%Z with (e-1)%Z by ring. apply Rle_trans with (Rabs z). now apply He. right; simpl; field. apply Rmult_lt_reg_l with (bpow 1). apply bpow_gt_0. rewrite <- bpow_plus. replace (1+(e-1))%Z with e by ring. apply Rle_lt_trans with (Rabs z). right; simpl; field. now apply He. apply Rle_ge; auto with real. unfold round, scaled_mantissa, F2R. rewrite H0; simpl. rewrite Rmult_comm, Rmult_assoc. apply f_equal2. apply f_equal, f_equal. replace (-(cexp z -1))%Z with (-cexp z +1)%Z by ring. rewrite bpow_plus. simpl; field. unfold Zminus; rewrite bpow_plus. simpl; field. Qed. `````` BOLDO Sylvie committed Nov 20, 2014 154 155 156 157 `````` Lemma FLT_ulp_le_id: forall u, bpow emin <= u -> ulp_flt u <= u. Proof with auto with typeclass_instances. intros u H. `````` BOLDO Sylvie committed Jul 29, 2015 158 159 ``````rewrite ulp_neq_0. 2: apply Rgt_not_eq, Rlt_le_trans with (2:=H), bpow_gt_0. `````` BOLDO Sylvie committed Nov 20, 2014 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 ``````case (Rle_or_lt (bpow (emin+prec-1)) u); intros Hu. unfold ulp; rewrite canonic_exp_FLT_FLX. unfold canonic_exp, FLX_exp. destruct (ln_beta radix2 u) as (e,He); simpl. apply Rle_trans with (bpow (e-1)). apply bpow_le. unfold Prec_gt_0 in prec_gt_0_; omega. rewrite <- (Rabs_right u). apply He. apply Rgt_not_eq, Rlt_gt. apply Rlt_le_trans with (2:=Hu). apply bpow_gt_0. apply Rle_ge, Rle_trans with (2:=Hu), bpow_ge_0. rewrite Rabs_right. assumption. apply Rle_ge, Rle_trans with (2:=Hu), bpow_ge_0. unfold ulp; rewrite canonic_exp_FLT_FIX. apply H. apply Rgt_not_eq, Rlt_gt. apply Rlt_le_trans with (2:=H). apply bpow_gt_0. rewrite Rabs_right. apply Rlt_le_trans with (1:=Hu). apply bpow_le; omega. apply Rle_ge, Rle_trans with (2:=H), bpow_ge_0. Qed. Lemma FLT_ulp_double: forall u, ulp_flt (2*u) <= 2*ulp_flt(u). intros u. `````` BOLDO Sylvie committed Jul 29, 2015 191 192 193 194 ``````case (Req_bool_spec u 0); intros Hu'. rewrite Hu', Rmult_0_r. rewrite <- (Rmult_1_l (ulp_flt 0)) at 1. apply Rmult_le_compat_r. `````` BOLDO Sylvie committed Sep 07, 2015 195 ``````apply ulp_ge_0. `````` BOLDO Sylvie committed Jul 29, 2015 196 197 198 199 200 ``````left; apply Rlt_plus_1. rewrite 2!ulp_neq_0; trivial. 2: apply Rmult_integral_contrapositive_currified; trivial. 2: apply Rgt_not_eq; apply Rlt_trans with (1:=Rlt_plus_1 _). 2: rewrite Rplus_0_l; apply Rlt_plus_1. `````` BOLDO Sylvie committed Nov 20, 2014 201 ``````pattern 2 at 2; replace 2 with (bpow 1) by reflexivity. `````` BOLDO Sylvie committed Jul 29, 2015 202 ``````rewrite <- bpow_plus. `````` BOLDO Sylvie committed Nov 20, 2014 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 ``````apply bpow_le. case (Rle_or_lt (bpow (emin+prec-1)) (Rabs u)); intros Hu. (* *) rewrite canonic_exp_FLT_FLX. rewrite canonic_exp_FLT_FLX; trivial. unfold canonic_exp, FLX_exp. replace 2 with (bpow 1) by reflexivity. rewrite Rmult_comm, ln_beta_mult_bpow. omega. intros H; contradict Hu. apply Rlt_not_le; rewrite H, Rabs_R0. apply bpow_gt_0. apply Rle_trans with (1:=Hu). rewrite Rabs_mult. pattern (Rabs u) at 1; rewrite <- (Rmult_1_l (Rabs u)). apply Rmult_le_compat_r. apply Rabs_pos. rewrite Rabs_right. now auto with real. apply Rle_ge; now auto with real. (* *) case (Req_dec u 0); intros K. rewrite K, Rmult_0_r. omega. rewrite canonic_exp_FLT_FIX. rewrite canonic_exp_FLT_FIX; trivial. unfold FIX_exp, canonic_exp; omega. apply Rlt_le_trans with (1:=Hu). apply bpow_le; omega. apply Rmult_integral_contrapositive_currified; trivial. apply Rgt_not_eq, Rlt_gt; now auto with real. rewrite Rabs_mult. rewrite Rabs_right. 2: apply Rle_ge; now auto with real. apply Rlt_le_trans with (2*bpow (emin + prec - 1)). apply Rmult_lt_compat_l. now auto with real. assumption. replace 2 with (bpow 1) by reflexivity. rewrite <- bpow_plus. apply bpow_le; omega. Qed. `````` BOLDO Sylvie committed Jun 05, 2015 247 ``````Lemma round_plus_small_id_aux: forall f h, format f -> (bpow (prec+emin) <= f) -> 0 < f `````` BOLDO Sylvie committed Jun 16, 2015 248 `````` -> Rabs h <= /4* ulp_flt f -> round_flt (f+h) = f. `````` BOLDO Sylvie committed Jun 05, 2015 249 250 251 252 253 254 ``````Proof with auto with typeclass_instances. intros f h Ff H1 H2 Hh. case (Rle_or_lt 0 h); intros H3;[destruct H3|idtac]. (* 0 < h *) rewrite Rabs_right in Hh. 2: now apply Rle_ge, Rlt_le. `````` BOLDO Sylvie committed Sep 08, 2015 255 ``````apply round_N_eq_DN_pt with (f+ ulp_flt f)... `````` BOLDO Sylvie committed Sep 09, 2015 256 ``````pattern f at 2; rewrite <- (round_DN_plus_eps_pos radix2 (FLT_exp emin prec) f) with (eps:=h); try assumption. `````` BOLDO Sylvie committed Jun 05, 2015 257 ``````apply round_DN_pt... `````` BOLDO Sylvie committed Jul 29, 2015 258 ``````now left. `````` BOLDO Sylvie committed Jun 05, 2015 259 260 ``````split. now left. `````` BOLDO Sylvie committed Jun 16, 2015 261 ``````apply Rle_lt_trans with (1:=Hh). `````` BOLDO Sylvie committed Jun 05, 2015 262 263 ``````rewrite <- (Rmult_1_l (ulp_flt f)) at 2. apply Rmult_lt_compat_r. `````` BOLDO Sylvie committed Jul 29, 2015 264 ``````rewrite ulp_neq_0; try now apply Rgt_not_eq. `````` BOLDO Sylvie committed Jun 05, 2015 265 266 ``````apply bpow_gt_0. fourier. `````` BOLDO Sylvie committed Sep 09, 2015 267 ``````rewrite <- (round_UP_plus_eps_pos radix2 (FLT_exp emin prec) f) with (eps:=h); try assumption. `````` BOLDO Sylvie committed Jun 05, 2015 268 ``````apply round_UP_pt... `````` BOLDO Sylvie committed Jul 29, 2015 269 ``````now left. `````` BOLDO Sylvie committed Jun 05, 2015 270 ``````split; trivial. `````` BOLDO Sylvie committed Jun 16, 2015 271 ``````apply Rle_trans with (1:=Hh). `````` BOLDO Sylvie committed Jun 05, 2015 272 273 ``````rewrite <- (Rmult_1_l (ulp_flt f)) at 2. apply Rmult_le_compat_r. `````` BOLDO Sylvie committed Sep 07, 2015 274 ``````apply ulp_ge_0. `````` BOLDO Sylvie committed Jun 05, 2015 275 276 277 278 ``````fourier. apply Rplus_lt_reg_l with (-f); ring_simplify. apply Rlt_le_trans with (/2*ulp_flt f). 2: right; field. `````` BOLDO Sylvie committed Jun 16, 2015 279 ``````apply Rle_lt_trans with (1:=Hh). `````` BOLDO Sylvie committed Jun 05, 2015 280 ``````apply Rmult_lt_compat_r. `````` BOLDO Sylvie committed Jul 29, 2015 281 ``````rewrite ulp_neq_0; try now apply Rgt_not_eq. `````` BOLDO Sylvie committed Jun 05, 2015 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 ``````apply bpow_gt_0. fourier. (* h = 0 *) rewrite <- H, Rplus_0_r. apply round_generic... (* h < 0 *) (* - assertions *) rewrite Rabs_left in Hh; try assumption. assert (0 < pred_flt f). apply Rlt_le_trans with (bpow emin). apply bpow_gt_0. apply le_pred_lt... apply FLT_format_bpow... omega. apply Rlt_le_trans with (2:=H1). apply bpow_lt. unfold Prec_gt_0 in prec_gt_0_; omega. `````` BOLDO Sylvie committed Jun 16, 2015 299 300 301 302 303 304 305 306 ``````assert (M:(prec + emin +1 <= ln_beta radix2 f)%Z). apply ln_beta_ge_bpow. replace (prec+emin+1-1)%Z with (prec+emin)%Z by ring. rewrite Rabs_right; try assumption. apply Rle_ge; now left. assert (T1:(ulp_flt (pred_flt f) = ulp_flt f) \/ ( ulp_flt (pred_flt f) = /2* ulp_flt f /\ f = bpow (ln_beta radix2 f -1))). `````` BOLDO Sylvie committed Sep 07, 2015 307 ``````generalize H; rewrite pred_eq_pos; [idtac|now left]. `````` BOLDO Sylvie committed Jul 29, 2015 308 ``````unfold pred_pos; case Req_bool_spec; intros K HH. `````` BOLDO Sylvie committed Jun 16, 2015 309 310 ``````(**) right; split; try assumption. `````` BOLDO Sylvie committed Jul 29, 2015 311 ``````rewrite ulp_neq_0;[idtac|now apply Rgt_not_eq]. `````` BOLDO Sylvie committed Jun 16, 2015 312 ``````apply trans_eq with (bpow (ln_beta radix2 f- prec -1)). `````` BOLDO Sylvie committed Jul 29, 2015 313 ``````apply f_equal. `````` BOLDO Sylvie committed Jun 16, 2015 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 ``````unfold canonic_exp. apply trans_eq with (FLT_exp emin prec (ln_beta radix2 f -1)%Z). apply f_equal. unfold FLT_exp. rewrite Z.max_l. 2: omega. apply ln_beta_unique. rewrite Rabs_right. split. apply Rplus_le_reg_l with (bpow (ln_beta radix2 f -1-prec)). ring_simplify. apply Rle_trans with (bpow (ln_beta radix2 f - 1 - 1) + bpow (ln_beta radix2 f - 1 - 1)). apply Rplus_le_compat_r. apply bpow_le. unfold Prec_gt_0 in prec_gt_0_; omega. apply Rle_trans with (bpow 1*bpow (ln_beta radix2 f - 1 - 1)). simpl; right; ring. rewrite <- bpow_plus. apply Rle_trans with (bpow (ln_beta radix2 f -1)). apply bpow_le; omega. rewrite <- K; now right. rewrite <- K. apply Rplus_lt_reg_l with (-f+bpow (ln_beta radix2 f-1-prec)); ring_simplify. apply bpow_gt_0. apply Rle_ge. rewrite K at 1. apply Rplus_le_reg_l with (bpow (ln_beta radix2 f - 1 - prec)). ring_simplify. apply bpow_le. unfold Prec_gt_0 in prec_gt_0_; omega. unfold FLT_exp. rewrite Z.max_l;[ring|omega]. replace (/2) with (bpow (-1)) by reflexivity. `````` BOLDO Sylvie committed Jul 29, 2015 347 348 ``````rewrite ulp_neq_0; try now apply Rgt_not_eq. rewrite <- bpow_plus. `````` BOLDO Sylvie committed Jun 16, 2015 349 350 351 352 353 354 355 356 ``````apply f_equal. unfold canonic_exp, FLT_exp. rewrite Z.max_l;[ring|omega]. (**) left. assert (bpow (ln_beta radix2 f -1) < f). destruct (ln_beta radix2 f); simpl in *. destruct a. `````` BOLDO Sylvie committed Jun 05, 2015 357 ``````now apply Rgt_not_eq. `````` BOLDO Sylvie committed Jun 16, 2015 358 359 360 361 362 363 ``````rewrite Rabs_right in H0. destruct H0; try assumption. contradict H0. now apply sym_not_eq. apply Rle_ge; now left. assert (bpow (ln_beta radix2 f -1) + ulp_flt (bpow (ln_beta radix2 f-1)) <= f). `````` BOLDO Sylvie committed Sep 07, 2015 364 ``````rewrite <- succ_eq_pos;[idtac|apply bpow_ge_0]. `````` BOLDO Sylvie committed Jul 29, 2015 365 ``````apply succ_le_lt... `````` BOLDO Sylvie committed Jun 16, 2015 366 367 368 369 370 371 372 ``````apply FLT_format_bpow... unfold Prec_gt_0 in prec_gt_0_;omega. rewrite ulp_bpow in H4. unfold FLT_exp in H4. rewrite Z.max_l in H4. 2: omega. replace (ln_beta radix2 f - 1 + 1 - prec)%Z with (ln_beta radix2 f - prec)%Z in H4 by ring. `````` BOLDO Sylvie committed Jul 29, 2015 373 374 375 ``````rewrite ulp_neq_0; try now apply Rgt_not_eq. rewrite ulp_neq_0 at 2; try now apply Rgt_not_eq. unfold canonic_exp. `````` BOLDO Sylvie committed Jun 16, 2015 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 ``````apply f_equal; apply f_equal. replace (ulp_flt f) with (bpow (ln_beta radix2 f -prec)). apply ln_beta_unique. rewrite Rabs_right. split. apply Rplus_le_reg_l with (bpow (ln_beta radix2 f -prec)). ring_simplify. apply Rle_trans with (2:=H4); right; ring. apply Rlt_trans with f. apply Rplus_lt_reg_l with (-f+bpow (ln_beta radix2 f - prec)). ring_simplify. apply bpow_gt_0. apply Rle_lt_trans with (1:=RRle_abs _). apply bpow_ln_beta_gt. apply Rle_ge. apply Rplus_le_reg_l with (bpow (ln_beta radix2 f - prec)). ring_simplify. left; apply Rle_lt_trans with (2:=H0). apply bpow_le. unfold Prec_gt_0 in prec_gt_0_;omega. `````` BOLDO Sylvie committed Jul 29, 2015 396 397 ``````rewrite ulp_neq_0; try now apply Rgt_not_eq. unfold canonic_exp, FLT_exp. `````` BOLDO Sylvie committed Jun 16, 2015 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 ``````rewrite Z.max_l. reflexivity. omega. assert (T: (ulp_flt (pred_flt f) = ulp_flt f \/ (ulp_flt (pred_flt f) = / 2 * ulp_flt f /\ - h < / 4 * ulp_flt f)) \/ (ulp_flt (pred_flt f) = / 2 * ulp_flt f /\ f = bpow (ln_beta radix2 f - 1) /\ - h = / 4 * ulp_flt f) ). destruct T1. left; now left. case Hh; intros P. left; right. split; try apply H0; assumption. right. split; try split; try apply H0; assumption. clear T1. `````` BOLDO Sylvie committed Jun 05, 2015 414 ``````(* - end of assertions *) `````` BOLDO Sylvie committed Jun 16, 2015 415 416 ``````destruct T. (* normal case *) `````` BOLDO Sylvie committed Sep 08, 2015 417 ``````apply round_N_eq_UP_pt with (pred_flt f)... `````` BOLDO Sylvie committed Sep 09, 2015 418 ``````rewrite <- (round_DN_minus_eps_pos radix2 (FLT_exp emin prec) f) with (eps:=-h); try assumption. `````` BOLDO Sylvie committed Jun 05, 2015 419 420 421 422 ``````replace (f--h) with (f+h) by ring. apply round_DN_pt... split. auto with real. `````` BOLDO Sylvie committed Jun 16, 2015 423 424 425 ``````apply Rle_trans with (1:=Hh). apply Rle_trans with (/2*ulp_flt f). apply Rmult_le_compat_r. `````` BOLDO Sylvie committed Sep 07, 2015 426 ``````apply ulp_ge_0. `````` BOLDO Sylvie committed Jun 16, 2015 427 428 429 430 ``````fourier. case H0. intros Y; rewrite Y. rewrite <- (Rmult_1_l (ulp_flt f)) at 2. `````` BOLDO Sylvie committed Jun 05, 2015 431 ``````apply Rmult_le_compat_r. `````` BOLDO Sylvie committed Sep 07, 2015 432 ``````apply ulp_ge_0. `````` BOLDO Sylvie committed Jun 05, 2015 433 ``````fourier. `````` BOLDO Sylvie committed Jun 16, 2015 434 ``````intros Y; rewrite (proj1 Y); now right. `````` BOLDO Sylvie committed Jun 05, 2015 435 ``````replace (f+h) with (pred_flt f + (f-pred_flt f+h)) by ring. `````` BOLDO Sylvie committed Sep 09, 2015 436 ``````pattern f at 4; rewrite <- (round_UP_pred_plus_eps_pos radix2 (FLT_exp emin prec) f) with (eps:=(f - pred_flt f + h)); try assumption. `````` BOLDO Sylvie committed Jun 05, 2015 437 438 439 440 ``````apply round_UP_pt... replace (f-pred_flt f) with (ulp_flt (pred_flt f)). split. apply Rplus_lt_reg_l with (-h); ring_simplify. `````` BOLDO Sylvie committed Jun 16, 2015 441 442 443 444 ``````case H0; [intros Y|intros (Y1,Y2)]. apply Rle_lt_trans with (1:=Hh). rewrite Y. rewrite <- (Rmult_1_l (ulp_flt f)) at 2. `````` BOLDO Sylvie committed Jun 05, 2015 445 ``````apply Rmult_lt_compat_r. `````` BOLDO Sylvie committed Jul 29, 2015 446 ``````rewrite ulp_neq_0;[apply bpow_gt_0|now apply Rgt_not_eq]. `````` BOLDO Sylvie committed Jun 05, 2015 447 ``````fourier. `````` BOLDO Sylvie committed Jun 16, 2015 448 449 450 ``````apply Rlt_le_trans with (1:=Y2). rewrite Y1. apply Rmult_le_compat_r. `````` BOLDO Sylvie committed Sep 07, 2015 451 ``````apply ulp_ge_0. `````` BOLDO Sylvie committed Jun 16, 2015 452 ``````fourier. `````` BOLDO Sylvie committed Jun 05, 2015 453 454 ``````apply Rplus_le_reg_l with (-ulp_flt (pred_flt f)); ring_simplify. now left. `````` BOLDO Sylvie committed Sep 07, 2015 455 ``````rewrite pred_eq_pos; try now left. `````` BOLDO Sylvie committed Jul 29, 2015 456 ``````pattern f at 2; rewrite <- (pred_pos_plus_ulp radix2 (FLT_exp emin prec) f)... `````` BOLDO Sylvie committed Jun 05, 2015 457 458 459 460 461 462 ``````ring. apply Rplus_lt_reg_l with (-f); ring_simplify. apply Rle_lt_trans with (-(/2 * ulp_flt (pred_flt f))). right. apply trans_eq with ((pred_flt f - f) / 2). field. `````` BOLDO Sylvie committed Sep 07, 2015 463 ``````rewrite pred_eq_pos; try now left. `````` BOLDO Sylvie committed Jul 29, 2015 464 ``````pattern f at 2; rewrite <- (pred_pos_plus_ulp radix2 (FLT_exp emin prec) f)... `````` BOLDO Sylvie committed Jun 05, 2015 465 466 467 ``````field. replace h with (--h) by ring. apply Ropp_lt_contravar. `````` BOLDO Sylvie committed Jun 16, 2015 468 469 470 471 ``````case H0;[intros Y|intros (Y1,Y2)]. apply Rle_lt_trans with (1:=Hh). rewrite Y. apply Rmult_lt_compat_r. `````` BOLDO Sylvie committed Jul 29, 2015 472 ``````rewrite ulp_neq_0; try apply bpow_gt_0; now apply Rgt_not_eq. `````` BOLDO Sylvie committed Jun 05, 2015 473 ``````fourier. `````` BOLDO Sylvie committed Jun 16, 2015 474 475 476 477 478 479 ``````apply Rlt_le_trans with (1:=Y2). rewrite Y1. right; field. (* complex case: even choosing *) elim H0; intros T1 (T2,T3); clear H0. assert (pred_flt f = bpow (ln_beta radix2 f - 1) - bpow (ln_beta radix2 f - 1 -prec)). `````` BOLDO Sylvie committed Sep 07, 2015 480 ``````rewrite pred_eq_pos; try now left. `````` BOLDO Sylvie committed Jul 29, 2015 481 ``````unfold pred_pos; case Req_bool_spec. `````` BOLDO Sylvie committed Jun 16, 2015 482 483 484 485 486 487 488 489 490 ``````intros _; rewrite <- T2. apply f_equal, f_equal. unfold FLT_exp. rewrite Z.max_l. ring. omega. intros Y; now contradict T2. assert (round radix2 (FLT_exp emin prec) Zfloor (f+h) = pred_flt f). replace (f+h) with (f-(-h)) by ring. `````` BOLDO Sylvie committed Sep 09, 2015 491 ``````apply round_DN_minus_eps_pos... `````` BOLDO Sylvie committed Jun 16, 2015 492 493 494 495 ``````split. auto with real. rewrite T3, T1. apply Rmult_le_compat_r. `````` BOLDO Sylvie committed Sep 07, 2015 496 ``````apply ulp_ge_0. `````` BOLDO Sylvie committed Jun 16, 2015 497 498 499 ``````fourier. assert (round radix2 (FLT_exp emin prec) Zceil (f+h) = f). replace (f+h) with (pred_flt f + /2*ulp_flt (pred_flt f)). `````` BOLDO Sylvie committed Sep 09, 2015 500 ``````apply round_UP_pred_plus_eps_pos... `````` BOLDO Sylvie committed Jun 16, 2015 501 502 503 ``````split. apply Rmult_lt_0_compat. fourier. `````` BOLDO Sylvie committed Jul 29, 2015 504 ``````rewrite ulp_neq_0; try now apply Rgt_not_eq. `````` BOLDO Sylvie committed Jun 16, 2015 505 506 507 ``````apply bpow_gt_0. rewrite <- (Rmult_1_l (ulp_flt (pred_flt f))) at 2. apply Rmult_le_compat_r. `````` BOLDO Sylvie committed Sep 07, 2015 508 ``````apply ulp_ge_0. `````` BOLDO Sylvie committed Jun 16, 2015 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 ``````fourier. rewrite T1, H0, <- T2. replace h with (--h) by ring; rewrite T3. replace (bpow (ln_beta radix2 f - 1 - prec)) with (/2*ulp_flt f). field. replace (/2) with (bpow (-1)) by reflexivity. rewrite T2 at 1. rewrite ulp_bpow, <- bpow_plus. apply f_equal; unfold FLT_exp. rewrite Z.max_l. ring. omega. assert ((Zeven (Zfloor (scaled_mantissa radix2 (FLT_exp emin prec) (f + h)))) = false). replace (Zfloor (scaled_mantissa radix2 (FLT_exp emin prec) (f + h))) with (Zpower radix2 prec -1)%Z. unfold Zminus; rewrite Zeven_plus. rewrite Zeven_opp. rewrite Zeven_Zpower. reflexivity. unfold Prec_gt_0 in prec_gt_0_; omega. apply eq_Z2R. rewrite <- scaled_mantissa_DN... 2: rewrite H4; assumption. rewrite H4. unfold scaled_mantissa. rewrite bpow_opp. `````` BOLDO Sylvie committed Jul 29, 2015 535 ``````rewrite <- ulp_neq_0; try now apply Rgt_not_eq. `````` BOLDO Sylvie committed Jun 16, 2015 536 537 538 ``````rewrite T1. rewrite Rinv_mult_distr. 2: apply Rgt_not_eq; fourier. `````` BOLDO Sylvie committed Jul 29, 2015 539 540 ``````2: apply Rgt_not_eq; rewrite ulp_neq_0; try apply bpow_gt_0. 2: now apply Rgt_not_eq. `````` BOLDO Sylvie committed Jun 16, 2015 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 ``````rewrite Rinv_involutive. 2: apply Rgt_not_eq; fourier. rewrite T2 at 2. rewrite ulp_bpow. rewrite <- bpow_opp. unfold FLT_exp at 2. rewrite Z.max_l. 2: omega. replace 2 with (bpow 1) by reflexivity. rewrite <- bpow_plus. rewrite H0. rewrite Rmult_minus_distr_r, <- 2!bpow_plus. rewrite Z2R_minus. apply f_equal2. rewrite Z2R_Zpower. apply f_equal. ring. unfold Prec_gt_0 in prec_gt_0_; omega. apply trans_eq with (bpow 0). reflexivity. apply f_equal. ring. rewrite round_N_middle. rewrite H5. rewrite H6. reflexivity. rewrite H5, H4. `````` BOLDO Sylvie committed Jul 29, 2015 568 ``````pattern f at 1; rewrite <- (pred_pos_plus_ulp radix2 (FLT_exp emin prec) f); try assumption. `````` BOLDO Sylvie committed Jun 16, 2015 569 ``````ring_simplify. `````` BOLDO Sylvie committed Sep 07, 2015 570 ``````rewrite <- pred_eq_pos;[idtac|now left]. `````` BOLDO Sylvie committed Jun 16, 2015 571 572 573 574 ``````rewrite T1. replace h with (--h) by ring. rewrite T3. field. `````` BOLDO Sylvie committed Jun 05, 2015 575 576 577 ``````Qed. Lemma round_plus_small_id: forall f h, format f -> (bpow (prec+emin) <= Rabs f) `````` BOLDO Sylvie committed Jun 16, 2015 578 `````` -> Rabs h <= /4* ulp_flt f -> round_flt (f+h) = f. `````` BOLDO Sylvie committed Jun 05, 2015 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 ``````intros f h Ff H1 H2. case (Rle_or_lt 0 f); intros V. case V; clear V; intros V. apply round_plus_small_id_aux; try assumption. rewrite Rabs_right in H1; try assumption. apply Rle_ge; now left. contradict H1. rewrite <- V, Rabs_R0. apply Rlt_not_le, bpow_gt_0. rewrite <- (Ropp_involutive f), <- (Ropp_involutive h). replace (--f + --h) with (-(-f+-h)) by ring. rewrite round_NE_opp. apply f_equal. apply round_plus_small_id_aux. now apply generic_format_opp. rewrite Rabs_left in H1; try assumption. auto with real. now rewrite Rabs_Ropp, ulp_opp. Qed. `````` BOLDO Sylvie committed Nov 20, 2014 600 601 602 603 604 605 606 607 608 609 `````` Definition average1 (x y : R) :=round_flt(round_flt(x+y)/2). Variables x y:R. Hypothesis Fx: format x. Hypothesis Fy: format y. Let a:=(x+y)/2. Let av:=average1 x y. `````` BOLDO Sylvie committed Jun 05, 2015 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 ``````Lemma average1_correct: av = round_flt a. Proof with auto with typeclass_instances. case (Rle_or_lt (bpow (prec + emin)) (Rabs (x+y))). (* normal case: division by 2 is exact *) intros H. unfold av,a,average1. rewrite round_generic... now apply sym_eq, FLT_round_half. apply FLT_format_half. apply generic_format_round... apply abs_round_ge_generic... apply FLT_format_bpow... unfold Prec_gt_0 in prec_gt_0_; omega. (* subnormal case: addition is exact, but division by 2 is not *) intros H. unfold av, average1. replace (round_flt (x + y)) with (x+y). reflexivity. apply sym_eq, round_generic... apply FLT_format_plus_small... left; assumption. Qed. `````` BOLDO Sylvie committed Nov 20, 2014 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 `````` Lemma average1_symmetry: forall u v, average1 u v = average1 v u. Proof. intros u v; unfold average1. rewrite Rplus_comm; reflexivity. Qed. Lemma average1_symmetry_Ropp: forall u v, average1 (-u) (-v) = - average1 u v. Proof. intros u v; unfold average1. replace (-u+-v) with (-(u+v)) by ring. rewrite round_NE_opp. replace (- round_flt (u + v) / 2) with (- (round_flt (u + v) / 2)) by (unfold Rdiv; ring). now rewrite round_NE_opp. Qed. Lemma average1_same_sign_1: 0 <= a -> 0 <= av. Proof with auto with typeclass_instances. intros H. `````` BOLDO Sylvie committed Jun 05, 2015 653 ``````rewrite average1_correct. `````` BOLDO Sylvie committed Nov 20, 2014 654 655 656 657 658 659 660 ``````apply round_ge_generic... apply generic_format_0. Qed. Lemma average1_same_sign_2: a <= 0-> av <= 0. Proof with auto with typeclass_instances. intros H. `````` BOLDO Sylvie committed Jun 05, 2015 661 ``````rewrite average1_correct. `````` BOLDO Sylvie committed Nov 20, 2014 662 663 664 665 666 667 ``````apply round_le_generic... apply generic_format_0. Qed. Lemma average1_between: Rmin x y <= av <= Rmax x y. Proof with auto with typeclass_instances. `````` BOLDO Sylvie committed Jun 05, 2015 668 669 ``````rewrite average1_correct. split. `````` BOLDO Sylvie committed Nov 20, 2014 670 ``````apply round_ge_generic... `````` BOLDO Sylvie committed Jun 05, 2015 671 ``````now apply P_Rmin. `````` BOLDO Sylvie committed Nov 20, 2014 672 673 ``````apply Rmult_le_reg_l with 2. auto with real. `````` BOLDO Sylvie committed Jun 05, 2015 674 675 676 677 678 679 ``````rewrite Rmult_plus_distr_r, Rmult_1_l. apply Rle_trans with (x+y);[idtac|right;unfold a; field]. apply Rplus_le_compat. apply Rmin_l. apply Rmin_r. (* *) `````` BOLDO Sylvie committed Nov 20, 2014 680 ``````apply round_le_generic... `````` BOLDO Sylvie committed Jun 05, 2015 681 ``````now apply Rmax_case. `````` BOLDO Sylvie committed Nov 20, 2014 682 683 ``````apply Rmult_le_reg_l with 2. auto with real. `````` BOLDO Sylvie committed Jun 05, 2015 684 685 686 687 688 ``````apply Rle_trans with (x+y);[right;unfold a; field|idtac]. rewrite Rmult_plus_distr_r, Rmult_1_l. apply Rplus_le_compat. apply Rmax_l. apply Rmax_r. `````` BOLDO Sylvie committed Nov 20, 2014 689 690 691 692 693 ``````Qed. Lemma average1_zero: a = 0 -> av = 0. Proof with auto with typeclass_instances. `````` BOLDO Sylvie committed Jun 05, 2015 694 ``````intros H1; rewrite average1_correct, H1. `````` BOLDO Sylvie committed Nov 20, 2014 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 ``````rewrite round_0... Qed. Lemma average1_no_underflow: (bpow emin) <= Rabs a -> av <> 0. Proof with auto with typeclass_instances. intros H. (* *) cut (bpow emin <= Rabs av). intros H1 H2. rewrite H2 in H1; rewrite Rabs_R0 in H1. contradict H1. apply Rlt_not_le. apply bpow_gt_0. (* *) `````` BOLDO Sylvie committed Jun 05, 2015 711 ``````rewrite average1_correct. `````` BOLDO Sylvie committed Nov 20, 2014 712 713 714 715 716 717 ``````apply abs_round_ge_generic... apply FLT_format_bpow... omega. Qed. `````` BOLDO Sylvie committed Jun 05, 2015 718 ``````Lemma average1_correct_weak1: Rabs (av -a) <= /2*ulp_flt a. `````` BOLDO Sylvie committed Nov 20, 2014 719 ``````Proof with auto with typeclass_instances. `````` BOLDO Sylvie committed Jun 05, 2015 720 ``````rewrite average1_correct. `````` BOLDO Sylvie committed Sep 08, 2015 721 ``````apply error_le_half_ulp... `````` BOLDO Sylvie committed Nov 20, 2014 722 723 ``````Qed. `````` BOLDO Sylvie committed Jun 05, 2015 724 ``````Lemma average1_correct_weak2: Rabs (av -a) <= 3/2*ulp_flt a. `````` BOLDO Sylvie committed Nov 20, 2014 725 ``````Proof with auto with typeclass_instances. `````` BOLDO Sylvie committed Jun 05, 2015 726 ``````apply Rle_trans with (1:=average1_correct_weak1). `````` BOLDO Sylvie committed Nov 20, 2014 727 ``````apply Rmult_le_compat_r. `````` BOLDO Sylvie committed Sep 07, 2015 728 ``````unfold ulp; apply ulp_ge_0. `````` BOLDO Sylvie committed Nov 20, 2014 729 730 731 732 733 734 735 736 737 738 739 740 741 742 ``````apply Rle_trans with (1/2); unfold Rdiv. right; ring. apply Rmult_le_compat_r. now auto with real. apply Rplus_le_reg_l with (-1); ring_simplify. now auto with real. Qed. (* Hypothesis diff_sign: (0 <= x /\ y <= 0) \/ (x <= 0 /\ 0 <= y). is useless for properties: only useful for preventing overflow *) `````` BOLDO Sylvie committed Jun 05, 2015 743 744 745 746 747 748 749 750 `````` Definition average2 (x y : R) :=round_flt(round_flt(x/2) + round_flt(y/2)). Let av2:=average2 x y. `````` BOLDO Sylvie committed Jun 16, 2015 751 ``````Lemma average2_correct: bpow (emin +prec+prec+1) <= Rabs x -> av2 = round_flt a. `````` BOLDO Sylvie committed Jun 05, 2015 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 ``````Proof with auto with typeclass_instances. intros Hx. assert (G:(0 < prec)%Z). unfold Prec_gt_0 in prec_gt_0_; assumption. unfold av2, average2. replace (round_flt (x/2)) with (x/2). 2: apply sym_eq, round_generic... 2: apply FLT_format_half; try assumption. 2: apply Rle_trans with (2:=Hx). 2: apply bpow_le; omega. case (Rle_or_lt (bpow (prec + emin)) (Rabs y)). (* y is big enough so that y/2 is correct *) intros Hy. replace (round_flt (y/2)) with (y/2). apply f_equal; unfold a; field. apply sym_eq, round_generic... apply FLT_format_half; assumption. (* y is a subnormal, then it is too small to impact the result *) intros Hy. assert (format (x/2)). apply FLT_format_half. assumption. apply Rle_trans with (2:=Hx). apply bpow_le. omega. assert (bpow (prec+emin) <= Rabs (x/2)). apply Rmult_le_reg_l with (bpow 1). apply bpow_gt_0. rewrite <- bpow_plus. apply Rle_trans with (Rabs x). apply Rle_trans with (2:=Hx). apply bpow_le. omega. rewrite <- (Rabs_right (bpow 1)). rewrite <- Rabs_mult. right; apply f_equal. simpl; field. apply Rle_ge, bpow_ge_0. `````` BOLDO Sylvie committed Jun 16, 2015 790 ``````assert (K1: Rabs (y / 2) <= bpow (prec+emin-1)). `````` BOLDO Sylvie committed Jun 05, 2015 791 792 793 794 795 796 797 ``````unfold Rdiv; rewrite Rabs_mult. unfold Zminus; rewrite bpow_plus. simpl; rewrite (Rabs_right (/2)). apply Rmult_le_compat_r. fourier. now left. fourier. `````` BOLDO Sylvie committed Jun 16, 2015 798 ``````assert (K2:bpow (prec+emin-1) <= / 4 * ulp_flt (x / 2)). `````` BOLDO Sylvie committed Jul 29, 2015 799 800 801 802 803 ``````assert (Z: x/2 <> 0). intros K; contradict H0. rewrite K, Rabs_R0. apply Rlt_not_le, bpow_gt_0. rewrite ulp_neq_0; trivial. `````` BOLDO Sylvie committed Jun 05, 2015 804 805 ``````replace (/4) with (bpow (-2)) by reflexivity. rewrite <- bpow_plus. `````` BOLDO Sylvie committed Jun 16, 2015 806 ``````apply bpow_le. `````` BOLDO Sylvie committed Jun 05, 2015 807 ``````unfold canonic_exp, FLT_exp. `````` BOLDO Sylvie committed Jun 16, 2015 808 ``````assert (emin+prec+prec+1 -1 < ln_beta radix2 (x/2))%Z. `````` BOLDO Sylvie committed Jun 05, 2015 809 810 811 812 813 814 815 816 817 818 819 ``````destruct (ln_beta radix2 (x/2)) as (e,He). simpl. apply lt_bpow with radix2. apply Rle_lt_trans with (Rabs (x/2)). unfold Rdiv; rewrite Rabs_mult. unfold Zminus; rewrite bpow_plus. simpl; rewrite (Rabs_right (/2)). apply Rmult_le_compat_r. fourier. exact Hx. fourier. `````` BOLDO Sylvie committed Jul 29, 2015 820 ``````apply He; trivial. `````` BOLDO Sylvie committed Jun 05, 2015 821 822 823 ``````rewrite Z.max_l. omega. omega. `````` BOLDO Sylvie committed Jun 16, 2015 824 825 826 827 828 829 830 ``````(* . *) apply trans_eq with (x/2). apply round_plus_small_id; try assumption. apply Rle_trans with (2:=K2). apply abs_round_le_generic... apply FLT_format_bpow... omega. `````` BOLDO Sylvie committed Jun 05, 2015 831 832 833 ``````unfold a; apply sym_eq. replace ((x+y)/2) with (x/2+y/2) by field. apply round_plus_small_id; try assumption. `````` BOLDO Sylvie committed Jun 16, 2015 834 ``````now apply Rle_trans with (2:=K2). `````` BOLDO Sylvie committed Jun 05, 2015 835 836 837 838 ``````Qed. `````` BOLDO Sylvie committed Nov 20, 2014 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 ``````End av1. Section av3. Notation bpow e := (bpow radix2 e). Variable emin prec : Z. Context { prec_gt_0_ : Prec_gt_0 prec }. Notation format := (generic_format radix2 (FLT_exp emin prec)). Notation round_flt :=(round radix2 (FLT_exp emin prec) ZnearestE). Notation ulp_flt :=(ulp radix2 (FLT_exp emin prec)). Notation cexp := (canonic_exp radix2 (FLT_exp emin prec)). Definition average3 (x y : R) :=round_flt(x+round_flt(round_flt(y-x)/2)). Variables x y:R. Hypothesis Fx: format x. Hypothesis Fy: format y. Let a:=(x+y)/2. Let av:=average3 x y. Lemma average3_symmetry_Ropp: forall u v, average3 (-u) (-v) = - average3 u v. intros u v; unfold average3. replace (-v--u) with (-(v-u)) by ring. rewrite round_NE_opp. replace (- round_flt (v-u) / 2) with (- (round_flt (v-u) / 2)) by (unfold Rdiv; ring). rewrite round_NE_opp. replace (- u + - round_flt (round_flt (v - u) / 2)) with (-(u+round_flt (round_flt (v - u) / 2))) by ring. apply round_NE_opp. Qed. Lemma average3_same_sign_1: 0 <= a -> 0 <= av. Proof with auto with typeclass_instances. intros H. apply round_ge_generic... apply generic_format_0. apply Rplus_le_reg_l with (-x). ring_simplify. apply round_ge_generic... now apply generic_format_opp. apply Rmult_le_reg_l with 2. auto with real. apply Rle_trans with (-(2*x)). right; ring. apply Rle_trans with (round_flt (y - x)). 2: right; field. apply round_ge_generic... apply generic_format_opp. now apply FLT_format_double... apply Rplus_le_reg_l with (2*x). apply Rmult_le_reg_r with (/2). auto with real. apply Rle_trans with 0;[right; ring|idtac]. apply Rle_trans with (1:=H). right; unfold a, Rdiv; ring. Qed. Lemma average3_same_sign_2: a <= 0-> av <= 0. Proof with auto with typeclass_instances. intros H. apply round_le_generic... apply generic_format_0. apply Rplus_le_reg_l with (-x). ring_simplify. apply round_le_generic... now apply generic_format_opp. apply Rmult_le_reg_l with 2. auto with real. apply Rle_trans with (-(2*x)). 2: right; ring. apply Rle_trans with (round_flt (y - x)). right; field. apply round_le_generic... apply generic_format_opp. now apply FLT_format_double... apply Rplus_le_reg_l with (2*x). apply Rmult_le_reg_r with (/2). auto with real. apply Rle_trans with 0;[idtac|right; ring]. apply Rle_trans with (2:=H). right; unfold a, Rdiv; ring. Qed. Lemma average3_between_aux: forall u v, format u -> format v -> u <= v -> u <= average3 u v <= v. Proof with auto with typeclass_instances. clear Fx Fy a av x y. intros x y Fx Fy M. split. (* . *) apply round_ge_generic... apply Rplus_le_reg_l with (-x). ring_simplify. apply round_ge_generic... apply generic_format_0. unfold Rdiv; apply Rmult_le_pos. apply round_ge_generic... apply generic_format_0. apply Rplus_le_reg_l with x. now ring_simplify. auto with real. (* . *) apply round_le_generic... assert (H:(0 <= round radix2 (FLT_exp emin prec) Zfloor (y-x))). apply round_ge_generic... apply generic_format_0. apply Rplus_le_reg_l with x. now ring_simplify. destruct H as [H|H]. (* .. *) pattern y at 2; replace y with (x + (y-x)) by ring. apply Rplus_le_compat_l. case (generic_format_EM radix2 (FLT_exp emin prec) (y-x)); intros K. apply round_le_generic... rewrite round_generic... apply Rmult_le_reg_l with 2. auto with real. apply Rplus_le_reg_l with (2*x-y). apply Rle_trans with x. right; field. apply Rle_trans with (1:=M). right; field. apply Rle_trans with (round radix2 (FLT_exp emin prec) Zfloor (y - x)). apply round_le_generic... apply generic_format_round... apply Rmult_le_reg_l with 2. auto with real. apply Rle_trans with (round_flt (y - x)). right; field. case (round_DN_or_UP radix2 (FLT_exp emin prec) ZnearestE (y-x)); intros H1; rewrite H1. apply Rplus_le_reg_l with (-round radix2 (FLT_exp emin prec) Zfloor (y - x)). ring_simplify. now left. `````` BOLDO Sylvie committed Sep 07, 2015 981 ``````rewrite round_UP_DN_ulp. `````` BOLDO Sylvie committed Nov 20, 2014 982 983 ``````apply Rplus_le_reg_l with (-round radix2 (FLT_exp emin prec) Zfloor (y - x)); ring_simplify. apply round_DN_pt... `````` BOLDO Sylvie committed Jul 29, 2015 984 ``````apply generic_format_ulp... `````` BOLDO Sylvie committed Nov 20, 2014 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 ``````case (Rle_or_lt (bpow (emin + prec - 1)) (y-x)); intros P. apply FLT_ulp_le_id... apply Rle_trans with (2:=P). apply bpow_le; unfold Prec_gt_0 in prec_gt_0_; omega. contradict K. apply FLT_format_plus_small... now apply generic_format_opp. rewrite Rabs_right. apply Rle_trans with (bpow (emin+prec-1)). left; exact P. apply bpow_le; omega. apply Rle_ge; apply Rplus_le_reg_l with x; now ring_simplify. assumption. apply round_DN_pt... (* .. *) case M; intros H1. 2: rewrite H1; replace (y-y) with 0 by ring. 2: rewrite round_0... 2: unfold Rdiv; rewrite Rmult_0_l. 2: rewrite round_0... 2: right; ring. apply Rle_trans with (x+0). 2: rewrite Rplus_0_r; assumption. apply Rplus_le_compat_l. replace 0 with (round_flt (bpow emin/2)). apply round_le... unfold Rdiv; apply Rmult_le_compat_r. auto with real. apply round_le_generic... apply FLT_format_bpow... omega. case (Rle_or_lt (y-x) (bpow emin)); trivial. intros H2. contradict H. apply Rlt_not_eq. apply Rlt_le_trans with (bpow emin). apply bpow_gt_0. apply round_DN_pt... apply FLT_format_bpow... omega. now left. replace (bpow emin /2) with (bpow (emin-1)). unfold round, scaled_mantissa, canonic_exp, FLT_exp. rewrite ln_beta_bpow. replace (emin - 1 + 1 - prec)%Z with (emin-prec)%Z by ring. rewrite Z.max_r. 2: unfold Prec_gt_0 in prec_gt_0_; omega. rewrite <- bpow_plus. replace (emin-1+-emin)%Z with (-1)%Z by ring. replace (ZnearestE (bpow (-1))) with 0%Z. unfold F2R; simpl; ring. simpl; unfold Znearest. replace (Zfloor (/2)) with 0%Z. rewrite Rcompare_Eq. reflexivity. simpl; ring. apply sym_eq, Zfloor_imp. simpl; split. auto with real. apply Rmult_lt_reg_l with 2. auto with real. apply Rle_lt_trans with 1. right; field. rewrite Rmult_1_r. auto with real. unfold Zminus; rewrite bpow_plus. reflexivity. Qed. Lemma average3_between: Rmin x y <= av <= Rmax x y. Proof with auto with typeclass_instances. case (Rle_or_lt x y); intros M. (* x <= y *) rewrite Rmin_left; try exact M. rewrite Rmax_right; try exact M. now apply average3_between_aux. (* y < x *) rewrite Rmin_right; try now left. rewrite Rmax_left; try now left. unfold av; rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive y). rewrite average3_symmetry_Ropp. split; apply Ropp_le_contravar. apply average3_between_aux. now apply generic_format_opp. now apply generic_format_opp. apply Ropp_le_contravar; now left. apply average3_between_aux. now apply generic_format_opp. now apply generic_format_opp. apply Ropp_le_contravar; now left. Qed. Lemma average3_zero: a = 0 -> av = 0. Proof with auto with typeclass_instances. intros H. assert (y=-x). apply Rplus_eq_reg_l with x. apply Rmult_eq_reg_r with (/2). apply trans_eq with a. reflexivity. rewrite H; ring. apply Rgt_not_eq, Rlt_gt. auto with real. unfold av, average3. rewrite H0. replace (-x-x) with (-(2*x)) by ring. rewrite round_generic with (x:=(-(2*x)))... replace (-(2*x)/2) with (-x) by field. rewrite round_generic with (x:=-x)... replace (x+-x) with 0 by ring. apply round_0... now apply generic_format_opp. apply generic_format_opp. now apply FLT_format_double. Qed. Lemma average3_no_underflow_aux_aux: forall z:Z, (0 < z)%Z -> (ZnearestE (Z2R z / 2) < z)%Z. Proof. intros z H1. case (Zle_lt_or_eq 1 z); [omega|intros H2|intros H2]. apply lt_Z2R. apply Rplus_lt_reg_r with (- ((Z2R z)/2)). apply Rle_lt_trans with (-(((Z2R z) /2) - Z2R (ZnearestE (Z2R z / 2)))). right; ring. apply Rle_lt_trans with (1:= RRle_abs _). rewrite Rabs_Ropp. apply Rle_lt_trans with (1:=Znearest_N (fun x => negb (Zeven x)) _). apply Rle_lt_trans with (1*/2);[right; ring|idtac]. apply Rlt_le_trans with ((Z2R z)*/2);[idtac|right; field]. apply Rmult_lt_compat_r. auto with real. replace 1 with (Z2R 1) by reflexivity. now apply Z2R_lt. rewrite <- H2. unfold Znearest; simpl. replace (Zfloor (1 / 2)) with 0%Z. rewrite Rcompare_Eq. simpl; omega. simpl; field. unfold Rdiv; rewrite Rmult_1_l. apply sym_eq, Zfloor_imp. simpl; split. auto with real. apply Rmult_lt_reg_l with 2. auto with real. apply Rle_lt_trans with 1. right; field. rewrite Rmult_1_r. auto with real. Qed. Lemma average3_no_underflow_aux1: forall f, format f -> 0 < f -> f <= round_flt (f/2) -> False. Proof with auto with typeclass_instances. intros f Ff Hf1 Hf2. apply FLT_format_generic in Ff... destruct Ff as (g, (H1,(H2,H3))). case (Zle_lt_or_eq emin (Fexp g)); try exact H3; intros H4. contradict Hf2. apply Rlt_not_le. rewrite round_generic... apply Rplus_lt_reg_l with (-(f/2)). apply Rle_lt_trans with 0;[right; ring|idtac]. apply Rlt_le_trans with (f*/2);[idtac|right;field]. apply Rmult_lt_0_compat; try assumption. auto with real. apply generic_format_FLT. exists (Float radix2 (Fnum g) (Fexp g-1)). split. rewrite H1; unfold F2R; simpl. unfold Zminus; rewrite bpow_plus. simpl; field. split. now simpl. simpl; omega. contradict Hf2; apply Rlt_not_le. unfold round, scaled_mantissa. replace (cexp (f/2)) with emin. rewrite H1; unfold F2R; simpl. rewrite <- H4. apply Rmult_lt_compat_r. apply bpow_gt_0. apply Z2R_lt. replace (Z2R (Fnum g) * bpow emin / 2 * bpow (- emin)) with (Z2R (Fnum g) /2). apply average3_no_underflow_aux_aux. apply lt_Z2R. apply Rmult_lt_reg_r with (bpow (Fexp g)). apply bpow_gt_0. rewrite Rmult_0_l. apply Rlt_le_trans with (1:=Hf1). right; rewrite H1; reflexivity. unfold Rdiv; apply trans_eq with (Z2R (Fnum g) * / 2 * (bpow (- emin)*bpow emin)). rewrite <- bpow_plus. ring_simplify (-emin+emin)%Z. simpl; ring. ring. apply sym_eq, canonic_exp_FLT_FIX. apply Rgt_not_eq, Rlt_gt. unfold Rdiv; apply Rmult_lt_0_compat; try assumption. auto with real. rewrite H1; unfold F2R, Rdiv; simpl. replace (/2) with (bpow (-1)) by reflexivity. rewrite Rmult_assoc, <- bpow_plus. rewrite Rabs_mult. rewrite (Rabs_right (bpow _)). 2: apply Rle_ge, bpow_ge_0. rewrite (Zplus_comm emin _). rewrite (bpow_plus _ prec _). apply Rmult_lt_compat. apply Rabs_pos. apply bpow_ge_0. rewrite <- Z2R_Zpower, <- Z2R_abs. now apply Z2R_lt. unfold Prec_gt_0 in prec_gt_0_; omega. rewrite <- H4; apply bpow_lt. omega. Qed. Lemma average3_no_underflow_aux2: forall u v, format u -> format v -> (0 <= u /\ 0 <= v) \/ (u <= 0 /\ v <= 0) -> u <= v -> (bpow emin) <= Rabs ((u+v)/2) -> average3 u v <> 0. Proof with auto with typeclass_instances. clear Fx Fy a av x y; intros x y Fx Fy same_sign xLey H; unfold average3. intros J. apply round_plus_eq_zero in J... 2: apply generic_format_round... assert (H1:x <= 0). apply Rplus_le_reg_r with (round_flt (round_flt (y - x) / 2)). rewrite J, Rplus_0_l. apply round_ge_generic... apply generic_format_0. unfold Rdiv; apply Rmult_le_pos. apply round_ge_generic... apply generic_format_0. apply Rplus_le_reg_l with x; now ring_simplify. auto with real. destruct H1 as [H1|H1]. (* *) destruct same_sign as [(H2,H3)|(_,H2)]. contradict H2; now apply Rlt_not_le. apply average3_no_underflow_aux1 with (-x). now apply generic_format_opp. rewrite <- Ropp_0; now apply Ropp_lt_contravar. apply Rle_trans with (round_flt (round_flt (y - x) / 2)). apply Rplus_le_reg_l with x. rewrite J; right; ring. apply round_le... unfold Rdiv; apply Rmult_le_compat_r. auto with real. apply round_le_generic... now apply generic_format_opp. apply Rplus_le_reg_l with x. now ring_simplify. (* *) rewrite H1 in J, H. rewrite Rplus_0_l in H. contradict J; apply Rgt_not_eq, Rlt_gt. rewrite Rplus_0_l. unfold Rminus; rewrite Ropp_0, Rplus_0_r. rewrite round_generic with (x:=y)... apply Rlt_le_trans with (bpow emin). apply bpow_gt_0. apply round_ge_generic... apply FLT_format_bpow... omega. apply Rle_trans with (1:=H). right; apply Rabs_right. apply Rle_ge; unfold Rdiv; apply Rmult_le_pos. rewrite <- H1; assumption. auto with real. Qed. Lemma average3_no_underflow_aux3: forall u v, format u -> format v -> (0 <= u /\ 0 <= v) \/ (u <= 0 /\ v <= 0) -> (bpow emin) <= Rabs ((u+v)/2) -> average3 u v <> 0. Proof with auto with typeclass_instances. clear Fx Fy a av x y; intros x y Fx Fy. intros same_sign H. case (Rle_or_lt x y); intros H1. now apply average3_no_underflow_aux2. rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive y). rewrite average3_symmetry_Ropp. apply Ropp_neq_0_compat. apply average3_no_underflow_aux2. now apply generic_format_opp. now apply generic_format_opp. rewrite <- Ropp_0; case same_sign; intros (T1,T2). right; split; now apply Ropp_le_contravar. left; split; now apply Ropp_le_contravar. apply Ropp_le_contravar; now left. apply Rle_trans with (1:=H). rewrite <- Rabs_Ropp. right; apply f_equal. unfold Rdiv; ring. Qed. Lemma average3_no_underflow: (0 <= x /\ 0 <= y) \/ (x <= 0 /\ y <= 0) -> (bpow emin) <= Rabs a -> av <> 0. Proof with auto with typeclass_instances. intros; now apply average3_no_underflow_aux3. Qed. Lemma average3_correct_aux: forall u v, format u -> format v -> u <= v -> (0 <= u /\ 0 <= v) \/ (u <= 0 /\ v <= 0) -> 0 < Rabs ((u+v)/2) < bpow emin -> Rabs (average3 u v -((u+v)/2)) <= 3/2 * ulp_flt ((u+v)/2). Proof with auto with typeclass_instances. `````` BOLDO Sylvie committed Jul 29, 2015 1302 1303 1304 ``````clear Fx Fy x y a av. intros u v Fu Fv uLev same_sign. pose (b:=(u+v)/2); fold b. `````` BOLDO Sylvie committed Nov 20, 2014 1305 1306 ``````(* mostly forward proof *) intros (H1,H2). `````` BOLDO Sylvie committed Jul 29, 2015 1307 1308 1309 1310 1311 1312 1313 ``````apply generic_format_FIX_FLT,FIX_format_generic in Fu. apply generic_format_FIX_FLT,FIX_format_generic in Fv. destruct Fu as ((nu,eu),(J1,J2)). destruct Fv as ((nv,ev),(J3,J4)); simpl in J2, J4. (* b is bpow emin /2 *) assert (b = Z2R (nu+nv) * bpow (emin-1)). unfold b; rewrite J1, J3; unfold F2R; rewrite J2,J4; simpl. `````` BOLDO Sylvie committed Nov 20, 2014 1314 ``````unfold Zminus; rewrite bpow_plus, Z2R_plus; simpl; field. `````` BOLDO Sylvie committed Jul 29, 2015 1315 1316 ``````assert (Z.abs (nu+nv) = 1)%Z. assert (0 < Z.abs (nu+nv) < 2)%Z;[idtac|omega]. `````` BOLDO Sylvie committed Nov 20, 2014 1317 1318 1319 1320 1321 1322 1323 1324 ``````split; apply lt_Z2R; simpl; rewrite Z2R_abs; apply Rmult_lt_reg_l with (bpow (emin-1)); try apply bpow_gt_0. rewrite Rmult_0_r. apply Rlt_le_trans with (1:=H1). right; rewrite H, Rabs_mult. rewrite (Rabs_right (bpow (emin -1))). ring. apply Rle_ge, bpow_ge_0. `````` BOLDO Sylvie committed Jul 29, 2015 1325 ``````apply Rle_lt_trans with (Rabs b). `````` BOLDO Sylvie committed Nov 20, 2014 1326 1327 1328 1329 1330 1331 1332 ``````right; rewrite H, Rabs_mult. rewrite (Rabs_right (bpow (emin -1))). ring. apply Rle_ge, bpow_ge_0. apply Rlt_le_trans with (1:=H2). right; unfold Zminus; rewrite bpow_plus. simpl; field. `````` BOLDO Sylvie committed Jul 29, 2015 1333 1334 1335 ``````(* only 2 possible values for u and v *) assert (((nu=0)/\ (nv=1)) \/ ((nu=-1)/\(nv=0)))%Z. assert (nu <= nv)%Z. `````` BOLDO Sylvie committed Nov 20, 2014 1336 1337 1338 ``````apply le_Z2R. apply Rmult_le_reg_r with (bpow emin). apply bpow_gt_0. `````` BOLDO Sylvie committed Jul 29, 2015 1339 ``````apply Rle_trans with u. `````` BOLDO Sylvie committed Nov 20, 2014 1340 ``````right; rewrite J1,J2; reflexivity. `````` BOLDO Sylvie committed Jul 29, 2015 1341 ``````apply Rle_trans with (1:=uLev). `````` BOLDO Sylvie committed Nov 20, 2014 1342 1343 ``````right; rewrite J3,J4; reflexivity. case same_sign; intros (L1,L2). `````` BOLDO Sylvie committed Jun 05, 2015 1344 1345 ``````rewrite J1 in L1; apply Fnum_ge_0_compat in L1; simpl in L1. rewrite J3 in L2; apply Fnum_ge_0_compat in L2; simpl in L2. `````` BOLDO Sylvie committed Nov 20, 2014 1346 1347 1348 1349 ``````left. rewrite Z.abs_eq in H0. omega. omega. `````` BOLDO Sylvie committed Jun 05, 2015 1350 1351 ``````rewrite J1 in L1; apply Fnum_le_0_compat in L1; simpl in L1. rewrite J3 in L2; apply Fnum_le_0_compat in L2; simpl in L2. `````` BOLDO Sylvie committed Nov 20, 2014 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 ``````right. rewrite Z.abs_neq in H0. omega. omega. (* look into the 2 possible cases *) assert (G1:(round_flt (bpow emin/2) = 0)). replace (bpow emin /2) with (bpow (emin-1)). unfold round, scaled_mantissa. rewrite canonic_exp_FLT_FIX. unfold canonic_exp, FIX_exp; simpl. rewrite <- bpow_plus. replace (bpow (emin - 1 + - emin)) with (/2). replace (ZnearestE (/ 2)) with 0%Z. unfold F2R; simpl; ring. unfold Znearest. replace (Zfloor (/2)) with 0%Z. rewrite Rcompare_Eq. reflexivity. simpl; ring. apply sym_eq, Zfloor_imp. simpl; split. auto with real. apply Rmult_lt_reg_l with 2. auto with real. apply Rle_lt_trans with 1. right; field. rewrite Rmult_1_r. auto with real. ring_simplify (emin-1+-emin)%Z; reflexivity. apply Rgt_not_eq, Rlt_gt, bpow_gt_0. rewrite Rabs_right. apply bpow_lt. unfold Prec_gt_0 in prec_gt_0_; omega. apply Rle_ge, bpow_ge_0. unfold Zminus; rewrite bpow_plus. reflexivity. case H3; intros (T1,T2). `````` BOLDO Sylvie committed Jul 29, 2015 1389 ``````unfold b, average3. `````` BOLDO Sylvie committed Nov 20, 2014 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 ``````rewrite J1,J3,J2,J4,T1,T2; unfold F2R; simpl. rewrite Rmult_0_l, Rmult_1_l, 2!Rplus_0_l. unfold Rminus; rewrite Ropp_0, Rplus_0_r. rewrite (round_generic _ _ _ (bpow (emin)))... 2: apply FLT_format_bpow... 2: omega. rewrite G1. rewrite round_0... rewrite Rplus_0_l, Rabs_Ropp. rewrite Rabs_right. 2: apply Rle_ge, Rmult_le_pos. 2: apply bpow_ge_0. 2: now auto with real. apply Rle_trans with ((3*ulp_flt (bpow emin / 2))/2);[idtac|right; unfold Rdiv; ring]. unfold Rdiv; apply Rmult_le_compat_r. now auto with real. apply Rle_trans with (3*bpow emin). apply Rle_trans with (1*bpow emin). right; ring. apply Rmult_le_compat_r. apply bpow_ge_0. apply Rplus_le_reg_l with (-1); ring_simplify. now auto with real. apply Rmult_le_compat_l. apply Fourier_util.Rle_zero_pos_plus1. now auto with real. `````` BOLDO Sylvie committed Jul 29, 2015 1416 1417 1418 1419 1420 ``````rewrite ulp_neq_0. 2: apply Rmult_integral_contrapositive_currified. 2: apply Rgt_not_eq, bpow_gt_0. 2: apply Rinv_neq_0_compat, Rgt_not_eq; fourier. apply bpow_le. `````` BOLDO Sylvie committed Nov 20, 2014 1421 1422 ``````unfold canonic_exp, FLT_exp. apply Z.le_max_r. `````` BOLDO Sylvie committed Jul 29, 2015 1423 ``````unfold b, average3. `````` BOLDO Sylvie committed Nov 20, 2014 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 ``````rewrite J1,J3,J2,J4,T1,T2; unfold F2R; simpl. rewrite Rmult_0_l, Rplus_0_r. replace (0 - -1 * bpow emin) with (bpow emin) by ring. rewrite (round_generic _ _ _ (bpow (emin)))... 2: apply FLT_format_bpow... 2: omega. rewrite G1. replace (-1 * bpow emin + 0) with (-bpow emin) by ring. rewrite round_generic... 2: apply generic_format_opp. 2: apply FLT_format_bpow... 2: omega. replace (- bpow emin - -1 * bpow emin / 2) with (-((bpow emin)/2)) by field. rewrite Rabs_Ropp. rewrite Rabs_right. replace (-1 * bpow emin / 2) with (-((bpow emin/2))) by field. rewrite ulp_opp. apply Rle_trans with ((3*ulp_flt (bpow emin / 2))/2);[idtac|right; unfold Rdiv; ring]. unfold Rdiv; apply Rmult_le_compat_r. now auto with real. apply Rle_trans with (3*bpow emin). apply Rle_trans with (1*bpow emin). right; ring. apply Rmult_le_compat_r. apply bpow_ge_0. apply Rplus_le_reg_l with (-1); ring_simplify. now auto with real. apply Rmult_le_compat_l. apply Fourier_util.Rle_zero_pos_plus1. now auto with real. `````` BOLDO Sylvie committed Jul 29, 2015 1454 1455 1456 1457 1458 ``````rewrite ulp_neq_0. 2: apply Rmult_integral_contrapositive_currified. 2: apply Rgt_not_eq, bpow_gt_0. 2: apply Rinv_neq_0_compat, Rgt_not_eq; fourier. apply bpow_le. `````` BOLDO Sylvie committed Nov 20, 2014 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 ``````unfold canonic_exp, FLT_exp. apply Z.le_max_r. apply Rle_ge, Rmult_le_pos. apply bpow_ge_0. now auto with real. Qed. Lemma average3_correct_aux2: forall u v, format u -> format v -> u <= v -> (0 <= u /\ 0 <= v) \/ (u <= 0 /\ v <= 0) -> Rabs (average3 u v -((u+v)/2)) <= 3/2 * ulp_flt ((u+v)/2). Proof with auto with typeclass_instances. clear Fx Fy a av x y. `````` BOLDO Sylvie committed Jul 29, 2015 1473 1474 ``````intros u v Fu Fv uLev same_sign. pose (b:=(u+v)/2); fold b. `````` BOLDO Sylvie committed Nov 20, 2014 1475 1476 1477 1478 ``````assert (T: forall z, Rabs (2*z) = 2* Rabs z). intros z; rewrite Rabs_mult. rewrite Rabs_right; try reflexivity. apply Rle_ge; now auto with real. `````` BOLDO Sylvie committed Jul 29, 2015 1479 1480 1481 1482 ``````destruct uLev as [uLtv|uEqv]. (* when u < v *) assert (B: u <= v) by now left. assert (K1: b <> 0). `````` BOLDO Sylvie committed Nov 20, 2014 1483 1484 1485 ``````apply Rmult_integral_contrapositive_currified. 2: apply Rgt_not_eq, Rlt_gt; now auto with real. intros L; case same_sign; intros (L1,L2). `````` BOLDO Sylvie committed Jul 29, 2015 1486 ``````absurd (0 <= u); try assumption. `````` BOLDO Sylvie committed Nov 20, 2014 1487 ``````apply Rlt_not_le. `````` BOLDO Sylvie committed Jul 29, 2015 1488 1489 ``````apply Rlt_le_trans with v; try assumption. apply Rplus_le_reg_l with u. `````` BOLDO Sylvie committed Nov 20, 2014 1490 ``````rewrite L, Rplus_0_r; assumption. `````` BOLDO Sylvie committed Jul 29, 2015 1491 ``````absurd (v <= 0); try assumption. `````` BOLDO Sylvie committed Nov 20, 2014 1492 ``````apply Rlt_not_le. `````` BOLDO Sylvie committed Jul 29, 2015 1493 1494 ``````apply Rle_lt_trans with u; try assumption. apply Rplus_le_reg_r with v. `````` BOLDO Sylvie committed Nov 20, 2014 1495 1496 ``````rewrite L, Rplus_0_l; assumption. (* . initial lemma *) `````` BOLDO Sylvie committed Jul 29, 2015 1497 1498 ``````assert (Y:(Rabs (round_flt (v - u) - (v-u)) <= ulp_flt b)). apply Rle_trans with (/2*ulp_flt (v-u)). `````` BOLDO Sylvie committed Sep 08, 2015 1499 ``````apply error_le_half_ulp... `````` BOLDO Sylvie committed Nov 20, 2014 1500 1501 1502 1503 ``````apply Rmult_le_reg_l with 2. now auto with real. rewrite <- Rmult_assoc, Rinv_r, Rmult_1_l. 2: apply Rgt_not_eq, Rlt_gt; now auto with real. `````` BOLDO Sylvie committed Jul 29, 2015 1504 ``````apply Rle_trans with (ulp_flt (2*b)). `````` BOLDO Sylvie committed Nov 20, 2014 1505 ``````case same_sign; intros (T1,T2). `````` BOLDO Sylvie committed Jul 29, 2015 1506 1507 1508 1509 ``````apply ulp_le_pos... apply Rplus_le_reg_l with u; ring_simplify; assumption. apply Rle_trans with (2*(b-u)). right; unfold b; field. `````` BOLDO Sylvie committed Nov 20, 2014 1510 1511 ``````apply Rmult_le_compat_l. now auto with real. `````` BOLDO Sylvie committed Jul 29, 2015 1512 1513 1514 1515 1516 1517 1518 ``````apply Rplus_le_reg_l with (-b+u); ring_simplify; assumption. rewrite <- (ulp_opp _ _ (2*b)). apply ulp_le_pos... apply Rplus_le_reg_l with u; ring_simplify; assumption. apply Rle_trans with (2*(v-b)). right; unfold b; field. apply Rle_trans with (2*(-b));[idtac|right; ring]. `````` BOLDO Sylvie committed Nov 20, 2014 1519 1520 ``````apply Rmult_le_compat_l. now auto with real. `````` BOLDO Sylvie committed Jul 29, 2015 1521 1522 1523 1524 ``````apply Rplus_le_reg_l with b; ring_simplify; assumption. rewrite 2!ulp_neq_0; trivial. 2: apply Rmult_integral_contrapositive_currified; trivial. 2: apply Rgt_not_eq; fourier. `````` BOLDO Sylvie committed Nov 20, 2014 1525 1526 1527 1528 1529 1530 ``````replace 2 with (bpow 1) by reflexivity. rewrite <- bpow_plus. apply bpow_le. unfold canonic_exp, FLT_exp. rewrite Rmult_comm, ln_beta_mult_bpow; trivial. rewrite <- Z.add_max_distr_l. `````` BOLDO Sylvie committed Jul 29, 2015 1531 ``````replace (ln_beta radix2 b + 1 - prec)%Z with (1 + (ln_beta radix2 b - prec))%Z by ring. `````` BOLDO Sylvie committed Nov 20, 2014 1532 1533 1534 ``````apply Z.max_le_compat_l. omega. (* . splitting case of av=0 *) `````` BOLDO Sylvie committed Jul 29, 2015 1535 ``````case (Rle_or_lt (bpow emin) (Rabs b)); intros D. `````` BOLDO Sylvie committed Nov 20, 2014 1536 1537 ``````(* . main proof *) unfold average3. `````` BOLDO Sylvie committed Jul 29, 2015 1538 1539 1540 ``````case (Rle_or_lt (bpow (prec+emin)) (v-u)); intros H1. (* .. v-u is big enough: division by 2 is exact *) cut (round_flt (round_flt (v - u) / 2) = round_flt (v - u) / 2). `````` BOLDO Sylvie committed Nov 20, 2014 1541 ``````intros Z; rewrite Z. `````` BOLDO Sylvie committed Jul 29, 2015 1542 1543 1544 ``````replace (round_flt (u + round_flt (v - u) / 2) - b) with ((round_flt (u + round_flt (v - u) / 2) - (u + round_flt (v - u) / 2)) +/2*(round_flt (v - u)-(v-u))). 2: unfold b; field. `````` BOLDO Sylvie committed Nov 20, 2014 1545 ``````apply Rle_trans with (1:=Rabs_triang _ _). `````` BOLDO Sylvie committed Jul 29, 2015 1546 ``````apply Rle_trans with (ulp_flt b+/2*ulp_flt b);[idtac|right; field]. `````` BOLDO Sylvie committed Nov 20, 2014 1547 ``````apply Rplus_le_compat. `````` BOLDO Sylvie committed Jul 29, 2015 1548 ``````apply Rle_trans with (/2*ulp_flt (u + round_flt (v - u) / 2)). `````` BOLDO Sylvie committed Sep 08, 2015 1549 ``````apply error_le_half_ulp... `````` BOLDO Sylvie committed Nov 20, 2014 1550 1551 1552 1553 1554 1555 ``````apply Rmult_le_reg_l with 2. auto with real. rewrite <- Rmult_assoc, Rinv_r, Rmult_1_l. 2: apply Rgt_not_eq, Rlt_gt; now auto with real. apply Rle_trans with (2:=FLT_ulp_double _ _ _). apply ulp_le... `````` BOLDO Sylvie committed Jul 29, 2015 1556 1557 1558 ``````replace (u + round_flt (v - u) / 2) with (b+/2*(round_flt (v - u) - (v - u))). 2: unfold b; field. rewrite (T b). `````` BOLDO Sylvie committed Nov 20, 2014 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 ``````rewrite Rmult_plus_distr_r, Rmult_1_l. apply Rle_trans with (1:=Rabs_triang _ _). apply Rplus_le_compat_l. rewrite Rabs_mult. rewrite Rabs_right. 2: apply Rle_ge; now auto with real. apply Rmult_le_reg_l with 2. now auto with real. rewrite <- Rmult_assoc, Rinv_r, Rmult_1_l. 2: apply Rgt_not_eq, Rlt_gt; now auto with real. apply Rle_trans with (1:=Y). `````` BOLDO Sylvie committed Jul 29, 2015 1570 ``````apply Rle_trans with (ulp_flt (2*b)). `````` BOLDO Sylvie committed Nov 20, 2014 1571 ``````apply ulp_le... `````` BOLDO Sylvie committed Jul 29, 2015 1572 1573 ``````rewrite <- (Rmult_1_l (Rabs b)). rewrite (T b). `````` BOLDO Sylvie committed Nov 20, 2014 1574 1575 1576 ``````apply Rmult_le_compat_r. apply Rabs_pos. now auto with real. `````` BOLDO Sylvie committed Jul 29, 2015 1577 ``````rewrite <- (T b). `````` BOLDO Sylvie committed Nov 20, 2014 1578 1579 ``````rewrite <- ulp_abs. apply FLT_ulp_le_id... `````` BOLDO Sylvie committed Jul 29, 2015 1580 1581 1582 1583 1584 1585 1586 ``````assert (H:generic_format radix2 (FIX_exp emin) (2*b)). replace (2*b) with (u+v). 2: unfold b; field. apply generic_format_FIX_FLT,FIX_format_generic in Fu. apply generic_format_FIX_FLT,FIX_format_generic in Fv. destruct Fu as (fu,(J1,J2)). destruct Fv as (fv,(J3,J4)). `````` BOLDO Sylvie committed Nov 20, 2014 1587 ``````apply generic_format_FIX. `````` BOLDO Sylvie committed Jul 29, 2015 1588 ``````exists (Float radix2 (Fnum fu+Fnum fv) emin). `````` BOLDO Sylvie committed Nov 20, 2014 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 ``````split;[idtac|reflexivity]. rewrite J1,J3; unfold F2R; simpl. rewrite J2,J4, Z2R_plus; ring. apply FIX_format_generic in H. destruct H as ((n,e),(J1,J2)). rewrite J1; unfold F2R; rewrite J2. simpl; rewrite Rabs_mult. pattern (bpow emin) at 1; rewrite <- (Rmult_1_l (bpow emin)). rewrite (Rabs_right (bpow emin)). 2: apply Rle_ge, bpow_ge_0. apply Rmult_le_compat_r. apply bpow_ge_0. rewrite <- Z2R_abs. replace 1 with (Z2R 1) by reflexivity. apply Z2R_le. assert (0 < Z.abs n)%Z;[idtac|omega]. apply Z.abs_pos. intros M; apply K1. apply Rmult_eq_reg_l with 2. 2: apply Rgt_not_eq, Rlt_gt; now auto with real. rewrite Rmult_0_r, J1,M; unfold F2R; simpl; ring. rewrite Rabs_mult. rewrite Rabs_right. 2: apply Rle_ge; auto with real. apply Rmult_le_compat_l. now auto with real. exact Y. apply round_generic... apply FLT_format_half... apply generic_format_round... apply abs_round_ge_generic... apply FLT_format_bpow... unfold Prec_gt_0 in prec_gt_0_; omega. rewrite Rabs_right; try assumption. `````` BOLDO Sylvie committed Jul 29, 2015 1623 1624 1625 ``````apply Rle_ge; left; apply Rplus_lt_reg_l with u; now ring_simplify. (* .. v-u is small: subtraction is exact *) cut ((round_flt (v - u)= (v-u))). `````` BOLDO Sylvie committed Nov 20, 2014 1626 ``````intros Z; rewrite Z. `````` BOLDO Sylvie committed Jul 29, 2015 1627 1628 1629 ``````replace (u + round_flt ((v-u) / 2)) with (b+((round_flt ((v-u) / 2) - (v-u)/2))). 2: unfold b; field. pose (eps:=(round_flt ((v - u) / 2) - (v - u) / 2)%R); fold eps. `````` BOLDO Sylvie committed Nov 20, 2014 1630 1631 ``````assert (Rabs eps <= /2*bpow emin). unfold eps. `````` BOLDO Sylvie committed Sep 08, 2015 1632 ``````apply Rle_trans with (1:=error_le_half_ulp _ _ _ _)... `````` BOLDO Sylvie committed Nov 20, 2014 1633 ``````right; apply f_equal. `````` BOLDO Sylvie committed Jul 29, 2015 1634 ``````apply ulp_FLT_small... `````` BOLDO Sylvie committed Nov 20, 2014 1635 1636 1637 1638 ``````rewrite Zplus_comm; apply Rle_lt_trans with (2:=H1). rewrite Rabs_right. apply Rmult_le_reg_l with 2. now auto with real. `````` BOLDO Sylvie committed Jul 29, 2015 1639 1640 ``````apply Rplus_le_reg_l with (-v+2*u). apply Rle_trans with u. `````` BOLDO Sylvie committed Nov 20, 2014 1641 1642 1643 ``````right; field. left; now ring_simplify. apply Rle_ge, Rmult_le_pos. `````` BOLDO Sylvie committed Jul 29, 2015 1644 ``````apply Rplus_le_reg_l with u; now ring_simplify. `````` BOLDO Sylvie committed Nov 20, 2014 1645 ``````now auto with real. `````` BOLDO Sylvie committed Jul 29, 2015 1646 ``````replace (round_flt (b + eps) - b) with ((round_flt (b+eps) -(b+eps)) + eps) by ring. `````` BOLDO Sylvie committed Nov 20, 2014 1647 ``````apply Rle_trans with (1:=Rabs_triang _ _). `````` BOLDO Sylvie committed Jul 29, 2015 1648 ``````apply Rle_trans with (/2*ulp_flt (b+eps) + /2*bpow emin). `````` BOLDO Sylvie committed Nov 20, 2014 1649 ``````apply Rplus_le_compat. `````` BOLDO Sylvie committed Sep 08, 2015 1650 ``````apply error_le_half_ulp... ``````