mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Algebraic.LinearIndependent

Goal state before:
pp: F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
H : ∀ (p : F[X]), (aeval x) p = 0 → p = 0
s : Finset F
m : F → F
i : F
hi : i ∈ s
hnz : ∀ (a : F), x - (algebraMap F E) a ≠ 0
b : E := ∏ j ∈ s, (x - (algebraMap F E) j)
h1 : ∀ i ∈ s, m i • (b * (x - (algebraMap F E) i)⁻¹) = m i • ∏ j ∈ s.erase i, (x - (algebraMap F E) j)
p : F[X] := ∑ i ∈ s, C (m i) * ∏ j ∈ s.erase i, (X - C j)
hm : (aeval i) p = (aeval i) 0
⊢ m i = 0

Tactic:
have h2 : ∀ j ∈ s.erase i, m j * ((s.erase j).prod fun x ↦ i - x) = 0 := fun j hj ↦ by
    have := Finset.mem_erase_of_ne_of_mem (Finset.ne_of_mem_erase hj).symm hi
    simp_rw [← (s.erase j).prod_erase_mul _ this, sub_self, mul_zero]

Goal state after:
F : Type u_1
E : Type u_2
inst✝² : Field F
inst✝¹ : Field E
inst✝ : Algebra F E
x : E
H : ∀ (p : F[X]), (aeval x) p = 0 → p = 0
s : Finset F
m : F → F
i : F
hi : i ∈ s
hnz : ∀ (a : F), x - (algebraMap F E) a ≠ 0
b : E := ∏ j ∈ s, (x - (algebraMap F E) j)
h1 : ∀ i ∈ s, m i • (b * (x - (algebraMap F E) i)⁻¹) = m i • ∏ j ∈ s.erase i, (x - (algebraMap F E) j)
p : F[X] := ∑ i ∈ s, C (m i) * ∏ j ∈ s.erase i, (X - C j)
hm : (aeval i) p = (aeval i) 0
h2 : ∀ j ∈ s.erase i, m j * ∏ x ∈ s.erase j, (i - x) = 0
⊢ m i = 0
Module: Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.Transcendental F E
⊢ Module.rank F E = #E

Tactic:
obtain ⟨ι, x, hx⟩ := exists_isTranscendenceBasis' F E

Goal state after:
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
inst✝ : Algebra.Transcendental F E
ι : Type v
x : ι → E
hx : IsTranscendenceBasis F x
⊢ Module.rank F E = #E
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: L : Type u_2
D : Type u_3
inst✝⁴ : Field L
inst✝³ : DivisionRing D
inst✝² : Algebra L D
inst✝¹ : Algebra.IsAlgebraic L D
inst✝ : IsCentral L D
hneq : ⊥ ≠ ⊤
hcenter : Subalgebra.center L D = ⊥
ntrivial : k ≠ ⊤
φ : ↥⊥ ≃ₐ[L] ↥(Subalgebra.center L D) := ⊥.equivOfEq (Subalgebra.center L D) ⋯
equiv : L ≃+* ↥k := ((botEquiv L D).symm.trans φ).toRingEquiv
x✝² : Algebra L ↥k := equiv.toRingHom.toAlgebra
x✝¹ : Algebra (↥k) L := equiv.symm.toRingHom.toAlgebra
x✝ : IsScalarTower L (↥k) D
x : ↥k
⊢ (algebraMap (↥k) D) x = (algebraMap L D) ((algebraMap (↥k) L) x)

Tactic:
IsScalarTower.algebraMap_apply L (center D)

Goal state after:
L : Type u_2
D : Type u_3
inst✝⁴ : Field L
inst✝³ : DivisionRing D
inst✝² : Algebra L D
inst✝¹ : Algebra.IsAlgebraic L D
inst✝ : IsCentral L D
hneq : ⊥ ≠ ⊤
hcenter : Subalgebra.center L D = ⊥
ntrivial : k ≠ ⊤
φ : ↥⊥ ≃ₐ[L] ↥(Subalgebra.center L D) := ⊥.equivOfEq (Subalgebra.center L D) ⋯
equiv : L ≃+* ↥k := ((botEquiv L D).symm.trans φ).toRingEquiv
x✝² : Algebra L ↥k := equiv.toRingHom.toAlgebra
x✝¹ : Algebra (↥k) L := equiv.symm.toRingHom.toAlgebra
x✝ : IsScalarTower L (↥k) D
x : ↥k
⊢ (algebraMap (↥k) D) x = (algebraMap (↥k) D) ((algebraMap L ↥k) ((algebraMap (↥k) L) x))
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case refine_3
G : Type u_1
inst✝¹ : 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
inst✝ : DecidablePred FiniteIndex
D : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ H i
t : (i : ι) → i ∈ s → (H i).FiniteIndex → Finset ↥(H i)
ht :
  ∀ (i : ι) (hi : i ∈ s) (hfi : (H i).FiniteIndex),
    IsComplement ↑(t i hi hfi) ↑(D.subgroupOf (H i)) ∧ ⋃ g ∈ t i hi hfi, ↑g • ↑D = ↑(H i)
κ : Type (max u_2 u_1) := (i : ↥s) × ↥(if h : (H ↑i).FiniteIndex then t ↑i ⋯ h else {1})
f : κ → G := fun k ↦ g ↑k.fst * ↑↑k.snd
K : κ → Subgroup G := fun k ↦ if (H ↑k.fst).FiniteIndex then D else H ↑k.fst
k : κ
hkfi : (H ↑k.fst).FiniteIndex
hk : K k = D
hcovers' : ⋃ i ∈ {x | K x = D}, f i • ↑D = Set.univ
hHD : ∀ (i : ι), ¬(H i).FiniteIndex → H i ≠ D
hdensity : ∑ i ∈ s, (↑(H i).index)⁻¹ = ↑{x | K x = D}.card * (↑D.index)⁻¹
h : {x | K x = D}.card = D.index
i : ι
hi : i ∈ s ∧ (H i).FiniteIndex
j : ι
hj : j ∈ s ∧ (H j).FiniteIndex
hij : i ≠ j
c : Set G
hi' : c ⊆ (fun i ↦ g i • ↑(H i)) i
hj' : c ⊆ (fun i ↦ g i • ↑(H i)) j
x : G
hx : x ∈ c
hdisjoint : (↑{x | K x = D}).PairwiseDisjoint fun x ↦ f x • ↑D
⊢ x ∈ ⊥

Tactic:
have hk' (i) (hi : i ∈ s ∧ (H i).FiniteIndex) (hi' : c ≤ g i • (H i : Set G)) :
        ∃ (k : κ), k.1.1 = i ∧ K k = D ∧ x ∈ f k • (D : Set G) := by
      rw [← (ht i hi.1 hi.2).2] at hi'
      suffices ∃ r : H i, r ∈ t i hi.1 hi.2 ∧ x ∈ (g i * r) • (D : Set G) by
        have ⟨r, hr, hxr⟩ := this
        refine ⟨⟨⟨i, hi.1⟩, ⟨r, dif_pos hi.2 ▸ hr⟩⟩, rfl, ?_⟩
        simpa [K, f, if_pos hi.2] using! hxr
      simpa [Set.mem_smul_set_iff_inv_smul_mem, smul_eq_mul, mul_assoc] using! hi' hx

Goal state after:
case refine_3
G : Type u_1
inst✝¹ : 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
inst✝ : DecidablePred FiniteIndex
D : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k
hD : D.FiniteIndex
hD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ H i
t : (i : ι) → i ∈ s → (H i).FiniteIndex → Finset ↥(H i)
ht :
  ∀ (i : ι) (hi : i ∈ s) (hfi : (H i).FiniteIndex),
    IsComplement ↑(t i hi hfi) ↑(D.subgroupOf (H i)) ∧ ⋃ g ∈ t i hi hfi, ↑g • ↑D = ↑(H i)
κ : Type (max u_2 u_1) := (i : ↥s) × ↥(if h : (H ↑i).FiniteIndex then t ↑i ⋯ h else {1})
f : κ → G := fun k ↦ g ↑k.fst * ↑↑k.snd
K : κ → Subgroup G := fun k ↦ if (H ↑k.fst).FiniteIndex then D else H ↑k.fst
k : κ
hkfi : (H ↑k.fst).FiniteIndex
hk : K k = D
hcovers' : ⋃ i ∈ {x | K x = D}, f i • ↑D = Set.univ
hHD : ∀ (i : ι), ¬(H i).FiniteIndex → H i ≠ D
hdensity : ∑ i ∈ s, (↑(H i).index)⁻¹ = ↑{x | K x = D}.card * (↑D.index)⁻¹
h : {x | K x = D}.card = D.index
i : ι
hi : i ∈ s ∧ (H i).FiniteIndex
j : ι
hj : j ∈ s ∧ (H j).FiniteIndex
hij : i ≠ j
c : Set G
hi' : c ⊆ (fun i ↦ g i • ↑(H i)) i
hj' : c ⊆ (fun i ↦ g i • ↑(H i)) j
x : G
hx : x ∈ c
hdisjoint : (↑{x | K x = D}).PairwiseDisjoint fun x ↦ f x • ↑D
hk' : ∀ (i : ι), i ∈ s ∧ (H i).FiniteIndex → c ⊆ g i • ↑(H i) → ∃ k, ↑k.fst = i ∧ K k = D ∧ x ∈ f k • ↑D
⊢ x ∈ ⊥
Module: Mathlib.FieldTheory.Minpoly.ConjRootClass

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x y : ConjRootClass K L
⊢ (∃ a ∈ x.carrier, ∃ b ∈ y.carrier, a + b = 0) ↔ x = -y

Tactic:
simp_rw [mem_carrier]
  constructor
  · rintro ⟨a, rfl, b, rfl, h⟩
    rw [mk_neg, mk_eq_mk, add_eq_zero_iff_eq_neg.mp h]
  · rintro rfl
    induction y with
    | h y => exact ⟨-y, mk_neg y, y, rfl, neg_add_cancel _⟩

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

Goal state before:
pp: case refine_1
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
⊢ ∀ (a₁ a₂ : ↥⊥) (b₁ b₂ : ↥A),
    (toSubmodule A).lTensorOne ((a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)) =
      (toSubmodule A).lTensorOne (a₁ ⊗ₜ[R] b₁) * (toSubmodule A).lTensorOne (a₂ ⊗ₜ[R] b₂)

Tactic:
rintro x y a b

Goal state after:
case refine_1
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
x y : ↥⊥
a b : ↥A
⊢ (toSubmodule A).lTensorOne ((x * y) ⊗ₜ[R] (a * b)) =
    (toSubmodule A).lTensorOne (x ⊗ₜ[R] a) * (toSubmodule A).lTensorOne (y ⊗ₜ[R] 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
n : ℕ
h : a ^ ringExpChar K ^ n ∈ (algebraMap K L).range
⊢ elemExponent K a ≤ n

Tactic:
let ⟨p, _⟩ := ExpChar.exists K
  rcases ‹ExpChar K p› with _ | ⟨hp⟩
  · exact elemExponent_eq_zero_of_charZero K a ▸ Nat.zero_le _
  · obtain ⟨y, hy⟩ := RingHom.mem_range.mp <| h
    let f := X ^ ringExpChar K ^ n - C y
    have hf₁ : f.aeval a = 0 := by rwa [map_sub, aeval_C, aeval_X_pow, sub_eq_zero, eq_comm]
    have hf₂ : f.Monic := monic_X_pow_sub_C y <| Nat.pos_iff_ne_zero.mp <| expChar_pow_pos K _ _
    have hf₃ : f.natDegree = ringExpChar K ^ n := by
      rw [natDegree_sub_C, natDegree_pow, natDegree_X, mul_one]
    exact (Nat.pow_le_pow_iff_right <| Nat.Prime.one_lt hp).mp <|
      ringExpChar.eq K p ▸ hf₃ ▸ minpoly_natDegree_eq K a ▸
      natDegree_le_natDegree (minpoly.min K a hf₂ hf₁)

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

Goal state before:
pp: case refine_1
F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
K : Type w
inst✝¹ : Field K
inst✝ : Algebra F K
i : E →ₐ[F] K
x : E
x✝ : ∃ n, i x ^ ringExpChar F ^ n ∈ (algebraMap F K).range
n : ℕ
y : F
h : (algebraMap F K) y = i x ^ ringExpChar F ^ n
⊢ i ((algebraMap F E) y) = i (x ^ ringExpChar F ^ n)

Tactic:
rwa [AlgHom.commutes, map_pow]

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

Goal state before:
pp: 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✝ : Algebra.IsSeparable F E
⊢ insepDegree F K = insepDegree E K

Tactic:
by
  rw [insepDegree, insepDegree, separableClosure.eq_restrictScalars_of_isSeparable F E K]
  rfl

Goal state after:
Module: Mathlib.Geometry.Convex.Hull

Goal state before:
pp: R : Type u_1
X : Type u_2
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R X
s : Set X
⊢ ((convexHull R) s).Nonempty ↔ s.Nonempty

Tactic:
by
  simp [nonempty_iff_ne_empty]

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 ≠ ⊥
⊢ Q₀ E ∈ lifts (algebraMap K[X] K⟮X⟯)

Tactic:
classical
  apply (Φ' E).isPrimitive_primPart.mul_map_mem_lifts_iff.mp
  rw [Q₀_mul_Φ h]
  exact ⟨_, rfl⟩

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 ≠ ⊥
this : Bivariate.swap (Q₁ h * Φ E) = Bivariate.swap (θ E)
⊢ (Bivariate.swap (Q₁ h)).natDegree = 0

Tactic:
rw [map_mul] at this

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
this : Bivariate.swap (Q₁ h) * Bivariate.swap (Φ E) = Bivariate.swap (θ E)
⊢ (Bivariate.swap (Q₁ h)).natDegree = 0
Module: Mathlib.Geometry.Euclidean.Altitude

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
n : ℕ
s : Simplex ℝ P n
S : AffineSubspace ℝ P
hS : affineSpan ℝ (Set.range s.points) ≤ S
i : Fin (n + 1)
⊢ (s.restrict S hS).altitude i = comap S.subtype (s.altitude i)

Tactic:
have := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
  rw [← s.map_altitude_restrict S hS, comap_map_eq_of_injective S.subtype_injective]

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

Goal state before:
pp: case mp.left
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)
p : P
h : vectorSpan ℝ {p, s.points i} = (s.altitude i).direction
⊢ p ≠ s.points i

Tactic:
intro heq
      rw [heq, Set.pair_eq_singleton, vectorSpan_singleton] at h
      have hd : finrank ℝ (s.altitude i).direction = 0 := by rw [← h, finrank_bot]
      simp at hd

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)
⊢ o.rotation 0 = LinearIsometryEquiv.refl ℝ V

Tactic:
by ext; simp [rotation]

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

Goal state before:
pp: case mp
𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace 𝕜 P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : s₁.direction.HasOrthogonalProjection
inst✝ : s₂.direction.HasOrthogonalProjection
p : P
h :
  ↑((orthogonalProjection s₁) p) -ᵥ ↑((orthogonalProjection s₂) p) +
      (↑((orthogonalProjection s₁) p) -ᵥ p - (↑((orthogonalProjection s₂) p) -ᵥ p)) =
    0
⊢ ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)

Tactic:
vsub_sub_vsub_cancel_right,

Goal state after:
case mp
𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s₁ s₂ : AffineSubspace 𝕜 P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : s₁.direction.HasOrthogonalProjection
inst✝ : s₂.direction.HasOrthogonalProjection
p : P
h :
  ↑((orthogonalProjection s₁) p) -ᵥ ↑((orthogonalProjection s₂) p) +
      (↑((orthogonalProjection s₁) p) -ᵥ ↑((orthogonalProjection s₂) p)) =
    0
⊢ ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

Goal state before:
pp: case neg
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
hx : ¬x = 0
hy : ¬y = 0
⊢ 2 • o.oangle (-x) y = 2 • o.oangle x y

Tactic:
simp [o.oangle_neg_left hx hy]

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

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s : AffineSubspace 𝕜 P
inst✝³ : Nonempty ↥s
s' : AffineSubspace 𝕜 ↥s
inst✝² : Nonempty ↥s'
inst✝¹ : s'.direction.HasOrthogonalProjection
inst✝ : (map s.subtype s').direction.HasOrthogonalProjection
p : ↥s
⊢ ↑((orthogonalProjection (map s.subtype s')) ↑p) = ↑↑((orthogonalProjection s') p)

Tactic:
have : (s'.map s.subtypeₐᵢ.toAffineMap).direction.HasOrthogonalProjection := by
    rw [subtypeₐᵢ_toAffineMap]
    infer_instance

Goal state after:
𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁸ : RCLike 𝕜
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace 𝕜 V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
s : AffineSubspace 𝕜 P
inst✝³ : Nonempty ↥s
s' : AffineSubspace 𝕜 ↥s
inst✝² : Nonempty ↥s'
inst✝¹ : s'.direction.HasOrthogonalProjection
inst✝ : (map s.subtype s').direction.HasOrthogonalProjection
p : ↥s
this : (map s.subtypeₐᵢ.toAffineMap s').direction.HasOrthogonalProjection
⊢ ↑((orthogonalProjection (map s.subtype s')) ↑p) = ↑↑((orthogonalProjection s') p)
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine

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
p₀ p : P
⊢ InnerProductGeometry.angle 0 (p -ᵥ p₀) = π / 2

Tactic:
exact angle_zero_left _

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)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : (o.oangle w x).sign = (o.oangle y z).sign
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : (o.oangle w x).sign = 0
hsyz : (o.oangle y z).sign = 0
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
hpi : π / 2 ≠ π
h0wx : w = 0 ∨ x = 0
⊢ y = 0 ∨ z = 0

Tactic:
have h0' := o.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero hsyz

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)
w x y z : V
h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z
hs : (o.oangle w x).sign = (o.oangle y z).sign
h0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0
hswx : (o.oangle w x).sign = 0
hsyz : (o.oangle y z).sign = 0
hwx : InnerProductGeometry.angle w x = π / 2
hyz : InnerProductGeometry.angle y z = π / 2
hpi : π / 2 ≠ π
h0wx : w = 0 ∨ x = 0
h0' : y = 0 ∨ z = 0 ∨ InnerProductGeometry.angle y z = 0 ∨ InnerProductGeometry.angle y z = π
⊢ y = 0 ∨ z = 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)
hs : (o.oangle x (x + y)).sign = 1
⊢ Real.sin (InnerProductGeometry.angle x (x + y)) * ‖x + y‖ = ‖y‖

Tactic:
InnerProductGeometry.sin_angle_add_mul_norm_of_inner_eq_zero
      (o.inner_eq_zero_of_oangle_eq_pi_div_two h)

Goal state after:
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)
hs : (o.oangle x (x + y)).sign = 1
⊢ ‖y‖ = ‖y‖
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).tan * ‖y‖ = ‖x‖

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

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

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 p₃ : P
h : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0
h0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ ≠ 0
⊢ Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₂ / dist p₁ p₃

Tactic:
rw [angle, dist_eq_norm_vsub V p₁ p₂, dist_eq_norm_vsub V p₁ p₃, ← vsub_add_vsub_cancel p₁ p₂ p₃,
    add_comm, sin_angle_add_of_inner_eq_zero h h0]

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 y (y - x)).sin = ‖x‖ / ‖y - x‖

Tactic:
by
  have hs : (o.oangle y (y - x)).sign = 1 := by
    rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
  rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
    InnerProductGeometry.sin_angle_sub_of_inner_eq_zero
      (o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)
      (Or.inl (o.right_ne_zero_of_oangle_eq_pi_div_two h))]

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

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
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')
hp : p ≠ p'
hpd : 0 < dist p p'
⊢ dist p ↑((orthogonalProjection s₁) p) ≤ dist p p'

Tactic:
rw [dist_orthogonalProjection_eq_infDist]

Goal state after:
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')
hp : p ≠ p'
hpd : 0 < dist p p'
⊢ Metric.infDist p ↑s₁ ≤ dist p p'
Module: Mathlib.Geometry.Euclidean.Angle.Bisector

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
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')
hp : p ≠ p'
hpd : 0 < dist p p'
⊢ dist p ↑((orthogonalProjection s₂) p) ≤ dist p p'

Tactic:
rw [dist_orthogonalProjection_eq_infDist]

Goal state after:
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')
hp : p ≠ p'
hpd : 0 < dist p p'
⊢ Metric.infDist p ↑s₂ ≤ dist p p'
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Affine

Goal state before:
pp: case neg
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 p₅ : P
hp₁p₂ : p₁ ≠ p₂
hp₃p₄ : p₃ ≠ p₄
hc : Collinear ℝ {p₁, p₂, p₃, p₄}
hr : SameRay ℝ (p₂ -ᵥ p₁) (p₄ -ᵥ p₃)
hc₅₁₂ : ¬Collinear ℝ {p₅, p₁, p₂}
s : Set (P × P × P) := (fun x ↦ (↑x.1, p₅, x.2 +ᵥ ↑x.1)) '' Set.univ ×ˢ {v | SameRay ℝ (p₂ -ᵥ p₁) v ∧ v ≠ 0}
hco : IsConnected s
hf : ContinuousOn (fun p ↦ ∡ p.1 p.2.1 p.2.2) s
⊢ (∡ p₁ p₅ p₂).sign = (∡ p₃ p₅ p₄).sign

Tactic:
have hsp : ∀ p : P × P × P, p ∈ s → ∡ p.1 p.2.1 p.2.2 ≠ 0 ∧ ∡ p.1 p.2.1 p.2.2 ≠ π := by
      intro p hp
      simp_rw [s, Set.mem_image, Set.mem_prod, Set.mem_ofPred, Set.mem_univ, true_and,
        Prod.ext_iff] at hp
      obtain ⟨q₁, q₅, q₂⟩ := p
      dsimp only at hp ⊢
      obtain ⟨⟨⟨q, hq⟩, v⟩, hv, rfl, rfl, rfl⟩ := hp
      dsimp only [Subtype.coe_mk, Set.mem_ofPred] at hv ⊢
      obtain ⟨hvr, hv0⟩ := hv
      rw [← exists_nonneg_left_iff_sameRay (vsub_ne_zero.2 hp₁p₂.symm)] at hvr
      obtain ⟨r, -, rfl⟩ := hvr
      change q ∈ line[ℝ, p₁, p₂] at hq
      rw [oangle_ne_zero_and_ne_pi_iff_affineIndependent]
      refine affineIndependent_of_ne_of_mem_of_notMem_of_mem ?_ hq
          (fun h => hc₅₁₂ ((collinear_insert_iff_of_mem_affineSpan h).2 (collinear_pair _ _ _))) ?_
      · rwa [← @vsub_ne_zero V, vsub_vadd_eq_vsub_sub, vsub_self, zero_sub, neg_ne_zero]
      · refine vadd_mem_of_mem_direction ?_ hq
        rw [direction_affineSpan]
        exact smul_vsub_rev_mem_vectorSpan_pair _ _ _

Goal state after:
case neg
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 p₅ : P
hp₁p₂ : p₁ ≠ p₂
hp₃p₄ : p₃ ≠ p₄
hc : Collinear ℝ {p₁, p₂, p₃, p₄}
hr : SameRay ℝ (p₂ -ᵥ p₁) (p₄ -ᵥ p₃)
hc₅₁₂ : ¬Collinear ℝ {p₅, p₁, p₂}
s : Set (P × P × P) := (fun x ↦ (↑x.1, p₅, x.2 +ᵥ ↑x.1)) '' Set.univ ×ˢ {v | SameRay ℝ (p₂ -ᵥ p₁) v ∧ v ≠ 0}
hco : IsConnected s
hf : ContinuousOn (fun p ↦ ∡ p.1 p.2.1 p.2.2) s
hsp : ∀ p ∈ s, ∡ p.1 p.2.1 p.2.2 ≠ 0 ∧ ∡ p.1 p.2.1 p.2.2 ≠ ↑π
⊢ (∡ p₁ p₅ p₂).sign = (∡ p₃ p₅ p₄).sign
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_wbtw : Wbtw ℝ a b c
h_inner : ⟪p -ᵥ a, c -ᵥ a⟫ = 0
⊢ dist p b ≤ dist p c

Tactic:
obtain ⟨t, ⟨ht0, ht1⟩, hb_eq⟩ := h_wbtw
  have h_sq_ineq : dist p b ^ 2 ≤ dist p c ^ 2 := by
    rw [← hb_eq, dist_sq_lineMap_of_inner_eq_zero t h_inner, dist_sq_of_inner_eq_zero h_inner]
    have ht_sq_le : t ^ 2 ≤ 1 := sq_le_one_iff₀ ht0 |>.mpr ht1
    nlinarith [sq_nonneg (dist p a), sq_nonneg (dist a c)]
  simpa only [Real.sqrt_sq dist_nonneg] using Real.sqrt_le_sqrt h_sq_ineq

Goal state after: