mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : HasZeroObject C
i j k : ι
j' : ι'
hj' : e.f j = j'
hi : c.prev j = i
hk : c.next j = k
hj : e.BoundaryGE j
hk' : e.f k = c'.next j'
⊢ (K.truncGE'XIsoOpcycles e hj' hj).inv ≫
      (truncGE'.homologyData K e i j k hk hj' hj).right.g' ≫ ((K.truncGE' e).extendXIso e hk').inv =
    ((shortComplexFunctor C c' j').obj K).fromOpcycles ≫ (e.liftExtend (K.restrictionToTruncGE' e) ⋯).f (c'.next j')

Tactic:
e.liftExtend_f _ _ hk',

Goal state after:
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁵ : Category.{v_1, u_3} C
inst✝⁴ : HasZeroMorphisms C
K L : HomologicalComplex C c'
φ : K ⟶ L
e : c.Embedding c'
inst✝³ : e.IsTruncGE
inst✝² : ∀ (i' : ι'), K.HasHomology i'
inst✝¹ : ∀ (i' : ι'), L.HasHomology i'
inst✝ : HasZeroObject C
i j k : ι
j' : ι'
hj' : e.f j = j'
hi : c.prev j = i
hk : c.next j = k
hj : e.BoundaryGE j
hk' : e.f k = c'.next j'
⊢ (K.truncGE'XIsoOpcycles e hj' hj).inv ≫
      (truncGE'.homologyData K e i j k hk hj' hj).right.g' ≫ ((K.truncGE' e).extendXIso e hk').inv =
    ((shortComplexFunctor C c' j').obj K).fromOpcycles ≫
      (K.restrictionXIso e hk').inv ≫ (K.restrictionToTruncGE' e).f k ≫ ((K.truncGE' e).extendXIso e hk').inv
Module: Mathlib.Algebra.Homology.Embedding.TruncGEHomology

Goal state before:
pp: case neg
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝² : e.IsTruncGE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : HasZeroObject C
j : ι
j' : ι'
hj' : e.f j = j'
hj : ¬e.BoundaryGE j
⊢ ShortComplex.QuasiIso ((shortComplexFunctor C c' j').map (K.πTruncGE e))

Tactic:
· let φ := (shortComplexFunctor C c' j').map (K.πTruncGE e)
    have : Epi φ.τ₁ := by
      by_cases hi : ∃ i, e.f i = c'.prev j'
      · obtain ⟨i, hi⟩ := hi
        dsimp [φ, πTruncGE]
        rw [e.epi_liftExtend_f_iff _ _ hi]
        infer_instance
      · apply IsZero.epi (isZero_extend_X _ _ _ (by simpa using hi))
    have : IsIso φ.τ₂ := by
      dsimp [φ, πTruncGE]
      rw [e.isIso_liftExtend_f_iff _ _ hj']
      exact K.isIso_restrictionToTruncGE' e j hj
    have : IsIso φ.τ₃ := by
      dsimp [φ, πTruncGE]
      have : c'.next j' = e.f (c.next j) := by simpa only [← hj'] using e.next_f rfl
      rw [e.isIso_liftExtend_f_iff _ _ this.symm]
      exact K.isIso_restrictionToTruncGE' e _ (e.not_boundaryGE_next' hj rfl)
    exact ShortComplex.quasiIso_of_epi_of_isIso_of_mono φ

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.TruncLEHomology

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝⁴ : Category.{v_1, u_3} C
inst✝³ : HasZeroMorphisms C
K : HomologicalComplex C c'
e : c.Embedding c'
inst✝² : e.IsTruncLE
inst✝¹ : ∀ (i' : ι'), K.HasHomology i'
inst✝ : HasZeroObject C
j : ι
j' : ι'
hj' : e.f j = j'
⊢ QuasiIsoAt (K.ιTruncLE e) j'

Tactic:
by
  have := K.op.quasiIsoAt_πTruncGE e.op hj'
  exact inferInstanceAs (QuasiIsoAt ((unopFunctor _ _).map (K.op.πTruncGE e.op).op) j')

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
K : CochainComplex C ℤ
n : ℤ
h : ∀ (i : ℤ), n < i → IsZero (K.X i)
i : ℤ
hi : ∀ (i_1 : ℕ), (embeddingUpIntLE n).f i_1 ≠ i
⊢ IsZero (K.X i)

Tactic:
rw [notMem_range_embeddingUpIntLE_iff] at hi

Goal state after:
case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
K : CochainComplex C ℤ
n : ℤ
h : ∀ (i : ℤ), n < i → IsZero (K.X i)
i : ℤ
hi : n < i
⊢ IsZero (K.X i)
Module: Mathlib.Algebra.Homology.Embedding.CochainComplex

Goal state before:
pp: case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
K : CochainComplex C ℤ
n : ℤ
h : ∀ (i : ℤ), n < i → ExactAt K i
i : ℤ
hi : ∀ (i_1 : ℕ), (embeddingUpIntLE n).f i_1 ≠ i
⊢ ExactAt K i

Tactic:
rw [notMem_range_embeddingUpIntLE_iff] at hi

Goal state after:
case mpr
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
K : CochainComplex C ℤ
n : ℤ
h : ∀ (i : ℤ), n < i → ExactAt K i
i : ℤ
hi : n < i
⊢ ExactAt K i
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.Basic

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
inst✝¹ : HasExt C
X Y : C
n : ℕ
inst✝ : HasDerivedCategory C
⊢ hom 0 = 0

Tactic:
by
  let β : Ext 0 Y n := 0
  have hβ : β.hom = 0 := by apply (Functor.map_isZero _ (isZero_zero C)).eq_of_src
  have : (0 : Ext X Y n) = (0 : Ext X 0 0).comp β (zero_add n) := by simp [β]
  rw [this, comp_hom, hβ, ShiftedHom.comp_zero]

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.ShortExact

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝¹ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w✝ : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' :
  π ≫ x = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w :
  a ≫ S.X₁.d n₁ (n₁ + 1) = 0 ∧
    a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ (mappingCone S.f).d n₀ n₁ ≫ (snd S.f).v n₁ n₁ ⋯ + b ≫ S.X₂.d n₀ n₁ = 0
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫
      HomologicalComplex.homologyπ (mappingCone S.f) n₀ ≫
        (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap (triangle S.f).mor₃ n₀ n₁ ⋯ =
    HomologicalComplex.liftCycles S.X₃ (x' ≫ (descShortComplex S).f n₀) n₁ ⋯ ⋯ ≫
      HomologicalComplex.homologyπ S.X₃ n₀ ≫ hS.δ n₀ n₁ h

Tactic:
d_snd_v _ _ _ h,

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
S : ShortComplex (CochainComplex C ℤ)
hS : S.ShortExact
n₀ n₁ : ℤ
h : n₀ + 1 = n₁
A : C
x : A ⟶ (HomologicalComplex.homologyFunctor C (up ℤ) n₀).obj (mappingCone S.f)
A' : C
π : A' ⟶ A
w✝¹ : Epi π
x' : A' ⟶ (mappingCone S.f).X n₀
w✝ : x' ≫ (mappingCone S.f).d n₀ n₁ = 0
hx' :
  π ≫ x = HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫ HomologicalComplex.homologyπ (mappingCone S.f) n₀
a : A' ⟶ S.X₁.X n₁
b : A' ⟶ S.X₂.X n₀
w :
  a ≫ S.X₁.d n₁ (n₁ + 1) = 0 ∧
    a ≫ (inl S.f).v n₁ n₀ ⋯ ≫ ((↑(fst S.f)).v n₀ n₁ h ≫ S.f.f n₁ + (snd S.f).v n₀ n₀ ⋯ ≫ S.X₂.d n₀ n₁) +
        b ≫ S.X₂.d n₀ n₁ =
      0
hab : x' = a ≫ (inl S.f).v n₁ n₀ ⋯ + b ≫ (inr S.f).f n₀
⊢ HomologicalComplex.liftCycles (mappingCone S.f) x' n₁ ⋯ w✝ ≫
      HomologicalComplex.homologyπ (mappingCone S.f) n₀ ≫
        (HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap (triangle S.f).mor₃ n₀ n₁ ⋯ =
    HomologicalComplex.liftCycles S.X₃ (x' ≫ (descShortComplex S).f n₀) n₁ ⋯ ⋯ ≫
      HomologicalComplex.homologyπ S.X₃ n₀ ≫ hS.δ n₀ n₁ h
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
inst✝² : HasDerivedCategory C
I : C
inst✝¹ : Injective I
K : CochainComplex C ℤ
i n : ℤ
hn : i < n
inst✝ : K.IsStrictlyGE n
L : CochainComplex C ℤ
w✝ : L.IsStrictlyGE i
g : K ⟶ L
ι : (CochainComplex.singleFunctor C i).obj I ⟶ L
h : IsIso (Q.map ι)
⊢ Q.map g ≫ inv (Q.map ι) = 0

Tactic:
have hπ : IsSplitMono ι := by
    rw [isIso_Q_map_iff_quasiIso] at h
    exact CochainComplex.isSplitMono_from_singleFunctor_obj_of_injective ι

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
inst✝² : HasDerivedCategory C
I : C
inst✝¹ : Injective I
K : CochainComplex C ℤ
i n : ℤ
hn : i < n
inst✝ : K.IsStrictlyGE n
L : CochainComplex C ℤ
w✝ : L.IsStrictlyGE i
g : K ⟶ L
ι : (CochainComplex.singleFunctor C i).obj I ⟶ L
h : IsIso (Q.map ι)
hπ : IsSplitMono ι
⊢ Q.map g ≫ inv (Q.map ι) = 0
Module: Mathlib.Algebra.Module.Presentation.Basic

Goal state before:
pp: A : Type u
inst✝ : Ring A
relations : Relations A
r : relations.R
⊢ (Submodule.span A (Set.range relations.relation)).mkQ (relations.relation r) = 0

Tactic:
Submodule.mkQ_apply,

Goal state after:
A : Type u
inst✝ : Ring A
relations : Relations A
r : relations.R
⊢ Submodule.Quotient.mk (relations.relation r) = 0
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.ExactSequences

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : Abelian C
inst✝ : HasExt C
S : ShortComplex C
hS : S.ShortExact
Y : C
n₀ n₁ : ℕ
h : 1 + n₀ = n₁
x✝ : ↑(AddCommGrpCat.of (Ext S.X₂ Y n₀))
⊢ hS.extClass.comp ((mk₀ S.f).comp x✝ ⋯) h = 0

Tactic:
simp only [ShortComplex.ShortExact.extClass_comp_assoc]

Goal state after:
Module: Mathlib.Algebra.CharP.Reduced

Goal state before:
pp: R : Type u_1
inst✝³ : Finite R
inst✝² : CommRing R
inst✝¹ : IsReduced R
inst✝ : CharP R 2
a : R
⊢ IsSquare a

Tactic:
cases nonempty_fintype R
  exact
    Exists.imp (fun b h => pow_two b ▸ Eq.symm h)
      (((Fintype.bijective_iff_injective_and_card _).mpr ⟨frobenius_inj R 2, rfl⟩).surjective a)

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

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
R : Type u_5
M : Type u_6
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : Fintype ι'
inst✝² : Fintype ι
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq ι'
b : Basis ι R M
v : ι' → M
e : ι ≃ ι'
⊢ (b.reindex e).toMatrix v = (reindexAlgEquiv R R e) (b.toMatrix (v ∘ ⇑e))

Tactic:
ext
  simp [Basis.toMatrix_apply]

Goal state after:
Module: Mathlib.RingTheory.RootsOfUnity.Basic

Goal state before:
pp: M✝ : Type u_1
N : Type u_2
G : Type u_3
R : Type u_4
S : Type u_5
F : Type u_6
inst✝³ : CommMonoid M✝
inst✝² : CommMonoid N
inst✝¹ : DivisionCommMonoid G
k l : ℕ
M : Type u_7
inst✝ : CommMonoid M
n : ℕ
ζ ζ' : ↥(rootsOfUnity n Mˣ)
⊢ ⟨↑↑(ζ * ζ'), ⋯⟩ = ⟨↑↑ζ, ⋯⟩ * ⟨↑↑ζ', ⋯⟩

Tactic:
by simp only [Subgroup.coe_mul, Units.val_mul, MulMemClass.mk_mul_mk]

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

Goal state before:
pp: n : Type u
inst✝³ : DecidableEq n
inst✝² : Fintype n
R : Type v
inst✝¹ : CommRing R
inst✝ : Nonempty n
r : Rˣ
⊢ ∃ a, det a = r

Tactic:
by
  obtain ⟨i⟩ := ‹Nonempty n›
  refine ⟨⟨diagonal fun j ↦ if j = i then r else 1, diagonal fun j ↦ if j = i then r⁻¹.1 else 1,
    ?_, ?_⟩, by simp [det]⟩
  <;> simp only [diagonal_mul_diagonal, mul_ite, ite_mul, Units.mul_inv, one_mul, mul_one,
      diagonal_eq_one]
  <;> funext j <;> split_ifs <;> simp

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

Goal state before:
pp: n : Type u
R : Type v
inst✝⁵ : DecidableEq n
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Fact (Even (Fintype.card n))
g : ↥(GLPos n R)
⊢ -↑g ∈ GLPos n R

Tactic:
mem_glpos,

Goal state after:
n : Type u
R : Type v
inst✝⁵ : DecidableEq n
inst✝⁴ : Fintype n
inst✝³ : CommRing R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : Fact (Even (Fintype.card n))
g : ↥(GLPos n R)
⊢ 0 < ↑(GeneralLinearGroup.det (-↑g))
Module: Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup

Goal state before:
pp: n : Type u
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
A : SpecialLinearGroup n R
⊢ Matrix.toLin' ↑A ∘ₗ Matrix.toLin' ↑A⁻¹ = LinearMap.id

Tactic:
← toLin'_mul,

Goal state after:
n : Type u
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
A : SpecialLinearGroup n R
⊢ Matrix.toLin' (↑A * ↑A⁻¹) = LinearMap.id
Module: Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup

Goal state before:
pp: n : Type u
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
A : SpecialLinearGroup n R
⊢ Matrix.toLin' ↑A⁻¹ ∘ₗ Matrix.toLin' ↑A = LinearMap.id

Tactic:
← toLin'_mul,

Goal state after:
n : Type u
inst✝² : DecidableEq n
inst✝¹ : Fintype n
R : Type v
inst✝ : CommRing R
A : SpecialLinearGroup n R
⊢ Matrix.toLin' (↑A⁻¹ * ↑A) = LinearMap.id
Module: Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup

Goal state before:
pp: ι : Type u_1
F : Type u_2
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : CommRing F
i j : ι
hij : i ≠ j
b : F
⊢ ↑(transvection hij b) *ᵥ Pi.single j 1 = Pi.single j 1 + b • Pi.single i 1

Tactic:
by
  rw [transvection_coe]
  simp [-mulVec_single, add_mulVec, single_mulVec_eq]

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
A : Type u_4
inst✝¹ : Ring A
inst✝ : Algebra R A
h2 : IsUnit 2
f : M →ₗ[R] A
h :
  ∀ (x x_1 : M),
    (((LinearMap.mul R A).compl₂ f ∘ₗ f + (LinearMap.mul R A).flip.compl₂ f ∘ₗ f) x) x_1 =
      ((LinearMap.compr₂ (QuadraticMap.polarBilin Q) (Algebra.linearMap R A)) x) x_1
x : M
⊢ f x * f x = (algebraMap R A) (Q x)

Tactic:
change ∀ x y : M, f x * f y + f y * f x = algebraMap R A (QuadraticMap.polar Q x y) at h

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
Q : QuadraticForm R M
A : Type u_4
inst✝¹ : Ring A
inst✝ : Algebra R A
h2 : IsUnit 2
f : M →ₗ[R] A
x : M
h : ∀ (x y : M), f x * f y + f y * f x = (algebraMap R A) (QuadraticMap.polar (⇑Q) x y)
⊢ f x * f x = (algebraMap R A) (Q x)
Module: Mathlib.Algebra.GradedMonoid

Goal state before:
pp: ι : Type u_1
α : Type u_2
A : ι → Type u_3
inst✝¹ : AddMonoid ι
inst✝ : GMonoid A
l : List α
f : α → GradedMonoid A
⊢ (List.map f l).prod = mk (l.dProdIndex fun i ↦ (f i).fst) (l.dProd (fun i ↦ (f i).fst) fun i ↦ (f i).snd)

Tactic:
rw [GradedMonoid.mk_list_dProd, GradedMonoid.mk]

Goal state after:
ι : Type u_1
α : Type u_2
A : ι → Type u_3
inst✝¹ : AddMonoid ι
inst✝ : GMonoid A
l : List α
f : α → GradedMonoid A
⊢ (List.map f l).prod = (List.map (fun a ↦ ⟨(f a).fst, (f a).snd⟩) l).prod
Module: Mathlib.LinearAlgebra.Matrix.SesquilinearForm

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
S₁ : Type u_3
R₂ : Type u_4
S₂ : Type u_5
N₂ : Type u_10
n : Type u_11
m : Type u_12
inst✝¹⁵ : CommSemiring R
inst✝¹⁴ : AddCommMonoid N₂
inst✝¹³ : Module R N₂
inst✝¹² : Semiring R₁
inst✝¹¹ : Semiring R₂
inst✝¹⁰ : Semiring S₁
inst✝⁹ : Semiring S₂
inst✝⁸ : Module S₁ N₂
inst✝⁷ : Module S₂ N₂
inst✝⁶ : SMulCommClass S₁ R N₂
inst✝⁵ : SMulCommClass S₂ R N₂
inst✝⁴ : SMulCommClass S₂ S₁ N₂
σ₁ : R₁ →+* S₁
σ₂ : R₂ →+* S₂
inst✝³ : Fintype n
inst✝² : Fintype m
inst✝¹ : DecidableEq n
inst✝ : DecidableEq m
M : Matrix n m N₂
x : n → R₁
y : m → R₂
⊢ (((toLinearMapₛₗ₂' R σ₁ σ₂) M) x) y = ∑ i, ∑ j, σ₁ (x i) •> σ₂ (y j) •> M i j

Tactic:
rw [toLinearMapₛₗ₂', toMatrixₛₗ₂', LinearEquiv.coe_symm_mk, toLinearMap₂'Aux, mk₂'ₛₗ_apply]

Goal state after:
R : Type u_1
R₁ : Type u_2
S₁ : Type u_3
R₂ : Type u_4
S₂ : Type u_5
N₂ : Type u_10
n : Type u_11
m : Type u_12
inst✝¹⁵ : CommSemiring R
inst✝¹⁴ : AddCommMonoid N₂
inst✝¹³ : Module R N₂
inst✝¹² : Semiring R₁
inst✝¹¹ : Semiring R₂
inst✝¹⁰ : Semiring S₁
inst✝⁹ : Semiring S₂
inst✝⁸ : Module S₁ N₂
inst✝⁷ : Module S₂ N₂
inst✝⁶ : SMulCommClass S₁ R N₂
inst✝⁵ : SMulCommClass S₂ R N₂
inst✝⁴ : SMulCommClass S₂ S₁ N₂
σ₁ : R₁ →+* S₁
σ₂ : R₂ →+* S₂
inst✝³ : Fintype n
inst✝² : Fintype m
inst✝¹ : DecidableEq n
inst✝ : DecidableEq m
M : Matrix n m N₂
x : n → R₁
y : m → R₂
⊢ ∑ i, ∑ j, σ₂ (y j) •> σ₁ (x i) •> M i j = ∑ i, ∑ j, σ₁ (x i) •> σ₂ (y j) •> M i j
Module: Mathlib.LinearAlgebra.Matrix.SesquilinearForm

Goal state before:
pp: n : Type u_11
m : Type u_12
n' : Type u_13
m' : Type u_14
R : Type u_16
inst✝⁸ : CommSemiring R
inst✝⁷ : Fintype n
inst✝⁶ : Fintype m
inst✝⁵ : DecidableEq n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype n'
inst✝² : Fintype m'
inst✝¹ : DecidableEq n'
inst✝ : DecidableEq m'
B : (n → R) →ₗ[R] (m → R) →ₗ[R] R
l : (n' → R) →ₗ[R] n → R
r : (m' → R) →ₗ[R] m → R
i : n'
j : m'
i' : n
⊢ ∑ i_1,
      ((Pi.basisFun R n).repr (l (Pi.single i 1))) i' •>
        ((Pi.basisFun R m).repr (r (Pi.single j 1))) i_1 •> (B ((Pi.basisFun R n) i')) ((Pi.basisFun R m) i_1) =
    ∑ x,
      of (fun i j ↦ l (Pi.single j 1) i) i' i * (B (Pi.single i' 1)) (Pi.single x 1) *
        of (fun i j ↦ r (Pi.single j 1) i) x j

pp: case h
n : Type u_11
m : Type u_12
n' : Type u_13
m' : Type u_14
R : Type u_16
inst✝⁸ : CommSemiring R
inst✝⁷ : Fintype n
inst✝⁶ : Fintype m
inst✝⁵ : DecidableEq n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype n'
inst✝² : Fintype m'
inst✝¹ : DecidableEq n'
inst✝ : DecidableEq m'
B : (n → R) →ₗ[R] (m → R) →ₗ[R] R
l : (n' → R) →ₗ[R] n → R
r : (m' → R) →ₗ[R] m → R
i : n'
j : m'
i' : n
⊢ ∀ (i_1 : m),
    ((Pi.basisFun R n).repr (l (Pi.single i 1))) i' •> 0 •> (B ((Pi.basisFun R n) i')) ((Pi.basisFun R m) i_1) = 0

Tactic:
· apply sum_congr rfl
      rintro j' -
      simp only [smul_eq_mul, Pi.basisFun_repr, mul_assoc, mul_comm, mul_left_comm,
        Pi.basisFun_apply, of_apply]

Goal state after:
case h
n : Type u_11
m : Type u_12
n' : Type u_13
m' : Type u_14
R : Type u_16
inst✝⁸ : CommSemiring R
inst✝⁷ : Fintype n
inst✝⁶ : Fintype m
inst✝⁵ : DecidableEq n
inst✝⁴ : DecidableEq m
inst✝³ : Fintype n'
inst✝² : Fintype m'
inst✝¹ : DecidableEq n'
inst✝ : DecidableEq m'
B : (n → R) →ₗ[R] (m → R) →ₗ[R] R
l : (n' → R) →ₗ[R] n → R
r : (m' → R) →ₗ[R] m → R
i : n'
j : m'
i' : n
⊢ ∀ (i_1 : m),
    ((Pi.basisFun R n).repr (l (Pi.single i 1))) i' •> 0 •> (B ((Pi.basisFun R n) i')) ((Pi.basisFun R m) i_1) = 0
Module: Mathlib.Algebra.DirectSum.Ring

Goal state before:
pp: ι : Type u_1
inst✝⁴ : DecidableEq ι
A : ι → Type u_2
R : Type u_3
inst✝³ : (i : ι) → AddCommMonoid (A i)
inst✝² : AddMonoid ι
inst✝¹ : GSemiring A
inst✝ : Semiring R
F : (⨁ (i : ι), A i) →+* R
⊢ (fun f ↦ toSemiring (fun x ↦ ↑f) ⋯ ⋯) ((fun F ↦ ⟨fun {i} ↦ (↑F).comp (of A i), ⋯⟩) F) = F

Tactic:
by
    apply RingHom.coe_addMonoidHom_injective
    refine DirectSum.addHom_ext' (fun xi ↦ AddMonoidHom.ext (fun xv ↦ ?_))
    simp only [DirectSum.toAddMonoid_of, AddMonoidHom.comp_apply, toSemiring_coe_addMonoidHom]

Goal state after:
Module: Mathlib.RingTheory.GradedAlgebra.Basic

Goal state before:
pp: case refine_2.refine_2
ι : Type u_1
R : Type u_2
A : Type u_3
σ : Type u_4
inst✝⁷ : Semiring A
inst✝⁶ : DecidableEq ι
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : PartialOrder ι
inst✝³ : CanonicallyOrderedAdd ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
i : ι
c : A
hc : c ∈ 𝒜 i
⊢ ∀ {i_1 : ι} (m : ↥(𝒜 i_1)),
    ↑(((decompose 𝒜) (↑⟨c, hc⟩ * ↑m)) 0) = ↑(((decompose 𝒜) ↑⟨c, hc⟩) 0) * ↑(((decompose 𝒜) ↑m) 0)

Tactic:
rintro j ⟨c', hc'⟩

Goal state after:
case refine_2.refine_2
ι : Type u_1
R : Type u_2
A : Type u_3
σ : Type u_4
inst✝⁷ : Semiring A
inst✝⁶ : DecidableEq ι
inst✝⁵ : AddCommMonoid ι
inst✝⁴ : PartialOrder ι
inst✝³ : CanonicallyOrderedAdd ι
inst✝² : SetLike σ A
inst✝¹ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝ : GradedRing 𝒜
i : ι
c : A
hc : c ∈ 𝒜 i
j : ι
c' : A
hc' : c' ∈ 𝒜 j
⊢ ↑(((decompose 𝒜) (↑⟨c, hc⟩ * ↑⟨c', hc'⟩)) 0) = ↑(((decompose 𝒜) ↑⟨c, hc⟩) 0) * ↑(((decompose 𝒜) ↑⟨c', hc'⟩) 0)
Module: Mathlib.RingTheory.GradedAlgebra.Basic

Goal state before:
pp: ι : Type u_1
A : Type u_3
σ : Type u_4
inst✝¹¹ : Semiring A
inst✝¹⁰ : DecidableEq ι
inst✝⁹ : AddCommMonoid ι
inst✝⁸ : PartialOrder ι
inst✝⁷ : CanonicallyOrderedAdd ι
inst✝⁶ : SetLike σ A
inst✝⁵ : AddSubmonoidClass σ A
𝒜 : ι → σ
inst✝⁴ : GradedRing 𝒜
a b : A
i : ι
inst✝³ : Sub ι
inst✝² : OrderedSub ι
inst✝¹ : AddLeftReflectLE ι
n : ι
inst✝ : Decidable (i ≤ n)
b_mem : b ∈ 𝒜 i
⊢ ↑(((decompose 𝒜) (a * b)) n) = if i ≤ n then ↑(((decompose 𝒜) a) (n - i)) * b else 0

Tactic:
lift b to 𝒜 i using b_mem
  rw [decompose_mul, decompose_coe, coe_mul_of_apply]

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

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
i j : ZMod 2
⊢ (⋃ i_1, ↑((ι Q).range ^ ↑i_1)) * ⋃ i, ↑((ι Q).range ^ ↑i) ⊆ ⋃ i_1, ↑((ι Q).range ^ ↑i_1)

Tactic:
Set.iUnion_mul,

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
i j : ZMod 2
⊢ ⋃ i_1, ↑((ι Q).range ^ ↑i_1) * ⋃ i, ↑((ι Q).range ^ ↑i) ⊆ ⋃ i_1, ↑((ι Q).range ^ ↑i_1)
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Grading

Goal state before:
pp: case mem.mem_mul.refine_1
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x' x✝ : CliffordAlgebra Q
i✝ i : ℕ
x : CliffordAlgebra Q
hx : x ∈ (ι Q).range ^ i
ih : ((lift Q) ⟨GradedAlgebra.ι Q, ⋯⟩) x = (DirectSum.of (fun i ↦ ↥(evenOdd Q i)) ↑i) ⟨x, ⋯⟩
w✝ : M
⊢ (GradedMonoid.mk (1 + ↑i) (GradedMonoid.GMul.mul ⟨(ι Q) w✝, ⋯⟩ ⟨x, ⋯⟩)).fst =
    (GradedMonoid.mk ↑i.succ ⟨(ι Q) w✝ * x, ⋯⟩).fst

Tactic:
dsimp only [GradedMonoid.mk, Subtype.coe_mk]

Goal state after:
case mem.mem_mul.refine_1
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x' x✝ : CliffordAlgebra Q
i✝ i : ℕ
x : CliffordAlgebra Q
hx : x ∈ (ι Q).range ^ i
ih : ((lift Q) ⟨GradedAlgebra.ι Q, ⋯⟩) x = (DirectSum.of (fun i ↦ ↥(evenOdd Q i)) ↑i) ⟨x, ⋯⟩
w✝ : M
⊢ 1 + ↑i = ↑i.succ
Module: Mathlib.LinearAlgebra.CliffordAlgebra.Grading

Goal state before:
pp: case mem.mem_mul.refine_2
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x' x✝ : CliffordAlgebra Q
i✝ i : ℕ
x : CliffordAlgebra Q
hx : x ∈ (ι Q).range ^ i
ih : ((lift Q) ⟨GradedAlgebra.ι Q, ⋯⟩) x = (DirectSum.of (fun i ↦ ↥(evenOdd Q i)) ↑i) ⟨x, ⋯⟩
w✝ : M
⊢ ↑(GradedMonoid.mk (1 + ↑i) (GradedMonoid.GMul.mul ⟨(ι Q) w✝, ⋯⟩ ⟨x, ⋯⟩)).snd =
    ↑(GradedMonoid.mk ↑i.succ ⟨(ι Q) w✝ * x, ⋯⟩).snd

Tactic:
dsimp only [GradedMonoid.mk, Subtype.coe_mk]

Goal state after:
case mem.mem_mul.refine_2
R : Type u_1
M : Type u_2
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
Q : QuadraticForm R M
x' x✝ : CliffordAlgebra Q
i✝ i : ℕ
x : CliffordAlgebra Q
hx : x ∈ (ι Q).range ^ i
ih : ((lift Q) ⟨GradedAlgebra.ι Q, ⋯⟩) x = (DirectSum.of (fun i ↦ ↥(evenOdd Q i)) ↑i) ⟨x, ⋯⟩
w✝ : M
⊢ ↑(GradedMonoid.GMul.mul ⟨(ι Q) w✝, ⋯⟩ ⟨x, ⋯⟩) = (ι Q) w✝ * x
Module: Mathlib.LinearAlgebra.ExteriorAlgebra.Basic

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
⊢ Set.range ⇑(ιMulti R n) ⊆ ↑(⋀[R]^n M)

Tactic:
Set.range_subset_iff

Goal state after:
R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
⊢ ∀ (y : Fin n → M), (ιMulti R n) y ∈ ↑(⋀[R]^n M)
Module: Mathlib.LinearAlgebra.ExteriorAlgebra.Basic

Goal state before:
pp: R : Type u1
inst✝² : CommRing R
M : Type u2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
n : ℕ
⊢ Submodule.span R (Set.range ⇑(ιMulti R n)) = ⋀[R]^n M

Tactic:
by
  refine le_antisymm (Submodule.span_le.2 (ιMulti_range R n)) ?_
  rw [exteriorPower, Submodule.pow_eq_span_pow_set, Submodule.span_le]
  refine fun u hu ↦ Submodule.subset_span ?_
  obtain ⟨f, rfl⟩ := Set.mem_pow.mp hu
  refine ⟨fun i => ιInv (f i).1, ?_⟩
  rw [ιMulti_apply]
  congr with i
  obtain ⟨v, hv⟩ := (f i).prop
  rw [← hv, ι_leftInverse]

Goal state after:
Module: Mathlib.Algebra.Category.ModuleCat.Free

Goal state before:
pp: ι : Type u_1
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
β : Type u_4
u : ι ⊕ β → ↑S.X₂
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
hv : ⊤ ≤ span R (range v)
hw : ⊤ ≤ span R (range (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr))
⊢ ⊤ ≤ span R (range u)

Tactic:
intro m _

Goal state after:
ι : Type u_1
R : Type u_3
inst✝ : Ring R
S : ShortComplex (ModuleCat R)
hS : S.Exact
v : ι → ↑S.X₁
β : Type u_4
u : ι ⊕ β → ↑S.X₂
huv : u ∘ Sum.inl = ⇑(ConcreteCategory.hom S.f) ∘ v
hv : ⊤ ≤ span R (range v)
hw : ⊤ ≤ span R (range (⇑(ConcreteCategory.hom S.g) ∘ u ∘ Sum.inr))
m : ↑S.X₂
a✝ : m ∈ ⊤
⊢ m ∈ span R (range u)