ErrorBounds.v 20 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 Feb 07, 2017 7 8 ``````Require Import Daisy.Infra.Abbrevs Daisy.Infra.RationalSimps Daisy.Infra.RealSimps Daisy.Infra.RealRationalProps. Require Import Daisy.Environments Daisy.Infra.ExpressionAbbrevs. `````` Heiko Becker committed Sep 18, 2016 9 `````` `````` = committed Apr 25, 2017 10 ``````(* TODO: absenv not used *) `````` = committed Apr 04, 2017 11 ``````Lemma const_abs_err_bounded (n:R) (nR:R) (nF:R) (E1 E2:env) (absenv:analysisResult) (m:mType) defVars: `````` = committed Apr 05, 2017 12 `````` eval_exp E1 (toREvalVars defVars) (Const M0 n) nR M0 -> `````` = committed Apr 04, 2017 13 `````` eval_exp E2 defVars (Const m n) nF m -> `````` Raphaël Monat committed Mar 01, 2017 14 `````` (Rabs (nR - nF) <= Rabs n * (Q2R (meps m)))%R. `````` Heiko Becker committed Aug 18, 2016 15 ``````Proof. `````` Heiko Becker committed Feb 08, 2017 16 `````` intros eval_real eval_float. `````` Heiko Becker committed Aug 18, 2016 17 `````` inversion eval_real; subst. `````` Heiko Becker committed Oct 13, 2016 18 `````` rewrite delta_0_deterministic; auto. `````` Heiko Becker committed Aug 18, 2016 19 20 `````` inversion eval_float; subst. unfold perturb; simpl. `````` Heiko Becker committed Jan 09, 2017 21 `````` rewrite Rabs_err_simpl, Rabs_mult. `````` Heiko Becker committed Aug 18, 2016 22 `````` apply Rmult_le_compat_l; [apply Rabs_pos | auto]. `````` Raphaël Monat committed Mar 01, 2017 23 24 `````` simpl (meps M0) in *. apply (Rle_trans _ (Q2R 0) _); try auto. `````` Raphaël Monat committed Mar 03, 2017 25 `````` rewrite Q2R0_is_0; lra. `````` Heiko Becker committed Aug 18, 2016 26 27 ``````Qed. `````` Heiko Becker committed Feb 17, 2017 28 29 30 31 32 ``````(* Lemma param_abs_err_bounded (P:precond) (n:nat) (nR:R) (nF:R) (E1 E2:env) (absenv:analysisResult): eval_exp 0%R E1 P (Param R n) nR -> eval_exp (Q2R machineEpsilon) E2 P (Param R n) nF -> (Rabs (nR - nF) <= * (Q2R machineEpsilon))%R. `````` Heiko Becker committed Aug 18, 2016 33 ``````Proof. `````` Heiko Becker committed Feb 08, 2017 34 `````` intros eval_real eval_float. `````` Heiko Becker committed Aug 18, 2016 35 `````` inversion eval_real; subst. `````` Heiko Becker committed Oct 13, 2016 36 `````` rewrite delta_0_deterministic; auto. `````` Heiko Becker committed Aug 18, 2016 37 38 `````` inversion eval_float; subst. unfold perturb; simpl. `````` Heiko Becker committed Feb 17, 2017 39 40 `````` exists v; split; try auto. rewrite H3 in H8; inversion H8. `````` Heiko Becker committed Aug 18, 2016 41 42 43 44 `````` rewrite Rabs_err_simpl. repeat rewrite Rabs_mult. apply Rmult_le_compat_l; [ apply Rabs_pos | auto]. Qed. `````` Heiko Becker committed Feb 17, 2017 45 ``````*) `````` Heiko Becker committed Aug 18, 2016 46 `````` `````` Heiko Becker committed Feb 01, 2017 47 ``````Lemma add_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) (e2F:R) `````` = committed Apr 04, 2017 48 `````` (vR:R) (vF:R) (E1 E2:env) (err1 err2 :Q) (m m1 m2:mType) defVars: `````` = committed Apr 05, 2017 49 `````` eval_exp E1 (toREvalVars defVars) (toREval (toRExp e1)) e1R M0 -> `````` = committed Apr 04, 2017 50 `````` eval_exp E2 defVars (toRExp e1) e1F m1-> `````` = committed Apr 05, 2017 51 `````` eval_exp E1 (toREvalVars defVars) (toREval (toRExp e2)) e2R M0 -> `````` = committed Apr 04, 2017 52 `````` eval_exp E2 defVars (toRExp e2) e2F m2 -> `````` = committed Apr 05, 2017 53 54 `````` eval_exp E1 (toREvalVars defVars) (toREval (Binop Plus (toRExp e1) (toRExp e2))) vR M0 -> eval_exp (updEnv 2 e2F (updEnv 1 e1F emptyEnv)) (fun n => if (n =? 2) then Some m2 else if (n =? 1) then Some m1 else defVars n) (Binop Plus (Var R 1) (Var R 2)) vF m-> `````` Heiko Becker committed Feb 13, 2017 55 56 `````` (Rabs (e1R - e1F) <= Q2R err1)%R -> (Rabs (e2R - e2F) <= Q2R err2)%R -> `````` Raphaël Monat committed Mar 01, 2017 57 `````` (Rabs (vR - vF) <= Q2R err1 + Q2R err2 + (Rabs (e1F + e2F) * (Q2R (meps m))))%R. `````` Heiko Becker committed Aug 18, 2016 58 ``````Proof. `````` Heiko Becker committed Feb 08, 2017 59 `````` intros e1_real e1_float e2_real e2_float plus_real plus_float bound_e1 bound_e2. `````` Heiko Becker committed Aug 18, 2016 60 61 `````` (* Prove that e1R and e2R are the correct values and that vR is e1R + e2R *) inversion plus_real; subst. `````` Raphaël Monat committed Mar 10, 2017 62 `````` destruct m0; destruct m3; inversion H2; `````` = committed Mar 28, 2017 63 `````` simpl in H3; rewrite Q2R0_is_0 in H3; auto. `````` Heiko Becker committed Oct 13, 2016 64 `````` rewrite delta_0_deterministic in plus_real; auto. `````` Heiko Becker committed Oct 31, 2016 65 66 `````` rewrite (delta_0_deterministic (evalBinop Plus v1 v2) delta); auto. unfold evalBinop in *; simpl in *. `````` = committed Mar 28, 2017 67 `````` clear delta H3. `````` Raphaël Monat committed Mar 10, 2017 68 69 70 71 `````` rewrite (meps_0_deterministic (toRExp e1) H5 e1_real); rewrite (meps_0_deterministic (toRExp e2) H6 e2_real). rewrite (meps_0_deterministic (toRExp e1) H5 e1_real) in plus_real. rewrite (meps_0_deterministic (toRExp e2) H6 e2_real) in plus_real. `````` = committed Mar 28, 2017 72 `````` clear H5 H6 H7 v1 v2. `````` Heiko Becker committed Aug 18, 2016 73 74 75 `````` (* Now unfold the float valued evaluation to get the deltas we need for the inequality *) inversion plus_float; subst. unfold perturb; simpl. `````` = committed Mar 28, 2017 76 `````` inversion H4; subst; inversion H7; subst. `````` Heiko Becker committed Aug 18, 2016 77 `````` unfold updEnv; simpl. `````` = committed Apr 04, 2017 78 79 80 `````` unfold updEnv in H1,H6; simpl in *. symmetry in H1,H6. inversion H1; inversion H6; subst. `````` Heiko Becker committed Aug 18, 2016 81 `````` (* We have now obtained all necessary values from the evaluations --> remove them for readability *) `````` = committed Apr 04, 2017 82 `````` clear plus_float H4 H7 plus_real e1_real e1_float e2_real e2_float H8 H6 H1. `````` Heiko Becker committed Aug 18, 2016 83 84 85 86 87 88 89 90 91 92 `````` 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))). `````` = committed Apr 04, 2017 93 `````` pose proof (Rplus_le_compat_l (Rabs (e1R + - e1F)) _ _ H1). `````` Heiko Becker committed Aug 18, 2016 94 `````` eapply Rle_trans. `````` = committed Apr 04, 2017 95 `````` apply H4. `````` Heiko Becker committed Aug 18, 2016 96 97 98 99 100 101 102 103 104 `````` 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. `````` Raphaël Monat committed Mar 10, 2017 105 `````` apply H3. `````` Heiko Becker committed Aug 18, 2016 106 107 108 109 110 111 `````` 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 112 ``````Lemma subtract_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) `````` = committed Apr 04, 2017 113 `````` (e2F:R) (vR:R) (vF:R) (E1 E2:env) err1 err2 (m m1 m2:mType) defVars: `````` = committed Apr 05, 2017 114 `````` eval_exp E1 (toREvalVars defVars) (toREval (toRExp e1)) e1R M0 -> `````` = committed Apr 04, 2017 115 `````` eval_exp E2 defVars (toRExp e1) e1F m1 -> `````` = committed Apr 05, 2017 116 `````` eval_exp E1 (toREvalVars defVars) (toREval (toRExp e2)) e2R M0 -> `````` = committed Apr 04, 2017 117 `````` eval_exp E2 defVars (toRExp e2) e2F m2 -> `````` = committed Apr 05, 2017 118 119 `````` eval_exp E1 (toREvalVars defVars) (toREval (Binop Sub (toRExp e1) (toRExp e2))) vR M0 -> eval_exp (updEnv 2 e2F (updEnv 1 e1F emptyEnv)) (fun n => if (n =? 2) then Some m2 else if (n =? 1) then Some m1 else defVars n) (Binop Sub (Var R 1) (Var R 2)) vF m -> `````` Heiko Becker committed Feb 08, 2017 120 121 `````` (Rabs (e1R - e1F) <= Q2R err1)%R -> (Rabs (e2R - e2F) <= Q2R err2)%R -> `````` Raphaël Monat committed Mar 01, 2017 122 `````` (Rabs (vR - vF) <= Q2R err1 + Q2R err2 + ((Rabs (e1F - e2F)) * (Q2R (meps m))))%R. `````` Heiko Becker committed Aug 18, 2016 123 ``````Proof. `````` Heiko Becker committed Feb 08, 2017 124 `````` intros e1_real e1_float e2_real e2_float sub_real sub_float bound_e1 bound_e2. `````` Heiko Becker committed Aug 18, 2016 125 `````` (* Prove that e1R and e2R are the correct values and that vR is e1R + e2R *) `````` Raphaël Monat committed Mar 01, 2017 126 `````` inversion sub_real; subst; `````` Raphaël Monat committed Mar 10, 2017 127 `````` destruct m0; destruct m3; inversion H2; `````` = committed Mar 28, 2017 128 `````` simpl in H3; rewrite Q2R0_is_0 in H3; auto. `````` Heiko Becker committed Oct 13, 2016 129 130 `````` rewrite delta_0_deterministic in sub_real; auto. rewrite delta_0_deterministic; auto. `````` Heiko Becker committed Oct 31, 2016 131 `````` unfold evalBinop in *; simpl in *. `````` = committed Mar 28, 2017 132 `````` clear delta H3. `````` Raphaël Monat committed Mar 10, 2017 133 134 135 136 `````` rewrite (meps_0_deterministic (toRExp e1) H5 e1_real); rewrite (meps_0_deterministic (toRExp e2) H6 e2_real). rewrite (meps_0_deterministic (toRExp e1) H5 e1_real) in sub_real. rewrite (meps_0_deterministic (toRExp e2) H6 e2_real) in sub_real. `````` = committed Mar 28, 2017 137 `````` clear H5 H6 H7 v1 v2. `````` Heiko Becker committed Aug 18, 2016 138 139 140 `````` (* Now unfold the float valued evaluation to get the deltas we need for the inequality *) inversion sub_float; subst. unfold perturb; simpl. `````` = committed Mar 28, 2017 141 `````` inversion H4; subst; inversion H7; subst. `````` Heiko Becker committed Aug 18, 2016 142 `````` unfold updEnv; simpl. `````` = committed Apr 04, 2017 143 144 145 146 `````` simpl in H0; simpl in H5; inversion H0; inversion H5; subst; clear H0 H5. symmetry in H6, H1. unfold updEnv in H6, H1; simpl in H6, H1. inversion H6; inversion H1; subst. `````` Heiko Becker committed Aug 18, 2016 147 `````` (* We have now obtained all necessary values from the evaluations --> remove them for readability *) `````` = committed Apr 04, 2017 148 `````` clear sub_float H4 H7 sub_real e1_real e1_float e2_real e2_float H8 H1 H6. `````` Heiko Becker committed Aug 18, 2016 149 150 151 152 153 `````` 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 154 `````` rewrite Ropp_involutive. `````` Heiko Becker committed Aug 18, 2016 155 156 `````` rewrite Rplus_assoc. eapply Rle_trans. `````` Heiko Becker committed Sep 21, 2016 157 `````` apply Rabs_triang. `````` Heiko Becker committed Aug 18, 2016 158 `````` eapply Rle_trans. `````` Heiko Becker committed Sep 21, 2016 159 160 `````` eapply