mathlib-initiative/mathlib-tactics huggingface dataset:
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.Bijective ⇑(liftAlgHom (X ^ n - C a) (Algebra.ofId K L) α ⋯)

Tactic:
refine ⟨(liftAlgHom (X ^ n - C a) _ α _).injective, ?_⟩

Goal state after:
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 ⋯)
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: G : Type u_1
inst✝ : Group G
ι : Type u_2
H : ι → Subgroup G
g : ι → G
s : Finset ι
hcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ
h : ∀ i ∈ s, (H i).FiniteIndex → s.card < (H i).index
hs : s.Nonempty
hs' : 0 < s.card
i : ι
hi : i ∈ s
x✝ : (H i).index = 0 ∨ (H i).index ≠ 0
⊢ (↑(H i).index)⁻¹ < (↑s.card)⁻¹

Tactic:
cases eq_or_ne (H i).index 0 with

Goal state after:
Module: Mathlib.LinearAlgebra.TensorProduct.Subalgebra

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring S
inst✝² : Algebra R S
inst✝¹ : Semiring T
inst✝ : Algebra R T
A : Subalgebra R S
⊢ ↥⊥ ⊗[R] ↥A ≃ₐ[R] ↥A

Tactic:
by
  refine Algebra.TensorProduct.algEquivOfLinearEquivTensorProduct (toSubmodule A).lTensorOne ?_ ?_
  · rintro x y a b
    obtain ⟨x', hx⟩ := Algebra.mem_bot.1 x.2
    replace hx : algebraMap R _ x' = x := Subtype.val_injective hx
    obtain ⟨y', hy⟩ := Algebra.mem_bot.1 y.2
    replace hy : algebraMap R _ y' = y := Subtype.val_injective hy
    rw [← hx, ← hy, ← map_mul, (toSubmodule A).lTensorOne_tmul x' a,
      (toSubmodule A).lTensorOne_tmul y' b, (toSubmodule A).lTensorOne_tmul (x' * y') (a * b),
      Algebra.mul_smul_comm, Algebra.smul_mul_assoc, smul_smul, mul_comm x' y']
  · exact Submodule.lTensorOne_one_tmul _

Goal state after:
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
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (toSubmodule B)) x = ((toSubmodule A).mulMap (toSubmodule B)) y
M' : Submodule R S
hM : M' ≤ toSubmodule A
hf : Module.Finite R ↥M'
h : {x, y} ⊆ ↑(LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hM)).range
s : Finset S
hs : Submodule.span R ↑s = M'
⊢ ↑s ⊆ ↑A

Tactic:
← hs,

Goal state after:
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
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (toSubmodule B)) x = ((toSubmodule A).mulMap (toSubmodule B)) y
M' : Submodule R S
hM✝ : M' ≤ toSubmodule A
hf : Module.Finite R ↥M'
h : {x, y} ⊆ ↑(LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hM✝)).range
s : Finset S
hM : Submodule.span R ↑s ≤ toSubmodule A
hs : Submodule.span R ↑s = M'
⊢ ↑s ⊆ ↑A
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
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (toSubmodule B)) x = ((toSubmodule A).mulMap (toSubmodule B)) y
M' : Submodule R S
hM : M' ≤ toSubmodule A
hf : Module.Finite R ↥M'
h : {x, y} ⊆ ↑(LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hM)).range
s : Finset S
hs : Submodule.span R ↑s = M'
hs' : ↑s ⊆ ↑A
A' : Subalgebra R S := Algebra.adjoin R ↑s
hf' : Module.Finite R ↥A'
hA : toSubmodule A' ≤ toSubmodule A
x✝ : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hx : x✝ ∈ {x, y}
⊢ M' ≤ toSubmodule A'

Tactic:
← hs,

Goal state after:
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
x y : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hxy : ((toSubmodule A).mulMap (toSubmodule B)) x = ((toSubmodule A).mulMap (toSubmodule B)) y
M' : Submodule R S
hM : M' ≤ toSubmodule A
hf : Module.Finite R ↥M'
h : {x, y} ⊆ ↑(LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hM)).range
s : Finset S
hs : Submodule.span R ↑s = M'
hs' : ↑s ⊆ ↑A
A' : Subalgebra R S := Algebra.adjoin R ↑s
hf' : Module.Finite R ↥A'
hA : toSubmodule A' ≤ toSubmodule A
x✝ : ↥(toSubmodule A) ⊗[R] ↥(toSubmodule B)
hx : x✝ ∈ {x, y}
⊢ Submodule.span R ↑s ≤ toSubmodule A'
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
hF : ExpChar F q
inst✝ : ExpChar E q
n : ℕ
a : E
hsep : IsSeparable F a
hai : IsIntegral F a
⊢ minpoly F ((iterateFrobenius E q n) a) = Polynomial.map (iterateFrobenius F q n) (minpoly F a)

Tactic:
have hapi : IsIntegral F (iterateFrobenius E q n a) := hai.pow _

Goal state after:
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
q : ℕ
hF : ExpChar F q
inst✝ : ExpChar E q
n : ℕ
a : E
hsep : IsSeparable F a
hai : IsIntegral F a
hapi : IsIntegral F ((iterateFrobenius E q n) a)
⊢ minpoly F ((iterateFrobenius E q n) a) = Polynomial.map (iterateFrobenius F q n) (minpoly F a)
Module: Mathlib.FieldTheory.Relrank

Goal state before:
pp: E : Type v
inst✝ : Field E
A B : Subfield E
⊢ Module.rank ↥(A ⊓ B) ↥(extendScalars ⋯) = 1 ↔ B ≤ A

Tactic:
IntermediateField.rank_eq_one_iff,

Goal state after:
E : Type v
inst✝ : Field E
A B : Subfield E
⊢ extendScalars ⋯ = ⊥ ↔ B ≤ A
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).coeff (generatorIndex h) ≠ 0

Tactic:
apply_fun algebraMap K[X] K⟮X⟯

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 (generatorIndex h)) ≠ (algebraMap K[X] K⟮X⟯) 0
Module: Mathlib.Geometry.Convex.Cone.Face.Basic

Goal state before:
pp: case refine_2
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : Semiring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
C F : PointedCone R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
sub : map f F ≤ map f C
x✝ y✝ : M
a✝ : R
hx : x✝ ∈ C
hy : y✝ ∈ C
ha : 0 < a✝
h : a✝ • x✝ + y✝ ∈ F
hF :
  ∀ {x y : N} {a : R},
    ∀ x_1 ∈ C, f x_1 = x → ∀ x_2 ∈ C, f x_2 = y → 0 < a → ∀ x_3 ∈ F, f x_3 = a • x + y → ∃ x_4 ∈ F, f x_4 = x
⊢ x✝ ∈ F

Tactic:
obtain ⟨_, ⟨hx', hhx'⟩⟩ := hF _ hx rfl _ hy rfl ha _ h (by simp)

Goal state after:
case refine_2
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : Semiring R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
C F : PointedCone R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
hf : Function.Injective ⇑f
sub : map f F ≤ map f C
x✝ y✝ : M
a✝ : R
hx : x✝ ∈ C
hy : y✝ ∈ C
ha : 0 < a✝
h : a✝ • x✝ + y✝ ∈ F
hF :
  ∀ {x y : N} {a : R},
    ∀ x_1 ∈ C, f x_1 = x → ∀ x_2 ∈ C, f x_2 = y → 0 < a → ∀ x_3 ∈ F, f x_3 = a • x + y → ∃ x_4 ∈ F, f x_4 = x
w✝ : M
hx' : w✝ ∈ F
hhx' : f w✝ = f x✝
⊢ x✝ ∈ F
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 = 0 ∨ o.oangle x y = ↑π ↔ x = 0 ∨ ∃ r, y = r • x

Tactic:
rw [oangle_eq_zero_iff_sameRay, oangle_eq_pi_iff_sameRay_neg]
  refine ⟨fun h => ?_, fun h => ?_⟩
  · rcases h with (h | ⟨-, -, h⟩)
    · by_cases hx : x = 0; · simp [hx]
      obtain ⟨r, -, rfl⟩ := h.exists_nonneg_left hx
      exact Or.inr ⟨r, rfl⟩
    · by_cases hx : x = 0; · simp [hx]
      obtain ⟨r, -, hy⟩ := h.exists_nonneg_left hx
      refine Or.inr ⟨-r, ?_⟩
      simp [hy]
  · rcases h with (rfl | ⟨r, rfl⟩); · simp
    by_cases hx : x = 0; · simp [hx]
    rcases lt_trichotomy r 0 with (hr | hr | hr)
    · rw [← neg_smul]
      exact Or.inr ⟨hx, smul_ne_zero hr.ne hx,
        SameRay.sameRay_pos_smul_right x (Left.neg_pos_iff.2 hr)⟩
    · simp [hr]
    · exact Or.inl (SameRay.sameRay_pos_smul_right x hr)

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
h : ∡ p₁ p₂ p₃ ≠ 0
⊢ p₁ ≠ p₂

Tactic:
rw [← @vsub_ne_zero V]; exact o.left_ne_zero_of_oangle_ne_zero h

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

Goal state before:
pp: case inl.inl
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
h : InnerProductGeometry.angle 0 x = InnerProductGeometry.angle y z
hs : (o.oangle 0 x).sign = (o.oangle y z).sign
⊢ (o.oangle 0 x).sign = 0 ∧ (o.oangle y z).sign = 0

pp: case inl.inr
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)
w y z : V
h : InnerProductGeometry.angle w 0 = InnerProductGeometry.angle y z
hs : (o.oangle w 0).sign = (o.oangle y z).sign
⊢ (o.oangle w 0).sign = 0 ∧ (o.oangle y z).sign = 0

pp: case inr.inl
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)
w x z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle 0 z
hs : (o.oangle w x).sign = (o.oangle 0 z).sign
⊢ (o.oangle w x).sign = 0 ∧ (o.oangle 0 z).sign = 0

pp: case inr.inr
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)
w x y : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y 0
hs : (o.oangle w x).sign = (o.oangle y 0).sign
⊢ (o.oangle w x).sign = 0 ∧ (o.oangle y 0).sign = 0

Tactic:
· simpa using hs.symm

Goal state after:
case inl.inr
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)
w y z : V
h : InnerProductGeometry.angle w 0 = InnerProductGeometry.angle y z
hs : (o.oangle w 0).sign = (o.oangle y z).sign
⊢ (o.oangle w 0).sign = 0 ∧ (o.oangle y z).sign = 0

case inr.inl
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)
w x z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle 0 z
hs : (o.oangle w x).sign = (o.oangle 0 z).sign
⊢ (o.oangle w x).sign = 0 ∧ (o.oangle 0 z).sign = 0

case inr.inr
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)
w x y : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y 0
hs : (o.oangle w x).sign = (o.oangle y 0).sign
⊢ (o.oangle w x).sign = 0 ∧ (o.oangle y 0).sign = 0
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: case inl.inr
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)
w y z : V
h : InnerProductGeometry.angle w 0 = InnerProductGeometry.angle y z
hs : (o.oangle w 0).sign = (o.oangle y z).sign
⊢ (o.oangle w 0).sign = 0 ∧ (o.oangle y z).sign = 0

pp: case inr.inl
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)
w x z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle 0 z
hs : (o.oangle w x).sign = (o.oangle 0 z).sign
⊢ (o.oangle w x).sign = 0 ∧ (o.oangle 0 z).sign = 0

pp: case inr.inr
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)
w x y : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y 0
hs : (o.oangle w x).sign = (o.oangle y 0).sign
⊢ (o.oangle w x).sign = 0 ∧ (o.oangle y 0).sign = 0

Tactic:
· simpa using hs.symm

Goal state after:
case inr.inl
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)
w x z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle 0 z
hs : (o.oangle w x).sign = (o.oangle 0 z).sign
⊢ (o.oangle w x).sign = 0 ∧ (o.oangle 0 z).sign = 0

case inr.inr
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)
w x y : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y 0
hs : (o.oangle w x).sign = (o.oangle y 0).sign
⊢ (o.oangle w x).sign = 0 ∧ (o.oangle y 0).sign = 0
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

Goal state before:
pp: 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 y : V
h : o.oangle x y = ↑(π / 2)
⊢ o.oangle (x + y) y = ↑(Real.arctan (‖x‖ / ‖y‖))

Tactic:
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
  rw [add_comm]
  exact (-o).oangle_add_right_eq_arctan_of_oangle_eq_pi_div_two h

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

Goal state before:
pp: 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 y : V
h : o.oangle x y = ↑(π / 2)
⊢ (o.oangle (x - y) x).tan * ‖x‖ = ‖y‖

Tactic:
by
  rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
  exact (-o).tan_oangle_sub_right_mul_norm_of_oangle_eq_pi_div_two h

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Bisector

Goal state before:
pp: case neg.refine_2
V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
p p' : P
s₁ s₂ : AffineSubspace ℝ P
inst✝¹ : s₁.direction.HasOrthogonalProjection
inst✝ : s₂.direction.HasOrthogonalProjection
hp'₁ : p' ∈ s₁
hp'₂ : p' ∈ s₂
this✝ : Nonempty ↥s₁
this : Nonempty ↥s₂
h' : p ∉ s₁ ∧ p ∉ s₂
h :
  Real.arcsin (dist p ↑((orthogonalProjection s₁) p) / dist p p') =
    Real.arcsin (dist p ↑((orthogonalProjection s₂) p) / dist p p')
⊢ dist p ↑((orthogonalProjection s₁) p) = dist p ↑((orthogonalProjection s₂) p)

Tactic:
· have hp : p ≠ p' := by
        rintro rfl
        exact h'.1 hp'₁
      have hpd : 0 < dist p p' := dist_pos.2 hp
      rw [Real.arcsin_inj (le_trans (by norm_num : (-1 : ℝ) ≤ 0) (by positivity))
        ((div_le_one hpd).2 ?_)
        (le_trans (by norm_num : (-1 : ℝ) ≤ 0) (by positivity)) ((div_le_one hpd).2 ?_)] at h
      · rwa [div_left_inj' hpd.ne'] at h
      · rw [dist_orthogonalProjection_eq_infDist]
        exact Metric.infDist_le_dist_of_mem (SetLike.mem_coe.1 hp'₁)
      · rw [dist_orthogonalProjection_eq_infDist]
        exact Metric.infDist_le_dist_of_mem (SetLike.mem_coe.1 hp'₂)

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

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
p₁ p₂ : P
h : perpBisector p₁ p₂ = ⊤
⊢ p₁ = p₂

Tactic:
← left_mem_perpBisector,

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
p₁ p₂ : P
h : perpBisector p₁ p₂ = ⊤
⊢ p₁ ∈ perpBisector p₁ p₂
Module: Mathlib.Geometry.Euclidean.PerpBisector

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
a b c p : P
h_inner : ⟪p -ᵥ a, c -ᵥ a⟫ = 0
t : ℝ
hb_eq : (AffineMap.lineMap a c) t = b
ht0 : 0 ≤ t
ht1 : t ≤ 1
⊢ dist p a ^ 2 + t ^ 2 * dist a c ^ 2 ≤ dist p c ^ 2

Tactic:
dist_sq_of_inner_eq_zero h_inner

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
a b c p : P
h_inner : ⟪p -ᵥ a, c -ᵥ a⟫ = 0
t : ℝ
hb_eq : (AffineMap.lineMap a c) t = b
ht0 : 0 ≤ t
ht1 : t ≤ 1
⊢ dist p a ^ 2 + t ^ 2 * dist a c ^ 2 ≤ dist p a ^ 2 + dist a c ^ 2
Module: Mathlib.Geometry.Euclidean.Sphere.Basic

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝³ : NormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : MetricSpace P
inst✝ : NormedAddTorsor V P
s : Sphere P
p₁ p₂ : P
h : p₁ ∈ s ∧ (Equiv.pointReflection s.center) p₁ = p₂
⊢ midpoint ℝ p₁ p₂ = s.center

Tactic:
simp [← h.2]

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁵ : NormedAddCommGroup V
inst✝⁴ : InnerProductSpace ℝ V
inst✝³ : MetricSpace P
inst✝² : NormedAddTorsor V P
S : AffineSubspace ℝ P
inst✝¹ : Nonempty ↥S
inst✝ : S.direction.HasOrthogonalProjection
ps : Set ↥S
⊢ Cospherical (Subtype.val '' ps) ↔ Cospherical ps

Tactic:
refine ⟨fun h => ?_, Cospherical.subtype_val⟩
  rcases ps.eq_empty_or_nonempty with rfl | ⟨p₀, hp₀⟩
  · exact cospherical_empty
  · rcases h with ⟨c, r, hr⟩
    let c' : S := orthogonalProjection S c
    refine ⟨c', dist p₀ c', fun p hp => ?_⟩
    have hp_dist : dist (p : P) c = r := by grind
    have hp₀_dist : dist (p₀ : P) c = r := by grind
    have hpp₀ : dist (p : P) (c : P) = dist (p₀ : P) (c : P) := hp_dist.trans hp₀_dist.symm
    exact (dist_eq_iff_dist_orthogonalProjection_eq (s := S) (p₃ := c) p.2 p₀.2).1 hpp₀

Goal state after:
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
hs : Cospherical s
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
hc : Collinear ℝ (Set.range p)
⊢ False

Tactic:
rw [collinear_iff_of_mem (Set.mem_range_self (0 : Fin 3))] at hc

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
hs : Cospherical s
p : Fin 3 → P
hps : Set.range p ⊆ s
hpi : Function.Injective p
hc : ∃ v, ∀ p_1 ∈ Set.range p, ∃ r, p_1 = r • v +ᵥ p 0
⊢ False
Module: Mathlib.Geometry.Euclidean.Basic

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
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥s.direction
hd : finrank ℝ ↥s.direction = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = s.direction
hv : ∀ v ∈ s.direction, ∃ t₁ t₂, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)
t₁ t₂ : ℝ
hpt : p -ᵥ p₁ = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)
hop : t₁ = 0
⊢ p = p₁ ∨ p = p₂

Tactic:
rw [hop, zero_smul, zero_add, ← eq_vadd_iff_vsub_eq] at hpt

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
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥s.direction
hd : finrank ℝ ↥s.direction = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = s.direction
hv : ∀ v ∈ s.direction, ∃ t₁ t₂, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)
t₁ t₂ : ℝ
hpt : p = t₂ • (p₂ -ᵥ p₁) +ᵥ p₁
hop : t₁ = 0
⊢ p = p₁ ∨ p = p₂
Module: Mathlib.Geometry.Euclidean.Sphere.Tangent

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 : Sphere P
p₁ p₂ q : P
as₁ as₂ : AffineSubspace ℝ P
h₁ : s.IsTangentAt p₁ as₁
h₂ : s.IsTangentAt p₂ as₂
hq_mem₁ : q ∈ as₁
hq_mem₂ : q ∈ as₂
⊢ dist q p₁ = dist q p₂

Tactic:
by
  have h1 := dist_sq_eq_of_mem h₁ hq_mem₁
  have h2 := dist_sq_eq_of_mem h₂ hq_mem₂
  rwa [h1, add_left_cancel_iff, sq_eq_sq₀ dist_nonneg dist_nonneg] at h2

Goal state after:
Module: Mathlib.Geometry.Euclidean.Angle.Incenter

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)
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
p : P
ha : AffineIndependent ℝ ![t.points i₁, t.points i₂, t.points i₃]
⊢ dist p ↑((faceOpposite t i₃).orthogonalProjectionSpan p) = dist p ↑((faceOpposite t i₂).orthogonalProjectionSpan p) ↔
    2 • ∡ (t.points i₂) (t.points i₁) p = 2 • ∡ p (t.points i₁) (t.points i₃)

Tactic:
rw [orthogonalProjectionSpan, orthogonalProjectionSpan,
    ← dist_orthogonalProjection_line_eq_iff_two_zsmul_oangle_eq ha]

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)
t : Triangle ℝ P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
p : P
ha : AffineIndependent ℝ ![t.points i₁, t.points i₂, t.points i₃]
⊢ dist p ↑((orthogonalProjection (affineSpan ℝ (Set.range (faceOpposite t i₃).points))) p) =
      dist p ↑((orthogonalProjection (affineSpan ℝ (Set.range (faceOpposite t i₂).points))) p) ↔
    dist p ↑((orthogonalProjection line[ℝ, t.points i₁, t.points i₂]) p) =
      dist p ↑((orthogonalProjection line[ℝ, t.points i₁, t.points i₃]) p)
Module: Mathlib.Geometry.Euclidean.Circumcenter

Goal state before:
pp: case intro.zero
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
ι : Type u_3
hne : Nonempty ι
inst✝ : Finite ι
p : ι → P
ha : AffineIndependent ℝ p
val✝ : Fintype ι
hn : Fintype.card ι = 0
⊢ ∃! cs, cs.center ∈ affineSpan ℝ (Set.range p) ∧ Set.range p ⊆ Metric.sphere cs.center cs.radius

Tactic:
| zero =>

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : ‖x‖ = 1
hy : ‖y‖ = 1
⊢ ⟪x, y⟫ * ⟪x, y⟫ ≤ 1

Tactic:
grw [← sq, sq_le_one_iff_abs_le_one, abs_real_inner_le_norm, hx, hy, one_mul]

Goal state after: