mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ ↑p'⁻¹ = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹

Tactic:
inv_inv,

Goal state after:
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
⊢ ↑(p⁻¹ - (↑(finrank ℝ E))⁻¹) = ↑p⁻¹ - (↑(finrank ℝ E))⁻¹
Module: Mathlib.Analysis.FunctionalSpaces.SobolevInequality

Goal state before:
pp: case e'_4.h
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
⊢ 0 ≤ 1 / ↑q - 1 / ↑p'

Tactic:
· rw [one_div, one_div]
          norm_cast
          rw [hp']
          simpa using hpq

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
ι₁ : Type u_6
ι₂ : Type u_7
b₁ : ι₁ → E
b₂ : ι₂ → F
hb₁ : Orthonormal 𝕜 b₁
hb₂ : Orthonormal 𝕜 b₂
⊢ Orthonormal 𝕜 fun i ↦ b₁ i.1 ⊗ₜ[𝕜] b₂ i.2

Tactic:
orthonormal_iff_ite

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace 𝕜 F
ι₁ : Type u_6
ι₂ : Type u_7
b₁ : ι₁ → E
b₂ : ι₂ → F
hb₁ : Orthonormal 𝕜 b₁
hb₂ : Orthonormal 𝕜 b₂
⊢ ∀ (i j : ι₁ × ι₂), inner 𝕜 (b₁ i.1 ⊗ₜ[𝕜] b₂ i.2) (b₁ j.1 ⊗ₜ[𝕜] b₂ j.2) = if i = j then 1 else 0
Module: Mathlib.Geometry.Euclidean.Volume.Measure

Goal state before:
pp: case e_f
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
hrank : finrank ℝ ↥s.directionᗮ = finrank ℝ ↥(mk' (↑(x +ᵥ p)) s.directionᗮ).direction
y : V
⊢ y ∈ Subtype.val '' Prod.mk x ⁻¹' ⇑(s.direction.measurableEquivProd ↑p).symm ⁻¹' t ↔
    y ∈ Subtype.val '' ⇑(IsometryEquiv.vaddConst ⟨↑(x +ᵥ p), hxp⟩) ⁻¹' u

Tactic:
simp [u, vadd_vadd, add_comm]

Goal state after:
Module: Mathlib.Topology.MetricSpace.HausdorffDimension

Goal state before:
pp: ι : Type u_1
inst✝ : Fintype ι
x : ι → ℝ
r : ℝ
hr : 0 < r
⊢ dimH (Metric.ball x r) = ↑(Fintype.card ι)

Tactic:
cases isEmpty_or_nonempty ι

Goal state after:
case inl
ι : Type u_1
inst✝ : Fintype ι
x : ι → ℝ
r : ℝ
hr : 0 < r
h✝ : IsEmpty ι
⊢ dimH (Metric.ball x r) = ↑(Fintype.card ι)

case inr
ι : Type u_1
inst✝ : Fintype ι
x : ι → ℝ
r : ℝ
hr : 0 < r
h✝ : Nonempty ι
⊢ dimH (Metric.ball x r) = ↑(Fintype.card ι)
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: 𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
f : U →L[𝕜] V
⊢ ↑((↑f).normDet ^ 2) = (ContinuousLinearMap.adjoint f ∘SL f).det

Tactic:
have : CompleteSpace U := FiniteDimensional.complete 𝕜 U

Goal state after:
𝕜 : Type u_1
U : Type u_2
V : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup U
inst✝⁴ : InnerProductSpace 𝕜 U
inst✝³ : FiniteDimensional 𝕜 U
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace 𝕜 V
inst✝ : CompleteSpace V
f : U →L[𝕜] V
this : CompleteSpace U
⊢ ↑((↑f).normDet ^ 2) = (ContinuousLinearMap.adjoint f ∘SL f).det
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m₁ m₂ : ℝ≥0∞ → ℝ≥0∞
c : ℝ≥0∞
hc : c ≠ ∞
h0 : c ≠ 0
hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂
s : Set X
⊢ (mkMetric m₁) s ≤ (c • mkMetric m₂) s

Tactic:
← OuterMeasure.coe_mkMetric,

Goal state after:
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
m₁ m₂ : ℝ≥0∞ → ℝ≥0∞
c : ℝ≥0∞
hc : c ≠ ∞
h0 : c ≠ 0
hle : m₁ ≤ᶠ[𝓝[≥] 0] c • m₂
s : Set X
⊢ (OuterMeasure.mkMetric m₁) s ≤ (c • mkMetric m₂) s
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: case neg
U : Type u_5
V : Type u_6
inst✝⁸ : NormedAddCommGroup U
inst✝⁷ : InnerProductSpace ℝ U
inst✝⁶ : FiniteDimensional ℝ U
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MeasurableSpace U
inst✝² : BorelSpace U
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : U →ₗ[ℝ] V
s : Set U
h : ¬f.ker = ⊥
⊢ μH[↑(finrank ℝ U)] (⇑f.range.subtypeₗᵢ '' ⇑f.rangeRestrict '' s) = 0

Tactic:
(LinearIsometry.isometry _).hausdorffMeasure_image (by simp)

Goal state after:
case neg
U : Type u_5
V : Type u_6
inst✝⁸ : NormedAddCommGroup U
inst✝⁷ : InnerProductSpace ℝ U
inst✝⁶ : FiniteDimensional ℝ U
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MeasurableSpace U
inst✝² : BorelSpace U
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : U →ₗ[ℝ] V
s : Set U
h : ¬f.ker = ⊥
⊢ μH[↑(finrank ℝ U)] (⇑f.rangeRestrict '' s) = 0
Module: Mathlib.MeasureTheory.Measure.Hausdorff

Goal state before:
pp: case a
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
x : X
t : ℕ → Set X
hst : {x} ⊆ ⋃ n, t n
⊢ (∀ (n : ℕ), ediam (t n) ≤ 1) → 1 ≤ ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), 1

Tactic:
_

Goal state after:
case a
X : Type u_2
inst✝² : EMetricSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
x : X
t : ℕ → Set X
hst : {x} ⊆ ⋃ n, t n
i✝ : ∀ (n : ℕ), ediam (t n) ≤ 1
⊢ 1 ≤ ∑' (n : ℕ), ⨆ (_ : (t n).Nonempty), 1
Module: Mathlib.Analysis.InnerProductSpace.NormDet

Goal state before:
pp: U : Type u_5
V : Type u_6
inst✝⁸ : NormedAddCommGroup U
inst✝⁷ : InnerProductSpace ℝ U
inst✝⁶ : FiniteDimensional ℝ U
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MeasurableSpace U
inst✝² : BorelSpace U
inst✝¹ : MeasurableSpace V
inst✝ : BorelSpace V
f : U →ₗ[ℝ] V
s : Set U
⊢ ↑(volume.addHaarScalarFactor μH[↑(finrank ℝ U)]) * (ENNReal.ofReal f.normDet * μH[↑(finrank ℝ U)] s) =
    ENNReal.ofReal f.normDet * (↑(volume.addHaarScalarFactor μH[↑(finrank ℝ U)]) * μH[↑(finrank ℝ U)] s)

Tactic:
exact mul_left_comm _ _ _

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

Goal state before:
pp: case mpr
H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
x : H
S : ClosedSubmodule ℝ H
h : ∀ y ∈ S, ⟪y, x⟫.im = 0
⊢ ∀ (u : H), -(I • u) ∈ S → inner ℝ u x = 0

Tactic:
_

Goal state after:
case mpr
H : Type u_1
inst✝ : NormedAddCommGroup H
ipc : InnerProductSpace ℂ H
x : H
S : ClosedSubmodule ℝ H
h : ∀ y ∈ S, ⟪y, x⟫.im = 0
u✝ : H
⊢ -(I • u✝) ∈ S → inner ℝ u✝ x = 0
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)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), ediam (t n i) ≤ 1 / ↑n
C : ∀ᶠ (n : ℕ) in atTop, (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i
a✝ : ℕ
hn : ∀ (i : γ a✝), ediam (t a✝ i) ≤ 1 / ↑a✝
i : γ a✝
⊢ i ∈ Finset.univ → ediam (t a✝ i) ^ Fintype.card ι ≤ (1 / ↑a✝) ^ Fintype.card ι

Tactic:
_

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
C : ∀ᶠ (n : ℕ) in atTop, (univ.pi fun i ↦ Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i
a✝ : ℕ
hn : ∀ (i : γ a✝), ediam (t a✝ i) ≤ 1 / ↑a✝
i : γ a✝
x✝ : i ∈ Finset.univ
⊢ ediam (t a✝ i) ^ Fintype.card ι ≤ (1 / ↑a✝) ^ Fintype.card ι
Module: Mathlib.Analysis.LocallyConvex.ContinuousOfBounded

Goal state before:
pp: case refine_1
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
n : ℕ
⊢ True → ∃ i', True ∧ c ^ i' • b i' ⊆ b n

Tactic:
_

Goal state after:
case refine_1
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
n : ℕ
a✝ : True
⊢ ∃ i', True ∧ c ^ i' • b i' ⊆ b n
Module: Mathlib.Analysis.LocallyConvex.ContinuousOfBounded

Goal state before:
pp: case refine_2
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
n : ℕ
⊢ True → c ^ n • b n ∈ 𝓝 0

Tactic:
_

Goal state after:
case refine_2
𝕜 : Type u_1
𝕜' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹¹ : AddCommGroup E
inst✝¹⁰ : TopologicalSpace E
inst✝⁹ : IsTopologicalAddGroup E
inst✝⁸ : AddCommGroup F
inst✝⁷ : TopologicalSpace F
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : Module 𝕜 E
inst✝⁴ : ContinuousSMul 𝕜 E
inst✝³ : NormedField 𝕜'
inst✝² : Module 𝕜' F
σ : 𝕜 →+* 𝕜'
inst✝¹ : RingHomIsometric σ
inst✝ : FirstCountableTopology E
f : E →ₛₗ[σ] F
hf : ∀ (s : Set E), IsVonNBounded 𝕜 s → IsVonNBounded 𝕜' (⇑f '' s)
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
c_ne : c ≠ 0
b : ℕ → Set E
bE1 : ∀ (i : ℕ), b i ∈ 𝓝 0 ∧ Balanced 𝕜 (b i)
bE : (𝓝 0).HasAntitoneBasis fun i ↦ b i
n : ℕ
a✝ : True
⊢ c ^ n • b n ∈ 𝓝 0
Module: Mathlib.Analysis.InnerProductSpace.TwoDim

Goal state before:
pp: case neg.«_@».Mathlib.Analysis.InnerProductSpace.TwoDim.2771185604._hygCtx._hyg.87.«1»
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)
a x : E
ha : ¬a = 0
⊢ (⟪a, x⟫ • (innerₛₗ ℝ) a + (o.areaForm a) x • o.areaForm a) ((o.basisRightAngleRotation a ha) ((fun i ↦ i) ⟨1, ⋯⟩)) =
    (‖a‖ ^ 2 • (innerₛₗ ℝ) x) ((o.basisRightAngleRotation a ha) ((fun i ↦ i) ⟨1, ⋯⟩))

Tactic:
simp [mul_comm, o.areaForm_swap a x]

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)
a x : E
⊢ ⟪a, x⟫ • (innerₛₗ ℝ) a + (o.areaForm a) x • o.areaForm a = ‖a‖ ^ 2 • (innerₛₗ ℝ) x

Tactic:
by
  by_cases ha : a = 0
  · simp [ha]
  apply (o.basisRightAngleRotation a ha).ext
  intro i
  fin_cases i
  · simp [mul_comm, real_inner_comm]
  · simp [mul_comm, o.areaForm_swap a x]

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

Goal state before:
pp: case neg.«_@».Mathlib.Analysis.InnerProductSpace.TwoDim.1235492232._hygCtx._hyg.87.«0»
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)
a x : E
ha : ¬a = 0
⊢ (⟪a, x⟫ • o.areaForm a - (o.areaForm a) x • (innerₛₗ ℝ) a) ((o.basisRightAngleRotation a ha) ((fun i ↦ i) ⟨0, ⋯⟩)) =
    (‖a‖ ^ 2 • o.areaForm x) ((o.basisRightAngleRotation a ha) ((fun i ↦ i) ⟨0, ⋯⟩))

Tactic:
simp [mul_comm, o.areaForm_swap a x]

Goal state after:
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
he : ∀ (f : StrongDual 𝕜 F), Continuous ⇑(e.dualMap ↑f)
he' : Continuous fun x ↦ (toWeakSpace 𝕜 F) (e ((toWeakSpace 𝕜 E).symm x))
⊢ ⇑e '' closure s ⊆ closure (⇑e '' s)

Tactic:
have h_convex : Convex ℝ (e '' s) := hs.linear_image (F := F) e

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
he : ∀ (f : StrongDual 𝕜 F), Continuous ⇑(e.dualMap ↑f)
he' : Continuous fun x ↦ (toWeakSpace 𝕜 F) (e ((toWeakSpace 𝕜 E).symm x))
h_convex : Convex ℝ (⇑e '' s)
⊢ ⇑e '' closure s ⊆ closure (⇑e '' 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' := ⋯ }
| Continuous[instTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun a ↦ z ((g.comp a) x)

Tactic:
1,

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' := ⋯ }
| fun a ↦ z ((g.comp a) x)
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
⊢ mellin (fun t ↦ f (t ^ a)) s = |a|⁻¹ • mellin f (s / ↑a)

Tactic:
simp_rw [mellin]

Goal state after:
case inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : a ≠ 0
⊢ ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f (t ^ a) = |a|⁻¹ • ∫ (t : ℝ) in Ioi 0, ↑t ^ (s / ↑a - 1) • f t
Module: Mathlib.Analysis.MellinTransform

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
⊢ mellin (fun t ↦ f (a * t)) s = ↑a ^ (-s) • mellin f s

Tactic:
simp_rw [mellin]

Goal state after:
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℝ → E
s : ℂ
a : ℝ
ha : 0 < a
⊢ ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f (a * t) = ↑a ^ (-s) • ∫ (t : ℝ) in Ioi 0, ↑t ^ (s - 1) • f t
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
x : ℝ
⊢ x ∈ uIcc 0 (π / 2) → HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x

Tactic:
_

Goal state after:
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
x : ℝ
a✝ : x ∈ uIcc 0 (π / 2)
⊢ HasDerivAt (fun y ↦ ↑(cos y) ^ n) (-↑n * ↑(sin x) * ↑(cos x) ^ (n - 1)) x
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: case e'_3
z : ℂ
hz : ‖z‖ < 1
⊢ log (1 + z) - z = log (1 + z) - logTaylor (1 + 1) z

Tactic:
simp [logTaylor_succ, logTaylor_zero, sub_eq_add_neg]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd

Goal state before:
pp: z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
x : ℝ
⊢ 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

Tactic:
_

Goal state after:
z : ℂ
n : ℕ
hn : 2 ≤ n
hz : z ≠ 0
x : ℝ
a✝ : 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
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: n : ℕ
z : ℂ
hz : ‖z‖ < 1
⊢ ‖log (1 - z)⁻¹ + logTaylor (n + 1) (-z)‖ ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (↑n + 1)

Tactic:
rw [sub_eq_add_neg,
    log_inv _ <| slitPlane_arg_ne_pi <| mem_slitPlane_of_norm_lt_one <| (norm_neg z).symm ▸ hz,
    ← sub_neg_eq_add, ← neg_sub', norm_neg]

Goal state after:
n : ℕ
z : ℂ
hz : ‖z‖ < 1
⊢ ‖log (1 + -z) - logTaylor (n + 1) (-z)‖ ≤ ‖z‖ ^ (n + 1) * (1 - ‖z‖)⁻¹ / (↑n + 1)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: case e'_3
z : ℂ
hz : ‖z‖ < 1
⊢ log (1 - z)⁻¹ - z = log (1 - z)⁻¹ + logTaylor (1 + 1) (-z)

Tactic:
simp [logTaylor_succ, logTaylor_zero, sub_eq_add_neg]

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

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
| Tendsto (fun x ↦ ∑ i ∈ Finset.range x, (-1) ^ (i + 1) * z ^ i / ↑i - log (1 + z)) atTop (nhds 0)

Tactic:
1,

Goal state after:
z : ℂ
hz : ‖z‖ < 1
| fun x ↦ ∑ i ∈ Finset.range x, (-1) ^ (i + 1) * z ^ i / ↑i - log (1 + z)
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: z : ℂ
hz : ‖z‖ < 1
| HasSum (fun n ↦ z ^ n / ↑n) (-log (1 - z))

Tactic:
1,

Goal state after:
z : ℂ
hz : ‖z‖ < 1
| fun n ↦ z ^ n / ↑n
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: g : ℝ → ℝ
t : ℝ
hg0 : Tendsto g atTop (𝓝 0)
hg : Tendsto (fun x ↦ ↑x * ↑(g x)) atTop (𝓝 ↑t)
⊢ (fun x ↦ ↑x * Complex.log (1 + ↑(g x))) =ᶠ[atTop] fun x ↦ ↑x * ↑(log (1 + g x))

Tactic:
filter_upwards [hg0.eventually_const_le (show (-1 : ℝ) < 0 by simp)] with x hg1

Goal state after:
g : ℝ → ℝ
t : ℝ
hg0 : Tendsto g atTop (𝓝 0)
hg : Tendsto (fun x ↦ ↑x * ↑(g x)) atTop (𝓝 ↑t)
x : ℝ
hg1 : -1 ≤ g x
⊢ ↑x * Complex.log (1 + ↑(g x)) = ↑x * ↑(log (1 + g x))
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogBounds

Goal state before:
pp: g : ℝ → ℝ
t : ℝ
hg0 : Tendsto g atTop (𝓝 0)
hg : Tendsto (fun x ↦ ↑x * ↑(g x)) atTop (𝓝 ↑t)
⊢ (fun x ↦ (1 + ↑(g x)) ^ ↑x) =ᶠ[atTop] fun x ↦ ↑((1 + g x) ^ x)

Tactic:
filter_upwards [hg0.eventually_const_le (show (-1 : ℝ) < 0 by simp)] with x hg1

Goal state after:
g : ℝ → ℝ
t : ℝ
hg0 : Tendsto g atTop (𝓝 0)
hg : Tendsto (fun x ↦ ↑x * ↑(g x)) atTop (𝓝 ↑t)
x : ℝ
hg1 : -1 ≤ g x
⊢ (1 + ↑(g x)) ^ ↑x = ↑((1 + g x) ^ x)
Module: Mathlib.Analysis.Meromorphic.LogDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
𝕜' : Type u_2
inst✝¹ : NontriviallyNormedField 𝕜'
inst✝ : NormedAlgebra 𝕜 𝕜'
ι : Type u_3
F : ι → 𝕜 → 𝕜'
hF : (mulSupport F).Finite
h : ∀ (i : ι), Meromorphic (F i)
h' : ∀ (i : ι) (x : 𝕜), meromorphicOrderAt (F i) x ≠ ⊤
⊢ logDeriv (∏ᶠ (i : ι), F i) =ᶠ[codiscrete 𝕜] ∑ᶠ (i : ι), logDeriv (F i)

Tactic:
by
  apply MeromorphicOn.logDeriv_finprod_eventuallyEq hF (fun i ↦ meromorphicOn_univ.mpr (h i))
  aesop

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

Goal state before:
pp: case a
k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v : V
⊢ ⨆ p, atTop • 𝓝 v +ᵥ pure p ≤ atTop • 𝓝 v

Tactic:
refine iSup_le fun u => ?_

Goal state after:
case a
k : Type u_1
V : Type u_2
inst✝⁹ : Field k
inst✝⁸ : LinearOrder k
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module k V
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace k
inst✝³ : OrderTopology k
inst✝² : IsStrictOrderedRing k
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul k V
v u : V
⊢ atTop • 𝓝 v +ᵥ pure u ≤ atTop • 𝓝 v