mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.DirichletCharacter.Basic

Goal state before:
pp: case pos
R : Type u_1
inst✝² : CommMonoidWithZero R
n : ℕ
inst✝¹ : Nontrivial R
m : ℕ
inst✝ : NeZero m
hm : n ∣ m
χ : DirichletCharacter R n
a : ℤ
ha : IsCoprime a ↑χ.conductor
⊢ (changeLevel ⋯) ((changeLevel hm) χ).primitiveCharacter = χ.primitiveCharacter

Tactic:
apply changeLevel_injective (χ.conductor_dvd_level.trans hm)

Goal state after:
case pos
R : Type u_1
inst✝² : CommMonoidWithZero R
n : ℕ
inst✝¹ : Nontrivial R
m : ℕ
inst✝ : NeZero m
hm : n ∣ m
χ : DirichletCharacter R n
a : ℤ
ha : IsCoprime a ↑χ.conductor
⊢ (changeLevel ⋯) ((changeLevel ⋯) ((changeLevel hm) χ).primitiveCharacter) = (changeLevel ⋯) χ.primitiveCharacter
Module: Mathlib.NumberTheory.DirichletCharacter.Basic

Goal state before:
pp: R : Type u_1
inst✝² : CommMonoidWithZero R
n : ℕ
inst✝¹ : Nontrivial R
m : ℕ
inst✝ : NeZero m
hm : n ∣ m
χ : DirichletCharacter R n
a : ℤ
ha : ¬IsCoprime a ↑χ.conductor
⊢ ¬IsCoprime a ↑((changeLevel hm) χ).conductor

Tactic:
conductor_changeLevel

Goal state after:
R : Type u_1
inst✝² : CommMonoidWithZero R
n : ℕ
inst✝¹ : Nontrivial R
m : ℕ
inst✝ : NeZero m
hm : n ∣ m
χ : DirichletCharacter R n
a : ℤ
ha : ¬IsCoprime a ↑χ.conductor
⊢ ¬IsCoprime a ↑χ.conductor
Module: Mathlib.NumberTheory.DirichletCharacter.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
⊢ χ⁻¹.conductor = χ.conductor

Tactic:
← zpow_neg_one

Goal state after:
R : Type u_1
inst✝ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
⊢ (χ ^ (-1)).conductor = χ.conductor
Module: Mathlib.NumberTheory.DirichletCharacter.Basic

Goal state before:
pp: R : Type u_1
inst✝ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
⊢ (χ⁻¹ ^ (-1)).conductor ∣ (χ ^ (-1)).conductor

Tactic:
← zpow_neg_one

Goal state after:
R : Type u_1
inst✝ : CommMonoidWithZero R
n : ℕ
χ : DirichletCharacter R n
⊢ ((χ ^ (-1)) ^ (-1)).conductor ∣ (χ ^ (-1)).conductor
Module: Mathlib.Analysis.Fourier.ZMod

Goal state before:
pp: N : ℕ
inst✝² : NeZero N
E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℂ E
Φ : ZMod N → E
k j : ZMod N
⊢ stdAddChar (-(j * k)) • Φ (-j) = stdAddChar (-((Equiv.neg (ZMod N)) j * -k)) • Φ ((Equiv.neg (ZMod N)) j)

Tactic:
rw [Equiv.neg_apply, neg_mul_neg]

Goal state after:
Module: Mathlib.Analysis.Fourier.ZMod

Goal state before:
pp: N : ℕ
inst✝² : NeZero N
E : Type u_1
inst✝¹ : AddCommGroup E
inst✝ : Module ℂ E
Φ : ZMod N → E
⊢ 𝓕 Φ 0 = ∑ j, Φ j

Tactic:
by
  simp only [dft_apply, mul_zero, neg_zero, map_zero_eq_one, one_smul]

Goal state after:
Module: Mathlib.Data.Finset.Grade

Goal state before:
pp: α : Type u_1
a : α
⊢ ∅ ⋖ {a}

Tactic:
by
  classical exact insert_empty_eq (β := Finset α) a ▸ covBy_insert <| notMem_empty a

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

Goal state before:
pp: case refine_3
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : Module.End 𝕜 E
hT : IsSymmetric T
p : Submodule 𝕜 E
hp₁ : p ∈ T.invtSubmodule
hp₂ : Module.Finite 𝕜 ↥p
q : Submodule 𝕜 E
hq₁ : q ∈ T.invtSubmodule
hq₂ : q ≤ p
⊢ q ⊔ qᗮ ⊓ p = p

Tactic:
suffices q ⊔ qᗮ = ⊤ by rw [← sup_inf_assoc_of_le _ hq₂, this, top_inf_eq p]
    replace hp₂ : Module.Finite 𝕜 q := Submodule.finiteDimensional_of_le hq₂
    exact Submodule.sup_orthogonal_of_hasOrthogonalProjection

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : FiniteDimensional 𝕜 E
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] E
hT : T.IsPositive
S : F →ₗ[𝕜] E
⊢ (LinearMap.adjoint S ∘ₗ T ∘ₗ S).IsPositive

Tactic:
by
  convert! hT.conj_adjoint S.adjoint
  rw [LinearMap.adjoint_adjoint]

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

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
ι : Type u_5
T : ι → E →ₗ[𝕜] E
hT : ∀ (i : ι), (T i).IsSymmetric
h : Pairwise (Commute on T)
x✝ : ι
⊢ ⨆ μ, maxGenEigenspace (T x✝) μ = ⊤

Tactic:
← orthogonal_eq_bot_iff,

Goal state after:
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
ι : Type u_5
T : ι → E →ₗ[𝕜] E
hT : ∀ (i : ι), (T i).IsSymmetric
h : Pairwise (Commute on T)
x✝ : ι
⊢ (⨆ μ, maxGenEigenspace (T x✝) μ)ᗮ = ⊥
Module: Mathlib.Analysis.InnerProductSpace.LaxMilgram

Goal state before:
pp: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
C : ℝ
C_pos : 0 < C
below_bound : ∀ (v : V), C * ‖v‖ ≤ ‖(continuousLinearMapOfBilin B) v‖
⊢ AntilipschitzWith C⁻¹.toNNReal ⇑(continuousLinearMapOfBilin B)

Tactic:
refine ContinuousLinearMap.antilipschitz_of_bound B♯ ?_

Goal state after:
V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
C : ℝ
C_pos : 0 < C
below_bound : ∀ (v : V), C * ‖v‖ ≤ ‖(continuousLinearMapOfBilin B) v‖
⊢ ∀ (x : V), ‖x‖ ≤ ↑C⁻¹.toNNReal * ‖(continuousLinearMapOfBilin B) x‖
Module: Mathlib.Analysis.InnerProductSpace.LaxMilgram

Goal state before:
pp: V : Type u
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : CompleteSpace V
B : V →L[ℝ] V →L[ℝ] ℝ
coercive : IsCoercive B
⊢ ∃ C, 0 < C ∧ AntilipschitzWith C ⇑(continuousLinearMapOfBilin B)

Tactic:
by
  rcases coercive.bounded_below with ⟨C, C_pos, below_bound⟩
  refine ⟨C⁻¹.toNNReal, Real.toNNReal_pos.mpr (inv_pos.mpr C_pos), ?_⟩
  refine ContinuousLinearMap.antilipschitz_of_bound B♯ ?_
  simp_rw [Real.coe_toNNReal', max_eq_left_of_lt (inv_pos.mpr C_pos), ←
    inv_mul_le_iff₀ (inv_pos.mpr C_pos)]
  simpa using below_bound

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

Goal state before:
pp: case hg_proj
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
f : E →L[𝕜] E
hf : ‖f‖ ≤ 1
x : E
⊢ ∀ (x : ↥((↑f).eqLocus 1)), ((↑f).eqLocus ↑1).orthogonalProjectionOnto ↑x = x

Tactic:
exact (f.eqLocus (1 : E →L[𝕜] E)).orthogonalProjectionOnto_mem_subspace_eq_self

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

Goal state before:
pp: case h
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
T : E →ₗ.[𝕜] F
inst✝ : CompleteSpace E
hT : Dense ↑T.domain
x : F × E
h : ∀ (a : E) (a_1 : a ∈ T.domain), ⟪↑T ⟨a, ⋯⟩, x.1⟫ = ⟪a, x.2⟫
a : E
ha : a ∈ T.domain
⊢ ⟪x.2, ↑⟨a, ha⟩⟫ = ⟪x.1, ↑T ⟨a, ha⟩⟫

Tactic:
rw [← inner_conj_symm, ← h a ha, inner_conj_symm]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : RCLike 𝕜
x✝ : 𝕜 ⊗[𝕜] 𝕜
⊢ (lidIsometry 𝕜 𝕜) x✝ = (ridIsometry 𝕜 𝕜) x✝

Tactic:
simp [lid_eq_rid]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case pos
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : μ (s \ t) = ∞
⊢ μ (s ∩ t) + ∞ ≤ μ s

Tactic:
add_top,

Goal state after:
case pos
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : μ (s \ t) = ∞
⊢ ∞ ≤ μ s
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: F : Type u_3
inst✝⁸ : NormedAddCommGroup F
inst✝⁷ : NormedSpace ℝ F
E : Type u_4
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : FiniteDimensional ℝ E
μ : Measure E
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : FiniteDimensional ℝ F
u : E → F
s : Set E
hu : ContDiff ℝ 1 u
h2u : support u ⊆ s
p q : ℝ≥0
hp : 1 ≤ p
h2p : p < ↑(finrank ℝ E)
hpq : ↑p⁻¹ - (↑(finrank ℝ E))⁻¹ ≤ (↑q)⁻¹
hs : Bornology.IsBounded s
hq0 : ¬q = 0
p' : ℝ≥0 := (p⁻¹ - (↑(finrank ℝ E))⁻¹)⁻¹
hp' : ↑p'⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
this : ↑q ≤ ↑p'
t : ℝ≥0 := (μ s).toNNReal ^ (1 / ↑q - 1 / ↑p')
C : ℝ≥0 := SNormLESNormFDerivOfEqConst F μ ↑p
⊢ eLpNorm u (↑p') μ * ↑t ≤ ↑C * eLpNorm (fderiv ℝ u) (↑p) μ * ↑t

Tactic:
have h2u' : HasCompactSupport u := by
          apply HasCompactSupport.of_support_subset_isCompact hs.isCompact_closure
          exact h2u.trans subset_closure
        rel [eLpNorm_le_eLpNorm_fderiv_of_eq μ hu h2u' hp (mod_cast h2p.pos) hp']

Goal state after:
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: 𝕜 : Type u_3
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
d : ℕ
r : 𝕜
hr : r ≠ 0
s : Set E
⊢ volume.addHaarScalarFactor μH[↑d] • ‖r‖₊ ^ ↑d • μH[↑d] s = ‖r‖₊ ^ d • volume.addHaarScalarFactor μH[↑d] • μH[↑d] s

Tactic:
smul_comm

Goal state after:
𝕜 : Type u_3
E : Type u_4
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
d : ℕ
r : 𝕜
hr : r ≠ 0
s : Set E
⊢ ‖r‖₊ ^ ↑d • volume.addHaarScalarFactor μH[↑d] • μH[↑d] s = ‖r‖₊ ^ d • volume.addHaarScalarFactor μH[↑d] • μH[↑d] s
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case neg
X : Type u_2
inst✝ : EMetricSpace X
μ : OuterMeasure X
hm : μ.IsMetric
t : Set X
ht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}
s : Set X
S : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}
Ssep : ∀ (n : ℕ), AreSeparated (S n) t
Ssep' : ∀ (n : ℕ), AreSeparated (S n) (s ∩ t)
S_sub : ∀ (n : ℕ), S n ⊆ s \ t
hSs : ∀ (n : ℕ), μ (s ∩ t) + μ (S n) ≤ μ s
iUnion_S : ⋃ n, S n = s \ t
htop : ¬μ (s \ t) = ∞
r n i j : ℕ
hj : i < j
A : (↑(2 * j + r))⁻¹ < (↑(2 * i + 1 + r))⁻¹
x : X
hx : x ∈ S (2 * i + 1 + r)
y : X
hy : y ∈ s \ S (2 * j + r)
this : infEDist y t < (↑(2 * j + r))⁻¹
z : X
hzt : z ∈ t
hyz : edist y z < (↑(2 * j + r))⁻¹
hxz : (↑(2 * i + 1 + r))⁻¹ ≤ edist x z
⊢ (↑(2 * i + 1 + r))⁻¹ - (↑(2 * j + r))⁻¹ + (↑(2 * j + r))⁻¹ = (↑(2 * i + 1 + r))⁻¹

Tactic:
rw [tsub_add_cancel_of_le A.le]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
s t : Set X
r : ℝ≥0∞
r0 : 0 < r
hr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y
ε : ℝ≥0∞
left✝ : 0 < ε
εr : ε < r
u : Set X
x : X
hxs : x ∈ s
hxu : x ∈ u
y : X
hyt : y ∈ t
hyu : y ∈ u
⊢ extend (fun s x ↦ m s) u = ∞

Tactic:
have : ε < ediam u := εr.trans_le ((hr x hxs y hyt).trans <| edist_le_ediam_of_mem hxu hyu)

Goal state after:
X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
s t : Set X
r : ℝ≥0∞
r0 : 0 < r
hr : ∀ x ∈ s, ∀ y ∈ t, r ≤ edist x y
ε : ℝ≥0∞
left✝ : 0 < ε
εr : ε < r
u : Set X
x : X
hxs : x ∈ s
hxu : x ∈ u
y : X
hyt : y ∈ t
hyu : y ∈ u
this : ε < ediam u
⊢ extend (fun s x ↦ m s) u = ∞
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: V : Type u_3
P : Type u_4
inst✝⁸ : NormedAddCommGroup V
inst✝⁷ : InnerProductSpace ℝ V
inst✝⁶ : MeasurableSpace V
inst✝⁵ : BorelSpace V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MetricSpace P
inst✝² : MeasurableSpace P
inst✝¹ : BorelSpace P
inst✝ : NormedAddTorsor V P
p : P
v : V
hv : v ≠ 0
t : Set P
ht : MeasurableSet t
hrank : finrank ℝ ↥(AffineSubspace.mk' p (ℝ ∙ v)).direction = 1
hrank' : finrank ℝ ↥(AffineSubspace.mk' p (ℝ ∙ v)).directionᗮ = finrank ℝ V - 1
f : ℝ ≃L[ℝ] ↥(AffineSubspace.mk' p (ℝ ∙ v)).direction :=
  (ContinuousLinearEquiv.toSpanNonzeroSingleton ℝ v hv).trans
    (ContinuousLinearEquiv.ofEq (ℝ ∙ v) (AffineSubspace.mk' p (ℝ ∙ v)).direction ⋯)
⊢ μHE[finrank ℝ V] t = ‖v‖ₑ * ∫⁻ (x : ℝ), μHE[finrank ℝ V - 1] (t ∩ ↑(AffineSubspace.mk' (x • v +ᵥ p) (ℝ ∙ v)ᗮ))

Tactic:
have hf : MeasurableEmbedding f := f.toHomeomorph.measurableEmbedding

Goal state after:
V : Type u_3
P : Type u_4
inst✝⁸ : NormedAddCommGroup V
inst✝⁷ : InnerProductSpace ℝ V
inst✝⁶ : MeasurableSpace V
inst✝⁵ : BorelSpace V
inst✝⁴ : FiniteDimensional ℝ V
inst✝³ : MetricSpace P
inst✝² : MeasurableSpace P
inst✝¹ : BorelSpace P
inst✝ : NormedAddTorsor V P
p : P
v : V
hv : v ≠ 0
t : Set P
ht : MeasurableSet t
hrank : finrank ℝ ↥(AffineSubspace.mk' p (ℝ ∙ v)).direction = 1
hrank' : finrank ℝ ↥(AffineSubspace.mk' p (ℝ ∙ v)).directionᗮ = finrank ℝ V - 1
f : ℝ ≃L[ℝ] ↥(AffineSubspace.mk' p (ℝ ∙ v)).direction :=
  (ContinuousLinearEquiv.toSpanNonzeroSingleton ℝ v hv).trans
    (ContinuousLinearEquiv.ofEq (ℝ ∙ v) (AffineSubspace.mk' p (ℝ ∙ v)).direction ⋯)
hf : MeasurableEmbedding ⇑f
⊢ μHE[finrank ℝ V] t = ‖v‖ₑ * ∫⁻ (x : ℝ), μHE[finrank ℝ V - 1] (t ∩ ↑(AffineSubspace.mk' (x • v +ᵥ p) (ℝ ∙ v)ᗮ))
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: case neg
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup U
inst✝⁶ : InnerProductSpace 𝕜 U
inst✝⁵ : FiniteDimensional 𝕜 U
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace 𝕜 V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : FiniteDimensional 𝕜 V
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
h : finrank 𝕜 U = finrank 𝕜 V
htop : ¬f.range = ⊤
⊢ (g ∘ₗ f).normDet = g.normDet * f.normDet

Tactic:
have hker : f.ker ≠ ⊥ := by
      simpa [ker_eq_bot_iff_range_eq_top_of_finrank_eq_finrank h] using htop

Goal state after:
case neg
𝕜 : Type u_1
U : Type u_2
V : Type u_3
W : Type u_4
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup U
inst✝⁶ : InnerProductSpace 𝕜 U
inst✝⁵ : FiniteDimensional 𝕜 U
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace 𝕜 V
inst✝² : NormedAddCommGroup W
inst✝¹ : InnerProductSpace 𝕜 W
inst✝ : FiniteDimensional 𝕜 V
f : U →ₗ[𝕜] V
g : V →ₗ[𝕜] W
h : finrank 𝕜 U = finrank 𝕜 V
htop : ¬f.range = ⊤
hker : f.ker ≠ ⊥
⊢ (g ∘ₗ f).normDet = g.normDet * f.normDet
Module: Mathlib.Analysis.InnerProductSpace.Reproducing

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
X : Type u_2
V : Type u_3
inst✝⁶ : NormedAddCommGroup V
inst✝⁵ : InnerProductSpace 𝕜 V
H : Type u_4
inst✝⁴ : NormedAddCommGroup H
inst✝³ : InnerProductSpace 𝕜 H
inst✝² : RKHS 𝕜 H X V
inst✝¹ : CompleteSpace H
inst✝ : CompleteSpace V
s : X →₀ V →L[𝕜] V
v : V
⊢ 0 ≤ ⟪((s.sum fun i xi ↦ s.sum fun j xj ↦ star xi * kernel H i j * xj) - 0) v, v⟫_𝕜

Tactic:
· simp [Finsupp.sum_apply'', Finsupp.sum_inner, star, adjoint_inner_left,
      kernel_inner, -inner_kerFun, -kerFun_inner]
    simp [← Finsupp.sum_inner, ← Finsupp.inner_sum, -kerFun_inner, -inner_kerFun]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedDivisionRing 𝕜
inst✝³ : Module 𝕜 E
inst✝² : NormSMulClass 𝕜 E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
d : ℝ
hd : 0 ≤ d
r : 𝕜
hr : r ≠ 0
s : Set E
this : ∀ {r : 𝕜} (s : Set E), μH[d] (r • s) ≤ ‖r‖₊ ^ d • μH[d] s
⊢ μH[d] s ≤ ↑(‖r⁻¹‖₊ ^ d) * μH[d] (r • s)

Tactic:
· refine Eq.trans_le ?_ (this (r • s))
      rw [inv_smul_smul₀ hr]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inl
X : Type u_2
Y : Type u_3
inst✝⁵ : EMetricSpace X
inst✝⁴ : EMetricSpace Y
inst✝³ : MeasurableSpace X
inst✝² : BorelSpace X
inst✝¹ : MeasurableSpace Y
inst✝ : BorelSpace Y
f : X → Y
d : ℝ
hd : 0 ≤ d
s : Set Y
hf : AntilipschitzWith 0 f
⊢ μH[d] (f ⁻¹' s) ≤ ↑0 ^ d * μH[d] s

Tactic:
rcases eq_empty_or_nonempty (f ⁻¹' s) with (hs | ⟨x, hx⟩)
    · simp only [hs, measure_empty, zero_le]
    have : f ⁻¹' s = {x} := by
      have : Subsingleton X := hf.subsingleton
      have : (f ⁻¹' s).Subsingleton := subsingleton_univ.anti (subset_univ _)
      exact (subsingleton_iff_singleton hx).1 this
    rw [this]
    rcases eq_or_lt_of_le hd with (rfl | h'd)
    · simp only [ENNReal.rpow_zero, one_mul]
      rw [hausdorffMeasure_zero_singleton]
      exact one_le_hausdorffMeasure_zero_of_nonempty ⟨f x, hx⟩
    · have := nullSingletonClass_hausdorff X h'd
      simp only [zero_le, measure_singleton]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n ↦ 1 / ↑n) atTop (𝓝 0)
⊢ μH[↑(Fintype.card ι)] (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i)) ≤ ∏ x, volume ((fun i ↦ Ioo ↑(a i) ↑(b i)) x)

Tactic:
have B : ∀ᶠ n in atTop, ∀ i : γ n, ediam (t n i) ≤ 1 / n := by
    refine eventually_atTop.2 ⟨1, fun n hn => ?_⟩
    intro f
    refine ediam_pi_le_of_le fun b => ?_
    simp only [Real.ediam_Icc, add_div, ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), le_refl,
      add_sub_add_left_eq_sub, add_sub_cancel_left, ENNReal.ofReal_one, ENNReal.ofReal_natCast]

Goal state after:
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n ↦ 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), ediam (t n i) ≤ 1 / ↑n
⊢ μH[↑(Fintype.card ι)] (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i)) ≤ ∏ x, volume ((fun i ↦ Ioo ↑(a i) ↑(b i)) x)
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: case refine_1.inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
h : 0 < ‖o.rightAngleRotationAux₁ x‖
⊢ ‖o.rightAngleRotationAux₁ x‖ * ‖o.rightAngleRotationAux₁ x‖ ≤ ‖x‖ * ‖o.rightAngleRotationAux₁ x‖

Tactic:
← real_inner_self_eq_norm_mul_norm,

Goal state after:
case refine_1.inr
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
x : E
h : 0 < ‖o.rightAngleRotationAux₁ x‖
⊢ ⟪o.rightAngleRotationAux₁ x, o.rightAngleRotationAux₁ x⟫ ≤ ‖x‖ * ‖o.rightAngleRotationAux₁ x‖
Module: Mathlib.Analysis.Matrix.LDL

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
this✝ : NormedAddCommGroup (n → 𝕜) := Sᵀ.toNormedAddCommGroup ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := Sᵀ.toInnerProductSpace ⋯
i j : n
⊢ gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i j = (⇑(gramSchmidtBasis (Pi.basisFun 𝕜 n)))ᵀᵀ i j

Tactic:
coe_gramSchmidtBasis

Goal state after:
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
n : Type u_2
inst✝³ : LinearOrder n
inst✝² : WellFoundedLT n
inst✝¹ : LocallyFiniteOrderBot n
S : Matrix n n 𝕜
inst✝ : Fintype n
hS : S.PosDef
this✝ : NormedAddCommGroup (n → 𝕜) := Sᵀ.toNormedAddCommGroup ⋯
this : InnerProductSpace 𝕜 (n → 𝕜) := Sᵀ.toInnerProductSpace ⋯
i j : n
⊢ gramSchmidt 𝕜 (⇑(Pi.basisFun 𝕜 n)) i j = (gramSchmidt 𝕜 ⇑(Pi.basisFun 𝕜 n))ᵀᵀ i j
Module: Mathlib.Analysis.LocallyConvex.WeakSpace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E ≃ₗ[𝕜] F
e_dual : StrongDual 𝕜 F ≃ StrongDual 𝕜 E
f : StrongDual 𝕜 E
he : ∀ (f : StrongDual 𝕜 F), ⇑(e_dual f) = ⇑(e.dualMap ↑f)
this : ∀ (g : StrongDual 𝕜 E), ⇑g = ⇑(e_dual.symm g) ∘ ⇑e
x : F
⊢ (e_dual.symm f) x = (e.symm.dualMap ↑f) x

Tactic:
conv_rhs => rw [LinearEquiv.dualMap_apply, ContinuousLinearMap.coe_coe, this]

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹⁶ : RCLike 𝕜
inst✝¹⁵ : AddCommGroup E
inst✝¹⁴ : Module 𝕜 E
inst✝¹³ : AddCommGroup F
inst✝¹² : Module 𝕜 F
inst✝¹¹ : Module ℝ E
inst✝¹⁰ : IsScalarTower ℝ 𝕜 E
inst✝⁹ : Module ℝ F
inst✝⁸ : IsScalarTower ℝ 𝕜 F
inst✝⁷ : TopologicalSpace E
inst✝⁶ : IsTopologicalAddGroup E
inst✝⁵ : ContinuousSMul 𝕜 E
inst✝⁴ : LocallyConvexSpace ℝ E
inst✝³ : TopologicalSpace F
inst✝² : IsTopologicalAddGroup F
inst✝¹ : ContinuousSMul 𝕜 F
inst✝ : LocallyConvexSpace ℝ F
s : Set E
hs : Convex ℝ s
e : E ≃ₗ[𝕜] F
e_dual : StrongDual 𝕜 F ≃ StrongDual 𝕜 E
f : StrongDual 𝕜 E
he : ∀ (f : StrongDual 𝕜 F), ⇑(e_dual f) = ⇑(e.dualMap ↑f)
this : ∀ (g : StrongDual 𝕜 E), ⇑g = ⇑(e_dual.symm g) ∘ ⇑e
x : F
⊢ (e_dual.symm f) x = (⇑(e_dual.symm f) ∘ ⇑e) (e.symm x)
Module: Mathlib.Analysis.MellinInversion

Goal state before:
pp: ⊢ rexp ∘ Neg.neg '' univ = Ioi 0

Tactic:
Set.image_comp,

Goal state after:
⊢ rexp '' Neg.neg '' univ = Ioi 0
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
f : E ≃ₗᵢ[ℝ] ℂ
hf : (map (Fin 2) f.toLinearEquiv) o = Complex.orientation
x y : E
⊢ (o.areaForm x) y = (o.areaForm (f.symm (f x))) (f.symm (f y))

Tactic:
iterate 2 rw [LinearIsometryEquiv.symm_apply_apply]

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace ℝ E
inst✝ : Fact (finrank ℝ E = 2)
o : Orientation ℝ E (Fin 2)
f : E ≃ₗᵢ[ℝ] ℂ
hf : (map (Fin 2) f.toLinearEquiv) o = Complex.orientation
x y : E
⊢ (o.kahler x) y = (o.kahler (f.symm (f x))) (f.symm (f y))

Tactic:
iterate 2 rw [LinearIsometryEquiv.symm_apply_apply]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : 1 - z ∈ slitPlane
⊢ log (1 - z)⁻¹ = z * ∫ (t : ℝ) in 0..1, (1 - t • z)⁻¹

Tactic:
by
  rw [sub_eq_add_neg 1 z] at hz ⊢
  rw [log_inv _ <| slitPlane_arg_ne_pi hz, neg_eq_iff_eq_neg, ← neg_mul]
  convert! log_eq_integral hz using 5
  rw [sub_eq_add_neg, smul_neg]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
t : ℝ
ht : t ∈ Set.Icc 0 1
⊢ HasDerivAt f (f' (0 + ↑t * z)) (0 + ↑t * z)

Tactic:
zero_add

Goal state after:
n : ℕ
z : ℂ
hz : ‖z‖ < 1
help : IntervalIntegrable (fun t ↦ t ^ n * (1 - ‖z‖)⁻¹) MeasureTheory.volume 0 1
f : ℂ → ℂ := fun z ↦ log (1 + z) - logTaylor (n + 1) z
f' : ℂ → ℂ := fun z ↦ (-z) ^ n * (1 + z)⁻¹
t : ℝ
ht : t ∈ Set.Icc 0 1
⊢ HasDerivAt f (f' (↑t * z)) (↑t * z)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.Deriv

Goal state before:
pp: n : ℕ
ih : ContinuousAt Gamma (-(↑n + 1))
⊢ ContinuousAt (fun s ↦ Gamma (s + 1)) (-↑n - 1)

Tactic:
← neg_add'

Goal state after:
n : ℕ
ih : ContinuousAt Gamma (-(↑n + 1))
⊢ ContinuousAt (fun s ↦ Gamma (s + 1)) (-(↑n + 1))
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case convert_1
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 : ∀ x ∈ uIcc 0 (π / 2), HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
⊢ IntervalIntegrable (fun x ↦ -↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) volume 0 (π / 2)

Tactic:
apply Continuous.intervalIntegrable (by fun_prop)

Goal state after: