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
h : i y = x ^ p ^ n
h' : i (Classical.choose ⋯).2 = x ^ p ^ (Classical.choose ⋯).1
⊢ (iterateFrobeniusEquiv M p (Classical.choose ⋯).1).symm (j (Classical.choose ⋯).2) =
    (iterateFrobeniusEquiv M p n).symm (j y)

Tactic:
set n' := (Classical.choose (lift_aux i p x)).1

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
h : i y = x ^ p ^ n
n' : ℕ := (Classical.choose ⋯).1
h' : i (Classical.choose ⋯).2 = x ^ p ^ n'
⊢ (iterateFrobeniusEquiv M p n').symm (j (Classical.choose ⋯).2) = (iterateFrobeniusEquiv M p n).symm (j y)
Module: Mathlib.FieldTheory.IsPerfectClosure

Goal state before:
pp: K : Type u_1
L : Type u_2
M : Type u_3
N : Type u_4
inst✝⁸ : CommRing K
inst✝⁷ : CommRing L
inst✝⁶ : CommRing M
inst✝⁵ : CommRing N
i : K →+* L
j : K →+* M
k : K →+* N
f : L →+* M
g : L →+* N
p : ℕ
inst✝⁴ : ExpChar M p
inst✝³ : ExpChar K p
inst✝² : PerfectRing M p
inst✝¹ : IsPRadical i p
inst✝ : ExpChar L p
⊢ i 0 = 0 ^ p ^ 0

Tactic:
by rw [pow_zero, pow_one, map_zero]

Goal state after:
Module: Mathlib.FieldTheory.PerfectClosure

Goal state before:
pp: K : Type u
inst✝⁵ : CommRing K
p : ℕ
inst✝⁴ : Fact (Nat.Prime p)
inst✝³ : CharP K p
L : Type v
inst✝² : CommSemiring L
inst✝¹ : CharP L p
inst✝ : PerfectRing L p
f : PerfectClosure K p →+* L
⊢ (fun f ↦
        { toFun := fun e ↦ e.liftOn (fun x ↦ (⇑(frobeniusEquiv L p).symm)^[x.1] (f x.2)) ⋯, map_one' := ⋯,
          map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ })
      ((fun f ↦ f.comp (of K p)) f) =
    f

Tactic:
ext ⟨n, x⟩

Goal state after:
case mk
K : Type u
inst✝⁵ : CommRing K
p : ℕ
inst✝⁴ : Fact (Nat.Prime p)
inst✝³ : CharP K p
L : Type v
inst✝² : CommSemiring L
inst✝¹ : CharP L p
inst✝ : PerfectRing L p
f : PerfectClosure K p →+* L
x✝ : PerfectClosure K p
n : ℕ
x : K
⊢ ((fun f ↦
          { toFun := fun e ↦ e.liftOn (fun x ↦ (⇑(frobeniusEquiv L p).symm)^[x.1] (f x.2)) ⋯, map_one' := ⋯,
            map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ })
        ((fun f ↦ f.comp (of K p)) f))
      (Quot.mk (R K p) (n, x)) =
    f (Quot.mk (R K p) (n, x))
Module: Mathlib.GroupTheory.CosetCover

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

Tactic:
let κ := ↥(s.filter (H · ≠ H j)) × Option ↥(s.filter (H · = H j))

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

Goal state before:
pp: D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
m : ℕ
hm2 : ∀ (n : ℕ), p ^ m ≤ n → (⇑((ad (↥k) D) a))^[n] = 0
⊢ ∃ n, 0 < n ∧ (⇑((ad (↥k) D) a))^[n] b ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0

Tactic:
have h_exist : ∃ n, 0 < n ∧ (ad k D a)^[n + 1] b = 0 := ⟨p ^ m,
      ⟨expChar_pow_pos D p m, by rw [hm2 (p ^ m + 1) (Nat.le_add_right _ _), Pi.zero_apply]⟩⟩

Goal state after:
D : Type u_1
inst✝¹ : DivisionRing D
inst✝ : Algebra.IsAlgebraic (↥k) D
H : k ≠ ⊤
p : ℕ
hp : ExpChar D p
insep : ∀ (x : D), IsSeparable (↥k) x → x ∈ k
a : D
ha : a ∉ k
ha₀ : a ≠ 0
b : D
hb1 : ((ad (↥k) D) a) b ≠ 0
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
⊢ ∃ n, 0 < n ∧ (⇑((ad (↥k) D) a))^[n] b ≠ 0 ∧ (⇑((ad (↥k) D) a))^[n + 1] b = 0
Module: Mathlib.RingTheory.TensorProduct.Nontrivial

Goal state before:
pp: R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : CommRing B
inst✝³ : Algebra R A
inst✝² : Algebra R B
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
inst✝¹ : IsDomain A
inst✝ : IsDomain B
⊢ Nontrivial (A ⊗[R] B)

Tactic:
by
  have := ha.isDomain _
  let FR := FractionRing R
  let FA := FractionRing A
  let FB := FractionRing B
  let fa : FR →ₐ[R] FA := IsFractionRing.liftAlgHom (g := Algebra.ofId R FA)
    ((IsFractionRing.injective A FA).comp ha)
  let fb : FR →ₐ[R] FB := IsFractionRing.liftAlgHom (g := Algebra.ofId R FB)
    ((IsFractionRing.injective B FB).comp hb)
  algebraize_only [fa.toRingHom, fb.toRingHom]
  let : CompatibleSMul FR R FA FB := CompatibleSMul.isScalarTower
  exact Algebra.TensorProduct.mapOfCompatibleSMul FR R R FA FB |>.comp
    (Algebra.TensorProduct.map (IsScalarTower.toAlgHom R A FA) (IsScalarTower.toAlgHom R B FB))
    |>.toRingHom.domain_nontrivial

Goal state after:
Module: Mathlib.FieldTheory.KummerExtension

Goal state before:
pp: case neg
K : Type u
inst✝¹ : Field K
n : ℕ
hζ : (primitiveRoots n K).Nonempty
a : K
H : Irreducible (X ^ n - C a)
inst✝ : NeZero n
σ : Gal(K[n√a]/K)
this : Fact (Irreducible (X ^ n - C a))
h : ¬root (X ^ n - C a) = 0
⊢ σ (root (X ^ n - C a)) / root (X ^ n - C a) * root (X ^ n - C a) = σ (root (X ^ n - C a))

Tactic:
· rw [div_mul_cancel₀ _ h]

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

Goal state before:
pp: case h
K : Type u_1
L : Type u_2
S : Type u_3
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Field S
inst✝³ : Algebra K L
inst✝² : Algebra K S
inst✝¹ : Algebra L S
inst✝ : IsScalarTower K L S
x✝ : L
⊢ - -mk K x✝ = mk K x✝

Tactic:
mk_neg,

Goal state after:
case h
K : Type u_1
L : Type u_2
S : Type u_3
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Field S
inst✝³ : Algebra K L
inst✝² : Algebra K S
inst✝¹ : Algebra L S
inst✝ : IsScalarTower K L S
x✝ : L
⊢ -mk K (-x✝) = mk K x✝
Module: Mathlib.FieldTheory.Minpoly.ConjRootClass

Goal state before:
pp: case h
K : Type u_1
L : Type u_2
S : Type u_3
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Field S
inst✝³ : Algebra K L
inst✝² : Algebra K S
inst✝¹ : Algebra L S
inst✝ : IsScalarTower K L S
x✝ : L
⊢ -mk K (-x✝) = mk K x✝

Tactic:
mk_neg,

Goal state after:
case h
K : Type u_1
L : Type u_2
S : Type u_3
inst✝⁶ : Field K
inst✝⁵ : Field L
inst✝⁴ : Field S
inst✝³ : Algebra K L
inst✝² : Algebra K S
inst✝¹ : Algebra L S
inst✝ : IsScalarTower K L S
x✝ : L
⊢ mk K (- -x✝) = mk K x✝
Module: Mathlib.FieldTheory.Minpoly.ConjRootClass

Goal state before:
pp: case mp
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
a b : L
h : a + b = 0
⊢ mk K a = -mk K b

Tactic:
mk_neg,

Goal state after:
case mp
K : Type u_1
L : Type u_2
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
a b : L
h : a + b = 0
⊢ mk K a = mk K (-b)
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Flat R A
inst✝² : Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this✝² : Field (A ⊗[R] B) := H.toField
a : A
hta : Transcendental R a
b : B
htb : Transcendental R b
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight
hfa : Function.Injective ⇑fa
hfb : Function.Injective ⇑fb
this✝¹ : IsDomain A
this✝ : IsDomain B
this : IsDomain R
⊢ False

Tactic:
have : Module.Flat R (toSubmodule fa.range) :=
    .of_linearEquiv (AlgEquiv.ofInjective fa hfa).symm.toLinearEquiv

Goal state after:
R : Type u
inst✝⁸ : CommRing R
A : Type v
inst✝⁷ : CommRing A
B : Type w
inst✝⁶ : CommRing B
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra R B
inst✝³ : Flat R A
inst✝² : Flat R B
inst✝¹ : Algebra.Transcendental R A
inst✝ : Algebra.Transcendental R B
H : IsField (A ⊗[R] B)
this✝³ : Field (A ⊗[R] B) := H.toField
a : A
hta : Transcendental R a
b : B
htb : Transcendental R b
ha : Function.Injective ⇑(algebraMap R A)
hb : Function.Injective ⇑(algebraMap R B)
fa : A →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeLeft
fb : B →ₐ[R] A ⊗[R] B := Algebra.TensorProduct.includeRight
hfa : Function.Injective ⇑fa
hfb : Function.Injective ⇑fb
this✝² : IsDomain A
this✝¹ : IsDomain B
this✝ : IsDomain R
this : Flat R ↥(toSubmodule fa.range)
⊢ False
Module: Mathlib.FieldTheory.PurelyInseparable.AdjoinPthRoots

Goal state before:
pp: k : Type u_1
inst✝¹ : Field k
p : ℕ
inst✝ : ExpChar k p
⊢ IsPurelyInseparable k (AdjoinPthRoots k)

Tactic:
obtain ⟨p, hp⟩ := ExpChar.exists k
  rw [isPurelyInseparable_iff_pow_mem k p]
  intro x
  use 1, (AdjoinPthRoots.root k).symm x
  simp [AdjoinPthRoots.algebraMap_root_symm p]

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

Goal state before:
pp: case prime
F : Type u_1
K : Type u_2
L : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : IsPurelyInseparable K L
inst✝ : FiniteDimensional K L
p : ℕ
h✝ : ExpChar K p
hp : Nat.Prime p
hchar✝ : CharP K p
⊢ HasExponent K L

Tactic:
let e := Nat.log (ringExpChar K) (Module.finrank K L)

Goal state after:
case prime
F : Type u_1
K : Type u_2
L : Type u_3
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : IsPurelyInseparable K L
inst✝ : FiniteDimensional K L
p : ℕ
h✝ : ExpChar K p
hp : Nat.Prime p
hchar✝ : CharP K p
e : ℕ := Nat.log (ringExpChar K) (Module.finrank K L)
⊢ HasExponent K L
Module: Mathlib.RingTheory.LinearDisjoint

Goal state before:
pp: R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
A B : Subalgebra R S
H : A.LinearDisjoint B
inst✝³ : Free R ↥A
inst✝² : Flat R ↥B
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
this : LinearIndependent (↥B) (⇑A.val ∘ ⇑(Free.chooseBasis R ↥A))
⊢ Module.rank ↥B ↥(Submodule.span (↥B) (Set.range (⇑A.val ∘ ⇑(Free.chooseBasis R ↥A)))) =
    Cardinal.mk (Free.ChooseBasisIndex R ↥A)

Tactic:
rank_span this,

Goal state after:
R : Type u
S : Type v
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
A B : Subalgebra R S
H : A.LinearDisjoint B
inst✝³ : Free R ↥A
inst✝² : Flat R ↥B
inst✝¹ : Nontrivial R
inst✝ : Nontrivial S
this : LinearIndependent (↥B) (⇑A.val ∘ ⇑(Free.chooseBasis R ↥A))
⊢ Cardinal.mk ↑(Set.range (⇑A.val ∘ ⇑(Free.chooseBasis R ↥A))) = Cardinal.mk (Free.ChooseBasisIndex R ↥A)
Module: Mathlib.FieldTheory.LinearDisjoint

Goal state before:
pp: F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
H :
  ∀ (K : Type (max v w)) [inst : Field K] [inst_1 : Algebra F K] (fa : A →ₐ[F] K) (fb : B →ₐ[F] K),
    fa.fieldRange.LinearDisjoint ↥fb.fieldRange
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
this : Field K := Ideal.Quotient.field M
i : A ⊗[F] B →ₐ[F] K := IsScalarTower.toAlgHom F (A ⊗[F] B) K
fa : A →ₐ[F] K := i.comp Algebra.TensorProduct.includeLeft
⊢ M = ⊥

Tactic:
let fb := i.comp (Algebra.TensorProduct.includeRight : B →ₐ[F] _)

Goal state after:
F : Type u
inst✝⁴ : Field F
A : Type v
inst✝³ : Field A
B : Type w
inst✝² : Field B
inst✝¹ : Algebra F A
inst✝ : Algebra F B
H :
  ∀ (K : Type (max v w)) [inst : Field K] [inst_1 : Algebra F K] (fa : A →ₐ[F] K) (fb : B →ₐ[F] K),
    fa.fieldRange.LinearDisjoint ↥fb.fieldRange
M : Ideal (A ⊗[F] B)
hM : M.IsMaximal
K : Type (max v w) := A ⊗[F] B ⧸ M
this : Field K := Ideal.Quotient.field M
i : A ⊗[F] B →ₐ[F] K := IsScalarTower.toAlgHom F (A ⊗[F] B) K
fa : A →ₐ[F] K := i.comp Algebra.TensorProduct.includeLeft
fb : B →ₐ[F] K := i.comp Algebra.TensorProduct.includeRight
⊢ M = ⊥
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✝ : IsPurelyInseparable F E
ι : Type u_1
v : ι → K
hsep : ∀ (i : ι), IsSeparable F (v i)
h : LinearIndependent F v
q : ℕ
h✝ : ExpChar F q
this : ExpChar K q
⊢ LinearIndependent E v

Tactic:
refine linearIndependent_iff.mpr fun l hl ↦ Finsupp.ext fun i ↦ ?_

Goal state after:
F : Type u
E : Type v
inst✝⁷ : Field F
inst✝⁶ : Field E
inst✝⁵ : Algebra F E
K : Type w
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : IsPurelyInseparable F E
ι : Type u_1
v : ι → K
hsep : ∀ (i : ι), IsSeparable F (v i)
h : LinearIndependent F v
q : ℕ
h✝ : ExpChar F q
this : ExpChar K q
l : ι →₀ E
hl : (Finsupp.linearCombination E v) l = 0
i : ι
⊢ l i = 0 i
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 v
inst✝⁴ : Field K
inst✝³ : Algebra F K
inst✝² : Algebra E K
inst✝¹ : IsScalarTower F E K
inst✝ : Algebra.IsAlgebraic F E
⊢ insepDegree F E * insepDegree E K = insepDegree F K

Tactic:
by
  simpa only [Cardinal.lift_id] using lift_insepDegree_mul_lift_insepDegree_of_isAlgebraic F E K

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

Goal state before:
pp: case refine_3
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
V₂ : Type u_3
P₂ : Type u_4
inst✝³ : NormedAddCommGroup V₂
inst✝² : InnerProductSpace ℝ V₂
inst✝¹ : MetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
n : ℕ
s : Simplex ℝ P n
f : P →ᵃⁱ[ℝ] P₂
i : Fin (n + 1)
hf : Function.Injective ⇑f.linear
⊢ (Submodule.map f.linear (vectorSpan ℝ (s.points '' {i}ᶜ)))ᗮ ⊓ vectorSpan ℝ (⇑f.toAffineMap '' Set.range s.points) =
    (Submodule.map f.linear (vectorSpan ℝ (s.points '' {i}ᶜ)))ᗮ ⊓
      (f.linear.range ⊓ Submodule.map f.linear (vectorSpan ℝ (Set.range s.points)))

Tactic:
← Submodule.map_top,

Goal state after:
case refine_3
V : Type u_1
P : Type u_2
inst✝⁷ : NormedAddCommGroup V
inst✝⁶ : InnerProductSpace ℝ V
inst✝⁵ : MetricSpace P
inst✝⁴ : NormedAddTorsor V P
V₂ : Type u_3
P₂ : Type u_4
inst✝³ : NormedAddCommGroup V₂
inst✝² : InnerProductSpace ℝ V₂
inst✝¹ : MetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
n : ℕ
s : Simplex ℝ P n
f : P →ᵃⁱ[ℝ] P₂
i : Fin (n + 1)
hf : Function.Injective ⇑f.linear
⊢ (Submodule.map f.linear (vectorSpan ℝ (s.points '' {i}ᶜ)))ᗮ ⊓ vectorSpan ℝ (⇑f.toAffineMap '' Set.range s.points) =
    (Submodule.map f.linear (vectorSpan ℝ (s.points '' {i}ᶜ)))ᗮ ⊓
      (Submodule.map f.linear ⊤ ⊓ Submodule.map f.linear (vectorSpan ℝ (Set.range s.points)))
Module: Mathlib.Geometry.Euclidean.Altitude

Goal state before:
pp: case mpr
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
⊢ p ≠ s.points i ∧
      p -ᵥ s.points i ∈ vectorSpan ℝ (Set.range s.points) ∧ p -ᵥ s.points i ∈ (vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ →
    vectorSpan ℝ {p, s.points i} = (s.altitude i).direction

Tactic:
rintro ⟨hne, h⟩

Goal state after:
case mpr
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
hne : p ≠ s.points i
h : p -ᵥ s.points i ∈ vectorSpan ℝ (Set.range s.points) ∧ p -ᵥ s.points i ∈ (vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ
⊢ vectorSpan ℝ {p, s.points i} = (s.altitude i).direction
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation

Goal state before:
pp: V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
hx : x ≠ 0
hy : y ≠ 0
θ : Real.Angle
⊢ o.oangle x ((o.rotation θ) y) = o.oangle x y + θ

Tactic:
simp only [oangle, o.kahler_rotation_right]
  rw [Complex.arg_mul_coe_angle, Real.Angle.arg_toCircle]
  · abel
  · exact Circle.coe_ne_zero _
  · exact o.kahler_ne_zero hx hy

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
r : ℝ
hr : r < 0
⊢ o.oangle (-(-r • x)) y = o.oangle (-x) y

Tactic:
← smul_neg,

Goal state after:
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
r : ℝ
hr : r < 0
⊢ o.oangle (-r • -x) y = o.oangle (-x) y
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

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

Tactic:
← smul_neg,

Goal state after:
V : Type u_1
inst✝² : NormedAddCommGroup V
inst✝¹ : InnerProductSpace ℝ V
inst✝ : Fact (finrank ℝ V = 2)
o : Orientation ℝ V (Fin 2)
x y : V
r : ℝ
hr : r < 0
⊢ o.oangle x (-r • -y) = o.oangle x (-y)
Module: Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

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

Tactic:
exact (o.two_zsmul_oangle_smul_right_of_ne_zero _ _ (Units.ne_zero _)).symm

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

Goal state before:
pp: case mpr
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
θ : Real.Angle
hp : 0 < ‖y‖ / ‖x‖
hye : y = (‖y‖ / ‖x‖) • (o.rotation θ) x
⊢ o.oangle x y = θ

Tactic:
rw [hye, o.oangle_smul_right_of_pos _ _ hp, o.oangle_rotation_self_right hx]

Goal state after:
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
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hij : i ≠ j
r : ℝ
hr : r ≠ 0
h : s.points j -ᵥ s.altitudeFoot j = r • (s.points i -ᵥ s.altitudeFoot i)
k : Fin (n + 1)
hki : k ≠ i
hkj : k ≠ j
hki' : s.points k ∈ Set.range (s.faceOpposite i).points
hkj' : s.points k ∈ Set.range (s.faceOpposite j).points
⊢ s.points '' Set.univ = s.points '' ({i} ∩ {j})ᶜ

Tactic:
rw [Set.inter_singleton_eq_empty.mpr ?_, Set.compl_empty]

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
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hij : i ≠ j
r : ℝ
hr : r ≠ 0
h : s.points j -ᵥ s.altitudeFoot j = r • (s.points i -ᵥ s.altitudeFoot i)
k : Fin (n + 1)
hki : k ≠ i
hkj : k ≠ j
hki' : s.points k ∈ Set.range (s.faceOpposite i).points
hkj' : s.points k ∈ Set.range (s.faceOpposite j).points
⊢ j ∉ {i}
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
h : o.oangle x y = ↑π
hx : ¬x = 0
hy : y = 0
⊢ x ≠ 0 ∧ y ≠ 0 ∧ o.oangle x (-y) = 0

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
h : o.oangle x y = ↑π
hx : ¬x = 0
hy : ¬y = 0
⊢ x ≠ 0 ∧ y ≠ 0 ∧ o.oangle x (-y) = 0

Tactic:
· simp [hy, Real.Angle.pi_ne_zero.symm] at h

Goal state after:
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
h : o.oangle x y = ↑π
hx : ¬x = 0
hy : ¬y = 0
⊢ x ≠ 0 ∧ y ≠ 0 ∧ o.oangle x (-y) = 0
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 p₅ p₆ : P
h₂ : p₂ ∉ line[ℝ, p₁, p₃]
h₄ : p₄ ∈ line[ℝ, p₁, p₃]
h₆ : p₆ ∈ line[ℝ, p₁, p₃]
h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]
h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]
hd : line[ℝ, p₆, p₄].direction ≤ line[ℝ, p₃, p₁].direction
r : ℝ
hr : r ≠ 0
h₅₄ : p₄ -ᵥ p₅ = -(r • (p₂ -ᵥ p₁))
h₆₅ : p₆ -ᵥ p₅ = r • (p₃ -ᵥ p₂)
⊢ o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = o.oangle (p₄ -ᵥ p₅) (p₆ -ᵥ p₅)

Tactic:
← smul_neg,

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 p₅ p₆ : P
h₂ : p₂ ∉ line[ℝ, p₁, p₃]
h₄ : p₄ ∈ line[ℝ, p₁, p₃]
h₆ : p₆ ∈ line[ℝ, p₁, p₃]
h₁₂₄₅ : line[ℝ, p₁, p₂] ∥ line[ℝ, p₄, p₅]
h₃₂₆₅ : line[ℝ, p₃, p₂] ∥ line[ℝ, p₆, p₅]
hd : line[ℝ, p₆, p₄].direction ≤ line[ℝ, p₃, p₁].direction
r : ℝ
hr : r ≠ 0
h₅₄ : p₄ -ᵥ p₅ = r • -(p₂ -ᵥ p₁)
h₆₅ : p₆ -ᵥ p₅ = r • (p₃ -ᵥ p₂)
⊢ o.oangle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = o.oangle (p₄ -ᵥ p₅) (p₆ -ᵥ p₅)
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)
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
h0yz : y = 0 ∨ z = 0
⊢ o.oangle w x = o.oangle y z

Tactic:
rcases h0wx with (h0wx | h0wx)

Goal state after:
case pos.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 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
h0yz : y = 0 ∨ z = 0
h0wx : w = 0
⊢ o.oangle w x = o.oangle y z

case pos.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 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
h0yz : y = 0 ∨ z = 0
h0wx : x = 0
⊢ o.oangle w x = o.oangle y z