mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int_iff :
  ¬¬∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν = ∞ ↔
    ¬Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν
h_int : ¬Integrable (llr μ ν) μ
⊢ (if μ ≪ ν ∧ Integrable (llr μ ν) μ then ENNReal.ofReal (∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν) else ∞) =
    if μ ≪ ν then ∫⁻ (x : α), ENNReal.ofReal (klFun (μ.rnDeriv ν x).toReal) ∂ν else ∞

Tactic:
Decidable.not_not

Goal state after:
case neg
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
hμν : μ ≪ ν
h_int_iff :
  ∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν = ∞ ↔
    ¬Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν
h_int : ¬Integrable (llr μ ν) μ
⊢ (if μ ≪ ν ∧ Integrable (llr μ ν) μ then ENNReal.ofReal (∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν) else ∞) =
    if μ ≪ ν then ∫⁻ (x : α), ENNReal.ofReal (klFun (μ.rnDeriv ν x).toReal) ∂ν else ∞
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
h_int_left : Integrable (llr (c⁻¹ • μ) ν) (c⁻¹ • μ)
h_int_right : Integrable (llr μ (c • ν)) μ
⊢ ENNReal.ofReal (↑c * (klDiv (c⁻¹ • μ) ν).toReal) = ↑c * klDiv (c⁻¹ • μ) ν

Tactic:
simp only [NNReal.zero_le_coe, ENNReal.ofReal_mul, ENNReal.ofReal_coe_nnreal]

Goal state after:
case pos
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : c ≠ 0
hc' : ↑c ≠ 0
hμ_smul : μ = c • c⁻¹ • μ
hν_smul : ν = c⁻¹ • c • ν
hμν : μ ≪ ν
hμν_right : μ ≪ c • ν
h_int : Integrable (llr μ ν) μ
h_int_left : Integrable (llr (c⁻¹ • μ) ν) (c⁻¹ • μ)
h_int_right : Integrable (llr μ (c • ν)) μ
⊢ ↑c * ENNReal.ofReal (klDiv (c⁻¹ • μ) ν).toReal = ↑c * klDiv (c⁻¹ • μ) ν
Module: Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable

Goal state before:
pp: α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hp : Fact (1 ≤ p)
hm : m ≤ m0
f g : ↥(lpMeasSubgroup F m p μ)
⊢ ‖f - g‖ = dist f g

Tactic:
dist_eq_norm

Goal state after:
α : Type u_1
F : Type u_2
p : ℝ≥0∞
inst✝ : NormedAddCommGroup F
m m0 : MeasurableSpace α
μ : Measure α
hp : Fact (1 ≤ p)
hm : m ≤ m0
f g : ↥(lpMeasSubgroup F m p μ)
⊢ ‖f - g‖ = ‖f - g‖
Module: Mathlib.InformationTheory.KullbackLeibler.Basic

Goal state before:
pp: α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : ¬c = 0
hc' : ↑c ≠ 0
hμ_smul : ∀ (μ : Measure α), μ = c⁻¹ • c • μ
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (llr (c • μ) (c • ν)) μ

Tactic:
rw [integrable_congr (llr_smul_nnreal_same hμν c hc)]

Goal state after:
α : Type u_1
mα : MeasurableSpace α
μ ν : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : IsFiniteMeasure ν
c : ℝ≥0
hc : ¬c = 0
hc' : ↑c ≠ 0
hμ_smul : ∀ (μ : Measure α), μ = c⁻¹ • c • μ
hμν : μ ≪ ν
h_int : Integrable (llr μ ν) μ
⊢ Integrable (llr μ ν) μ
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 = ∞
⊢ ‖condExpIndL1 hm μ s x‖ ≤ μ.real s * ‖x‖

Tactic:
condExpIndL1_of_measure_eq_top hμs x,

Goal state after:
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‖
Module: Mathlib.InformationTheory.KullbackLeibler.ChainRule

Goal state before:
pp: 𝓧 : Type u_1
𝓨 : Type u_2
m𝓧 : MeasurableSpace 𝓧
m𝓨 : MeasurableSpace 𝓨
μ ν : Measure 𝓧
κ η : Kernel 𝓧 𝓨
inst✝³ : IsFiniteMeasure μ
inst✝² : IsFiniteMeasure ν
inst✝¹ : IsMarkovKernel κ
inst✝ : IsMarkovKernel η
h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η
h_int : Integrable (llr (μ ⊗ₘ κ) (ν ⊗ₘ η)) (μ ⊗ₘ κ)
h_ac_μν : μ ≪ ν
h_ac_κη : μ ⊗ₘ κ ≪ μ ⊗ₘ η
h_int_μν : Integrable (llr μ ν) μ
h_int_κη : Integrable (llr (μ ⊗ₘ κ) (μ ⊗ₘ η)) (μ ⊗ₘ κ)
this :
  ∫ (p : 𝓧 × 𝓨), llr (μ ⊗ₘ κ) (ν ⊗ₘ η) p ∂μ ⊗ₘ κ =
    ∫ (a : 𝓧), llr μ ν a ∂μ + ∫ (p : 𝓧 × 𝓨), llr (μ ⊗ₘ κ) (μ ⊗ₘ η) p ∂μ ⊗ₘ κ
⊢ ENNReal.ofReal (∫ (x : 𝓧 × 𝓨), llr (μ ⊗ₘ κ) (ν ⊗ₘ η) x ∂μ ⊗ₘ κ + (ν univ).toReal - (μ univ).toReal) =
    ENNReal.ofReal
      (∫ (x : 𝓧), llr μ ν x ∂μ + (ν univ).toReal - (μ univ).toReal + ∫ (x : 𝓧 × 𝓨), llr (μ ⊗ₘ κ) (μ ⊗ₘ η) x ∂μ ⊗ₘ κ)

Tactic:
rw [this]; ring_nf

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

Goal state before:
pp: α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
h_int_restrict : Integrable (t.indicator (μ.restrict s)[f | m]) (μ.restrict s)
⊢ IntegrableOn (t.indicator (μ.restrict s)[f | m]) s μ

Tactic:
IntegrableOn

Goal state after:
α : Type u_1
E : Type u_2
m m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
s : Set α
hm : m ≤ m0
inst✝ : SigmaFinite (μ.trim hm)
hs_m : MeasurableSet s
hf_int : Integrable f μ
this : SigmaFinite ((μ.restrict s).trim hm)
t : Set α
ht : MeasurableSet t
a✝ : μ t < ∞
h_int_restrict : Integrable (t.indicator (μ.restrict s)[f | m]) (μ.restrict s)
⊢ Integrable (t.indicator (μ.restrict s)[f | m]) (μ.restrict s)
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
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
ι : Type u_5
inst✝ : Countable ι
f : ι → α → E
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
A : ∀ᵐ (a : α) ∂μ, ∀ (i : ι), μ[f i | m] a = ↑↑(condExpL1 hm μ (f i)) a
⊢ ∑' (n : ι), ‖condExpL1 hm μ (f n)‖ₑ ≠ ∞

Tactic:
apply (lt_of_le_of_lt ?_ hf'.lt_top).ne

Goal state after:
α : Type u_1
E : Type u_3
m m₀ : MeasurableSpace α
μ : Measure α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
ι : Type u_5
inst✝ : Countable ι
f : ι → α → E
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hm : m ≤ m₀
hμm : SigmaFinite (μ.trim hm)
A : ∀ᵐ (a : α) ∂μ, ∀ (i : ι), μ[f i | m] a = ↑↑(condExpL1 hm μ (f i)) a
⊢ ∑' (n : ι), ‖condExpL1 hm μ (f n)‖ₑ ≤ ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ
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✝ : 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 =
    ∫ (y : 𝓨) in t, ((Measure.map g μ).rnDeriv (Measure.map g ν) y).toReal ∂Measure.map g ν

Tactic:
rw [setIntegral_map ht _ hg.aemeasurable]
      exact Measurable.aestronglyMeasurable (by fun_prop)

Goal state after:
Module: Mathlib.LinearAlgebra.Basis.Flag

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Semiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
n : ℕ
inst✝ : Nontrivial R
b : Basis (Fin n) R M
x✝ : Fin n
⊢ b.flag x✝.castSucc < b.flag x✝.succ

Tactic:
by simp [flag_succ]

Goal state after:
Module: Mathlib.LinearAlgebra.BilinearForm.Isometry

Goal state before:
pp: R : Type u_1
M : Type u_2
M₁ : Type u_3
M₂ : Type u_4
M₃ : Type u_5
M₄ : Type u_6
N : Type u_7
inst✝¹³ : CommSemiring R
inst✝¹² : AddCommMonoid M
inst✝¹¹ : AddCommMonoid M₁
inst✝¹⁰ : AddCommMonoid M₂
inst✝⁹ : AddCommMonoid M₃
inst✝⁸ : AddCommMonoid M₄
inst✝⁷ : AddCommMonoid N
inst✝⁶ : Module R M
inst✝⁵ : Module R M₁
inst✝⁴ : Module R M₂
inst✝³ : Module R M₃
inst✝² : Module R M₄
inst✝¹ : Module R N
B₁ : LinearMap.BilinForm R M₁
B₂ : LinearMap.BilinForm R M₂
B₃ : LinearMap.BilinForm R M₃
B₄ : LinearMap.BilinForm R M₄
inst✝ : Subsingleton M₁
x y : M₁
⊢ (B₂ ((toAddHom 0).toFun x)) ((toAddHom 0).toFun y) = (B₁ x) y

Tactic:
Subsingleton.elim x 0,

Goal state after:
R : Type u_1
M : Type u_2
M₁ : Type u_3
M₂ : Type u_4
M₃ : Type u_5
M₄ : Type u_6
N : Type u_7
inst✝¹³ : CommSemiring R
inst✝¹² : AddCommMonoid M
inst✝¹¹ : AddCommMonoid M₁
inst✝¹⁰ : AddCommMonoid M₂
inst✝⁹ : AddCommMonoid M₃
inst✝⁸ : AddCommMonoid M₄
inst✝⁷ : AddCommMonoid N
inst✝⁶ : Module R M
inst✝⁵ : Module R M₁
inst✝⁴ : Module R M₂
inst✝³ : Module R M₃
inst✝² : Module R M₄
inst✝¹ : Module R N
B₁ : LinearMap.BilinForm R M₁
B₂ : LinearMap.BilinForm R M₂
B₃ : LinearMap.BilinForm R M₃
B₄ : LinearMap.BilinForm R M₄
inst✝ : Subsingleton M₁
x y : M₁
⊢ (B₂ ((toAddHom 0).toFun 0)) ((toAddHom 0).toFun y) = (B₁ 0) y
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift

Goal state before:
pp: case inl
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
i : ι
w : ι → k
hw : ∑ i, w i = 1
h✝ : Subsingleton k
⊢ (affineCombination k univ p) w ∈ (affineSpan k (p '' {i}ᶜ)).shift (p i) (1 - w i)

Tactic:
suffices (affineSpan k <| p '' {i}ᶜ) = ⊤ by simp [this]

Goal state after:
case inl
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
i : ι
w : ι → k
hw : ∑ i, w i = 1
h✝ : Subsingleton k
⊢ affineSpan k (p '' {i}ᶜ) = ⊤
Module: Mathlib.RingTheory.TensorProduct.IsBaseChangeFree

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommSemiring R
V : Type u_2
inst✝⁵ : AddCommMonoid V
inst✝⁴ : Module R V
A : Type u_3
inst✝³ : CommSemiring A
inst✝² : Algebra A R
inst✝¹ : Module A V
inst✝ : IsScalarTower A R V
ι : Type u_4
b : Module.Basis ι R V
⊢ IsBaseChange R (Finsupp.linearCombination A ⇑b)

Tactic:
by
  classical
  let j := TensorProduct.finsuppScalarRight A R R ι
  refine of_equiv ?_ ?_
  · apply LinearEquiv.ofBijective (Finsupp.linearCombination R b ∘ₗ j)
    rw [LinearMap.coe_comp, LinearEquiv.coe_toLinearMap, j.bijective.of_comp_iff]
    simp [Function.Bijective,
        ← span_range_eq_top_iff_surjective_finsuppLinearCombination,
        ← linearIndependent_iff_injective_finsuppLinearCombination,
        Module.Basis.span_eq, b.linearIndependent]
  · intro x
    suffices (j (1 ⊗ₜ[A] x)) = x.mapRange (algebraMap A R) (by simp) by
      simp [this, Finsupp.linearCombination_apply, Finsupp.sum_mapRange_index]
    ext i
    simp [j, Algebra.algebraMap_eq_smul_one]

Goal state after:
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
v : ιa ⊕ ιb → Mᵢ
σ : Perm (ιa ⊕ ιb)
sl : Perm ιa
sr : Perm ιb
i' j' : ιa
hv : v (σ (Sum.inl i')) = v (σ (Sum.inl j'))
hij : σ (Sum.inl i') ≠ σ (Sum.inl j')
hσ✝ : swap (σ (Sum.inl i')) (σ (Sum.inl j')) • Quotient.mk'' σ = Quotient.mk'' σ
hσ : (Perm.sumCongrHom ιa ιb) (sl, sr) = ((fun x ↦ swap (σ (Sum.inl i')) (σ (Sum.inl j')) • x) σ)⁻¹ * σ
⊢ (Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) v = 0

Tactic:
suffices (a fun i ↦ v (σ (Sum.inl i))) = 0 by simp_all

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
v : ιa ⊕ ιb → Mᵢ
σ : Perm (ιa ⊕ ιb)
sl : Perm ιa
sr : Perm ιb
i' j' : ιa
hv : v (σ (Sum.inl i')) = v (σ (Sum.inl j'))
hij : σ (Sum.inl i') ≠ σ (Sum.inl j')
hσ✝ : swap (σ (Sum.inl i')) (σ (Sum.inl j')) • Quotient.mk'' σ = Quotient.mk'' σ
hσ : (Perm.sumCongrHom ιa ιb) (sl, sr) = ((fun x ↦ swap (σ (Sum.inl i')) (σ (Sum.inl j')) • x) σ)⁻¹ * σ
⊢ (a fun i ↦ v (σ (Sum.inl i))) = 0
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case mpr
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 = ↑e - LinearMap.id
⊢ ∃ f v, ↑e = LinearMap.transvection f v

Tactic:
eq_sub_iff_add_eq

Goal state after:
case mpr
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
⊢ ∃ f v, ↑e = LinearMap.transvection f v
Module: Mathlib.LinearAlgebra.QuadraticForm.TensorProduct

Goal state before:
pp: R : Type uR
A : Type uA
M₁ : Type uM₁
M₂ : Type uM₂
N₁ : Type uN₁
N₂ : Type uN₂
inst✝¹⁷ : CommRing R
inst✝¹⁶ : CommRing A
inst✝¹⁵ : AddCommGroup M₁
inst✝¹⁴ : AddCommGroup M₂
inst✝¹³ : AddCommGroup N₁
inst✝¹² : AddCommGroup N₂
inst✝¹¹ : Algebra R A
inst✝¹⁰ : Module R M₁
inst✝⁹ : Module A M₁
inst✝⁸ : Module R N₁
inst✝⁷ : Module A N₁
inst✝⁶ : SMulCommClass R A M₁
inst✝⁵ : IsScalarTower R A M₁
inst✝⁴ : IsScalarTower R A N₁
inst✝³ : Module R M₂
inst✝² : Module R N₂
inst✝¹ : Invertible 2
inst✝ : Invertible 2
Q₁ : QuadraticMap A M₁ N₁
Q₂ : QuadraticMap R M₂ N₂
⊢ associated (Q₁.tmul Q₂) = (associated Q₁).tmul (associated Q₂)

Tactic:
let : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm
  rw [QuadraticMap.tmul, BilinMap.tmul]
  have : Subsingleton (Invertible (2 : A)) := inferInstance
  convert!
    associated_left_inverse A
      (LinearMap.BilinMap.tmul_isSymm (QuadraticMap.associated_isSymm A Q₁)
        (QuadraticMap.associated_isSymm R Q₂))

Goal state after:
Module: Mathlib.LinearAlgebra.QuadraticForm.TensorProduct

Goal state before:
pp: R : Type uR
A : Type uA
M₁ : Type uM₁
M₂ : Type uM₂
N₁ : Type uN₁
N₂ : Type uN₂
inst✝¹⁷ : CommRing R
inst✝¹⁶ : CommRing A
inst✝¹⁵ : AddCommGroup M₁
inst✝¹⁴ : AddCommGroup M₂
inst✝¹³ : AddCommGroup N₁
inst✝¹² : AddCommGroup N₂
inst✝¹¹ : Algebra R A
inst✝¹⁰ : Module R M₁
inst✝⁹ : Module A M₁
inst✝⁸ : Module R N₁
inst✝⁷ : Module A N₁
inst✝⁶ : SMulCommClass R A M₁
inst✝⁵ : IsScalarTower R A M₁
inst✝⁴ : IsScalarTower R A N₁
inst✝³ : Module R M₂
inst✝² : Module R N₂
inst✝¹ : Invertible 2
inst✝ : Invertible 2
Q₁ : QuadraticMap A M₁ N₁
Q₂ : QuadraticMap R M₂ N₂
⊢ associated (Q₁.tmul Q₂) = (associated Q₁).tmul (associated Q₂)

Tactic:
by
  let : Invertible (2 : A) := (Invertible.map (algebraMap R A) 2).copy 2 (map_ofNat _ _).symm
  rw [QuadraticMap.tmul, BilinMap.tmul]
  have : Subsingleton (Invertible (2 : A)) := inferInstance
  convert!
    associated_left_inverse A
      (LinearMap.BilinMap.tmul_isSymm (QuadraticMap.associated_isSymm A Q₁)
        (QuadraticMap.associated_isSymm R Q₂))

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:
rw [← hu, LinearMap.transvection.apply, add_comm]

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
⊢ (LinearMap.id + u) x = x + (b.coord default ∘ₗ u.rangeRestrict) x • ↑(b default)
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
this✝ : Invertible 2 := (Invertible.map (algebraMap R A) 2).copy 2 ⋯
this : Invertible 2 := (Invertible.map (algebraMap R (A ⊗[R] CliffordAlgebra Q)) 2).copy 2 ⋯
v w : V
⊢ (1 * 1) ⊗ₜ[R] ((ι Q) v * (ι Q) w) + (1 * 1) ⊗ₜ[R] ((ι Q) w * (ι Q) v) =
    ((QuadraticMap.polarBilin (QuadraticForm.baseChange A Q)) (1 ⊗ₜ[R] v)) (1 ⊗ₜ[R] w) ⊗ₜ[R] 1

Tactic:
rw [← TensorProduct.tmul_add, CliffordAlgebra.ι_mul_ι_add_swap,
      QuadraticForm.polarBilin_baseChange, LinearMap.BilinForm.baseChange_tmul, one_mul,
      TensorProduct.smul_tmul, Algebra.algebraMap_eq_smul_one, QuadraticMap.polarBilin_apply_apply]

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

Goal state before:
pp: case h
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
x✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1
he : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1
he' : e.fixedReduce = 1
he_one : ¬e = 1
hefixed_ne_top : (↑e).fixedSubmodule ≠ ⊤
w : V
hw : w ∉ (↑e).fixedSubmodule
f : Dual K V
hfw : f w ≠ 0
hf : map f (↑e).fixedSubmodule = ⊥
hf' : (↑e).fixedSubmodule = ker f
eq_top : (↑e).fixedSubmodule ⊔ K ∙ w = ⊤
v : V := ⋯
hfv : f v = 0
⊢ e = transvection hfv

Tactic:
← LinearEquiv.toLinearMap_inj,

Goal state after:
case h
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
x✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1
he : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1
he' : e.fixedReduce = 1
he_one : ¬e = 1
hefixed_ne_top : (↑e).fixedSubmodule ≠ ⊤
w : V
hw : w ∉ (↑e).fixedSubmodule
f : Dual K V
hfw : f w ≠ 0
hf : map f (↑e).fixedSubmodule = ⊥
hf' : (↑e).fixedSubmodule = ker f
eq_top : (↑e).fixedSubmodule ⊔ K ∙ w = ⊤
v : V := (f w)⁻¹ • (e w - w)
hfv : f v = 0
⊢ ↑e = ↑(transvection hfv)
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 : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
  ((Algebra.lmul R (CliffordAlgebra Q)).toLinearMap ∘ₗ
        (ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘ₗ LinearMap.snd R M R)).compl₂
    (ι Q ∘ₗ LinearMap.fst R M R - Algebra.linearMap R (CliffordAlgebra Q) ∘ₗ LinearMap.snd R M R)
f_apply :
  ∀ (x y : M × R),
    (f x) y =
      ((ι Q) x.1 + (algebraMap R (CliffordAlgebra Q)) x.2) * ((ι Q) y.1 - (algebraMap R (CliffordAlgebra Q)) y.2)
⊢ ↥(even (Q' Q)) →ₐ[R] CliffordAlgebra Q

Tactic:
haveI hc : ∀ (r : R) (x : CliffordAlgebra Q), Commute (algebraMap _ _ r) x := Algebra.commutes

Goal state after:
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
f : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=
  ((Algebra.lmul R (CliffordAlgebra Q)).toLinearMap ∘ₗ
        (ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘ₗ LinearMap.snd R M R)).compl₂
    (ι Q ∘ₗ LinearMap.fst R M R - Algebra.linearMap R (CliffordAlgebra Q) ∘ₗ LinearMap.snd R M R)
f_apply :
  ∀ (x y : M × R),
    (f x) y =
      ((ι Q) x.1 + (algebraMap R (CliffordAlgebra Q)) x.2) * ((ι Q) y.1 - (algebraMap R (CliffordAlgebra Q)) y.2)
hc : ∀ (r : R) (x : CliffordAlgebra Q), Commute ((algebraMap R (CliffordAlgebra Q)) r) x
⊢ ↥(even (Q' Q)) →ₐ[R] CliffordAlgebra Q
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Prod

Goal state before:
pp: case mem
R : Type u_1
M₁ : Type u_2
M₂ : Type u_3
N : Type u_4
inst✝⁶ : CommRing R
inst✝⁵ : AddCommGroup M₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : AddCommGroup N
inst✝² : Module R M₁
inst✝¹ : Module R M₂
inst✝ : Module R N
Q₁ : QuadraticForm R M₁
Q₂ : QuadraticForm R M₂
Qₙ : QuadraticForm R N
f₁ : Q₁ →qᵢ Qₙ
f₂ : Q₂ →qᵢ Qₙ
hf : ∀ (x : M₁) (y : M₂), QuadraticMap.IsOrtho Qₙ (f₁ x) (f₂ y)
m₁ : CliffordAlgebra Q₁
m₂ : CliffordAlgebra Q₂
i₁ i₂ : ZMod 2
hm₂ : m₂ ∈ evenOdd Q₂ i₂
i₁' : { n // ↑n = i₁ }
m₁' : CliffordAlgebra Q₁
hm₁ : m₁' ∈ (ι Q₁).range ^ ↑i₁'
⊢ (map f₁) m₁' * (map f₂) m₂ = (-1) ^ (i₂ * i₁) • ((map f₂) m₂ * (map f₁) m₁')

Tactic:
obtain ⟨i₁n, rfl⟩ := i₁'
    dsimp only at *
    induction hm₁ using Submodule.pow_induction_on_left' with
    | algebraMap =>
      rw [AlgHom.commutes, Nat.cast_zero, mul_zero, uzpow_zero, one_smul, Algebra.commutes]
    | add _ _ _ _ _ ihx ihy =>
      rw [map_add, add_mul, mul_add, ihx, ihy, smul_add]
    | mem_mul m₁ hm₁ i x₁ _hx₁ ih₁ =>
      obtain ⟨v₁, rfl⟩ := hm₁
      -- This is the first interesting goal.
      rw [map_mul, mul_assoc, ih₁, mul_smul_comm, map_apply_ι, Nat.cast_succ, mul_add_one,
        uzpow_add, mul_smul, ← mul_assoc, ← mul_assoc, ← smul_mul_assoc ((-1) ^ i₂)]
      clear ih₁
      congr 2
      induction hm₂ using Submodule.iSup_induction' with
      | zero => rw [map_zero, zero_mul, mul_zero, smul_zero]
      | add _ _ _ _ ihx ihy => rw [map_add, add_mul, mul_add, ihx, ihy, smul_add]
      | mem i₂' m₂' hm₂ =>
        clear m₂
        obtain ⟨i₂n, rfl⟩ := i₂'
        dsimp only at *
        induction hm₂ using Submodule.pow_induction_on_left' with
        | algebraMap =>
          rw [AlgHom.commutes, Nat.cast_zero, uzpow_zero, one_smul, Algebra.commutes]
        | add _ _ _ _ _ ihx ihy =>
          rw [map_add, add_mul, mul_add, ihx, ihy, smul_add]
        | mem_mul m₂ hm₂ i x₂ _hx₂ ih₂ =>
          obtain ⟨v₂, rfl⟩ := hm₂
          -- This is the second interesting goal.
          rw [map_mul, map_apply_ι, Nat.cast_succ, ← mul_assoc,
            ι_mul_ι_comm_of_isOrtho (hf _ _), neg_mul, mul_assoc, ih₂, mul_smul_comm,
            ← mul_assoc, ← Units.neg_smul, uzpow_add, uzpow_one, mul_neg_one]

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
⊢ LinearIndependent R fun (x : ↑↑⟨s, hs⟩) ↦ M'.mkQ ↑x

Tactic:
LinearIndepOn,

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 : LinearIndependent R fun (x : ↑s) ↦ id ↑x
⊢ LinearIndependent R fun (x : ↑↑⟨s, hs✝⟩) ↦ M'.mkQ ↑x
Module: Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup

Goal state before:
pp: R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
y : (CliffordAlgebra Q)ˣ
hy₂ : ↑y = x
_hx₂ : star x * x = 1
hx₃ : x * star x = 1
hy₁ : y ∈ lipschitzGroup Q
⊢ star y ∈ lipschitzGroup Q

Tactic:
← hy₂

Goal state after:
R : Type u_1
inst✝² : CommRing R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x : CliffordAlgebra Q
y : (CliffordAlgebra Q)ˣ
hy₂ : ↑y = x
_hx₂ : star x * x = 1
hx₃ : ↑y * star ↑y = 1
hy₁ : y ∈ lipschitzGroup Q
⊢ star y ∈ lipschitzGroup Q