mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : FiniteDimensional 𝕜 ↥K
hK : FiniteDimensional 𝕜 ↥Kᗮ
e : (↥K × ↥Kᗮ) ≃ₗ[𝕜] E := K.prodEquivOfIsCompl Kᗮ ⋯
b : Basis (Fin (finrank 𝕜 ↥K) ⊕ Fin (finrank 𝕜 ↥Kᗮ)) 𝕜 (↥K × ↥Kᗮ) := (finBasis 𝕜 ↥K).prod (finBasis 𝕜 ↥Kᗮ)
this : (LinearMap.toMatrix b b) (↑e.symm ∘ₗ ↑K.reflection.toLinearEquiv ∘ₗ ↑e.symm.symm) = Matrix.fromBlocks 1 0 0 (-1)
⊢ (Matrix.fromBlocks 1 0 0 (-1)).det = (-1) ^ finrank 𝕜 ↥Kᗮ

Tactic:
Matrix.det_fromBlocks_zero₂₁,

Goal state after:
case pos
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : FiniteDimensional 𝕜 ↥K
hK : FiniteDimensional 𝕜 ↥Kᗮ
e : (↥K × ↥Kᗮ) ≃ₗ[𝕜] E := K.prodEquivOfIsCompl Kᗮ ⋯
b : Basis (Fin (finrank 𝕜 ↥K) ⊕ Fin (finrank 𝕜 ↥Kᗮ)) 𝕜 (↥K × ↥Kᗮ) := (finBasis 𝕜 ↥K).prod (finBasis 𝕜 ↥Kᗮ)
this : (LinearMap.toMatrix b b) (↑e.symm ∘ₗ ↑K.reflection.toLinearEquiv ∘ₗ ↑e.symm.symm) = Matrix.fromBlocks 1 0 0 (-1)
⊢ Matrix.det 1 * (-1).det = (-1) ^ finrank 𝕜 ↥Kᗮ
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
K : Submodule 𝕜 E
n : ℕ
h_dim : finrank 𝕜 ↥K + n = finrank 𝕜 E
⊢ finrank 𝕜 ↥K + finrank 𝕜 ↥Kᗮ = finrank 𝕜 ↥K + n

Tactic:
simp [Submodule.finrank_add_finrank_orthogonal, h_dim]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
⊢ (↑U.starProjection).range = U

Tactic:
ext x
  exact ⟨fun ⟨y, hy⟩ ↦ hy ▸ coe_mem (U.orthogonalProjectionOnto y),
    fun h ↦ ⟨x, starProjection_eq_self_iff.mpr h⟩⟩

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
K : Submodule 𝕜 E
inst✝ : K.HasOrthogonalProjection
⊢ ContinuousLinearMap.id 𝕜 E = K.starProjection + Kᗮ.starProjection

Tactic:
by
  ext w
  exact (K.starProjection_add_starProjection_orthogonal w).symm

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: case inr
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : y ≠ 0
⊢ ∃ x, dist (f x) y ≤ 1 / 2 * ‖y‖ ∧ ‖x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖

Tactic:
have hc' : 1 < ‖σ c‖ := by simp only [RingHomIsometric.norm_map, hc]
    rcases rescale_to_shell hc' (half_pos εpos) hy with ⟨d, hd, ydlt, -, dinv⟩
    let δ := ‖d‖ * ‖y‖ / 4
    have δpos : 0 < δ := by positivity
    have : a + d • y ∈ ball a ε := by
      simp [dist_eq_norm, lt_of_le_of_lt ydlt.le (half_lt_self εpos)]
    rcases Metric.mem_closure_iff.1 (H this) _ δpos with ⟨z₁, z₁im, h₁⟩
    rcases (mem_image _ _ _).1 z₁im with ⟨x₁, hx₁, xz₁⟩
    rw [← xz₁] at h₁
    rw [mem_ball, dist_eq_norm, sub_zero] at hx₁
    have : a ∈ ball a ε := by
      simp only [mem_ball, dist_self]
      exact εpos
    rcases Metric.mem_closure_iff.1 (H this) _ δpos with ⟨z₂, z₂im, h₂⟩
    rcases (mem_image _ _ _).1 z₂im with ⟨x₂, hx₂, xz₂⟩
    rw [← xz₂] at h₂
    rw [mem_ball, dist_eq_norm, sub_zero] at hx₂
    let x := x₁ - x₂
    have I : ‖f x - d • y‖ ≤ 2 * δ :=
      calc
        ‖f x - d • y‖ = ‖f x₁ - (a + d • y) - (f x₂ - a)‖ := by
          congr 1
          simp only [x, f.map_sub]
          abel
        _ ≤ ‖f x₁ - (a + d • y)‖ + ‖f x₂ - a‖ := norm_sub_le _ _
        _ ≤ 2 * δ := by grind [dist_eq_norm']
    have J : ‖f (σ' d⁻¹ • x) - y‖ ≤ 1 / 2 * ‖y‖ :=
      calc
        ‖f (σ' d⁻¹ • x) - y‖ = ‖d⁻¹ • f x - (d⁻¹ * d) • y‖ := by
          rwa [f.map_smulₛₗ _, inv_mul_cancel₀, one_smul, map_inv₀, map_inv₀,
            RingHomCompTriple.comp_apply, RingHom.id_apply]
        _ = ‖d⁻¹ • (f x - d • y)‖ := by rw [mul_smul, smul_sub]
        _ = ‖d‖⁻¹ * ‖f x - d • y‖ := by rw [norm_smul, norm_inv]
        _ ≤ ‖d‖⁻¹ * (2 * δ) := by gcongr
        _ = 1 / 2 * ‖y‖ := by simp [δ, field]; norm_num
    rw [← dist_eq_norm] at J
    have K : ‖σ' d⁻¹ • x‖ ≤ (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖ :=
      calc
        ‖σ' d⁻¹ • x‖ = ‖d‖⁻¹ * ‖x₁ - x₂‖ := by rw [norm_smul, RingHomIsometric.norm_map, norm_inv]
        _ ≤ (ε / 2)⁻¹ * ‖c‖ * ‖y‖ * (n + n) := by
          gcongr
          · simpa using dinv
          · exact le_trans (norm_sub_le _ _) (by gcongr)
        _ = (ε / 2)⁻¹ * ‖c‖ * 2 * ↑n * ‖y‖ := by ring
    exact ⟨σ' d⁻¹ • x, J, K⟩

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝⁴ : RingHomInvPair σ σ'
inst✝³ : RingHomIsometric σ
inst✝² : RingHomIsometric σ'
inst✝¹ : CompleteSpace F
inst✝ : CompleteSpace E
surj : Surjective ⇑f
C : ℝ
C0 : C ≥ 0
g : F → E
hg : ∀ (y : F), dist (f (g y)) y ≤ 1 / 2 * ‖y‖ ∧ ‖g y‖ ≤ C * ‖y‖
h : F → F := fun y ↦ y - f (g y)
hle : ∀ (y : F), ‖h y‖ ≤ 1 / 2 * ‖y‖
y : F
hnle : ∀ (n : ℕ), ‖h^[n] y‖ ≤ (1 / 2) ^ n * ‖y‖
u : ℕ → E := fun n ↦ g (h^[n] y)
ule : ∀ (n : ℕ), ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖)
sNu : Summable fun n ↦ ‖u n‖
su : Summable u
x : E := tsum u
⊢ (∑' (n : ℕ), (1 / 2) ^ n) * (C * ‖y‖) = 2 * C * ‖y‖

Tactic:
by rw [tsum_geometric_two, mul_assoc]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:

Tactic:
edist x.snd y.snd ≤ (edist x.snd y.snd ^ p.toReal) ^ (1 / p.toReal) := by
          simp only [← ENNReal.rpow_mul, cancel, ENNReal.rpow_one, le_refl]
        _ ≤ (edist x.fst y.fst ^ p.toReal + edist x.snd y.snd ^ p.toReal) ^ (1 / p.toReal) := by
          gcongr
          simp only [self_le_add_left]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v : Set E
hv : Orthonormal 𝕜 Subtype.val
x : E
hx' : x ∈ (span 𝕜 v)ᗮ
hx : x ≠ 0
e : E := (↑‖x‖)⁻¹ • x
he : ‖e‖ = 1
he' : e ∈ (span 𝕜 v)ᗮ
hev : e ∈ v
this : e ∈ span 𝕜 v ⊓ (span 𝕜 v)ᗮ
⊢ e = 0

Tactic:
simpa [(span 𝕜 v).inf_orthogonal_eq_bot] using this

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: p : ℝ≥0∞
α : Type u_2
β : Type u_3
hp : Fact (1 ≤ p)
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
x y : WithLp p (α × β)
h : 1 ≤ p.toReal
pos : 0 < p.toReal
nonneg : 0 ≤ 1 / p.toReal
cancel : p.toReal * (1 / p.toReal) = 1
⊢ (edist x.ofLp y.ofLp ^ p.toReal + edist x.ofLp y.ofLp ^ p.toReal) ^ (1 / p.toReal) =
    ↑(2 ^ (1 / p.toReal)) * edist x.ofLp y.ofLp

Tactic:
simp only [← two_mul, ENNReal.mul_rpow_of_nonneg _ _ nonneg, ← ENNReal.rpow_mul, cancel,
          ENNReal.rpow_one, ENNReal.coe_rpow_of_nonneg _ nonneg, coe_ofNat]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp✝ : Fact (1 ≤ p)
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Module 𝕜 α
inst✝² : Module 𝕜 β
inst✝¹ : IsBoundedSMul 𝕜 α
inst✝ : IsBoundedSMul 𝕜 β
c : 𝕜
f : WithLp p (α × β)
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
⊢ ‖c • f‖₊ ≤ ‖c‖₊ * ‖f‖₊

Tactic:
have hpt : p ≠ ⊤ := p.toReal_pos_iff_ne_top.mp hp0

Goal state after:
case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp✝ : Fact (1 ≤ p)
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Module 𝕜 α
inst✝² : Module 𝕜 β
inst✝¹ : IsBoundedSMul 𝕜 α
inst✝ : IsBoundedSMul 𝕜 β
c : 𝕜
f : WithLp p (α × β)
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ‖c • f‖₊ ≤ ‖c‖₊ * ‖f‖₊
Module: Mathlib.Analysis.Normed.Lp.ProdLp

Goal state before:
pp: case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp✝ : Fact (1 ≤ p)
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Module 𝕜 α
inst✝² : Module 𝕜 β
inst✝¹ : NormSMulClass 𝕜 α
inst✝ : NormSMulClass 𝕜 β
c : 𝕜
f : WithLp p (α × β)
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
⊢ ‖c • f‖₊ = ‖c‖₊ * ‖f‖₊

Tactic:
have hpt : p ≠ ⊤ := p.toReal_pos_iff_ne_top.mp hp0

Goal state after:
case inr
p : ℝ≥0∞
𝕜 : Type u_1
α : Type u_2
β : Type u_3
hp✝ : Fact (1 ≤ p)
inst✝⁶ : SeminormedAddCommGroup α
inst✝⁵ : SeminormedAddCommGroup β
inst✝⁴ : SeminormedRing 𝕜
inst✝³ : Module 𝕜 α
inst✝² : Module 𝕜 β
inst✝¹ : NormSMulClass 𝕜 α
inst✝ : NormSMulClass 𝕜 β
c : 𝕜
f : WithLp p (α × β)
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ‖c • f‖₊ = ‖c‖₊ * ‖f‖₊
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝¹ : Fintype ι
inst✝ : (i : ι) → SeminormedAddCommGroup (β i)
x y : PiLp p β
⊢ dist x y = ‖-x + y‖

Tactic:
rcases p.dichotomy with (rfl | h)
    · simp only [dist_eq_iSup, norm_eq_ciSup, dist_eq_norm, add_apply, neg_apply, norm_neg_add]
    · have : p ≠ ∞ := by
        intro hp
        rw [hp, ENNReal.toReal_top] at h
        linarith
      simp only [dist_eq_sum (zero_lt_one.trans_le h), norm_eq_sum (zero_lt_one.trans_le h),
        dist_eq_norm, add_apply, neg_apply, norm_neg_add]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case inr
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp✝ : Fact (1 ≤ p)
inst✝⁴ : Fintype ι
inst✝³ : SeminormedRing 𝕜
inst✝² : (i : ι) → SeminormedAddCommGroup (β i)
inst✝¹ : (i : ι) → Module 𝕜 (β i)
inst✝ : ∀ (i : ι), IsBoundedSMul 𝕜 (β i)
c : 𝕜
f : PiLp p β
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
⊢ ‖c • f‖₊ ≤ ‖c‖₊ * ‖f‖₊

Tactic:
have hpt : p ≠ ⊤ := p.toReal_pos_iff_ne_top.mp hp0

Goal state after:
case inr
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp✝ : Fact (1 ≤ p)
inst✝⁴ : Fintype ι
inst✝³ : SeminormedRing 𝕜
inst✝² : (i : ι) → SeminormedAddCommGroup (β i)
inst✝¹ : (i : ι) → Module 𝕜 (β i)
inst✝ : ∀ (i : ι), IsBoundedSMul 𝕜 (β i)
c : 𝕜
f : PiLp p β
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ‖c • f‖₊ ≤ ‖c‖₊ * ‖f‖₊
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case inr
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp✝ : Fact (1 ≤ p)
inst✝⁴ : Fintype ι
inst✝³ : SeminormedRing 𝕜
inst✝² : (i : ι) → SeminormedAddCommGroup (β i)
inst✝¹ : (i : ι) → Module 𝕜 (β i)
inst✝ : ∀ (i : ι), NormSMulClass 𝕜 (β i)
c : 𝕜
f : PiLp p β
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
⊢ ‖c • f‖₊ = ‖c‖₊ * ‖f‖₊

Tactic:
have hpt : p ≠ ⊤ := p.toReal_pos_iff_ne_top.mp hp0

Goal state after:
case inr
p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
α : ι → Type u_3
β : ι → Type u_4
hp✝ : Fact (1 ≤ p)
inst✝⁴ : Fintype ι
inst✝³ : SeminormedRing 𝕜
inst✝² : (i : ι) → SeminormedAddCommGroup (β i)
inst✝¹ : (i : ι) → Module 𝕜 (β i)
inst✝ : ∀ (i : ι), NormSMulClass 𝕜 (β i)
c : 𝕜
f : PiLp p β
hp : 1 ≤ p.toReal
hp0 : 0 < p.toReal
hpt : p ≠ ∞
⊢ ‖c • f‖₊ = ‖c‖₊ * ‖f‖₊
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: a : ℝ
h : a ≠ 0
⊢ ofReal |a| • Measure.map (fun x ↦ a * x) volume = volume

Tactic:
by
  refine (Real.measure_ext_Ioo_rat fun p q => ?_).symm
  rcases lt_or_gt_of_ne h with h | h
  · simp only [Real.volume_Ioo, Measure.smul_apply, ← ENNReal.ofReal_mul (le_of_lt <| neg_pos.2 h),
      Measure.map_apply (measurable_const_mul a) measurableSet_Ioo, neg_sub_neg, neg_mul,
      preimage_const_mul_Ioo_of_neg _ _ h, abs_of_neg h, mul_sub, smul_eq_mul,
      mul_div_cancel₀ _ (ne_of_lt h)]
  · simp only [Real.volume_Ioo, Measure.smul_apply, ← ENNReal.ofReal_mul (le_of_lt h),
      Measure.map_apply (measurable_const_mul a) measurableSet_Ioo, preimage_const_mul_Ioo₀ _ _ h,
      abs_of_pos h, mul_sub, mul_div_cancel₀ _ (ne_of_gt h), smul_eq_mul]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
b : OrthonormalBasis ι ℝ ℝ
⊢ parallelepiped ⇑b = Icc 0 1 ∨ parallelepiped ⇑b = Icc (-1) 0

Tactic:
by
  have e : ι ≃ Fin 1 := by
    apply Fintype.equivFinOfCardEq
    simp only [← finrank_eq_card_basis b.toBasis, finrank_self]
  have B : parallelepiped (b.reindex e) = parallelepiped b := by
    convert! parallelepiped_comp_equiv b e.symm
    ext i
    simp only [OrthonormalBasis.coe_reindex]
  rw [← B]
  let F : ℝ → Fin 1 → ℝ := fun t => fun _i => t
  have A : Icc (0 : Fin 1 → ℝ) 1 = F '' Icc (0 : ℝ) 1 := by
    apply Subset.antisymm
    · intro x hx
      refine ⟨x 0, ⟨hx.1 0, hx.2 0⟩, ?_⟩
      ext j
      simp only [F, Subsingleton.elim j 0]
    · rintro x ⟨y, hy, rfl⟩
      exact ⟨fun _j => hy.1, fun _j => hy.2⟩
  rcases orthonormalBasis_one_dim (b.reindex e) with (H | H)
  · left
    simp_rw [parallelepiped, H, A, smul_eq_mul, mul_one]
    simp only [F, Finset.univ_unique, Fin.default_eq_zero, Finset.sum_singleton,
      ← image_comp, Function.comp_apply, image_id']
  · right
    simp_rw [H, parallelepiped, smul_eq_mul, A]
    simp only [F, Finset.univ_unique, Fin.default_eq_zero, mul_neg, mul_one, Finset.sum_neg_distrib,
      Finset.sum_singleton, ← image_comp, Function.comp, image_neg_eq_neg, neg_Icc, neg_zero]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case coe
ι : Type u_2
β : ι → Type u_4
inst✝² : Fintype ι
inst✝¹ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝ : DecidableEq ι
i : ι
b : β i
this : Nonempty ι
p : ℝ≥0
hp : Fact (1 ≤ ↑p)
hp0 : ↑p ≠ 0
⊢ ‖b‖₊ ^ 1 = ‖b‖₊

pp: case coe
ι : Type u_2
β : ι → Type u_4
inst✝² : Fintype ι
inst✝¹ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝ : DecidableEq ι
i : ι
b : β i
this : Nonempty ι
p : ℝ≥0
hp : Fact (1 ≤ ↑p)
hp0 : ↑p ≠ 0
⊢ ∀ (x : ι), x ≠ i → ‖(single (↑p) i b).ofLp x‖₊ ^ ↑p = 0

Tactic:
NNReal.rpow_one

Goal state after:
case coe
ι : Type u_2
β : ι → Type u_4
inst✝² : Fintype ι
inst✝¹ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝ : DecidableEq ι
i : ι
b : β i
this : Nonempty ι
p : ℝ≥0
hp : Fact (1 ≤ ↑p)
hp0 : ↑p ≠ 0
⊢ ‖b‖₊ = ‖b‖₊

case coe
ι : Type u_2
β : ι → Type u_4
inst✝² : Fintype ι
inst✝¹ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝ : DecidableEq ι
i : ι
b : β i
this : Nonempty ι
p : ℝ≥0
hp : Fact (1 ≤ ↑p)
hp0 : ↑p ≠ 0
⊢ ∀ (x : ι), x ≠ i → ‖(single (↑p) i b).ofLp x‖₊ ^ ↑p = 0
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
hp : Fact (1 ≤ p)
inst✝² : Fintype ι
inst✝¹ : (i : ι) → SeminormedAddCommGroup (β i)
inst✝ : DecidableEq ι
i : ι
b₁ b₂ : β i
⊢ edist (single p i b₁) (single p i b₂) = edist b₁ b₂

Tactic:
by
  simp only [edist_nndist, nndist_single_same p β i b₁ b₂]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: case hmul
ι : Type u_1
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
M : Matrix ι ι ℝ
hM : M.det ≠ 0
⊢ ∀ (A B : Matrix ι ι ℝ),
    A.det ≠ 0 →
      B.det ≠ 0 →
        Measure.map (⇑(toLin' A)) volume = ofReal |A.det⁻¹| • volume →
          Measure.map (⇑(toLin' B)) volume = ofReal |B.det⁻¹| • volume →
            Measure.map (⇑(toLin' (A * B))) volume = ofReal |(A * B).det⁻¹| • volume

Tactic:
· intro A B _ _ IHA IHB
    rw [toLin'_mul, det_mul, LinearMap.coe_comp, ← Measure.map_map, IHB, Measure.map_smul, IHA,
      smul_smul, ← ENNReal.ofReal_mul (abs_nonneg _), ← abs_mul, mul_comm, mul_inv]
    · apply Continuous.measurable
      apply LinearMap.continuous_on_pi
    · apply Continuous.measurable
      apply LinearMap.continuous_on_pi

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s : Set E
u : ℕ → E
sb : Bornology.IsBounded s
hu : Bornology.IsBounded (range u)
hs : Pairwise (Disjoint on fun n ↦ {u n} + s)
h's : MeasurableSet s
h : ¬μ s = 0
⊢ ∞ < ∞

Tactic:
calc
    ∞ = ∑' _ : ℕ, μ s := (ENNReal.tsum_const_eq_top_of_ne_zero h).symm
    _ = ∑' n : ℕ, μ ({u n} + s) := by
      congr 1; ext1 n; simp only [image_add_left, measure_preimage_add, singleton_add]
    _ = μ (⋃ n, {u n} + s) := Eq.symm <| measure_iUnion hs fun n => by
      simpa only [image_add_left, singleton_add] using! measurable_id.const_add _ h's
    _ = μ (range u + s) := by rw [← iUnion_add, iUnion_singleton_eq_range]
    _ < ∞ := (hu.add sb).measure_lt_top

Goal state after:
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℝ (ι → ℝ)
⊢ volume (fundamentalDomain b) = ENNReal.ofReal |(Matrix.of ⇑b).det|

Tactic:
by
  rw [measure_fundamentalDomain b volume (b₀ := Pi.basisFun ℝ ι), fundamentalDomain_pi_basisFun,
    volume_pi, Measure.pi_pi, Real.volume_Ico, sub_zero, ENNReal.ofReal_one, Finset.prod_const_one,
    mul_one, ← Matrix.det_transpose]
  rfl

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I : Box ι
π₁ π₂ : Prepartition I
πi₁ πi₂ : (J : Box ι) → Prepartition J
h : π₁ = π₂
hi : ∀ J ∈ π₁, πi₁ J = πi₂ J
⊢ π₁.biUnion πi₁ = π₂.biUnion πi₂

Tactic:
subst π₂
  ext J
  simp only [mem_biUnion]
  constructor <;> exact fun ⟨J', h₁, h₂⟩ => ⟨J', h₁, hi J' h₁ ▸ h₂⟩

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I : Box ι
π : Prepartition I
πi : (J : Box ι) → Prepartition J
πi' : Box ι → (J : Box ι) → Prepartition J
⊢ (π.biUnion fun J ↦ (πi J).biUnion (πi' J)) = (π.biUnion πi).biUnion fun J ↦ πi' (π.biUnionIndex πi J) J

Tactic:
ext J
  simp only [mem_biUnion]
  constructor
  · rintro ⟨J₁, hJ₁, J₂, hJ₂, hJ⟩
    refine ⟨J₂, ⟨J₁, hJ₁, hJ₂⟩, ?_⟩
    rwa [π.biUnionIndex_of_mem hJ₁ hJ₂]
  · rintro ⟨J₁, ⟨J₂, hJ₂, hJ₁⟩, hJ⟩
    refine ⟨J₂, hJ₂, J₁, hJ₁, ?_⟩
    rwa [π.biUnionIndex_of_mem hJ₂ hJ₁] at hJ

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.Basic

Goal state before:
pp: ι : Type u_1
I : Box ι
boxes : Finset (WithBot (Box ι))
le_of_mem : ∀ J ∈ boxes, J ≤ ↑I
pairwise_disjoint : (↑boxes).Pairwise Disjoint
⊢ ⋃ J, ⋃ (_ : ↑J ∈ boxes), ↑J = ⋃ J ∈ boxes, ↑J

Tactic:
simp only [← Box.biUnion_coe_eq_coe, @iUnion_comm _ _ (Box ι), @iUnion_comm _ _ (@Eq _ _ _),
    iUnion_iUnion_eq_right]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
p : Box ι → Prop
I : Box ι
H_ind : ∀ J ≤ I, (∀ J' ∈ splitCenter J, p J') → p J
H_nhds :
  ∀ z ∈ Box.Icc I,
    ∃ U ∈ 𝓝[Box.Icc I] z,
      ∀ J ≤ I,
        ∀ (m : ℕ),
          z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J
⊢ p I

Tactic:
by
  refine subbox_induction_on' I (fun J hle hs => H_ind J hle fun J' h' => ?_) H_nhds
  rcases mem_splitCenter.1 h' with ⟨s, rfl⟩
  exact hs s

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: ι : Type u_1
I : Box ι
s : Finset (ι × ℝ)
p : ι × ℝ
⊢ (s.inf fun p ↦ split I p.1 p.2) ⊓ split I p.1 p.2 = splitMany I s ⊓ split I p.1 p.2

Tactic:
splitMany

Goal state after:
ι : Type u_1
I : Box ι
s : Finset (ι × ℝ)
p : ι × ℝ
⊢ (s.inf fun p ↦ split I p.1 p.2) ⊓ split I p.1 p.2 = (s.inf fun p ↦ split I p.1 p.2) ⊓ split I p.1 p.2
Module: Mathlib.Analysis.BoxIntegral.Partition.Split

Goal state before:
pp: case refine_2
ι : Type u_1
I : Box ι
inst✝ : Finite ι
π : Prepartition I
t : Finset (ι × ℝ)
ht : ∀ (I_1 J : Box ι), J ∈ π.boxes → ∀ J' ∈ splitMany I_1 t, ¬Disjoint ↑J ↑J' → J' ≤ J
J : Box ι
hJ : J ∈ (splitMany I t).filter fun J ↦ ↑J ⊆ π.iUnion
⊢ ∃ I' ∈ π, J ≤ I'

Tactic:
mem_filter

Goal state after:
case refine_2
ι : Type u_1
I : Box ι
inst✝ : Finite ι
π : Prepartition I
t : Finset (ι × ℝ)
ht : ∀ (I_1 J : Box ι), J ∈ π.boxes → ∀ J' ∈ splitMany I_1 t, ¬Disjoint ↑J ↑J' → J' ≤ J
J : Box ι
hJ : J ∈ splitMany I t ∧ ↑J ⊆ π.iUnion
⊢ ∃ I' ∈ π, J ≤ I'
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: K : Type u_1
inst✝⁴ : NormedField K
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace K E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace K F
L : Submodule ℤ E
hL : DiscreteTopology ↥L
e : F →ₗ[K] E
he₁ : Continuous ⇑e
he₂ : Function.Injective ⇑e
⊢ DiscreteTopology ↥(ZLattice.comap K L e)

Tactic:
exact DiscreteTopology.preimage_of_continuous_injective L he₁ he₂

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.Partition.Measure

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
I : Box ι
⊢ volume.real (univ.pi fun i ↦ Ioc (I.lower i) (I.upper i)) = ∏ i, (I.upper i - I.lower i)

Tactic:
measureReal_def,

Goal state after:
ι : Type u_1
inst✝ : Fintype ι
I : Box ι
⊢ (volume (univ.pi fun i ↦ Ioc (I.lower i) (I.upper i))).toReal = ∏ i, (I.upper i - I.lower i)
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
n : ι
⊢ gramSchmidt 𝕜 f n = f n - ∑ i < n, (𝕜 ∙ gramSchmidt 𝕜 f i).starProjection (f n)

Tactic:
← sum_attach,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
n : ι
⊢ gramSchmidt 𝕜 f n = f n - ∑ x ∈ (Iio n).attach, (𝕜 ∙ gramSchmidt 𝕜 f ↑x).starProjection (f n)
Module: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
n : ι
⊢ gramSchmidt 𝕜 f n = f n - ∑ x, (𝕜 ∙ gramSchmidt 𝕜 f ↑x).starProjection (f n)

Tactic:
gramSchmidt

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_3
inst✝² : LinearOrder ι
inst✝¹ : LocallyFiniteOrderBot ι
inst✝ : WellFoundedLT ι
f : ι → E
n : ι
⊢ f n - ∑ i, (𝕜 ∙ gramSchmidt 𝕜 f ↑i).starProjection (f n) = f n - ∑ x, (𝕜 ∙ gramSchmidt 𝕜 f ↑x).starProjection (f n)
Module: Mathlib.Analysis.BoxIntegral.UnitPartition

Goal state before:
pp: case refine_4
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
this : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))
x✝ : ι → ℝ
hx : x✝ ∈ s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ F x✝ / ↑n ^ card ι = volume.real ↑(box n (index n x✝)) * F ((prepartition n B).tag (box n (index n x✝)))

Tactic:
measureReal_def,

Goal state after:
case refine_4
ι : Type u_1
n : ℕ
inst✝¹ : NeZero n
s : Set (ι → ℝ)
F : (ι → ℝ) → ℝ
inst✝ : Fintype ι
B : Box ι
hB : hasIntegralVertices B
hs₀ : s ⊆ ↑B
this : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))
x✝ : ι → ℝ
hx : x✝ ∈ s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))
⊢ F x✝ / ↑n ^ card ι = (volume ↑(box n (index n x✝))).toReal * F ((prepartition n B).tag (box n (index n x✝)))