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

Goal state before:
pp: K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
| fun x ↦ Bivariate.swap ((monomial x) ((Φ E).coeff x))

Tactic:
ext

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
x✝ : ℕ
| Bivariate.swap ((monomial x✝) ((Φ E).coeff x✝))
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
hx : x ∉ (convexHull R) (s \ {x})
⊢ IsConvexSet R (s \ {x})

Tactic:
by
    rw [← convexHull_subset_self]
    rintro y hy
    exact ⟨convexHull_min sdiff_subset hs hy, by rintro rfl; exact hx hy⟩

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Face.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : DivisionRing R
inst✝³ : LinearOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
C F : PointedCone R M
hF : F.IsFaceOf C
⊢ F.lineal = C.lineal

Tactic:
ext

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : DivisionRing R
inst✝³ : LinearOrder R
inst✝² : IsOrderedRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
C F : PointedCone R M
hF : F.IsFaceOf C
x✝ : M
⊢ x✝ ∈ F.lineal ↔ x✝ ∈ C.lineal
Module: Mathlib.Geometry.Convex.Cone.Face.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : DivisionRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
C₁ F₁ : PointedCone R M
C₂ F₂ : PointedCone R N
hF₁ : F₁.IsFaceOf C₁
hF₂ : F₂.IsFaceOf C₂
⊢ ∀ (a : M) (b : N) (a_1 : M) (b_1 : N) {a_2 : R},
    a ∈ C₁ → b ∈ C₂ → a_1 ∈ C₁ → b_1 ∈ C₂ → 0 < a_2 → a_2 • a + a_1 ∈ F₁ → a_2 • b + b_1 ∈ F₂ → a ∈ F₁ ∧ b ∈ F₂

Tactic:
intro _ _ _ _ _ xc₁ xc₂ yc₁ yc₂ a0 hab₁ hab₂

Goal state after:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : DivisionRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
C₁ F₁ : PointedCone R M
C₂ F₂ : PointedCone R N
hF₁ : F₁.IsFaceOf C₁
hF₂ : F₂.IsFaceOf C₂
a✝² : M
b✝¹ : N
a✝¹ : M
b✝ : N
a✝ : R
xc₁ : a✝² ∈ C₁
xc₂ : b✝¹ ∈ C₂
yc₁ : a✝¹ ∈ C₁
yc₂ : b✝ ∈ C₂
a0 : 0 < a✝
hab₁ : a✝ • a✝² + a✝¹ ∈ F₁
hab₂ : a✝ • b✝¹ + b✝ ∈ F₂
⊢ a✝² ∈ F₁ ∧ b✝¹ ∈ F₂
Module: Mathlib.Geometry.Convex.Cone.Face.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : DivisionRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
C₁ : PointedCone R M
C₂ : PointedCone R N
F : PointedCone R (M × N)
hF : F.IsFaceOf (prod C₁ C₂)
⊢ (map (LinearMap.fst R M N) F).IsFaceOf C₁

Tactic:
constructor

Goal state after:
case le
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : DivisionRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
C₁ : PointedCone R M
C₂ : PointedCone R N
F : PointedCone R (M × N)
hF : F.IsFaceOf (prod C₁ C₂)
⊢ map (LinearMap.fst R M N) F ≤ C₁

case mem_of_smul_add_mem
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : DivisionRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
C₁ : PointedCone R M
C₂ : PointedCone R N
F : PointedCone R (M × N)
hF : F.IsFaceOf (prod C₁ C₂)
⊢ ∀ {x y : M} {a : R},
    x ∈ C₁ → y ∈ C₁ → 0 < a → a • x + y ∈ map (LinearMap.fst R M N) F → x ∈ map (LinearMap.fst R M N) F
Module: Mathlib.Geometry.Convex.Cone.Face.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : DivisionRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
C₁ : PointedCone R M
C₂ : PointedCone R N
F : PointedCone R (M × N)
hF : F.IsFaceOf (prod C₁ C₂)
this : (map (↑(LinearEquiv.prodComm R M N)) F).IsFaceOf (map (↑(LinearEquiv.prodComm R M N)) (prod C₁ C₂))
⊢ map (LinearMap.snd R M N) F =
    map (LinearMap.fst R N M) { carrier := (fun a ↦ a.swap) '' ↑F, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }

Tactic:
ext

Goal state after:
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : DivisionRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
C₁ : PointedCone R M
C₂ : PointedCone R N
F : PointedCone R (M × N)
hF : F.IsFaceOf (prod C₁ C₂)
this : (map (↑(LinearEquiv.prodComm R M N)) F).IsFaceOf (map (↑(LinearEquiv.prodComm R M N)) (prod C₁ C₂))
x✝ : N
⊢ x✝ ∈ map (LinearMap.snd R M N) F ↔
    x✝ ∈ map (LinearMap.fst R N M) { carrier := (fun a ↦ a.swap) '' ↑F, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }
Module: Mathlib.Geometry.Convex.Cone.Face.Basic

Goal state before:
pp: R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : DivisionRing R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsOrderedRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup N
inst✝ : Module R N
C₁ : PointedCone R M
C₂ : PointedCone R N
F : PointedCone R (M × N)
hF : F.IsFaceOf (prod C₁ C₂)
⊢ (map (LinearMap.snd R M N) F).IsFaceOf C₂

Tactic:
by
  have := hF.map _ (LinearEquiv.prodComm R M N).injective
  convert IsFaceOf.fst (by simpa [PointedCone.map, Submodule.map])
  ext; simp

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 ≠ ⊥
F : Type u_1 := AlgebraicClosure K
H : ¬(Polynomial.map (algebraMap K F) (Q₂ h)).degree ≤ 0
α : F
hα : (aeval α) (Q₂ h) = 0
this :
  (aeval (Polynomial.C α)) (θ E) =
    (aeval (Polynomial.C α)) (Polynomial.map Polynomial.C (Q₂ h)) * (aeval (Polynomial.C α)) (Φ E)
⊢ (algebraMap K[X] F[X]).comp Polynomial.C = Polynomial.C.comp (algebraMap K F)

Tactic:
ext

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
F : Type u_1 := AlgebraicClosure K
H : ¬(Polynomial.map (algebraMap K F) (Q₂ h)).degree ≤ 0
α : F
hα : (aeval α) (Q₂ h) = 0
this :
  (aeval (Polynomial.C α)) (θ E) =
    (aeval (Polynomial.C α)) (Polynomial.map Polynomial.C (Q₂ h)) * (aeval (Polynomial.C α)) (Φ E)
x✝ : K
n✝ : ℕ
⊢ (((algebraMap K[X] F[X]).comp Polynomial.C) x✝).coeff n✝ = ((Polynomial.C.comp (algebraMap K F)) x✝).coeff n✝
Module: Mathlib.FieldTheory.RatFunc.Luroth

Goal state before:
pp: K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
F : Type u_1 := AlgebraicClosure K
H : ¬(Polynomial.map (algebraMap K F) (Q₂ h)).degree ≤ 0
α : F
hα : (aeval α) (Q₂ h) = 0
this :
  (algebraMap K[X] F[X]) (g E) * (aeval (Polynomial.C α)) (Polynomial.map Polynomial.C (f E)) -
      (algebraMap K[X] F[X]) (f E) * (aeval (Polynomial.C α)) (Polynomial.map Polynomial.C (g E)) =
    0
⊢ (algebraMap K[X] F[X]).comp Polynomial.C = Polynomial.C.comp (algebraMap K F)

Tactic:
ext

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
F : Type u_1 := AlgebraicClosure K
H : ¬(Polynomial.map (algebraMap K F) (Q₂ h)).degree ≤ 0
α : F
hα : (aeval α) (Q₂ h) = 0
this :
  (algebraMap K[X] F[X]) (g E) * (aeval (Polynomial.C α)) (Polynomial.map Polynomial.C (f E)) -
      (algebraMap K[X] F[X]) (f E) * (aeval (Polynomial.C α)) (Polynomial.map Polynomial.C (g E)) =
    0
x✝ : K
n✝ : ℕ
⊢ (((algebraMap K[X] F[X]).comp Polynomial.C) x✝).coeff n✝ = ((Polynomial.C.comp (algebraMap K F)) x✝).coeff n✝
Module: Mathlib.FieldTheory.RatFunc.Luroth

Goal state before:
pp: K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
F : Type u_1 := AlgebraicClosure K
H : ¬(Polynomial.map (algebraMap K F) (Q₂ h)).degree ≤ 0
α : F
hα : (aeval α) (Q₂ h) = 0
this :
  (algebraMap K[X] F[X]) (g E) * Polynomial.C ((aeval α) (f E)) -
      (algebraMap K[X] F[X]) (f E) * (aeval (Polynomial.C α)) (Polynomial.map Polynomial.C (g E)) =
    0
⊢ (algebraMap K[X] F[X]).comp Polynomial.C = Polynomial.C.comp (algebraMap K F)

Tactic:
ext

Goal state after:
K : Type u_1
inst✝ : Field K
E : IntermediateField K K⟮X⟯
h : E ≠ ⊥
F : Type u_1 := AlgebraicClosure K
H : ¬(Polynomial.map (algebraMap K F) (Q₂ h)).degree ≤ 0
α : F
hα : (aeval α) (Q₂ h) = 0
this :
  (algebraMap K[X] F[X]) (g E) * Polynomial.C ((aeval α) (f E)) -
      (algebraMap K[X] F[X]) (f E) * (aeval (Polynomial.C α)) (Polynomial.map Polynomial.C (g E)) =
    0
x✝ : K
n✝ : ℕ
⊢ (((algebraMap K[X] F[X]).comp Polynomial.C) x✝).coeff n✝ = ((Polynomial.C.comp (algebraMap K F)) x✝).coeff n✝
Module: Mathlib.Geometry.Diffeology.Basic

Goal state before:
pp: X : Type u_1
d : DiffeologicalSpace X
t : TopologicalSpace X
h : dTopology = t
⊢ d.replaceDTopology t h = d

Tactic:
ext

Goal state after:
X : Type u_1
d : DiffeologicalSpace X
t : TopologicalSpace X
h : dTopology = t
x✝¹ : ℕ
x✝ : EuclideanSpace ℝ (Fin x✝¹) → X
⊢ Diffeology.IsPlot x✝ ↔ Diffeology.IsPlot x✝
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
⊢ ↑s ∩ ↑(mk' p s.directionᗮ) = {↑((orthogonalProjection s) p)}

Tactic:
obtain ⟨q, hq⟩ := inter_eq_singleton_of_nonempty_of_isCompl (nonempty_subtype.mp ‹_›)
    (mk'_nonempty p s.directionᗮ)
    (by
      rw [direction_mk' p s.directionᗮ]
      exact s.direction.isCompl_orthogonal)

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 q : P
hq : ↑s ∩ ↑(mk' p s.directionᗮ) = {q}
⊢ ↑s ∩ ↑(mk' p s.directionᗮ) = {↑((orthogonalProjection s) p)}
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 q : P
hq : ↑s ∩ ↑(mk' p s.directionᗮ) = {q}
⊢ ↑s ∩ ↑(mk' p s.directionᗮ) = {↑((orthogonalProjection s) p)}

Tactic:
rwa [Set.eq_singleton_iff_nonempty_unique_mem.1 hq |>.2 _
    ⟨orthogonalProjection_mem _, orthogonalProjection_mem_orthogonal _ _⟩]

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
⊢ ↑((orthogonalProjection s) p) = p ↔ p ∈ s

Tactic:
constructor

Goal state after:
case mp
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
⊢ ↑((orthogonalProjection s) p) = p → p ∈ s

case mpr
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
⊢ p ∈ s → ↑((orthogonalProjection s) p) = p
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 : ↥s
⊢ (orthogonalProjection s) ↑p = p

Tactic:
ext

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 : ↥s
⊢ ↑((orthogonalProjection s) ↑p) = ↑p
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 q : P
⊢ ↑((orthogonalProjection s) p) = q ↔ q ∈ s ∧ p -ᵥ q ∈ s.directionᗮ

Tactic:
constructor

Goal state after:
case mp
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 q : P
⊢ ↑((orthogonalProjection s) p) = q → q ∈ s ∧ p -ᵥ q ∈ s.directionᗮ

case mpr
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 q : P
⊢ q ∈ s ∧ p -ᵥ q ∈ s.directionᗮ → ↑((orthogonalProjection s) p) = q
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
hp : p ∈ s
v : V
hv : v ∈ s.directionᗮ
⊢ (orthogonalProjection s) (v +ᵥ p) = ⟨p, hp⟩

Tactic:
ext

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
hp : p ∈ s
v : V
hv : v ∈ s.directionᗮ
⊢ ↑((orthogonalProjection s) (v +ᵥ p)) = ↑⟨p, hp⟩
Module: Mathlib.Geometry.Euclidean.Projection

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace 𝕜 V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace 𝕜 P
inst✝ : s.direction.HasOrthogonalProjection
p₁ p₂ : P
hp₂ : p₂ ∈ s
this : Nonempty ↥s
⊢ dist p₁ ↑((orthogonalProjection s) p₁) = dist p₁ p₂ ↔ ↑((orthogonalProjection s) p₁) = p₂

Tactic:
constructor

Goal state after:
case mp
𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace 𝕜 V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace 𝕜 P
inst✝ : s.direction.HasOrthogonalProjection
p₁ p₂ : P
hp₂ : p₂ ∈ s
this : Nonempty ↥s
⊢ dist p₁ ↑((orthogonalProjection s) p₁) = dist p₁ p₂ → ↑((orthogonalProjection s) p₁) = p₂

case mpr
𝕜 : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace 𝕜 V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace 𝕜 P
inst✝ : s.direction.HasOrthogonalProjection
p₁ p₂ : P
hp₂ : p₂ ∈ s
this : Nonempty ↥s
⊢ ↑((orthogonalProjection s) p₁) = p₂ → dist p₁ ↑((orthogonalProjection s) p₁) = dist p₁ p₂
Module: Mathlib.Geometry.Euclidean.Altitude

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i : Fin (n + 1)
p : P
⊢ vectorSpan ℝ {p, s.points i} = (s.altitude i).direction ↔
    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

Tactic:
constructor

Goal state after:
case mp
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
⊢ vectorSpan ℝ {p, s.points i} = (s.altitude i).direction →
    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

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
Module: Mathlib.Geometry.Euclidean.Altitude

Goal state before:
pp: case mp
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 ∧
    p -ᵥ s.points i ∈ vectorSpan ℝ (Set.range s.points) ∧ p -ᵥ s.points i ∈ (vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ

Tactic:
constructor

Goal state after:
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

case mp.right
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 -ᵥ s.points i ∈ vectorSpan ℝ (Set.range s.points) ∧ p -ᵥ s.points i ∈ (vectorSpan ℝ (s.points '' {i}ᶜ))ᗮ
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)
θ : Real.Angle
⊢ (o.rotation θ).symm = o.rotation (-θ)

Tactic:
ext

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)
θ : Real.Angle
x✝ : V
⊢ (o.rotation θ).symm x✝ = (o.rotation (-θ)) x✝
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:
ext

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✝ : V
⊢ (o.rotation 0) x✝ = (LinearIsometryEquiv.refl ℝ V) x✝
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
⊢ (reflection s).symm = reflection s

Tactic:
ext

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
x✝ : P
⊢ (reflection s).symm x✝ = (reflection s) x✝
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
s₁ s₂ : AffineSubspace 𝕜 P
inst✝³ : Nonempty ↥s₁
inst✝² : Nonempty ↥s₂
inst✝¹ : s₁.direction.HasOrthogonalProjection
inst✝ : s₂.direction.HasOrthogonalProjection
p : P
⊢ (↑((orthogonalProjection s₁) p) -ᵥ p) +ᵥ ↑((orthogonalProjection s₁) p) =
      (↑((orthogonalProjection s₂) p) -ᵥ p) +ᵥ ↑((orthogonalProjection s₂) p) ↔
    ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)

Tactic:
constructor

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
⊢ (↑((orthogonalProjection s₁) p) -ᵥ p) +ᵥ ↑((orthogonalProjection s₁) p) =
      (↑((orthogonalProjection s₂) p) -ᵥ p) +ᵥ ↑((orthogonalProjection s₂) p) →
    ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p)

case mpr
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
⊢ ↑((orthogonalProjection s₁) p) = ↑((orthogonalProjection s₂) p) →
    (↑((orthogonalProjection s₁) p) -ᵥ p) +ᵥ ↑((orthogonalProjection s₁) p) =
      (↑((orthogonalProjection s₂) p) -ᵥ p) +ᵥ ↑((orthogonalProjection s₂) p)
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
hx : x ≠ 0
θ : Real.Angle
⊢ (o.rotation θ) x = x ↔ θ = 0

Tactic:
constructor

Goal state after:
case mp
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
hx : x ≠ 0
θ : Real.Angle
⊢ (o.rotation θ) x = x → θ = 0

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 : V
hx : x ≠ 0
θ : Real.Angle
⊢ θ = 0 → (o.rotation θ) x = x
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
⊢ (o.rotation (o.oangle x y)) x = y ↔ ‖x‖ = ‖y‖

Tactic:
constructor

Goal state after:
case mp
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.rotation (o.oangle x y)) x = y → ‖x‖ = ‖y‖

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
⊢ ‖x‖ = ‖y‖ → (o.rotation (o.oangle x y)) 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 y : V
hx : x ≠ 0
hy : y ≠ 0
θ : Real.Angle
hp : 0 < ‖y‖ / ‖x‖
⊢ o.oangle x y = θ ↔ y = (‖y‖ / ‖x‖) • (o.rotation θ) x

Tactic:
constructor

Goal state after:
case mp
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‖
⊢ o.oangle x y = θ → y = (‖y‖ / ‖x‖) • (o.rotation θ) x

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‖
⊢ y = (‖y‖ / ‖x‖) • (o.rotation θ) x → o.oangle x y = θ
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 z : V
r : ℝˣ
⊢ 2 • o.oangle x z = 2 • o.oangle (r • x) z

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

Goal state after: