by exact

Carleson/DoublingMeasure.lean

@@ -111,7 +111,7 @@ lemma measure_ball_le_pow_two {x : X} {r : ℝ} {n : ℕ} :
       _ ≤ A * μ.real (ball x (2 ^ m * r)) := by
         rw[mul_assoc]; norm_cast; exact measure_real_ball_two_le_same _ _
       _ ≤ A * (↑(A ^ m) * μ.real (ball x r)) := by
-        exact mul_le_mul_of_nonneg_left hm (by exact zero_le_coe)
+        exact mul_le_mul_of_nonneg_left hm zero_le_coe
       _ = A^(m.succ) * μ.real (ball x r) := by rw [NNReal.coe_pow,← mul_assoc, pow_succ']
 
 lemma measure_ball_le_pow_two' {x : X} {r : ℝ} {n : ℕ} :

Carleson/GridStructure.lean

@@ -114,7 +114,7 @@ postfix:max "ᵒ" => Grid.int
 /-- An auxiliary measure used in the well-foundedness of `Ω` in Lemma `tile_structure`. -/
 def opSize (i : Grid X) : ℕ := (S - s i).toNat
 
-lemma int_subset : i.int ⊆ i := by exact ball_subset_Grid
+lemma int_subset : i.int ⊆ i := ball_subset_Grid
 
 end Grid
 end Generic

Carleson/RealInterpolation.lean

@@ -1693,7 +1693,7 @@ lemma rpow_le_rpow_of_exponent_le_base_le {a b t γ : ℝ} (ht : t > 0) (htγ :
 lemma rpow_le_rpow_of_exponent_le_base_ge {a b t γ : ℝ} (hγ : γ > 0) (htγ : γ ≤ t) (hab : a ≤ b) :
     ENNReal.ofReal (t ^ a) ≤ ENNReal.ofReal (t ^ b) * ENNReal.ofReal (γ ^ (a - b)) := by
   rw [mul_comm]
-  have t_pos : 0 < t := by exact gt_of_ge_of_gt htγ hγ
+  have t_pos : 0 < t := gt_of_ge_of_gt htγ hγ
   rw [Real.rpow_sub hγ]
   refine (ENNReal.mul_le_mul_left (a := ENNReal.ofReal (γ ^ (-a) )) ?_ coe_ne_top).mp ?_
   · exact (ofReal_pos.mpr (Real.rpow_pos_of_pos hγ (-a))).ne.symm
@@ -1799,7 +1799,7 @@ lemma estimate_eLpNorm_trunc {p q : ℝ≥0∞}
   have p_ne_top : p ≠ ⊤ := LT.lt.ne_top hpq.2
   have : p ≠ 0 := hpq.1.ne.symm
   have : q ≠ 0 := (lt_trans hpq.1 hpq.2).ne.symm
-  have q_toReal_pos : q.toReal > 0 := by exact toReal_pos this hq
+  have q_toReal_pos : q.toReal > 0 := toReal_pos this hq
   split_ifs
   · contradiction
   · calc
@@ -1939,7 +1939,7 @@ lemma res'comp (j : Bool) (β : ℝ) (hβ : β > 0) :
       exact ⟨lt_trans hβ h, fun _ ↦ h⟩
   · ext x
     simp only [Ioi_diff_Ici, mem_Ioo]
-  · have : j = false := by exact eq_false_of_ne_true h₀
+  · have : j = false := eq_false_of_ne_true h₀
     rw [this] at h₂
     simp at h₂
 
@@ -2649,7 +2649,7 @@ lemma estimate_trnc {p₀ q₀ q : ℝ} {spf : ScaledPowerFunction} {j : Bool}
   unfold eLpNorm eLpNorm'
   let tc := spf_to_tc spf
   split_ifs with is_p₀pos is_p₀top
-  · have : p₀ ≤ 0 := by exact ofReal_eq_zero.mp is_p₀pos
+  · have : p₀ ≤ 0 := ofReal_eq_zero.mp is_p₀pos
     contrapose! this; exact hp₀
   · contrapose! is_p₀top; exact coe_ne_top
   · have coe_p₀ : (ENNReal.ofReal p₀).toReal = p₀ := by
@@ -3515,22 +3515,22 @@ lemma estimate_norm_rpow_range_operator'
     apply setLIntegral_mono' measurableSet_Ioi
     intro s (s_pos : s > 0)
     simp only [is_q₀top, mem_Ioi, false_or] at hq₀'
-    have : q₀ = ⊤ := by exact not_lt_top.mp is_q₀top
+    have : q₀ = ⊤ := not_lt_top.mp is_q₀top
     rw [hq₀' this s s_pos, zero_mul, add_zero]
     gcongr
     apply weaktype_estimate_trunc p_pos <;> try assumption
   · rw [one_mul, zero_mul, zero_add]
     apply setLIntegral_mono' measurableSet_Ioi
     intro s (s_pos : s > 0)
     simp only [is_q₁top, mem_Ioi, false_or] at hq₁'
-    have : q₁ = ⊤ := by exact not_lt_top.mp is_q₁top
+    have : q₁ = ⊤ := not_lt_top.mp is_q₁top
     rw [hq₁' this s s_pos, zero_mul, zero_add]
     gcongr
     apply weaktype_estimate_trunc_compl (p₀ := p₀) <;> try assumption
     · exact LT.lt.ne_top hp₁p
     · exact tc.ran_ton s s_pos
   · simp only [zero_mul, add_zero, nonpos_iff_eq_zero]
-    have : ∫⁻ (_ : ℝ) in Ioi 0, 0 = 0 := by exact lintegral_zero
+    have : ∫⁻ (_ : ℝ) in Ioi 0, 0 = 0 := lintegral_zero
     rw [← this]
     apply lintegral_congr_ae
     filter_upwards [self_mem_ae_restrict measurableSet_Ioi] with s (s_pos)
@@ -3562,7 +3562,7 @@ lemma simplify_factor₀ {D : ℝ}
   have q₁pos : q₁ > 0 := lt_of_lt_of_le hp₁.1 hp₁.2
   have p₀ne_top : p₀ ≠ ⊤ := ne_top_of_le_ne_top hq₀' hp₀.2
   have hp' : p ∈ Ioo 0 ⊤ := interp_exp_in_Ioo_zero_top ht p₀pos p₁pos (by left; exact p₀ne_top) hp
-  have p_toReal_pos : p.toReal > 0 := by exact toReal_pos_of_Ioo hp'
+  have p_toReal_pos : p.toReal > 0 := toReal_pos_of_Ioo hp'
   rw [hD]
   unfold d
   rw [← ofReal_rpow_of_pos]
@@ -3811,7 +3811,7 @@ lemma combine_estimates₀ {A : ℝ} (hA : A > 0)
   have q₀pos : q₀ > 0 := lt_of_lt_of_le hp₀.1 hp₀.2
   have p₁pos : p₁ > 0 := hp₁.1
   have q₁pos : q₁ > 0 := lt_of_lt_of_le hp₁.1 hp₁.2
-  have p_pos : p > 0 := by exact interpolated_pos' one_le_p₀ one_le_p1 hp
+  have p_pos : p > 0 := interpolated_pos' one_le_p₀ one_le_p1 hp
   have : SigmaFinite (μ.restrict (Function.support f)) := by
     apply support_sigma_finite_from_Memℒp (p := p) hf
     · exact interp_exp_ne_top (ne_of_lt hp₀p₁) ht hp
@@ -3997,7 +3997,7 @@ lemma simplify_factor₃ [NormedAddCommGroup E₁] (hp₀ : p₀ > 0) (hp₀' :
     (hp : p⁻¹ = (1 - (ENNReal.ofReal t)) * p₀⁻¹ + (ENNReal.ofReal t) * p₁⁻¹) (hp₀p₁ : p₀ = p₁) :
     C₀ ^ q₀.toReal * (eLpNorm f p μ ^ p.toReal) ^ (q₀.toReal / p₀.toReal) =
     (↑C₀ * eLpNorm f p μ) ^ q₀.toReal := by
-  have hp' : p = p₀ := by exact Eq.symm (interp_exp_eq hp₀p₁ ht hp) -- TODO abstract
+  have hp' : p = p₀ := (interp_exp_eq hp₀p₁ ht hp).symm -- TODO abstract
   rw [hp']
   rw [ENNReal.mul_rpow_of_nonneg, ← ENNReal.rpow_mul, ← mul_div_assoc, mul_div_cancel_left₀] <;>
       try positivity
@@ -4597,7 +4597,7 @@ lemma Subadditive_trunc_from_SubadditiveOn_Lp₀p₁ {p₀ p₁ p : ℝ≥0∞}
       · exact interpolated_pos' hp₀ hp₁ hp
       · exact (interp_exp_between hp₀ hp₁ p₀lt_p₁ ht hp).2
     · left
-      have : p₀ = p := by exact interp_exp_eq p₀eq_p₁ ht hp
+      have : p₀ = p := interp_exp_eq p₀eq_p₁ ht hp
       rw [this]
       exact trunc_preserves_Lp hf
     · left
@@ -4613,7 +4613,7 @@ lemma Subadditive_trunc_from_SubadditiveOn_Lp₀p₁ {p₀ p₁ p : ℝ≥0∞}
         · exact hp₀
         · exact (interp_exp_between hp₀ hp₁ p₀lt_p₁ ht hp).1
     · left
-      have : p₀ = p := by exact interp_exp_eq p₀eq_p₁ ht hp
+      have : p₀ = p := interp_exp_eq p₀eq_p₁ ht hp
       rw [this]
       exact trunc_compl_preserves_Lp hf
     · right

Carleson/TileExistence.lean

@@ -399,7 +399,7 @@ mutual
       rw [I2,dif_pos hk_s]
       exact measurableSet_ball
     else
-      let hk'' : -S < k := by exact lt_of_le_of_ne hk fun a_1 ↦ hk_s (id (Eq.symm a_1))
+      let hk'' : -S < k := lt_of_le_of_ne hk fun a_1 ↦ hk_s (id (Eq.symm a_1))
       have : (S + (k-1)).toNat < (S + k).toNat := by
         rw [Int.lt_toNat,Int.toNat_of_nonneg (by linarith)]
         linarith
@@ -534,7 +534,7 @@ mutual
     else
       simp_rw [dif_neg hk_s]
       intro x hx
-      have : -S < k := by exact lt_of_le_of_ne hk fun a_1 ↦ hk_s (id (Eq.symm a_1))
+      have : -S < k := lt_of_le_of_ne hk fun a_1 ↦ hk_s (id (Eq.symm a_1))
       have : ((2 * (S + (k - 1))).toNat : ℤ) + 1 < 2 * (S + k) := by
         rw [Int.toNat_of_nonneg (by linarith)]
         linarith
@@ -1583,8 +1583,8 @@ lemma boundary_measure {k:ℤ} (hk:-S ≤ k) (y:Yk X k) {t:ℝ≥0} (ht:t∈ Set
           simp only at this
           nth_rw 1 [NNReal.val_eq_coe] at this
           simp_rw [← Real.rpow_intCast] at this
-          · rw [← ENNReal.ofReal_le_ofReal_iff (Real.rpow_nonneg (realD_nonneg) _), ENNReal.ofReal_mul (by exact ht.left.le),
-              ENNReal.ofReal_coe_nnreal,
+          · rw [← ENNReal.ofReal_le_ofReal_iff (Real.rpow_nonneg (realD_nonneg) _),
+              ENNReal.ofReal_mul (by exact ht.left.le), ENNReal.ofReal_coe_nnreal,
               ← ENNReal.ofReal_rpow_of_pos (defaultD_pos a),← ENNReal.ofReal_rpow_of_pos (defaultD_pos a),
               ENNReal.ofReal_natCast, ENNReal.rpow_intCast, ENNReal.rpow_intCast] at this
             exact this
@@ -2137,7 +2137,7 @@ lemma Ω_RFD {p q : 𝔓 X} (h𝓘 : 𝓘 p ≤ 𝓘 q) : Disjoint (Ω p) (Ω q)
     rw [mem_iUnion] at zi; obtain ⟨a, ma⟩ := zi -- Paper's `q'` is `⟨J, a⟩`
     have nmaxJ : ¬IsMax J := by
       by_contra maxJ; rw [Grid.isMax_iff] at maxJ
-      rw [maxJ, show s topCube = S by exact s_topCube (X := X)] at sJ
+      rw [maxJ, show s topCube = S from s_topCube (X := X)] at sJ
       have : 𝔰 q ≤ S := (range_s_subset ⟨q.1, rfl⟩).2
       omega
     have succJ : J.succ = q.1 := (Grid.succ_def nmaxJ).mpr ⟨ubJ, by change 𝔰 q = _; omega⟩

Carleson/ToMathlib/Misc.lean

@@ -66,13 +66,13 @@ section ENNReal
 
 lemma tsum_one_eq' {α : Type*} (s : Set α) : ∑' (_:s), (1 : ℝ≥0∞) = s.encard := by
   if hfin : s.Finite then
-    have hfin' : Finite s := by exact hfin
+    have hfin' : Finite s := hfin
     rw [tsum_def]
     simp only [ENNReal.summable, ↓reduceDIte]
     have hsup: support (fun (_ : s) ↦ (1 : ℝ≥0∞)) = Set.univ := by
       ext i
       simp only [mem_support, ne_eq, one_ne_zero, not_false_eq_true, mem_univ]
-    have hsupfin: (Set.univ : Set s).Finite := by exact finite_univ
+    have hsupfin: (Set.univ : Set s).Finite := finite_univ
     rw [← hsup] at hsupfin
     rw [if_pos hsupfin]
     rw [hfin.encard_eq_coe_toFinset_card]
@@ -94,7 +94,7 @@ lemma tsum_one_eq' {α : Type*} (s : Set α) : ∑' (_:s), (1 : ℝ≥0∞) = s.
       simp only [Finite.mem_toFinset, mem_support, ne_eq, one_ne_zero, not_false_eq_true,
         exists_const]
   else
-  have : Infinite s := by exact infinite_coe_iff.mpr hfin
+  have : Infinite s := infinite_coe_iff.mpr hfin
   rw [ENNReal.tsum_const_eq_top_of_ne_zero (by norm_num)]
   rw [Set.encard_eq_top_iff.mpr hfin]
   simp only [ENat.toENNReal_top]