mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.IntegralClosure.Algebra.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : IsDomain A
M : Type u_5
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module A M
inst✝¹ : IsScalarTower R A M
inst✝ : Module.IsTorsionFree A M
N : Submodule R M
hN : N ≠ ⊥
hN' : N.FG
x : A
hx : ∀ n ∈ N, x • n ∈ N
A' : Subalgebra R A :=
  { carrier := {x | ∀ n ∈ N, x • n ∈ N}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯,
    algebraMap_mem' := ⋯ }
f : ↥A' →ₐ[R] Module.End R ↥N :=
  AlgHom.ofLinearMap { toFun := fun x ↦ (DistribSMul.toLinearMap R M x).restrict ⋯, map_add' := ⋯, map_smul' := ⋯ } ⋯ ⋯
a : M
ha₁ : a ∈ N
ha₂ : a ≠ 0
⊢ f.toLinearMap.ker ≤ ⊥

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : IsDomain A
M : Type u_5
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module A M
inst✝¹ : IsScalarTower R A M
inst✝ : Module.IsTorsionFree A M
N : Submodule R M
hN : N ≠ ⊥
hN' : N.FG
x : A
hx : ∀ n ∈ N, x • n ∈ N
A' : Subalgebra R A :=
  { carrier := {x | ∀ n ∈ N, x • n ∈ N}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯,
    algebraMap_mem' := ⋯ }
f : ↥A' →ₐ[R] Module.End R ↥N :=
  AlgHom.ofLinearMap { toFun := fun x ↦ (DistribSMul.toLinearMap R M x).restrict ⋯, map_add' := ⋯, map_smul' := ⋯ } ⋯ ⋯
a : M
ha₁ : a ∈ N
ha₂ : a ≠ 0
⊢ f.toLinearMap.ker ≤ ⊥
Module: Mathlib.RingTheory.IntegralClosure.Algebra.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : IsDomain A
M : Type u_5
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module A M
inst✝¹ : IsScalarTower R A M
inst✝ : Module.IsTorsionFree A M
N : Submodule R M
hN : N ≠ ⊥
hN' : N.FG
x : A
hx : ∀ n ∈ N, x • n ∈ N
A' : Subalgebra R A :=
  { carrier := {x | ∀ n ∈ N, x • n ∈ N}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯,
    algebraMap_mem' := ⋯ }
f : ↥A' →ₐ[R] Module.End R ↥N :=
  AlgHom.ofLinearMap { toFun := fun x ↦ (DistribSMul.toLinearMap R M x).restrict ⋯, map_add' := ⋯, map_smul' := ⋯ } ⋯ ⋯
a : M
ha₁ : a ∈ N
ha₂ : a ≠ 0
this : Function.Injective ⇑f
⊢ IsIntegral R (f ⟨x, hx⟩)

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : IsDomain A
M : Type u_5
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module A M
inst✝¹ : IsScalarTower R A M
inst✝ : Module.IsTorsionFree A M
N : Submodule R M
hN : N ≠ ⊥
hN' : N.FG
x : A
hx : ∀ n ∈ N, x • n ∈ N
A' : Subalgebra R A :=
  { carrier := {x | ∀ n ∈ N, x • n ∈ N}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯,
    algebraMap_mem' := ⋯ }
f : ↥A' →ₐ[R] Module.End R ↥N :=
  AlgHom.ofLinearMap { toFun := fun x ↦ (DistribSMul.toLinearMap R M x).restrict ⋯, map_add' := ⋯, map_smul' := ⋯ } ⋯ ⋯
a : M
ha₁ : a ∈ N
ha₂ : a ≠ 0
this : Function.Injective ⇑f
⊢ IsIntegral R (f ⟨x, hx⟩)
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det
q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det
ht : t_inv * t = 1
hp : toLaurentAlg M.charpoly = p
hq : toLaurentAlg M.charpolyRev = q
this : t_inv ^ Fintype.card n * p = invert q
⊢ T (-1) ^ Fintype.card n * p = invert q

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det
q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det
ht : t_inv * t = 1
hp : toLaurentAlg M.charpoly = p
hq : toLaurentAlg M.charpolyRev = q
this : t_inv ^ Fintype.card n * p = invert q
⊢ T (-1) ^ Fintype.card n * p = invert q
Module: Mathlib.RingTheory.IntegralClosure.Algebra.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : IsDomain A
M : Type u_5
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module A M
inst✝¹ : IsScalarTower R A M
inst✝ : Module.IsTorsionFree A M
N : Submodule R M
hN : N ≠ ⊥
hN' : N.FG
x : A
hx : ∀ n ∈ N, x • n ∈ N
A' : Subalgebra R A :=
  { carrier := {x | ∀ n ∈ N, x • n ∈ N}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯,
    algebraMap_mem' := ⋯ }
f : ↥A' →ₐ[R] Module.End R ↥N :=
  AlgHom.ofLinearMap { toFun := fun x ↦ (DistribSMul.toLinearMap R M x).restrict ⋯, map_add' := ⋯, map_smul' := ⋯ } ⋯ ⋯
a : M
ha₁ : a ∈ N
ha₂ : a ≠ 0
this : Function.Injective ⇑f
⊢ N.FG

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝⁸ : CommRing R
inst✝⁷ : CommRing A
inst✝⁶ : Algebra R A
inst✝⁵ : IsDomain A
M : Type u_5
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : Module A M
inst✝¹ : IsScalarTower R A M
inst✝ : Module.IsTorsionFree A M
N : Submodule R M
hN : N ≠ ⊥
hN' : N.FG
x : A
hx : ∀ n ∈ N, x • n ∈ N
A' : Subalgebra R A :=
  { carrier := {x | ∀ n ∈ N, x • n ∈ N}, mul_mem' := ⋯, one_mem' := ⋯, add_mem' := ⋯, zero_mem' := ⋯,
    algebraMap_mem' := ⋯ }
f : ↥A' →ₐ[R] Module.End R ↥N :=
  AlgHom.ofLinearMap { toFun := fun x ↦ (DistribSMul.toLinearMap R M x).restrict ⋯, map_add' := ⋯, map_smul' := ⋯ } ⋯ ⋯
a : M
ha₁ : a ∈ N
ha₂ : a ≠ 0
this : Function.Injective ⇑f
⊢ N.FG
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det
q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det
ht : t_inv * t = 1
hp : toLaurentAlg M.charpoly = p
hq : toLaurentAlg M.charpolyRev = q
⊢ (1 - t_inv • M.map ⇑LaurentPolynomial.C).det = (invert.mapMatrix (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C)).det

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
a✝ : Nontrivial R
t : R[T;T⁻¹] := T 1
t_inv : R[T;T⁻¹] := T (-1)
p : R[T;T⁻¹] := ((scalar n) t - M.map ⇑LaurentPolynomial.C).det
q : R[T;T⁻¹] := (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C).det
ht : t_inv * t = 1
hp : toLaurentAlg M.charpoly = p
hq : toLaurentAlg M.charpolyRev = q
⊢ (1 - t_inv • M.map ⇑LaurentPolynomial.C).det = (invert.mapMatrix (1 - (scalar n) t * M.map ⇑LaurentPolynomial.C)).det
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
A : Matrix n n R
h : IsUnit A
this : Invertible A
⊢ C (A⁻¹.det * A.det) * ((scalar n) X - C.mapMatrix A⁻¹).det =
    C A⁻¹.det * C A.det * ((scalar n) X - C.mapMatrix A⁻¹).det

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
A : Matrix n n R
h : IsUnit A
this : Invertible A
⊢ C (A⁻¹.det * A.det) * ((scalar n) X - C.mapMatrix A⁻¹).det =
    C A⁻¹.det * C A.det * ((scalar n) X - C.mapMatrix A⁻¹).det
Module: Mathlib.RingTheory.IntegralClosure.Algebra.Basic

Goal state before:
pp: R : Type u_1
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
x y z : S
hx : f.IsIntegralElem x
hy : f.IsIntegralElem y
hz : z ∈ Subring.closure {x, y}
this✝ : Algebra R S := f.toAlgebra
this : (Subalgebra.toSubmodule R[x, y]).FG
⊢ f.IsIntegralElem z

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_4
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
x y z : S
hx : f.IsIntegralElem x
hy : f.IsIntegralElem y
hz : z ∈ Subring.closure {x, y}
this✝ : Algebra R S := f.toAlgebra
this : (Subalgebra.toSubmodule R[x, y]).FG
⊢ f.IsIntegralElem z
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
A : Matrix n n R
h : IsUnit A
this : Invertible A
⊢ C A⁻¹.det * (C.mapMatrix A * (scalar n) X - C.mapMatrix (A * A⁻¹)).det =
    C A⁻¹.det * (C.mapMatrix A * (scalar n) X - 1).det

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
A : Matrix n n R
h : IsUnit A
this : Invertible A
⊢ C A⁻¹.det * (C.mapMatrix A * (scalar n) X - C.mapMatrix (A * A⁻¹)).det =
    C A⁻¹.det * (C.mapMatrix A * (scalar n) X - 1).det
Module: Mathlib.RingTheory.IntegralClosure.Algebra.Basic

Goal state before:
pp: B : Type u_3
S : Type u_4
inst✝⁶ : Ring B
inst✝⁵ : CommRing S
R : Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : Algebra R B
inst✝² : Algebra S B
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R S B
x : B
r : R
hx : IsIntegral S x
⊢ (algebraMap R S) r • x ∈ S[x]

Tactic:
]

Goal state after:
B : Type u_3
S : Type u_4
inst✝⁶ : Ring B
inst✝⁵ : CommRing S
R : Type u_5
inst✝⁴ : CommSemiring R
inst✝³ : Algebra R B
inst✝² : Algebra S B
inst✝¹ : Algebra R S
inst✝ : IsScalarTower R S B
x : B
r : R
hx : IsIntegral S x
⊢ (algebraMap R S) r • x ∈ S[x]
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
A : Matrix n n R
h : IsUnit A
this : Invertible A
⊢ C A⁻¹.det * (C.mapMatrix A * ((scalar n) X - C.mapMatrix A⁻¹)).det =
    C A⁻¹.det * (C.mapMatrix A * (scalar n) X - 1).det

Tactic:
by rw [mul_sub, ← map_mul]; simp

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.Algebra.Basic

Goal state before:
pp: R : Type u_1
B : Type u_3
inst✝² : CommRing R
inst✝¹ : Ring B
inst✝ : Algebra R B
n : ℤ
⊢ IsIntegral R (?m.25 ↑n)

pp: R : Type u_1
B : Type u_3
inst✝² : CommRing R
inst✝¹ : Ring B
inst✝ : Algebra R B
n : ℤ
⊢ R →+* B

Tactic:
]

Goal state after:
R : Type u_1
B : Type u_3
inst✝² : CommRing R
inst✝¹ : Ring B
inst✝ : Algebra R B
n : ℤ
⊢ IsIntegral R (?m.25 ↑n)

R : Type u_1
B : Type u_3
inst✝² : CommRing R
inst✝¹ : Ring B
inst✝ : Algebra R B
n : ℤ
⊢ R →+* B
Module: Mathlib.RingTheory.IntegralClosure.Algebra.Basic

Goal state before:
pp: R : Type u_1
B : Type u_3
inst✝² : CommRing R
inst✝¹ : Ring B
inst✝ : Algebra R B
a : ℕ
⊢ IsIntegral R ↑↑a

Tactic:
]

Goal state after:
R : Type u_1
B : Type u_3
inst✝² : CommRing R
inst✝¹ : Ring B
inst✝ : Algebra R B
a : ℕ
⊢ IsIntegral R ↑↑a
Module: Mathlib.RingTheory.IntegralClosure.Algebra.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Ring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
x : A
y : B
h : IsIntegral R y
⊢ IsIntegral A (x • 1 ⊗ₜ[R] y)

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommRing R
inst✝³ : CommRing A
inst✝² : Ring B
inst✝¹ : Algebra R A
inst✝ : Algebra R B
x : A
y : B
h : IsIntegral R y
⊢ IsIntegral A (x • 1 ⊗ₜ[R] y)
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
A : Matrix n n R
h : IsUnit A
this : Invertible A
⊢ C A⁻¹.det * ((-1) ^ Fintype.card n * (1 - (scalar n) X * C.mapMatrix A).det) =
    C A⁻¹.det * ((-1) ^ Fintype.card n * (1 - (scalar n) X * C.mapMatrix A).det)

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ ((evalRingHom 0).mapMatrix (1 - X • M.map ⇑C)).det = det 1

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
⊢ ((evalRingHom 0).mapMatrix (1 - X • M.map ⇑C)).det = det 1
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
k : ℕ
hk : M ^ k = 0
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
k : ℕ
hk : 1 - X • M.map ⇑C ∣ 1
⊢ M.charpolyRev ∣ det 1

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
k : ℕ
hk : 1 - X • M.map ⇑C ∣ 1
⊢ M.charpolyRev ∣ det 1
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
k : ℕ
f : R[X]
n : ℕ
⊢ (if n + k - k = n then ↑((n + k).choose k) * f.coeff (n + k) else 0) = ↑((n + k).choose k) * f.coeff (n + k)

pp: case h₀
R : Type u_1
inst✝ : Semiring R
k : ℕ
f : R[X]
n : ℕ
⊢ ∀ b ∈ f.support, b ≠ n + k → ((monomial (b - k)) (↑(b.choose k) * f.coeff b)).coeff n = 0

pp: case h₁
R : Type u_1
inst✝ : Semiring R
k : ℕ
f : R[X]
n : ℕ
⊢ n + k ∉ f.support → ((monomial (n + k - k)) (↑((n + k).choose k) * f.coeff (n + k))).coeff n = 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : Semiring R
k : ℕ
f : R[X]
n : ℕ
⊢ (if n + k - k = n then ↑((n + k).choose k) * f.coeff (n + k) else 0) = ↑((n + k).choose k) * f.coeff (n + k)

case h₀
R : Type u_1
inst✝ : Semiring R
k : ℕ
f : R[X]
n : ℕ
⊢ ∀ b ∈ f.support, b ≠ n + k → ((monomial (b - k)) (↑(b.choose k) * f.coeff b)).coeff n = 0

case h₁
R : Type u_1
inst✝ : Semiring R
k : ℕ
f : R[X]
n : ℕ
⊢ n + k ∉ f.support → ((monomial (n + k - k)) (↑((n + k).choose k) * f.coeff (n + k))).coeff n = 0
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
hM : IsNilpotent M
a✝ : Nontrivial R
p : R[X] := M.charpolyRev
hp : p - 1 = X * (p /ₘ X)
this : IsNilpotent (p /ₘ X)
aux : (M.charpoly - X ^ Fintype.card n).natDegree ≤ M.charpoly.natDegree
⊢ IsNilpotent (M.charpolyRev - reflect M.charpoly.natDegree (X ^ Fintype.card n))

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
hM : IsNilpotent M
a✝ : Nontrivial R
p : R[X] := M.charpolyRev
hp : p - 1 = X * (p /ₘ X)
this : IsNilpotent (p /ₘ X)
aux : (M.charpoly - X ^ Fintype.card n).natDegree ≤ M.charpoly.natDegree
⊢ IsNilpotent (M.charpolyRev - reflect M.charpoly.natDegree (X ^ Fintype.card n))
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case h₀
R : Type u_1
inst✝ : Semiring R
k : ℕ
f : R[X]
n i : ℕ
_hi : i ∈ f.support
hink : i ≠ n + k
⊢ (if i - k = n then ↑(i.choose k) * f.coeff i else 0) = 0

Tactic:
]

Goal state after:
case h₀
R : Type u_1
inst✝ : Semiring R
k : ℕ
f : R[X]
n i : ℕ
_hi : i ∈ f.support
hink : i ≠ n + k
⊢ (if i - k = n then ↑(i.choose k) * f.coeff i else 0) = 0
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
s : Finset n
e : ↥s ⊕ { a // a ∉ s } ≃ n := Equiv.sumCompl fun x ↦ x ∈ s
A : Matrix n n R := of (s.piecewise M 1)
⊢ (A.submatrix ⇑e ⇑e).det = (M.submatrix Subtype.val Subtype.val).det

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
s : Finset n
e : ↥s ⊕ { a // a ∉ s } ≃ n := Equiv.sumCompl fun x ↦ x ∈ s
A : Matrix n n R := of (s.piecewise M 1)
⊢ (A.submatrix ⇑e ⇑e).det = (M.submatrix Subtype.val Subtype.val).det
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h : p.natDegree < n
⊢ ∑ n_1 ∈ p.support, (monomial (n_1 - n)) (↑(n_1.choose n) * p.coeff n_1) = 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : Semiring R
p : R[X]
n : ℕ
h : p.natDegree < n
⊢ ∑ n_1 ∈ p.support, (monomial (n_1 - n)) (↑(n_1.choose n) * p.coeff n_1) = 0
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Semiring R
k n : ℕ
r : R
i : ℕ
hnik : ¬n = i + k
⊢ 0 = if n - k = i then ↑(n.choose k) * r else 0

Tactic:
]

Goal state after:
case neg
R : Type u_1
inst✝ : Semiring R
k n : ℕ
r : R
i : ℕ
hnik : ¬n = i + k
⊢ 0 = if n - k = i then ↑(n.choose k) * r else 0
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
k n : ℕ
r : R
i : ℕ
hnik : ¬n - k = i
hkn : k ≤ n
⊢ 0 = if n - k = i then ↑(n.choose k) * r else 0

Tactic:
]

Goal state after:
case pos
R : Type u_1
inst✝ : Semiring R
k n : ℕ
r : R
i : ℕ
hnik : ¬n - k = i
hkn : k ≤ n
⊢ 0 = if n - k = i then ↑(n.choose k) * r else 0
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
k n : ℕ
r : R
i : ℕ
hnik : ¬n - k = i
hkn : k ≤ n
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Semiring R
k n : ℕ
r : R
i : ℕ
hnik : ¬n = i + k
hkn : n < k
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
k : ℕ
r : R
hk : 0 < k
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
k : ℕ
hk : 0 < k
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
k : ℕ
hk : 1 < k
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case zero
R : Type u_1
inst✝ : Semiring R
k : ℕ
⊢ id = id

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case succ
R : Type u_1
inst✝ : Semiring R
k✝ k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ (((k + 1)! • hasseDeriv (k + 1)) f).coeff n = (derivative ((k ! • hasseDeriv k) f)).coeff n

Tactic:
]

Goal state after:
case succ
R : Type u_1
inst✝ : Semiring R
k✝ k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ (((k + 1)! • hasseDeriv (k + 1)) f).coeff n = (derivative ((k ! • hasseDeriv k) f)).coeff n
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case succ
R : Type u_1
inst✝ : Semiring R
k✝ k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ ↑((k + 1) * k !) * (↑((n + k + 1).choose (k + 1)) * f.coeff (n + k + 1)) =
    ↑k ! * (↑((n + k + 1).choose (n + 1)) * (↑(n + 1) * f.coeff (n + k + 1)))

Tactic:
]

Goal state after:
case succ
R : Type u_1
inst✝ : Semiring R
k✝ k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ ↑((k + 1) * k !) * (↑((n + k + 1).choose (k + 1)) * f.coeff (n + k + 1)) =
    ↑k ! * (↑((n + k + 1).choose (n + 1)) * (↑(n + 1) * f.coeff (n + k + 1)))
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case succ.e_a
R : Type u_1
inst✝ : Semiring R
k✝ k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ k ! * ((k + 1) * (n + k + 1).choose (k + 1)) = k ! * ((n + k + 1).choose (n + 1) * (n + 1))

Tactic:
]

Goal state after:
case succ.e_a
R : Type u_1
inst✝ : Semiring R
k✝ k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
⊢ k ! * ((k + 1) * (n + k + 1).choose (k + 1)) = k ! * ((n + k + 1).choose (n + 1) * (n + 1))
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
s : Finset n
e : ↥s ⊕ { a // a ∉ s } ≃ n := Equiv.sumCompl fun x ↦ x ∈ s
A : Matrix n n R := of (s.piecewise M 1)
h_blocks :
  A.submatrix ⇑e ⇑e = fromBlocks (M.submatrix Subtype.val Subtype.val) (M.submatrix Subtype.val Subtype.val) 0 1
⊢ (M.submatrix Subtype.val Subtype.val).det = (M.submatrix Subtype.val Subtype.val).det

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case succ.e_a.e_a
R : Type u_1
inst✝ : Semiring R
k✝ k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
this : n + k + 1 = n + (k + 1)
⊢ (n + k + 1).choose n * (k + 1) = (n + k + 1).choose n * (n + k + 1 - n)

Tactic:
]

