mathlib-initiative/mathlib-tactics huggingface dataset:
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
α : 𝕜
A B : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
hB : B.IsSymmetric
hAB : Commute A B
⊢ ⨆ γ, eigenspace A α ⊓ eigenspace B γ = eigenspace A α

Tactic:
conv_rhs => rw [← (eigenspace A α).map_subtype_top]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
α : 𝕜
A B : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
hB : B.IsSymmetric
hAB : Commute A B
⊢ ⨆ γ, eigenspace A α ⊓ eigenspace B γ = map (eigenspace A α).subtype ⊤
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
this : CompleteSpace ↑↑(↑(continuousLinearMapOfBilin B)).range
v w : V
mem_w_orthogonal : w ∈ (↑(continuousLinearMapOfBilin B)).rangeᗮ
C : ℝ
C_pos : 0 < C
coercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u
⊢ w = 0

Tactic:
rw [← norm_eq_zero, ← mul_self_eq_zero, ← mul_right_inj' C_pos.ne', mul_zero, ←
      mul_assoc]

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
this : CompleteSpace ↑↑(↑(continuousLinearMapOfBilin B)).range
v w : V
mem_w_orthogonal : w ∈ (↑(continuousLinearMapOfBilin B)).rangeᗮ
C : ℝ
C_pos : 0 < C
coercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u
⊢ C * ‖w‖ * ‖w‖ = 0
Module: Mathlib.Analysis.InnerProductSpace.Positive

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : Type u_4
inst✝ : Finite n
M : Matrix n n 𝕜
⊢ M.PosSemidef ↔ ∃ m v, M = ∑ i, vecMulVec (v i) (star (v i))

Tactic:
have := Fintype.ofFinite n

Goal state after:
𝕜 : Type u_1
inst✝¹ : RCLike 𝕜
n : Type u_4
inst✝ : Finite n
M : Matrix n n 𝕜
this : Fintype n
⊢ M.PosSemidef ↔ ∃ m v, M = ∑ i, vecMulVec (v i) (star (v i))
Module: Mathlib.Topology.Algebra.Module.LinearPMap

Goal state before:
pp: R : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup E
inst✝⁴ : AddCommGroup F
inst✝³ : Module R E
inst✝² : Module R F
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace F
f : E →ₗ.[R] F
hf : f.toFun.ker = ⊥
⊢ f.inverse.IsClosed ↔ f.IsClosed

Tactic:
by
  rw [IsClosed, inverse_graph hf]
  exact (ContinuousLinearEquiv.prodComm R E F).isClosed_image

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
⊢ eLpNorm' u (↑p') μ < ∞

Tactic:
← eLpNorm_nnreal_eq_eLpNorm' hp'0

Goal state after:
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
⊢ eLpNorm u (↑p') μ < ∞
Module: Mathlib.Analysis.InnerProductSpace.SingularValues

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : RCLike 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
inst✝³ : FiniteDimensional 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace 𝕜 F
inst✝ : FiniteDimensional 𝕜 F
T : E →ₗ[𝕜] F
⊢ Function.Injective ⇑T ↔ ∀ i < finrank 𝕜 E, 0 < T.singularValues i

Tactic:
have := (adjoint T ∘ₗ T).not_hasEigenvalue_zero_tfae.out 4 0
  rw [← adjoint_comp_self_injective_iff, ← coe_comp, ← ker_eq_bot, ← not_iff_not, this.not_left]
  push Not
  constructor
  · intro h
    obtain ⟨i, hi⟩ := T.isSymmetric_adjoint_comp_self.exists_eigenvalues_eq rfl h
    use i, i.isLt
    simp [RCLike.ofReal_eq_zero.mp hi, T.singularValues_fin rfl]
  · intro ⟨i, h, hz⟩
    convert! T.isSymmetric_adjoint_comp_self.hasEigenvalue_eigenvalues rfl ⟨i, h⟩
    rw [← sq_singularValues_of_lt, le_antisymm hz (T.singularValues_nonneg i)]
    simp

Goal state after:
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case e'_4
E : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F' : Type u_5
inst✝¹ : NormedAddCommGroup F'
inst✝ : InnerProductSpace ℝ F'
u : E → F'
hu : ContDiff ℝ 1 u
h2u : HasCompactSupport u
p p' : ℝ≥0
hp✝ : 1 ≤ p
hp'0 : ¬p' = 0
n : ℕ := finrank ℝ E
hn✝ : 0 < n
hp' : (↑p')⁻¹ = ↑p⁻¹ - (↑n)⁻¹
n' : ℝ≥0 := (↑n).conjExponent
h2p : ↑p < ↑n
h0n : 2 ≤ n
hn : (↑n).HolderConjugate n'
h1n : 1 ≤ ↑n
h2n : 0 < ↑n - 1
hnp : 0 < ↑n - ↑p
hp : 1 < p
q : ℝ := (↑p).conjExponent
hq : (↑p).HolderConjugate q
h0p : p ≠ 0
h1p : ↑p ≠ 1
h3p : ↑p - 1 ≠ 0
h2q : 1 / ↑n' - 1 / q = 1 / ↑p'
γ : ℝ≥0 := NNReal.mk (↑p * (↑n - 1) / (↑n - ↑p)) ⋯
h0γ : ↑γ = ↑p * (↑n - 1) / (↑n - ↑p)
h1γ : 1 < ↑γ
h2γ : γ * n' = p'
h3γ : (↑γ - 1) * q = ↑p'
h4γ : ↑γ ≠ 0
h3u : ¬∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ = 0
h4u : ∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ ≠ ∞
h5u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ 0
h6u : (∫⁻ (x : E), ‖u x‖ₑ ^ ↑p' ∂μ) ^ (1 / q) ≠ ∞
h7u : Continuous u
v : E → ℝ := fun x ↦ ‖u x‖ ^ ↑γ
hv : ContDiff ℝ 1 v
h2v : HasCompactSupport v
C : ℝ≥0 := eLpNormLESNormFDerivOneConst μ ↑n'
x✝ : E
⊢ ‖u x✝‖ₑ ^ ↑p' = (‖u x✝‖ₑ ^ (↑γ - 1)) ^ (↑p).conjExponent

Tactic:
simp [γ, n, q, ← ENNReal.rpow_mul, ← h3γ]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
H : Type u_5
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : InnerProductSpace 𝕜 E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : InnerProductSpace 𝕜 F
inst✝³ : NormedAddCommGroup G
inst✝² : InnerProductSpace 𝕜 G
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
f₁ f₂ : E →L[𝕜] F
g : G →L[𝕜] H
⊢ mapL (f₁ + f₂) g = mapL f₁ g + mapL f₂ g

Tactic:
ext; simp [map_add_left]

Goal state after:
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
s : AffineSubspace ℝ P
hs : Nonempty ↥s
t : Set P
ht : MeasurableSet t
p : ↥s
x : ↥s.direction
u : Set ↥(mk' (↑(x +ᵥ p)) s.directionᗮ) := Subtype.val ⁻¹' (t ∩ ↑(mk' (↑(x +ᵥ p)) s.directionᗮ))
hu : MeasurableSet u
hinter : t ∩ ↑(mk' (↑(x +ᵥ p)) s.directionᗮ) = Subtype.val '' u
hxp : ↑(x +ᵥ p) ∈ mk' (↑(x +ᵥ p)) s.directionᗮ
⊢ finrank ℝ ↥s.directionᗮ = finrank ℝ ↥(mk' (↑(x +ᵥ p)) s.directionᗮ).direction

Tactic:
rw [direction_mk']

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

Goal state before:
pp: X : Type u_2
inst✝ : EMetricSpace X
m : Set X → ℝ≥0∞
k l : ℕ
h : k ≤ l
x✝ : Set X
hs : ediam x✝ ≤ (↑l)⁻¹
⊢ (↑l)⁻¹ ≤ (↑k)⁻¹

Tactic:
by simpa

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

Goal state before:
pp: 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 = ⊤
this : (g.domRestrict f.range).normDet * id.normDet = g.normDet
⊢ (g.domRestrict f.range).normDet = g.normDet

Tactic:
by simpa

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

Goal state before:
pp: case inl.inr.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
r : ℝ≥0
f : X → Y
s : Set X
hr : 0 < r
h : HolderOnWith 0 r f s
x : X
hx : x ∈ s
this : f '' s = {f x}
hd : 0 ≤ 0
⊢ μH[0] {f x} ≤ μH[0] s

Tactic:
rw [hausdorffMeasure_zero_singleton]

Goal state after:
case inl.inr.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
r : ℝ≥0
f : X → Y
s : Set X
hr : 0 < r
h : HolderOnWith 0 r f s
x : X
hx : x ∈ s
this : f '' s = {f x}
hd : 0 ≤ 0
⊢ 1 ≤ μH[0] s
Module: Mathlib.Analysis.InnerProductSpace.StarOrder

Goal state before:
pp: case mpr
𝕜 : Type u_1
H : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
inst✝² : Algebra ℝ (H →L[𝕜] H)
inst✝¹ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)
inst✝ : ContinuousFunctionalCalculus ℝ (H →L[𝕜] H) IsSelfAdjoint
f p : H →L[𝕜] H
hp : p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
⊢ f ≤ f + p

Tactic:
le_def,

Goal state after:
case mpr
𝕜 : Type u_1
H : Type u_2
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : InnerProductSpace 𝕜 H
inst✝³ : CompleteSpace H
inst✝² : Algebra ℝ (H →L[𝕜] H)
inst✝¹ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)
inst✝ : ContinuousFunctionalCalculus ℝ (H →L[𝕜] H) IsSelfAdjoint
f p : H →L[𝕜] H
hp : p ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)
⊢ (f + p - f).IsPositive
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case inl.inr.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
s : Set Y
hf : AntilipschitzWith 0 f
x : X
hx : x ∈ f ⁻¹' s
this : f ⁻¹' s = {x}
hd : 0 ≤ 0
⊢ μH[0] {x} ≤ μH[0] s

Tactic:
rw [hausdorffMeasure_zero_singleton]

Goal state after:
case inl.inr.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
s : Set Y
hf : AntilipschitzWith 0 f
x : X
hx : x ∈ f ⁻¹' s
this : f ⁻¹' s = {x}
hd : 0 ≤ 0
⊢ 1 ≤ μH[0] s
Module: Mathlib.Analysis.LocallyConvex.WeakOperatorTopology

Goal state before:
pp: 𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁸ : NormedField 𝕜₁
inst✝¹⁷ : NormedField 𝕜₂
inst✝¹⁶ : NormedField 𝕜₃
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
inst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : TopologicalSpace E
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Module 𝕜₃ G
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousConstSMul 𝕜₂ F
inst✝³ : IsTopologicalAddGroup G
inst✝² : ContinuousConstSMul 𝕜₃ G
inst✝¹ : RingHomSurjective σ₂₃
inst✝ : RingHomIsometric σ₂₃
g : F →SWOT[σ₂₃] G
x : E
z : StrongDual 𝕜₃ G
σ_bij : Function.Bijective ⇑σ₂₃
σ_equiv : 𝕜₂ ≃+* 𝕜₃ := RingEquiv.ofBijective σ₂₃ σ_bij
invPair : RingHomInvPair σ₂₃ ↑σ_equiv.symm := RingHomInvPair.of_ringEquiv σ_equiv
invPair_symm : RingHomInvPair (↑σ_equiv.symm) σ₂₃ := RingHomInvPair.symm σ₂₃ ↑σ_equiv.symm
σ_li : 𝕜₂ ≃ₛₗᵢ[σ₂₃] 𝕜₃ :=
  let __LinearEquiv := LinearEquiv.ofBijective σ₂₃.toSemilinearMap σ_bij;
  { toLinearEquiv := __LinearEquiv, norm_map' := ⋯ }
a : E →SWOT[σ₁₂] F
| σ_li (σ_li.symm (z (g (a x))))

Tactic:
← toCLM_apply

Goal state after:
𝕜₁ : Type u_5
𝕜₂ : Type u_6
𝕜₃ : Type u_7
E : Type u_9
F : Type u_10
G : Type u_11
inst✝¹⁸ : NormedField 𝕜₁
inst✝¹⁷ : NormedField 𝕜₂
inst✝¹⁶ : NormedField 𝕜₃
σ₁₂ : 𝕜₁ →+* 𝕜₂
σ₁₃ : 𝕜₁ →+* 𝕜₃
σ₂₃ : 𝕜₂ →+* 𝕜₃
inst✝¹⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹⁴ : AddCommGroup E
inst✝¹³ : TopologicalSpace E
inst✝¹² : Module 𝕜₁ E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : Module 𝕜₂ F
inst✝⁸ : AddCommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Module 𝕜₃ G
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : ContinuousConstSMul 𝕜₂ F
inst✝³ : IsTopologicalAddGroup G
inst✝² : ContinuousConstSMul 𝕜₃ G
inst✝¹ : RingHomSurjective σ₂₃
inst✝ : RingHomIsometric σ₂₃
g : F →SWOT[σ₂₃] G
x : E
z : StrongDual 𝕜₃ G
σ_bij : Function.Bijective ⇑σ₂₃
σ_equiv : 𝕜₂ ≃+* 𝕜₃ := RingEquiv.ofBijective σ₂₃ σ_bij
invPair : RingHomInvPair σ₂₃ ↑σ_equiv.symm := RingHomInvPair.of_ringEquiv σ_equiv
invPair_symm : RingHomInvPair (↑σ_equiv.symm) σ₂₃ := RingHomInvPair.symm σ₂₃ ↑σ_equiv.symm
σ_li : 𝕜₂ ≃ₛₗᵢ[σ₂₃] 𝕜₃ :=
  let __LinearEquiv := LinearEquiv.ofBijective σ₂₃.toSemilinearMap σ_bij;
  { toLinearEquiv := __LinearEquiv, norm_map' := ⋯ }
a : E →SWOT[σ₁₂] F
| σ_li (σ_li.symm (z (g.toCLM (a x))))
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ (↑n / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n) -
      (↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2) =
    ∫ (x : ℝ) in 0..π / 2,
      -((↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) * (-Complex.cos (2 * z * ↑x) / (2 * z)))

pp: case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ n - 1 ≠ 0

Tactic:
← integral_const_mul,

Goal state after:
case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ (∫ (x : ℝ) in 0..π / 2, ↑n / (2 * z) * (Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n)) -
      (↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2) =
    ∫ (x : ℝ) in 0..π / 2,
      -((↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) * (-Complex.cos (2 * z * ↑x) / (2 * z)))

case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ n - 1 ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ (∫ (x : ℝ) in 0..π / 2, ↑n / (2 * z) * (Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n)) -
      (↑n - 1) / (2 * z) * ∫ (x : ℝ) in 0..π / 2, Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2) =
    ∫ (x : ℝ) in 0..π / 2,
      -((↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) * (-Complex.cos (2 * z * ↑x) / (2 * z)))

pp: case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ n - 1 ≠ 0

Tactic:
← integral_const_mul,

Goal state after:
case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ (∫ (x : ℝ) in 0..π / 2, ↑n / (2 * z) * (Complex.cos (2 * z * ↑x) * ↑(cos x) ^ n)) -
      ∫ (x : ℝ) in 0..π / 2, (↑n - 1) / (2 * z) * (Complex.cos (2 * z * ↑x) * ↑(cos x) ^ (n - 2)) =
    ∫ (x : ℝ) in 0..π / 2,
      -((↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) * (-Complex.cos (2 * z * ↑x) / (2 * z)))

case e'_3
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
der1 :
  ∀ x ∈ uIcc 0 (π / 2),
    HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x
⊢ n - 1 ≠ 0
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: ⊢ ConvexOn ℝ (Ioi 0) Γ

Tactic:
refine
    ((convexOn_exp.subset (subset_univ _) ?_).comp convexOn_log_Gamma
          (exp_monotone.monotoneOn _)).congr
      fun x hx => exp_log (Gamma_pos_of_pos hx)

Goal state after:
⊢ Convex ℝ (log ∘ Γ '' Ioi 0)
Module: Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup

Goal state before:
pp: ⊢ Γ (3 / 2) < 1

Tactic:
have A : (0 : ℝ) < 3 / 2 := by simp

Goal state after:
A : 0 < 3 / 2
⊢ Γ (3 / 2) < 1
Module: Mathlib.LinearAlgebra.AffineSpace.Ceva

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
ι : Type u_4
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
p : ι → P
hp : AffineIndependent k p
s : Set ι
hs : s.Nonempty
fs : ↑s → Finset ι
hfs : ∀ (i : ↑s), ↑i ∈ fs i
w : ↑s → ι → k
hw : ∀ (i : ↑s), ∑ j ∈ fs i, w i j = 1
i : ↑s
r : k
hp' :
  ∀ (i_1 : ↑s),
    ∃ r_1,
      (AffineMap.lineMap (p ↑i_1) ((Finset.affineCombination k (fs i_1) p) (w i_1))) r_1 =
        (AffineMap.lineMap (p ↑i) ((Finset.affineCombination k (fs i) p) (w i))) r
⊢ ∃ r_1,
    (Finset.affineCombination k (fs i) p) ((AffineMap.lineMap (Pi.single (↑i) 1) (w i)) r_1) =
      (AffineMap.lineMap (p ↑i) ((Finset.affineCombination k (fs i) p) (w i))) r

Tactic:
exact ⟨r, by simp [hfs]⟩

Goal state after:
Module: Mathlib.Topology.Algebra.AsymptoticCone

Goal state before:
pp: k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s✝ s : Set V
hs : Convex k s
hs' : IsClosed[inst✝²] s
⊢ Convex k (asymptoticCone k s)

Tactic:
rcases s.eq_empty_or_nonempty with rfl | ⟨p, hp⟩

Goal state after:
case inl
k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s : Set V
hs : Convex k ∅
hs' : IsClosed[inst✝²] ∅
⊢ Convex k (asymptoticCone k ∅)

case inr
k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : IsStrictOrderedRing k
inst✝⁶ : TopologicalSpace k
inst✝⁵ : OrderTopology k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
s✝ s : Set V
hs : Convex k s
hs' : IsClosed[inst✝²] s
p : V
hp : p ∈ s
⊢ Convex k (asymptoticCone k s)
Module: Mathlib.Analysis.Normed.Affine.ContinuousAffineMap

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_3
W : Type u_4
inst✝⁴ : SeminormedAddCommGroup V
inst✝³ : SeminormedAddCommGroup W
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 V
inst✝ : NormedSpace 𝕜 W
f : V →ᴬ[𝕜] W
h : f 0 = 0
⊢ max ‖f 0‖ ‖f.contLinear‖ = max 0 ‖f.contLinear‖

Tactic:
by rw [h, norm_zero]

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.MazurUlam

Goal state before:
pp: E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  ((f.trans (pointReflection ℝ (midpoint ℝ (f x) (f y))).toIsometryEquiv).trans f.symm).trans
    (pointReflection ℝ (midpoint ℝ x y)).toIsometryEquiv
hx : e x = x
hy : e y = y
hm :
  f.symm ((pointReflection ℝ (midpoint ℝ (f x) (f y))).toIsometryEquiv (f (midpoint ℝ x y))) =
    (pointReflection ℝ (midpoint ℝ x y)).toIsometryEquiv (midpoint ℝ x y)
⊢ f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)

Tactic:
coe_toIsometryEquiv,

Goal state after:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  ((f.trans (pointReflection ℝ (midpoint ℝ (f x) (f y))).toIsometryEquiv).trans f.symm).trans
    (pointReflection ℝ (midpoint ℝ x y)).toIsometryEquiv
hx : e x = x
hy : e y = y
hm :
  f.symm ((pointReflection ℝ (midpoint ℝ (f x) (f y))) (f (midpoint ℝ x y))) =
    (pointReflection ℝ (midpoint ℝ x y)).toIsometryEquiv (midpoint ℝ x y)
⊢ f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)
Module: Mathlib.Analysis.Normed.Affine.MazurUlam

Goal state before:
pp: E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  ((f.trans (pointReflection ℝ (midpoint ℝ (f x) (f y))).toIsometryEquiv).trans f.symm).trans
    (pointReflection ℝ (midpoint ℝ x y)).toIsometryEquiv
hx : e x = x
hy : e y = y
hm :
  f.symm ((pointReflection ℝ (midpoint ℝ (f x) (f y))) (f (midpoint ℝ x y))) =
    (pointReflection ℝ (midpoint ℝ x y)).toIsometryEquiv (midpoint ℝ x y)
⊢ f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)

Tactic:
coe_toIsometryEquiv,

Goal state after:
E : Type u_1
PE : Type u_2
F : Type u_3
PF : Type u_4
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MetricSpace PE
inst✝⁴ : NormedAddTorsor E PE
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : MetricSpace PF
inst✝ : NormedAddTorsor F PF
f : PE ≃ᵢ PF
x y : PE
e : PE ≃ᵢ PE :=
  ((f.trans (pointReflection ℝ (midpoint ℝ (f x) (f y))).toIsometryEquiv).trans f.symm).trans
    (pointReflection ℝ (midpoint ℝ x y)).toIsometryEquiv
hx : e x = x
hy : e y = y
hm :
  f.symm ((pointReflection ℝ (midpoint ℝ (f x) (f y))) (f (midpoint ℝ x y))) =
    (pointReflection ℝ (midpoint ℝ x y)) (midpoint ℝ x y)
⊢ f (midpoint ℝ x y) = midpoint ℝ (f x) (f y)