mathlib-initiative/mathlib-tactics huggingface dataset:
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
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
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
⊢ φ ∘ μ[f | m] ≤ᵐ[μ] μ[φ ∘ f | m]

Tactic:
apply (isCountablySpanning_spanningSets (μ.trim hm)).null_of_forall_restrict_null <;>
    rintro - ⟨n, rfl⟩

Goal state after:
case hm
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
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
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
n : ℕ
⊢ MeasurableSet (spanningSets (μ.trim hm) n)

case ht
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
hm : m ≤ mα
inst✝ : SigmaFinite (μ.trim hm)
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
n : ℕ
⊢ (μ.restrict (spanningSets (μ.trim hm) n)) {x | (fun x ↦ (φ ∘ μ[f | m]) x ≤ μ[φ ∘ f | m] x) x}ᶜ = 0
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
⊢ klDiv (Measure.map g μ) (Measure.map g ν) =
    ENNReal.ofReal (∫ (x : 𝓧), klFun (ν[fun x ↦ ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] x) ∂ν)

Tactic:
klDiv_eq_integral_klFun,

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
⊢ (if Measure.map g μ ≪ Measure.map g ν ∧ Integrable (llr (Measure.map g μ) (Measure.map g ν)) (Measure.map g μ) then
      ENNReal.ofReal (∫ (x : 𝓨), klFun ((∂Measure.map g μ/∂Measure.map g ν) x).toReal ∂Measure.map g ν)
    else ∞) =
    ENNReal.ofReal (∫ (x : 𝓧), klFun (ν[fun x ↦ ((∂μ/∂ν) x).toReal | MeasurableSpace.comap g m𝓨] x) ∂ν)
Module: Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift

Goal state before:
pp: case inr
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✝ : Nontrivial k
j : ι
hj : j ≠ i
⊢ -(w i • (p i -ᵥ p j)) +ᵥ (affineCombination k univ p) w ∈ affineSpan k (p '' {i}ᶜ)

Tactic:
rw [← affineCombination_piSingle k _ p (mem_univ i),
    ← affineCombination_piSingle k _ p (mem_univ j), affineCombination_vsub, ← map_smul, ← map_neg,
    weightedVSub_vadd_affineCombination]

Goal state after:
case inr
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✝ : Nontrivial k
j : ι
hj : j ≠ i
⊢ (affineCombination k univ p) (-(w i • (Pi.single i 1 - Pi.single j 1)) + w) ∈ affineSpan k (p '' {i}ᶜ)
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
i : ι
w : ι → k
hw : ∑ i, w i = 1
h✝ : Nontrivial k
j : ι
hj : j ≠ i
i' : ι
x✝ : i' ∈ univ
hi : i' ∉ {i}ᶜ
⊢ (-(w i • (Pi.single i 1 - Pi.single j 1)) + w) i' = 0

Tactic:
by aesop

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

Goal state before:
pp: 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 : (affineCombination k univ s.points) w ∈ s.closedInterior
hshift : (affineCombination k univ s.points) w ∈ ↑((affineSpan k (s.points '' {i}ᶜ)).shift (s.points i) 0)
⊢ (affineCombination k univ s.points) w ∈ {s.points i}

Tactic:
suffices w = Pi.single i 1 by simp [this]

Goal state after:
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 : (affineCombination k univ s.points) w ∈ s.closedInterior
hshift : (affineCombination k univ s.points) w ∈ ↑((affineSpan k (s.points '' {i}ᶜ)).shift (s.points i) 0)
⊢ w = Pi.single i 1
Module: Mathlib.RingTheory.TensorProduct.IsBaseChangeHom

Goal state before:
pp: R : Type u_1
inst✝¹⁶ : CommSemiring R
S : Type u_2
inst✝¹⁵ : CommSemiring S
inst✝¹⁴ : Algebra R S
M : Type u_3
inst✝¹³ : AddCommMonoid M
inst✝¹² : Module R M
N : Type u_4
inst✝¹¹ : AddCommMonoid N
inst✝¹⁰ : Module R N
P : Type u_5
inst✝⁹ : AddCommMonoid P
inst✝⁸ : Module R P
Q : Type u_6
inst✝⁷ : AddCommMonoid Q
inst✝⁶ : Module R Q
inst✝⁵ : Module S P
inst✝⁴ : IsScalarTower R S P
inst✝³ : Module S Q
inst✝² : IsScalarTower R S Q
inst✝¹ : Free R M
inst✝ : Module.Finite R M
α : M →ₗ[R] P
j : IsBaseChange S α
β : N →ₗ[R] Q
k : IsBaseChange S β
⊢ IsBaseChange S (j.linearMapLeftRightHom β)

Tactic:
by
  apply of_equiv <|
      (k.linearMapRight M).equiv ≪≫ₗ liftBaseChangeEquiv S ≪≫ₗ LinearEquiv.congrLeft Q S j.equiv
  intro f
  ext p
  simp [IsBaseChange.equiv_tmul, LinearEquiv.congrLeft, linearMapLeftRightHom_apply]

Goal state after:
Module: Mathlib.LinearAlgebra.FixedSubmodule

Goal state before:
pp: R : Type u_1
inst✝⁴ : Semiring R
U : Type u_2
V : Type u_3
inst✝³ : AddCommMonoid U
inst✝² : AddCommMonoid V
inst✝¹ : Module R U
inst✝ : Module R V
e : V ≃ₗ[R] V
f : V →ₗ[R] V
x y : V
hx : x ∈ {x | f x = x}
hy : y ∈ {x | f x = x}
⊢ x + y ∈ {x | f x = x}

Tactic:
by aesop

Goal state after:
Module: Mathlib.LinearAlgebra.FixedSubmodule

Goal state before:
pp: R : Type u_1
inst✝⁴ : Semiring R
U : Type u_2
V : Type u_3
inst✝³ : AddCommMonoid U
inst✝² : AddCommMonoid V
inst✝¹ : Module R U
inst✝ : Module R V
e : V ≃ₗ[R] V
f : V →ₗ[R] V
r : R
x : V
hx : x ∈ {x | f x = x}
⊢ r • x ∈ {x | f x = x}

Tactic:
by aesop

Goal state after:
Module: Mathlib.LinearAlgebra.FixedSubmodule

Goal state before:
pp: R✝ : Type u_1
inst✝⁷ : Semiring R✝
U : Type u_2
V✝ : Type u_3
inst✝⁶ : AddCommMonoid U
inst✝⁵ : AddCommMonoid V✝
inst✝⁴ : Module R✝ U
inst✝³ : Module R✝ V✝
e : V✝ ≃ₗ[R✝] V✝
P : Submodule R✝ U
Q : Submodule R✝ V✝
R : Type u_4
V : Type u_5
inst✝² : Ring R
inst✝¹ : AddCommGroup V
inst✝ : Module R V
W : Submodule R V
⊢ Quotient.equiv W W ↑1 ⋯ = 1

Tactic:
by aesop

Goal state after:
Module: Mathlib.LinearAlgebra.Center

Goal state before:
pp: case inr
R : Type u_1
V : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
f : V →ₗ[R] V
hV : Nontrivial V
h_allEq : ∀ (i j : ι), (b.coord i) (f (b i)) = (b.coord j) (f (b j))
hcomm : ∀ (i : ι) (r : R), r • f (b i) = (b.coord i) (f (b i)) • r • b i
⊢ ∃ x, ∃ (h : ∀ (g : R), g * x = x * g), f = ⟨x, ⋯⟩ • 1

Tactic:
let i : ι := Classical.ofNonempty

Goal state after:
case inr
R : Type u_1
V : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
ι : Type u_3
inst✝ : Nontrivial ι
b : Basis ι R V
f : V →ₗ[R] V
hV : Nontrivial V
h_allEq : ∀ (i j : ι), (b.coord i) (f (b i)) = (b.coord j) (f (b j))
hcomm : ∀ (i : ι) (r : R), r • f (b i) = (b.coord i) (f (b i)) • r • b i
i : ι := Classical.ofNonempty
⊢ ∃ x, ∃ (h : ∀ (g : R), g * x = x * g), f = ⟨x, ⋯⟩ • 1
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
⊢ refl R V = transvection ⋯

Tactic:
by aesop

Goal state after:
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
h : ∀ (v : V), ¬LinearIndependent R ![v, f v]
h' : ∀ (i j : ι), i ≠ j → ∀ (r : R), (b.coord i) (f (b i)) * r = r * (b.coord j) (f (b j))
feq : ∀ (i j : ι), f (b j) = (b.coord i) (f (b i)) • b j
⊢ ∃ a, f = a • 1

Tactic:
let i : ι := Classical.ofNonempty

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
h : ∀ (v : V), ¬LinearIndependent R ![v, f v]
h' : ∀ (i j : ι), i ≠ j → ∀ (r : R), (b.coord i) (f (b i)) * r = r * (b.coord j) (f (b j))
feq : ∀ (i j : ι), f (b j) = (b.coord i) (f (b i)) • b j
i : ι := Classical.ofNonempty
⊢ ∃ a, f = a • 1
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) + (1 + ↑(-h.unit⁻¹) * f v)) = 0

Tactic:
rw [← add_assoc, add_comm _ 1, add_assoc]

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 * (1 + (↑(-h.unit⁻¹) + ↑(-h.unit⁻¹) * f v)) = 0
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case refine_2
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 : e ∈ dilatransvections K V
he' : e.fixedReduce = 1
⊢ e ∈ transvections K V

Tactic:
by_cases he_one : e = 1

Goal state after:
case pos
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 : e ∈ dilatransvections K V
he' : e.fixedReduce = 1
he_one : e = 1
⊢ e ∈ transvections K V

case neg
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 : e ∈ dilatransvections K V
he' : e.fixedReduce = 1
he_one : ¬e = 1
⊢ e ∈ transvections K V
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
x✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1
he : e ∈ dilatransvections K V
he' : e.fixedReduce = 1
he_one : e = 1
⊢ e = transvection ⋯

Tactic:
by aesop

Goal state after:
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
⊢ { f // ∀ (m : V), f m * f m = (algebraMap R (CliffordAlgebra (QuadraticForm.baseChange A Q))) (Q m) }

Tactic:
by
    refine ⟨(ι (Q.baseChange A)).restrictScalars R ∘ₗ TensorProduct.mk R A V 1, fun v => ?_⟩
    refine (CliffordAlgebra.ι_sq_scalar (Q.baseChange A) (1 ⊗ₜ v)).trans ?_
    rw [QuadraticForm.baseChange_tmul, one_mul, ← Algebra.algebraMap_eq_smul_one,
      ← IsScalarTower.algebraMap_apply]

Goal state after:
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Contraction

Goal state before:
pp: case ι_mul
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
B : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
d : Module.Dual R M
x✝ : CliffordAlgebra Q
m✝ : M
hx : (changeForm h) ((contractLeft d) x✝) = (contractLeft d) ((changeForm h) x✝)
⊢ d m✝ • (changeForm h) x✝ - (ι Q') m✝ * (changeForm h) ((contractLeft d) x✝) +
      (contractLeft (B m✝)) ((changeForm h) ((contractLeft d) x✝)) =
    d m✝ • (changeForm h) x✝ - (ι Q') m✝ * (changeForm h) ((contractLeft d) x✝) -
      -(contractLeft (B m✝)) ((contractLeft d) ((changeForm h) x✝))

Tactic:
sub_neg_eq_add,

Goal state after:
case ι_mul
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q Q' : QuadraticForm R M
B : BilinForm R M
h : BilinMap.toQuadraticMap B = Q' - Q
d : Module.Dual R M
x✝ : CliffordAlgebra Q
m✝ : M
hx : (changeForm h) ((contractLeft d) x✝) = (contractLeft d) ((changeForm h) x✝)
⊢ d m✝ • (changeForm h) x✝ - (ι Q') m✝ * (changeForm h) ((contractLeft d) x✝) +
      (contractLeft (B m✝)) ((changeForm h) ((contractLeft d) x✝)) =
    d m✝ • (changeForm h) x✝ - (ι Q') m✝ * (changeForm h) ((contractLeft d) x✝) +
      (contractLeft (B m✝)) ((contractLeft d) ((changeForm h) x✝))
Module: Mathlib.LinearAlgebra.Transvection.Basic

Goal state before:
pp: case inr
R : Type u_3
V : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : Free R V
inst✝ : Module.Finite R V
f : Dual R V
v : V
hR : Nontrivial R
n : ℕ := finrank R V
b : Basis (Fin n) R V
⊢ LinearMap.det (transvection f v) = 1 + f v

Tactic:
let S := MvPolynomial (Fin n ⊕ Fin n) ℤ

Goal state after:
case inr
R : Type u_3
V : Type u_4
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : Free R V
inst✝ : Module.Finite R V
f : Dual R V
v : V
hR : Nontrivial R
n : ℕ := finrank R V
b : Basis (Fin n) R V
S : Type := MvPolynomial (Fin n ⊕ Fin n) ℤ
⊢ LinearMap.det (transvection f v) = 1 + f v
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.External

Goal state before:
pp: R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
x y : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
i₁ : ι
a₁ : 𝒜 i₁
j₁ : ι
b₁ : ℬ j₁
i₂ : ι
a₂ : 𝒜 i₂
j₂ : ι
b₂ : ℬ j₂
⊢ ↑↑((-1) ^ (j₁ * i₂)) •
      (gradedComm R 𝒜 ℬ) (((lof R ι 𝒜 i₁) a₁ * (lof R ι 𝒜 i₂) a₂) ⊗ₜ[R] ((lof R ι ℬ j₁) b₁ * (lof R ι ℬ j₂) b₂)) =
    ↑↑((-1) ^ (j₂ * i₂)) •
      (↑↑((-1) ^ (j₁ * i₁)) • (gradedMul R ℬ 𝒜) ((lof R ι ℬ j₁) b₁ ⊗ₜ[R] (lof R ι 𝒜 i₁) a₁))
        ((lof R ι ℬ j₂) b₂ ⊗ₜ[R] (lof R ι 𝒜 i₂) a₂)

Tactic:
LinearMap.smul_apply

Goal state after:
R : Type u_1
ι : Type u_2
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : Module ι (Additive ℤˣ)
inst✝⁹ : DecidableEq ι
𝒜 : ι → Type u_3
ℬ : ι → Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)
inst✝⁶ : (i : ι) → AddCommGroup (ℬ i)
inst✝⁵ : (i : ι) → Module R (𝒜 i)
inst✝⁴ : (i : ι) → Module R (ℬ i)
inst✝³ : DirectSum.GRing 𝒜
inst✝² : DirectSum.GRing ℬ
inst✝¹ : DirectSum.GAlgebra R 𝒜
inst✝ : DirectSum.GAlgebra R ℬ
x y : DirectSum ι 𝒜 ⊗[R] DirectSum ι ℬ
i₁ : ι
a₁ : 𝒜 i₁
j₁ : ι
b₁ : ℬ j₁
i₂ : ι
a₂ : 𝒜 i₂
j₂ : ι
b₂ : ℬ j₂
⊢ ↑↑((-1) ^ (j₁ * i₂)) •
      (gradedComm R 𝒜 ℬ) (((lof R ι 𝒜 i₁) a₁ * (lof R ι 𝒜 i₂) a₂) ⊗ₜ[R] ((lof R ι ℬ j₁) b₁ * (lof R ι ℬ j₂) b₂)) =
    ↑↑((-1) ^ (j₂ * i₂)) •
      ↑↑((-1) ^ (j₁ * i₁)) •
        ((gradedMul R ℬ 𝒜) ((lof R ι ℬ j₁) b₁ ⊗ₜ[R] (lof R ι 𝒜 i₁) a₁)) ((lof R ι ℬ j₂) b₂ ⊗ₜ[R] (lof R ι 𝒜 i₂) a₂)
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) = star x

Tactic:
by simp only [hy₂, Units.coe_star]

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
r : R
x : 𝒜 ᵍ⊗[R] ℬ
⊢ r • x = (auxEquiv R 𝒜 ℬ).symm (r • (auxEquiv R 𝒜 ℬ) x)

Tactic:
rw [LinearEquiv.map_smul]

Goal state after:
R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝⁹ : CommSemiring ι
inst✝⁸ : DecidableEq ι
inst✝⁷ : CommRing R
inst✝⁶ : Ring A
inst✝⁵ : Ring B
inst✝⁴ : Algebra R A
inst✝³ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝² : GradedAlgebra 𝒜
inst✝¹ : GradedAlgebra ℬ
inst✝ : Module ι (Additive ℤˣ)
r : R
x : 𝒜 ᵍ⊗[R] ℬ
⊢ r • x = r • (auxEquiv R 𝒜 ℬ).symm ((auxEquiv R 𝒜 ℬ) x)
Module: Mathlib.LinearAlgebra.TensorProduct.Graded.Internal

Goal state before:
pp: R : Type u_1
ι : Type u_2
A : Type u_3
B : Type u_4
inst✝¹¹ : CommSemiring ι
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : CommRing R
inst✝⁸ : Ring A
inst✝⁷ : Ring B
inst✝⁶ : Algebra R A
inst✝⁵ : Algebra R B
𝒜 : ι → Submodule R A
ℬ : ι → Submodule R B
inst✝⁴ : GradedAlgebra 𝒜
inst✝³ : GradedAlgebra ℬ
inst✝² : Module ι (Additive ℤˣ)
C : Type ?u.46
inst✝¹ : Ring C
inst✝ : Algebra R C
F : 𝒜 ᵍ⊗[R] ℬ →ₐ[R] C
⊢ (fun fg ↦ lift 𝒜 ℬ (↑fg).1 (↑fg).2 ⋯) ((fun F ↦ ⟨(F.comp (includeLeft 𝒜 ℬ), F.comp (includeRight 𝒜 ℬ)), ⋯⟩) F) = F

Tactic:
apply AlgHom.toLinearMap_injective
    ext
    dsimp
    rw [← map_mul, tmul_one_mul_one_tmul]

Goal state after:
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✝ : AddMonoid M
φ : { φ // ∀ (g : G), n • φ g = 0 }
⊢ (fun f ↦ ⟨f.comp (QuotientAddGroup.mk' ((LinearMap.lsmul ℤ G) ↑n).range.toAddSubgroup), ⋯⟩)
      ((fun φ ↦ QuotientAddGroup.lift ((LinearMap.lsmul ℤ G) ↑n).range.toAddSubgroup ↑φ ⋯) φ) =
    φ

Tactic:
by aesop

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Determinant.Misc

Goal state before:
pp: case succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
M : Matrix (Fin (n + 1)) (Fin n) R
hv : ∑ j, M j = 0
j₁ j₂ : Fin (n + 1)
i : Fin n
h_ind : (M.submatrix i.castSucc.succAbove id).det = (↑↑i.castSucc).negOnePow • (M.submatrix (Fin.succAbove 0) id).det
⊢ ((M.submatrix i.succ.succAbove id).updateRow i (∑ k, -1 • M.submatrix i.succ.succAbove id k)).det =
    (↑↑i.castSucc).negOnePow • (M.submatrix (Fin.succAbove 0) id).det

Tactic:
← h_ind

Goal state after:
case succ
R : Type u_1
inst✝ : CommRing R
n : ℕ
M : Matrix (Fin (n + 1)) (Fin n) R
hv : ∑ j, M j = 0
j₁ j₂ : Fin (n + 1)
i : Fin n
h_ind : (M.submatrix i.castSucc.succAbove id).det = (↑↑i.castSucc).negOnePow • (M.submatrix (Fin.succAbove 0) id).det
⊢ ((M.submatrix i.succ.succAbove id).updateRow i (∑ k, -1 • M.submatrix i.succ.succAbove id k)).det =
    (M.submatrix i.castSucc.succAbove id).det
Module: Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular

Goal state before:
pp: case mp
m : Type u_1
n : Type u_3
R : Type u_5
inst✝ : CommRing R
A : Matrix m n R
⊢ (∀ (k : ℕ) (f : Fin k → m) (g : Fin k → n), (A.submatrix f g).det ∈ Set.range SignType.cast) →
    ∀ (ι : Type w) [inst : Fintype ι] [inst_1 : DecidableEq ι] (f : ι → m) (g : ι → n),
      (A.submatrix f g).det ∈ Set.range SignType.cast

Tactic:
intro hA ι _ _ f g
    specialize hA (Fintype.card ι) (f ∘ (Fintype.equivFin ι).symm) (g ∘ (Fintype.equivFin ι).symm)
    rwa [← submatrix_submatrix, det_submatrix_equiv_self] at hA

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Echelon.Pivot

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
inst✝² : Zero R
A : Matrix m n R
l : m → WithTop n
inst✝¹ : LinearOrder n
inst✝ : PartialOrder m
hA : A.IsPivotedBy l
⊢ Monotone l

Tactic:
by
  refine monotone_iff_forall_lt.mpr ?_
  intro i₁ i₂ hlt
  by_cases h₁ : l i₁ = ⊤
  · simp [hA.eq_top_iff.mpr (hA.isRowEchelon.row_eq_zero_of_lt hlt (hA.eq_top_iff.mp h₁))]
  · exact (hA.lt_of_lt_of_ne_top hlt h₁).le

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Echelon.Pivot

Goal state before:
pp: m : Type u_1
n : Type u_2
R : Type u_3
inst✝⁵ : Fintype m
inst✝⁴ : Fintype n
inst✝³ : LinearOrder m
inst✝² : LinearOrder n
inst✝¹ : CommRing R
inst✝ : IsDomain R
A : Matrix m n R
l : m → WithTop n
hA : A.IsPivotedBy l
i : m
hi : i ∉ ↑{i | l i ≠ ⊤}
⊢ l i = ⊤

Tactic:
by aesop

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.MvPolynomial

Goal state before:
pp: case pos
m : Type u_1
k : Type u_2
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Field k
inst✝ : Infinite k
p q : MvPolynomial (m × m) k
h : ∀ (g : GL m k), (eval fun ij ↦ ↑g ij.1 ij.2) p = (eval fun ij ↦ ↑g ij.1 ij.2) q
s : m × m → k
hs_det : (Matrix.of fun i j ↦ s (i, j)).det = 0
⊢ ((eval s) p - (eval s) q) * (Matrix.of fun i j ↦ s (i, j)).det = 0

pp: case neg
m : Type u_1
k : Type u_2
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Field k
inst✝ : Infinite k
p q : MvPolynomial (m × m) k
h : ∀ (g : GL m k), (eval fun ij ↦ ↑g ij.1 ij.2) p = (eval fun ij ↦ ↑g ij.1 ij.2) q
s : m × m → k
hs_det : ¬(Matrix.of fun i j ↦ s (i, j)).det = 0
⊢ ((eval s) p - (eval s) q) * (Matrix.of fun i j ↦ s (i, j)).det = 0

Tactic:
· rw [hs_det, mul_zero]

Goal state after:
case neg
m : Type u_1
k : Type u_2
inst✝³ : Fintype m
inst✝² : DecidableEq m
inst✝¹ : Field k
inst✝ : Infinite k
p q : MvPolynomial (m × m) k
h : ∀ (g : GL m k), (eval fun ij ↦ ↑g ij.1 ij.2) p = (eval fun ij ↦ ↑g ij.1 ij.2) q
s : m × m → k
hs_det : ¬(Matrix.of fun i j ↦ s (i, j)).det = 0
⊢ ((eval s) p - (eval s) q) * (Matrix.of fun i j ↦ s (i, j)).det = 0
Module: Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Card

Goal state before:
pp: 𝔽 : Type u_1
inst✝¹ : Field 𝔽
inst✝ : Fintype 𝔽
n : ℕ
⊢ Nat.card { s // LinearIndependent 𝔽 s } = ∏ i, (q ^ n - q ^ ↑i)

Tactic:
card_linearIndependent,

Goal state after:
𝔽 : Type u_1
inst✝¹ : Field 𝔽
inst✝ : Fintype 𝔽
n : ℕ
⊢ ∏ i, (q ^ finrank 𝔽 (Fin n → 𝔽) - q ^ ↑i) = ∏ i, (q ^ n - q ^ ↑i)

𝔽 : Type u_1
inst✝¹ : Field 𝔽
inst✝ : Fintype 𝔽
n : ℕ
⊢ n ≤ finrank 𝔽 (Fin n → 𝔽)
Module: Mathlib.LinearAlgebra.Matrix.HadamardMatrix

Goal state before:
pp: n : Type u_2
R : Type u_3
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : StarRing R
A : Matrix n n R
hA : A.IsHadamard
⊢ A.det * star A.det = ↑(Fintype.card n) ^ Fintype.card n

Tactic:
have := congr_arg det hA.mul_conjTranspose

Goal state after:
n : Type u_2
R : Type u_3
inst✝³ : Fintype n
inst✝² : DecidableEq n
inst✝¹ : CommRing R
inst✝ : StarRing R
A : Matrix n n R
hA : A.IsHadamard
this : (A * Aᴴ).det = (↑(Fintype.card n) • 1).det
⊢ A.det * star A.det = ↑(Fintype.card n) ^ Fintype.card n
Module: Mathlib.LinearAlgebra.Matrix.Unique

Goal state before:
pp: m : Type u_1
n : Type u_2
A : Type u_3
R : Type u_4
inst✝⁴ : Unique m
inst✝³ : Unique n
inst✝² : Semiring A
inst✝¹ : CommSemiring R
inst✝ : Algebra R A
r : R
⊢ uniqueRingEquiv.toFun ((algebraMap R (Matrix m m A)) r) = (algebraMap R A) r

Tactic:
by aesop

Goal state after:
Module: Mathlib.RingTheory.PiTensorProduct

Goal state before:
pp: ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁶ : CommSemiring R'
inst✝⁵ : CommSemiring R
inst✝⁴ : (i : ι) → Semiring (A i)
inst✝³ : Algebra R' R
inst✝² : (i : ι) → Algebra R (A i)
inst✝¹ : (i : ι) → Algebra R' (A i)
inst✝ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
x : ⨂[R] (i : ι), A i
⊢ (r • mul 1) x = r • (mul x) 1

Tactic:
LinearMap.smul_apply

Goal state after:
ι : Type u_1
R' : Type u_2
R : Type u_3
A : ι → Type u_4
inst✝⁶ : CommSemiring R'
inst✝⁵ : CommSemiring R
inst✝⁴ : (i : ι) → Semiring (A i)
inst✝³ : Algebra R' R
inst✝² : (i : ι) → Algebra R (A i)
inst✝¹ : (i : ι) → Algebra R' (A i)
inst✝ : ∀ (i : ι), IsScalarTower R' R (A i)
r : R'
x : ⨂[R] (i : ι), A i
⊢ r • (mul 1) x = r • (mul x) 1