Rplus_le_compat_l. apply Rabs_triang. `````` Heiko Becker committed Aug 18, 2016 161 `````` rewrite <- Rplus_assoc. `````` Heiko Becker committed Sep 21, 2016 162 `````` setoid_rewrite Rplus_comm at 4. `````` Heiko Becker committed Aug 18, 2016 163 164 `````` repeat rewrite <- Rsub_eq_Ropp_Rplus. rewrite Rabs_Ropp. `````` Heiko Becker committed Sep 21, 2016 165 166 167 168 `````` 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 169 170 ``````Qed. `````` Heiko Becker committed Feb 03, 2017 171 ``````Lemma mult_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) (e2F:R) `````` = committed Apr 04, 2017 172 `````` (vR:R) (vF:R) (E1 E2:env) (m m1 m2:mType) defVars: `````` = committed Apr 05, 2017 173 `````` eval_exp E1 (toREvalVars defVars) (toREval (toRExp e1)) e1R M0 -> `````` = committed Apr 04, 2017 174 `````` eval_exp E2 defVars (toRExp e1) e1F m1 -> `````` = committed Apr 05, 2017 175 `````` eval_exp E1 (toREvalVars defVars) (toREval (toRExp e2)) e2R M0 -> `````` = committed Apr 04, 2017 176 `````` eval_exp E2 defVars (toRExp e2) e2F m2 -> `````` = committed Apr 05, 2017 177 178 `````` eval_exp E1 (toREvalVars defVars) (toREval (Binop Mult (toRExp e1) (toRExp e2))) vR M0 -> eval_exp (updEnv 2 e2F (updEnv 1 e1F emptyEnv)) (fun n => if (n =? 2) then Some m2 else if (n =? 1) then Some m1 else defVars n) (Binop Mult (Var R 1) (Var R 2)) vF m-> `````` Raphaël Monat committed Mar 01, 2017 179 `````` (Rabs (vR - vF) <= Rabs (e1R * e2R - e1F * e2F) + Rabs (e1F * e2F) * (Q2R (meps m)))%R. `````` Heiko Becker committed Aug 23, 2016 180 ``````Proof. `````` Heiko Becker committed Feb 08, 2017 181 `````` intros e1_real e1_float e2_real e2_float mult_real mult_float. `````` Heiko Becker committed Oct 07, 2016 182 `````` (* Prove that e1R and e2R are the correct values and that vR is e1R * e2R *) `````` Raphaël Monat committed Mar 01, 2017 183 `````` inversion mult_real; subst; `````` Raphaël Monat committed Mar 10, 2017 184 `````` destruct m0; destruct m3; inversion H2; `````` = committed Mar 28, 2017 185 `````` simpl in H3; rewrite Q2R0_is_0 in H3; auto. `````` Heiko Becker committed Oct 13, 2016 186 187 `````` rewrite delta_0_deterministic in mult_real; auto. rewrite delta_0_deterministic; auto. `````` Heiko Becker committed Oct 31, 2016 188 `````` unfold evalBinop in *; simpl in *. `````` = committed Mar 28, 2017 189 `````` clear delta H3. `````` Raphaël Monat committed Mar 10, 2017 190 191 192 193 `````` rewrite (meps_0_deterministic (toRExp e1) H5 e1_real); rewrite (meps_0_deterministic (toRExp e2) H6 e2_real). rewrite (meps_0_deterministic (toRExp e1) H5 e1_real) in mult_real. rewrite (meps_0_deterministic (toRExp e2) H6 e2_real) in mult_real. `````` = committed Mar 28, 2017 194 `````` clear H5 H6 v1 v2 H7 H2. `````` Heiko Becker committed Aug 23, 2016 195 196 197 `````` (* Now unfold the float valued evaluation to get the deltas we need for the inequality *) inversion mult_float; subst. unfold perturb; simpl. `````` = committed Mar 28, 2017 198 `````` inversion H3; subst; inversion H6; subst. `````` Heiko Becker committed Feb 17, 2017 199 `````` unfold updEnv in *; simpl in *. `````` = committed Apr 04, 2017 200 201 `````` inversion H6; inversion H1; subst. simpl in H8; simpl in H9; intros; inversion H5; subst. `````` Heiko Becker committed Aug 23, 2016 202 `````` (* We have now obtained all necessary values from the evaluations --> remove them for readability *) `````` = committed Apr 04, 2017 203 `````` clear mult_float H7 mult_real e1_real e1_float e2_real e2_float H6 H1. `````` Heiko Becker committed Aug 23, 2016 204 205 206 207 `````` 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 208 209 `````` rewrite <- Rplus_assoc. setoid_rewrite <- Rsub_eq_Ropp_Rplus at 2. `````` Heiko Becker committed Aug 23, 2016 210 211 212 `````` eapply Rle_trans. eapply Rabs_triang. eapply Rplus_le_compat_l. `````` Heiko Becker committed Aug 24, 2016 213 `````` rewrite Rabs_Ropp. `````` Heiko Becker committed Aug 23, 2016 214 `````` repeat rewrite Rabs_mult. `````` Heiko Becker committed Aug 24, 2016 215 `````` eapply Rmult_le_compat_l; auto. `````` Heiko Becker committed Aug 23, 2016 216 217 `````` rewrite <- Rabs_mult. apply Rabs_pos. `````` Heiko Becker committed Oct 07, 2016 218 219 ``````Qed. `````` Heiko Becker committed Feb 03, 2017 220 ``````Lemma div_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) (e2F:R) `````` = committed Apr 04, 2017 221 `````` (vR:R) (vF:R) (E1 E2:env) (m m1 m2:mType) defVars: `````` = committed Apr 05, 2017 222 `````` eval_exp E1 (toREvalVars defVars) (toREval (toRExp e1)) e1R M0 -> `````` = committed Apr 04, 2017 223 `````` eval_exp E2 defVars (toRExp e1) e1F m1 -> `````` = committed Apr 05, 2017 224 `````` eval_exp E1 (toREvalVars defVars) (toREval (toRExp e2)) e2R M0 -> `````` = committed Apr 04, 2017 225 `````` eval_exp E2 defVars (toRExp e2) e2F m2 -> `````` = committed Apr 05, 2017 226 227 `````` eval_exp E1 (toREvalVars defVars) (toREval (Binop Div (toRExp e1) (toRExp e2))) vR M0 -> eval_exp (updEnv 2 e2F (updEnv 1 e1F emptyEnv)) (fun n => if (n =? 2) then Some m2 else if (n =? 1) then Some m1 else defVars n) (Binop Div (Var R 1) (Var R 2)) vF m -> `````` Raphaël Monat committed Mar 01, 2017 228 `````` (Rabs (vR - vF) <= Rabs (e1R / e2R - e1F / e2F) + Rabs (e1F / e2F) * (Q2R (meps m)))%R. `````` Heiko Becker committed Oct 07, 2016 229 ``````Proof. `````` Heiko Becker committed Feb 08, 2017 230 `````` intros e1_real e1_float e2_real e2_float div_real div_float. `````` Heiko Becker committed Oct 07, 2016 231 `````` (* Prove that e1R and e2R are the correct values and that vR is e1R * e2R *) `````` Raphaël Monat committed Mar 01, 2017 232 `````` inversion div_real; subst; `````` Raphaël Monat committed Mar 10, 2017 233 `````` destruct m0; destruct m3; inversion H2; `````` = committed Mar 28, 2017 234 `````` simpl in H3; rewrite Q2R0_is_0 in H3; auto. `````` Heiko Becker committed Oct 13, 2016 235 236 `````` rewrite delta_0_deterministic in div_real; auto. rewrite delta_0_deterministic; auto. `````` Heiko Becker committed Oct 31, 2016 237 `````` unfold evalBinop in *; simpl in *. `````` = committed Mar 28, 2017 238 `````` clear delta H3 H2. `````` Raphaël Monat committed Mar 10, 2017 239 240 241 242 `````` rewrite (meps_0_deterministic (toRExp e1) H5 e1_real); rewrite (meps_0_deterministic (toRExp e2) H6 e2_real). rewrite (meps_0_deterministic (toRExp e1) H5 e1_real) in div_real. rewrite (meps_0_deterministic (toRExp e2) H6 e2_real) in div_real. `````` = committed Mar 28, 2017 243 `````` (* clear H5 H6 v1 v2. *) `````` Heiko Becker committed Oct 07, 2016 244 245 246 `````` (* Now unfold the float valued evaluation to get the deltas we need for the inequality *) inversion div_float; subst. unfold perturb; simpl. `````` = committed Mar 28, 2017 247 `````` inversion H3; subst; inversion H9; subst. `````` Heiko Becker committed Feb 17, 2017 248 `````` unfold updEnv in *; simpl in *. `````` = committed Apr 04, 2017 249 `````` inversion H8; inversion H1; subst. `````` Heiko Becker committed Oct 07, 2016 250 `````` (* We have now obtained all necessary values from the evaluations --> remove them for readability *) `````` = committed Apr 04, 2017 251 `````` clear div_float H0 H1 div_real e1_real e1_float e2_real e2_float. `````` Heiko Becker committed Oct 07, 2016 252 253 254 255 256 257 258 259 260 261 262 263 264 `````` 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 265 266 ``````Qed. `````` Heiko Becker committed Jan 13, 2017 267 268 269 270 271 272 273 274 ``````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 275 ``````Proof. `````` Heiko Becker committed Jan 13, 2017 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 `````` 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 406 `````` rewrite equal_naming_inv; try lra. `````` Heiko Becker committed Jan 13, 2017 407 408 `````` assert (nR + - (nR + err) = - err)%R as simpl_up by lra. rewrite simpl_up. `````` Heiko Becker committed Nov 08, 2016 409 `````` unfold Rdiv. `````` Heiko Becker committed Jan 13, 2017 410 `````` rewrite <- Ropp_mult_distr_l, <- Ropp_mult_distr_r, <- Ropp_inv_permute. `````` Heiko Becker committed Nov 08, 2016 411 412 413 `````` { 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 414 415 416 417 418 419 420 421 422 423 424 425 426 427 `````` - 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 428 429 ``````Qed. `````` Heiko Becker committed Jan 13, 2017 430 431 432 433 434 435 436 437 ``````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 438 ``````Proof. `````` Heiko Becker committed Jan 13, 2017 439 440 441 442 443 444 `````` 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. `````` Raphaël Monat committed Mar 03, 2017 445 `````` `````` = committed Apr 04, 2017 446 ``````Lemma round_abs_err_bounded (e:exp R) (nR nF1 nF:R) (E1 E2: env) (err:R) (machineEpsilon m:mType) defVars: `````` = committed Apr 05, 2017 447 `````` eval_exp E1 (toREvalVars defVars) (toREval e) nR M0 -> `````` = committed Apr 04, 2017 448 `````` eval_exp E2 defVars e nF1 m -> `````` = committed Apr 05, 2017 449 `````` eval_exp (updEnv 1 nF1 emptyEnv) (fun n => if n =? 1 then Some m else defVars n) (toRExp (Downcast machineEpsilon (Var Q 1))) nF machineEpsilon-> `````` Raphaël Monat committed Mar 03, 2017 450 451 452 453 454 455 456 457 458 459 460 461 `````` (Rabs (nR - nF1) <= err)%R -> (Rabs (nR - nF) <= err + (Rabs nF1) * Q2R (meps machineEpsilon))%R. Proof. intros eval_real eval_float eval_float_rnd err_bounded. replace (nR - nF)%R with ((nR - nF1) + (nF1 - nF))%R by lra. eapply Rle_trans. apply (Rabs_triang (nR - nF1) (nF1 - nF)). apply (Rle_trans _ (err + Rabs (nF1 - nF)) _). - apply Rplus_le_compat_r; assumption. - apply Rplus_le_compat_l. inversion eval_float_rnd; subst. inversion H5; subst. `````` = committed Apr 04, 2017 462 `````` inversion H0; subst. `````` Raphaël Monat committed Mar 03, 2017 463 `````` unfold perturb; simpl. `````` = committed Apr 04, 2017 464 `````` unfold updEnv in H3; simpl in H3; inversion H3; subst. `````` Raphaël Monat committed Mar 03, 2017 465 466 467 468 469 `````` replace (v1 - v1 * (1 + delta))%R with (- (v1 * delta))%R by lra. replace (Rabs (-(v1*delta))) with (Rabs (v1*delta)) by (symmetry; apply Rabs_Ropp). rewrite Rabs_mult. apply Rmult_le_compat_l. + apply Rabs_pos. `````` = committed Apr 25, 2017 470 `````` + auto. `````` Raphaël Monat committed Mar 03, 2017 471 ``Qed.``