chore(Topology/Order): rename 2 lemmas (#19261)

urkud · urkud · commit 43c93172bdc3 · 2024-11-22T02:00:10.000Z
... to allow dot notation.
diff --git a/Mathlib/Algebra/Order/Floor/Prime.lean b/Mathlib/Algebra/Order/Floor/Prime.lean
@@ -39,7 +39,7 @@ theorem exists_prime_mul_pow_div_factorial_lt_one [LinearOrderedField K] [FloorR
   letI := Preorder.topology K
   haveI : OrderTopology K := ⟨rfl⟩
   ((Filter.frequently_atTop.mpr Nat.exists_infinite_primes).and_eventually
-    (eventually_lt_of_tendsto_lt zero_lt_one
+    (Filter.Tendsto.eventually_lt_const zero_lt_one
       (FloorSemiring.tendsto_mul_pow_div_factorial_sub_atTop a c 1))).forall_exists_of_atTop
     (n + 1)
 
diff --git a/Mathlib/Analysis/Asymptotics/TVS.lean b/Mathlib/Analysis/Asymptotics/TVS.lean
@@ -125,7 +125,7 @@ lemma isLittleOTVS_one [ContinuousSMul 𝕜 E] {f : α → E} {l : Filter α} :
     rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
     obtain ⟨ε, hε₀, hε⟩ : ∃ ε : ℝ≥0, 0 < ε ∧ (ε * ‖c‖₊ / r : ℝ≥0∞) < 1 := by
       apply Eventually.exists_gt
-      refine eventually_lt_of_tendsto_lt zero_lt_one <| Continuous.tendsto' ?_ _ _ (by simp)
+      refine Continuous.tendsto' ?_ _ _ (by simp) |>.eventually_lt_const zero_lt_one
       fun_prop (disch := intros; first | apply ENNReal.coe_ne_top | positivity)
     filter_upwards [hr ε hε₀.ne'] with x hx
     refine mem_of_egauge_lt_one hUb (hx.trans_lt ?_)
diff --git a/Mathlib/Analysis/BoundedVariation.lean b/Mathlib/Analysis/BoundedVariation.lean
@@ -180,7 +180,7 @@ theorem lowerSemicontinuous_aux {ι : Type*} {F : ι → α → E} {p : Filter 
       (𝓝 (∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)))) := by
     apply tendsto_finset_sum
     exact fun i _ => Tendsto.edist (Ffs (u i.succ) (us i.succ)) (Ffs (u i) (us i))
-  exact (eventually_gt_of_tendsto_gt hlt this).mono fun i h => h.trans_le (sum_le (F i) n um us)
+  exact (this.eventually_const_lt hlt).mono fun i h => h.trans_le (sum_le (F i) n um us)
 
 /-- The map `(eVariationOn · s)` is lower semicontinuous for pointwise convergence *on `s`*.
 Pointwise convergence on `s` is encoded here as uniform convergence on the family consisting of the
diff --git a/Mathlib/Analysis/Normed/Lp/lpSpace.lean b/Mathlib/Analysis/Normed/Lp/lpSpace.lean
@@ -183,7 +183,7 @@ theorem of_exponent_ge {p q : ℝ≥0∞} {f : ∀ i, E i} (hfq : Memℓp f q) (
     have hf' := hfq.summable hq
     refine .of_norm_bounded_eventually _ hf' (@Set.Finite.subset _ { i | 1 ≤ ‖f i‖ } ?_ _ ?_)
     · have H : { x : α | 1 ≤ ‖f x‖ ^ q.toReal }.Finite := by
-        simpa using eventually_lt_of_tendsto_lt (by norm_num) hf'.tendsto_cofinite_zero
+        simpa using hf'.tendsto_cofinite_zero.eventually_lt_const (by norm_num)
       exact H.subset fun i hi => Real.one_le_rpow hi hq.le
     · show ∀ i, ¬|‖f i‖ ^ p.toReal| ≤ ‖f i‖ ^ q.toReal → 1 ≤ ‖f i‖
       intro i hi
diff --git a/Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean b/Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
@@ -592,7 +592,7 @@ theorem tendsto_one_plus_div_rpow_exp (t : ℝ) :
   have h₁ : (1 : ℝ) / 2 < 1 := by norm_num
   have h₂ : Tendsto (fun x : ℝ => 1 + t / x) atTop (𝓝 1) := by
     simpa using (tendsto_inv_atTop_zero.const_mul t).const_add 1
-  refine (eventually_ge_of_tendsto_gt h₁ h₂).mono fun x hx => ?_
+  refine (h₂.eventually_const_le h₁).mono fun x hx => ?_
   have hx' : 0 < 1 + t / x := by linarith
   simp [mul_comm x, exp_mul, exp_log hx']
 
diff --git a/Mathlib/Analysis/SpecificLimits/Normed.lean b/Mathlib/Analysis/SpecificLimits/Normed.lean
@@ -601,7 +601,7 @@ theorem summable_of_ratio_test_tendsto_lt_one {α : Type*} [NormedAddCommGroup 
     (h : Tendsto (fun n ↦ ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)) : Summable f := by
   rcases exists_between hl₁ with ⟨r, hr₀, hr₁⟩
   refine summable_of_ratio_norm_eventually_le hr₁ ?_
-  filter_upwards [eventually_le_of_tendsto_lt hr₀ h, hf] with _ _ h₁
+  filter_upwards [h.eventually_le_const hr₀, hf] with _ _ h₁
   rwa [← div_le_iff₀ (norm_pos_iff.mpr h₁)]
 
 theorem not_summable_of_ratio_norm_eventually_ge {α : Type*} [SeminormedAddCommGroup α] {f : ℕ → α}
@@ -629,12 +629,12 @@ theorem not_summable_of_ratio_test_tendsto_gt_one {α : Type*} [SeminormedAddCom
     {f : ℕ → α} {l : ℝ} (hl : 1 < l) (h : Tendsto (fun n ↦ ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)) :
     ¬Summable f := by
   have key : ∀ᶠ n in atTop, ‖f n‖ ≠ 0 := by
-    filter_upwards [eventually_ge_of_tendsto_gt hl h] with _ hn hc
+    filter_upwards [h.eventually_const_le hl] with _ hn hc
     rw [hc, _root_.div_zero] at hn
     linarith
   rcases exists_between hl with ⟨r, hr₀, hr₁⟩
   refine not_summable_of_ratio_norm_eventually_ge hr₀ key.frequently ?_
-  filter_upwards [eventually_ge_of_tendsto_gt hr₁ h, key] with _ _ h₁
+  filter_upwards [h.eventually_const_le hr₁, key] with _ _ h₁
   rwa [← le_div_iff₀ (lt_of_le_of_ne (norm_nonneg _) h₁.symm)]
 
 section NormedDivisionRing
diff --git a/Mathlib/Computability/AkraBazzi/AkraBazzi.lean b/Mathlib/Computability/AkraBazzi/AkraBazzi.lean
@@ -544,9 +544,9 @@ lemma tendsto_atTop_sumCoeffsExp : Tendsto (fun (p : ℝ) => ∑ i, a i * (b i)
 lemma one_mem_range_sumCoeffsExp : 1 ∈ Set.range (fun (p : ℝ) => ∑ i, a i * (b i) ^ p) := by
   refine mem_range_of_exists_le_of_exists_ge R.continuous_sumCoeffsExp ?le_one ?ge_one
   case le_one =>
-    exact Eventually.exists <| eventually_le_of_tendsto_lt zero_lt_one R.tendsto_zero_sumCoeffsExp
+    exact R.tendsto_zero_sumCoeffsExp.eventually_le_const zero_lt_one |>.exists
   case ge_one =>
-    exact Eventually.exists <| R.tendsto_atTop_sumCoeffsExp.eventually_ge_atTop _
+    exact R.tendsto_atTop_sumCoeffsExp.eventually_ge_atTop _ |>.exists
 
 /-- The function x ↦ ∑ a_i b_i^x is injective. This implies the uniqueness of `p`. -/
 lemma injective_sumCoeffsExp : Function.Injective (fun (p : ℝ) => ∑ i, a i * (b i) ^ p) :=
diff --git a/Mathlib/Data/Real/Pi/Irrational.lean b/Mathlib/Data/Real/Pi/Irrational.lean
@@ -283,8 +283,8 @@ private lemma not_irrational_exists_rep {x : ℝ} :
     rwa [lt_div_iff₀ (by positivity), zero_mul] at this
   have k (n : ℕ) : 0 < (a : ℝ) ^ (2 * n + 1) / n ! := by positivity
   have j : ∀ᶠ n : ℕ in atTop, (a : ℝ) ^ (2 * n + 1) / n ! * I n (π / 2) < 1 := by
-    have := eventually_lt_of_tendsto_lt (show (0 : ℝ) < 1 / 2 by norm_num)
-              (tendsto_pow_div_factorial_at_top_aux a)
+    have := (tendsto_pow_div_factorial_at_top_aux a).eventually_lt_const
+      (show (0 : ℝ) < 1 / 2 by norm_num)
     filter_upwards [this] with n hn
     rw [lt_div_iff₀ (zero_lt_two : (0 : ℝ) < 2)] at hn
     exact hn.trans_le' (mul_le_mul_of_nonneg_left (I_le _) (by positivity))
diff --git a/Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean b/Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
@@ -193,12 +193,12 @@ theorem ae_le_of_forall_setLIntegral_le_of_sigmaFinite₀ [SigmaFinite μ]
       have : Tendsto (fun n => g x + u n) atTop (𝓝 (g x + (0 : ℝ≥0))) :=
         tendsto_const_nhds.add (ENNReal.tendsto_coe.2 u_lim)
       simp only [ENNReal.coe_zero, add_zero] at this
-      exact eventually_le_of_tendsto_lt hx this
+      exact this.eventually_le_const hx
     have L2 : ∀ᶠ n : ℕ in (atTop : Filter ℕ), g x ≤ (n : ℝ≥0) :=
-      haveI : Tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) atTop (𝓝 ∞) := by
+      have : Tendsto (fun n : ℕ => ((n : ℝ≥0) : ℝ≥0∞)) atTop (𝓝 ∞) := by
         simp only [ENNReal.coe_natCast]
         exact ENNReal.tendsto_nat_nhds_top
-      eventually_ge_of_tendsto_gt (hx.trans_le le_top) this
+      this.eventually_const_le (hx.trans_le le_top)
     apply Set.mem_iUnion.2
     exact ((L1.and L2).and (eventually_mem_spanningSets μ x)).exists
   refine le_antisymm ?_ bot_le
diff --git a/Mathlib/MeasureTheory/Function/LpSpace.lean b/Mathlib/MeasureTheory/Function/LpSpace.lean
@@ -673,8 +673,7 @@ theorem exists_eLpNorm_indicator_le (hp : p ≠ ∞) (c : E) {ε : ℝ≥0∞} (
       convert (NNReal.continuousAt_rpow_const (Or.inr hp₀')).tendsto.const_mul _
       simp [hp₀''.ne']
     have hε' : 0 < ε := hε.bot_lt
-    obtain ⟨δ, hδ, hδε'⟩ :=
-      NNReal.nhds_zero_basis.eventually_iff.mp (eventually_le_of_tendsto_lt hε' this)
+    obtain ⟨δ, hδ, hδε'⟩ := NNReal.nhds_zero_basis.eventually_iff.mp (this.eventually_le_const hε')
     obtain ⟨η, hη, hηδ⟩ := exists_between hδ
     refine ⟨η, hη, ?_⟩
     rw [← ENNReal.coe_rpow_of_nonneg _ hp₀', ← ENNReal.coe_mul]
diff --git a/Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean b/Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
@@ -388,9 +388,7 @@ theorem AECover.iSup_lintegral_eq_of_countably_generated [Nonempty ι] [l.NeBot]
   have := hφ.lintegral_tendsto_of_countably_generated hfm
   refine ciSup_eq_of_forall_le_of_forall_lt_exists_gt
     (fun i => lintegral_mono' Measure.restrict_le_self le_rfl) fun w hw => ?_
-  rcases exists_between hw with ⟨m, hm₁, hm₂⟩
-  rcases (eventually_ge_of_tendsto_gt hm₂ this).exists with ⟨i, hi⟩
-  exact ⟨i, lt_of_lt_of_le hm₁ hi⟩
+  exact (this.eventually_const_lt hw).exists
 
 end Lintegral
 
diff --git a/Mathlib/MeasureTheory/Measure/Portmanteau.lean b/Mathlib/MeasureTheory/Measure/Portmanteau.lean
@@ -278,14 +278,14 @@ theorem FiniteMeasure.limsup_measure_closed_le_of_tendsto {Ω ι : Type*} {L : F
     HasOuterApproxClosed.tendsto_lintegral_apprSeq F_closed (μ : Measure Ω)
   have room₁ : (μ : Measure Ω) F < (μ : Measure Ω) F + ε / 2 :=
     ENNReal.lt_add_right (measure_lt_top (μ : Measure Ω) F).ne ε_pos'
-  obtain ⟨M, hM⟩ := eventually_atTop.mp <| eventually_lt_of_tendsto_lt room₁ key₁
+  obtain ⟨M, hM⟩ := eventually_atTop.mp <| key₁.eventually_lt_const room₁
   have key₂ := FiniteMeasure.tendsto_iff_forall_lintegral_tendsto.mp μs_lim (fs M)
   have room₂ :
     (lintegral (μ : Measure Ω) fun a ↦ fs M a) <
       (lintegral (μ : Measure Ω) fun a ↦ fs M a) + ε / 2 :=
     ENNReal.lt_add_right (ne_of_lt ((fs M).lintegral_lt_top_of_nnreal _)) ε_pos'
-  have ev_near := Eventually.mono (eventually_lt_of_tendsto_lt room₂ key₂) fun n ↦ le_of_lt
-  have ev_near' := Eventually.mono ev_near
+  have ev_near := key₂.eventually_le_const room₂
+  have ev_near' := ev_near.mono
     (fun n ↦ le_trans (HasOuterApproxClosed.measure_le_lintegral F_closed (μs n) M))
   apply (Filter.limsup_le_limsup ev_near').trans
   rw [limsup_const]
diff --git a/Mathlib/MeasureTheory/Measure/WithDensity.lean b/Mathlib/MeasureTheory/Measure/WithDensity.lean
@@ -669,7 +669,7 @@ lemma IsLocallyFiniteMeasure.withDensity_coe {f : α → ℝ≥0} (hf : Continuo
     IsLocallyFiniteMeasure (μ.withDensity fun x ↦ f x) := by
   refine ⟨fun x ↦ ?_⟩
   rcases (μ.finiteAt_nhds x).exists_mem_basis ((nhds_basis_opens' x).restrict_subset
-    (eventually_le_of_tendsto_lt (lt_add_one _) (hf.tendsto x))) with ⟨U, ⟨⟨hUx, hUo⟩, hUf⟩, hμU⟩
+    ((hf.tendsto x).eventually_le_const (lt_add_one _))) with ⟨U, ⟨⟨hUx, hUo⟩, hUf⟩, hμU⟩
   refine ⟨U, hUx, ?_⟩
   rw [withDensity_apply _ hUo.measurableSet]
   exact setLIntegral_lt_top_of_bddAbove hμU.ne ⟨f x + 1, forall_mem_image.2 hUf⟩
diff --git a/Mathlib/Topology/Order/IntermediateValue.lean b/Mathlib/Topology/Order/IntermediateValue.lean
@@ -122,21 +122,20 @@ theorem IsPreconnected.intermediate_value {s : Set X} (hs : IsPreconnected s) {a
 theorem IsPreconnected.intermediate_value_Ico {s : Set X} (hs : IsPreconnected s) {a : X}
     {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α}
     (ht : Tendsto f l (𝓝 v)) : Ico (f a) v ⊆ f '' s := fun _ h =>
-  hs.intermediate_value₂_eventually₁ ha hl hf continuousOn_const h.1
-    (eventually_ge_of_tendsto_gt h.2 ht)
+  hs.intermediate_value₂_eventually₁ ha hl hf continuousOn_const h.1 (ht.eventually_const_le h.2)
 
 theorem IsPreconnected.intermediate_value_Ioc {s : Set X} (hs : IsPreconnected s) {a : X}
     {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s) {v : α}
     (ht : Tendsto f l (𝓝 v)) : Ioc v (f a) ⊆ f '' s := fun _ h =>
   (hs.intermediate_value₂_eventually₁ ha hl continuousOn_const hf h.2
-    (eventually_le_of_tendsto_lt h.1 ht)).imp fun _ h => h.imp_right Eq.symm
+    (ht.eventually_le_const h.1)).imp fun _ h => h.imp_right Eq.symm
 
 theorem IsPreconnected.intermediate_value_Ioo {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
     [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
     {v₁ v₂ : α} (ht₁ : Tendsto f l₁ (𝓝 v₁)) (ht₂ : Tendsto f l₂ (𝓝 v₂)) :
     Ioo v₁ v₂ ⊆ f '' s := fun _ h =>
   hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const
-    (eventually_le_of_tendsto_lt h.1 ht₁) (eventually_ge_of_tendsto_gt h.2 ht₂)
+    (ht₁.eventually_le_const h.1) (ht₂.eventually_const_le h.2)
 
 theorem IsPreconnected.intermediate_value_Ici {s : Set X} (hs : IsPreconnected s) {a : X}
     {l : Filter X} (ha : a ∈ s) [NeBot l] (hl : l ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
@@ -153,19 +152,19 @@ theorem IsPreconnected.intermediate_value_Ioi {s : Set X} (hs : IsPreconnected s
     [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
     {v : α} (ht₁ : Tendsto f l₁ (𝓝 v)) (ht₂ : Tendsto f l₂ atTop) : Ioi v ⊆ f '' s := fun y h =>
   hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const
-    (eventually_le_of_tendsto_lt h ht₁) (tendsto_atTop.1 ht₂ y)
+    (ht₁.eventually_le_const h) (ht₂.eventually_ge_atTop y)
 
 theorem IsPreconnected.intermediate_value_Iio {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
     [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
     {v : α} (ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ (𝓝 v)) : Iio v ⊆ f '' s := fun y h =>
-  hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (tendsto_atBot.1 ht₁ y)
-    (eventually_ge_of_tendsto_gt h ht₂)
+  hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (ht₁.eventually_le_atBot y)
+    (ht₂.eventually_const_le h)
 
 theorem IsPreconnected.intermediate_value_Iii {s : Set X} (hs : IsPreconnected s) {l₁ l₂ : Filter X}
     [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f : X → α} (hf : ContinuousOn f s)
     (ht₁ : Tendsto f l₁ atBot) (ht₂ : Tendsto f l₂ atTop) : univ ⊆ f '' s := fun y _ =>
-  hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (tendsto_atBot.1 ht₁ y)
-    (tendsto_atTop.1 ht₂ y)
+  hs.intermediate_value₂_eventually₂ hl₁ hl₂ hf continuousOn_const (ht₁.eventually_le_atBot y)
+    (ht₂.eventually_ge_atTop y)
 
 /-- **Intermediate Value Theorem** for continuous functions on connected spaces. -/
 theorem intermediate_value_univ [PreconnectedSpace X] (a b : X) {f : X → α} (hf : Continuous f) :
diff --git a/Mathlib/Topology/Order/OrderClosed.lean b/Mathlib/Topology/Order/OrderClosed.lean
@@ -213,14 +213,20 @@ theorem Ici_mem_nhds (h : a < b) : Ici a ∈ 𝓝 b :=
 
 theorem eventually_ge_nhds (hab : b < a) : ∀ᶠ x in 𝓝 a, b ≤ x := Ici_mem_nhds hab
 
-theorem eventually_gt_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)
+theorem Filter.Tendsto.eventually_const_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)
     (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u < f a :=
   h.eventually <| eventually_gt_nhds hv
 
-theorem eventually_ge_of_tendsto_gt {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)
+@[deprecated (since := "2024-11-17")]
+alias eventually_gt_of_tendsto_gt := Filter.Tendsto.eventually_const_lt
+
+theorem Filter.Tendsto.eventually_const_le {l : Filter γ} {f : γ → α} {u v : α} (hv : u < v)
     (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, u ≤ f a :=
   h.eventually <| eventually_ge_nhds hv
 
+@[deprecated (since := "2024-11-17")]
+alias eventually_ge_of_tendsto_gt := Filter.Tendsto.eventually_const_le
+
 protected theorem Dense.exists_gt [NoMaxOrder α] {s : Set α} (hs : Dense s) (x : α) :
     ∃ y ∈ s, x < y :=
   hs.exists_mem_open isOpen_Ioi (exists_gt x)
@@ -451,14 +457,20 @@ theorem Iic_mem_nhds (h : a < b) : Iic b ∈ 𝓝 a :=
 
 theorem eventually_le_nhds (hab : a < b) : ∀ᶠ x in 𝓝 a, x ≤ b := Iic_mem_nhds hab
 
-theorem eventually_lt_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)
+theorem Filter.Tendsto.eventually_lt_const {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)
     (h : Filter.Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a < u :=
   h.eventually <| eventually_lt_nhds hv
 
-theorem eventually_le_of_tendsto_lt {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)
+@[deprecated (since := "2024-11-17")]
+alias eventually_lt_of_tendsto_lt := Filter.Tendsto.eventually_lt_const
+
+theorem Filter.Tendsto.eventually_le_const {l : Filter γ} {f : γ → α} {u v : α} (hv : v < u)
     (h : Tendsto f l (𝓝 v)) : ∀ᶠ a in l, f a ≤ u :=
   h.eventually <| eventually_le_nhds hv
 
+@[deprecated (since := "2024-11-17")]
+alias eventually_le_of_tendsto_lt := Filter.Tendsto.eventually_le_const
+
 protected theorem Dense.exists_lt [NoMinOrder α] {s : Set α} (hs : Dense s) (x : α) :
     ∃ y ∈ s, y < x :=
   hs.orderDual.exists_gt x
