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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : SeminormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝¹ : (i : ι) → (x : G i) → Decidable (x ≠ 0)
inst✝ : DecidableEq ι
l : ⨁ (i : ι), G i
i : ι
v : G i
⊢ (DFinsupp.sum l fun j w ↦ if i = j then ⟪(V i) v, (V j) w⟫ else 0) = ⟪v, l i⟫

Tactic:
by
      simp only [DFinsupp.sum, Finset.sum_ite_eq,
        DFinsupp.mem_support_toFun]
      split_ifs with h
      · simp only [LinearIsometry.inner_map_map]
      · simp only [of_not_not h, inner_zero_right]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
F : Type u_6
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
⊢ ‖y‖ * ‖x‖ = ‖x‖ * ‖y‖

Tactic:
mul_comm

Goal state after:
𝕜 : Type u_4
E : Type u_5
F : Type u_6
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
x : E
y : F
⊢ ‖x‖ * ‖y‖ = ‖x‖ * ‖y‖
Module: Mathlib.Analysis.RCLike.Lemmas

Goal state before:
pp: K : Type u_1
E : Type u_2
inst✝³ : RCLike K
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace K E
inst✝ : FiniteDimensional K E
this✝ : NormedSpace ℝ E
this : FiniteDimensional ℝ E
⊢ ProperSpace E

Tactic:
infer_instance

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Orthogonal

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
s t : Set E
⊢ (∀ i ∈ s, 𝕜 ∙ i ⟂ ⨆ x ∈ t, 𝕜 ∙ x) ↔ ∀ ⦃u : E⦄, u ∈ s → ∀ ⦃v : E⦄, v ∈ t → ⟪u, v⟫ = 0

Tactic:
isOrtho_iSup_right,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
s t : Set E
⊢ (∀ i ∈ s, ∀ i_2 ∈ t, 𝕜 ∙ i ⟂ 𝕜 ∙ i_2) ↔ ∀ ⦃u : E⦄, u ∈ s → ∀ ⦃v : E⦄, v ∈ t → ⟪u, v⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Reflection

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
⊢ Kᗮ.reflection = K.reflection.trans (LinearIsometryEquiv.neg 𝕜)

Tactic:
by
  ext; apply reflection_orthogonal_apply

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝¹ : Preorder ι
U : ι → Submodule 𝕜 E
inst✝ : ∀ (t : ι), (U t).HasOrthogonalProjection
hU : Monotone U
x : E
hU' : ⊤ ≤ (⨆ t, U t).topologicalClosure
⊢ ⊤.HasOrthogonalProjection

Tactic:
infer_instance

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

Goal state before:
pp: 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
hc' : 1 < ‖σ c‖
d : 𝕜'
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖σ c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
δpos : 0 < δ
this✝ : a + d • y ∈ ball a ε
z₁ : F
z₁im : z₁ ∈ ⇑f '' ball 0 ↑n
x₁ : E
h₁ : dist (a + d • y) (f x₁) < δ
hx₁ : ‖x₁‖ < ↑n
xz₁ : f x₁ = z₁
this : a ∈ ball a ε
z₂ : F
z₂im : z₂ ∈ ⇑f '' ball 0 ↑n
x₂ : E
h₂ : dist a (f x₂) < δ
hx₂ : ‖x₂‖ < ↑n
xz₂ : f x₂ = z₂
x : E := x₁ - x₂
I : ‖f x - d • y‖ ≤ 2 * δ
⊢ ‖f (σ' d⁻¹ • x) - y‖ = ‖d⁻¹ • f x - (d⁻¹ * d) • y‖

Tactic:
rwa [f.map_smulₛₗ _, inv_mul_cancel₀, one_smul, map_inv₀, map_inv₀,
            RingHomCompTriple.comp_apply, RingHom.id_apply]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K : Submodule 𝕜 E
inst✝ : CompleteSpace ↥K
v : E
⊢ ∃ w ∈ K, v - w ∈ Kᗮ

Tactic:
by
    rcases K.exists_norm_eq_iInf_of_complete_subspace (completeSpace_coe_iff_isComplete.mp ‹_›) v
      with ⟨w, hwK, hw⟩
    refine ⟨w, hwK, (K.mem_orthogonal' _).2 ?_⟩
    rwa [← K.norm_eq_iInf_iff_inner_eq_zero hwK]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : InnerProductSpace ℝ F
K✝ : Submodule 𝕜 E
K : ClosedSubmodule 𝕜 E
inst✝ : CompleteSpace E
this : IsClosed (↑K).carrier := K.isClosed'
⊢ (↑K).HasOrthogonalProjection

Tactic:
infer_instance

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
⊢ Uᗮ.starProjection = ContinuousLinearMap.id 𝕜 E - U.starProjection

Tactic:
by
  ext
  simp only [starProjection, ContinuousLinearMap.comp_apply,
    orthogonalProjectionOnto_orthogonal]
  simp

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
⊢ ‖K.orthogonalProjectionOnto‖ ≤ 1

Tactic:
refine K.orthogonalProjectionOnto.opNorm_le_bound zero_le_one ?_
  simp [orthogonalProjectionOnto, projectionOntoL, norm_projection_orthogonal_le]

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

Goal state before:
pp: case of.inl
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝¹ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
i : ι
inst✝ : CompleteSpace ↥(V i)
x : ↥(V i)
⊢ (V i).orthogonalProjectionOnto ↑x = (DFinsupp.single i x) i

Tactic:
orthogonalProjectionOnto_mem_subspace_eq_self,

Goal state after:
case of.inl
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝¹ : DecidableEq ι
V : ι → Submodule 𝕜 E
hV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ
i : ι
inst✝ : CompleteSpace ↥(V i)
x : ↥(V i)
⊢ x = (DFinsupp.single i x) i
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
h : ∀ x ∈ (span 𝕜 v)ᗮ, x = 0
u : Set E
huv : v ⊆ u
hu : Orthonormal 𝕜 Subtype.val
x : E
hxu : x ∈ u
hxv : x ∉ v
y : E
hy : y ∈ span 𝕜 v
hxv' : ⟨x, hxu⟩ ∉ Subtype.val ⁻¹' v
⊢ ∃ l ∈ supported 𝕜 𝕜 (Subtype.val ⁻¹' v), (linearCombination 𝕜 Subtype.val) l = y

Tactic:
rw [← Finsupp.mem_span_image_iff_linearCombination]

Goal state after:
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
h : ∀ x ∈ (span 𝕜 v)ᗮ, x = 0
u : Set E
huv : v ⊆ u
hu : Orthonormal 𝕜 Subtype.val
x : E
hxu : x ∈ u
hxv : x ∉ v
y : E
hy : y ∈ span 𝕜 v
hxv' : ⟨x, hxu⟩ ∉ Subtype.val ⁻¹' v
⊢ y ∈ span 𝕜 (Subtype.val '' Subtype.val ⁻¹' v)
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
⊢ ‖c • f‖₊ ≤ ‖c‖₊ * ‖f‖₊

Tactic:
have hp0 : 0 < p.toReal := zero_lt_one.trans_le hp

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
⊢ ‖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
⊢ ‖c • f‖₊ = ‖c‖₊ * ‖f‖₊

Tactic:
have hp0 : 0 < p.toReal := zero_lt_one.trans_le hp

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
⊢ ‖c • f‖₊ = ‖c‖₊ * ‖f‖₊
Module: Mathlib.LinearAlgebra.UnitaryGroup

Goal state before:
pp: n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
α : Type v
inst✝¹ : CommRing α
inst✝ : StarRing α
U : Matrix n n α
⊢ Uᵀ ∈ unitaryGroup n α ↔ U ∈ unitaryGroup n α

Tactic:
conv_rhs => rw [mem_unitaryGroup_iff']
  rw [mem_unitaryGroup_iff, show star Uᵀ = (star U)ᵀ by rfl, ← transpose_mul, ← transpose_inj]
  simp

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

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : WithLp p ((i : ι) → β i)
⊢ edist x.ofLp y.ofLp ≤ edist x y

Tactic:
simp_rw [edist_pi_def, Finset.sup_le_iff, Finset.mem_univ, forall_true_left]

Goal state after:
p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
x y : WithLp p ((i : ι) → β i)
⊢ ∀ (b : ι), edist (x.ofLp b) (y.ofLp b) ≤ edist x y
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
⊢ ‖c • f‖₊ ≤ ‖c‖₊ * ‖f‖₊

Tactic:
have hp0 : 0 < p.toReal := zero_lt_one.trans_le hp

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
⊢ ‖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
⊢ ‖c • f‖₊ = ‖c‖₊ * ‖f‖₊

Tactic:
have hp0 : 0 < p.toReal := zero_lt_one.trans_le hp

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
⊢ ‖c • f‖₊ = ‖c‖₊ * ‖f‖₊
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: s : Set ℝ
⊢ volume s ≤ ediam s

Tactic:
by_cases hs : Bornology.IsBounded s

Goal state after:
case pos
s : Set ℝ
hs : Bornology.IsBounded s
⊢ volume s ≤ ediam s

case neg
s : Set ℝ
hs : ¬Bornology.IsBounded s
⊢ volume s ≤ ediam s
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: case inr
ι : Type u_1
inst✝ : Fintype ι
b : OrthonormalBasis ι ℝ ℝ
e : ι ≃ Fin 1
B : parallelepiped ⇑(b.reindex e) = parallelepiped ⇑b
F : ℝ → Fin 1 → ℝ := fun t _i ↦ t
A : Icc 0 1 = F '' Icc 0 1
H : ⇑(b.reindex e) = fun x ↦ -1
⊢ parallelepiped ⇑(b.reindex e) = Icc 0 1 ∨ parallelepiped ⇑(b.reindex e) = Icc (-1) 0

Tactic:
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.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
⊢ MeasureSpace ℝ

Tactic:
infer_instance

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 : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n ↦ c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
⊢ -(c ^ m • x) + y ∈ s → -(c ^ n • x) + y ∉ s

Tactic:
hym

Goal state after:
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 : Submodule ℝ E
hs : s ≠ ⊤
x : E
hx : x ∉ s
c : ℝ
cpos : 0 < c
cone : c < 1
A : Bornology.IsBounded (range fun n ↦ c ^ n • x)
m n : ℕ
hmn : m ≠ n
y : E
hym : -(c ^ m • x) + y ∈ s
⊢ -(c ^ n • x) + y ∉ s
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
f : E →ₗ[ℝ] E
hf : LinearMap.det f ≠ 0
ι : Type := Fin (finrank ℝ E)
⊢ FiniteDimensional ℝ (ι → ℝ)

Tactic:
infer_instance

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

Goal state before:
pp: case neg
μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x
T : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2
u : Set ℝ := ⋃ i, T i
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
x : ℝ
hx : x ∈ s
h'x : x ∉ ⋃ i, T i
⊢ x ∈ s \ u ∪ ⋃ p ∈ A, s ∩ T p

Tactic:
exact Or.inl ⟨hx, h'x⟩

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

Goal state before:
pp: case refine_2.h
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
a✝ b✝ : E
hxy : fract b a✝ = fract b b✝
⊢ ⟦a✝⟧ = ⟦b✝⟧

Tactic:
apply Quotient.sound'

Goal state after:
case refine_2.h
E : Type u_1
ι : Type u_2
K : Type u_3
inst✝⁶ : NormedField K
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
b : Basis ι K E
inst✝³ : LinearOrder K
inst✝² : IsStrictOrderedRing K
inst✝¹ : FloorRing K
inst✝ : Fintype ι
a✝ b✝ : E
hxy : fract b a✝ = fract b b✝
⊢ (span ℤ (Set.range ⇑b)).quotientRel a✝ b✝
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: case left
μ : Measure ℝ
inst✝ : NullSingletonClass μ
s : Set ℝ
p : ℝ → Prop
h : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x
T : ↑s × ↑s → Set ℝ := ⋯
u : Set ℝ := ⋯
hfinite : (s \ u).Finite
A : Set (↑s × ↑s)
A_count : A.Countable
hA : ⋃ i ∈ A, T i = ⋃ i, T i
this : s ⊆ s \ u ∪ ⋃ p ∈ A, s ∩ T p
⊢ ∀ᵐ (x : ℝ) ∂μ.restrict (s \ u), p x

Tactic:
have : μ.restrict (s \ u) = 0 := by simp only [restrict_eq_zero, hfinite.measure_zero]
    simp only [this, ae_zero, eventually_bot]

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

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : OpensMeasurableSpace E
inst✝ : Finite ι
val✝ : Fintype ι
this : FiniteDimensional ℝ E
D : Set (ι → ℝ) := Set.univ.pi fun x ↦ Set.Ico 0 1
x✝ : E
⊢ x✝ ∈ fundamentalDomain b ↔ x✝ ∈ ⇑↑b.equivFun ⁻¹' D

Tactic:
simp only [D, fundamentalDomain, Set.mem_Ico, Set.mem_ofPred_eq, LinearEquiv.coe_coe,
      Set.mem_preimage, Basis.equivFun_apply, Set.mem_pi, Set.mem_univ, forall_true_left]

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)
⊢ ENNReal.ofReal |(Pi.basisFun ℝ ι).det ⇑b| = ENNReal.ofReal |(Matrix.of ⇑b).det|

Tactic:
← Matrix.det_transpose

Goal state after:
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
b : Basis ι ℝ (ι → ℝ)
⊢ ENNReal.ofReal |(Pi.basisFun ℝ ι).det ⇑b| = ENNReal.ofReal |(Matrix.of ⇑b).transpose.det|
Module: Mathlib.Algebra.Module.ZLattice.Basic

Goal state before:
pp: E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Fintype ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
this✝ : FiniteDimensional ℝ E
this : parallelepiped ⇑b \ fundamentalDomain b ⊆ ⋃ i, ↑(AffineSubspace.mk' (b i) (span ℝ (⇑b '' (Set.univ \ {i}))))
i : ι
⊢ 0 -ᵥ b i ∉ span ℝ (⇑b '' (Set.univ \ {i}))

Tactic:
simp_rw [vsub_eq_sub, zero_sub, neg_mem_iff]

Goal state after:
E : Type u_1
ι : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
b : Basis ι ℝ E
inst✝³ : Fintype ι
inst✝² : MeasurableSpace E
μ : Measure E
inst✝¹ : BorelSpace E
inst✝ : μ.IsAddHaarMeasure
this✝ : FiniteDimensional ℝ E
this : parallelepiped ⇑b \ fundamentalDomain b ⊆ ⋃ i, ↑(AffineSubspace.mk' (b i) (span ℝ (⇑b '' (Set.univ \ {i}))))
i : ι
⊢ b i ∉ span ℝ (⇑b '' (Set.univ \ {i}))
Module: Mathlib.Analysis.BoxIntegral.Box.Basic

Goal state before:
pp: case neg
ι : Type u_1
l u : ι → ℝ
h : ¬∀ (i : ι), l i < u i
⊢ ↑⊥ = univ.pi fun i ↦ Ioc (l i) (u i)

Tactic:
· rcases not_forall.mp h with ⟨i, hi⟩
    rw [coe_bot, univ_pi_eq_empty]
    exact Ioc_eq_empty hi

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

Goal state before:
pp: E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : FiniteDimensional ℝ E
μ : Measure E
inst✝⁷ : μ.IsAddHaarMeasure
s : Set E
ι : Type u_2
G : Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : DecidableEq ι
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : FiniteDimensional ℝ G
n : ℕ
_i : Fact (finrank ℝ G = n)
ω : G [⋀^Fin n]→ₗ[ℝ] ℝ
⊢ (‖ω ⇑(finBasisOfFinrankEq ℝ G ⋯)‖₊ • (finBasisOfFinrankEq ℝ G ⋯).addHaar).IsAddLeftInvariant

Tactic:
infer_instance

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

Goal state before:
pp: E : Type u_1
inst✝¹² : NormedAddCommGroup E
inst✝¹¹ : NormedSpace ℝ E
inst✝¹⁰ : MeasurableSpace E
inst✝⁹ : BorelSpace E
inst✝⁸ : FiniteDimensional ℝ E
μ : Measure E
inst✝⁷ : μ.IsAddHaarMeasure
s : Set E
ι : Type u_2
G : Type u_3
inst✝⁶ : Fintype ι
inst✝⁵ : DecidableEq ι
inst✝⁴ : NormedAddCommGroup G
inst✝³ : NormedSpace ℝ G
inst✝² : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : FiniteDimensional ℝ G
n : ℕ
_i : Fact (finrank ℝ G = n)
ω : G [⋀^Fin n]→ₗ[ℝ] ℝ
⊢ IsLocallyFiniteMeasure (‖ω ⇑(finBasisOfFinrankEq ℝ G ⋯)‖₊ • (finBasisOfFinrankEq ℝ G ⋯).addHaar)

Tactic:
infer_instance

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

Goal state before:
pp: n : ℕ
I : Box (Fin (n + 1))
i : Fin (n + 1)
x : ℝ
hx : x ∈ Ioc (I.lower i) (I.upper i)
⊢ MapsTo (i.insertNth x) ↑(I.face i) ↑I

Tactic:
by
  intro y hy
  simp_rw [mem_coe, mem_def, i.forall_iff_succAbove, Fin.insertNth_apply_same,
    Fin.insertNth_apply_succAbove]
  exact ⟨hx, hy⟩

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

Goal state before:
pp: K : Type u_1
inst✝¹⁰ : NormedField K
inst✝⁹ : LinearOrder K
inst✝⁸ : IsStrictOrderedRing K
inst✝⁷ : HasSolidNorm K
inst✝⁶ : FloorRing K
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace K E
inst✝³ : FiniteDimensional K E
inst✝² : ProperSpace E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice K L
this✝¹ : Module.Finite ℤ ↥L
this✝ : Module ℚ E
this : IsAddTorsionFree E
⊢ Free ℤ ↥L

Tactic:
infer_instance

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

Goal state before:
pp: K : Type u_1
inst✝¹³ : NormedField K
inst✝¹² : LinearOrder K
inst✝¹¹ : IsStrictOrderedRing K
inst✝¹⁰ : HasSolidNorm K
inst✝⁹ : FloorRing K
E✝ : Type u_2
inst✝⁸ : NormedAddCommGroup E✝
inst✝⁷ : NormedSpace K E✝
inst✝⁶ : FiniteDimensional K E✝
inst✝⁵ : ProperSpace E✝
L✝ : Submodule ℤ E✝
inst✝⁴ : DiscreteTopology ↥L✝
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝ : DiscreteTopology ↥L
this✝ : Module ℚ E
this : IsAddTorsionFree E
⊢ Free ℤ ↥L

Tactic:
infer_instance

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

Goal state before:
pp: ι : Type u_1
I J J₁ : Box ι
π : Prepartition I
⊢ J₁ ∈ π.restrict J ↔ ∃ J' ∈ π, ↑J₁ = ↑J ∩ ↑J'

Tactic:
simp only [mem_restrict, ← Box.withBotCoe_inj, Box.coe_inf, Box.coe_coe]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: case e'_2.a
ι : Type u_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
x y : E
this : ∑ x_1, ((innerSL 𝕜) x) ((b.repr y).ofLp x_1 • b x_1) = ((innerSL 𝕜) x) y
x✝ : ι
a✝ : x✝ ∈ Finset.univ
⊢ ⟪x, b x✝⟫ * ⟪b x✝, y⟫ = ⟪b x✝, y⟫ • ((innerSL 𝕜) x) (b x✝)

Tactic:
mul_comm

Goal state after:
case e'_2.a
ι : Type u_1
𝕜 : Type u_3
inst✝³ : RCLike 𝕜
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : Fintype ι
b : OrthonormalBasis ι 𝕜 E
x y : E
this : ∑ x_1, ((innerSL 𝕜) x) ((b.repr y).ofLp x_1 • b x_1) = ((innerSL 𝕜) x) y
x✝ : ι
a✝ : x✝ ∈ Finset.univ
⊢ ⟪b x✝, y⟫ * ⟪x, b x✝⟫ = ⟪b x✝, y⟫ • ((innerSL 𝕜) x) (b x✝)
Module: Mathlib.Analysis.InnerProductSpace.PiL2

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
E : Type u_4
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : Fintype ι
inst✝ : Fintype ι'
b : OrthonormalBasis ι 𝕜 E
e : ι ≃ ι'
x : E
i' : ι'
⊢ ((b.reindex e).repr x).ofLp i' = (b.repr x).ofLp (e.symm i')

Tactic:
by
  rw [OrthonormalBasis.repr_apply_apply, b.repr_apply_apply, OrthonormalBasis.coe_reindex,
    comp_apply]

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

Goal state before:
pp: K : Type u_1
inst✝¹⁴ : NormedField K
inst✝¹³ : LinearOrder K
inst✝¹² : IsStrictOrderedRing K
inst✝¹¹ : HasSolidNorm K
inst✝¹⁰ : FloorRing K
E✝ : Type u_2
inst✝⁹ : NormedAddCommGroup E✝
inst✝⁸ : NormedSpace K E✝
inst✝⁷ : FiniteDimensional K E✝
inst✝⁶ : ProperSpace E✝
L✝ : Submodule ℤ E✝
inst✝⁵ : DiscreteTopology ↥L✝
ι : Type u_3
hs : IsZLattice K L✝
b : Basis ι ℤ ↥L✝
E : Type u_4
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
L : Submodule ℤ E
inst✝¹ : DiscreteTopology ↥L
inst✝ : IsZLattice ℝ L
⊢ Countable ↥(span ℤ (Set.range ⇑(Basis.ofZLatticeBasis ℝ L (Free.chooseBasis ℤ ↥L))))

Tactic:
infer_instance

Goal state after: