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

Goal state before:
pp: case refine_4
M : Type u_2
inst✝⁴ : AddCommGroup M
K : Type u_3
inst✝³ : Field K
inst✝² : Module K M
f g : End K M
inst✝¹ : FiniteDimensional K M
inst✝ : PerfectField K
comm : Commute f g
hf : f.IsSemisimple
hg : g.IsSemisimple
R : Type u_3 := K[X] ⧸ Ideal.span {minpoly K f}
S : Type u_3 := AdjoinRoot (map (algebraMap K R) (minpoly K g))
this✝⁵ : Module.Finite K R
this✝⁴ : Module.Finite R S
this✝³ : Module.Finite K S
this✝² : IsArtinianRing R
this✝¹ : IsReduced R
this✝ : IsReduced S
φ : S →ₐ[K] End K M :=
  Ideal.Quotient.liftₐ (Ideal.span {map (algebraMap K R) (minpoly K g)})
    (eval₂AlgHom (Ideal.Quotient.liftₐ (Ideal.span {minpoly K f}) (aeval f) ⋯) g ⋯) ⋯
this : K[f, g] ≤ φ.range
p : S
ha : φ p ∈ K[f, g]
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Dynamics.Newton

Goal state before:
pp: case e'_4
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
d : S
hd :
  (aeval
        (P.newtonMap^[n] x +
          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))
      P =
    (aeval (P.newtonMap^[n] x)) P +
        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *
          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +
      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2
this : IsUnit ((aeval (P.newtonMap^[n] x)) (derivative P))
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: case h
n : ℕ
IH :
  ∀ m < n,
    ∀ {R : Type u} [inst : CommRing R] [Nontrivial R] {p : R[X]},
      p.Monic →
        p.natDegree = m →
          ∃ S x x_1,
            ∃ (_ : Module.Finite R S) (_ : Free R S) (_ : Nontrivial S), (Polynomial.map (algebraMap R S) p).Splits
R : Type u
inst✝¹ : CommRing R
inst✝ : Nontrivial R
p : R[X]
hp : p.Monic
hn : p.natDegree = n
⊢ ∃ S x x_1, ∃ (_ : Module.Finite R S) (_ : Free R S) (_ : Nontrivial S), (Polynomial.map (algebraMap R S) p).Splits

Tactic:
| h n IH =>

Goal state after:
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ :
  ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i =
    ∑ ij ∈ R2N ×ˢ R2N with ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2)
z₁ : ∑ ij ∈ R2N ×ˢ R2N with ¬ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 + x.2 ≤ 2 * N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ :
  ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i =
    ∑ ij ∈ R2N ×ˢ R2N with ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2)
z₁ : ∑ ij ∈ R2N ×ˢ R2N with ¬ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 + x.2 ≤ 2 * N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i
Module: Mathlib.LinearAlgebra.JordanChevalley

Goal state before:
pp: case h
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
P : K[X]
k : ℕ
sep : P.Separable
ff : ↥K[f] := ⟨f, ⋯⟩
P' : K[X] := derivative P
q : K[X]
hq : P ^ k = minpoly K f * q
⊢ ↑((aeval ff) (P ^ k)) = ↑0

Tactic:
]

Goal state after:
case h
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
P : K[X]
k : ℕ
sep : P.Separable
ff : ↥K[f] := ⟨f, ⋯⟩
P' : K[X] := derivative P
q : K[X]
hq : P ^ k = minpoly K f * q
⊢ ↑((aeval ff) (P ^ k)) = ↑0
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ : ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₁ : ∑ ij ∈ R2N ×ˢ R2N with ¬ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 + x.2 ≤ 2 * N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ : ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₁ : ∑ ij ∈ R2N ×ˢ R2N with ¬ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 + x.2 ≤ 2 * N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i
Module: Mathlib.LinearAlgebra.JordanChevalley

Goal state before:
pp: case refine_2
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
P : K[X]
k✝ : ℕ
sep : P.Separable
nil : minpoly K f ∣ P ^ k✝
ff : ↥K[f] := ⟨f, ⋯⟩
P' : K[X] := derivative P
nil' : IsNilpotent ((aeval ff) P)
sep' : IsUnit ((aeval ff) P')
s : End K V
mem : s ∈ K[f]
hss : (aeval ⟨s, mem⟩) P = 0
k : ℕ
hk : ↑((ff - ⟨s, mem⟩) ^ k) = ↑0
⊢ (f - s) ^ k = 0

