(** Proofs of general bounds on the error of arithmetic expressions. This shortens soundness proofs later. Bounds are explained in section 5, Deriving Computable Error Bounds **) Require Import Coq.Reals.Reals Coq.micromega.Psatz Coq.QArith.QArith Coq.QArith.Qreals. Require Import Daisy.Infra.Abbrevs Daisy.Infra.RationalSimps Daisy.Infra.RealSimps Daisy.Infra.RealRationalProps. Require Import Daisy.Environments Daisy.Infra.ExpressionAbbrevs. (* TODO: absenv not used *) Lemma const_abs_err_bounded (n:R) (nR:R) (nF:R) (E1 E2:env) (absenv:analysisResult) (m:mType) defVars: eval_exp E1 (toREvalVars defVars) (Const M0 n) nR M0 -> eval_exp E2 defVars (Const m n) nF m -> (Rabs (nR - nF) <= Rabs n * (Q2R (meps m)))%R. Proof. intros eval_real eval_float. inversion eval_real; subst. rewrite delta_0_deterministic; auto. inversion eval_float; subst. unfold perturb; simpl. rewrite Rabs_err_simpl, Rabs_mult. apply Rmult_le_compat_l; [apply Rabs_pos | auto]. simpl (meps M0) in *. apply (Rle_trans _ (Q2R 0) _); try auto. rewrite Q2R0_is_0; lra. Qed. (* 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. Proof. intros eval_real eval_float. inversion eval_real; subst. rewrite delta_0_deterministic; auto. inversion eval_float; subst. unfold perturb; simpl. exists v; split; try auto. rewrite H3 in H8; inversion H8. rewrite Rabs_err_simpl. repeat rewrite Rabs_mult. apply Rmult_le_compat_l; [ apply Rabs_pos | auto]. Qed. *) Lemma add_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) (e2F:R) (vR:R) (vF:R) (E1 E2:env) (err1 err2 :Q) (m m1 m2:mType) defVars: eval_exp E1 (toREvalVars defVars) (toREval (toRExp e1)) e1R M0 -> eval_exp E2 defVars (toRExp e1) e1F m1-> eval_exp E1 (toREvalVars defVars) (toREval (toRExp e2)) e2R M0 -> eval_exp E2 defVars (toRExp e2) e2F m2 -> 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-> (Rabs (e1R - e1F) <= Q2R err1)%R -> (Rabs (e2R - e2F) <= Q2R err2)%R -> (Rabs (vR - vF) <= Q2R err1 + Q2R err2 + (Rabs (e1F + e2F) * (Q2R (meps m))))%R. Proof. intros e1_real e1_float e2_real e2_float plus_real plus_float bound_e1 bound_e2. (* Prove that e1R and e2R are the correct values and that vR is e1R + e2R *) inversion plus_real; subst. destruct m0; destruct m3; inversion H2; simpl in H3; rewrite Q2R0_is_0 in H3; auto. rewrite delta_0_deterministic in plus_real; auto. rewrite (delta_0_deterministic (evalBinop Plus v1 v2) delta); auto. unfold evalBinop in *; simpl in *. clear delta H3. 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. clear H5 H6 H7 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 H7; subst. unfold updEnv; simpl. unfold updEnv in H1,H6; simpl in *. symmetry in H1,H6. inversion H1; inversion H6; subst. (* We have now obtained all necessary values from the evaluations --> remove them for readability *) clear plus_float H4 H7 plus_real e1_real e1_float e2_real e2_float H8 H6 H1. 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)) _ _ H1). eapply Rle_trans. apply H4. 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 H3. apply Req_le; auto. Qed. (** Copy-Paste proof with minor differences, was easier then manipulating the evaluations and then applying the lemma **) Lemma subtract_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) (e2F:R) (vR:R) (vF:R) (E1 E2:env) err1 err2 (m m1 m2:mType) defVars: eval_exp E1 (toREvalVars defVars) (toREval (toRExp e1)) e1R M0 -> eval_exp E2 defVars (toRExp e1) e1F m1 -> eval_exp E1 (toREvalVars defVars) (toREval (toRExp e2)) e2R M0 -> eval_exp E2 defVars (toRExp e2) e2F m2 -> 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 -> (Rabs (e1R - e1F) <= Q2R err1)%R -> (Rabs (e2R - e2F) <= Q2R err2)%R -> (Rabs (vR - vF) <= Q2R err1 + Q2R err2 + ((Rabs (e1F - e2F)) * (Q2R (meps m))))%R. Proof. intros e1_real e1_float e2_real e2_float sub_real sub_float bound_e1 bound_e2. (* Prove that e1R and e2R are the correct values and that vR is e1R + e2R *) inversion sub_real; subst; destruct m0; destruct m3; inversion H2; simpl in H3; rewrite Q2R0_is_0 in H3; auto. rewrite delta_0_deterministic in sub_real; auto. rewrite delta_0_deterministic; auto. unfold evalBinop in *; simpl in *. clear delta H3. 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. clear H5 H6 H7 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 H7; subst. unfold updEnv; simpl. 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. (* We have now obtained all necessary values from the evaluations --> remove them for readability *) clear sub_float H4 H7 sub_real e1_real e1_float e2_real e2_float H8 H1 H6. 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. rewrite Ropp_involutive. rewrite Rplus_assoc. eapply Rle_trans. apply Rabs_triang. eapply Rle_trans. eapply Rplus_le_compat_l. apply Rabs_triang. rewrite <- Rplus_assoc. setoid_rewrite Rplus_comm at 4. repeat rewrite <- Rsub_eq_Ropp_Rplus. rewrite Rabs_Ropp. 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]. Qed. Lemma mult_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) (e2F:R) (vR:R) (vF:R) (E1 E2:env) (m m1 m2:mType) defVars: eval_exp E1 (toREvalVars defVars) (toREval (toRExp e1)) e1R M0 -> eval_exp E2 defVars (toRExp e1) e1F m1 -> eval_exp E1 (toREvalVars defVars) (toREval (toRExp e2)) e2R M0 -> eval_exp E2 defVars (toRExp e2) e2F m2 -> 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-> (Rabs (vR - vF) <= Rabs (e1R * e2R - e1F * e2F) + Rabs (e1F * e2F) * (Q2R (meps m)))%R. Proof. intros e1_real e1_float e2_real e2_float mult_real mult_float. (* Prove that e1R and e2R are the correct values and that vR is e1R * e2R *) inversion mult_real; subst; destruct m0; destruct m3; inversion H2; simpl in H3; rewrite Q2R0_is_0 in H3; auto. rewrite delta_0_deterministic in mult_real; auto. rewrite delta_0_deterministic; auto. unfold evalBinop in *; simpl in *. clear delta H3. 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. clear H5 H6 v1 v2 H7 H2. (* Now unfold the float valued evaluation to get the deltas we need for the inequality *) inversion mult_float; subst. unfold perturb; simpl. inversion H3; subst; inversion H6; subst. unfold updEnv in *; simpl in *. inversion H6; inversion H1; subst. simpl in H8; simpl in H9; intros; inversion H5; subst. (* We have now obtained all necessary values from the evaluations --> remove them for readability *) clear mult_float H7 mult_real e1_real e1_float e2_real e2_float H6 H1. 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. rewrite <- Rabs_mult. apply Rabs_pos. Qed. Lemma div_abs_err_bounded (e1:exp Q) (e1R:R) (e1F:R) (e2:exp Q) (e2R:R) (e2F:R) (vR:R) (vF:R) (E1 E2:env) (m m1 m2:mType) defVars: eval_exp E1 (toREvalVars defVars) (toREval (toRExp e1)) e1R M0 -> eval_exp E2 defVars (toRExp e1) e1F m1 -> eval_exp E1 (toREvalVars defVars) (toREval (toRExp e2)) e2R M0 -> eval_exp E2 defVars (toRExp e2) e2F m2 -> 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 -> (Rabs (vR - vF) <= Rabs (e1R / e2R - e1F / e2F) + Rabs (e1F / e2F) * (Q2R (meps m)))%R. Proof. intros e1_real e1_float e2_real e2_float div_real div_float. (* Prove that e1R and e2R are the correct values and that vR is e1R * e2R *) inversion div_real; subst; destruct m0; destruct m3; inversion H2; simpl in H3; rewrite Q2R0_is_0 in H3; auto. rewrite delta_0_deterministic in div_real; auto. rewrite delta_0_deterministic; auto. unfold evalBinop in *; simpl in *. clear delta H3 H2. 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. (* clear H5 H6 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 H3; subst; inversion H9; subst. unfold updEnv in *; simpl in *. inversion H8; inversion H1; subst. (* We have now obtained all necessary values from the evaluations --> remove them for readability *) clear div_float H0 H1 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. Qed. 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. Proof. 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. 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_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). Qed. 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. Proof. 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. Lemma round_abs_err_bounded (e:exp R) (nR nF1 nF:R) (E1 E2: env) (err:R) (machineEpsilon m:mType) defVars: eval_exp E1 (toREvalVars defVars) (toREval e) nR M0 -> eval_exp E2 defVars e nF1 m -> 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-> (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. inversion H0; subst. unfold perturb; simpl. unfold updEnv in H3; simpl in H3; inversion H3; subst. 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. + auto. Qed.