Goal state after:
case succ.e_a.e_a
R : Type u_1
inst✝ : Semiring R
k✝ k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
this : n + k + 1 = n + (k + 1)
⊢ (n + k + 1).choose n * (k + 1) = (n + k + 1).choose n * (n + k + 1 - n)
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case succ.e_a.e_a.e_a
R : Type u_1
inst✝ : Semiring R
k✝ k : ℕ
ih : ⇑(k ! • hasseDeriv k) = (⇑derivative)^[k]
f : R[X]
n : ℕ
this : n + k + 1 = n + (k + 1)
⊢ k + 1 = k + 1

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff

Goal state before:
pp: R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
k : ℕ
D : (n → R[X]) [⋀^n]→ₗ[R[X]] R[X] := detRowAlternating
⊢ (detRowAlternating (X • M.map ⇑C + 1)).coeff k =
    ∑ x ∈ powersetCard k univ, detRowAlternating (M.submatrix Subtype.val Subtype.val)

Tactic:
]

Goal state after:
R : Type u
inst✝² : CommRing R
n : Type v
inst✝¹ : DecidableEq n
inst✝ : Fintype n
M : Matrix n n R
k : ℕ
D : (n → R[X]) [⋀^n]→ₗ[R[X]] R[X] := detRowAlternating
⊢ (detRowAlternating (X • M.map ⇑C + 1)).coeff k =
    ∑ x ∈ powersetCard k univ, detRowAlternating (M.submatrix Subtype.val Subtype.val)
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
k l i : ℕ
⊢ (monomial (i - (k + l))) ↑((i - l).choose k * i.choose l) =
    (monomial (i - (k + l))) ↑((k + l).choose k * i.choose (k + l))

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
⊢ (monomial (i - (k + l))) ↑((i - l).choose k * i.choose l) =
    (monomial (i - (k + l))) ↑((k + l).choose k * i.choose (k + l))
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : i < k + l
⊢ (i - l).choose k * i.choose l = 0

Tactic:
]

Goal state after:
case pos
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : i < k + l
⊢ (i - l).choose k * i.choose l = 0
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : i < k + l
hil : i < l
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : i - l < k
hil : l ≤ i
⊢ (i - l).choose k * i.choose l = 0

Tactic:
]

Goal state after:
case neg
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : i - l < k
hil : l ≤ i
⊢ (i - l).choose k * i.choose l = 0
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : i - l < k
hil : l ≤ i
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
h3 : k ≤ k + l
⊢ ↑(i - l)! / (↑k ! * ↑(i - l - k)!) * (↑i ! / (↑l ! * ↑(i - l)!)) =
    ↑(k + l)! / (↑k ! * ↑(k + l - k)!) * (↑i ! / (↑(k + l)! * ↑(i - (k + l))!))

Tactic:
]

Goal state after:
case neg
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
h3 : k ≤ k + l
⊢ ↑(i - l)! / (↑k ! * ↑(i - l - k)!) * (↑i ! / (↑l ! * ↑(i - l)!)) =
    ↑(k + l)! / (↑k ! * ↑(k + l - k)!) * (↑i ! / (↑(k + l)! * ↑(i - (k + l))!))
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
h3 : k ≤ k + l
⊢ i - (l + k) = i - l - k

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
h3 : k ≤ k + l
⊢ i - (l + k) = i - l - k
Module: Mathlib.Algebra.Polynomial.HasseDeriv

Goal state before:
pp: case neg
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
h3 : k ≤ k + l
⊢ ↑(i - l)! / (↑k ! * ↑(i - l - k)!) * (↑i ! / (↑l ! * ↑(i - l)!)) =
    ↑(k + l)! / (↑k ! * ↑(k + l - k)!) * (↑i ! / (↑(k + l)! * ↑(i - l - k)!))

Tactic:
]

Goal state after:
case neg
R : Type u_1
inst✝ : Semiring R
k l i : ℕ
hikl : k + l ≤ i
h1 : l ≤ i
h2 : k ≤ i - l
h3 : k ≤ k + l
⊢ ↑(i - l)! / (↑k ! * ↑(i - l - k)!) * (↑i ! / (↑l ! * ↑(i - l)!)) =
    ↑(k + l)! / (↑k ! * ↑(k + l - k)!) * (↑i ! / (↑(k + l)! * ↑(i - l - k)!))