Tactic:
]

Goal state after:
case refine_2
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
P : K[X]
k✝ : ℕ
sep : P.Separable
nil : minpoly K f ∣ P ^ k✝
ff : ↥K[f] := ⟨f, ⋯⟩
P' : K[X] := derivative P
nil' : IsNilpotent ((aeval ff) P)
sep' : IsUnit ((aeval ff) P')
s : End K V
mem : s ∈ K[f]
hss : (aeval ⟨s, mem⟩) P = 0
k : ℕ
hk : ↑((ff - ⟨s, mem⟩) ^ k) = ↑0
⊢ (f - s) ^ k = 0
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: case inl
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ : ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₁ : ∑ ij ∈ R2N ×ˢ R2N with ¬ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 + x.2 ≤ 2 * N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
i : ℕ × ℕ
hi : i ∈ R2N ×ˢ R2N ∧ (i.1 ≤ N → N < i.2)
h : N + 1 ≤ i.1
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.JordanChevalley

Goal state before:
pp: K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
P : K[X]
k✝ : ℕ
sep : P.Separable
nil : minpoly K f ∣ P ^ k✝
ff : ↥K[f] := ⟨f, ⋯⟩
P' : K[X] := derivative P
nil' : IsNilpotent ((aeval ff) P)
sep' : IsUnit ((aeval ff) P')
s : End K V
mem : s ∈ K[f]
hss : (aeval s) P = ↑0
k : ℕ
hk : (ff - ⟨s, mem⟩) ^ k = 0
⊢ (aeval s) P = 0

Tactic:
]

Goal state after:
K : Type u_1
V : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
f : End K V
P : K[X]
k✝ : ℕ
sep : P.Separable
nil : minpoly K f ∣ P ^ k✝
ff : ↥K[f] := ⟨f, ⋯⟩
P' : K[X] := derivative P
nil' : IsNilpotent ((aeval ff) P)
sep' : IsUnit ((aeval ff) P')
s : End K V
mem : s ∈ K[f]
hss : (aeval s) P = ↑0
k : ℕ
hk : (ff - ⟨s, mem⟩) ^ k = 0
⊢ (aeval s) P = 0
Module: Mathlib.Dynamics.Newton

Goal state before:
pp: case succ.refine_2
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
d : S
hd :
  (aeval
        (P.newtonMap^[n] x +
          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))
      P =
    (aeval (P.newtonMap^[n] x)) P +
        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *
          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +
      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2
⊢ ((aeval x) P ^ 2 ^ n) ^ 2 ∣
    Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) ^ 2 * (aeval (P.newtonMap^[n] x)) P ^ 2

Tactic:
]

Goal state after:
case succ.refine_2
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
ih : (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
d : S
hd :
  (aeval
        (P.newtonMap^[n] x +
          -Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P))
      P =
    (aeval (P.newtonMap^[n] x)) P +
        eval (P.newtonMap^[n] x) (derivative (map (algebraMap R S) P)) *
          (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) +
      d * (-Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P) ^ 2
⊢ ((aeval x) P ^ 2 ^ n) ^ 2 ∣
    Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) ^ 2 * (aeval (P.newtonMap^[n] x)) P ^ 2
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: case inr
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ : ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₁ : ∑ ij ∈ R2N ×ˢ R2N with ¬ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 + x.2 ≤ 2 * N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
i : ℕ × ℕ
hi : i ∈ R2N ×ˢ R2N ∧ (i.1 ≤ N → N < i.2)
h : i.1 < N + 1
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.JordanChevalley

Goal state before:
pp: K : Type u_1
V : Type u_2
inst✝⁴ : Field K
inst✝³ : AddCommGroup V
inst✝² : Module K V
f : End K V
inst✝¹ : FiniteDimensional K V
inst✝ : PerfectField K
g : K[X]
k : ℕ
sep : g.Separable
nil : minpoly K f ∣ g ^ k
⊢ ∃ n ∈ K[f], ∃ s ∈ K[f], IsNilpotent n ∧ s.IsSemisimple ∧ f = n + s

Tactic:
]

Goal state after:
K : Type u_1
V : Type u_2
inst✝⁴ : Field K
inst✝³ : AddCommGroup V
inst✝² : Module K V
f : End K V
inst✝¹ : FiniteDimensional K V
inst✝ : PerfectField K
g : K[X]
k : ℕ
sep : g.Separable
nil : minpoly K f ∣ g ^ k
⊢ ∃ n ∈ K[f], ∃ s ∈ K[f], IsNilpotent n ∧ s.IsSemisimple ∧ f = n + s
Module: Mathlib.Dynamics.Newton

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
hn : (aeval x) P ^ n = 0
⊢ (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P

Tactic:
]

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
n : ℕ
hn : (aeval x) P ^ n = 0
⊢ (aeval x) P ^ 2 ^ n ∣ (aeval (P.newtonMap^[n] x)) P
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ : ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₁ : ∑ ij ∈ R2N ×ˢ R2N with ¬ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 + x.2 ≤ 2 * N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₂ : ∑ ij ∈ R2N ×ˢ R2N with ¬(ij.1 ≤ N ∧ ij.2 ≤ N), ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₂ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 ≤ N ∧ x.2 ≤ N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ : ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₁ : ∑ ij ∈ R2N ×ˢ R2N with ¬ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 + x.2 ≤ 2 * N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₂ : ∑ ij ∈ R2N ×ˢ R2N with ¬(ij.1 ≤ N ∧ ij.2 ≤ N), ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₂ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 ≤ N ∧ x.2 ≤ N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i
Module: Mathlib.Algebra.Lie.AdjointAction.JordanChevalley

Goal state before:
pp: K : Type u_1
V : Type u_2
inst✝⁴ : Field K
inst✝³ : PerfectField K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
n s : Module.End K V
hc : Commute n s
hn : IsNilpotent n
hs : s.IsSemisimple
n' : Module.End K (Module.End K V)
hn'_adj : n' ∈ K[(ad K (Module.End K V)) (n + s)]
s' : Module.End K (Module.End K V)
hs'_adj : (ad K (Module.End K V)) s ∈ K[(ad K (Module.End K V)) (n + s)]
hn'_nil : IsNilpotent n'
hs'_ss : s'.IsSemisimple
h_jc : (ad K (Module.End K V)) (n + s) = n' + s'
hc' : Commute n' s'
hs_eq : s' = (ad K (Module.End K V)) s
⊢ (ad K (Module.End K V)) s ∈ K[(ad K (Module.End K V)) (n + s)]

Tactic:
]

Goal state after:
K : Type u_1
V : Type u_2
inst✝⁴ : Field K
inst✝³ : PerfectField K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
n s : Module.End K V
hc : Commute n s
hn : IsNilpotent n
hs : s.IsSemisimple
n' : Module.End K (Module.End K V)
hn'_adj : n' ∈ K[(ad K (Module.End K V)) (n + s)]
s' : Module.End K (Module.End K V)
hs'_adj : (ad K (Module.End K V)) s ∈ K[(ad K (Module.End K V)) (n + s)]
hn'_nil : IsNilpotent n'
hs'_ss : s'.IsSemisimple
h_jc : (ad K (Module.End K V)) (n + s) = n' + s'
hc' : Commute n' s'
hs_eq : s' = (ad K (Module.End K V)) s
⊢ (ad K (Module.End K V)) s ∈ K[(ad K (Module.End K V)) (n + s)]
Module: Mathlib.Algebra.Lie.AdjointAction.JordanChevalley

Goal state before:
pp: K : Type u_1
V : Type u_2
inst✝⁴ : Field K
inst✝³ : PerfectField K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
n s : Module.End K V
hc : Commute n s
hn : IsNilpotent n
hs : s.IsSemisimple
this : (ad K (Module.End K V)) n = (ad K (Module.End K V)) (n + s) - (ad K (Module.End K V)) s
⊢ (ad K (Module.End K V)) (n + s) - (ad K (Module.End K V)) s ∈ K[(ad K (Module.End K V)) (n + s)]

Tactic:
]

Goal state after:
K : Type u_1
V : Type u_2
inst✝⁴ : Field K
inst✝³ : PerfectField K
inst✝² : AddCommGroup V
inst✝¹ : Module K V
inst✝ : FiniteDimensional K V
n s : Module.End K V
hc : Commute n s
hn : IsNilpotent n
hs : s.IsSemisimple
this : (ad K (Module.End K V)) n = (ad K (Module.End K V)) (n + s) - (ad K (Module.End K V)) s
⊢ (ad K (Module.End K V)) (n + s) - (ad K (Module.End K V)) s ∈ K[(ad K (Module.End K V)) (n + s)]
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ :
  ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i =
    ∑ x ∈ R2N ×ˢ R2N with x.1 ≤ N ∧ x.2 ≤ N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₁ : ∑ ij ∈ R2N ×ˢ R2N with ¬ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 + x.2 ≤ 2 * N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₂ : ∑ ij ∈ R2N ×ˢ R2N with ¬(ij.1 ≤ N ∧ ij.2 ≤ N), ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₂ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 ≤ N ∧ x.2 ≤ N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i

Tactic:
]

Goal state after:
A : Type u_1
inst✝¹ : Ring A
inst✝ : Module ℚ A
a b : A
h₁ : Commute a b
n₁ : ℕ
hn₁ : a ^ n₁ = 0
n₂ : ℕ
hn₂ : b ^ n₂ = 0
N : ℕ := max n₁ n₂
h₄ : a ^ (N + 1) = 0
h₅ : b ^ (N + 1) = 0
R2N : Finset ℕ := range (2 * N + 1)
hR2N : R2N = range (2 * N + 1)
RN : Finset ℕ := range (N + 1)
hRN : RN = range (N + 1)
s₁ :
  ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i =
    ∑ x ∈ R2N ×ˢ R2N with x.1 ≤ N ∧ x.2 ≤ N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₁ : ∑ ij ∈ R2N ×ˢ R2N with ¬ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₁ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 + x.2 ≤ 2 * N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
z₂ : ∑ ij ∈ R2N ×ˢ R2N with ¬(ij.1 ≤ N ∧ ij.2 ≤ N), ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2) = 0
split₂ :
  ∑ x ∈ R2N ×ˢ R2N with x.1 ≤ N ∧ x.2 ≤ N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2) =
    ∑ x ∈ R2N ×ˢ R2N, ((↑x.1!)⁻¹ * (↑x.2!)⁻¹) • (a ^ x.1 * b ^ x.2)
⊢ ∑ i ∈ R2N, (↑i !)⁻¹ • (a + b) ^ i = (∑ i ∈ RN, (↑i !)⁻¹ • a ^ i) * ∑ i ∈ RN, (↑i !)⁻¹ • b ^ i
Module: Mathlib.LinearAlgebra.JordanChevalley

Goal state before:
pp: K : Type u_1
V : Type u_2
inst✝⁴ : Field K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : FiniteDimensional K V
inst✝ : PerfectField K
n₁ s₁ n₂ s₂ : End K V
hn₁ : IsNilpotent n₁
hs₁ : s₁.IsSemisimple
hn₂ : IsNilpotent n₂
hs₂ : s₂.IsSemisimple
hc₁ : Commute n₁ s₁
hc₂ : Commute n₂ s₂
h : n₁ + s₁ = n₂ + s₂
n₀ : End K V
hn₀ : n₀ ∈ K[n₁ + s₁]
s₀ : End K V
hs₀ : s₀ ∈ K[n₁ + s₁]
hn₀_nil : IsNilpotent n₀
hs₀_ss : s₀.IsSemisimple
h₀ : n₁ + s₁ = n₀ + s₀
n s : End K V
hn : IsNilpotent n
hs : s.IsSemisimple
hc : Commute n s
heq : n₁ + s₁ = n + s
hsf : Commute s (n₁ + s₁)
hnf : Commute n (n₁ + s₁)
⊢ IsNilpotent (n₀ - n)

Tactic:
]

Goal state after:
K : Type u_1
V : Type u_2
inst✝⁴ : Field K
inst✝³ : AddCommGroup V
inst✝² : Module K V
inst✝¹ : FiniteDimensional K V
inst✝ : PerfectField K
n₁ s₁ n₂ s₂ : End K V
hn₁ : IsNilpotent n₁
hs₁ : s₁.IsSemisimple
hn₂ : IsNilpotent n₂
hs₂ : s₂.IsSemisimple
hc₁ : Commute n₁ s₁
hc₂ : Commute n₂ s₂
h : n₁ + s₁ = n₂ + s₂
n₀ : End K V
hn₀ : n₀ ∈ K[n₁ + s₁]
s₀ : End K V
hs₀ : s₀ ∈ K[n₁ + s₁]
hn₀_nil : IsNilpotent n₀
hs₀_ss : s₀.IsSemisimple
h₀ : n₁ + s₁ = n₀ + s₀
n s : End K V
hn : IsNilpotent n
hs : s.IsSemisimple
hc : Commute n s
heq : n₁ + s₁ = n + s
hsf : Commute s (n₁ + s₁)
hnf : Commute n (n₁ + s₁)
⊢ IsNilpotent (n₀ - n)
Module: Mathlib.Dynamics.Newton

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
r₁ r₂ : S
x✝¹ : IsNilpotent (r₁ - x) ∧ (aeval r₁) P = 0
x✝ : IsNilpotent (r₂ - x) ∧ (aeval r₂) P = 0
hr₁ : IsNilpotent (r₁ - x)
hr₁' : (aeval r₁) P = 0
hr₂ : IsNilpotent (r₂ - x)
hr₂' : (aeval r₂) P = 0
u : S
hu : r₁ = r₂
this : IsUnit ((aeval r₁) (derivative P) + u * (r₂ - r₁))
⊢ r₁ = r₂

Tactic:
]

Goal state after:
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
r₁ r₂ : S
x✝¹ : IsNilpotent (r₁ - x) ∧ (aeval r₁) P = 0
x✝ : IsNilpotent (r₂ - x) ∧ (aeval r₂) P = 0
hr₁ : IsNilpotent (r₁ - x)
hr₁' : (aeval r₁) P = 0
hr₂ : IsNilpotent (r₂ - x)
hr₂' : (aeval r₂) P = 0
u : S
hu : r₁ = r₂
this : IsUnit ((aeval r₁) (derivative P) + u * (r₂ - r₁))
⊢ r₁ = r₂
Module: Mathlib.Dynamics.Newton

Goal state before:
pp: case refine_2
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
r₁ r₂ : S
x✝¹ : IsNilpotent (r₁ - x) ∧ (aeval r₁) P = 0
x✝ : IsNilpotent (r₂ - x) ∧ (aeval r₂) P = 0
hr₁ : IsNilpotent (r₁ - x)
hr₁' : (aeval r₁) P = 0
hr₂ : IsNilpotent (r₂ - x)
hr₂' : (aeval r₂) P = 0
u : S
hu :
  eval (r₁ + (r₂ - r₁)) (map (algebraMap R S) P) =
    eval r₁ (map (algebraMap R S) P) + eval r₁ (derivative (map (algebraMap R S) P)) * (r₂ - r₁) + u * (r₂ - r₁) ^ 2
this : IsUnit ((aeval r₁) (derivative P))
⊢ IsUnit ((aeval r₁) (derivative P) + u * (r₂ - x - (r₁ - x)))

Tactic:
]

Goal state after:
case refine_2
R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P : R[X]
x : S
h : IsNilpotent ((aeval x) P)
h' : IsUnit ((aeval x) (derivative P))
r₁ r₂ : S
x✝¹ : IsNilpotent (r₁ - x) ∧ (aeval r₁) P = 0
x✝ : IsNilpotent (r₂ - x) ∧ (aeval r₂) P = 0
hr₁ : IsNilpotent (r₁ - x)
hr₁' : (aeval r₁) P = 0
hr₂ : IsNilpotent (r₂ - x)
hr₂' : (aeval r₂) P = 0
u : S
hu :
  eval (r₁ + (r₂ - r₁)) (map (algebraMap R S) P) =
    eval r₁ (map (algebraMap R S) P) + eval r₁ (derivative (map (algebraMap R S) P)) * (r₂ - r₁) + u * (r₂ - r₁) ^ 2
this : IsUnit ((aeval r₁) (derivative P))
⊢ IsUnit ((aeval r₁) (derivative P) + u * (r₂ - x - (r₁ - x)))
Module: Mathlib.Algebra.Ring.Divisibility.Lemmas

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : SMul M R
inst✝¹ : Semigroup R
inst✝ : SMulCommClass M R R
x y : R
m : M
h : x ∣ y
k : R
hk : y = x * k
⊢ m • y = m • y

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Ring.Divisibility.Lemmas

Goal state before:
pp: R : Type u_1
x y : R
n m p : ℕ
inst✝ : Semiring R
hp : n + m ≤ p + 1
h_comm : Commute x y
hy : y ^ n = 0
⊢ x ^ m ∣ ∑ m ∈ Finset.antidiagonal p, p.choose m.1 • (x ^ m.1 * y ^ m.2)

Tactic:
]

Goal state after:
R : Type u_1
x y : R
n m p : ℕ
inst✝ : Semiring R
hp : n + m ≤ p + 1
h_comm : Commute x y
hy : y ^ n = 0
⊢ x ^ m ∣ ∑ m ∈ Finset.antidiagonal p, p.choose m.1 • (x ^ m.1 * y ^ m.2)