{l} (islim : IsLimit x l) (epsilon : Qpos Q) : x (M (epsilon / 2)) - ' (' epsilon) < l < x (M (epsilon / 2)) + ' (' epsilon). Proof. assert (lim_close := is_limit x (epsilon / 2)); strip_truncations. destruct lim_close as [N isclose']. set (n := Nat.Core.nat_max (M (epsilon / 2)) N). assert (leNn := le_nat_max_r (M (epsilon / 2)) N : N ≤ n). assert (isclose := isclose' n leNn). clear isclose'. assert (leMn := le_nat_max_l (M (epsilon / 2)) N : M (epsilon / 2) ≤ n). assert (leMM : M (epsilon / 2) ≤ M (epsilon / 2) ) by apply (Nat.Core.leq_refl). assert (x_close := cauchy_convergence x (epsilon/2) n (M (epsilon / 2)) leMn leMM). cbn in isclose, x_close. rewrite (@preserves_mult Q F _ _ _ _ _ _ _ _ _ _ _ _) in isclose, x_close. assert (eq22 : ' 2 = 2). { rewrite (@preserves_plus Q F _ _ _ _ _ _ _ _ _ _ _ _). rewrite (@preserves_1 Q F _ _ _ _ _ _ _ _ _ _). reflexivity. } set (ap20 := positive_apart_zero 2 lt_0_2 : 2 ≶ 0). assert (ap20' : ' 2 ≶ 0). { rewrite eq22; exact ap20. } rewrite (dec_recip_to_recip 2 ap20') in isclose, x_close. assert (eq_recip_22 : recip' (' 2) ap20' = recip' 2 ap20). { apply recip_proper_alt. exact eq22. } unfold recip' in eq_recip_22. rewrite eq_recip_22 in isclose, x_close. clear eq22 ap20' eq_recip_22. rewrite <- (field_split2 (' (' epsilon))). set (eps_recip_2 := (' (' epsilon) * recip' 2 ap20)). fold ap20. change (' (' epsilon) * recip' 2 ap20) with eps_recip_2. unfold recip' in eps_recip_2. set (xMeps2 := x (M (epsilon / 2))). fold xMeps2 in x_close. rewrite negate_plus_distr. split. - apply (strictly_order_reflecting (+ (- x n))). refine (transitivity _ (fst isclose)). clear isclose. fold eps_recip_2. fold eps_recip_2 in x_close. apply fst, flip_lt_minus_r in x_close. rewrite plus_comm in x_close. apply flip_lt_minus_l in x_close. rewrite plus_comm in x_close. apply flip_lt_minus_l in x_close. rewrite <-(plus_assoc xMeps2 _ (- x n)). rewrite (plus_comm _ (- x n)). rewrite (plus_assoc xMeps2 (- x n) _). apply (strictly_order_reflecting (+ eps_recip_2)). apply (strictly_order_reflecting (+ eps_recip_2)). rewrite plus_negate_l, plus_0_l. rewrite <- (plus_assoc (xMeps2 - x n) _ _). rewrite <- (plus_assoc (-eps_recip_2) _ _). rewrite plus_negate_l, plus_0_r. rewrite <- (plus_assoc (xMeps2 - x n) _ _). rewrite plus_negate_l, plus_0_r. assumption. - apply (strictly_order_reflecting (+ (- x n))). refine (transitivity (snd isclose) _). clear isclose. fold eps_recip_2. fold eps_recip_2 in x_close. apply snd in x_close. apply flip_lt_minus_l in x_close. rewrite plus_comm in x_close. apply (strictly_order_reflecting (+ x n)). rewrite <- (plus_assoc _ (-x n) (x n)). rewrite plus_negate_l, plus_0_r. rewrite (plus_comm eps_recip_2 (x n)). rewrite (plus_assoc xMeps2 _ _). apply (strictly_order_preserving (+ eps_recip_2)). assumption. Qed.
From HoTT.Classes Require Import