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

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module L M
inst✝⁴ : Module L N
inst✝³ : Module K M
inst✝² : Module K N
inst✝¹ : IsScalarTower K L M
p : M →ₗ[L] N →ₗ[L] L
inst✝ : p.IsPerfPair
M' : Submodule K M
N' : Submodule K N
hM : span L ↑M' = ⊤
hN : span L ↑N' = ⊤
hp : ∀ x ∈ M', ∀ y ∈ N', (p x) y ∈ (algebraMap K L).range
this✝¹ : IsReflexive L M
this✝ : IsReflexive L N
v : Set M
hv₁ : v ⊆ ↑M'
hv₂ : span L v = ⊤
hv₃ : LinearIndependent L Subtype.val
b : Basis { x // x ∈ v } L M := Basis.mk hv₃ ⋯
this : Fintype ↑v
v' : ↑v → ↥M' := fun i ↦ ⟨↑i, ⋯⟩
hv' : LinearIndependent (ι := ↑v) K v'
m : M
hm : m ∈ M'
w : Set N
hw₁ : w ⊆ ↑N'
hw₂ : span L w = ⊤
hw₃ : LinearIndependent L Subtype.val
⊢ m ∈ span K v

Tactic:
]

Goal state after:
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module L M
inst✝⁴ : Module L N
inst✝³ : Module K M
inst✝² : Module K N
inst✝¹ : IsScalarTower K L M
p : M →ₗ[L] N →ₗ[L] L
inst✝ : p.IsPerfPair
M' : Submodule K M
N' : Submodule K N
hM : span L ↑M' = ⊤
hN : span L ↑N' = ⊤
hp : ∀ x ∈ M', ∀ y ∈ N', (p x) y ∈ (algebraMap K L).range
this✝¹ : IsReflexive L M
this✝ : IsReflexive L N
v : Set M
hv₁ : v ⊆ ↑M'
hv₂ : span L v = ⊤
hv₃ : LinearIndependent L Subtype.val
b : Basis { x // x ∈ v } L M := Basis.mk hv₃ ⋯
this : Fintype ↑v
v' : ↑v → ↥M' := fun i ↦ ⟨↑i, ⋯⟩
hv' : LinearIndependent (ι := ↑v) K v'
m : M
hm : m ∈ M'
w : Set N
hw₁ : w ⊆ ↑N'
hw₂ : span L w = ⊤
hw₃ : LinearIndependent L Subtype.val
⊢ m ∈ span K v
Module: Mathlib.RingTheory.Localization.NumDen

Goal state before:
pp: A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
a' b' c' : A
no_factor : IsRelPrime a' b'
b_nonzero : c' * b' ∈ nonZeroDivisors A
left✝ : c' ∈ nonZeroDivisors A
b'_nonzero : b' ∈ nonZeroDivisors A
hab : (algebraMap A K) c' * (algebraMap A K) a' = (algebraMap A K) c' * (algebraMap A K) b' * x
⊢ (algebraMap A K) c' * (algebraMap A K) a' = (algebraMap A K) c' * (algebraMap A K) a'

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Localization.NumDen

Goal state before:
pp: A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
⊢ mk' K (num A x) (den A x) = x

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
⊢ mk' K (num A x) (den A x) = x
Module: Mathlib.LinearAlgebra.PerfectPairing.Restrict

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module L M
inst✝⁴ : Module L N
inst✝³ : Module K M
inst✝² : Module K N
inst✝¹ : IsScalarTower K L M
p : M →ₗ[L] N →ₗ[L] L
inst✝ : p.IsPerfPair
M' : Submodule K M
N' : Submodule K N
hM : span L ↑M' = ⊤
hN : span L ↑N' = ⊤
hp : ∀ x ∈ M', ∀ y ∈ N', (p x) y ∈ (algebraMap K L).range
this✝¹ : IsReflexive L M
this✝ : IsReflexive L N
v : Set M
hv₁ : v ⊆ ↑M'
hv₂ : span L v = ⊤
hv₃ : LinearIndependent L Subtype.val
b : Basis { x // x ∈ v } L M := Basis.mk hv₃ ⋯
this : Fintype ↑v
v' : ↑v → ↥M' := fun i ↦ ⟨↑i, ⋯⟩
hv' : LinearIndependent (ι := ↑v) K v'
m : M
hm : m ∈ M'
w : Set N
hw₁ : w ⊆ ↑N'
hw₂ : span L w = ⊤
hw₃ : LinearIndependent L Subtype.val
⊢ span L w ≤ span L w

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Localization.NumDen

Goal state before:
pp: A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x y : K
h : x = y
⊢ y = y

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Localization.NumDen

Goal state before:
pp: A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x y : K
h : x = y
⊢ num A y * ↑(den A y) = num A y * ↑(den A y)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
k : ℕ
ih : (∏ x ∈ Finset.range k, (2 * ↑x + 1)) * eval 1 ((⇑derivative)^[k] (T R n)) = ∏ x ∈ Finset.range k, (↑n ^ 2 - ↑x ^ 2)
⊢ (↑n ^ 2 - ↑k ^ 2) * ∏ x ∈ Finset.range k, (↑n ^ 2 - ↑x ^ 2) =
    (↑n ^ 2 - ↑k ^ 2) * ∏ x ∈ Finset.range k, (↑n ^ 2 - ↑x ^ 2)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Localization.NumDen

Goal state before:
pp: A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
h : num A x = 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Localization.NumDen

Goal state before:
pp: case h
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
d : Aˣ
hd : ↑d = ↑(den A x)
d_ne_zero : (algebraMap A K) ↑(den A x) ≠ 0
⊢ ((algebraMap A K) ↑(den A x))⁻¹ * (algebraMap A K) (num A x) = mk' K (num A x) (den A x)

Tactic:
]

Goal state after:
case h
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
d : Aˣ
hd : ↑d = ↑(den A x)
d_ne_zero : (algebraMap A K) ↑(den A x) ≠ 0
⊢ ((algebraMap A K) ↑(den A x))⁻¹ * (algebraMap A K) (num A x) = mk' K (num A x) (den A x)
Module: Mathlib.RingTheory.Localization.NumDen

Goal state before:
pp: case h
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
d : Aˣ
hd : ↑d = ↑(den A x)
d_ne_zero : (algebraMap A K) ↑(den A x) ≠ 0
⊢ (algebraMap A K) (num A x) = (algebraMap A K) (num A x)

Tactic:
]

Goal state after:
Module: Mathlib.Algebra.Lie.Basis.Basic

Goal state before:
pp: ι : Type u_1
R : Type u_2
L : Type u_3
inst✝⁸ : Finite ι
inst✝⁷ : CommRing R
inst✝⁶ : LieRing L
inst✝⁵ : LieAlgebra R L
H : LieSubalgebra R L
b : Basis ι H
inst✝⁴ : Fintype ι
inst✝³ : IsDomain R
inst✝² : CharZero R
inst✝¹ : IsTorsionFree R L
inst✝ : IsNoetherian R L
this : IsLieAbelian ↥H := isLieAbelian_cartan b
⊢ rootSpace H 0 = rootSpace H 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Localization.NumDen

Goal state before:
pp: case h
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
h✝ : IsInteger A x
v : A
h : (algebraMap A K) v = x
⊢ (algebraMap A K) (num A x) / (algebraMap A K) ↑(den A x) = x

pp: case h
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
h✝ : IsInteger A x
v : A
h : (algebraMap A K) v = x
⊢ (algebraMap A K) ↑(den A x) ≠ 0

Tactic:
]

Goal state after:
case h
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
h✝ : IsInteger A x
v : A
h : (algebraMap A K) v = x
⊢ (algebraMap A K) (num A x) / (algebraMap A K) ↑(den A x) = x

case h
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
h✝ : IsInteger A x
v : A
h : (algebraMap A K) v = x
⊢ (algebraMap A K) ↑(den A x) ≠ 0
Module: Mathlib.LinearAlgebra.PerfectPairing.Restrict

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module L M
inst✝⁴ : Module L N
inst✝³ : Module K M
inst✝² : Module K N
inst✝¹ : IsScalarTower K L M
p : M →ₗ[L] N →ₗ[L] L
inst✝ : p.IsPerfPair
M' : Submodule K M
N' : Submodule K N
hM : span L ↑M' = ⊤
hN : span L ↑N' = ⊤
hp : ∀ x ∈ M', ∀ y ∈ N', (p x) y ∈ (algebraMap K L).range
this✝² : IsReflexive L M
this✝¹ : IsReflexive L N
v : Set M
hv₁ : v ⊆ ↑M'
hv₂ : span L v = ⊤
hv₃ : LinearIndependent L Subtype.val
b : Basis { x // x ∈ v } L M := Basis.mk hv₃ ⋯
this✝ : Fintype ↑v
v' : ↑v → ↥M' := fun i ↦ ⟨↑i, ⋯⟩
hv' : LinearIndependent (ι := ↑v) K v'
m : M
hm : m ∈ M'
w : Set N
hw₁ : w ⊆ ↑N'
hw₂ : span L w = ⊤
hw₃ : LinearIndependent L Subtype.val
bN : Basis { x // x ∈ w } L N := Basis.mk hw₃ ⋯
this : Fintype ↑w
⊢ finrank L N = finrank L N

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Localization.NumDen

Goal state before:
pp: A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
hx : x ≠ 0
this : Associated (num A x) ↑(den A x⁻¹)
⊢ Associated (num A x) ↑(den A x⁻¹)

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
hx : x ≠ 0
this : Associated (num A x) ↑(den A x⁻¹)
⊢ Associated (num A x) ↑(den A x⁻¹)
Module: Mathlib.RingTheory.Polynomial.RationalRoot

Goal state before:
pp: case e
A : Type u_1
S : Type u_4
inst✝³ : CommRing A
inst✝² : CommRing S
M : Submonoid A
inst✝¹ : Algebra A S
inst✝ : IsLocalization M S
p : A[X]
r : A
s : ↥M
hr : (aeval (mk' S r s)) p = 0
x✝ : A[X]
⊢ eval₂ (algebraMap A S) ((algebraMap A S) r) x✝ = eval₂ (algebraMap A S) ((algebraMap A S) r) x✝

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.RationalRoot

Goal state before:
pp: A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : UniqueFactorizationMonoid A
p : A[X]
x : K
hr : (aeval x) p = 0
⊢ (aeval x) p = 0

Tactic:
]

Goal state after:
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsFractionRing A K
inst✝¹ : IsDomain A
inst✝ : UniqueFactorizationMonoid A
p : A[X]
x : K
hr : (aeval x) p = 0
⊢ (aeval x) p = 0
Module: Mathlib.RingTheory.Localization.NumDen

Goal state before:
pp: A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
n : A
d : ↥(nonZeroDivisors A)
pr : IsRelPrime n ↑d
h : n * ↑(den A x) = num A x * ↑d
⊢ Associated (num A x) n ∧ Associated ↑(den A x) ↑d

Tactic:
]

Goal state after:
A : Type u_1
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
n : A
d : ↥(nonZeroDivisors A)
pr : IsRelPrime n ↑d
h : n * ↑(den A x) = num A x * ↑d
⊢ Associated (num A x) n ∧ Associated ↑(den A x) ↑d
Module: Mathlib.RingTheory.DedekindDomain.Basic

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing A
hA : DimensionLEOne A
e : R ≃+* A
P : Ideal R
hP_ne : P ≠ ⊥
hP_prime : P.IsPrime
⊢ (comap e.symm P).IsMaximal

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝¹ : CommRing R
inst✝ : CommRing A
hA : DimensionLEOne A
e : R ≃+* A
P : Ideal R
hP_ne : P ≠ ⊥
hP_prime : P.IsPrime
⊢ (comap e.symm P).IsMaximal
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
n : ℤ
k : ℕ
ih :
  (∏ x ∈ Finset.range k, (2 * ↑x + 3)) * eval 1 ((⇑derivative)^[k] (U R n)) =
    (∏ x ∈ Finset.range k, ((↑n + 1) ^ 2 - (↑x + 1) ^ 2)) * (↑n + 1)
⊢ ((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * ((∏ x ∈ Finset.range k, ((↑n + 1) ^ 2 - (↑x + 1) ^ 2)) * (↑n + 1)) =
    ((↑n + 1) ^ 2 - (↑k + 1) ^ 2) * ((∏ x ∈ Finset.range k, ((↑n + 1) ^ 2 - (↑x + 1) ^ 2)) * (↑n + 1))

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: 𝔽 : Type u_3
inst✝¹ : Field 𝔽
inst✝ : CharZero 𝔽
n : ℤ
k : ℕ
⊢ ↑(∏ l ∈ Finset.range k, (n ^ 2 - ↑l ^ 2)) = ↑(∏ l ∈ Finset.range k, (n ^ 2 - ↑l ^ 2))

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: 𝔽 : Type u_3
inst✝¹ : Field 𝔽
inst✝ : CharZero 𝔽
n : ℤ
k : ℕ
⊢ ↑(∏ l ∈ Finset.range k, ((n + 1) ^ 2 - (↑l + 1) ^ 2)) * (↑n + 1) =
    ↑(∏ l ∈ Finset.range k, ((n + 1) ^ 2 - (↑l + 1) ^ 2)) * (↑n + 1)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.RationalRoot

Goal state before:
pp: case e'_4
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
r : K
hr : (aeval r) p = 0
⊢ (p.scaleRoots ↑(den A r)).coeff 0 = (p.scaleRoots ↑(den A r)).coeff 0

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Chebyshev

Goal state before:
pp: case one
R : Type u_1
inst✝ : CommRing R
m : ℤ
⊢ 2 * T R m * X = 2 * X * T R m - T R (m - 1) + T R (m - 1)

Tactic:
]

Goal state after:
case one
R : Type u_1
inst✝ : CommRing R
m : ℤ
⊢ 2 * T R m * X = 2 * X * T R m - T R (m - 1) + T R (m - 1)
Module: Mathlib.RingTheory.Polynomial.RationalRoot

Goal state before:
pp: A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
r : K
hr : (aeval r) p = 0
⊢ ↑(den A r) ∣ (p.scaleRoots ?s).coeff p.natDegree * num A r ^ p.natDegree

pp: case s
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
r : K
hr : (aeval r) p = 0
⊢ A

Tactic:
]

Goal state after:
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
r : K
hr : (aeval r) p = 0
⊢ ↑(den A r) ∣ (p.scaleRoots ?s).coeff p.natDegree * num A r ^ p.natDegree

case s
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
r : K
hr : (aeval r) p = 0
⊢ A
Module: Mathlib.RingTheory.Polynomial.RationalRoot

Goal state before:
pp: case pos
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
r : K
hr : (aeval r) p = 0
j : ℕ
hj : j ≠ p.natDegree
h : j < p.natDegree
⊢ ↑(den A r) ∣ p.coeff j * ↑(den A r) ^ (p.natDegree - j) * num A r ^ j

Tactic:
]

Goal state after:
case pos
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
r : K
hr : (aeval r) p = 0
j : ℕ
hj : j ≠ p.natDegree
h : j < p.natDegree
⊢ ↑(den A r) ∣ p.coeff j * ↑(den A r) ^ (p.natDegree - j) * num A r ^ j
Module: Mathlib.RingTheory.Polynomial.RationalRoot

Goal state before:
pp: case neg
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
r : K
hr : (aeval r) p = 0
j : ℕ
hj : j ≠ (p.scaleRoots ↑(den A r)).natDegree
h : (p.scaleRoots ↑(den A r)).natDegree ≤ j
⊢ ↑(den A r) ∣ (p.scaleRoots ↑(den A r)).coeff j * num A r ^ j

Tactic:
]

Goal state after:
case neg
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
r : K
hr : (aeval r) p = 0
j : ℕ
hj : j ≠ (p.scaleRoots ↑(den A r)).natDegree
h : (p.scaleRoots ↑(den A r)).natDegree ≤ j
⊢ ↑(den A r) ∣ (p.scaleRoots ↑(den A r)).coeff j * num A r ^ j
Module: Mathlib.RingTheory.Polynomial.RationalRoot

Goal state before:
pp: case neg
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
r : K
hr : (aeval r) p = 0
j : ℕ
hj : j ≠ (p.scaleRoots ↑(den A r)).natDegree
h : (p.scaleRoots ↑(den A r)).natDegree ≤ j
⊢ ↑(den A r) ∣ 0

Tactic:
]

Goal state after:
case neg
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
r : K
hr : (aeval r) p = 0
j : ℕ
hj : j ≠ (p.scaleRoots ↑(den A r)).natDegree
h : (p.scaleRoots ↑(den A r)).natDegree ≤ j
⊢ ↑(den A r) ∣ 0
Module: Mathlib.RingTheory.Ideal.Quotient.PowTransition

Goal state before:
pp: case refine_1
R : Type u_1
inst✝² : Ring R
I J : Ideal R
H : I ≤ J
inst✝¹ : I.IsTwoSided
inst✝ : J.IsTwoSided
x : R ⧸ I
r : R
h : (mk I) r ∈ RingHom.ker (factor H)
hr : (mk I) r = x
⊢ (mk I) r ∈ map (mk I) J

Tactic:
]

Goal state after:
case refine_1
R : Type u_1
inst✝² : Ring R
I J : Ideal R
H : I ≤ J
inst✝¹ : I.IsTwoSided
inst✝ : J.IsTwoSided
x : R ⧸ I
r : R
h : (mk I) r ∈ RingHom.ker (factor H)
hr : (mk I) r = x
⊢ (mk I) r ∈ map (mk I) J
Module: Mathlib.RingTheory.Polynomial.RationalRoot

Goal state before:
pp: case left
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
hp : p.Monic
r : K
hr : (aeval r) p = 0
inv : A
h_inv : 1 = ↑(den A r) * inv
d_ne_zero : (algebraMap A K) ↑(den A r) ≠ 0
⊢ (algebraMap A K) (num A r) / (algebraMap A K) ↑(den A r) = (algebraMap A K) (num A r * inv)

Tactic:
]

Goal state after:
case left
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
hp : p.Monic
r : K
hr : (aeval r) p = 0
inv : A
h_inv : 1 = ↑(den A r) * inv
d_ne_zero : (algebraMap A K) ↑(den A r) ≠ 0
⊢ (algebraMap A K) (num A r) / (algebraMap A K) ↑(den A r) = (algebraMap A K) (num A r * inv)
Module: Mathlib.RingTheory.Ideal.Quotient.PowTransition

Goal state before:
pp: case zero
R : Type u_1
inst✝² : Ring R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p : ℕ → Submodule R M
hp : Antitone p
x : (n : ℕ) → M ⧸ p n
h : ∀ (m : ℕ), x m = (factor ⋯) (x (m + 1))
m n : ℕ
g : m ≤ n
this : n = m
hmn : n - m = 0
⊢ x m = (factor ⋯) (x n)

Tactic:
]

Goal state after:
case zero
R : Type u_1
inst✝² : Ring R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p : ℕ → Submodule R M
hp : Antitone p
x : (n : ℕ) → M ⧸ p n
h : ∀ (m : ℕ), x m = (factor ⋯) (x (m + 1))
m n : ℕ
g : m ≤ n
this : n = m
hmn : n - m = 0
⊢ x m = (factor ⋯) (x n)
Module: Mathlib.RingTheory.Polynomial.RationalRoot

Goal state before:
pp: case left
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : UniqueFactorizationMonoid A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
p : A[X]
hp : p.Monic
r : K
hr : (aeval r) p = 0
inv : A
h_inv : 1 = ↑(den A r) * inv
d_ne_zero : (algebraMap A K) ↑(den A r) ≠ 0
⊢ (algebraMap A K) (num A r) = (algebraMap A K) (num A r)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.Ideal.Quotient.PowTransition

Goal state before:
pp: case succ
R : Type u_1
inst✝² : Ring R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p : ℕ → Submodule R M
hp : Antitone p
x : (n : ℕ) → M ⧸ p n
h : ∀ (m : ℕ), x m = (factor ⋯) (x (m + 1))
k : ℕ
ih : ∀ {m n : ℕ} (g : m ≤ n), n = m + (n - m) → n - m = k → x m = (factor ⋯) (x n)
m n : ℕ
g : m ≤ n
this : n = m + k + 1
hmn : n - m = k + 1
⊢ x m = (factor ⋯) (x n)

Tactic:
]

Goal state after:
case succ
R : Type u_1
inst✝² : Ring R
M : Type u_2
inst✝¹ : AddCommGroup M
inst✝ : Module R M
p : ℕ → Submodule R M
hp : Antitone p
x : (n : ℕ) → M ⧸ p n
h : ∀ (m : ℕ), x m = (factor ⋯) (x (m + 1))
k : ℕ
ih : ∀ {m n : ℕ} (g : m ≤ n), n = m + (n - m) → n - m = k → x m = (factor ⋯) (x n)
m n : ℕ
g : m ≤ n
this : n = m + k + 1
hmn : n - m = k + 1
⊢ x m = (factor ⋯) (x n)
Module: Mathlib.LinearAlgebra.PerfectPairing.Restrict

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module L M
inst✝⁴ : Module L N
inst✝³ : Module K M
inst✝² : Module K N
inst✝¹ : IsScalarTower K L M
p : M →ₗ[L] N →ₗ[L] L
inst✝ : p.IsPerfPair
M' : Submodule K M
N' : Submodule K N
hM : span L ↑M' = ⊤
hN : span L ↑N' = ⊤
this✝² : IsReflexive L M
this✝¹ : IsReflexive L N
v : Set M
hv₁ : v ⊆ ↑M'
hv₂ : span L v = ⊤
hv₃ : LinearIndependent L Subtype.val
b : Basis { x // x ∈ v } L M := Basis.mk hv₃ ⋯
this✝ : Fintype ↑v
v' : ↑v → ↥M' := fun i ↦ ⟨↑i, ⋯⟩
hv' : LinearIndependent (ι := ↑v) K v'
m : M
hm : m ∈ M'
w : Set N
hw₁ : w ⊆ ↑N'
hw₂ : span L w = ⊤
hw₃ : LinearIndependent L Subtype.val
bN : Basis { x // x ∈ w } L N := Basis.mk hw₃ ⋯
this : Fintype ↑w
e : ↑v ≃ ↑w
bM : Basis { x // x ∈ w } L M := bN.dualBasis.map p.toPerfPair.symm
hbM : ∀ (j : ↑w), ∀ x ∈ M', (bM.repr x) j = (p x) ↑j
hj : ∀ (j : ↑w), (bM.repr m) j ∈ (algebraMap K L).range
hp : ∀ (i : ↑w) (j : ↑v), (bN.dualBasis.map p.toPerfPair.symm).toMatrix (⇑b) i j ∈ (algebraMap K L).fieldRange
hA : ∀ (i : { x // x ∈ v }) (j : { x // x ∈ w }), b.toMatrix (⇑bM) i j ∈ (algebraMap K L).range
h_span : span K v = span K (Set.range ⇑b)
⊢ ∀ (i : { x // x ∈ v }), (b.toMatrix ⇑bM).mulVec (⇑(bM.repr m)) i ∈ Set.range ⇑(algebraMap K L)

Tactic:
]

Goal state after:
K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝¹⁰ : Field K
inst✝⁹ : Field L
inst✝⁸ : Algebra K L
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup N
inst✝⁵ : Module L M
inst✝⁴ : Module L N
inst✝³ : Module K M
inst✝² : Module K N
inst✝¹ : IsScalarTower K L M
p : M →ₗ[L] N →ₗ[L] L
inst✝ : p.IsPerfPair
M' : Submodule K M
N' : Submodule K N
hM : span L ↑M' = ⊤
hN : span L ↑N' = ⊤
this✝² : IsReflexive L M
this✝¹ : IsReflexive L N
v : Set M
hv₁ : v ⊆ ↑M'
hv₂ : span L v = ⊤
hv₃ : LinearIndependent L Subtype.val
b : Basis { x // x ∈ v } L M := Basis.mk hv₃ ⋯
this✝ : Fintype ↑v
v' : ↑v → ↥M' := fun i ↦ ⟨↑i, ⋯⟩
hv' : LinearIndependent (ι := ↑v) K v'
m : M
hm : m ∈ M'
w : Set N
hw₁ : w ⊆ ↑N'
hw₂ : span L w = ⊤
hw₃ : LinearIndependent L Subtype.val
bN : Basis { x // x ∈ w } L N := Basis.mk hw₃ ⋯
this : Fintype ↑w
e : ↑v ≃ ↑w
bM : Basis { x // x ∈ w } L M := bN.dualBasis.map p.toPerfPair.symm
hbM : ∀ (j : ↑w), ∀ x ∈ M', (bM.repr x) j = (p x) ↑j
hj : ∀ (j : ↑w), (bM.repr m) j ∈ (algebraMap K L).range
hp : ∀ (i : ↑w) (j : ↑v), (bN.dualBasis.map p.toPerfPair.symm).toMatrix (⇑b) i j ∈ (algebraMap K L).fieldRange
hA : ∀ (i : { x // x ∈ v }) (j : { x // x ∈ w }), b.toMatrix (⇑bM) i j ∈ (algebraMap K L).range
h_span : span K v = span K (Set.range ⇑b)
⊢ ∀ (i : { x // x ∈ v }), (b.toMatrix ⇑bM).mulVec (⇑(bM.repr m)) i ∈ Set.range ⇑(algebraMap K L)