mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.ModelTheory.Algebra.Field.IsAlgClosed

Goal state before:
pp: p : ℕ
κ : Cardinal.{u_2}
hκ : ℵ₀ < κ
M : Type u_2
struc✝¹ : Language.ring.Structure M
is_model✝¹ : M ⊨ Theory.ACF p
nonempty'✝¹ : Nonempty M
N : Type u_2
struc✝ : Language.ring.Structure N
is_model✝ : N ⊨ Theory.ACF p
nonempty'✝ : Nonempty N
hM : #↑{ Carrier := M, struc := struc✝¹, is_model := is_model✝¹, nonempty' := nonempty'✝¹ } = κ
hN : #↑{ Carrier := N, struc := struc✝, is_model := is_model✝, nonempty' := nonempty'✝ } = κ
x✝¹ : Field M := fieldOfModelACF p M
this✝ : M ⊨ Theory.field
x✝ : CompatibleRing M := compatibleRingOfModelField M
this : IsAlgClosed M
⊢ Nonempty
    (↑{ Carrier := M, struc := struc✝¹, is_model := is_model✝¹, nonempty' := nonempty'✝¹ } ≃[Language.ring]
      ↑{ Carrier := N, struc := struc✝, is_model := is_model✝, nonempty' := nonempty'✝ })

Tactic:
have := charP_of_model_fieldOfChar p M

Goal state after:
p : ℕ
κ : Cardinal.{u_2}
hκ : ℵ₀ < κ
M : Type u_2
struc✝¹ : Language.ring.Structure M
is_model✝¹ : M ⊨ Theory.ACF p
nonempty'✝¹ : Nonempty M
N : Type u_2
struc✝ : Language.ring.Structure N
is_model✝ : N ⊨ Theory.ACF p
nonempty'✝ : Nonempty N
hM : #↑{ Carrier := M, struc := struc✝¹, is_model := is_model✝¹, nonempty' := nonempty'✝¹ } = κ
hN : #↑{ Carrier := N, struc := struc✝, is_model := is_model✝, nonempty' := nonempty'✝ } = κ
x✝¹ : Field M := fieldOfModelACF p M
this✝¹ : M ⊨ Theory.field
x✝ : CompatibleRing M := compatibleRingOfModelField M
this✝ : IsAlgClosed M
this : CharP M p
⊢ Nonempty
    (↑{ Carrier := M, struc := struc✝¹, is_model := is_model✝¹, nonempty' := nonempty'✝¹ } ≃[Language.ring]
      ↑{ Carrier := N, struc := struc✝, is_model := is_model✝, nonempty' := nonempty'✝ })
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: case pos
R : Type u_1
inst✝ : CommSemiring R
p : ℕ
x : R
hp : 1 < p
x✝ : x ∈ nilradical R
n : ℕ
h : x ^ n ∈ 0
⊢ x ^ p ^ n = 0

Tactic:
rw [← Nat.sub_add_cancel ((n.lt_pow_self hp).le), pow_add, h, mul_zero]

Goal state after:
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
e : PerfectClosure K p
x✝ : ℕ × K
n : ℕ
x : K
⊢ ((0, 0).1 + (n, x).1, (⇑(frobenius K p))^[(n, x).1] (0, 0).2 + (⇑(frobenius K p))^[(0, 0).1] (n, x).2) = (n, x)

Tactic:
by
          simp only [iterate_map_zero, iterate_zero_apply, zero_add]

Goal state after:
Module: Mathlib.GroupTheory.CosetCover

Goal state before:
pp: case right
G : Type u_1
inst✝ : Group G
ι : Type u_2
s : Finset ι
H : Subgroup G
g : ι → G
hcovers : ⋃ i ∈ s, g i • ↑H = Set.univ
hind : H.index = s.card
this : Fintype (G ⧸ H)
⊢ Fintype.card ↥s = Fintype.card (G ⧸ H)

Tactic:
simp only [Fintype.card_coe, ← hind, index_eq_card, Nat.card_eq_fintype_card]

Goal state after:
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
hm : ∑ i ∈ s, m i • (x - (algebraMap F E) i)⁻¹ = 0
i✝ : F
hi✝ : i✝ ∈ s
hnz : ∀ (a : F), x - (algebraMap F E) a ≠ 0
b : E := ∏ j ∈ s, (x - (algebraMap F E) j)
i : F
hi : i ∈ s
⊢ m i • ((∏ j ∈ s.erase i, (x - (algebraMap F E) j)) * (x - (algebraMap F E) i) * (x - (algebraMap F E) i)⁻¹) =
    m i • ∏ j ∈ s.erase i, (x - (algebraMap F E) j)

Tactic:
mul_inv_cancel_right₀ (hnz i)

Goal state after:
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
hm : ∑ i ∈ s, m i • (x - (algebraMap F E) i)⁻¹ = 0
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)
⊢ m i = 0

Tactic:
replace hm := congr(b * $(hm))

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)
hm : b * ∑ i ∈ s, m i • (x - (algebraMap F E) i)⁻¹ = b * 0
⊢ m i = 0
Module: Mathlib.FieldTheory.JacobsonNoether

Goal state before:
pp: case neg
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
m : ℕ
hm2 : ∀ (n : ℕ), p ^ m ≤ n → (⇑((ad (↥k) D) a))^[n] = 0
h_exist : ∃ n, 0 < n ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
t : ℕ := Nat.find h_exist - 1
ht : t = Nat.find h_exist - 1
h_pos : t ≤ 0
⊢ (⇑((ad (↥k) D) a))^[Nat.find h_exist] b ≠ 0

Tactic:
suffices h_find : Nat.find h_exist = 1 by
        rwa [h_find]
      rw [Nat.le_zero, ht, Nat.sub_eq_zero_iff_le] at h_pos
      linarith [(Nat.find_spec h_exist).1]

Goal state after:
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : Ring S
inst✝ : Algebra R S
A B : Subalgebra R S
ι : Type u_1
a : ι → ↥A
i : (ι →₀ ↥B) →ₗ[R] S := Submodule.mulLeftMap (toSubmodule B) a
j : (ι →₀ ↥B) →ₗ[R] S :=
  ↑(MulOpposite.opLinearEquiv R).symm ∘ₗ
    ↑R (Finsupp.linearCombination (↥B.op) (MulOpposite.op ∘ ⇑A.val ∘ a)) ∘ₗ
      ↑(Finsupp.mapRange.linearEquiv B.linearEquivOp)
this : i = j
⊢ Function.Injective ⇑(Submodule.mulLeftMap (toSubmodule B) a) ↔
    Function.Injective ⇑(Finsupp.linearCombination (↥B.op) (MulOpposite.op ∘ ⇑A.val ∘ a))

Tactic:
by
    change Function.Injective i ↔ _
    simp_rw [this, j, LinearMap.coe_comp, LinearEquiv.coe_coe, EquivLike.comp_injective,
      EquivLike.injective_comp, LinearMap.coe_restrictScalars]

Goal state after:
Module: Mathlib.FieldTheory.Minpoly.ConjRootClass

Goal state before:
pp: case h
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
x x✝ : L
⊢ (aeval x) (mk K x✝).minpoly = 0 ↔ mk K x = mk K x✝

Tactic:
simpa [← isConjRoot_iff_aeval_eq_zero (Algebra.IsIntegral.isIntegral _)] using comm

Goal state after:
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: F : Type u
E : Type v
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Algebra F E
A B : IntermediateField F E
inst✝² : IsGalois F ↥A
inst✝¹ : FiniteDimensional F ↥A
inst✝ : FiniteDimensional F ↥B
h : A ⊓ B = ⊥
C : IntermediateField F E := A ⊔ B
A' : IntermediateField F ↥C := restrict ⋯
B' : IntermediateField F ↥C := restrict ⋯
⊢ A.LinearDisjoint ↥B

Tactic:
have hA : IntermediateField.map C.val A' = A := lift_restrict le_sup_left

Goal state after:
F : Type u
E : Type v
inst✝⁵ : Field F
inst✝⁴ : Field E
inst✝³ : Algebra F E
A B : IntermediateField F E
inst✝² : IsGalois F ↥A
inst✝¹ : FiniteDimensional F ↥A
inst✝ : FiniteDimensional F ↥B
h : A ⊓ B = ⊥
C : IntermediateField F E := A ⊔ B
A' : IntermediateField F ↥C := restrict ⋯
B' : IntermediateField F ↥C := restrict ⋯
hA : IntermediateField.map C.val A' = A
⊢ A.LinearDisjoint ↥B
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
A : IntermediateField F E
L : Type w
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Algebra L E
inst✝ : IsScalarTower F L E
H : A.LinearDisjoint L
halg : Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L
⊢ Module.rank ↥A ↥(extendScalars ⋯) = Module.rank F ↥(IsScalarTower.toAlgHom F L E).range

Tactic:
← Subalgebra.LinearDisjoint.adjoin_rank_eq_rank_right H

Goal state after:
F : Type u
E : Type v
inst✝⁶ : Field F
inst✝⁵ : Field E
inst✝⁴ : Algebra F E
A : IntermediateField F E
L : Type w
inst✝³ : Field L
inst✝² : Algebra F L
inst✝¹ : Algebra L E
inst✝ : IsScalarTower F L E
H : A.LinearDisjoint L
halg : Algebra.IsAlgebraic F ↥A ∨ Algebra.IsAlgebraic F L
⊢ Module.rank ↥A ↥(extendScalars ⋯) =
    Module.rank ↥A.toSubalgebra ↥(Algebra.adjoin ↥A.toSubalgebra ↑(IsScalarTower.toAlgHom F L E).range)
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}
⊢ x✝ ∈ ↑(LinearMap.rTensor (↥(toSubmodule B)) (Submodule.inclusion hA)).range

Tactic:
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)

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

Goal state before:
pp: case pos
K : Type u
inst✝ : Field K
x : K⟮X⟯
hx : x = 0
⊢ (-x).intDegree = x.intDegree

pp: case neg
K : Type u
inst✝ : Field K
x : K⟮X⟯
hx : ¬x = 0
⊢ (-x).intDegree = x.intDegree

Tactic:
· rw [hx, neg_zero]

Goal state after:
case neg
K : Type u
inst✝ : Field K
x : K⟮X⟯
hx : ¬x = 0
⊢ (-x).intDegree = x.intDegree
Module: Mathlib.FieldTheory.PurelyInseparable.Tower

Goal state before:
pp: F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
f : F[X]
hsep : f.Separable
hirr : Irreducible f
inst✝ : IsPurelyInseparable F E
K : Type v := AlgebraicClosure E
x : K
hx : (aeval x) f = 0
ha : Associated f (minpoly F x)
ha' : Associated (Polynomial.map (algebraMap F E) f) (Polynomial.map (algebraMap F E) (minpoly F x))
heq : Polynomial.map (algebraMap F E) (minpoly F x) = minpoly E x
⊢ Irreducible (Polynomial.map (algebraMap F E) f)

Tactic:
rw [ha'.irreducible_iff, heq]

Goal state after:
F : Type u
E : Type v
inst✝³ : Field F
inst✝² : Field E
inst✝¹ : Algebra F E
f : F[X]
hsep : f.Separable
hirr : Irreducible f
inst✝ : IsPurelyInseparable F E
K : Type v := AlgebraicClosure E
x : K
hx : (aeval x) f = 0
ha : Associated f (minpoly F x)
ha' : Associated (Polynomial.map (algebraMap F E) f) (Polynomial.map (algebraMap F E) (minpoly F x))
heq : Polynomial.map (algebraMap F E) (minpoly F x) = minpoly E x
⊢ Irreducible (minpoly E x)
Module: Mathlib.Geometry.Convex.Cone.DualFinite

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
S T : Finset M
⊢ DualFG p (dual p ↑S ⊓ dual p ↑T)

Tactic:
use S ∪ T

Goal state after:
case h
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : PartialOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
p : M →ₗ[R] N →ₗ[R] R
S T : Finset M
⊢ dual p ↑(S ∪ T) = dual p ↑S ⊓ dual p ↑T
Module: Mathlib.FieldTheory.Relrank

Goal state before:
pp: E : Type v
inst✝¹ : Field E
A B : Subfield E
L : Type v
inst✝ : Field L
f : L →+* E
h : B ≤ f.fieldRange
⊢ (comap f A).relrank (comap f B) = A.relrank B

Tactic:
by
  simpa only [lift_id] using lift_relrank_comap_comap_eq_lift_relrank_of_le A B f h

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
x : X
hs : IsConvexSet R s
hsx : IsConvexSet R (s \ {x})
hx : x ∈ (convexHull R) (s \ {x})
⊢ False

Tactic:
rw [hsx.convexHull_eq_self] at hx

Goal state after:
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
x : X
hs : IsConvexSet R s
hsx : IsConvexSet R (s \ {x})
hx : x ∈ s \ {x}
⊢ False
Module: Mathlib.FieldTheory.Relrank

Goal state before:
pp: F : Type u
E : Type v
inst✝⁴ : Field F
inst✝³ : Field E
inst✝² : Algebra F E
A B : IntermediateField F E
L : Type v
inst✝¹ : Field L
inst✝ : Algebra F L
f : L →ₐ[F] E
h : B ≤ f.fieldRange
⊢ (comap f A).relrank (comap f B) = A.relrank B

Tactic:
by
  simpa only [lift_id] using lift_relrank_comap_comap_eq_lift_relrank_of_le A B f h

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✝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 : P
x : P := ↑(Classical.arbitrary ↥s)
v : V := s.direction.starProjection (p -ᵥ x)
⊢ (2 • s.direction.starProjection (p -ᵥ x) - (p -ᵥ x)) +ᵥ x = ((v +ᵥ x) -ᵥ p) +ᵥ v +ᵥ x

Tactic:
two_smul,

Goal state after:
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 : P
x : P := ↑(Classical.arbitrary ↥s)
v : V := s.direction.starProjection (p -ᵥ x)
⊢ (s.direction.starProjection (p -ᵥ x) + s.direction.starProjection (p -ᵥ x) - (p -ᵥ x)) +ᵥ x =
    ((v +ᵥ x) -ᵥ p) +ᵥ v +ᵥ x
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
⊢ SameRay ℝ x y ∨ x ≠ 0 ∧ y ≠ 0 ∧ SameRay ℝ x (-y) ↔ x = 0 ∨ ∃ r, y = r • x

Tactic:
refine ⟨fun h => ?_, fun h => ?_⟩

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
h : SameRay ℝ x y ∨ x ≠ 0 ∧ y ≠ 0 ∧ SameRay ℝ x (-y)
⊢ x = 0 ∨ ∃ r, y = r • x

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
h : x = 0 ∨ ∃ r, y = r • x
⊢ SameRay ℝ x y ∨ x ≠ 0 ∧ y ≠ 0 ∧ SameRay ℝ x (-y)
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
⊢ ⟪x, (o.rotation ↑(π / 2)) x⟫ = 0

Tactic:
by
  rw [real_inner_comm, inner_rotation_pi_div_two_left]

Goal state after:
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
p₁ p₂ p₃ : P
⊢ Collinear ℝ {p₁, p₂, p₃} ↔ p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π

Tactic:
refine ⟨fun h => ?_, fun h => ?_⟩

Goal state after:
case 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
p₁ p₂ p₃ : P
h : Collinear ℝ {p₁, p₂, p₃}
⊢ p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π

case refine_2
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 : p₁ = p₂ ∨ p₃ = p₂ ∨ ∠ p₁ p₂ p₃ = 0 ∨ ∠ p₁ p₂ p₃ = π
⊢ Collinear ℝ {p₁, p₂, p₃}
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.sin (angle x (x - y)) * ‖x - y‖ = ‖y‖

Tactic:
rw [sub_eq_add_neg, sin_angle_add_mul_norm_of_inner_eq_zero h, norm_neg]

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
hx : x ≠ 0
hy : y ≠ 0
⊢ o.oangle x y = 0 ↔ InnerProductGeometry.angle x y = 0

Tactic:
refine ⟨fun h => ?_, fun h => ?_⟩

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
hx : x ≠ 0
hy : y ≠ 0
h : o.oangle x y = 0
⊢ InnerProductGeometry.angle x y = 0

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
hx : x ≠ 0
hy : y ≠ 0
h : InnerProductGeometry.angle x y = 0
⊢ o.oangle x y = 0
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
⊢ o.oangle x y = ↑π ↔ InnerProductGeometry.angle x y = π

Tactic:
refine ⟨fun h => ?_, fun h => ?_⟩

Goal state after:
case neg.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
hx : ¬x = 0
hy : ¬y = 0
h : o.oangle x y = ↑π
⊢ InnerProductGeometry.angle x y = π

case neg.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
hx : ¬x = 0
hy : ¬y = 0
h : InnerProductGeometry.angle x y = π
⊢ o.oangle x y = ↑π
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
⊢ o.oangle x y = ↑(π / 2) ∨ o.oangle x y = ↑(-π / 2) ↔ InnerProductGeometry.angle x y = π / 2

Tactic:
refine ⟨fun h => ?_, fun h => ?_⟩

Goal state after:
case neg.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
hx : ¬x = 0
hy : ¬y = 0
h : o.oangle x y = ↑(π / 2) ∨ o.oangle x y = ↑(-π / 2)
⊢ InnerProductGeometry.angle x y = π / 2

case neg.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
hx : ¬x = 0
hy : ¬y = 0
h : InnerProductGeometry.angle x y = π / 2
⊢ o.oangle x y = ↑(π / 2) ∨ o.oangle x y = ↑(-π / 2)
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 = ↑π ↔ Sbtw ℝ p₁ p₂ p₃

Tactic:
by
  rw [oangle_eq_pi_iff_angle_eq_pi, angle_eq_pi_iff_sbtw]

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
h₁₂ : p₁ ≠ p₂
h₃₂ : p₃ ≠ p₂
⊢ (AffineEquiv.pointReflection ℝ p₂) p₃ ≠ p₂

Tactic:
conv_rhs => rw [← AffineEquiv.pointReflection_self ℝ p₂]

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
h₁₂ : p₁ ≠ p₂
h₃₂ : p₃ ≠ p₂
⊢ (AffineEquiv.pointReflection ℝ p₂) p₃ ≠ (AffineEquiv.pointReflection ℝ p₂) p₂
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
r : ℝ
⊢ (∃ g, g 0 • ![x, r • x + y] 0 + g 1 • ![x, r • x + y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)) ↔
    ∃ g, g 0 • ![x, y] 0 + g 1 • ![x, y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)

Tactic:
refine ⟨fun h => ?_, fun h => ?_⟩

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 : ∃ g, g 0 • ![x, r • x + y] 0 + g 1 • ![x, r • x + y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)
⊢ ∃ g, g 0 • ![x, y] 0 + g 1 • ![x, y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)

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 : ∃ g, g 0 • ![x, y] 0 + g 1 • ![x, y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)
⊢ ∃ g, g 0 • ![x, r • x + y] 0 + g 1 • ![x, r • x + y] 1 = 0 ∧ (g 0 ≠ 0 ∨ g 1 ≠ 0)
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle

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 = ↑(π / 2)
⊢ ∡ p₃ p₁ p₂ = ↑(Real.arccos (dist p₁ p₂ / dist p₁ p₃))

Tactic:
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
  rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm,
    angle_eq_arccos_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h),
    dist_comm p₁ p₃]

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

Goal state before:
pp: case inr
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 ∈ s₂
⊢ dist p ↑((orthogonalProjection s₁) p) = dist p ↑((orthogonalProjection s₂) p) ↔
    ∠ p p' ↑((orthogonalProjection s₁) p) = ∠ p p' ↑((orthogonalProjection s₂) p)

Tactic:
· nth_rw 1 [eq_comm]
    nth_rw 2 [eq_comm]
    exact dist_orthogonalProjection_eq_iff_angle_eq_aux₁ hp'₂ hp'₁ 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
hz : z ≠ 0
hy : ¬y = 0
hr : ¬SameRay ℝ x z
hn : o.oangle x y ≠ o.oangle y z
he : InnerProductGeometry.angle x y = InnerProductGeometry.angle y z
⊢ ¬(o.oangle x y).sign = (o.oangle y z).sign →
    ¬(o.oangle x y).sign = -(o.oangle y z).sign → (o.oangle x y).sign = 0 ∨ (o.oangle y z).sign = 0

Tactic:
generalize (o.oangle x y).sign = sxy

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 z : V
hx : x ≠ 0
hz : z ≠ 0
hy : ¬y = 0
hr : ¬SameRay ℝ x z
hn : o.oangle x y ≠ o.oangle y z
he : InnerProductGeometry.angle x y = InnerProductGeometry.angle y z
sxy : SignType
⊢ ¬sxy = (o.oangle y z).sign → ¬sxy = -(o.oangle y z).sign → sxy = 0 ∨ (o.oangle y z).sign = 0