mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.Multilinear.Basic

Goal state before:
pp: R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : CommSemiring R
inst✝⁴ : (i : ι) → AddCommMonoid (M₁ i)
inst✝³ : AddCommMonoid M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : DecidableEq ι
c : ι → R
m : (i : ι) → M₁ i
s✝ : Finset ι
j : ι
s : Finset ι
j_notMem_s : j ∉ s
Hrec : f (s.piecewise (fun i ↦ c i • m i) m) = (∏ i ∈ s, c i) • f m
A : update (s.piecewise (fun i ↦ c i • m i) m) j (m j) = s.piecewise (fun i ↦ c i • m i) m
⊢ c j • (∏ i ∈ s, c i) • f m = (∏ i ∈ insert j s, c i) • f m

Tactic:
]

Goal state after:
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : CommSemiring R
inst✝⁴ : (i : ι) → AddCommMonoid (M₁ i)
inst✝³ : AddCommMonoid M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : DecidableEq ι
c : ι → R
m : (i : ι) → M₁ i
s✝ : Finset ι
j : ι
s : Finset ι
j_notMem_s : j ∉ s
Hrec : f (s.piecewise (fun i ↦ c i • m i) m) = (∏ i ∈ s, c i) • f m
A : update (s.piecewise (fun i ↦ c i • m i) m) j (m j) = s.piecewise (fun i ↦ c i • m i) m
⊢ c j • (∏ i ∈ s, c i) • f m = (∏ i ∈ insert j s, c i) • f m
Module: Mathlib.LinearAlgebra.Matrix.Determinant.Basic

Goal state before:
pp: m : Type u_1
n : Type u_2
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
R : Type v
inst✝ : CommRing R
A : Matrix m m R
C : Matrix n m R
D : Matrix n n R
⊢ A.det * D.det = A.det * D.det

Tactic:
]

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

Goal state before:
pp: R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
⊢ ∑ σ ∈ Finset.map decomposeFin.symm.toEmbedding (univ ×ˢ univ), sign σ • ∏ i, A (σ i) i =
    ∑ i, (-1) ^ ↑i * A i 0 * (A.submatrix i.succAbove Fin.succ).det

Tactic:
]

Goal state after:
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
⊢ ∑ σ ∈ Finset.map decomposeFin.symm.toEmbedding (univ ×ˢ univ), sign σ • ∏ i, A (σ i) i =
    ∑ i, (-1) ^ ↑i * A i 0 * (A.submatrix i.succAbove Fin.succ).det
Module: Mathlib.Algebra.Category.MonCat.FilteredColimits

Goal state before:
pp: J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ CommMonCat
i : J
x : ↑((F ⋙ forget₂ CommMonCat MonCat).obj i)
j : J
y : ↑((F ⋙ forget₂ CommMonCat MonCat).obj j)
k : J := IsFiltered.max i j
f : i ⟶ IsFiltered.max i j := IsFiltered.leftToMax i j
g : j ⟶ IsFiltered.max i j := IsFiltered.rightToMax i j
⊢ MonCat.FilteredColimits.M.mk (F ⋙ forget₂ CommMonCat MonCat)
      ⟨k,
        (ConcreteCategory.hom ((F ⋙ forget₂ CommMonCat MonCat).map f)) ⟨i, x⟩.snd *
          (ConcreteCategory.hom ((F ⋙ forget₂ CommMonCat MonCat).map g)) ⟨j, y⟩.snd⟩ =
    MonCat.FilteredColimits.M.mk (F ⋙ forget₂ CommMonCat MonCat)
      ⟨k,
        (ConcreteCategory.hom ((F ⋙ forget₂ CommMonCat MonCat).map g)) ⟨j, y⟩.snd *
          (ConcreteCategory.hom ((F ⋙ forget₂ CommMonCat MonCat).map f)) ⟨i, x⟩.snd⟩

Tactic:
]

Goal state after:
J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ CommMonCat
i : J
x : ↑((F ⋙ forget₂ CommMonCat MonCat).obj i)
j : J
y : ↑((F ⋙ forget₂ CommMonCat MonCat).obj j)
k : J := IsFiltered.max i j
f : i ⟶ IsFiltered.max i j := IsFiltered.leftToMax i j
g : j ⟶ IsFiltered.max i j := IsFiltered.rightToMax i j
⊢ MonCat.FilteredColimits.M.mk (F ⋙ forget₂ CommMonCat MonCat)
      ⟨k,
        (ConcreteCategory.hom ((F ⋙ forget₂ CommMonCat MonCat).map f)) ⟨i, x⟩.snd *
          (ConcreteCategory.hom ((F ⋙ forget₂ CommMonCat MonCat).map g)) ⟨j, y⟩.snd⟩ =
    MonCat.FilteredColimits.M.mk (F ⋙ forget₂ CommMonCat MonCat)
      ⟨k,
        (ConcreteCategory.hom ((F ⋙ forget₂ CommMonCat MonCat).map g)) ⟨j, y⟩.snd *
          (ConcreteCategory.hom ((F ⋙ forget₂ CommMonCat MonCat).map f)) ⟨i, x⟩.snd⟩
Module: Mathlib.Algebra.Category.MonCat.FilteredColimits

Goal state before:
pp: J : Type v
inst✝¹ : SmallCategory J
inst✝ : IsFiltered J
F : J ⥤ CommMonCat
i : J
x : ↑((F ⋙ forget₂ CommMonCat MonCat).obj i)
j : J
y : ↑((F ⋙ forget₂ CommMonCat MonCat).obj j)
k : J := IsFiltered.max i j
f : i ⟶ IsFiltered.max i j := IsFiltered.leftToMax i j
g : j ⟶ IsFiltered.max i j := IsFiltered.rightToMax i j
⊢ MonCat.FilteredColimits.M.mk (F ⋙ forget₂ CommMonCat MonCat) ⟨k, (Hom.hom (F.map g)) y * (Hom.hom (F.map f)) x⟩ =
    MonCat.FilteredColimits.M.mk (F ⋙ forget₂ CommMonCat MonCat) ⟨k, (Hom.hom (F.map g)) y * (Hom.hom (F.map f)) x⟩

Tactic:
]

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

Goal state before:
pp: R : Type uR
ι : Type uι
M₂ : Type v₂
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M₂
inst✝ : Fintype ι
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: R : Type uR
ι : Type uι
M₂ : Type v₂
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M₂
inst✝¹ : Module R M₂
inst✝ : Fintype ι
z : M₂
⊢ z = 0 ↔ z = 0

Tactic:
]

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

Goal state before:
pp: case refine_2
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i✝ : Fin n.succ
x✝ : i✝ ∈ univ
i : Fin n
this : (-1) ^ ↑i = ↑↑(sign i.cycleRange)
⊢ ∑ y, sign (decomposeFin.symm (i.succ, y)) • ∏ x, A ((decomposeFin.symm (i.succ, y)) x) x =
    ∑ i_1,
      -1 *
        (A i.succ 0 *
          sign i_1 • ∏ i_2, (of fun i_3 j ↦ A (i.succ.succAbove i_3) j.succ).submatrix (⇑i.cycleRange) id (i_1 i_2) i_2)

Tactic:
]

Goal state after:
case refine_2
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i✝ : Fin n.succ
x✝ : i✝ ∈ univ
i : Fin n
this : (-1) ^ ↑i = ↑↑(sign i.cycleRange)
⊢ ∑ y, sign (decomposeFin.symm (i.succ, y)) • ∏ x, A ((decomposeFin.symm (i.succ, y)) x) x =
    ∑ i_1,
      -1 *
        (A i.succ 0 *
          sign i_1 • ∏ i_2, (of fun i_3 j ↦ A (i.succ.succAbove i_3) j.succ).submatrix (⇑i.cycleRange) id (i_1 i_2) i_2)
Module: Mathlib.LinearAlgebra.Multilinear.Basic

Goal state before:
pp: R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → AddCommGroup (M₁ i)
inst✝³ : AddCommGroup M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : DecidableEq ι
m : (i : ι) → M₁ i
i : ι
x : M₁ i
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → AddCommGroup (M₁ i)
inst✝³ : AddCommGroup M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : DecidableEq ι
m : (i : ι) → M₁ i
i : ι
x y : M₁ i
⊢ f (update m i x) + -f (update m i y) = f (update m i x) + -f (update m i y)

Tactic:
]

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

Goal state before:
pp: case refine_2
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i✝ : Fin n.succ
x✝¹ : i✝ ∈ univ
i : Fin n
this : (-1) ^ ↑i = ↑↑(sign i.cycleRange)
σ : Perm (Fin n)
x✝ : σ ∈ univ
⊢ (-1 * sign σ) • ∏ x, A ((decomposeFin.symm (i.succ, σ)) x) x =
    -1 *
      (A i.succ 0 *
        sign σ • ∏ i_1, (of fun i_2 j ↦ A (i.succ.succAbove i_2) j.succ).submatrix (⇑i.cycleRange) id (σ i_1) i_1)

Tactic:
]

Goal state after:
case refine_2
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i✝ : Fin n.succ
x✝¹ : i✝ ∈ univ
i : Fin n
this : (-1) ^ ↑i = ↑↑(sign i.cycleRange)
σ : Perm (Fin n)
x✝ : σ ∈ univ
⊢ (-1 * sign σ) • ∏ x, A ((decomposeFin.symm (i.succ, σ)) x) x =
    -1 *
      (A i.succ 0 *
        sign σ • ∏ i_1, (of fun i_2 j ↦ A (i.succ.succAbove i_2) j.succ).submatrix (⇑i.cycleRange) id (σ i_1) i_1)
Module: Mathlib.LinearAlgebra.Multilinear.Basic

Goal state before:
pp: R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → AddCommGroup (M₁ i)
inst✝³ : AddCommGroup M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : DecidableEq ι
x : (i : ι) → M₁ i
i : ι
v : M₁ i
⊢ f (update x i v) = f (update x i v)

Tactic:
]

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

Goal state before:
pp: case empty
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → AddCommGroup (M₁ i)
inst✝³ : AddCommGroup M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : LinearOrder ι
a b : (i : ι) → M₁ i
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
⊢ ∑ i, (-1) ^ ↑i * Aᵀ i 0 * (Aᵀ.submatrix i.succAbove Fin.succ).det =
    ∑ j, (-1) ^ ↑j * A 0 j * (A.submatrix Fin.succ j.succAbove).det

Tactic:
]

Goal state after:
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
⊢ ∑ i, (-1) ^ ↑i * Aᵀ i 0 * (Aᵀ.submatrix i.succAbove Fin.succ).det =
    ∑ j, (-1) ^ ↑j * A 0 j * (A.submatrix Fin.succ j.succAbove).det
Module: Mathlib.Algebra.Category.Ring.FilteredColimits

Goal state before:
pp: J : Type v
inst✝¹ : SmallCategory J
F : J ⥤ SemiRingCat
inst✝ : IsFiltered J
j₁ : J
x : ((F ⋙ forget₂ SemiRingCat MonCat) ⋙ forget MonCat).obj j₁
j₂ : J
y : ((F ⋙ forget₂ SemiRingCat MonCat) ⋙ forget MonCat).obj j₂
j₃ : J
z : ((F ⋙ forget₂ SemiRingCat MonCat) ⋙ forget MonCat).obj j₃
k : J := IsFiltered.max₃ j₁ j₂ j₃
f : j₁ ⟶ IsFiltered.max₃ j₁ j₂ j₃ := IsFiltered.firstToMax₃ j₁ j₂ j₃
g : j₂ ⟶ IsFiltered.max₃ j₁ j₂ j₃ := IsFiltered.secondToMax₃ j₁ j₂ j₃
h : j₃ ⟶ IsFiltered.max₃ j₁ j₂ j₃ := IsFiltered.thirdToMax₃ j₁ j₂ j₃
⊢ MonCat.FilteredColimits.M.mk (F ⋙ forget₂ SemiRingCat MonCat)
      ⟨k,
        (ConcreteCategory.hom (F.map f)) x * (ConcreteCategory.hom (F.map g)) y +
          (ConcreteCategory.hom (F.map f)) x * (ConcreteCategory.hom (F.map h)) z⟩ =
    AddMonCat.FilteredColimits.M.mk ((F ⋙ forget₂ SemiRingCat AddCommMonCat) ⋙ forget₂ AddCommMonCat AddMonCat)
      ⟨k,
        (ConcreteCategory.hom (𝟙 (((F ⋙ forget₂ SemiRingCat AddCommMonCat) ⋙ forget₂ AddCommMonCat AddMonCat).obj k)))
            ((ConcreteCategory.hom ((F ⋙ forget₂ SemiRingCat MonCat).map f)) x *
              (ConcreteCategory.hom ((F ⋙ forget₂ SemiRingCat MonCat).map g)) y) +
          (ConcreteCategory.hom (𝟙 (((F ⋙ forget₂ SemiRingCat AddCommMonCat) ⋙ forget₂ AddCommMonCat AddMonCat).obj k)))
            ((ConcreteCategory.hom ((F ⋙ forget₂ SemiRingCat MonCat).map f)) x *
              (ConcreteCategory.hom ((F ⋙ forget₂ SemiRingCat MonCat).map h)) z)⟩

Tactic:
]

Goal state after:
J : Type v
inst✝¹ : SmallCategory J
F : J ⥤ SemiRingCat
inst✝ : IsFiltered J
j₁ : J
x : ((F ⋙ forget₂ SemiRingCat MonCat) ⋙ forget MonCat).obj j₁
j₂ : J
y : ((F ⋙ forget₂ SemiRingCat MonCat) ⋙ forget MonCat).obj j₂
j₃ : J
z : ((F ⋙ forget₂ SemiRingCat MonCat) ⋙ forget MonCat).obj j₃
k : J := IsFiltered.max₃ j₁ j₂ j₃
f : j₁ ⟶ IsFiltered.max₃ j₁ j₂ j₃ := IsFiltered.firstToMax₃ j₁ j₂ j₃
g : j₂ ⟶ IsFiltered.max₃ j₁ j₂ j₃ := IsFiltered.secondToMax₃ j₁ j₂ j₃
h : j₃ ⟶ IsFiltered.max₃ j₁ j₂ j₃ := IsFiltered.thirdToMax₃ j₁ j₂ j₃
⊢ MonCat.FilteredColimits.M.mk (F ⋙ forget₂ SemiRingCat MonCat)
      ⟨k,
        (ConcreteCategory.hom (F.map f)) x * (ConcreteCategory.hom (F.map g)) y +
          (ConcreteCategory.hom (F.map f)) x * (ConcreteCategory.hom (F.map h)) z⟩ =
    AddMonCat.FilteredColimits.M.mk ((F ⋙ forget₂ SemiRingCat AddCommMonCat) ⋙ forget₂ AddCommMonCat AddMonCat)
      ⟨k,
        (ConcreteCategory.hom (𝟙 (((F ⋙ forget₂ SemiRingCat AddCommMonCat) ⋙ forget₂ AddCommMonCat AddMonCat).obj k)))
            ((ConcreteCategory.hom ((F ⋙ forget₂ SemiRingCat MonCat).map f)) x *
              (ConcreteCategory.hom ((F ⋙ forget₂ SemiRingCat MonCat).map g)) y) +
          (ConcreteCategory.hom (𝟙 (((F ⋙ forget₂ SemiRingCat AddCommMonCat) ⋙ forget₂ AddCommMonCat AddMonCat).obj k)))
            ((ConcreteCategory.hom ((F ⋙ forget₂ SemiRingCat MonCat).map f)) x *
              (ConcreteCategory.hom ((F ⋙ forget₂ SemiRingCat MonCat).map h)) z)⟩
Module: Mathlib.LinearAlgebra.Multilinear.Basic

Goal state before:
pp: case insert
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → AddCommGroup (M₁ i)
inst✝³ : AddCommGroup M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : LinearOrder ι
a b : (i : ι) → M₁ i
k : ι
s : Finset ι
hk : ∀ x ∈ s, k < x
ih : f a - f (s.piecewise b a) = ∑ i ∈ s, f fun j ↦ if j ∈ s → j < i then a j else if i = j then a j - b j else b j
⊢ (f (update (s.piecewise b a) k (s.piecewise b a k - b k)) +
      ∑ i ∈ s, f fun j ↦ if j ∈ s → j < i then a j else if i = j then a j - b j else b j) =
    (f fun j ↦ if j ∈ insert k s → j < k then a j else if k = j then a j - b j else b j) +
      ∑ x ∈ s, f fun j ↦ if j ∈ insert k s → j < x then a j else if x = j then a j - b j else b j

Tactic:
]

Goal state after:
case insert
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → AddCommGroup (M₁ i)
inst✝³ : AddCommGroup M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : LinearOrder ι
a b : (i : ι) → M₁ i
k : ι
s : Finset ι
hk : ∀ x ∈ s, k < x
ih : f a - f (s.piecewise b a) = ∑ i ∈ s, f fun j ↦ if j ∈ s → j < i then a j else if i = j then a j - b j else b j
⊢ (f (update (s.piecewise b a) k (s.piecewise b a k - b k)) +
      ∑ i ∈ s, f fun j ↦ if j ∈ s → j < i then a j else if i = j then a j - b j else b j) =
    (f fun j ↦ if j ∈ insert k s → j < k then a j else if k = j then a j - b j else b j) +
      ∑ x ∈ s, f fun j ↦ if j ∈ insert k s → j < x then a j else if x = j then a j - b j else b j
Module: Mathlib.LinearAlgebra.Matrix.Determinant.Basic

Goal state before:
pp: R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i : Fin n.succ
x✝ : i ∈ univ
⊢ (-1) ^ ↑i * Aᵀ i 0 * (Aᵀ.submatrix i.succAbove Fin.succ)ᵀ.det =
    (-1) ^ ↑i * A 0 i * (A.submatrix Fin.succ i.succAbove).det

Tactic:
]

Goal state after:
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i : Fin n.succ
x✝ : i ∈ univ
⊢ (-1) ^ ↑i * Aᵀ i 0 * (Aᵀ.submatrix i.succAbove Fin.succ)ᵀ.det =
    (-1) ^ ↑i * A 0 i * (A.submatrix Fin.succ i.succAbove).det
Module: Mathlib.CategoryTheory.Limits.Types.Images

Goal state before:
pp: α β : Type u
f : α ⟶ β
F' : MonoFactorisation f
x : Image f
⊢ ↑x = (ConcreteCategory.hom (ι f)).toFun x

Tactic:
]

Goal state after:
α β : Type u
f : α ⟶ β
F' : MonoFactorisation f
x : Image f
⊢ ↑x = (ConcreteCategory.hom (ι f)).toFun x
Module: Mathlib.CategoryTheory.Limits.Types.Images

Goal state before:
pp: α β : Type u
f✝ : α ⟶ β
f g : Arrow (Type u)
st : f ⟶ g
x : (monoFactorisation f.hom).I
⊢ (ConcreteCategory.hom (Arrow.Hom.right st)) ((ConcreteCategory.hom f.hom) (Classical.choose ⋯)) =
    (ConcreteCategory.hom (Arrow.Hom.right st)) ↑x

Tactic:
]

Goal state after:
α β : Type u
f✝ : α ⟶ β
f g : Arrow (Type u)
st : f ⟶ g
x : (monoFactorisation f.hom).I
⊢ (ConcreteCategory.hom (Arrow.Hom.right st)) ((ConcreteCategory.hom f.hom) (Classical.choose ⋯)) =
    (ConcreteCategory.hom (Arrow.Hom.right st)) ↑x
Module: Mathlib.CategoryTheory.Limits.Types.Images

Goal state before:
pp: α β : Type u
f✝ : α ⟶ β
f g : Arrow (Type u)
st : f ⟶ g
x : (monoFactorisation f.hom).I
⊢ (ConcreteCategory.hom (Arrow.Hom.right st)) ↑x = (ConcreteCategory.hom (Arrow.Hom.right st)) ↑x

Tactic:
]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.Images

Goal state before:
pp: case refl
F : ℕᵒᵖ ⥤ Type u
hF : ∀ (n : ℕ), Function.Surjective ⇑(ConcreteCategory.hom (F.map (homOfLE ⋯).op))
a : F.obj (Opposite.op 0)
n m : ℕ
⊢ (fun x ↦
        match x with
        | Opposite.op n => preimage hF a n)
      (Opposite.op m) =
    (fun x ↦
        match x with
        | Opposite.op n => preimage hF a n)
      (Opposite.op m)

Tactic:
]

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

Goal state before:
pp: R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i : Fin n.succ
this : A.det = (-1) ^ ↑i * ↑↑(sign i.cycleRange⁻¹) * A.det
⊢ (-1) ^ ↑i * (↑↑(sign i.cycleRange⁻¹) * A.det) =
    (-1) ^ ↑i * ∑ i_1, (-1) ^ ↑i_1 * (A i i_1 * (A.submatrix i.succAbove i_1.succAbove).det)

Tactic:
]

Goal state after:
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i : Fin n.succ
this : A.det = (-1) ^ ↑i * ↑↑(sign i.cycleRange⁻¹) * A.det
⊢ (-1) ^ ↑i * (↑↑(sign i.cycleRange⁻¹) * A.det) =
    (-1) ^ ↑i * ∑ i_1, (-1) ^ ↑i_1 * (A i i_1 * (A.submatrix i.succAbove i_1.succAbove).det)
Module: Mathlib.CategoryTheory.Limits.Types.Images

Goal state before:
pp: case step
F : ℕᵒᵖ ⥤ Type u
hF : ∀ (n : ℕ), Function.Surjective ⇑(ConcreteCategory.hom (F.map (homOfLE ⋯).op))
a : F.obj (Opposite.op 0)
n m p : ℕ
h : m.le p
ih :
  (ConcreteCategory.hom (F.map (Opposite.op { down := { down := h } })))
      ((fun x ↦
          match x with
          | Opposite.op n => preimage hF a n)
        (Opposite.op p)) =
    (fun x ↦
        match x with
        | Opposite.op n => preimage hF a n)
      (Opposite.op m)
⊢ (ConcreteCategory.hom (F.map (Opposite.op { down := { down := ⋯ } })))
      ((fun x ↦
          match x with
          | Opposite.op n => preimage hF a n)
        (Opposite.op p.succ)) =
    (ConcreteCategory.hom (F.map (Opposite.op { down := { down := h } })))
      ((fun x ↦
          match x with
          | Opposite.op n => preimage hF a n)
        (Opposite.op p))

Tactic:
]

Goal state after:
case step
F : ℕᵒᵖ ⥤ Type u
hF : ∀ (n : ℕ), Function.Surjective ⇑(ConcreteCategory.hom (F.map (homOfLE ⋯).op))
a : F.obj (Opposite.op 0)
n m p : ℕ
h : m.le p
ih :
  (ConcreteCategory.hom (F.map (Opposite.op { down := { down := h } })))
      ((fun x ↦
          match x with
          | Opposite.op n => preimage hF a n)
        (Opposite.op p)) =
    (fun x ↦
        match x with
        | Opposite.op n => preimage hF a n)
      (Opposite.op m)
⊢ (ConcreteCategory.hom (F.map (Opposite.op { down := { down := ⋯ } })))
      ((fun x ↦
          match x with
          | Opposite.op n => preimage hF a n)
        (Opposite.op p.succ)) =
    (ConcreteCategory.hom (F.map (Opposite.op { down := { down := h } })))
      ((fun x ↦
          match x with
          | Opposite.op n => preimage hF a n)
        (Opposite.op p))
Module: Mathlib.LinearAlgebra.Matrix.Determinant.Basic

Goal state before:
pp: case e_a
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i : Fin n.succ
this : A.det = (-1) ^ ↑i * ↑↑(sign i.cycleRange⁻¹) * A.det
⊢ ∑ j,
      (-1) ^ ↑j * A.submatrix (⇑i.cycleRange⁻¹) id 0 j *
        ((A.submatrix (⇑i.cycleRange⁻¹) id).submatrix Fin.succ j.succAbove).det =
    ∑ i_1, (-1) ^ ↑i_1 * (A i i_1 * (A.submatrix i.succAbove i_1.succAbove).det)

Tactic:
]

Goal state after:
case e_a
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i : Fin n.succ
this : A.det = (-1) ^ ↑i * ↑↑(sign i.cycleRange⁻¹) * A.det
⊢ ∑ j,
      (-1) ^ ↑j * A.submatrix (⇑i.cycleRange⁻¹) id 0 j *
        ((A.submatrix (⇑i.cycleRange⁻¹) id).submatrix Fin.succ j.succAbove).det =
    ∑ i_1, (-1) ^ ↑i_1 * (A i i_1 * (A.submatrix i.succAbove i_1.succAbove).det)
Module: Mathlib.CategoryTheory.Limits.Types.Images

Goal state before:
pp: case step
F : ℕᵒᵖ ⥤ Type u
hF : ∀ (n : ℕ), Function.Surjective ⇑(ConcreteCategory.hom (F.map (homOfLE ⋯).op))
a : F.obj (Opposite.op 0)
n m p : ℕ
h : m.le p
ih :
  (ConcreteCategory.hom (F.map (Opposite.op { down := { down := h } })))
      ((fun x ↦
          match x with
          | Opposite.op n => preimage hF a n)
        (Opposite.op p)) =
    (fun x ↦
        match x with
        | Opposite.op n => preimage hF a n)
      (Opposite.op m)
h' : m ≤ p
⊢ (ConcreteCategory.hom (F.map (homOfLE h').op)) (preimage hF a p) =
    (ConcreteCategory.hom (F.map (Opposite.op { down := { down := h } })))
      ((fun x ↦
          match x with
          | Opposite.op n => preimage hF a n)
        (Opposite.op p))

Tactic:
]

Goal state after:
case step
F : ℕᵒᵖ ⥤ Type u
hF : ∀ (n : ℕ), Function.Surjective ⇑(ConcreteCategory.hom (F.map (homOfLE ⋯).op))
a : F.obj (Opposite.op 0)
n m p : ℕ
h : m.le p
ih :
  (ConcreteCategory.hom (F.map (Opposite.op { down := { down := h } })))
      ((fun x ↦
          match x with
          | Opposite.op n => preimage hF a n)
        (Opposite.op p)) =
    (fun x ↦
        match x with
        | Opposite.op n => preimage hF a n)
      (Opposite.op m)
h' : m ≤ p
⊢ (ConcreteCategory.hom (F.map (homOfLE h').op)) (preimage hF a p) =
    (ConcreteCategory.hom (F.map (Opposite.op { down := { down := h } })))
      ((fun x ↦
          match x with
          | Opposite.op n => preimage hF a n)
        (Opposite.op p))
Module: Mathlib.LinearAlgebra.Matrix.Determinant.Basic

Goal state before:
pp: case e_a
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i : Fin n.succ
this : A.det = (-1) ^ ↑i * ↑↑(sign i.cycleRange⁻¹) * A.det
j : Fin n.succ
x✝ : j ∈ univ
⊢ (-1) ^ ↑j * (A (i.cycleRange⁻¹ 0) j * (A.submatrix (fun x ↦ i.cycleRange⁻¹ x.succ) j.succAbove).det) =
    (-1) ^ ↑j * (A i j * (A.submatrix i.succAbove j.succAbove).det)

Tactic:
]

Goal state after:
case e_a
R : Type v
inst✝ : CommRing R
n : ℕ
A : Matrix (Fin n.succ) (Fin n.succ) R
i : Fin n.succ
this : A.det = (-1) ^ ↑i * ↑↑(sign i.cycleRange⁻¹) * A.det
j : Fin n.succ
x✝ : j ∈ univ
⊢ (-1) ^ ↑j * (A (i.cycleRange⁻¹ 0) j * (A.submatrix (fun x ↦ i.cycleRange⁻¹ x.succ) j.succAbove).det) =
    (-1) ^ ↑j * (A i j * (A.submatrix i.succAbove j.succAbove).det)
Module: Mathlib.LinearAlgebra.Multilinear.Basic

Goal state before:
pp: case pos
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → AddCommGroup (M₁ i)
inst✝³ : AddCommGroup M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : LinearOrder ι
a b : (i : ι) → M₁ i
k : ι
s : Finset ι
hk : ∀ x ∈ s, k < x
ih : f a - f (s.piecewise b a) = ∑ i ∈ s, f fun j ↦ if j ∈ s → j < i then a j else if i = j then a j - b j else b j
i : ι
h₁ : i = k ∨ i ∈ s → i < k
⊢ a i = a i

pp: case pos.hi
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → AddCommGroup (M₁ i)
inst✝³ : AddCommGroup M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : LinearOrder ι
a b : (i : ι) → M₁ i
k : ι
s : Finset ι
hk : ∀ x ∈ s, k < x
ih : f a - f (s.piecewise b a) = ∑ i ∈ s, f fun j ↦ if j ∈ s → j < i then a j else if i = j then a j - b j else b j
i : ι
h₁ : i = k ∨ i ∈ s → i < k
⊢ i ∉ s

pp: case pos.h
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → AddCommGroup (M₁ i)
inst✝³ : AddCommGroup M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : LinearOrder ι
a b : (i : ι) → M₁ i
k : ι
s : Finset ι
hk : ∀ x ∈ s, k < x
ih : f a - f (s.piecewise b a) = ∑ i ∈ s, f fun j ↦ if j ∈ s → j < i then a j else if i = j then a j - b j else b j
i : ι
h₁ : i = k ∨ i ∈ s → i < k
⊢ i ≠ k

Tactic:
]

Goal state after:
case pos.hi
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → AddCommGroup (M₁ i)
inst✝³ : AddCommGroup M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : LinearOrder ι
a b : (i : ι) → M₁ i
k : ι
s : Finset ι
hk : ∀ x ∈ s, k < x
ih : f a - f (s.piecewise b a) = ∑ i ∈ s, f fun j ↦ if j ∈ s → j < i then a j else if i = j then a j - b j else b j
i : ι
h₁ : i = k ∨ i ∈ s → i < k
⊢ i ∉ s

case pos.h
R : Type uR
ι : Type uι
M₁ : ι → Type v₁
M₂ : Type v₂
inst✝⁵ : Semiring R
inst✝⁴ : (i : ι) → AddCommGroup (M₁ i)
inst✝³ : AddCommGroup M₂
inst✝² : (i : ι) → Module R (M₁ i)
inst✝¹ : Module R M₂
f : MultilinearMap R M₁ M₂
inst✝ : LinearOrder ι
a b : (i : ι) → M₁ i
k : ι
s : Finset ι
hk : ∀ x ∈ s, k < x
ih : f a - f (s.piecewise b a) = ∑ i ∈ s, f fun j ↦ if j ∈ s → j < i then a j else if i = j then a j - b j else b j
i : ι
h₁ : i = k ∨ i ∈ s → i < k
⊢ i ≠ k