mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Localization.HomEquiv

Goal state before:
pp: C : Type u_1
D₁ : Type u_5
D₂ : Type u_6
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_5, u_5} D₁
inst✝² : Category.{v_6, u_6} D₂
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝¹ : L₁.IsLocalization W
L₂ : C ⥤ D₂
inst✝ : L₂.IsLocalization W
X Y : C
f : Y ⟶ X
hf : W f
⊢ (homEquiv W L₁ L₂) (isoOfHom L₁ W f hf).inv ≫ (homEquiv W L₁ L₂) (L₁.map f) = 𝟙 (L₂.obj X)

Tactic:
← homEquiv_comp,

Goal state after:
C : Type u_1
D₁ : Type u_5
D₂ : Type u_6
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Category.{v_5, u_5} D₁
inst✝² : Category.{v_6, u_6} D₂
W : MorphismProperty C
L₁ : C ⥤ D₁
inst✝¹ : L₁.IsLocalization W
L₂ : C ⥤ D₂
inst✝ : L₂.IsLocalization W
X Y : C
f : Y ⟶ X
hf : W f
⊢ (homEquiv W L₁ L₂) ((isoOfHom L₁ W f hf).inv ≫ L₁.map f) = 𝟙 (L₂.obj X)
Module: Mathlib.CategoryTheory.Localization.SmallHom

Goal state before:
pp: C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
W : MorphismProperty C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
L : C ⥤ D
inst✝³ : L.IsLocalization W
X Y Z : C
inst✝² : HasSmallLocalizedHom W X Y
inst✝¹ : HasSmallLocalizedHom W Y Z
inst✝ : HasSmallLocalizedHom W X Z
α : SmallHom W X Y
β : SmallHom W Y Z
this : Small.{w, max u₁ v₁} (W.Q.obj X ⟶ W.Q.obj Y) := small_of_hasSmallLocalizedHom W W.Q X Y
⊢ (equiv W L) (α.comp β) = (equiv W L) α ≫ (equiv W L) β

Tactic:
let := small_of_hasSmallLocalizedHom.{w} W W.Q Y Z

Goal state after:
C : Type u₁
inst✝⁵ : Category.{v₁, u₁} C
W : MorphismProperty C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
L : C ⥤ D
inst✝³ : L.IsLocalization W
X Y Z : C
inst✝² : HasSmallLocalizedHom W X Y
inst✝¹ : HasSmallLocalizedHom W Y Z
inst✝ : HasSmallLocalizedHom W X Z
α : SmallHom W X Y
β : SmallHom W Y Z
this✝ : Small.{w, max u₁ v₁} (W.Q.obj X ⟶ W.Q.obj Y) := small_of_hasSmallLocalizedHom W W.Q X Y
this : Small.{w, max u₁ v₁} (W.Q.obj Y ⟶ W.Q.obj Z) := small_of_hasSmallLocalizedHom W W.Q Y Z
⊢ (equiv W L) (α.comp β) = (equiv W L) α ≫ (equiv W L) β
Module: Mathlib.Algebra.Homology.HomologySequenceLemmas

Goal state before:
pp: case neg
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
j : ι
h₁ : Mono (HomologicalComplex.homologyMap S.g j)
h₂ : ∀ (i : ι), c.Rel i j → Epi (HomologicalComplex.homologyMap S.g i)
hj : ∀ (i : ι), ¬c.Rel i j
this✝ : Mono S.f
this : Mono (HomologicalComplex.homologyMap S.f j)
⊢ IsZero (S.X₁.homology j)

Tactic:
rw [IsZero.iff_id_eq_zero,
      ← cancel_mono (HomologicalComplex.homologyMap S.f j),
      ← cancel_mono (HomologicalComplex.homologyMap S.g j)]

Goal state after:
case neg
C : Type u_1
ι : Type u_2
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
c : ComplexShape ι
S : ShortComplex (HomologicalComplex C c)
hS : S.ShortExact
j : ι
h₁ : Mono (HomologicalComplex.homologyMap S.g j)
h₂ : ∀ (i : ι), c.Rel i j → Epi (HomologicalComplex.homologyMap S.g i)
hj : ∀ (i : ι), ¬c.Rel i j
this✝ : Mono S.f
this : Mono (HomologicalComplex.homologyMap S.f j)
⊢ (𝟙 (S.X₁.homology j) ≫ HomologicalComplex.homologyMap S.f j) ≫ HomologicalComplex.homologyMap S.g j =
    (0 ≫ HomologicalComplex.homologyMap S.f j) ≫ HomologicalComplex.homologyMap S.g j
Module: Mathlib.CategoryTheory.Abelian.DiagramLemmas.Four

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
R₁ R₂ : ComposableArrows C 3
φ : R₁ ⟶ R₂
hR₁ :
  (mk₂ (R₁.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)
      (R₁.map' 2 3 mono_of_epi_of_mono_of_mono'._proof_4 mono_of_epi_of_mono_of_mono'._proof_7)).Exact
hR₂ :
  (mk₂ (R₂.map' 0 1 mono_of_epi_of_mono_of_mono'._proof_9 mono_of_epi_of_mono_of_mono'._proof_11)
      (R₂.map' 1 2 mono_of_epi_of_mono_of_mono'._proof_6 mono_of_epi_of_mono_of_mono'._proof_4)).Exact
hR₂' : R₂.map' 1 3 mono_of_epi_of_mono_of_mono'._proof_11 mono_of_epi_of_mono_of_mono'._proof_7 = 0
h₀ : Epi (app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13)
h₂✝ : Epi (app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4)
h₃✝ : Mono (app' φ 3 mono_of_epi_of_mono_of_mono'._proof_7)
A : C
g₁ : A ⟶ R₂.obj' 1 mono_of_epi_of_mono_of_mono'._proof_11
A₁ : C
π₁ : A₁ ⟶ A
w✝³ : Epi π₁
f₂ : A₁ ⟶ R₁.obj' 2 mono_of_epi_of_mono_of_mono'._proof_4
h₁ : π₁ ≫ g₁ ≫ R₂.map' 1 2 ⋯ ⋯ = f₂ ≫ app' φ 2 mono_of_epi_of_mono_of_mono'._proof_4
h₂ : f₂ ≫ R₁.map' 2 3 ⋯ ⋯ = 0
A₂ : C
π₂ : A₂ ⟶ A₁
w✝² : Epi π₂
f₁ : A₂ ⟶ R₁.obj 1
h₃ : π₂ ≫ f₂ = f₁ ≫ R₁.map (homOfLE ⋯)
h₄ : (π₂ ≫ π₁ ≫ g₁ - f₁ ≫ app' φ 1 mono_of_epi_of_mono_of_mono'._proof_11) ≫ R₂.map' 1 2 ⋯ ⋯ = 0
A₃ : C
π₃ : A₃ ⟶ A₂
w✝¹ : Epi π₃
g₀ : A₃ ⟶ R₂.obj 0
h₅ : π₃ ≫ π₂ ≫ π₁ ≫ g₁ - π₃ ≫ f₁ ≫ φ.app 1 = g₀ ≫ R₂.map (homOfLE ⋯)
A₄ : C
π₄ : A₄ ⟶ A₃
w✝ : Epi π₄
f₀ : A₄ ⟶ R₁.obj' 0 mono_of_epi_of_mono_of_mono'._proof_13
h₆ : π₄ ≫ g₀ = f₀ ≫ app' φ 0 mono_of_epi_of_mono_of_mono'._proof_13
⊢ R₁.obj' 0 mono_of_epi_of_mono_of_mono'._proof_13 ⟶ R₁.obj 1

Tactic:
by exact R₁.map' 0 1

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
n : ℤ
φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅ (triangleh S.f) n (n + 1) ⋯).δlast ⟶
  (composableArrows₅ hS n (n + 1) ⋯).δlast :=
  homMk₄ ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₂)
    ((homologyFunctorFactors C (up ℤ) n).hom.app (triangle S.f).obj₃ ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₁)
    ((homologyFunctorFactors C (up ℤ) (n + 1)).hom.app (triangle S.f).obj₂) ⋯ ⋯ ⋯ ⋯
this :
  IsIso
    ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
      HomologicalComplex.homologyMap (descShortComplex S) n)
⊢ IsIso (HomologicalComplex.homologyMap (descShortComplex S) n)

Tactic:
apply IsIso.of_isIso_comp_left ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f))

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Basic

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
⊢ ∀ (a b : ℤ), -(a + b) = -a + -b

Tactic:
intros

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
a✝ b✝ : ℤ
⊢ -(a✝ + b✝) = -a✝ + -b✝
Module: Mathlib.CategoryTheory.Shift.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁶ : Category.{v_1, u_1} C
inst✝⁵ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝⁴ : AddMonoid A
inst✝³ : HasShift C A
inst✝² : HasShift D A
inst✝¹ : F.CommShift A
inst✝ : G.CommShift A
x✝ : NatTrans.CommShift adj.unit A
a : A
⊢ (Functor.commShiftIso (G ⋙ F) a).hom ≫ Functor.whiskerRight adj.counit (shiftFunctor D a) =
    (shiftFunctor D a).whiskerLeft adj.counit ≫ (Functor.commShiftIso (𝟭 D) a).hom

Tactic:
ext
    simp only [Functor.comp_obj, Functor.id_obj, NatTrans.comp_app,
      Functor.commShiftIso_comp_hom_app, Functor.whiskerRight_app, assoc, Functor.whiskerLeft_app,
      Functor.commShiftIso_id_hom_app, comp_id]
    refine (compatibilityCounit_of_compatibilityUnit adj _ _ (fun X ↦ ?_) _).symm
    simpa [Functor.commShiftIso_comp_hom_app] using NatTrans.shift_app_comm adj.unit a X

Goal state after:
Module: Mathlib.CategoryTheory.Shift.Adjunction

Goal state before:
pp: C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : G.CommShift A
a b : A
⊢ iso adj (a + b) = Functor.CommShift.isoAdd (iso adj a) (iso adj b)

Tactic:
refine CommShift.compatibilityUnit_unique_left adj _ _ (G.commShiftIso (a + b))
      (compatibilityUnit_iso adj (a + b)) ?_

Goal state after:
C : Type u_1
D : Type u_2
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Category.{v_2, u_2} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
A : Type u_3
inst✝³ : AddGroup A
inst✝² : HasShift C A
inst✝¹ : HasShift D A
inst✝ : G.CommShift A
a b : A
⊢ CommShift.CompatibilityUnit adj (Functor.CommShift.isoAdd (iso adj a) (iso adj b)) (Functor.commShiftIso G (a + b))
Module: Mathlib.CategoryTheory.Triangulated.Yoneda

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
B : C
n a a' : ℤ
h : n + a = a'
⊢ ∀ {X Y : Cᵒᵖ} (f : X ⟶ Y),
    (shiftFunctor Cᵒᵖ n ⋙ preadditiveYoneda.obj ((shiftFunctor C a).obj B)).map f ≫
        (let __Equiv := Quiver.Hom.opEquiv.trans (ShiftedHom.opEquiv' n a a' h).symm;
            { toEquiv := __Equiv, map_add' := ⋯ }).toAddCommGrpIso.hom =
      (let __Equiv := Quiver.Hom.opEquiv.trans (ShiftedHom.opEquiv' n a a' h).symm;
            { toEquiv := __Equiv, map_add' := ⋯ }).toAddCommGrpIso.hom ≫
        (preadditiveYoneda.obj ((shiftFunctor C a').obj B)).map f

Tactic:
intros

Goal state after:
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : Preadditive C
inst✝¹ : HasShift C ℤ
inst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive
B : C
n a a' : ℤ
h : n + a = a'
X✝ Y✝ : Cᵒᵖ
f✝ : X✝ ⟶ Y✝
⊢ (shiftFunctor Cᵒᵖ n ⋙ preadditiveYoneda.obj ((shiftFunctor C a).obj B)).map f✝ ≫
      (let __Equiv := Quiver.Hom.opEquiv.trans (ShiftedHom.opEquiv' n a a' h).symm;
          { toEquiv := __Equiv, map_add' := ⋯ }).toAddCommGrpIso.hom =
    (let __Equiv := Quiver.Hom.opEquiv.trans (ShiftedHom.opEquiv' n a a' h).symm;
          { toEquiv := __Equiv, map_add' := ⋯ }).toAddCommGrpIso.hom ≫
      (preadditiveYoneda.obj ((shiftFunctor C a').obj B)).map f✝
Module: Mathlib.CategoryTheory.Triangulated.Opposite.Triangle

Goal state before:
pp: case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
T : Triangle Cᵒᵖ
⊢ T.mor₃ ≫
      (opShiftFunctorEquivalence C 1).counitIso.inv.app ((shiftFunctor Cᵒᵖ 1).obj T.obj₁) ≫
        (shiftFunctor Cᵒᵖ 1).map ((opShiftFunctorEquivalence C 1).unitIso.inv.app T.obj₁) =
    T.mor₃

Tactic:
opShiftFunctorEquivalence_counitIso_inv_app_shift,

Goal state after:
case refine_3
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasShift C ℤ
T : Triangle Cᵒᵖ
⊢ T.mor₃ ≫
      (shiftFunctor Cᵒᵖ 1).map ((opShiftFunctorEquivalence C 1).unitIso.hom.app T.obj₁) ≫
        (shiftFunctor Cᵒᵖ 1).map ((opShiftFunctorEquivalence C 1).unitIso.inv.app T.obj₁) =
    T.mor₃
Module: Mathlib.CategoryTheory.Center.Linear

Goal state before:
pp: R : Type w
inst✝² : Ring R
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
φ : R →+* CatCenter C
X✝ Y✝ : C
this : (X Y : C) → Module R (X ⟶ Y) := homModuleOfRingMorphism φ
X Y Z : C
f : X ⟶ Y
r : R
g : Y ⟶ Z
⊢ f ≫ (r • g) = r • f ≫ g

Tactic:
by simp only [smulOfRingMorphism_smul_eq', assoc]

Goal state after:
Module: Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughInjectives

Goal state before:
pp: C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
I : C
inst✝² : Injective I
L : CochainComplex C ℤ
i : ℤ
ι : (singleFunctor C i).obj I ⟶ L
inst✝¹ : L.IsStrictlyGE i
inst✝ : QuasiIsoAt ι i
e : L.X i ≅ opcycles L i := pOpcyclesIso L (i - 1) i ⋯ ⋯
α : I ⟶ L.X i := (singleObjHomologySelfIso (ComplexShape.up ℤ) i I).inv ≫ homologyMap ι i ≫ homologyι L i ≫ e.inv
⊢ ι.f i ≫ pOpcycles L i = ((single C (ComplexShape.up ℤ) i).obj I).pOpcycles i ≫ opcyclesMap ι i

Tactic:
p_opcyclesMap

Goal state after:
C : Type u
inst✝⁴ : Category.{v, u} C
inst✝³ : Abelian C
I : C
inst✝² : Injective I
L : CochainComplex C ℤ
i : ℤ
ι : (singleFunctor C i).obj I ⟶ L
inst✝¹ : L.IsStrictlyGE i
inst✝ : QuasiIsoAt ι i
e : L.X i ≅ opcycles L i := pOpcyclesIso L (i - 1) i ⋯ ⋯
α : I ⟶ L.X i := (singleObjHomologySelfIso (ComplexShape.up ℤ) i I).inv ≫ homologyMap ι i ≫ homologyι L i ≫ e.inv
⊢ ι.f i ≫ pOpcycles L i = ι.f i ≫ pOpcycles L i
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:
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
S : Type u_1
inst✝ : CommRing S
f : R →+* S
u : Rˣ
⊢ (map f) ((scalar n) u) = (scalar n) ((Units.map ↑f) u)

Tactic:
by
  ext
  simp [Matrix.diagonal_apply]
  split <;> simp

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

Goal state before:
pp: n : Type u_1
inst✝² : Fintype n
A : Type u_4
inst✝¹ : CommRing A
inst✝ : IsDomain A
M N : Matrix n n A
h : N.det ≠ 0
⊢ M.det * N.det ≠ 0 ↔ M.det ≠ 0

Tactic:
exact mul_ne_zero_iff_right h

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

Goal state before:
pp: n✝ : Type u_1
inst✝⁵ : Fintype n✝
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
S : Type u_2
inst✝² : CommRing S
inst✝¹ : LinearOrder S
inst✝ : IsStrictOrderedRing S
A : Matrix n n S
h1 : Pairwise fun i j ↦ A i j < 0
h✝ : Nonempty n
h2 : A.det = 0
v : n → S
h_vnz : v ≠ 0
h_vA : v ᵥ* A = 0
h_sup : 0 < Finset.univ.sup' ⋯ v
j₀ : n
h_j₀ : Finset.univ.sup' ⋯ v = v j₀
⊢ ∑ i, v j₀ * A i j₀ ≤ 0

Tactic:
show 0 = ∑ i, v i * A i j₀ from (congrFun h_vA j₀).symm

Goal state after:
n✝ : Type u_1
inst✝⁵ : Fintype n✝
n : Type u_1
inst✝⁴ : Fintype n
inst✝³ : DecidableEq n
S : Type u_2
inst✝² : CommRing S
inst✝¹ : LinearOrder S
inst✝ : IsStrictOrderedRing S
A : Matrix n n S
h1 : Pairwise fun i j ↦ A i j < 0
h✝ : Nonempty n
h2 : A.det = 0
v : n → S
h_vnz : v ≠ 0
h_vA : v ᵥ* A = 0
h_sup : 0 < Finset.univ.sup' ⋯ v
j₀ : n
h_j₀ : Finset.univ.sup' ⋯ v = v j₀
⊢ ∑ i, v j₀ * A i j₀ ≤ ∑ i, v i * A i j₀
Module: Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup

Goal state before:
pp: F : Type u_1
inst✝ : Field F
a : F
ha : a ≠ 0
⊢ ↑(diag2 a ha * diag2 a⁻¹ ⋯) = ↑1

Tactic:
by
  simp [diag2_coe, funext_iff, mul_inv_cancel₀ ha, inv_mul_cancel₀ ha]

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

Goal state before:
pp: F : Type u_1
inst✝ : Field F
a : F
ha : a ≠ 0
hasq : a ^ 2 ≠ 1
c b c' : F
hc : c' = b * (a ^ 2 - 1)
⊢ diag2 a⁻¹ ⋯ * SpecialLinearGroup.transvection ⋯ b * diag2 a⁻¹⁻¹ ⋯ * (SpecialLinearGroup.transvection ⋯ b)⁻¹ =
    SpecialLinearGroup.transvection ⋯ c'

Tactic:
SpecialLinearGroup.transvection_inv one_ne_zero b

Goal state after:
F : Type u_1
inst✝ : Field F
a : F
ha : a ≠ 0
hasq : a ^ 2 ≠ 1
c b c' : F
hc : c' = b * (a ^ 2 - 1)
⊢ diag2 a⁻¹ ⋯ * SpecialLinearGroup.transvection ⋯ b * diag2 a⁻¹⁻¹ ⋯ * SpecialLinearGroup.transvection ⋯ (-b) =
    SpecialLinearGroup.transvection ⋯ c'
Module: Mathlib.LinearAlgebra.Matrix.SesquilinearForm

Goal state before:
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:
intros

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✝ : m
⊢ ((Pi.basisFun R n).repr (l (Pi.single i 1))) i' •> 0 •> (B ((Pi.basisFun R n) i')) ((Pi.basisFun R m) i✝) = 0
Module: Mathlib.LinearAlgebra.Matrix.SesquilinearForm

Goal state before:
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),
    (((Pi.basisFun R m).repr (r (Pi.single j 1))).sum fun j yj ↦
        0 •> yj •> (B ((Pi.basisFun R n) i)) ((Pi.basisFun R m) j)) =
      0

Tactic:
intros

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
⊢ (((Pi.basisFun R m).repr (r (Pi.single j 1))).sum fun j yj ↦
      0 •> yj •> (B ((Pi.basisFun R n) i✝)) ((Pi.basisFun R m) j)) =
    0
Module: Mathlib.Algebra.DirectSum.Ring

Goal state before:
pp: ι : Type u_1
inst✝³ : DecidableEq ι
A : ι → Type u_2
inst✝² : Add ι
inst✝¹ : (i : ι) → AddCommMonoid (A i)
inst✝ : GNonUnitalNonAssocSemiring A
x✝² x✝¹ x✝ : ⨁ (i : ι), A i
⊢ x✝² * (x✝¹ + x✝) = x✝² * x✝¹ + x✝² * x✝

Tactic:
simp only [Mul.mul, HMul.hMul, map_add]

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

Goal state before:
pp: R : Type u_1
M₁ : Type u_6
M₂ : Type u_7
N₂ : Type u_10
n : Type u_11
m : Type u_12
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M₁
inst✝⁸ : Module R M₁
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Module R M₂
inst✝⁵ : AddCommMonoid N₂
inst✝⁴ : Module R N₂
σ₁ σ₂ : R →+* R
inst✝³ : Fintype n
inst✝² : Fintype m
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
b₁ : Basis n R M₁
b₂ : Basis m R M₂
B : M₁ →ₛₗ[σ₁] M₂ →ₛₗ[σ₂] N₂
i : n
j : m
⊢ (toMatrix₂Aux R ⇑b₁ ⇑b₂) B i j = (toMatrix₂ b₁ b₂) B i j

Tactic:
LinearMap.toMatrix₂_apply,

Goal state after:
R : Type u_1
M₁ : Type u_6
M₂ : Type u_7
N₂ : Type u_10
n : Type u_11
m : Type u_12
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M₁
inst✝⁸ : Module R M₁
inst✝⁷ : AddCommMonoid M₂
inst✝⁶ : Module R M₂
inst✝⁵ : AddCommMonoid N₂
inst✝⁴ : Module R N₂
σ₁ σ₂ : R →+* R
inst✝³ : Fintype n
inst✝² : Fintype m
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
b₁ : Basis n R M₁
b₂ : Basis m R M₂
B : M₁ →ₛₗ[σ₁] M₂ →ₛₗ[σ₂] N₂
i : n
j : m
⊢ (toMatrix₂Aux R ⇑b₁ ⇑b₂) B i j = (B (b₁ i)) (b₂ j)
Module: Mathlib.LinearAlgebra.Matrix.SesquilinearForm

Goal state before:
pp: R : Type u_1
N₂ : Type u_10
n : Type u_11
m : Type u_12
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid N₂
inst✝⁴ : Module R N₂
inst✝³ : Fintype n
inst✝² : Fintype m
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
B : (n → R) →ₗ[R] (m → R) →ₗ[R] N₂
i✝ : n
j✝ : m
⊢ (toMatrix₂ (Pi.basisFun R n) (Pi.basisFun R m)) B i✝ j✝ = (toMatrix₂' R) B i✝ j✝

Tactic:
LinearMap.toMatrix₂_apply,

Goal state after:
R : Type u_1
N₂ : Type u_10
n : Type u_11
m : Type u_12
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid N₂
inst✝⁴ : Module R N₂
inst✝³ : Fintype n
inst✝² : Fintype m
inst✝¹ : DecidableEq m
inst✝ : DecidableEq n
B : (n → R) →ₗ[R] (m → R) →ₗ[R] N₂
i✝ : n
j✝ : m
⊢ (B ((Pi.basisFun R n) i✝)) ((Pi.basisFun R m) j✝) = (toMatrix₂' R) B i✝ j✝
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: S : Type u_1
T : Type u_2
R : Type u_3
M : Type u_4
N : Type u_5
P : Type u_6
A : Type u_7
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
Q Q' x y : QuadraticMap R M N
h : x.toFun = y.toFun
⊢ x = y

Tactic:
cases x; cases y; congr

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

Goal state before:
pp: R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
Q : QuadraticMap R M N
x y z : M
B : BilinMap R M N
h : ∀ (x y : M), Q (x + y) = Q x + Q y + (B x) y
⊢ Q (x + y + z) + (Q x + Q y + Q z) = Q (x + y) + Q (y + z) + Q (x + z)

Tactic:
simp only [h, LinearMap.map_add₂]

Goal state after:
R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
Q : QuadraticMap R M N
x y z : M
B : BilinMap R M N
h : ∀ (x y : M), Q (x + y) = Q x + Q y + (B x) y
⊢ Q x + Q y + (B x) y + Q z + ((B x) z + (B y) z) + (Q x + Q y + Q z) =
    Q x + Q y + (B x) y + (Q y + Q z + (B y) z) + (Q x + Q z + (B x) z)
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
Q : QuadraticMap R M N
x : M
⊢ Q (2 • x) = 4 • Q x

Tactic:
Q.map_smul,

Goal state after:
R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
Q : QuadraticMap R M N
x : M
⊢ (2 * 2) • Q x = 4 • Q x
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
Q : QuadraticMap R M N
⊢ Q (0 • 0) = 0

Tactic:
Q.map_smul,

Goal state after:
R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : Module R M
inst✝¹ : AddCommMonoid N
inst✝ : Module R N
Q : QuadraticMap R M N
⊢ (0 * 0) • Q 0 = 0
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: S : Type u_1
R : Type u_3
M : Type u_4
N : Type u_5
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid N
inst✝⁶ : Module R N
Q : QuadraticMap R M N
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S R
inst✝³ : SMul S M
inst✝² : IsScalarTower S R M
inst✝¹ : Module S N
inst✝ : IsScalarTower S R N
a : S
x : M
⊢ Q ((algebraMap S R) a • x) = (a * a) • Q x

Tactic:
Q.map_smul,

Goal state after:
S : Type u_1
R : Type u_3
M : Type u_4
N : Type u_5
inst✝¹⁰ : CommSemiring R
inst✝⁹ : AddCommMonoid M
inst✝⁸ : Module R M
inst✝⁷ : AddCommMonoid N
inst✝⁶ : Module R N
Q : QuadraticMap R M N
inst✝⁵ : CommSemiring S
inst✝⁴ : Algebra S R
inst✝³ : SMul S M
inst✝² : IsScalarTower S R M
inst✝¹ : Module S N
inst✝ : IsScalarTower S R N
a : S
x : M
⊢ ((algebraMap S R) a * (algebraMap S R) a) • Q x = (a * a) • Q x
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
Q : QuadraticMap R M N
x : M
⊢ Q (-1 • x) = Q x

Tactic:
Q.map_smul,

Goal state after:
R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
Q : QuadraticMap R M N
x : M
⊢ (-1 * -1) • Q x = Q x
Module: Mathlib.LinearAlgebra.QuadraticForm.Basic

Goal state before:
pp: R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
Q : QuadraticMap R M N
a : R
x y : M
B : BilinMap R M N
h : ∀ (x y : M), Q (x + y) = Q x + Q y + (B x) y
⊢ Q (a • x) + Q y + (B (a • x)) y - Q (a • x) - Q y = a • (Q x + Q y + (B x) y - Q x - Q y)

Tactic:
Q.map_smul,

Goal state after:
R : Type u_3
M : Type u_4
N : Type u_5
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : AddCommGroup N
inst✝¹ : Module R M
inst✝ : Module R N
Q : QuadraticMap R M N
a : R
x y : M
B : BilinMap R M N
h : ∀ (x y : M), Q (x + y) = Q x + Q y + (B x) y
⊢ (a * a) • Q x + Q y + (B (a • x)) y - (a * a) • Q x - Q y = a • (Q x + Q y + (B x) y - Q x - Q y)