mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2

Goal state before:
pp: case refine_2
α : Type u_1
E : Type u_2
𝕜 : Type u_7
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
f : ↥(Lp E 2 μ)
c : E
h_mem_Lp : MemLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) 2 μ
h_eq : ↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) h_mem_Lp) =ᵐ[μ] fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ∫ (x : α) in s, ↑↑↑((condExpL2 𝕜 𝕜 hm) (MemLp.toLp (fun a ↦ ⟪c, ↑↑f a⟫) ⋯)) x ∂μ =
    ∫ (x : α) in s, ↑↑(MemLp.toLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) h_mem_Lp) x ∂μ

Tactic:
rw [integral_condExpL2_eq_of_fin_meas_real _ hs hμs.ne,
      integral_congr_ae (ae_restrict_of_ae h_eq), ←
      L2.inner_indicatorConstLp_eq_setIntegral_inner 𝕜 (↑(condExpL2 E 𝕜 hm f)) (hm s hs) c hμs.ne,
      ← inner_condExpL2_left_eq_right, condExpL2_indicator_of_measurable _ hs,
      L2.inner_indicatorConstLp_eq_setIntegral_inner 𝕜 f (hm s hs) c hμs.ne,
      setIntegral_congr_ae (hm s hs)
        ((MemLp.coeFn_toLp ((Lp.memLp f).const_inner c)).mono fun x hx _ => hx)]

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
s : Set α
inst✝ : CompleteSpace F'
f : ↥(Lp F' 1 μ)
hs : MeasurableSet s
S : ℕ → Set α := spanningSets (μ.trim hm)
⊢ ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ

Tactic:
have hS_meas : ∀ i, MeasurableSet[m] (S i) := measurableSet_spanningSets (μ.trim hm)

Goal state after:
α : Type u_1
F' : Type u_3
inst✝³ : NormedAddCommGroup F'
inst✝² : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝¹ : SigmaFinite (μ.trim hm)
s : Set α
inst✝ : CompleteSpace F'
f : ↥(Lp F' 1 μ)
hs : MeasurableSet s
S : ℕ → Set α := spanningSets (μ.trim hm)
hS_meas : ∀ (i : ℕ), MeasurableSet (S i)
⊢ ∫ (x : α) in s, ↑↑((condExpL1CLM F' hm μ) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case pos.h12
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
hs : MeasurableSet s
a✝ : (P.trim hm) s < ∞
⊢ ∫⁻ (x : Ω) in s, P⁻[X | mΩ] x + P⁻[Y | mΩ] x ∂P.trim hm ≤ ∫⁻ (x : Ω) in s, P⁻[X + Y | mΩ] x ∂P.trim hm

Tactic:
lintegral_add_left (by fun_prop)

Goal state after:
case pos.h12
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
P : Measure Ω
X Y : Ω → ℝ≥0∞
hm : mΩ ≤ mΩ₀
hσ : SigmaFinite (P.trim hm)
s : Set Ω
hs : MeasurableSet s
a✝ : (P.trim hm) s < ∞
⊢ ∫⁻ (a : Ω) in s, P⁻[X | mΩ] a ∂P.trim hm + ∫⁻ (a : Ω) in s, P⁻[Y | mΩ] a ∂P.trim hm ≤
    ∫⁻ (x : Ω) in s, P⁻[X + Y | mΩ] x ∂P.trim hm
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
hμν : μ ≪ ν
g : 𝓧 → 𝓨
hg : Measurable g
inst✝ : SigmaFinite (Measure.map g ν)
⊢ (fun a ↦ ((Measure.map g μ).rnDeriv (Measure.map g ν) (g a)).toReal) =ᵐ[ν.trim ⋯]
    ν[fun a ↦ (μ.rnDeriv ν a).toReal | MeasurableSpace.comap g m𝓨]

Tactic:
by
  rw [StronglyMeasurable.ae_eq_trim_iff]
  · exact toReal_rnDeriv_map hμν hg
  · refine Measurable.stronglyMeasurable fun s hs ↦ ?_
    refine ⟨(fun a ↦ ((μ.map g).rnDeriv (ν.map g) a).toReal) ⁻¹' s, hs.preimage (by fun_prop), ?_⟩
    rw [← Set.preimage_comp]
    rfl
  · fun_prop

Goal state after:
Module: Mathlib.InformationTheory.KullbackLeibler.DataProcessing

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
h_int : Integrable (llr μ ν) μ
⊢ ∫ (x : 𝓨), klFun ((∂Measure.map g μ/∂Measure.map g ν) x).toReal ∂Measure.map g ν =
    ∫ (x : 𝓧), klFun (ν[fun x ↦ ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] x) ∂ν

Tactic:
← toReal_klDiv_eq_integral_klFun (hμν.map hg),

Goal state after:
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
g : 𝓧 → 𝓨
hμν : μ ≪ ν
hg : Measurable g
h_int : Integrable (llr μ ν) μ
⊢ (klDiv (Measure.map g μ) (Measure.map g ν)).toReal =
    ∫ (x : 𝓧), klFun (ν[fun x ↦ ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] x) ∂ν
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: case h
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)
⊢ swap i j • Quotient.mk'' σ = Quotient.mk'' σ → (summand a b (Quotient.mk'' σ)) v = 0

Tactic:
| _ σ
   => 

Goal state after:
case h
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)
⊢ swap i j • Quotient.mk'' σ = Quotient.mk'' σ → (summand a b (Quotient.mk'' σ)) v = 0
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: case h.inl.inl
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)
hσ✝ : swap i j • Quotient.mk'' σ = Quotient.mk'' σ
sl : Perm ιa
sr : Perm ιb
hσ : (Perm.sumCongrHom ιa ιb) (sl, sr) = ((fun x ↦ swap i j • x) σ)⁻¹ * σ
i' : ιa
hi : σ⁻¹ i = Sum.inl i'
j' : ιa
hj : σ⁻¹ j = Sum.inl j'
⊢ (Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) v = 0

Tactic:
rw [Perm.inv_eq_iff_eq] at hi hj

Goal state after:
case h.inl.inl
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)
hσ✝ : swap i j • Quotient.mk'' σ = Quotient.mk'' σ
sl : Perm ιa
sr : Perm ιb
hσ : (Perm.sumCongrHom ιa ιb) (sl, sr) = ((fun x ↦ swap i j • x) σ)⁻¹ * σ
i' : ιa
hi : i = σ (Sum.inl i')
j' : ιa
hj : j = σ (Sum.inl j')
⊢ (Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) v = 0
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: case h.inl.inr
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)
hσ✝ : swap i j • Quotient.mk'' σ = Quotient.mk'' σ
sl : Perm ιa
sr : Perm ιb
hσ : (Perm.sumCongrHom ιa ιb) (sl, sr) = ((fun x ↦ swap i j • x) σ)⁻¹ * σ
i' : ιa
hi : σ⁻¹ i = Sum.inl i'
j' : ιb
hj : σ⁻¹ j = Sum.inr j'
⊢ (Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) v = 0

Tactic:
rw [Perm.inv_eq_iff_eq] at hi hj

Goal state after:
case h.inl.inr
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)
hσ✝ : swap i j • Quotient.mk'' σ = Quotient.mk'' σ
sl : Perm ιa
sr : Perm ιb
hσ : (Perm.sumCongrHom ιa ιb) (sl, sr) = ((fun x ↦ swap i j • x) σ)⁻¹ * σ
i' : ιa
hi : i = σ (Sum.inl i')
j' : ιb
hj : j = σ (Sum.inr j')
⊢ (Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) v = 0
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: case h.inr.inl
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)
hσ✝ : swap i j • Quotient.mk'' σ = Quotient.mk'' σ
sl : Perm ιa
sr : Perm ιb
hσ : (Perm.sumCongrHom ιa ιb) (sl, sr) = ((fun x ↦ swap i j • x) σ)⁻¹ * σ
i' : ιb
hi : σ⁻¹ i = Sum.inr i'
j' : ιa
hj : σ⁻¹ j = Sum.inl j'
⊢ (Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) v = 0

Tactic:
rw [Perm.inv_eq_iff_eq] at hi hj

Goal state after:
case h.inr.inl
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)
hσ✝ : swap i j • Quotient.mk'' σ = Quotient.mk'' σ
sl : Perm ιa
sr : Perm ιb
hσ : (Perm.sumCongrHom ιa ιb) (sl, sr) = ((fun x ↦ swap i j • x) σ)⁻¹ * σ
i' : ιb
hi : i = σ (Sum.inr i')
j' : ιa
hj : j = σ (Sum.inl j')
⊢ (Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) v = 0
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: case h.inr.inr
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)
hσ✝ : swap i j • Quotient.mk'' σ = Quotient.mk'' σ
sl : Perm ιa
sr : Perm ιb
hσ : (Perm.sumCongrHom ιa ιb) (sl, sr) = ((fun x ↦ swap i j • x) σ)⁻¹ * σ
i' : ιb
hi : σ⁻¹ i = Sum.inr i'
j' : ιb
hj : σ⁻¹ j = Sum.inr j'
⊢ (Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) v = 0

Tactic:
rw [Perm.inv_eq_iff_eq] at hi hj

Goal state after:
case h.inr.inr
ιa : Type u_1
ιb : Type u_2
inst✝¹⁰ : Fintype ιa
inst✝⁹ : Fintype ιb
R' : Type u_3
Mᵢ : Type u_4
N₁ : Type u_5
N₂ : Type u_6
inst✝⁸ : CommSemiring R'
inst✝⁷ : AddCommGroup N₁
inst✝⁶ : Module R' N₁
inst✝⁵ : AddCommGroup N₂
inst✝⁴ : Module R' N₂
inst✝³ : AddCommMonoid Mᵢ
inst✝² : Module R' Mᵢ
inst✝¹ : DecidableEq ιa
inst✝ : DecidableEq ιb
a : Mᵢ [⋀^ιa]→ₗ[R'] N₁
b : Mᵢ [⋀^ιb]→ₗ[R'] N₂
v : ιa ⊕ ιb → Mᵢ
i j : ιa ⊕ ιb
hv : v i = v j
hij : i ≠ j
σ : Perm (ιa ⊕ ιb)
hσ✝ : swap i j • Quotient.mk'' σ = Quotient.mk'' σ
sl : Perm ιa
sr : Perm ιb
hσ : (Perm.sumCongrHom ιa ιb) (sl, sr) = ((fun x ↦ swap i j • x) σ)⁻¹ * σ
i' : ιb
hi : i = σ (Sum.inr i')
j' : ιb
hj : j = σ (Sum.inr j')
⊢ (Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) v = 0
Module: Mathlib.RingTheory.TensorProduct.IsBaseChangeHom

Goal state before:
pp: case add
R : Type u_1
inst✝¹⁰ : CommSemiring R
S : Type u_2
inst✝⁹ : CommSemiring S
inst✝⁸ : Algebra R S
M : Type u_3
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
N : Type u_4
inst✝⁵ : AddCommMonoid N
inst✝⁴ : Module R N
P : Type u_5
inst✝³ : AddCommMonoid P
inst✝² : Module R P
inst✝¹ : Module S P
inst✝ : IsScalarTower R S P
ε : N →ₗ[R] P
s : S
x y : S ⊗[R] (M →ₗ[R] N)
hx :
  (TensorProduct.lift { toFun := fun s ↦ s • compRight R ε, map_add' := ⋯, map_smul' := ⋯ }) (s • x) =
    s • (TensorProduct.lift { toFun := fun s ↦ s • compRight R ε, map_add' := ⋯, map_smul' := ⋯ }) x
hy :
  (TensorProduct.lift { toFun := fun s ↦ s • compRight R ε, map_add' := ⋯, map_smul' := ⋯ }) (s • y) =
    s • (TensorProduct.lift { toFun := fun s ↦ s • compRight R ε, map_add' := ⋯, map_smul' := ⋯ }) y
⊢ (TensorProduct.lift { toFun := fun s ↦ s • compRight R ε, map_add' := ⋯, map_smul' := ⋯ }) (s • (x + y)) =
    s • (TensorProduct.lift { toFun := fun s ↦ s • compRight R ε, map_add' := ⋯, map_smul' := ⋯ }) (x + y)

Tactic:
simp [smul_add, hx, hy]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift

Goal state before:
pp: case neg
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁷ : Ring k
inst✝⁶ : PartialOrder k
inst✝⁵ : IsOrderedAddMonoid k
inst✝⁴ : AddCommGroup V
inst✝³ : AffineSpace V P
inst✝² : Module k V
n : ℕ
inst✝¹ : NeZero n
s : Simplex k P n
i : Fin (n + 1)
inst✝ : ZeroLEOneClass k
w : Fin (n + 1) → k
hw : ∑ i, w i = 1
x✝ :
  (affineCombination k univ s.points) w ∈ s.closedInterior ∩ ↑((affineSpan k (s.points '' {i}ᶜ)).shift (s.points i) 0)
hp : ∀ (i : Fin (n + 1)), w i ∈ Set.Icc 0 1
hshift : w i = 1
j : Fin (n + 1)
hj : ¬j = i
⊢ w j = Pi.single i 1 j

Tactic:
rw [← univ.sum_erase_add w (mem_univ i), hshift, add_eq_right,
    sum_eq_zero_iff_of_nonneg fun j _ ↦ (hp j).1] at hw

Goal state after:
case neg
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁷ : Ring k
inst✝⁶ : PartialOrder k
inst✝⁵ : IsOrderedAddMonoid k
inst✝⁴ : AddCommGroup V
inst✝³ : AffineSpace V P
inst✝² : Module k V
n : ℕ
inst✝¹ : NeZero n
s : Simplex k P n
i : Fin (n + 1)
inst✝ : ZeroLEOneClass k
w : Fin (n + 1) → k
hw : ∀ i_1 ∈ univ.erase i, w i_1 = 0
x✝ :
  (affineCombination k univ s.points) w ∈ s.closedInterior ∩ ↑((affineSpan k (s.points '' {i}ᶜ)).shift (s.points i) 0)
hp : ∀ (i : Fin (n + 1)), w i ∈ Set.Icc 0 1
hshift : w i = 1
j : Fin (n + 1)
hj : ¬j = i
⊢ w j = Pi.single i 1 j
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift

Goal state before:
pp: case refine_2
k : Type u_1
inst✝² : Field k
inst✝¹ : LinearOrder k
inst✝ : IsOrderedRing k
n : ℕ
i : Fin n
x : k
hxpos : 0 < x
hx1 : x ≤ 1
w : Fin n → k
hw : ∑ i, w i = 1
hi : x = 1 - w i
hj : ∀ (j : Fin n), w j ∈ Set.Icc 0 1
j : Fin n
hji : j ≠ i
⊢ x⁻¹ * w j ≤ 1

Tactic:
rw [inv_mul_le_one₀ hxpos, hi, le_sub_iff_add_le, ← hw]

Goal state after:
case refine_2
k : Type u_1
inst✝² : Field k
inst✝¹ : LinearOrder k
inst✝ : IsOrderedRing k
n : ℕ
i : Fin n
x : k
hxpos : 0 < x
hx1 : x ≤ 1
w : Fin n → k
hw : ∑ i, w i = 1
hi : x = 1 - w i
hj : ∀ (j : Fin n), w j ∈ Set.Icc 0 1
j : Fin n
hji : j ≠ i
⊢ w j + w i ≤ ∑ i, w i
Module: Mathlib.LinearAlgebra.Dual.BaseChange

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommSemiring R
V : Type u_2
inst✝⁷ : AddCommMonoid V
inst✝⁶ : Module R V
W : Type u_3
inst✝⁵ : AddCommMonoid W
inst✝⁴ : Module R W
A : Type u_4
inst✝³ : CommSemiring A
inst✝² : Algebra R A
inst✝¹ : Module A W
inst✝ : IsScalarTower R A W
j : V →ₗ[R] W
ibc : IsBaseChange A j
a : A
f : Dual R V
v : V
⊢ (ibc.toDualBaseChangeAux (a ⊗ₜ[R] f)) (j v) = a * (algebraMap R A) (f v)

Tactic:
by
  simp [toDualBaseChangeAux, toDual_comp_apply]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift

Goal state before:
pp: case pos
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁶ : Field k
inst✝⁵ : LinearOrder k
inst✝⁴ : IsOrderedRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
i : Fin (n + 1)
x : k
hx : x ∈ Set.Icc 0 1
hxpos : 0 < x
w : Fin (n + 1) → k
hw : ∑ i, w i = 1
hp : (affineCombination k univ s.points) w ∈ affineSpan k (Set.range s.points)
⊢ (∀ (i : Fin (n + 1)), w i ∈ Set.Icc 0 1) ∧ w i = 1 - x ↔
    ∃ x_1 ∈ (s.faceOpposite i).closedInterior, (homothety (s.points i) x) x_1 = (affineCombination k univ s.points) w

Tactic:
simp_rw [AffineMap.homothety_eq_iff_of_mul_eq_one (mul_inv_cancel₀ hxpos.ne.symm),
      univ.homothety_affineCombination _ _ (mem_univ i)]

Goal state after:
case pos
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁶ : Field k
inst✝⁵ : LinearOrder k
inst✝⁴ : IsOrderedRing k
inst✝³ : AddCommGroup V
inst✝² : Module k V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex k P n
i : Fin (n + 1)
x : k
hx : x ∈ Set.Icc 0 1
hxpos : 0 < x
w : Fin (n + 1) → k
hw : ∑ i, w i = 1
hp : (affineCombination k univ s.points) w ∈ affineSpan k (Set.range s.points)
⊢ (∀ (i : Fin (n + 1)), w i ∈ Set.Icc 0 1) ∧ w i = 1 - x ↔
    ∃ x_1 ∈ (s.faceOpposite i).closedInterior,
      (affineCombination k univ s.points) ((lineMap (Pi.single i 1) w) x⁻¹) = x_1
Module: Mathlib.LinearAlgebra.Center

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝⁴ : Ring R
inst✝³ : IsDomain R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
f : V →ₗ[R] V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
i j : ι
h✝ : ∀ (v : V), ∃ x x_1, ∃ (_ : x • v + x_1 • f v = 0), ¬(x = 0 ∧ x_1 = 0)
s t : R
h : s • b i + t • f (b i) = 0
h' : ¬(s = 0 ∧ t = 0)
hj : ¬i = j
⊢ t = 0 ∨ (b.repr (f (b i))) j = 0

Tactic:
rw [b.ext_elem_iff] at h

Goal state after:
R : Type u_1
V : Type u_2
inst✝⁴ : Ring R
inst✝³ : IsDomain R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
f : V →ₗ[R] V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
i j : ι
h✝ : ∀ (v : V), ∃ x x_1, ∃ (_ : x • v + x_1 • f v = 0), ¬(x = 0 ∧ x_1 = 0)
s t : R
h : ∀ (i_1 : ι), (b.repr (s • b i + t • f (b i))) i_1 = (b.repr 0) i_1
h' : ¬(s = 0 ∧ t = 0)
hj : ¬i = j
⊢ t = 0 ∨ (b.repr (f (b i))) j = 0
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case pos
V : Type u_2
inst✝² : AddCommGroup V
K : Type u_3
inst✝¹ : DivisionRing K
inst✝ : Module K V
e : V ≃ₗ[K] V
u : V →ₗ[K] V := ↑e - LinearMap.id
he : Module.rank K ↥u.range ≤ 1
hu : u + LinearMap.id = ↑e
hr : Module.rank K ↥u.range = 0
⊢ ∃ f v, ↑e = LinearMap.transvection f v

Tactic:
use 0, 0
      ext x
      suffices u x = 0 by simp [← hu, this]
      rw [rank_zero_iff] at hr
      simpa [← Subtype.coe_inj] using Subsingleton.allEq (⟨u x , mem_range_self u x⟩ : range u) 0

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case h
V : Type u_2
inst✝² : AddCommGroup V
K : Type u_3
inst✝¹ : DivisionRing K
inst✝ : Module K V
e : V ≃ₗ[K] V
u : V →ₗ[K] V := ↑e - LinearMap.id
hu : u + LinearMap.id = ↑e
hr : 1 ≤ Module.rank K ↥u.range
b : Basis Unit K ↥u.range
x : V
⊢ ↑e x = ((b.coord default ∘ₗ u.rangeRestrict).transvection ↑(b default)) x

Tactic:
← hu,

Goal state after:
case h
V : Type u_2
inst✝² : AddCommGroup V
K : Type u_3
inst✝¹ : DivisionRing K
inst✝ : Module K V
e : V ≃ₗ[K] V
u : V →ₗ[K] V := ↑e - LinearMap.id
hu : u + LinearMap.id = ↑e
hr : 1 ≤ Module.rank K ↥u.range
b : Basis Unit K ↥u.range
x : V
⊢ (u + LinearMap.id) x = ((b.coord default ∘ₗ u.rangeRestrict).transvection ↑(b default)) x
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: V : Type u_2
inst✝³ : AddCommGroup V
K : Type u_3
inst✝² : DivisionRing K
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
⊢ finrank K ↥(↑e - LinearMap.id).range ≤ 1 ↔ finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1

Tactic:
← (quotKerEquivRange _).finrank_eq,

Goal state after:
V : Type u_2
inst✝³ : AddCommGroup V
K : Type u_3
inst✝² : DivisionRing K
inst✝¹ : Module K V
inst✝ : Module.Finite K V
e : V ≃ₗ[K] V
⊢ finrank K (V ⧸ (↑e - LinearMap.id).ker) ≤ 1 ↔ finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1
Module: Mathlib.LinearAlgebra.CliffordAlgebra.BaseChange

Goal state before:
pp: R : Type u_1
A : Type u_2
V : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : CommRing A
inst✝³ : AddCommGroup V
inst✝² : Algebra R A
inst✝¹ : Module R V
inst✝ : Invertible 2
Q : QuadraticForm R V
⊢ (AlgHom.op (toBaseChange A Q)).comp reverseOp =
    (↑(Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q))).comp
      ((Algebra.TensorProduct.map (↑(AlgEquiv.toOpposite A A)) reverseOp).comp (toBaseChange A Q))

Tactic:
ext v
  change op (toBaseChange A Q (reverse (ι (Q.baseChange A) (1 ⊗ₜ[R] v)))) =
    Algebra.TensorProduct.opAlgEquiv R A A (CliffordAlgebra Q)
      (Algebra.TensorProduct.map (AlgEquiv.toOpposite A A).toAlgHom (reverseOp (Q := Q))
        (toBaseChange A Q (ι (Q.baseChange A) (1 ⊗ₜ[R] v))))
  rw [toBaseChange_ι, reverse_ι, toBaseChange_ι, Algebra.TensorProduct.map_tmul,
    Algebra.TensorProduct.opAlgEquiv_tmul, reverseOp_ι]
  rfl

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case neg.inl
K : Type u_3
V : Type u_4
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : Dual K V
v : V
hfv : f v = 0
hv : ¬v = 0
hf : ¬f = 0
ι : Set V
b : Basis (↑ι) K V
i j : ↑ι
hi : i ≠ j
hj : v = b i ∧ f = b.coord j
this : Fintype ↑ι
x y : ↑ι
h : ¬(i = x ∧ j = y)
h' : ¬i = x
⊢ (Finsupp.single i ((Finsupp.single y 1) j)) x = Matrix.single i j 1 x y

Tactic:
simp [Finsupp.single_eq_of_ne' h',
            Finsupp.single_eq_of_ne h',
            Matrix.single_apply_of_ne (h := h)]

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case neg.inr
K : Type u_3
V : Type u_4
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : Dual K V
v : V
hfv : f v = 0
hv : ¬v = 0
hf : ¬f = 0
ι : Set V
b : Basis (↑ι) K V
i j : ↑ι
hi : i ≠ j
hj : v = b i ∧ f = b.coord j
this : Fintype ↑ι
x y : ↑ι
h : ¬(i = x ∧ j = y)
h' : ¬j = y
⊢ (Finsupp.single i ((Finsupp.single y 1) j)) x = Matrix.single i j 1 x y

Tactic:
simp [Finsupp.single_eq_of_ne' h',
            Finsupp.single_eq_of_ne h',
            Matrix.single_apply_of_ne (h := h)]

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case neg
K : Type u_3
V : Type u_4
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : Dual K V
v : V
hfv : f v = 0
hv : ¬v = 0
hf : ¬f = 0
ι : Set V
b : Basis (↑ι) K V
i j : ↑ι
hi : i ≠ j
hj : v = b i ∧ f = b.coord j
this : Fintype ↑ι
x y : ↑ι
h : ¬(i = x ∧ j = y)
⊢ (Finsupp.single i ((Finsupp.single y 1) j)) x = Matrix.single i j 1 x y

Tactic:
rcases not_and_or.mp h with h' | h' <;>
          simp [Finsupp.single_eq_of_ne' h',
            Finsupp.single_eq_of_ne h',
            Matrix.single_apply_of_ne (h := h)]

Goal state after:
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case neg
K : Type u_3
V : Type u_4
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : Dual K V
v : V
hfv : ¬f v = 0
ι : Set V
b : Basis (↑ι) K V
i : ↑ι
hv : v = b i
hf : f = f v • b.coord i
this : Fintype ↑ι
x y : ↑ι
⊢ (Finsupp.single y 1) x + f (b y) * (b.repr (b i)) x = if x = y then Function.update 1 i (1 + f (b i)) x else 0

Tactic:
Function.update_apply,

Goal state after:
case neg
K : Type u_3
V : Type u_4
inst✝³ : Field K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
f : Dual K V
v : V
hfv : ¬f v = 0
ι : Set V
b : Basis (↑ι) K V
i : ↑ι
hv : v = b i
hf : f = f v • b.coord i
this : Fintype ↑ι
x y : ↑ι
⊢ (Finsupp.single y 1) x + f (b y) * (b.repr (b i)) x = if x = y then if x = i then 1 + f (b i) else 1 x else 0