mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.ModelTheory.Definability

Goal state before:
pp: case mp
M : Type w
A : Set M
L : Language
inst✝ : L.Structure M
α : Type u₁
m : ℕ
φ : L[[↑A]].Formula (α ⊕ Fin m)
x : α → M
⊢ (∃ x_1, φ.Realize x_1 ∧ x_1 ∘ Sum.inl = x) → ∃ xs, BoundedFormula.Realize φ (Sum.elim x xs) (xs ∘ Fin.natAdd m)

pp: case mpr
M : Type w
A : Set M
L : Language
inst✝ : L.Structure M
α : Type u₁
m : ℕ
φ : L[[↑A]].Formula (α ⊕ Fin m)
x : α → M
⊢ (∃ xs, BoundedFormula.Realize φ (Sum.elim x xs) (xs ∘ Fin.natAdd m)) → ∃ x_1, φ.Realize x_1 ∧ x_1 ∘ Sum.inl = x

Tactic:
· rintro ⟨y, hy, rfl⟩
    exact
      ⟨y ∘ Sum.inr, (congr (congr rfl (Sum.elim_comp_inl_inr y).symm) (funext finZeroElim)).mp hy⟩

Goal state after:
case mpr
M : Type w
A : Set M
L : Language
inst✝ : L.Structure M
α : Type u₁
m : ℕ
φ : L[[↑A]].Formula (α ⊕ Fin m)
x : α → M
⊢ (∃ xs, BoundedFormula.Realize φ (Sum.elim x xs) (xs ∘ Fin.natAdd m)) → ∃ x_1, φ.Realize x_1 ∧ x_1 ∘ Sum.inl = x
Module: Mathlib.ModelTheory.Skolem

Goal state before:
pp: case H
L : Language
n : ℕ
⊢ lift.{max u v, u} #(L.Functions n) ≤ #(L.BoundedFormula Empty (n + 1))

Tactic:
← lift_le.{_, max u v},

Goal state after:
case H
L : Language
n : ℕ
⊢ lift.{?u.62, max u v} (lift.{max u v, u} #(L.Functions n)) ≤ lift.{?u.62, max u v} #(L.BoundedFormula Empty (n + 1))
Module: Mathlib.ModelTheory.Satisfiability

Goal state before:
pp: L : Language
T : L.Theory
α : Type w
φ : L.Formula α
h : T ⊨ᵇ φ
M : Type u_1
inst✝² : L.Structure M
inst✝¹ : M ⊨ T
inst✝ : Nonempty M
v : α → M
⊢ φ.Realize v

Tactic:
models_formula_iff_onTheory_models_equivSentence

Goal state after:
L : Language
T : L.Theory
α : Type w
φ : L.Formula α
h : (L.lhomWithConstants α).onTheory T ⊨ᵇ Formula.equivSentence φ
M : Type u_1
inst✝² : L.Structure M
inst✝¹ : M ⊨ T
inst✝ : Nonempty M
v : α → M
⊢ φ.Realize v
Module: Mathlib.FieldTheory.Differential.Basic

Goal state before:
pp: case succ.inr
R : Type u_1
inst✝¹ : Field R
inst✝ : Differential R
a : R
n : ℕ
h2 : logDeriv (a ^ n) = ↑n * logDeriv a
hb : a ≠ 0
⊢ logDeriv (a ^ n * a) = logDeriv (a ^ n) + logDeriv a

Tactic:
logDeriv_mul

Goal state after:
case succ.inr
R : Type u_1
inst✝¹ : Field R
inst✝ : Differential R
a : R
n : ℕ
h2 : logDeriv (a ^ n) = ↑n * logDeriv a
hb : a ≠ 0
⊢ logDeriv (a ^ n) + logDeriv a = logDeriv (a ^ n) + logDeriv a

case succ.inr.ha
R : Type u_1
inst✝¹ : Field R
inst✝ : Differential R
a : R
n : ℕ
h2 : logDeriv (a ^ n) = ↑n * logDeriv a
hb : a ≠ 0
⊢ a ^ n ≠ 0

case succ.inr.hb
R : Type u_1
inst✝¹ : Field R
inst✝ : Differential R
a : R
n : ℕ
h2 : logDeriv (a ^ n) = ↑n * logDeriv a
hb : a ≠ 0
⊢ a ≠ 0
Module: Mathlib.FieldTheory.Differential.Basic

Goal state before:
pp: R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : Differential R
a b : R
F : Type u_2
inst✝⁴ : Field F
inst✝³ : Differential F
inst✝² : CharZero F
p : F[X]
inst✝¹ : Fact (Irreducible p)
inst✝ : Fact p.Monic
q : F[X]
⊢ p ∣
    (implicitDeriv
        ((AdjoinRoot.modByMonicHom ⋯)
          (-(aeval (AdjoinRoot.root p)) (mapCoeffs p) / (aeval (AdjoinRoot.root p)) (derivative p))))
      p

Tactic:
rw [← AdjoinRoot.mk_eq_zero]

Goal state after:
R : Type u_1
inst✝⁶ : Field R
inst✝⁵ : Differential R
a b : R
F : Type u_2
inst✝⁴ : Field F
inst✝³ : Differential F
inst✝² : CharZero F
p : F[X]
inst✝¹ : Fact (Irreducible p)
inst✝ : Fact p.Monic
q : F[X]
⊢ (AdjoinRoot.mk p)
      ((implicitDeriv
          ((AdjoinRoot.modByMonicHom ⋯)
            (-(aeval (AdjoinRoot.root p)) (mapCoeffs p) / (aeval (AdjoinRoot.root p)) (derivative p))))
        p) =
    0
Module: Mathlib.FieldTheory.Finite.Extension

Goal state before:
pp: k : Type u_1
inst✝¹ : Field k
inst✝ : Finite k
f : k[X]
hi : Irreducible f
n : ℕ
hdvd : f.natDegree ∣ n
⊢ f ∣ X ^ Nat.card k ^ f.natDegree - X

Tactic:
let a := AdjoinRoot.root f

Goal state after:
k : Type u_1
inst✝¹ : Field k
inst✝ : Finite k
f : k[X]
hi : Irreducible f
n : ℕ
hdvd : f.natDegree ∣ n
a : AdjoinRoot f := AdjoinRoot.root f
⊢ f ∣ X ^ Nat.card k ^ f.natDegree - X
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x1 x2 y : ℕ × K
H : R K p x1 x2
n : ℕ
x : K
⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 + (⇑(frobenius K p))^[(n, x).1] y.2)
    ((n + 1, (frobenius K p) x).1 + y.1,
      (frobenius K p) ((⇑(frobenius K p))^[y.1] x) + (frobenius K p) ((⇑(frobenius K p))^[n] y.2))

Tactic:
← map_add,

Goal state after:
K : Type u
inst✝² : CommRing K
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : CharP K p
x1 x2 y : ℕ × K
H : R K p x1 x2
n : ℕ
x : K
⊢ R K p ((n, x).1 + y.1, (⇑(frobenius K p))^[y.1] (n, x).2 + (⇑(frobenius K p))^[(n, x).1] y.2)
    ((n + 1, (frobenius K p) x).1 + y.1, (frobenius K p) ((⇑(frobenius K p))^[y.1] x + (⇑(frobenius K p))^[n] y.2))
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : CommSemiring K
inst✝² : CommSemiring L
i : K →+* L
p : ℕ
inst✝¹ : ExpChar L p
inst✝ : PerfectRing L p
x : K
h✝ : x ∈ pNilradical K p
n : ℕ
h : (iterateFrobeniusEquiv L p n).symm ((iterateFrobenius L p n) (i x)) = (iterateFrobeniusEquiv L p n).symm (i 0)
⊢ x ∈ ker i

Tactic:
← iterateFrobeniusEquiv_apply,

Goal state after:
K : Type u_1
L : Type u_2
inst✝³ : CommSemiring K
inst✝² : CommSemiring L
i : K →+* L
p : ℕ
inst✝¹ : ExpChar L p
inst✝ : PerfectRing L p
x : K
h✝ : x ∈ pNilradical K p
n : ℕ
h : (iterateFrobeniusEquiv L p n).symm ((iterateFrobeniusEquiv L p n) (i x)) = (iterateFrobeniusEquiv L p n).symm (i 0)
⊢ x ∈ ker i
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : CommSemiring K
inst✝³ : CommSemiring L
i : K →+* L
p : ℕ
inst✝² : IsPRadical i p
inst✝¹ : ExpChar L p
inst✝ : PerfectRing L p
x : L
⊢ (iterateFrobeniusEquiv L p (Classical.choose ⋯).1).symm ((iterateFrobenius L p (Classical.choose ⋯).1) x) = x

Tactic:
← iterateFrobeniusEquiv_apply,

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁴ : CommSemiring K
inst✝³ : CommSemiring L
i : K →+* L
p : ℕ
inst✝² : IsPRadical i p
inst✝¹ : ExpChar L p
inst✝ : PerfectRing L p
x : L
⊢ (iterateFrobeniusEquiv L p (Classical.choose ⋯).1).symm ((iterateFrobeniusEquiv L p (Classical.choose ⋯).1) x) = x
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
M : Type u_3
inst✝³ : CommSemiring K
inst✝² : CommSemiring M
j : K →+* M
p : ℕ
inst✝¹ : ExpChar M p
inst✝ : PerfectRing M p
x : K
this : (Classical.choose ⋯).2 = x ^ p ^ (Classical.choose ⋯).1
⊢ (iterateFrobeniusEquiv M p (Classical.choose ⋯).1).symm ((iterateFrobenius M p (Classical.choose ⋯).1) (j x)) = j x

Tactic:
← iterateFrobeniusEquiv_apply,

Goal state after:
K : Type u_1
M : Type u_3
inst✝³ : CommSemiring K
inst✝² : CommSemiring M
j : K →+* M
p : ℕ
inst✝¹ : ExpChar M p
inst✝ : PerfectRing M p
x : K
this : (Classical.choose ⋯).2 = x ^ p ^ (Classical.choose ⋯).1
⊢ (iterateFrobeniusEquiv M p (Classical.choose ⋯).1).symm ((iterateFrobeniusEquiv M p (Classical.choose ⋯).1) (j x)) =
    j x
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : CommRing K
inst✝⁷ : CommRing L
inst✝⁶ : CommRing M
inst✝⁵ : CommRing N
i : K →+* L
j : K →+* M
k : K →+* N
f : L →+* M
g : L →+* N
p : ℕ
inst✝⁴ : ExpChar M p
inst✝³ : ExpChar K p
inst✝² : PerfectRing M p
inst✝¹ : IsPRadical i p
inst✝ : ExpChar L p
x1 x2 : L
n1 : ℕ
y1 : K
h1 : i y1 = x1 ^ p ^ n1
n2 : ℕ
y2 : K
h2 : i y2 = x2 ^ p ^ n2
⊢ i (y1 ^ p ^ n2 * y2 ^ p ^ n1) = (x1 * x2) ^ p ^ (n1 + n2)

Tactic:
by rw [map_mul,
        map_pow, map_pow, h1, h2, ← pow_mul, ← pow_add, ← pow_mul, ← pow_add,
        add_comm n2, mul_pow]

Goal state after:
Module: Mathlib.FieldTheory.IsRealClosed.Basic

Goal state before:
pp: R : Type u
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
isSquare_of_nonneg : ∀ {x : R}, 0 ≤ x → IsSquare x
exists_isRoot_of_odd_natDegree : ∀ {f : R[X]}, Odd f.natDegree → ∃ x, f.IsRoot x
x : R
⊢ IsSquare x ∨ IsSquare (-x)

Tactic:
rcases le_total x 0 with (neg | pos)

Goal state after:
case inl
R : Type u
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
isSquare_of_nonneg : ∀ {x : R}, 0 ≤ x → IsSquare x
exists_isRoot_of_odd_natDegree : ∀ {f : R[X]}, Odd f.natDegree → ∃ x, f.IsRoot x
x : R
neg : x ≤ 0
⊢ IsSquare x ∨ IsSquare (-x)

case inr
R : Type u
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
isSquare_of_nonneg : ∀ {x : R}, 0 ≤ x → IsSquare x
exists_isRoot_of_odd_natDegree : ∀ {f : R[X]}, Odd f.natDegree → ∃ x, f.IsRoot x
x : R
pos : 0 ≤ x
⊢ IsSquare x ∨ IsSquare (-x)
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
  ∀ m < n,
    ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
      ⋃ i ∈ s, g i • ↑(H i) = Set.univ →
        ∀ j ∈ s,
          ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →
            m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
j : ι
hj : j ∈ s
hcovers' : ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ i ∈ s, H i = H j → ↑(g i) ≠ ↑x
y z : G
hz : z ∈ y • ↑(H j)
⊢ ∃ i, ∃ (_ : i ∈ s ∧ H i ≠ H j), z ∈ (y * x⁻¹ * g i) • ↑(H i)

Tactic:
have ⟨i, hi, hmem⟩ : ∃ i ∈ s, x * (y⁻¹ * z) ∈ g i • (H i : Set G) := by
        simpa using Set.eq_univ_iff_forall.mp hcovers (x * (y⁻¹ * z))

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : DecidableEq (Subgroup G)
n : ℕ
ih :
  ∀ m < n,
    ∀ {ι : Type u_2} {H : ι → Subgroup G} {g : ι → G} {s : Finset ι},
      ⋃ i ∈ s, g i • ↑(H i) = Set.univ →
        ∀ j ∈ s,
          ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ →
            m = (Finset.image H s).card → ∃ i ∈ s, H i ≠ H j ∧ (H i).FiniteIndex
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
j : ι
hj : j ∈ s
hcovers' : ⋃ i ∈ {x ∈ s | H x = H j}, g i • ↑(H i) ≠ Set.univ
hn : n = (Finset.image H s).card
x : G
hx : ∀ i ∈ s, H i = H j → ↑(g i) ≠ ↑x
y z : G
hz : z ∈ y • ↑(H j)
i : ι
hi : i ∈ s
hmem : x * (y⁻¹ * z) ∈ g i • ↑(H i)
⊢ ∃ i, ∃ (_ : i ∈ s ∧ H i ≠ H j), z ∈ (y * x⁻¹ * g i) • ↑(H i)
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case pos
K : Type u
inst✝ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
this✝ : Fact (Irreducible (X ^ n - C a)) := { out := H }
this : Algebra K K[n√a] := inferInstance
hn : 0 < n
hn' : n = 1
⊢ (X ^ n - C a).Separable

Tactic:
rw [hn', pow_one]

Goal state after:
case pos
K : Type u
inst✝ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
this✝ : Fact (Irreducible (X ^ n - C a)) := { out := H }
this : Algebra K K[n√a] := inferInstance
hn : 0 < n
hn' : n = 1
⊢ (X - C a).Separable
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
hc : a * c = c * a
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc' : c⁻¹ * a = a * c⁻¹
c_eq : a * (⇑((ad (↥k) D) a))^[n - 1] b - (⇑((ad (↥k) D) a))^[n - 1] b * a = c
eq1 : c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b - c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = 1
deq : d - a⁻¹ * d * a = 1
⊢ False

Tactic:
sub_eq_iff_eq_add

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
n : ℕ
hn : 0 < n
c : D := (⇑((ad (↥k) D) a))^[n] b
hb : c ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
hc_def : c = (⇑((ad (↥k) D) a))^[n] b
x✝ : Invertible c := { invOf := c⁻¹, invOf_mul_self := ⋯, mul_invOf_self := ⋯ }
hc : a * c = c * a
d : D := c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hd_def : d = c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b
hc' : c⁻¹ * a = a * c⁻¹
c_eq : a * (⇑((ad (↥k) D) a))^[n - 1] b - (⇑((ad (↥k) D) a))^[n - 1] b * a = c
eq1 : c⁻¹ * a * (⇑((ad (↥k) D) a))^[n - 1] b - c⁻¹ * (⇑((ad (↥k) D) a))^[n - 1] b * a = 1
deq : d = 1 + a⁻¹ * d * a
⊢ False
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: K : Type u
inst✝⁴ : Field K
n : ℕ
a : K
H : Irreducible (X ^ n - C a)
L : Type u_1
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : IsSplittingField K L (X ^ n - C a)
α : L
hα : α ^ n = (algebraMap K L) a
inst✝ : NeZero n
ζ : K
hζ : IsPrimitiveRoot ζ n
m : ℤ
hn : 0 < n
⊢ ((autEquivZmod H L hζ).symm (Multiplicative.ofAdd ↑m)) α = ζ ^ m • α

Tactic:
← autEquivRootsOfUnity_smul ⟨ζ, (mem_primitiveRoots hn).mpr hζ⟩ H L hα

Goal state after:
K : Type u
inst✝⁴ : Field K
n : ℕ
a : K
H : Irreducible (X ^ n - C a)
L : Type u_1
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : IsSplittingField K L (X ^ n - C a)
α : L
hα : α ^ n = (algebraMap K L) a
inst✝ : NeZero n
ζ : K
hζ : IsPrimitiveRoot ζ n
m : ℤ
hn : 0 < n
⊢ (autEquivRootsOfUnity ⋯ H L) ((autEquivZmod H L hζ).symm (Multiplicative.ofAdd ↑m)) • α = ζ ^ m • α
Module: Mathlib.FieldTheory.Minpoly.ConjRootClass

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsAlgebraic K L
c : ConjRootClass K L
x : L
⊢ c.minpoly ≠ 0 ∧ (aeval x) c.minpoly = 0 ↔ mk K x = c

Tactic:
aeval_minpoly_iff x c

Goal state after:
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsAlgebraic K L
c : ConjRootClass K L
x : L
⊢ c.minpoly ≠ 0 ∧ mk K x = c ↔ mk K x = c
Module: Mathlib.FieldTheory.Minpoly.ConjRootClass

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
c : ConjRootClass K L
inst✝ : Fintype ↑c.carrier
⊢ c.minpoly.aroots L = c.carrier.toFinset.val

Tactic:
← rootSet_minpoly_eq_carrier,

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
c : ConjRootClass K L
inst✝ : Fintype ↑c.carrier
⊢ c.minpoly.aroots L = (c.minpoly.rootSet L).toFinset.val
Module: Mathlib.FieldTheory.Minpoly.ConjRootClass

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
c : ConjRootClass K L
inst✝ : Fintype ↑c.carrier
⊢ c.minpoly.aroots L = (↑(c.minpoly.aroots L).toFinset).toFinset.val

Tactic:
Finset.toFinset_coe,

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
c : ConjRootClass K L
inst✝ : Fintype ↑c.carrier
⊢ c.minpoly.aroots L = (c.minpoly.aroots L).toFinset.val
Module: Mathlib.FieldTheory.Minpoly.ConjRootClass

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
inst✝¹ : Normal K L
c : ConjRootClass K L
inst✝ : Fintype ↑c.carrier
⊢ map (algebraMap K L) c.minpoly = ∏ x ∈ c.carrier.toFinset, (X - C x)

Tactic:
← rootSet_minpoly_eq_carrier,

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
inst✝¹ : Normal K L
c : ConjRootClass K L
inst✝ : Fintype ↑c.carrier
⊢ map (algebraMap K L) c.minpoly = ∏ x ∈ (c.minpoly.rootSet L).toFinset, (X - C x)
Module: Mathlib.FieldTheory.Minpoly.ConjRootClass

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
inst✝¹ : Normal K L
c : ConjRootClass K L
inst✝ : Fintype ↑c.carrier
⊢ map (algebraMap K L) c.minpoly = (Multiset.map (fun x ↦ X - C x) (↑(c.minpoly.aroots L).toFinset).toFinset.val).prod

Tactic:
Finset.toFinset_coe,

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : Algebra.IsSeparable K L
inst✝¹ : Normal K L
c : ConjRootClass K L
inst✝ : Fintype ↑c.carrier
⊢ map (algebraMap K L) c.minpoly = (Multiset.map (fun x ↦ X - C x) (c.minpoly.aroots L).toFinset.val).prod
Module: Mathlib.FieldTheory.NormalizedTrace

Goal state before:
pp: F : Type u_1
K : Type u_2
inst✝⁴ : Field F
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : CharZero F
inst✝ : Algebra.IsIntegral F K
a b : K
E : IntermediateField F K := F⟮a⟯ ⊔ F⟮b⟯
this✝ : FiniteDimensional F ↥F⟮a⟯
this : FiniteDimensional F ↥F⟮b⟯
ha : a ∈ E
hb : b ∈ E
hab : a + b ∈ E
a' : ↥E := ⟨a, ha⟩
b' : ↥E := ⟨b, hb⟩
ab' : ↥E := ⟨a + b, hab⟩
⊢ (↑(Module.finrank F ↥E))⁻¹ • (trace F ↥E) ab' = (↑(Module.finrank F ↥E))⁻¹ • ((trace F ↥E) a' + (trace F ↥E) b')

Tactic:
← map_add,

Goal state after:
F : Type u_1
K : Type u_2
inst✝⁴ : Field F
inst✝³ : Field K
inst✝² : Algebra F K
inst✝¹ : CharZero F
inst✝ : Algebra.IsIntegral F K
a b : K
E : IntermediateField F K := F⟮a⟯ ⊔ F⟮b⟯
this✝ : FiniteDimensional F ↥F⟮a⟯
this : FiniteDimensional F ↥F⟮b⟯
ha : a ∈ E
hb : b ∈ E
hab : a + b ∈ E
a' : ↥E := ⟨a, ha⟩
b' : ↥E := ⟨b, hb⟩
ab' : ↥E := ⟨a + b, hab⟩
⊢ (↑(Module.finrank F ↥E))⁻¹ • (trace F ↥E) ab' = (↑(Module.finrank F ↥E))⁻¹ • (trace F ↥E) (a' + b')
Module: Mathlib.FieldTheory.PurelyInseparable.Exponent

Goal state before:
pp: K : Type u_2
L : Type u_3
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsPurelyInseparable K L
a : L
⊢ (minpoly K a).natDegree = ringExpChar K ^ elemExponent K a

Tactic:
by
  rw [minpoly_eq K a, natDegree_sub_C, natDegree_pow, natDegree_X, mul_one]

Goal state after:
Module: Mathlib.FieldTheory.PurelyInseparable.PerfectClosure

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝ : Algebra.IsSeparable F E
h : Submodule.span F (Set.range v) = ⊤
⊢ Submodule.span F (Set.range fun x ↦ v x ^ q ^ n) = Subalgebra.toSubmodule ⊤.toSubalgebra

Tactic:
← adjoin_univ,

Goal state after:
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
q n : ℕ
hF : ExpChar F q
ι : Type u_1
v : ι → E
inst✝ : Algebra.IsSeparable F E
h : Submodule.span F (Set.range v) = ⊤
⊢ Submodule.span F (Set.range fun x ↦ v x ^ q ^ n) = Subalgebra.toSubmodule (adjoin F Set.univ).toSubalgebra
Module: Mathlib.FieldTheory.PurelyInseparable.Tower

Goal state before:
pp: case neg
F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : IsPurelyInseparable F E
ι : Type u_1
v : ι → K
hsep : ∀ (i : ι), IsSeparable F (v i)
h : LinearIndependent F v
q : ℕ
h✝ : ExpChar F q
this✝ : ExpChar K q
l : ι →₀ E
hl : (Finsupp.linearCombination E v) l = 0
i✝ : ι
f : ι → ℕ
hf : ∀ (i : ι), l i ^ q ^ f i ∈ (algebraMap F E).range
n : ℕ := l.support.sup f
this : q ^ n ≠ 0
i : ι
hs : i ∉ l.support
⊢ l i ^ q ^ n ∈ (algebraMap F E).range

Tactic:
exact ⟨0, by rw [map_zero, Finsupp.notMem_support_iff.1 hs, zero_pow this]⟩

Goal state after:
Module: Mathlib.FieldTheory.RatFunc.Luroth

Goal state before:
pp: K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
⊢ (φ E).natDegree = Module.finrank (↥E) K⟮X⟯

Tactic:
by
  rw [← (IntermediateField.adjoinXEquiv E).toLinearEquiv.finrank_eq,
    adjoin.finrank (IntermediateField.isAlgebraic_X h).isIntegral]

Goal state after:
Module: Mathlib.FieldTheory.Relrank

Goal state before:
pp: E : Type v
inst✝ : Field E
A B C : Subfield E
h1 : A ≤ B
h2 : B ≤ C
h3 : A ≤ C
⊢ Module.rank ↥A ↥(extendScalars h1) * Module.rank ↥B ↥(extendScalars h2) = A.relrank C

Tactic:
relrank_eq_rank_of_le h3

Goal state after:
E : Type v
inst✝ : Field E
A B C : Subfield E
h1 : A ≤ B
h2 : B ≤ C
h3 : A ≤ C
⊢ Module.rank ↥A ↥(extendScalars h1) * Module.rank ↥B ↥(extendScalars h2) = Module.rank ↥A ↥(extendScalars h3)
Module: Mathlib.Geometry.Euclidean.Projection

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s₁ s₂ : AffineSubspace 𝕜 P
p₁ p₂ : P
inst✝¹ : Nonempty ↥s₁
inst✝ : s₁.direction.HasOrthogonalProjection
h : s₁ = s₂
hp : p₁ = p₂
⊢ ↑((orthogonalProjection s₁) p₁) = ↑((orthogonalProjection s₂) p₂)

Tactic:
by
  subst h hp
  rfl

Goal state after:
Module: Mathlib.Geometry.Euclidean.Projection

Goal state before:
pp: case mpr
𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s : AffineSubspace 𝕜 P
inst✝¹ : Nonempty ↥s
inst✝ : s.direction.HasOrthogonalProjection
p q : P
hqs : q ∈ s
hpq : p -ᵥ q ∈ s.directionᗮ
hq : q ∈ mk' p s.directionᗮ
⊢ q ∈ {↑((orthogonalProjection s) p)}

Tactic:
← inter_eq_singleton_orthogonalProjection

Goal state after:
case mpr
𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace 𝕜 V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
s : AffineSubspace 𝕜 P
inst✝¹ : Nonempty ↥s
inst✝ : s.direction.HasOrthogonalProjection
p q : P
hqs : q ∈ s
hpq : p -ᵥ q ∈ s.directionᗮ
hq : q ∈ mk' p s.directionᗮ
⊢ q ∈ ↑s ∩ ↑(mk' p s.directionᗮ)
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation

Goal state before:
pp: V : Type u_1
V' : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : NormedAddCommGroup V'
inst✝³ : InnerProductSpace ℝ V
inst✝² : InnerProductSpace ℝ V'
inst✝¹ : Fact (finrank ℝ V = 2)
inst✝ : Fact (finrank ℝ V' = 2)
o : Orientation ℝ V (Fin 2)
θ : Real.Angle
⊢ (θ.cos • LinearMap.id - θ.sin • ↑o.rightAngleRotation.toLinearEquiv) ∘ₗ (o.rotationAux θ).toLinearMap = LinearMap.id

Tactic:
by
      ext x
      convert! congr_arg (fun t : ℝ => t • x) θ.cos_sq_add_sin_sq using 1
      · simp only [o.rightAngleRotation_rightAngleRotation, o.rotationAux_apply,
          Function.comp_apply, id, LinearEquiv.coe_coe, LinearIsometry.coe_toLinearMap,
          LinearIsometryEquiv.coe_toLinearEquiv, map_add, map_smul, LinearMap.coe_comp,
          LinearMap.id_coe, LinearMap.smul_apply, LinearMap.sub_apply]
        module
      · simp

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
⊢ (o.kahler x) ((o.rotation θ) y) = ↑θ.toCircle * (o.kahler x) y

Tactic:
simp only [o.rotation_apply, map_add, map_smulₛₗ, RingHom.id_apply, real_smul,
    kahler_rightAngleRotation_right, Real.Angle.coe_toCircle]

Goal state after:
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
θ : Real.Angle
⊢ ↑θ.cos * (o.kahler x) y + ↑θ.sin * (I * (o.kahler x) y) = (↑θ.cos + ↑θ.sin * I) * (o.kahler x) y
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
⊢ o.oangle (-x) y + o.oangle (-y) x = 0

Tactic:
by
  rw [oangle_neg_left_eq_neg_right, oangle_rev, neg_add_cancel]

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
v₁ v₂ v₃ v : V
⊢ ∠ (v₁ - v) (v₂ - v) (v₃ - v) = ∠ v₁ v₂ v₃

Tactic:
by
  simpa only [vsub_eq_sub] using angle_vsub_const v₁ v₂ v₃ v

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
r₁ r₂ : ℝ
⊢ ⟪r₂ • x, r₁ • (o.rotation ↑(π / 2)) x⟫ = 0

Tactic:
rw [real_inner_comm, inner_smul_rotation_pi_div_two_smul_left]

Goal state after: