mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Probability.Kernel.Composition.KernelLemmas

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
T : Type u_4
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
mT : MeasurableSpace T
κ : Kernel X Y
X' : Type u_5
mX' : MeasurableSpace X'
η : Kernel X' Z
inst✝¹ : IsSFiniteKernel η
ξ : Kernel Z T
inst✝ : IsSFiniteKernel ξ
⊢ ξ ∥ₖ Kernel.id ∘ₖ (η ∥ₖ κ) = ξ ∘ₖ η ∥ₖ κ

Tactic:
suffices swap T Y ∘ₖ (ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ) = swap T Y ∘ₖ ((ξ ∘ₖ η) ∥ₖ κ) by
    calc ξ ∥ₖ Kernel.id ∘ₖ (η ∥ₖ κ)
    _ = swap Y T ∘ₖ (swap T Y ∘ₖ (ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ)) := by
      simp_rw [← comp_assoc, swap_swap, id_comp]
    _ = swap Y T ∘ₖ (swap T Y ∘ₖ ((ξ ∘ₖ η) ∥ₖ κ)) := by rw [this]
    _ = ξ ∘ₖ η ∥ₖ κ := by simp_rw [← comp_assoc, swap_swap, id_comp]

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
T : Type u_4
mX : MeasurableSpace X
mY : MeasurableSpace Y
mZ : MeasurableSpace Z
mT : MeasurableSpace T
κ : Kernel X Y
X' : Type u_5
mX' : MeasurableSpace X'
η : Kernel X' Z
inst✝¹ : IsSFiniteKernel η
ξ : Kernel Z T
inst✝ : IsSFiniteKernel ξ
⊢ swap T Y ∘ₖ (ξ ∥ₖ Kernel.id) ∘ₖ (η ∥ₖ κ) = swap T Y ∘ₖ (ξ ∘ₖ η ∥ₖ κ)
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ ν : Measure α
κ : Kernel α β
⊢ (κ ∘ₖ Kernel.const Unit μ + κ ∘ₖ Kernel.const Unit ν) () =
    (κ ∘ₖ Kernel.const Unit μ) () + (κ ∘ₖ Kernel.const Unit ν) ()

Tactic:
_root_.add_apply

Goal state after:
Module: Mathlib.Probability.Kernel.Composition.MeasureComp

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
μ : Measure α
κ η : Kernel α β
⊢ (κ ∘ₖ Kernel.const Unit μ + η ∘ₖ Kernel.const Unit μ) () =
    (κ ∘ₖ Kernel.const Unit μ) () + (η ∘ₖ Kernel.const Unit μ) ()

Tactic:
_root_.add_apply

Goal state after:
Module: Mathlib.Probability.Kernel.MeasurableIntegral

Goal state before:
pp: α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : IsSFiniteKernel κ
f : α → β → E
hf : StronglyMeasurable (uncurry f)
⊢ MeasurableSet {x | Integrable (f x) (κ x)}

Tactic:
by
  simp_rw [Integrable, hf.of_uncurry_left.aestronglyMeasurable, true_and]
  exact measurableSet_lt (Measurable.lintegral_kernel_prod_right hf.enorm) measurable_const

Goal state after:
Module: Mathlib.Probability.Kernel.MeasurableIntegral

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : β → E
hf : StronglyMeasurable f
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : TopologicalSpace.SeparableSpace ↑(range f ∪ {0})
s : ℕ → SimpleFunc β E := SimpleFunc.approxOn f ⋯ (range f ∪ {0}) 0 ⋯
f' : ℕ → α → E := fun n ↦ {x | Integrable f (κ x)}.indicator fun x ↦ SimpleFunc.integral (κ x) (s n)
x : α
hfx : ¬Integrable f (κ x)
⊢ Tendsto (fun i ↦ f' i x) atTop (𝓝 (∫ (y : β), f y ∂κ x))

Tactic:
simp [f', hfx, integral_undef]

Goal state after:
Module: Mathlib.Probability.Kernel.MeasurableIntegral

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
mβ : MeasurableSpace β
κ : Kernel α β
E : Type u_4
inst✝² : NormedAddCommGroup E
inst✝¹ : IsSFiniteKernel κ
inst✝ : NormedSpace ℝ E
f : α → β → E
hf : StronglyMeasurable (uncurry f)
hE : CompleteSpace E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
this : TopologicalSpace.SeparableSpace ↑(range (uncurry f) ∪ {0})
s : ℕ → SimpleFunc (α × β) E := SimpleFunc.approxOn (uncurry f) ⋯ (range (uncurry f) ∪ {0}) 0 ⋯
s' : ℕ → α → SimpleFunc β E := fun n x ↦ (s n).comp (Prod.mk x) ⋯
f' : ℕ → α → E := fun n ↦ {x | Integrable (f x) (κ x)}.indicator fun x ↦ SimpleFunc.integral (κ x) (s' n x)
hf' : ∀ (n : ℕ), StronglyMeasurable (f' n)
x : α
hfx : ¬Integrable (f x) (κ x)
⊢ Tendsto (fun i ↦ f' i x) atTop (𝓝 (∫ (y : β), f x y ∂κ x))

Tactic:
simp [f', hfx, integral_undef]

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

Goal state before:
pp: case neg.e_a
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
c : ℝ
h0 : NeZero μ
h_univ : ¬μ Set.univ = ∞
⊢ ENNReal.ofReal ((μ.real Set.univ)⁻¹ * ((rexp c)⁻¹ * rexp c)) = (μ Set.univ)⁻¹

Tactic:
inv_mul_cancel₀ (exp_pos _).ne',

Goal state after:
case neg.e_a
α : Type u_1
mα : MeasurableSpace α
μ : Measure α
c : ℝ
h0 : NeZero μ
h_univ : ¬μ Set.univ = ∞
⊢ ENNReal.ofReal ((μ.real Set.univ)⁻¹ * 1) = (μ Set.univ)⁻¹
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1

Goal state before:
pp: case neg
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
x : G
hs : ¬MeasurableSet s
⊢ 0 ≤ μ.real s * ‖x‖

Tactic:
exact mul_nonneg ENNReal.toReal_nonneg (norm_nonneg _)

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

Goal state before:
pp: case pos
α : Type u_1
G : Type u_4
inst✝² : NormedAddCommGroup G
m m0 : MeasurableSpace α
μ : Measure α
s : Set α
inst✝¹ : NormedSpace ℝ G
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
x : G
hs : MeasurableSet s
hμs : μ s = ∞
⊢ 0 ≤ μ.real s * ‖x‖

Tactic:
exact mul_nonneg ENNReal.toReal_nonneg (norm_nonneg _)

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.InfiniteSum

Goal state before:
pp: X : Type u_1
E : Type u_2
x✝ : MeasurableSpace X
μ : Measure X
inst✝¹ : NormedAddCommGroup E
p : ℝ≥0∞
hp : Fact (1 ≤ p)
inst✝ : CompleteSpace E
f : ℕ → ↥(Lp E p μ)
hf : ∑' (n : ℕ), ‖f n‖ₑ ≠ ∞
⊢ ∀ᵐ (a : X) ∂μ, HasSum (fun n ↦ ↑↑(f n) a) (↑↑(∑' (n : ℕ), f n) a)

Tactic:
have A : ∀ᵐ a ∂μ, Summable (fun n ↦ ‖f n a‖) := by
    apply summable_norm_of_tsum_eLpNorm_ne_top hp.out (fun n ↦ Lp.aestronglyMeasurable (f n))
    simpa [enorm_def] using hf
  have B : ∀ᵐ x ∂μ, ∀ n, ⇑(∑ i ∈ range n, f i) x = ∑ i ∈ range n, f i x := by
    rw [ae_all_iff]
    exact fun i ↦ coeFn_fun_finsetSum _ _
  obtain ⟨ns, hns, nslim⟩ : ∃ ns : ℕ → ℕ, StrictMono ns ∧ ∀ᵐ x ∂μ,
      Tendsto (fun i ↦ (∑ j ∈ range (ns i), f j : Lp E p μ) x) atTop (𝓝 ((∑' n, f n) x)) := by
    have : Tendsto (fun i ↦ (∑ j ∈ range i, f j)) atTop (𝓝 (∑' n, f n)) :=
      Summable.tendsto_sum_tsum_nat (Summable.of_enorm hf)
    exact (tendstoInMeasure_of_tendsto_Lp this).exists_seq_tendsto_ae
  filter_upwards [A, B, nslim] with x S h'x h''x
  apply hasSum_of_subseq_of_summable S (tendsto_finset_range.comp hns.tendsto_atTop)
  simpa only [h'x, Function.comp] using h''x

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

Goal state before:
pp: α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm_not : ¬SigmaFinite (μ.trim hm)
⊢ μ[f | m] = 0

Tactic:
rw [condExp, dif_pos hm, dif_neg]

Goal state after:
case hnc
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
f : α → E
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
hm : m ≤ m₀
hμm_not : ¬SigmaFinite (μ.trim hm)
⊢ ¬(SigmaFinite (μ.trim hm) ∧ Integrable f μ)
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator

Goal state before:
pp: case pos.refine_5
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hs : MeasurableSet s
hf : f =ᵐ[μ.restrict s] 0
hm : m ≤ m0
hμm this✝ : SigmaFinite (μ.trim hm)
this : SigmaFinite ((μ.restrict s).trim hm)
hf_int : Integrable f μ
⊢ AEStronglyMeasurable 0 (μ.restrict s)

Tactic:
· exact stronglyMeasurable_zero.aestronglyMeasurable

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

Goal state before:
pp: α : Type u_1
F' : Type u_3
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m m0 : MeasurableSpace α
μ : Measure α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
f : α → F'
hF' : CompleteSpace ↥(Lp F' 1 μ)
hf : Integrable f μ
hf' : ¬f =ᵐ[μ] 0
⊢ ¬f =ᵐ[μ] 0

Tactic:
exact hf'

Goal state after:
Module: Mathlib.MeasureTheory.Function.ConditionalLExpectation

Goal state before:
pp: case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
hm : mΩ ≤ mΩ₀
P : Measure Ω
hσ : SigmaFinite (P.trim hm)
X : Ω → ℝ≥0∞
s : Set Ω
hs : MeasurableSet s
hX : ¬Measurable X
h : (P.withDensity X).trim hm ≪ P.trim hm
this : SFinite ((P.withDensity X).trim hm)
⊢ ∫⁻ (ω : Ω) in s, P⁻[X | mΩ] ω ∂P = ∫⁻ (ω : Ω) in s, X ω ∂P

Tactic:
condLExp_of_not_sub_sigma_measurable hm _ hX,

Goal state after:
case neg
Ω : Type u_1
mΩ₀ mΩ : MeasurableSpace Ω
hm : mΩ ≤ mΩ₀
P : Measure Ω
hσ : SigmaFinite (P.trim hm)
X : Ω → ℝ≥0∞
s : Set Ω
hs : MeasurableSet s
hX : ¬Measurable X
h : (P.withDensity X).trim hm ≪ P.trim hm
this : SFinite ((P.withDensity X).trim hm)
⊢ ∫⁻ (ω : Ω) in s, (∂(P.withDensity X).trim hm/∂P.trim hm) ω ∂P = ∫⁻ (ω : Ω) in s, X ω ∂P
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hf_int : Integrable f μ
hs : IsClosed s
hc : Convex ℝ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
lem2 : ∀ᵐ (a : α) ∂μ, fY a ∈ ⇑Y.subtypeL ⁻¹' s
⊢ ∀ᵐ (a : α) ∂μ, μ[f | m] a ∈ s

Tactic:
have hfY_int : Integrable fY μ := by
    refine (hf_int.congr lem1).mono ?_ (by simp [fX])
    obtain ⟨g, hg1, hg2, hg3⟩ := hf_int.1.exists_stronglyMeasurable_range_subset
      ((Submodule.isClosed_topologicalClosure _).measurableSet) Nonempty.of_subtype lem0
    refine ⟨codRestrict g Y hg2, (hg1.measurable.codRestrict hg2).stronglyMeasurable, ?_⟩
    filter_upwards [hg3] with a ha
    have : g a ∈ Y := hg2 a
    simp_all [fY, codRestrict]

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hf_int : Integrable f μ
hs : IsClosed s
hc : Convex ℝ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
lem2 : ∀ᵐ (a : α) ∂μ, fY a ∈ ⇑Y.subtypeL ⁻¹' s
hfY_int : Integrable fY μ
⊢ ∀ᵐ (a : α) ∂μ, μ[f | m] a ∈ s
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym

Goal state before:
pp: case refine_2
𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
inst✝ : IsFiniteMeasure μ
hμν : μ ≪ ν
g : 𝓧 → 𝓨
hg : Measurable g
hσ : SigmaFinite (Measure.map g ν)
this✝ : SigmaFinite (ν.trim ⋯)
this : SigmaFinite ν
t : Set 𝓨
ht : MeasurableSet t
a✝ : ν (g ⁻¹' t) < ∞
⊢ ∫ (x : 𝓧) in g ⁻¹' t, ((Measure.map g μ).rnDeriv (Measure.map g ν) (g x)).toReal ∂ν =
    ∫ (x : 𝓧) in g ⁻¹' t, (μ.rnDeriv ν x).toReal ∂ν

Tactic:
calc ∫ x in g ⁻¹' t, ((μ.map g).rnDeriv (ν.map g) (g x)).toReal ∂ν
    _ = ∫ y in t, ((μ.map g).rnDeriv (ν.map g) y).toReal ∂(ν.map g) := by
      rw [setIntegral_map ht _ hg.aemeasurable]
      exact Measurable.aestronglyMeasurable (by fun_prop)
    _ = ∫ x in g ⁻¹' t, (μ.rnDeriv ν x).toReal ∂ν := by
      rw [Measure.setIntegral_toReal_rnDeriv (hμν.map hg),
        Measure.setIntegral_toReal_rnDeriv hμν, measureReal_def, Measure.map_apply hg ht,
        measureReal_def]

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

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
⊢ φ ∘ μ[f | m] ≤ᵐ[μ] μ[φ ∘ f | m]

Tactic:
have hfY_int : Integrable fY μ := by
    refine (hf_int.congr lem1).mono ?_ (by simp [fX])
    obtain ⟨g, hg1, hg2, hg3⟩ := hf_int.1.exists_stronglyMeasurable_range_subset
      ((Submodule.isClosed_topologicalClosure _).measurableSet) Nonempty.of_subtype lem0
    refine ⟨codRestrict g Y hg2, (hg1.measurable.codRestrict hg2).stronglyMeasurable, ?_⟩
    filter_upwards [hg3] with a ha
    have : g a ∈ Y := hg2 a
    simp_all [fY, codRestrict]

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
α : Type u_2
f : α → E
φ : E → ℝ
m mα : MeasurableSpace α
μ : Measure α
s : Set E
inst✝ : IsFiniteMeasure μ
hm : m ≤ mα
hφ_cvx : ConvexOn ℝ s φ
hφ_cont : LowerSemicontinuousOn φ s
hf : ∀ᵐ (a : α) ∂μ, f a ∈ s
hs : IsClosed s
hf_int : Integrable f μ
hφ_int : Integrable (φ ∘ f) μ
this✝² : MeasurableSpace E := borel E
this✝¹ : BorelSpace E
t : Set E
ht : TopologicalSpace.IsSeparable t
htt : ∀ᵐ (x : α) ∂μ, f x ∈ t
Y : Submodule ℝ E := (Submodule.span ℝ t).topologicalClosure
this✝ : CompleteSpace ↥Y
this : SecondCountableTopology ↥Y
φY : ↥Y → ℝ := φ ∘ ⇑Y.subtypeL
fY : α → ↥Y := fun a ↦ if h : f a ∈ Y then ⟨f a, h⟩ else 0
fX : α → E := ⇑Y.subtypeL ∘ fY
lem0 : ∀ᵐ (a : α) ∂μ, f a ∈ Y
lem1 : f =ᵐ[μ] fX
hfY_int : Integrable fY μ
⊢ φ ∘ μ[f | m] ≤ᵐ[μ] μ[φ ∘ f | m]
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner

Goal state before:
pp: 𝓧 : Type u_1
m m𝓧 : MeasurableSpace 𝓧
hm : m ≤ m𝓧
μ : Measure 𝓧
inst✝ : SigmaFinite (μ.trim hm)
s : Set 𝓧
hs : MeasurableSet s
h's : μ s ≠ ∞
⊢ IntegrableOn 1 s μ

Tactic:
apply integrableOn_const h's

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Matrix

Goal state before:
pp: ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁵ : AddCommGroup V
inst✝⁴ : AffineSpace V P
inst✝³ : Ring k
inst✝² : Module k V
b : AffineBasis ι k P
inst✝¹ : Fintype ι
b₂ : AffineBasis ι k P
inst✝ : DecidableEq ι
l m : ι
⊢ (b.toMatrix ⇑b₂ * b₂.toMatrix ⇑b) l m = 1 l m

Tactic:
change (b.coords (b₂ l) ᵥ* b₂.toMatrix b) m = _

Goal state after:
ι : Type u₁
k : Type u₂
V : Type u₃
P : Type u₄
inst✝⁵ : AddCommGroup V
inst✝⁴ : AffineSpace V P
inst✝³ : Ring k
inst✝² : Module k V
b : AffineBasis ι k P
inst✝¹ : Fintype ι
b₂ : AffineBasis ι k P
inst✝ : DecidableEq ι
l m : ι
⊢ (b.coords (b₂ l) ᵥ* b₂.toMatrix ⇑b) m = 1 l m
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring k
inst✝⁴ : AddCommGroup V
inst✝³ : AddTorsor V P
inst✝² : Module k V
ι : Type u_4
inst✝¹ : Fintype ι
inst✝ : Nontrivial ι
p : ι → P
h✝ : AffineIndependent k p
i : ι
w : ι → k
hw : ∑ i, w i = 1
c : k
h : w i = 1 - c
⊢ (affineCombination k univ p) w ∈ (affineSpan k (p '' {i}ᶜ)).shift (p i) c

Tactic:
by simpa [h] using affineCombination_mem_shift p i hw

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⁻¹) ((affineCombination k univ s.points) w) = x_1

Tactic:
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.Alternating.DomCoprod

Goal state before:
pp: 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
⊢ (alternatization ↑a).domCoprod (alternatization ↑b) =
    ((Fintype.card ιa).factorial * (Fintype.card ιb).factorial) • a.domCoprod b

Tactic:
coe_alternatization,

Goal state after:
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
⊢ ((Fintype.card ιa).factorial • a).domCoprod (alternatization ↑b) =
    ((Fintype.card ιa).factorial * (Fintype.card ιb).factorial) • a.domCoprod b
Module: Mathlib.LinearAlgebra.Alternating.DomCoprod

Goal state before:
pp: 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
⊢ ((Fintype.card ιa).factorial • a).domCoprod (alternatization ↑b) =
    ((Fintype.card ιa).factorial * (Fintype.card ιb).factorial) • a.domCoprod b

Tactic:
coe_alternatization,

Goal state after:
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
⊢ ((Fintype.card ιa).factorial • a).domCoprod ((Fintype.card ιb).factorial • b) =
    ((Fintype.card ιa).factorial * (Fintype.card ιb).factorial) • a.domCoprod b
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: R : Type u_1
V : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup V
inst✝ : Module R V
f : Dual R V
v : V
h : IsUnit (1 + f v)
x : V
⊢ f x * ↑(-h.unit⁻¹) + (f x + f x * ↑(-h.unit⁻¹) * f v) = 0

Tactic:
nth_rewrite 2 [← mul_one (f x)]

Goal state after:
R : Type u_1
V : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup V
inst✝ : Module R V
f : Dual R V
v : V
h : IsUnit (1 + f v)
x : V
⊢ f x * ↑(-h.unit⁻¹) + (f x * 1 + f x * ↑(-h.unit⁻¹) * f v) = 0
Module: Mathlib.LinearAlgebra.Center

Goal state before:
pp: case pos
R : Type u_1
V : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : Free R V
f : V →ₗ[R] V
h : ∀ (v : V), ¬LinearIndependent R ![v, f v]
hV1 : Nonempty (Basis Unit R V)
b : Basis Unit R V := Classical.ofNonempty
⊢ ∃ a, f = a • 1

Tactic:
use b.coord () (f (b ()))

Goal state after:
case h
R : Type u_1
V : Type u_2
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : Free R V
f : V →ₗ[R] V
h : ∀ (v : V), ¬LinearIndependent R ![v, f v]
hV1 : Nonempty (Basis Unit R V)
b : Basis Unit R V := ⋯
⊢ f = (b.coord ()) (f (b ())) • 1
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
⊢ Module.rank K ↥(↑e - LinearMap.id).range ≤ 1 ↔ finrank K ↥(↑e - LinearMap.id).range ≤ 1

Tactic:
finrank,

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
⊢ Module.rank K ↥(↑e - LinearMap.id).range ≤ 1 ↔ toNat (Module.rank K ↥(↑e - LinearMap.id).range) ≤ 1
Module: Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
h : Q' = -Q
h' : Q = -Q'
⊢ (evenToNeg Q' Q h').comp (evenToNeg Q Q' h) = AlgHom.id R ↥(even Q)

Tactic:
by
  ext m₁ m₂ : 4
  simp [evenToNeg_ι]

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.Torsion.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
M' : Submodule R M
hN : M' ≤ torsion R M
a✝ : Nontrivial R
x✝ : { s // LinearIndepOn R id s }
s : Set M
hs✝ : LinearIndepOn R id s
hs : ∀ (s_1 : Finset ↑s) (g : ↑s → R), ∑ i ∈ s_1, g i • id ↑i = 0 → ∀ i ∈ s_1, g i = 0
t : Finset ↑s
g : ↑s → R
hg✝ : ∑ x ∈ t, g x • ↑x ∈ M'
i : ↑s
hi : i ∈ t
r : ↥(nonZeroDivisors R)
hg : r • ∑ x ∈ t, g x • ↑x = 0
⊢ g i = 0

Tactic:
Finset.smul_sum,

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
M' : Submodule R M
hN : M' ≤ torsion R M
a✝ : Nontrivial R
x✝ : { s // LinearIndepOn R id s }
s : Set M
hs✝ : LinearIndepOn R id s
hs : ∀ (s_1 : Finset ↑s) (g : ↑s → R), ∑ i ∈ s_1, g i • id ↑i = 0 → ∀ i ∈ s_1, g i = 0
t : Finset ↑s
g : ↑s → R
hg✝ : ∑ x ∈ t, g x • ↑x ∈ M'
i : ↑s
hi : i ∈ t
r : ↥(nonZeroDivisors R)
hg : ∑ x ∈ t, r • g x • ↑x = 0
⊢ g i = 0
Module: Mathlib.LinearAlgebra.FreeModule.ModN

Goal state before:
pp: G : Type u_1
H✝ : Type u_2
M : Type u_3
inst✝¹ : AddCommGroup G
n : ℕ
inst✝ : NeZero n
ι : Type u_4
b : Basis ι ℤ G
nG : Submodule ℤ G := ((LinearMap.lsmul ℤ G) ↑n).range
H : Type u_1 := G ⧸ nG
φ : G →ₗ[ℤ] H := nG.mkQ
mod : (ι →₀ ℤ) →ₗ[ℤ] ι →₀ ZMod n := mapRange.linearMap (Int.castAddHom (ZMod n)).toIntLinearMap
⊢ Basis ι (ZMod n) (ModN G n)

Tactic:
let f : G →ₗ[ℤ] (ι →₀ ℤ) := b.repr

Goal state after:
G : Type u_1
H✝ : Type u_2
M : Type u_3
inst✝¹ : AddCommGroup G
n : ℕ
inst✝ : NeZero n
ι : Type u_4
b : Basis ι ℤ G
nG : Submodule ℤ G := ((LinearMap.lsmul ℤ G) ↑n).range
H : Type u_1 := G ⧸ nG
φ : G →ₗ[ℤ] H := nG.mkQ
mod : (ι →₀ ℤ) →ₗ[ℤ] ι →₀ ZMod n := mapRange.linearMap (Int.castAddHom (ZMod n)).toIntLinearMap
f : G →ₗ[ℤ] ι →₀ ℤ := ↑b.repr
⊢ Basis ι (ZMod n) (ModN G n)