@@ -29,7 +29,7 @@ private lemma lemma_0 (p : ℕ) (B₁ B₂ A : Finset G) (f : G → ℝ) :
2929 refine Fintype.sum_equiv (Equiv.neg _) _ _ fun s ↦ ?_
3030 rw [← smul_mul_assoc, mul_smul_mul_comm, card_smul_mu_apply, card_smul_mu_apply,
3131 indicate_inter_apply, indicate_inter_apply, mul_mul_mul_comm, prod_mul_distrib]
32- simp [c, indicate_inf_apply, ← translate_indicate, sub_eq_add_neg, mul_assoc, add_comm]
32+ simp [c, indicate_inf_apply, sub_eq_add_neg, mul_assoc, add_comm]
3333
3434private lemma sum_c (p : ℕ) (B A : Finset G) : ∑ s, #(B ∩ c p A s) = #A ^ p * #B := by
3535 simp only [card_eq_sum_indicate, indicate_inter_apply, c, indicate_inf_apply, mul_sum, sum_mul,
@@ -91,7 +91,7 @@ lemma drc (hp₂ : 2 ≤ p) (f : G → ℝ≥0) (hf : ∃ x, x ∈ B₁ - B₂
9191 have : 0 ≤ μ_[ℝ] B₁ ○ μ B₂ * (𝟭 A ○ 𝟭 A) ^ p * (↑) ∘ f :=
9292 mul_nonneg (mul_nonneg (dconv_nonneg mu_nonneg mu_nonneg) <| pow_nonneg
9393 (dconv_nonneg indicate_nonneg indicate_nonneg) _) fun _ ↦ by simp -- positivity
94- refine Fintype.sum_pos <| this.gt_iff_ne .2 <| support_nonempty_iff.1 ?_
94+ refine Fintype.sum_pos <| this.lt_iff_ne' .2 <| support_nonempty_iff.1 ?_
9595 simp only [support_comp_eq, Set.Nonempty, and_assoc, support_mul', support_dconv,
9696 indicate_nonneg, mu_nonneg, support_indicate, support_mu, NNReal.coe_eq_zero, iff_self,
9797 forall_const, Set.mem_inter_iff, ← coe_sub, mem_coe, support_pow' _ hp₀, hf]
@@ -160,7 +160,7 @@ lemma drc (hp₂ : 2 ≤ p) (f : G → ℝ≥0) (hf : ∃ x, x ∈ B₁ - B₂
160160 _ ≤ M * (sqrt (∑ s, #(A₁ s)) * sqrt (∑ s, #(A₂ s))) := by
161161 gcongr; exact sum_sqrt_mul_sqrt_le _ fun i ↦ by positivity fun i ↦ by positivity
162162 _ = _ := ?_
163- · simp only [mem_filter, mem_univ, true_and, Nat.cast_nonneg, and_imp]
163+ · simp only [mem_filter, mem_univ, true_and, and_imp]
164164 exact fun s hsM hs ↦ mul_lt_mul_of_pos_right ((sqrt_lt' hM).2 hsM) <|
165165 sqrt_pos.2 <| (hg _).lt_of_ne' hs
166166 rw [sum_cast_c, sum_cast_c, sqrt_mul', sqrt_mul', mul_mul_mul_comm (sqrt _), mul_self_sqrt,
@@ -187,17 +187,16 @@ lemma sifting (B₁ B₂ : Finset G) (hε : 0 < ε) (hε₁ : ε ≤ 1) (hδ : 0
187187 c.Nonempty := by
188188 simp_rw [nonempty_iff_ne_empty]
189189 rintro rfl
190- simp [pow_mul', (zero_lt_four' ℝ).not_ge, inv_mul_le_iff₀ (zero_lt_four' ℝ), mul_assoc,
191- div_nonpos_iff, mul_nonpos_iff, (pow_pos (dLpNorm_conv_pos hp₀.ne' hB hA) 2 ).not_ge,
192- hp₀, hp₀.ne', hA.ne_empty] at h
190+ simp [pow_mul', inv_mul_le_iff₀ (zero_lt_four' ℝ), div_nonpos_iff,
191+ (pow_pos (dLpNorm_conv_pos hp₀.ne' hB hA) 2 ).not_ge, hp₀.ne', hA.ne_empty] at h
193192 have hA₁ : A₁.Nonempty := aux _ _ hcard₁
194193 have hA₂ : A₂.Nonempty := aux _ _ hcard₂
195194 clear hcard₁ hcard₂ aux
196195 rw [sub_le_comm]
197196 calc
198197 _ = ∑ x ∈ (s p ε B₁ B₂ A)ᶜ, (μ A₁ ○ μ A₂) x := ?_
199198 _ = ⟪μ_[ℝ] A₁ ○ μ A₂, (↑) ∘ 𝟭_[ℝ≥0 ] ((s (↑p) ε B₁ B₂ A)ᶜ)⟫_[ℝ] := by
200- simp [wInner_one_eq_sum, -mem_compl, -mem_s, apply_ite NNReal.toReal, indicate_apply]
199+ simp [wInner_one_eq_sum, -mem_compl, -mem_s, indicate_apply]
201200 _ ≤ _ := (le_div_iff₀ <| dLpNorm_conv_pos hp₀.ne' hB hA).2 h
202201 _ ≤ _ := ?_
203202 · simp_rw [sub_eq_iff_eq_add', sum_add_sum_compl, sum_dconv, map_mu]
@@ -252,8 +251,7 @@ lemma sifting_cor (hε : 0 < ε) (hε₁ : ε ≤ 1) (hδ : 0 < δ) (hp : Even p
252251 rw [nnratCast_dens, le_div_iff₀, ← mul_div_right_comm]
253252 calc
254253 _ = (‖𝟭_[ℝ] A ○ 𝟭 A‖_[1 , μ univ] : ℝ) := by
255- simp [mu, wLpNorm_smul_right, hp₀, dL1Norm_dconv, card_univ, inv_mul_eq_div]
256-
254+ simp [mu, wLpNorm_smul_right, dL1Norm_dconv, card_univ, inv_mul_eq_div]
257255 _ ≤ _ := wLpNorm_mono_right (one_le_two.trans <| by norm_cast) _ _
258256 · exact Nat.cast_pos.2 hA.card_pos
259257 obtain ⟨A₁, -, A₂, -, h, hcard₁, hcard₂⟩ :=
0 commit comments