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

Goal state before:
pp: case refine_1
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
a✝ : End K M
ha : a✝ ∈ K[f, g]
R : Type u_3 := K[X] ⧸ Ideal.span {minpoly K f}
S : Type u_3 := AdjoinRoot (map (algebraMap K R) (minpoly K g))
this✝⁴ : Module.Finite K R
this✝³ : Module.Finite R S
this✝² : Module.Finite K S
this✝¹ : IsArtinianRing R
this✝ : IsReduced R
this : IsReduced S
a : K[X]
⊢ a ∈ RingHom.ker (aeval f) → (aeval f) a = 0

Tactic:
]

Goal state after:
case refine_1
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
a✝ : End K M
ha : a✝ ∈ K[f, g]
R : Type u_3 := K[X] ⧸ Ideal.span {minpoly K f}
S : Type u_3 := AdjoinRoot (map (algebraMap K R) (minpoly K g))
this✝⁴ : Module.Finite K R
this✝³ : Module.Finite R S
this✝² : Module.Finite K S
this✝¹ : IsArtinianRing R
this✝ : IsReduced R
this : IsReduced S
a : K[X]
⊢ a ∈ RingHom.ker (aeval f) → (aeval f) a = 0
Module: Mathlib.Algebra.Lie.Derivation.Basic

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
D : LieDerivation R L L
n : ℕ
a b : L
⊢ ∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅(⇑D)^[ij.1] a, (⇑D)^[ij.2] b⁆ =
    ∑ i ∈ range (n + 1), n.choose i • ⁅(⇑D)^[i] a, (⇑D)^[n - i] b⁆

Tactic:
]

Goal state after:
R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
D : LieDerivation R L L
n : ℕ
a b : L
⊢ ∑ ij ∈ antidiagonal n, n.choose ij.1 • ⁅(⇑D)^[ij.1] a, (⇑D)^[ij.2] b⁆ =
    ∑ i ∈ range (n + 1), n.choose i • ⁅(⇑D)^[i] a, (⇑D)^[n - i] b⁆
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)
i : ℕ
hi : i ∈ R2N
j : ℕ
hj : j ≤ i
this : (↑i !)⁻¹ * ↑(i.choose j) = (↑j !)⁻¹ * (↑(i - j)!)⁻¹
⊢ i.choose j • (↑i !)⁻¹ • (a ^ j * b ^ (i - j)) = ↑(i.choose j) • (↑i !)⁻¹ • (a ^ j * b ^ (i - j))

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)
i : ℕ
hi : i ∈ R2N
j : ℕ
hj : j ≤ i
this : (↑i !)⁻¹ * ↑(i.choose j) = (↑j !)⁻¹ * (↑(i - j)!)⁻¹
⊢ i.choose j • (↑i !)⁻¹ • (a ^ j * b ^ (i - j)) = ↑(i.choose j) • (↑i !)⁻¹ • (a ^ j * b ^ (i - j))
Module: Mathlib.RingTheory.Nilpotent.Exp

Goal state before:
pp: case calc_2
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)
i : ℕ
hi : i ∈ R2N
j : ℕ
hj : j ≤ i
⊢ (↑i !)⁻¹ * (↑i ! / ↑(j ! * (i - j)!)) = (↑j !)⁻¹ * (↑(i - j)!)⁻¹

Tactic:
]

Goal state after:
case calc_2
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)
i : ℕ
hi : i ∈ R2N
j : ℕ
hj : j ≤ i
⊢ (↑i !)⁻¹ * (↑i ! / ↑(j ! * (i - j)!)) = (↑j !)⁻¹ * (↑(i - j)!)⁻¹
Module: Mathlib.Algebra.Lie.AdjointAction.Derivation

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : L
⊢ -(⁅⁅z, x⁆, y⁆ + -⁅⁅z, y⁆, x⁆) = - -⁅⁅z, y⁆, x⁆ - - -⁅⁅z, x⁆, y⁆

Tactic:
]

Goal state after:
R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x y z : L
⊢ -(⁅⁅z, x⁆, y⁆ + -⁅⁅z, y⁆, x⁆) = - -⁅⁅z, y⁆, x⁆ - - -⁅⁅z, x⁆, y⁆
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
K : Type u_5
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
pb : PowerBasis R S
I : Ideal R
x : R
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case calc_3
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)
⊢ ∑ x ∈ (range (2 * N + 1)).sigma fun i ↦ range (i + 1),
      ((↑x.snd !)⁻¹ * (↑(x.fst - x.snd)!)⁻¹) • (a ^ x.snd * b ^ (x.fst - x.snd)) =
    ∑ ij ∈ range (2 * N + 1) ×ˢ range (2 * N + 1) with ij.1 + ij.2 ≤ 2 * N,
      ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2)

Tactic:
]

Goal state after:
case calc_3
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)
⊢ ∑ x ∈ (range (2 * N + 1)).sigma fun i ↦ range (i + 1),
      ((↑x.snd !)⁻¹ * (↑(x.fst - x.snd)!)⁻¹) • (a ^ x.snd * b ^ (x.fst - x.snd)) =
    ∑ ij ∈ range (2 * N + 1) ×ˢ range (2 * N + 1) with ij.1 + ij.2 ≤ 2 * N,
      ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2)
Module: Mathlib.Algebra.Lie.AdjointAction.Derivation

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x : L
⊢ (ad R L) x = 0 ↔ ∀ (m : L), ⁅x, m⁆ = 0

Tactic:
]

Goal state after:
R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
x : L
⊢ (ad R L) x = 0 ↔ ∀ (m : L), ⁅x, m⁆ = 0
Module: Mathlib.Algebra.Lie.AdjointAction.Derivation

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
h : LieAlgebra.center R L = ⊥
⊢ ⊥ = ⊥

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
K : Type u_5
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
pb : PowerBasis R S
I : Ideal R
x : R
⊢ Ideal.map (algebraMap R (AdjoinRoot (minpoly R pb.gen))) I =
    Ideal.map (algebraMap R (AdjoinRoot (minpoly R pb.gen))) I

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Lie.AdjointAction.Derivation

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
D : LieDerivation R L L
x : L
⊢ ⁅D, (ad R L) x⁆ = ⁅D, (ad R L) x⁆

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Lie.AdjointAction.Derivation

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
h : LieAlgebra.center R L = ⊥
D : LieDerivation R L L
hD : ∀ (x : L), ⁅x, D⁆ = 0
x : L
⊢ (ad R L) (D x) = 0

Tactic:
]

Goal state after:
R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
h : LieAlgebra.center R L = ⊥
D : LieDerivation R L L
hD : ∀ (x : L), ⁅x, D⁆ = 0
x : L
⊢ (ad R L) (D x) = 0
Module: Mathlib.Algebra.Lie.AdjointAction.Derivation

Goal state before:
pp: R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
h : LieAlgebra.center R L = ⊥
D : LieDerivation R L L
hD : D ∈ LieModule.maxTrivSubmodule R L (LieDerivation R L L)
x : L
this : D x = 0
⊢ D x = 0 x

Tactic:
]

Goal state after:
R : Type u_1
L : Type u_2
inst✝² : CommRing R
inst✝¹ : LieRing L
inst✝ : LieAlgebra R L
h : LieAlgebra.center R L = ⊥
D : LieDerivation R L L
hD : D ∈ LieModule.maxTrivSubmodule R L (LieDerivation R L L)
x : L
this : D x = 0
⊢ D x = 0 x
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
U : Type u_4
K : Type u_5
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
pb : PowerBasis R S
I : Ideal R
x : R
⊢ (algebraMap R (AdjoinRoot (minpoly R pb.gen) ⧸ Ideal.map (of (minpoly R pb.gen)) I)) x =
    (algebraMap R (AdjoinRoot (minpoly R pb.gen) ⧸ Ideal.map (of (minpoly R pb.gen)) I)) x

Tactic:
]

Goal state after:
Module: Mathlib.Dynamics.Newton

Goal state before:
pp: case succ
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)
n : ℕ
ih : IsNilpotent (P.newtonMap^[n] x - x)
⊢ IsNilpotent
    (P.newtonMap^[n] x - x - Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P)

Tactic:
]

Goal state after:
case succ
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)
n : ℕ
ih : IsNilpotent (P.newtonMap^[n] x - x)
⊢ IsNilpotent
    (P.newtonMap^[n] x - x - Ring.inverse ((aeval (P.newtonMap^[n] x)) (derivative P)) * (aeval (P.newtonMap^[n] x)) P)
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 : (aeval x) P = 0
⊢ x = x

Tactic:
]

Goal state after:
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 : IsUnit ((aeval x) (derivative P))
⊢ (aeval x) P = 0 ↔ (aeval x) P = 0

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Semisimple

Goal state before:
pp: case inl
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
a : End K M
ha : a ∈ K[f, g]
R : Type u_3 := K[X] ⧸ Ideal.span {minpoly K f}
S : Type u_3 := AdjoinRoot (map (algebraMap K R) (minpoly K g))
this✝⁴ : Module.Finite K R
this✝³ : Module.Finite R S
this✝² : Module.Finite K S
this✝¹ : IsArtinianRing R
this✝ : 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 ⋯) ⋯
x : End K M
hx : x = f
⊢ f ∈ ↑φ.range

Tactic:
]

Goal state after:
case inl
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
a : End K M
ha : a ∈ K[f, g]
R : Type u_3 := K[X] ⧸ Ideal.span {minpoly K f}
S : Type u_3 := AdjoinRoot (map (algebraMap K R) (minpoly K g))
this✝⁴ : Module.Finite K R
this✝³ : Module.Finite R S
this✝² : Module.Finite K S
this✝¹ : IsArtinianRing R
this✝ : 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 ⋯) ⋯
x : End K M
hx : x = f
⊢ f ∈ ↑φ.range
Module: Mathlib.Dynamics.Newton

Goal state before:
pp: case succ
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 + 1) ∣ (aeval (P.newtonMap^[n + 1] x)) P

Tactic:
]

Goal state after:
case succ
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 + 1) ∣ (aeval (P.newtonMap^[n + 1] x)) P
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
pb : PowerBasis R S
I : Ideal R
g : R[X]
⊢ (Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R pb.gen)}))
      (Polynomial.map (Ideal.Quotient.mk I) g) =
    (Ideal.Quotient.mk (span {Polynomial.map (Ideal.Quotient.mk I) (minpoly R pb.gen)}))
      (Polynomial.map (Ideal.Quotient.mk I) g)

Tactic:
]

Goal state after:
Module: Mathlib.LinearAlgebra.Semisimple

Goal state before:
pp: case inr
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
a : End K M
ha : a ∈ K[f, g]
R : Type u_3 := K[X] ⧸ Ideal.span {minpoly K f}
S : Type u_3 := AdjoinRoot (map (algebraMap K R) (minpoly K g))
this✝⁴ : Module.Finite K R
this✝³ : Module.Finite R S
this✝² : Module.Finite K S
this✝¹ : IsArtinianRing R
this✝ : 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 ⋯) ⋯
x : End K M
hx : x ∈ {g}
⊢ g ∈ ↑φ.range

Tactic:
]

Goal state after:
case inr
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
a : End K M
ha : a ∈ K[f, g]
R : Type u_3 := K[X] ⧸ Ideal.span {minpoly K f}
S : Type u_3 := AdjoinRoot (map (algebraMap K R) (minpoly K g))
this✝⁴ : Module.Finite K R
this✝³ : Module.Finite R S
this✝² : Module.Finite K S
this✝¹ : IsArtinianRing R
this✝ : 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 ⋯) ⋯
x : End K M
hx : x ∈ {g}
⊢ g ∈ ↑φ.range
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)
i : ℕ × ℕ
hi : i ∈ R2N ×ˢ R2N ∧ ¬i.1 + i.2 ≤ 2 * N
⊢ ((↑i.1!)⁻¹ * (↑i.2!)⁻¹) • (a ^ i.1 * b ^ i.2) = 0

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)
i : ℕ × ℕ
hi : i ∈ R2N ×ˢ R2N ∧ ¬i.1 + i.2 ≤ 2 * N
⊢ ((↑i.1!)⁻¹ * (↑i.2!)⁻¹) • (a ^ i.1 * b ^ i.2) = 0
Module: Mathlib.Dynamics.Newton

Goal state before:
pp: case succ
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 + 1) ∣
    (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

Tactic:
]

Goal state after:
case succ
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 + 1) ∣
    (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
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 =
    ∑ ij ∈ R2N ×ˢ R2N with ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2)
i : ℕ × ℕ
hi : i ∈ R2N ×ˢ R2N ∧ ¬i.1 + i.2 ≤ 2 * N
h : N + 1 ≤ i.1
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: 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
hpu : ¬IsUnit p
q : (AdjoinRoot p)[X]
hq : Polynomial.map (algebraMap R (AdjoinRoot p)) p = (X - C (AdjoinRoot.root p)) * q
hqm : q.Monic
this✝¹ : Free R (AdjoinRoot p)
this✝ : Module.Finite R (AdjoinRoot p)
this : Nontrivial (AdjoinRoot p)
⊢ q.natDegree < (X - C (AdjoinRoot.root p)).natDegree + q.natDegree

Tactic:
]

Goal state after:
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
hpu : ¬IsUnit p
q : (AdjoinRoot p)[X]
hq : Polynomial.map (algebraMap R (AdjoinRoot p)) p = (X - C (AdjoinRoot.root p)) * q
hqm : q.Monic
this✝¹ : Free R (AdjoinRoot p)
this✝ : Module.Finite R (AdjoinRoot p)
this : Nontrivial (AdjoinRoot p)
⊢ q.natDegree < (X - C (AdjoinRoot.root p)).natDegree + q.natDegree
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 =
    ∑ ij ∈ R2N ×ˢ R2N with ij.1 + ij.2 ≤ 2 * N, ((↑ij.1!)⁻¹ * (↑ij.2!)⁻¹) • (a ^ ij.1 * b ^ ij.2)
i : ℕ × ℕ
hi : i ∈ R2N ×ˢ R2N ∧ ¬i.1 + i.2 ≤ 2 * N
h✝ : i.1 < N + 1
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.AdjoinRoot

Goal state before:
pp: case neg
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
hpu : ¬IsUnit p
q : (AdjoinRoot p)[X]
hq : Polynomial.map (algebraMap R (AdjoinRoot p)) p = (X - C (AdjoinRoot.root p)) * q
hqm : q.Monic
this✝¹ : Free R (AdjoinRoot p)
this✝ : Module.Finite R (AdjoinRoot p)
this : Nontrivial (AdjoinRoot p)
S : Type u
w✝⁴ : CommRing S
w✝³ : Algebra (AdjoinRoot p) S
w✝² : Module.Finite (AdjoinRoot p) S
w✝¹ : Free (AdjoinRoot p) S
w✝ : Nontrivial S
hS : (Polynomial.map (algebraMap (AdjoinRoot p) S) q).Splits
algInst✝ : Algebra R S := ((algebraMap (AdjoinRoot p) S).comp (algebraMap R (AdjoinRoot p))).toAlgebra
scalarTowerInst✝ : IsScalarTower R (AdjoinRoot p) S := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap R S))
⊢ (Polynomial.map (algebraMap (AdjoinRoot p) S) (X - C (AdjoinRoot.root p)) *
      Polynomial.map (algebraMap (AdjoinRoot p) S) q).Splits

Tactic:
]

Goal state after:
case neg
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
hpu : ¬IsUnit p
q : (AdjoinRoot p)[X]
hq : Polynomial.map (algebraMap R (AdjoinRoot p)) p = (X - C (AdjoinRoot.root p)) * q
hqm : q.Monic
this✝¹ : Free R (AdjoinRoot p)
this✝ : Module.Finite R (AdjoinRoot p)
this : Nontrivial (AdjoinRoot p)
S : Type u
w✝⁴ : CommRing S
w✝³ : Algebra (AdjoinRoot p) S
w✝² : Module.Finite (AdjoinRoot p) S
w✝¹ : Free (AdjoinRoot p) S
w✝ : Nontrivial S
hS : (Polynomial.map (algebraMap (AdjoinRoot p) S) q).Splits
algInst✝ : Algebra R S := ((algebraMap (AdjoinRoot p) S).comp (algebraMap R (AdjoinRoot p))).toAlgebra
scalarTowerInst✝ : IsScalarTower R (AdjoinRoot p) S := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap R S))
⊢ (Polynomial.map (algebraMap (AdjoinRoot p) S) (X - C (AdjoinRoot.root p)) *
      Polynomial.map (algebraMap (AdjoinRoot p) S) q).Splits