mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁶ : CommRing K
inst✝⁵ : CommRing L
inst✝⁴ : CommRing M
i : K →+* L
j : K →+* M
p : ℕ
inst✝³ : ExpChar M p
inst✝² : ExpChar K p
inst✝¹ : PerfectRing M p
inst✝ : IsPRadical i p
x : L
n : ℕ
y : K
n' : ℕ := (Classical.choose ⋯).1
h' : i (Classical.choose ⋯).2 = x ^ p ^ n'
h✝ : (Classical.choose ⋯).2 ^ p ^ n - y ^ p ^ n' ∈ RingHom.ker i
m : ℕ
h : ((Classical.choose ⋯).2 ^ p ^ n - y ^ p ^ n') ^ p ^ m = 0
⊢ j (((Classical.choose ⋯).2 ^ p ^ n) ^ p ^ m - y ^ (p ^ n' * p ^ m)) = 0

Tactic:
pow_mul,

Goal state after:
K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁶ : CommRing K
inst✝⁵ : CommRing L
inst✝⁴ : CommRing M
i : K →+* L
j : K →+* M
p : ℕ
inst✝³ : ExpChar M p
inst✝² : ExpChar K p
inst✝¹ : PerfectRing M p
inst✝ : IsPRadical i p
x : L
n : ℕ
y : K
n' : ℕ := (Classical.choose ⋯).1
h' : i (Classical.choose ⋯).2 = x ^ p ^ n'
h✝ : (Classical.choose ⋯).2 ^ p ^ n - y ^ p ^ n' ∈ RingHom.ker i
m : ℕ
h : ((Classical.choose ⋯).2 ^ p ^ n - y ^ p ^ n') ^ p ^ m = 0
⊢ j (((Classical.choose ⋯).2 ^ p ^ n) ^ p ^ m - (y ^ p ^ n') ^ p ^ m) = 0
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
inst✝⁷ : CommRing K
inst✝⁶ : CommRing L
inst✝⁵ : CommRing M
i : K →+* L
j : K →+* M
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
x : K
⊢ i x = i x ^ p ^ 0

Tactic:
rw [pow_zero, pow_one]

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

Goal state before:
pp: case inr
R : Type u
inst✝¹ : Field R
inst✝ : IsRealClosed R
x : R
hx : IsSquare x
n : ℕ
hk : -↑n ≠ 0
⊢ ∃ r, x = r ^ (-↑n)

Tactic:
rcases exists_eq_pow_of_isSquare hx (by simpa using hk) with ⟨r, hr⟩
    exact ⟨r⁻¹, by simpa using hr⟩

Goal state after:
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case prime_mul
p n : ℕ
hp : Nat.Prime p
IH :
  ∀ {K : Type u} [inst : Field K],
    Odd n → ∀ {a : K}, (∀ (p : ℕ), Nat.Prime p → p ∣ n → ∀ (b : K), b ^ p ≠ a) → Irreducible (X ^ n - C a)
K : Type u
inst✝² : Field K
hn : Odd (p * n)
a : K
ha : ∀ (p_1 : ℕ), Nat.Prime p_1 → p_1 ∣ p * n → ∀ (b : K), b ^ p_1 ≠ a
E : Type u
inst✝¹ : Field E
inst✝ : Algebra K E
x : E
hx : minpoly K x = X ^ p - C a
this : IsIntegral K x
⊢ Irreducible (X ^ n - C (AdjoinSimple.gen K x))

Tactic:
apply IH (Nat.odd_mul.mp hn).2

Goal state after:
case prime_mul
p n : ℕ
hp : Nat.Prime p
IH :
  ∀ {K : Type u} [inst : Field K],
    Odd n → ∀ {a : K}, (∀ (p : ℕ), Nat.Prime p → p ∣ n → ∀ (b : K), b ^ p ≠ a) → Irreducible (X ^ n - C a)
K : Type u
inst✝² : Field K
hn : Odd (p * n)
a : K
ha : ∀ (p_1 : ℕ), Nat.Prime p_1 → p_1 ∣ p * n → ∀ (b : K), b ^ p_1 ≠ a
E : Type u
inst✝¹ : Field E
inst✝ : Algebra K E
x : E
hx : minpoly K x = X ^ p - C a
this : IsIntegral K x
⊢ ∀ (p : ℕ), Nat.Prime p → p ∣ n → ∀ (b : ↥K⟮x⟯), b ^ p ≠ AdjoinSimple.gen K x
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: K : Type u
inst✝³ : Field K
n : ℕ
hζ✝ : (primitiveRoots n K).Nonempty
a : K
H✝ : Irreducible (X ^ n - C a)
L : Type u_1
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsSplittingField K L (X ^ n - C a)
α : L
hα✝ : α ^ n = (algebraMap K L) a
hζ : (primitiveRoots n K).Nonempty
H : Irreducible (X ^ n - C a)
hα : α ^ n = (algebraMap K L) a
this✝ : Fact (Irreducible (X ^ n - C a))
this : IsSplittingField K K[n√a] (X ^ n - C a) := isSplittingField_AdjoinRoot_X_pow_sub_C hζ H
⊢ Function.Surjective ⇑(liftAlgHom (X ^ n - C a) (Algebra.ofId K L) α ⋯)

Tactic:
rw [← AlgHom.range_eq_top, ← IsSplittingField.adjoin_rootSet _ (X ^ n - C a),
      eq_comm, Splits.adjoin_rootSet_eq_range, IsSplittingField.adjoin_rootSet]

Goal state after:
case hf
K : Type u
inst✝³ : Field K
n : ℕ
hζ✝ : (primitiveRoots n K).Nonempty
a : K
H✝ : Irreducible (X ^ n - C a)
L : Type u_1
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : IsSplittingField K L (X ^ n - C a)
α : L
hα✝ : α ^ n = (algebraMap K L) a
hζ : (primitiveRoots n K).Nonempty
H : Irreducible (X ^ n - C a)
hα : α ^ n = (algebraMap K L) a
this✝ : Fact (Irreducible (X ^ n - C a))
this : IsSplittingField K K[n√a] (X ^ n - C a) := isSplittingField_AdjoinRoot_X_pow_sub_C hζ H
⊢ (Polynomial.map (algebraMap K K[n√a]) (X ^ n - C a)).Splits
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case refine_1
K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : IsPrimitiveRoot ζ (finrank K L)
σ : Gal(L/K)
hσ : Function.Surjective fun x ↦ σ ^ x
hσ'✝ : orderOf σ = Nat.card Gal(L/K)
this : (minpoly K σ.toLinearMap).IsRoot ζ
v : L
hv : End.HasEigenvector σ.toLinearMap ζ v
hv' : ∀ (n : ℕ), (σ ^ n) v = ζ ^ n • v
n : ℕ
hσ' : σ ^ n ∈ OrderDual.ofDual ⊥
⊢ ζ ^ (finrank K L * n) • v ^ finrank K L = v ^ finrank K L

Tactic:
pow_mul,

Goal state after:
case refine_1
K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : IsPrimitiveRoot ζ (finrank K L)
σ : Gal(L/K)
hσ : Function.Surjective fun x ↦ σ ^ x
hσ'✝ : orderOf σ = Nat.card Gal(L/K)
this : (minpoly K σ.toLinearMap).IsRoot ζ
v : L
hv : End.HasEigenvector σ.toLinearMap ζ v
hv' : ∀ (n : ℕ), (σ ^ n) v = ζ ^ n • v
n : ℕ
hσ' : σ ^ n ∈ OrderDual.ofDual ⊥
⊢ (ζ ^ finrank K L) ^ n • v ^ finrank K L = v ^ finrank K L
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case refine_2
K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : IsPrimitiveRoot ζ (finrank K L)
σ : Gal(L/K)
hσ : Function.Surjective fun x ↦ σ ^ x
hσ'✝ : orderOf σ = Nat.card Gal(L/K)
this✝ : (minpoly K σ.toLinearMap).IsRoot ζ
v : L
hv : End.HasEigenvector σ.toLinearMap ζ v
hv' : ∀ (n : ℕ), (σ ^ n) v = ζ ^ n • v
k : ℕ
hσ' : σ ^ (finrank K L * k) ∈ IsGalois.intermediateFieldEquivSubgroup K⟮v⟯
this : ζ ^ (finrank K L * k) • v = 1 • v
⊢ σ ^ (finrank K L * k) ∈ ⊤

Tactic:
pow_mul,

Goal state after:
case refine_2
K : Type u
inst✝⁵ : Field K
L : Type u_1
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : FiniteDimensional K L
hK : (primitiveRoots (finrank K L) K).Nonempty
inst✝¹ : IsGalois K L
inst✝ : IsCyclic Gal(L/K)
ζ : K
hζ : IsPrimitiveRoot ζ (finrank K L)
σ : Gal(L/K)
hσ : Function.Surjective fun x ↦ σ ^ x
hσ'✝ : orderOf σ = Nat.card Gal(L/K)
this✝ : (minpoly K σ.toLinearMap).IsRoot ζ
v : L
hv : End.HasEigenvector σ.toLinearMap ζ v
hv' : ∀ (n : ℕ), (σ ^ n) v = ζ ^ n • v
k : ℕ
hσ' : σ ^ (finrank K L * k) ∈ IsGalois.intermediateFieldEquivSubgroup K⟮v⟯
this : ζ ^ (finrank K L * k) • v = 1 • v
⊢ (σ ^ finrank K L) ^ k ∈ ⊤
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommSemiring R
inst✝¹ : CommSemiring S
inst✝ : Algebra R S
A B : Subalgebra R S
H : A.LinearDisjoint B
H' : A ⊔ B = ⊤
ι : Type u_1
b : Basis ι R ↥B
x : ↥B
i : ι
⊢ (algebraMap (↥A) S) (((H.basisOfBasisRight H' b).repr ↑x) i) = (algebraMap R S) ((b.repr x) i)

Tactic:
simp [basisOfBasisRight, Algebra.algebraMap_eq_smul_one]

Goal state after:
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.IsSeparable K L
c : ConjRootClass K L
⊢ c.minpoly.Separable

Tactic:
by
  induction c
  exact Algebra.IsSeparable.isSeparable K _

Goal state after:
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
h : a ∈ (algebraMap K L).range
⊢ ∃ y, minpoly K a = X ^ ringExpChar K ^ 0 - C y

Tactic:
rw [pow_zero, pow_one]

Goal state after:
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
h : a ∈ (algebraMap K L).range
⊢ ∃ y, minpoly K a = X - C y
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
A B : Subalgebra R S
inst✝ : Algebra.IsIntegral R ↥A
H : ∀ A' ≤ A, ∀ [Module.Finite R ↥A'], A'.LinearDisjoint B
⊢ A.LinearDisjoint B

Tactic:
by
  rw [linearDisjoint_iff, Submodule.linearDisjoint_iff]
  intro x y hxy
  obtain ⟨M', hM, hf, h⟩ :=
    TensorProduct.exists_finite_submodule_left_of_setFinite' {x, y} (Set.toFinite _)
  obtain ⟨s, hs⟩ : M'.FG := .of_finite
  have hs' : (s : Set S) ⊆ A := by rwa [← hs, Submodule.span_le] at hM
  let A' := Algebra.adjoin R (s : Set S)
  have hf' : Submodule.FG (toSubmodule A') := fg_adjoin_of_finite s.finite_toSet fun x hx ↦
    (isIntegral_algHom_iff A.val Subtype.val_injective).2
      (Algebra.IsIntegral.isIntegral (R := R) (A := A) ⟨x, hs' hx⟩)
  replace hf' : Module.Finite R A' := .of_fg hf'
  have hA : toSubmodule A' ≤ toSubmodule A := Algebra.adjoin_le_iff.2 hs'
  replace h : {x, y} ⊆ (LinearMap.range (LinearMap.rTensor (toSubmodule B)
      (Submodule.inclusion hA)) : Set _) := fun _ hx ↦ by
    have : Submodule.inclusion hM = Submodule.inclusion hA ∘ₗ Submodule.inclusion
      (show M' ≤ toSubmodule A' by
        rw [← hs, Submodule.span_le]; exact Algebra.adjoin_le_iff.1 (le_refl _)) := rfl
    rw [this, LinearMap.rTensor_comp] at h
    exact LinearMap.range_comp_le_range _ _ (h hx)
  obtain ⟨x', hx'⟩ := h (show x ∈ {x, y} by simp)
  obtain ⟨y', hy'⟩ := h (show y ∈ {x, y} by simp)
  rw [← hx', ← hy']; congr
  exact (H A' hA).injective (by simp [← Submodule.mulMap_comp_rTensor _ hA, hx', hy', hxy])

Goal state after:
Module: Mathlib.FieldTheory.Relrank

Goal state before:
pp: E : Type v
inst✝ : Field E
A : Subfield E
⊢ A.relfinrank A = 1

Tactic:
simp [relfinrank_eq_toNat_relrank]

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 ≠ ⊥
⊢ (c E).denom = 1

Tactic:
← Φ_coeff_φ_natDegree h

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
⊢ ((algebraMap K[X] K⟮X⟯) ((Φ E).coeff (φ E).natDegree)).denom = 1
Module: Mathlib.FieldTheory.RatFunc.Luroth

Goal state before:
pp: case e'_4.e'_3
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
⊢ (f E).natDegree = (Polynomial.C (g E) * Polynomial.map Polynomial.C (f E)).natDegree

pp: case e'_4.e'_4
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
⊢ (g E).natDegree = (Polynomial.C (f E) * Polynomial.map Polynomial.C (g E)).natDegree

Tactic:
· rw [natDegree_mul (C_ne_zero.mpr (generator E).denom_ne_zero)
      (Polynomial.map_ne_zero (num_ne_zero (generator_ne_zero h))), natDegree_C, zero_add,
      natDegree_map]

Goal state after:
case e'_4.e'_4
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
⊢ (g E).natDegree = (Polynomial.C (f E) * Polynomial.map Polynomial.C (g E)).natDegree
Module: Mathlib.Geometry.Euclidean.Altitude

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
h :
  finrank ℝ ↥(vectorSpan ℝ (s.points '' {i}ᶜ)) +
      finrank ℝ ↥((vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)) =
    finrank ℝ ↥(vectorSpan ℝ (Set.range s.points))
⊢ finrank ℝ ↥((vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)) = 1

Tactic:
have hn : (n - 1) + 1 = n := by
    have := NeZero.ne n
    cases n <;> lia

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
h :
  finrank ℝ ↥(vectorSpan ℝ (s.points '' {i}ᶜ)) +
      finrank ℝ ↥((vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)) =
    finrank ℝ ↥(vectorSpan ℝ (Set.range s.points))
hn : n - 1 + 1 = n
⊢ finrank ℝ ↥((vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ ⊓ vectorSpan ℝ (Set.range s.points)) = 1
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 : V
r : ℝ
⊢ 2 • o.oangle (r • x) x = 0

Tactic:
by
  rcases lt_or_ge r 0 with (h | h) <;> simp [h]

Goal state after:
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 : V
r : ℝ
⊢ 2 • o.oangle x (r • x) = 0

Tactic:
by
  rcases lt_or_ge r 0 with (h | h) <;> simp [h]

Goal state after:
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 z : V
hx : x ≠ 0
hy : y ≠ 0
hz : z ≠ 0
⊢ o.oangle x (-y) + o.oangle y (-z) + o.oangle z (-x) = ↑π

Tactic:
simp_rw [← oangle_neg_left_eq_neg_right, o.oangle_add_cyc3_neg_left hx hy hz]

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
⊢ ∡ p₁ p₂ p₃ ≠ 0 ∧ ∡ p₁ p₂ p₃ ≠ ↑π ↔ AffineIndependent ℝ ![p₁, p₂, p₃]

Tactic:
rw [oangle, o.oangle_ne_zero_and_ne_pi_iff_linearIndependent,
    affineIndependent_iff_linearIndependent_vsub ℝ _ (1 : Fin 3), ←
    linearIndependent_equiv (finSuccAboveEquiv (1 : Fin 3))]

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ : P
⊢ LinearIndependent ℝ ![p₁ -ᵥ p₂, p₃ -ᵥ p₂] ↔
    LinearIndependent ℝ ((fun i ↦ ![p₁, p₂, p₃] ↑i -ᵥ ![p₁, p₂, p₃] 1) ∘ ⇑(finSuccAboveEquiv 1))
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, y⟫ = 0
⊢ Real.tan (angle x (x + y)) = ‖y‖ / ‖x‖

Tactic:
by_cases h0 : x = 0; · simp [h0]
  rw [angle_add_eq_arctan_of_inner_eq_zero h h0, Real.tan_arctan]

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
h : ⟪x, -y⟫ = 0
⊢ angle x (x - y) = Real.arccos (‖x‖ / ‖x - y‖)

Tactic:
rw [sub_eq_add_neg, angle_add_eq_arccos_of_inner_eq_zero h]

Goal state after:
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
h : o.oangle x y = ↑(π / 2)
⊢ ⟪y, x⟫ = 0

Tactic:
by rw [real_inner_comm, o.inner_eq_zero_of_oangle_eq_pi_div_two h]

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
p p' : P
s : AffineSubspace ℝ P
inst✝ : s.direction.HasOrthogonalProjection
h : p' ∈ s
⊢ ∠ p (↑((orthogonalProjection s) p)) p' = π / 2

Tactic:
have : Nonempty s := ⟨p', h⟩
  rw [angle, ← InnerProductGeometry.inner_eq_zero_iff_angle_eq_pi_div_two]
  exact Submodule.inner_left_of_mem_orthogonal (K := s.direction)
    (AffineSubspace.vsub_mem_direction h (orthogonalProjection_mem _))
    (vsub_orthogonalProjection_mem_direction_orthogonal _ _)

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

Goal state before:
pp: case pos
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
r : ℝ
h : o.oangle x y = 0 ∨ o.oangle x y = ↑π
⊢ o.oangle x (r • x + y) = 0 ∨ o.oangle x (r • x + y) = ↑π

Tactic:
oangle_smul_add_right_eq_zero_or_eq_pi_iff

Goal state after:
case pos
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
r : ℝ
h : o.oangle x y = 0 ∨ o.oangle x y = ↑π
⊢ o.oangle x y = 0 ∨ o.oangle x y = ↑π
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: case refine_1
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
r : ℝ
h : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)
h' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π
s : Set (V × V) := (fun r' ↦ (x, r' • x + y)) '' Set.univ
hc : IsConnected s
z : V × V
hz : z ∈ s
⊢ z.1 ≠ 0

Tactic:
simp_rw [s, Set.mem_image] at hz
      obtain ⟨r', -, rfl⟩ := hz
      simp only
      intro hz

Goal state after:
case refine_1
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
r : ℝ
h : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)
h' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π
s : Set (V × V) := (fun r' ↦ (x, r' • x + y)) '' Set.univ
hc : IsConnected s
r' : ℝ
hz : x = 0
⊢ False
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: case inl
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
hd2 : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x : V
h : x ≠ 0
r : ℝ
hr : r < 0
⊢ o.oangle x (x + r • (o.rotation ↑(π / 2)) x) = ↑(Real.arctan r)

Tactic:
have ha : o.oangle x (r • o.rotation (π / 2 : ℝ) x) = -(π / 2 : ℝ) := by
      rw [o.oangle_smul_right_of_neg _ _ hr, o.oangle_neg_right h, o.oangle_rotation_self_right h, ←
        sub_eq_zero, add_comm, sub_neg_eq_add, ← Real.Angle.coe_add, ← Real.Angle.coe_add,
        add_assoc, add_halves, ← two_mul, Real.Angle.coe_two_pi]
      simpa using h
    rw [← neg_inj, ← oangle_neg_orientation_eq_neg, neg_neg] at ha
    rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj, oangle_rev,
      (-o).oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two ha, norm_smul,
      LinearIsometryEquiv.norm_map, mul_div_assoc, div_self (norm_ne_zero_iff.2 h), mul_one,
      Real.norm_eq_abs, abs_of_neg hr, Real.arctan_neg, Real.Angle.coe_neg, neg_neg]

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

Goal state before:
pp: case refine_2
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
r : ℝ
h : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)
h' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π
s : Set (V × V) := (fun r' ↦ (x, r' • x + y)) '' Set.univ
hc : IsConnected s
z : V × V
hz : z ∈ s
⊢ z.2 ≠ 0

Tactic:
simp_rw [s, Set.mem_image] at hz
      obtain ⟨r', -, rfl⟩ := hz
      simp only
      intro hz

Goal state after:
case refine_2
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
r : ℝ
h : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)
h' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π
s : Set (V × V) := (fun r' ↦ (x, r' • x + y)) '' Set.univ
hc : IsConnected s
r' : ℝ
hz : r' • x + y = 0
⊢ False
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) x).sign = (o.oangle x y).sign

Tactic:
rw [oangle_sign_sub_left_eq_neg, o.oangle_rev y x, Real.Angle.sign_neg]

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
ι₁ : Type u_3
s₁ : Finset ι₁
w₁ : ι₁ → ℝ
p₁ : ι₁ → P
h₁ : ∑ i ∈ s₁, w₁ i = 0
ι₂ : Type u_4
s₂ : Finset ι₂
w₂ : ι₂ → ℝ
p₂ : ι₂ → P
h₂ : ∑ i ∈ s₂, w₂ i = 0
⊢ (-∑ x ∈ s₁, ∑ x_1 ∈ s₂, w₁ x * w₂ x_1 * (‖p₁ x -ᵥ p₂ x_1‖ * ‖p₁ x -ᵥ p₂ x_1‖)) / 2 =
    (-∑ i₁ ∈ s₁, ∑ i₂ ∈ s₂, w₁ i₁ * w₂ i₂ * (dist (p₁ i₁) (p₂ i₂) * dist (p₁ i₁) (p₂ i₂))) / 2

Tactic:
rcongr (i₁ i₂) <;> rw [dist_eq_norm_vsub V (p₁ i₁) (p₂ i₂)]

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
v : V
p₁ p₂ : P
hv : v ≠ 0
r : ℝ
hvi : ⟪v, v⟫ ≠ 0
hd : discrim ⟪v, v⟫ (2 * ⟪v, p₁ -ᵥ p₂⟫) 0 = 2 * ⟪v, p₁ -ᵥ p₂⟫ * (2 * ⟪v, p₁ -ᵥ p₂⟫)
⊢ r = 0 ∨ r = (-2 + -2) * ⟪v, p₁ -ᵥ p₂⟫ / (2 * ⟪v, v⟫) ↔ r = 0 ∨ r = -2 * ⟪v, p₁ -ᵥ p₂⟫ / ⟪v, v⟫

Tactic:
← mul_two,

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
v : V
p₁ p₂ : P
hv : v ≠ 0
r : ℝ
hvi : ⟪v, v⟫ ≠ 0
hd : discrim ⟪v, v⟫ (2 * ⟪v, p₁ -ᵥ p₂⟫) 0 = 2 * ⟪v, p₁ -ᵥ p₂⟫ * (2 * ⟪v, p₁ -ᵥ p₂⟫)
⊢ r = 0 ∨ r = -2 * 2 * ⟪v, p₁ -ᵥ p₂⟫ / (2 * ⟪v, v⟫) ↔ r = 0 ∨ r = -2 * ⟪v, p₁ -ᵥ p₂⟫ / ⟪v, v⟫
Module: Mathlib.Geometry.Euclidean.Sphere.Basic

Goal state before:
pp: E : Type u_3
F : Type u_4
inst✝¹ : MetricSpace E
inst✝ : MetricSpace F
f : E → F
hf : Isometry f
ps : Set E
c : E
r : ℝ
hc : ∀ p ∈ ps, dist p c = r
p : E
hp : p ∈ ps
⊢ dist (f p) (f c) = r

Tactic:
hf.dist_eq,

Goal state after:
E : Type u_3
F : Type u_4
inst✝¹ : MetricSpace E
inst✝ : MetricSpace F
f : E → F
hf : Isometry f
ps : Set E
c : E
r : ℝ
hc : ∀ p ∈ ps, dist p c = r
p : E
hp : p ∈ ps
⊢ dist p c = r
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ : P
h₁₂ : p₁ ≠ p₂
ha : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄
hs : line[ℝ, p₁, p₂].SOppSide p₃ p₄
h₃₂ : p₃ ≠ p₂
h₄₂ : p₄ ≠ p₂
⊢ ∡ p₁ p₂ ((AffineEquiv.pointReflection ℝ p₂) p₃) = ∡ p₃ p₂ p₄ + ↑π

Tactic:
oangle_pointReflection_right h₁₂ h₃₂

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
hd2 : Fact (finrank ℝ V = 2)
inst✝ : Oriented ℝ V (Fin 2)
p₁ p₂ p₃ p₄ : P
h₁₂ : p₁ ≠ p₂
ha : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄
hs : line[ℝ, p₁, p₂].SOppSide p₃ p₄
h₃₂ : p₃ ≠ p₂
h₄₂ : p₄ ≠ p₂
⊢ ∡ p₁ p₂ p₃ + ↑π = ∡ p₃ p₂ p₄ + ↑π
Module: Mathlib.Geometry.Euclidean.Sphere.Basic

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
⊢ False

Tactic:
have hsd : ∀ i, dist (f i • v +ᵥ p 0) c = r := by
    intro i
    rw [← hf]
    exact hs' i

Goal state after:
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Set P
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
v : V
hv0 : v ≠ 0
c : P
r : ℝ
hs : ∀ p ∈ s, dist p c = r
hs' : ∀ (i : Fin 3), dist (p i) c = r
f : Fin 3 → ℝ
hf : ∀ (i : Fin 3), p i = f i • v +ᵥ p 0
hsd : ∀ (i : Fin 3), dist (f i • v +ᵥ p 0) c = r
⊢ False
Module: Mathlib.Geometry.Euclidean.Sphere.Tangent

Goal state before:
pp: case refine_2.refine_1
V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : InnerProductSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
as : AffineSubspace ℝ P
hr : s.radius = 0
hm : s.center ∈ as
⊢ s.center ∈ s

Tactic:
rw [center_mem_iff, hr]

Goal state after: