ErrorBounds.v 17.8 KB
 Heiko Becker committed Aug 18, 2016 1 ``````(** `````` Heiko Becker committed Sep 18, 2016 2 3 ``````Proofs of general bounds on the error of arithmetic expressions. This shortens soundness proofs later. `````` Heiko Becker committed Oct 06, 2016 4 ``````Bounds are explained in section 5, Deriving Computable Error Bounds `````` Heiko Becker committed Aug 18, 2016 5 ``````**) `````` Heiko Becker committed Nov 08, 2016 6 ``````Require Import Coq.Reals.Reals Coq.micromega.Psatz Coq.QArith.QArith Coq.QArith.Qreals. `````` Heiko Becker committed Nov 17, 2016 7 ``````Require Import Daisy.Infra.Abbrevs Daisy.Infra.RationalSimps Daisy.Infra.RealSimps Daisy.Infra.RealRationalProps Daisy.Expressions. `````` Heiko Becker committed Feb 01, 2017 8 ``````Require Import Daisy.Environments. `````` Heiko Becker committed Sep 18, 2016 9 `````` `````` Heiko Becker committed Feb 01, 2017 10 11 12 ``````Lemma const_abs_err_bounded (P:precond) (n:R) (nR:R) (nF:R) (VarEnv ParamEnv:env): eval_exp 0%R VarEnv ParamEnv P (Const n) nR -> eval_exp (Q2R machineEpsilon) VarEnv ParamEnv P (Const n) nF -> `````` Heiko Becker committed Nov 17, 2016 13 `````` (Rabs (nR - nF) <= Rabs n * (Q2R machineEpsilon))%R. `````` Heiko Becker committed Aug 18, 2016 14 ``````Proof. `````` Heiko Becker committed Feb 01, 2017 15 `````` intros eval_real eval_float. `````` Heiko Becker committed Aug 18, 2016 16 `````` inversion eval_real; subst. `````` Heiko Becker committed Oct 13, 2016 17 `````` rewrite delta_0_deterministic; auto. `````` Heiko Becker committed Aug 18, 2016 18 19 `````` inversion eval_float; subst. unfold perturb; simpl. `````` Heiko Becker committed Jan 09, 2017 20 `````` rewrite Rabs_err_simpl, Rabs_mult. `````` Heiko Becker committed Aug 18, 2016 21 22 23 `````` apply Rmult_le_compat_l; [apply Rabs_pos | auto]. Qed. `````` Heiko Becker committed Feb 01, 2017 24 25 26 27 ``````Lemma param_abs_err_bounded (P:precond) (n:nat) (nR:R) (nF:R) (VarEnv ParamEnv:env): eval_exp 0%R VarEnv ParamEnv P (Param R n) nR -> eval_exp (Q2R machineEpsilon) VarEnv ParamEnv P (Param R n) nF -> (Rabs (nR - nF) <= Rabs (ParamEnv n) * (Q2R machineEpsilon))%R. `````` Heiko Becker committed Aug 18, 2016 28 29 30 ``````Proof. intros eval_real eval_float. inversion eval_real; subst. `````` Heiko Becker committed Oct 13, 2016 31 `````` rewrite delta_0_deterministic; auto. `````` Heiko Becker committed Aug 18, 2016 32 33 34 35 36 37 38 `````` inversion eval_float; subst. unfold perturb; simpl. rewrite Rabs_err_simpl. repeat rewrite Rabs_mult. apply Rmult_le_compat_l; [ apply Rabs_pos | auto]. Qed. `````` Heiko Becker committed Feb 01, 2017 39 40 41 42 43 44 45 46 47 48 49 50 ``````Lemma add_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) (e2F:R) (vR:R) (vF:R) (VarEnv1 VarEnv2 ParamEnv:env) (P:precond) absenv: approxEnv VarEnv1 absenv VarEnv2 -> eval_exp 0%R VarEnv1 ParamEnv P (toRExp e1) e1R -> eval_exp (Q2R machineEpsilon) VarEnv2 ParamEnv P (toRExp e1) e1F -> eval_exp 0%R VarEnv1 ParamEnv P (toRExp e2) e2R -> eval_exp (Q2R machineEpsilon) VarEnv2 ParamEnv P (toRExp e2) e2F -> eval_exp 0%R VarEnv1 ParamEnv P (Binop Plus (toRExp e1) (toRExp e2)) vR -> eval_exp (Q2R machineEpsilon) (updEnv 2 e2F (updEnv 1 e1F VarEnv2)) ParamEnv P (Binop Plus (Var R 1) (Var R 2)) vF -> (Rabs (e1R - e1F) <= Q2R (snd (absenv e1)))%R -> (Rabs (e2R - e2F) <= Q2R (snd (absenv e2)))%R -> (Rabs (vR - vF) <= Q2R (snd(absenv e1)) + Q2R (snd (absenv e2)) + (Rabs (e1F + e2F) * (Q2R machineEpsilon)))%R. `````` Heiko Becker committed Aug 18, 2016 51 ``````Proof. `````` Heiko Becker committed Feb 01, 2017 52 `````` intros approxCEnv e1_real e1_float e2_real e2_float plus_real plus_float bound_e1 bound_e2. `````` Heiko Becker committed Aug 18, 2016 53 54 `````` (* Prove that e1R and e2R are the correct values and that vR is e1R + e2R *) inversion plus_real; subst. `````` Heiko Becker committed Oct 13, 2016 55 `````` rewrite delta_0_deterministic in plus_real; auto. `````` Heiko Becker committed Oct 31, 2016 56 57 `````` rewrite (delta_0_deterministic (evalBinop Plus v1 v2) delta); auto. unfold evalBinop in *; simpl in *. `````` Heiko Becker committed Aug 18, 2016 58 `````` clear delta H2. `````` Heiko Becker committed Oct 13, 2016 59 60 61 62 `````` rewrite (meps_0_deterministic H4 e1_real); rewrite (meps_0_deterministic H5 e2_real). rewrite (meps_0_deterministic H4 e1_real) in plus_real. rewrite (meps_0_deterministic H5 e2_real) in plus_real. `````` Heiko Becker committed Aug 18, 2016 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 98 99 `````` clear H4 H5 v1 v2. (* Now unfold the float valued evaluation to get the deltas we need for the inequality *) inversion plus_float; subst. unfold perturb; simpl. inversion H4; subst; inversion H5; subst. unfold updEnv; simpl. (* We have now obtained all necessary values from the evaluations --> remove them for readability *) clear plus_float H4 H5 plus_real e1_real e1_float e2_real e2_float. repeat rewrite Rmult_plus_distr_l. rewrite Rmult_1_r. rewrite Rsub_eq_Ropp_Rplus. repeat rewrite Ropp_plus_distr. rewrite plus_bounds_simplify. pose proof (Rabs_triang (e1R + - e1F) ((e2R + - e2F) + - ((e1F + e2F) * delta))). rewrite Rplus_assoc. eapply Rle_trans. apply H. pose proof (Rabs_triang (e2R + - e2F) (- ((e1F + e2F) * delta))). pose proof (Rplus_le_compat_l (Rabs (e1R + - e1F)) _ _ H0). eapply Rle_trans. apply H1. rewrite <- Rplus_assoc. repeat rewrite <- Rsub_eq_Ropp_Rplus. rewrite Rabs_Ropp. eapply Rplus_le_compat. - eapply Rplus_le_compat; auto. - rewrite Rabs_mult. eapply Rle_trans. eapply Rmult_le_compat_l. apply Rabs_pos. apply H2. apply Req_le; auto. Qed. (** Copy-Paste proof with minor differences, was easier then manipulating the evaluations and then applying the lemma **) `````` Heiko Becker committed Feb 03, 2017 100 101 102 103 104 105 106 107 108 109 110 111 ``````Lemma subtract_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) (e2F:R) (vR:R) (vF:R) (VarEnv1 VarEnv2 ParamEnv:nat->R) P absenv: approxEnv VarEnv1 absenv VarEnv2 -> eval_exp 0%R VarEnv1 ParamEnv P (toRExp e1) e1R -> eval_exp (Q2R machineEpsilon) VarEnv2 ParamEnv P (toRExp e1) e1F -> eval_exp 0%R VarEnv1 ParamEnv P (toRExp e2) e2R -> eval_exp (Q2R machineEpsilon) VarEnv2 ParamEnv P (toRExp e2) e2F -> eval_exp 0%R VarEnv1 ParamEnv P (Binop Sub (toRExp e1) (toRExp e2)) vR -> eval_exp (Q2R machineEpsilon) (updEnv 2 e2F (updEnv 1 e1F VarEnv2)) ParamEnv P (Binop Sub (Var R 1) (Var R 2)) vF -> (Rabs (e1R - e1F) <= Q2R (snd (absenv e1)))%R -> (Rabs (e2R - e2F) <= Q2R (snd (absenv e2)))%R -> (Rabs (vR - vF) <= Q2R (snd (absenv e1)) + Q2R (snd (absenv e2)) + ((Rabs (e1F - e2F)) * (Q2R machineEpsilon)))%R. `````` Heiko Becker committed Aug 18, 2016 112 ``````Proof. `````` Heiko Becker committed Feb 03, 2017 113 `````` intros approxCEnv e1_real e1_float e2_real e2_float sub_real sub_float bound_e1 bound_e2. `````` Heiko Becker committed Aug 18, 2016 114 115 `````` (* Prove that e1R and e2R are the correct values and that vR is e1R + e2R *) inversion sub_real; subst. `````` Heiko Becker committed Oct 13, 2016 116 117 `````` rewrite delta_0_deterministic in sub_real; auto. rewrite delta_0_deterministic; auto. `````` Heiko Becker committed Oct 31, 2016 118 `````` unfold evalBinop in *; simpl in *. `````` Heiko Becker committed Aug 18, 2016 119 `````` clear delta H2. `````` Heiko Becker committed Oct 13, 2016 120 121 122 123 `````` rewrite (meps_0_deterministic H4 e1_real); rewrite (meps_0_deterministic H5 e2_real). rewrite (meps_0_deterministic H4 e1_real) in sub_real. rewrite (meps_0_deterministic H5 e2_real) in sub_real. `````` Heiko Becker committed Aug 18, 2016 124 125 126 127 128 129 130 131 132 133 134 135 136 `````` clear H4 H5 v1 v2. (* Now unfold the float valued evaluation to get the deltas we need for the inequality *) inversion sub_float; subst. unfold perturb; simpl. inversion H4; subst; inversion H5; subst. unfold updEnv; simpl. (* We have now obtained all necessary values from the evaluations --> remove them for readability *) clear sub_float H4 H5 sub_real e1_real e1_float e2_real e2_float. repeat rewrite Rmult_plus_distr_l. rewrite Rmult_1_r. repeat rewrite Rsub_eq_Ropp_Rplus. repeat rewrite Ropp_plus_distr. rewrite plus_bounds_simplify. `````` Heiko Becker committed Sep 21, 2016 137 `````` rewrite Ropp_involutive. `````` Heiko Becker committed Aug 18, 2016 138 139 `````` rewrite Rplus_assoc. eapply Rle_trans. `````` Heiko Becker committed Sep 21, 2016 140 `````` apply Rabs_triang. `````` Heiko Becker committed Aug 18, 2016 141 `````` eapply Rle_trans. `````` Heiko Becker committed Sep 21, 2016 142 143 `````` eapply Rplus_le_compat_l. apply Rabs_triang. `````` Heiko Becker committed Aug 18, 2016 144 `````` rewrite <- Rplus_assoc. `````` Heiko Becker committed Sep 21, 2016 145 `````` setoid_rewrite Rplus_comm at 4. `````` Heiko Becker committed Aug 18, 2016 146 147 `````` repeat rewrite <- Rsub_eq_Ropp_Rplus. rewrite Rabs_Ropp. `````` Heiko Becker committed Sep 21, 2016 148 149 150 151 `````` rewrite Rabs_minus_sym in bound_e2. apply Rplus_le_compat; [apply Rplus_le_compat; auto | ]. rewrite Rabs_mult. eapply Rmult_le_compat_l; [apply Rabs_pos | auto]. `````` Heiko Becker committed Aug 23, 2016 152 153 ``````Qed. `````` Heiko Becker committed Feb 03, 2017 154 155 156 157 158 159 160 161 162 ``````Lemma mult_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) (e2F:R) (vR:R) (vF:R) (VarEnv1 VarEnv2 ParamEnv:env) (P:precond) absenv: approxEnv VarEnv1 absenv VarEnv2 -> eval_exp 0%R VarEnv1 ParamEnv P (toRExp e1) e1R -> eval_exp (Q2R machineEpsilon) VarEnv2 ParamEnv P (toRExp e1) e1F -> eval_exp 0%R VarEnv1 ParamEnv P (toRExp e2) e2R -> eval_exp (Q2R machineEpsilon) VarEnv2 ParamEnv P (toRExp e2) e2F -> eval_exp 0%R VarEnv1 ParamEnv P (Binop Mult (toRExp e1) (toRExp e2)) vR -> eval_exp (Q2R machineEpsilon) (updEnv 2 e2F (updEnv 1 e1F VarEnv2)) ParamEnv P (Binop Mult (Var R 1) (Var R 2)) vF -> `````` Heiko Becker committed Nov 17, 2016 163 `````` (Rabs (vR - vF) <= Rabs (e1R * e2R - e1F * e2F) + Rabs (e1F * e2F) * (Q2R machineEpsilon))%R. `````` Heiko Becker committed Aug 23, 2016 164 ``````Proof. `````` Heiko Becker committed Feb 03, 2017 165 `````` intros approxCEnv e1_real e1_float e2_real e2_float mult_real mult_float. `````` Heiko Becker committed Oct 07, 2016 166 `````` (* Prove that e1R and e2R are the correct values and that vR is e1R * e2R *) `````` Heiko Becker committed Aug 23, 2016 167 `````` inversion mult_real; subst. `````` Heiko Becker committed Oct 13, 2016 168 169 `````` rewrite delta_0_deterministic in mult_real; auto. rewrite delta_0_deterministic; auto. `````` Heiko Becker committed Oct 31, 2016 170 `````` unfold evalBinop in *; simpl in *. `````` Heiko Becker committed Aug 23, 2016 171 `````` clear delta H2. `````` Heiko Becker committed Oct 13, 2016 172 173 174 175 `````` rewrite (meps_0_deterministic H4 e1_real); rewrite (meps_0_deterministic H5 e2_real). rewrite (meps_0_deterministic H4 e1_real) in mult_real. rewrite (meps_0_deterministic H5 e2_real) in mult_real. `````` Heiko Becker committed Aug 23, 2016 176 177 178 179 180 181 182 183 184 185 186 187 `````` clear H4 H5 v1 v2. (* Now unfold the float valued evaluation to get the deltas we need for the inequality *) inversion mult_float; subst. unfold perturb; simpl. inversion H4; subst; inversion H5; subst. unfold updEnv; simpl. (* We have now obtained all necessary values from the evaluations --> remove them for readability *) clear mult_float H4 H5 mult_real e1_real e1_float e2_real e2_float. repeat rewrite Rmult_plus_distr_l. rewrite Rmult_1_r. rewrite Rsub_eq_Ropp_Rplus. rewrite Ropp_plus_distr. `````` Heiko Becker committed Aug 24, 2016 188 189 `````` rewrite <- Rplus_assoc. setoid_rewrite <- Rsub_eq_Ropp_Rplus at 2. `````` Heiko Becker committed Aug 23, 2016 190 191 192 `````` eapply Rle_trans. eapply Rabs_triang. eapply Rplus_le_compat_l. `````` Heiko Becker committed Aug 24, 2016 193 `````` rewrite Rabs_Ropp. `````` Heiko Becker committed Aug 23, 2016 194 `````` repeat rewrite Rabs_mult. `````` Heiko Becker committed Aug 24, 2016 195 `````` eapply Rmult_le_compat_l; auto. `````` Heiko Becker committed Aug 23, 2016 196 197 `````` rewrite <- Rabs_mult. apply Rabs_pos. `````` Heiko Becker committed Oct 07, 2016 198 199 ``````Qed. `````` Heiko Becker committed Feb 03, 2017 200 201 202 203 204 205 206 207 208 ``````Lemma div_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) (e2F:R) (vR:R) (vF:R) (VarEnv1 VarEnv2 ParamEnv:env) (P:precond) absenv: approxEnv VarEnv1 absenv VarEnv2 -> eval_exp 0%R VarEnv1 ParamEnv P (toRExp e1) e1R -> eval_exp (Q2R machineEpsilon) VarEnv2 ParamEnv P (toRExp e1) e1F -> eval_exp 0%R VarEnv1 ParamEnv P (toRExp e2) e2R -> eval_exp (Q2R machineEpsilon) VarEnv2 ParamEnv P (toRExp e2) e2F -> eval_exp 0%R VarEnv1 ParamEnv P (Binop Div (toRExp e1) (toRExp e2)) vR -> eval_exp (Q2R machineEpsilon) (updEnv 2 e2F (updEnv 1 e1F VarEnv2)) ParamEnv P (Binop Div (Var R 1) (Var R 2)) vF -> `````` Heiko Becker committed Nov 17, 2016 209 `````` (Rabs (vR - vF) <= Rabs (e1R / e2R - e1F / e2F) + Rabs (e1F / e2F) * (Q2R machineEpsilon))%R. `````` Heiko Becker committed Oct 07, 2016 210 ``````Proof. `````` Heiko Becker committed Feb 03, 2017 211 `````` intros approxCenv e1_real e1_float e2_real e2_float div_real div_float. `````` Heiko Becker committed Oct 07, 2016 212 213 `````` (* Prove that e1R and e2R are the correct values and that vR is e1R * e2R *) inversion div_real; subst. `````` Heiko Becker committed Oct 13, 2016 214 215 `````` rewrite delta_0_deterministic in div_real; auto. rewrite delta_0_deterministic; auto. `````` Heiko Becker committed Oct 31, 2016 216 `````` unfold evalBinop in *; simpl in *. `````` Heiko Becker committed Oct 07, 2016 217 `````` clear delta H2. `````` Heiko Becker committed Oct 13, 2016 218 219 220 221 `````` rewrite (meps_0_deterministic H4 e1_real); rewrite (meps_0_deterministic H5 e2_real). rewrite (meps_0_deterministic H4 e1_real) in div_real. rewrite (meps_0_deterministic H5 e2_real) in div_real. `````` Heiko Becker committed Oct 07, 2016 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 `````` clear H4 H5 v1 v2. (* Now unfold the float valued evaluation to get the deltas we need for the inequality *) inversion div_float; subst. unfold perturb; simpl. inversion H4; subst; inversion H5; subst. unfold updEnv; simpl. (* We have now obtained all necessary values from the evaluations --> remove them for readability *) clear div_float H4 H5 div_real e1_real e1_float e2_real e2_float. repeat rewrite Rmult_plus_distr_l. rewrite Rmult_1_r. rewrite Rsub_eq_Ropp_Rplus. rewrite Ropp_plus_distr. rewrite <- Rplus_assoc. setoid_rewrite <- Rsub_eq_Ropp_Rplus at 2. eapply Rle_trans. eapply Rabs_triang. eapply Rplus_le_compat_l. rewrite Rabs_Ropp. repeat rewrite Rabs_mult. eapply Rmult_le_compat_l; auto. apply Rabs_pos. `````` Heiko Becker committed Nov 08, 2016 243 244 ``````Qed. `````` Heiko Becker committed Jan 13, 2017 245 246 247 248 249 250 251 252 ``````Lemma err_prop_inversion_pos_real nF nR err elo ehi (float_iv_pos : (0 < elo - err)%R) (real_iv_pos : (0 < elo)%R) (err_bounded : (Rabs (nR - nF) <= err)%R) (valid_bounds_e2 : (elo <= nR <= ehi)%R) (valid_bounds_e2_err : (elo - err <= nF <= ehi + err)%R) (err_pos : (0 <= err)%R): (Rabs (/nR - / nF) <= err * / ((elo - err) * (elo- err)))%R. `````` Heiko Becker committed Nov 08, 2016 253 ``````Proof. `````` Heiko Becker committed Jan 13, 2017 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 `````` rewrite Rabs_case_inverted in err_bounded. assert (0 < nF)%R as nF_pos by lra. destruct err_bounded as [ [diff_pos err_bounded] | [diff_neg err_bounded]]. - cut (0 < /nF - / nR)%R. + intros abs_neg. rewrite Rabs_left; try lra. rewrite Rsub_eq_Ropp_Rplus, Ropp_plus_distr, Ropp_involutive. rewrite Ropp_inv_permute; try lra. apply (Rle_trans _ (/ - nR + / (nR - err))). * apply Rplus_le_compat_l. apply Rinv_le_contravar; lra. * rewrite equal_naming_inv; try lra. assert (- nR + (nR - err) = - err)%R as simplify_up by lra. rewrite simplify_up. unfold Rdiv. repeat(rewrite <- Ropp_mult_distr_l); rewrite <- Ropp_inv_permute. { rewrite <- Ropp_mult_distr_r, Ropp_involutive. apply Rmult_le_compat_l; try lra. apply Rinv_le_contravar. - apply Rmult_0_lt_preserving; lra. - apply Rmult_le_compat; lra. } { assert (0 < nR * (nR - err))%R by (apply Rmult_0_lt_preserving; lra); lra. } + cut (/ nR < /nF)%R. * intros; lra. * apply Rinv_lt_contravar; try lra. apply Rmult_0_lt_preserving; lra. - cut (0 <= /nR - /nF)%R. + intros abs_pos. rewrite Rabs_right; try lra. rewrite Rsub_eq_Ropp_Rplus, Ropp_plus_distr, Ropp_involutive in err_bounded. rewrite Rsub_eq_Ropp_Rplus. apply (Rle_trans _ (/nR - / (nR + err))). * apply Rplus_le_compat_l. apply Ropp_le_contravar. apply Rinv_le_contravar; lra. * rewrite Rsub_eq_Ropp_Rplus, Ropp_inv_permute; try lra. rewrite equal_naming_inv; try lra. assert (nR + - (nR + err) = - err)%R as simpl_up by lra. rewrite simpl_up. unfold Rdiv. rewrite <- Ropp_mult_distr_l, <- Ropp_mult_distr_r, <- Ropp_inv_permute. { rewrite <- Ropp_mult_distr_r. rewrite Ropp_involutive. apply Rmult_le_compat_l; try auto. apply Rinv_le_contravar. - apply Rmult_0_lt_preserving; lra. - apply Rmult_le_compat; lra. } { assert (0 < nR * (nR + err))%R by (apply Rmult_0_lt_preserving; lra); lra. } + cut (/nF <= /nR)%R. * intros; lra. * apply Rinv_le_contravar; try lra. Qed. Lemma err_prop_inversion_pos nF nR err (elo ehi:Q) (float_iv_pos : (Q2R 0 < Q2R (elo - err))%R) (real_iv_pos : (Q2R 0 < Q2R elo)%R) (err_bounded : (Rabs (nR - nF) <= Q2R err)%R) (valid_bounds_e2 : (Q2R elo <= nR <= Q2R ehi)%R) (valid_bounds_e2_err : (Q2R elo - Q2R err <= nF <= Q2R ehi + Q2R err)%R) (err_pos : (0 <= Q2R err)%R): (Rabs (/nR - / nF) <= Q2R err * / ((Q2R elo- Q2R err) * (Q2R elo- Q2R err)))%R. Proof. eapply err_prop_inversion_pos_real; try rewrite <- Q2R0_is_0; eauto. rewrite <- Q2R_minus; auto. rewrite Q2R0_is_0; auto. Qed. Lemma err_prop_inversion_neg_real nF nR err elo ehi (float_iv_neg : (ehi + err < 0)%R) (real_iv_neg : (ehi < 0)%R) (err_bounded : (Rabs (nR - nF) <= err)%R) (valid_bounds_e : (elo <= nR <= ehi)%R) (valid_bounds_e_err : (elo - err <= nF <= ehi + err)%R) (err_pos : (0 <= err)%R): (Rabs (/nR - / nF) <= err * / ((ehi + err) * (ehi + err)))%R. Proof. rewrite Rabs_case_inverted in err_bounded. assert (nF < 0)%R as nF_neg by lra. destruct err_bounded as [ [diff_pos err_bounded] | [diff_neg err_bounded]]. - cut (0 < /nF - / nR)%R. + intros abs_neg. rewrite Rabs_left; try lra. rewrite Rsub_eq_Ropp_Rplus, Ropp_plus_distr, Ropp_involutive. rewrite Ropp_inv_permute; try lra. apply (Rle_trans _ (/ - nR + / (nR - err))). * apply Rplus_le_compat_l. assert (0 < - nF)%R by lra. assert (0 < - (nR - err))%R by lra. assert (nR - err <= nF)%R as nR_lower by lra. apply Ropp_le_contravar in nR_lower. apply Rinv_le_contravar in nR_lower; try lra. repeat (rewrite <- Ropp_inv_permute in nR_lower; try lra). * rewrite equal_naming_inv; try lra. assert (- nR + (nR - err) = - err)%R as simplify_up by lra. rewrite simplify_up. unfold Rdiv. repeat(rewrite <- Ropp_mult_distr_l); rewrite <- Ropp_inv_permute. { rewrite <- Ropp_mult_distr_r, Ropp_involutive. apply Rmult_le_compat_l; try lra. apply Rinv_le_contravar. - apply Rmult_lt_0_inverting; lra. - eapply Rle_trans. eapply Rmult_le_compat_neg_l; try lra. instantiate (1 := (nR - err)%R); try lra. setoid_rewrite Rmult_comm. eapply Rmult_le_compat_neg_l; lra. } { assert (0 < nR * (nR - err))%R by (apply Rmult_lt_0_inverting; lra); lra. } + cut (/ nR < /nF)%R. * intros; lra. * apply Rinv_lt_contravar; try lra. apply Rmult_lt_0_inverting; lra. - cut (0 <= /nR - /nF)%R. + intros abs_pos. rewrite Rabs_right; try lra. rewrite Rsub_eq_Ropp_Rplus, Ropp_plus_distr, Ropp_involutive in err_bounded. rewrite Rsub_eq_Ropp_Rplus. apply (Rle_trans _ (/nR - / (nR + err))). * apply Rplus_le_compat_l. apply Ropp_le_contravar. assert (0 < - nF)%R by lra. assert (0 < - (nR + err))%R by lra. assert (nF <= nR + err)%R as nR_upper by lra. apply Ropp_le_contravar in nR_upper. apply Rinv_le_contravar in nR_upper; try lra. repeat (rewrite <- Ropp_inv_permute in nR_upper; try lra). * rewrite Rsub_eq_Ropp_Rplus, Ropp_inv_permute; try lra. `````` Heiko Becker committed Nov 08, 2016 384 `````` rewrite equal_naming_inv; try lra. `````` Heiko Becker committed Jan 13, 2017 385 386 `````` assert (nR + - (nR + err) = - err)%R as simpl_up by lra. rewrite simpl_up. `````` Heiko Becker committed Nov 08, 2016 387 `````` unfold Rdiv. `````` Heiko Becker committed Jan 13, 2017 388 `````` rewrite <- Ropp_mult_distr_l, <- Ropp_mult_distr_r, <- Ropp_inv_permute. `````` Heiko Becker committed Nov 08, 2016 389 390 391 `````` { rewrite <- Ropp_mult_distr_r. rewrite Ropp_involutive. apply Rmult_le_compat_l; try auto. apply Rinv_le_contravar. `````` Heiko Becker committed Jan 13, 2017 392 393 394 395 396 397 398 399 400 401 402 403 404 405 `````` - apply Rmult_lt_0_inverting; lra. - eapply Rle_trans. eapply Rmult_le_compat_neg_l; try lra. instantiate (1:= (nR + err)%R); try lra. setoid_rewrite Rmult_comm. eapply Rmult_le_compat_neg_l; lra. } { assert (0 < nR * (nR + err))%R by (apply Rmult_lt_0_inverting; lra); lra. } + cut (/nF <= /nR)%R. * intros; lra. * assert (nR <= nF)%R by lra. assert (- nF <= - nR)%R as le_inv by lra. apply Rinv_le_contravar in le_inv; try lra. repeat (rewrite <- Ropp_inv_permute in le_inv; try lra). `````` Heiko Becker committed Nov 08, 2016 406 407 ``````Qed. `````` Heiko Becker committed Jan 13, 2017 408 409 410 411 412 413 414 415 ``````Lemma err_prop_inversion_neg nF nR err (elo ehi:Q) (float_iv_neg : (Q2R (ehi + err) < Q2R 0)%R) (real_iv_neg : (Q2R ehi < Q2R 0)%R) (err_bounded : (Rabs (nR - nF) <= Q2R err)%R) (valid_bounds_e : (Q2R elo <= nR <= Q2R ehi)%R) (valid_bounds_e_err : (Q2R elo - Q2R err <= nF <= Q2R ehi + Q2R err)%R) (err_pos : (0 <= Q2R err)%R): (Rabs (/nR - / nF) <= Q2R err * / ((Q2R ehi + Q2R err) * (Q2R ehi + Q2R err)))%R. `````` Heiko Becker committed Nov 08, 2016 416 ``````Proof. `````` Heiko Becker committed Jan 13, 2017 417 418 419 420 421 422 `````` eapply err_prop_inversion_neg_real; try rewrite <- Q2R0_is_0; try lra. rewrite <- Q2R_plus ; auto. apply valid_bounds_e. auto. rewrite Q2R0_is_0; auto. Qed.``````