mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hx : ‖x‖ = 1
hy : ‖y‖ = 1
hz : ‖z‖ = 1
⊢ angle x z ≠ π →
    angle x z = angle x y + angle y z → Real.sin (angle x z) • y = Real.sin (angle y z) • x + Real.sin (angle x y) • z

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Congruence

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoEMetricSpace P₁
inst✝ : PseudoEMetricSpace P₂
⊢ Congruent v₁ v₂ ↔ Pairwise fun i₁ i₂ ↦ edist (v₁ i₁) (v₁ i₂) = edist (v₂ i₁) (v₂ i₂)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Congruence

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
P₁ : Type u_3
P₂ : Type u_4
inst✝² : PseudoEMetricSpace P₁
inst✝¹ : PseudoEMetricSpace P₂
E : Type u_7
inst✝ : EquivLike E ι' ι
f : E
v₁ : ι → P₁
v₂ : ι → P₂
⊢ v₁ ∘ ⇑f ≅ v₂ ∘ ⇑f ↔ v₁ ≅ v₂

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Congruence

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝² : PseudoEMetricSpace P₁
inst✝¹ : PseudoEMetricSpace P₂
inst✝ : Subsingleton ι
i j : ι
⊢ edist (v₁ i) (v₁ j) = edist (v₂ i) (v₂ j)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Congruence

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
P₃ : Type u_5
v₁ : ι → P₁
v₂ : ι → P₂
inst✝² : PseudoEMetricSpace P₁
inst✝¹ : PseudoEMetricSpace P₂
inst✝ : PseudoEMetricSpace P₃
f : P₂ → P₃
hf : Isometry f
⊢ v₁ ≅ f ∘ v₂ ↔ v₁ ≅ v₂

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Congruence

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
P₃ : Type u_5
P₄ : Type u_6
v₁ : ι → P₁
v₂ : ι → P₂
inst✝⁷ : PseudoEMetricSpace P₁
inst✝⁶ : PseudoEMetricSpace P₂
inst✝⁵ : PseudoEMetricSpace P₃
inst✝⁴ : PseudoEMetricSpace P₄
F₁ : Type u_7
F₂ : Type u_8
inst✝³ : FunLike F₁ P₁ P₃
inst✝² : DilationClass F₁ P₁ P₃
inst✝¹ : FunLike F₂ P₂ P₄
inst✝ : DilationClass F₂ P₂ P₄
f₁ : F₁
f₂ : F₂
h : v₁ ≅ v₂
hf : Dilation.ratio f₁ = Dilation.ratio f₂
i j : ι
⊢ edist ((⇑f₁ ∘ v₁) i) ((⇑f₁ ∘ v₁) j) = edist ((⇑f₂ ∘ v₂) i) ((⇑f₂ ∘ v₂) j)

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
signs : Finset (Fin (n + 1))
i : Fin (n + 1)
⊢ s.touchpointWeights signs i i = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Congruence

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
x✝¹ x✝ : ι
⊢ edist (v₁ x✝¹) (v₁ x✝) = edist (v₂ x✝¹) (v₂ x✝) ↔ nndist (v₁ x✝¹) (v₁ x✝) = nndist (v₂ x✝¹) (v₂ x✝)

Tactic:
by

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

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
n : ℕ
inst✝ : NeZero n
s : Simplex ℝ P n
i j : Fin (n + 1)
hne : i ≠ j
⊢ 0 < s.touchpointWeights ∅ i j

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
⊢ angle x z ≤ angle x y + angle y z

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Congruence

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
⊢ Congruent v₁ v₂ ↔ Pairwise fun i₁ i₂ ↦ nndist (v₁ i₁) (v₁ i₂) = nndist (v₂ i₁) (v₂ i₂)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Congruence

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
x✝¹ x✝ : ι
⊢ nndist (v₁ x✝¹) (v₁ x✝) = nndist (v₂ x✝¹) (v₂ x✝) ↔ dist (v₁ x✝¹) (v₁ x✝) = dist (v₂ x✝¹) (v₂ x✝)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Congruence

Goal state before:
pp: ι : Type u_1
P₁ : Type u_3
P₂ : Type u_4
v₁ : ι → P₁
v₂ : ι → P₂
inst✝¹ : PseudoMetricSpace P₁
inst✝ : PseudoMetricSpace P₂
⊢ Congruent v₁ v₂ ↔ Pairwise fun i₁ i₂ ↦ dist (v₁ i₁) (v₁ i₂) = dist (v₂ i₁) (v₂ i₂)

Tactic:
by

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

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
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i : Fin (n + 1)
⊢ s.touchpoint ∅ i ∈ (s.faceOpposite i).interior

Tactic:
by

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

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
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hne : i ≠ j
⊢ SignType.sign (s.touchpointWeights {i} i j) = 1

Tactic:
by

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

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
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hne : i ≠ j
⊢ 0 < s.touchpointWeights {i} i j

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x z : V
kx kz : ℝ≥0
hy : kx • x + kz • z ≠ 0
hkx : 0 < kx
hkz : 0 < kz
hz : z = 0
⊢ x ≠ 0

Tactic:
by

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

Goal state before:
pp: P₁ : Type u_4
P₂ : Type u_5
inst✝¹ : MetricSpace P₁
inst✝ : MetricSpace P₂
a b c : P₁
a' b' c' : P₂
hd₁ : dist a b = dist a' b'
hd₂ : dist b c = dist b' c'
hd₃ : dist c a = dist c' a'
⊢ ![a, b, c] ≅ ![a', b', c']

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
h_mem : y ∈ Submodule.span ℝ≥0 {x, z}
⊢ angle x z = angle x y + angle y z

Tactic:
by

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

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
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i : Fin (n + 1)
⊢ s.touchpoint {i} i ∈ (s.faceOpposite i).interior

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
kx ky kz : ℝ≥0
hy : ky ≠ 0
hlincomb : ky • y = kx • x + kz • z
⊢ ky • y ∈ Submodule.span ℝ≥0 {x, z}

Tactic:
by

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

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ∠ a b c = ∠ a' b' c'
hd₁ : dist a b = dist a' b'
hd₂ : dist b c = dist b' c'
⊢ 0 ≤ dist a c

Tactic:
by

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

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ∠ a b c = ∠ a' b' c'
hd₁ : dist a b = dist a' b'
hd₂ : dist b c = dist b' c'
⊢ 0 ≤ dist a' c'

Tactic:
by

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

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
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hne : i ≠ j
⊢ SignType.sign (s.touchpointWeights {i} j i) = -1

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
kx ky kz : ℝ≥0
hy : ky ≠ 0
hlincomb : ky • y = kx • x + kz • z
⊢ y ∈ Submodule.span ℝ≥0 {x, z}

Tactic:
by

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

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
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
i j : Fin (n + 1)
hne : i ≠ j
⊢ s.touchpointWeights {i} j i < 0

Tactic:
by

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

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ∠ a b c = ∠ a' b' c'
hd₁ : dist a b = dist a' b'
hd₂ : dist b c = dist b' c'
⊢ ![a, b, c] ≅ ![a', b', c']

Tactic:
by

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

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ¬Collinear ℝ {a, b, c}
ha₁ : ∠ a b c = ∠ a' b' c'
hd : dist b c = dist b' c'
ha₂ : ∠ b c a = ∠ b' c' a'
⊢ ¬Collinear ℝ {a', b', c'}

Tactic:
by

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

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ¬Collinear ℝ {a, b, c}
ha₁ : ∠ a b c = ∠ a' b' c'
hd : dist b c = dist b' c'
ha₂ : ∠ b c a = ∠ b' c' a'
h' : ¬Collinear ℝ {a', b', c'}
ha₃' : ∠ c' a' b' + ∠ a' b' c' + ∠ b' c' a' = Real.pi
ha₃ : ∠ c a b = ∠ c' a' b'
⊢ ¬Collinear ℝ {b, a, c}

Tactic:
by

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

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
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
he : Pi.single j 1 = s.touchpointWeights signs i
this : 1 < n
⊢ 2 < n + 1

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
h : angle x z = angle x y + angle y z
hxz₁ : ¬angle x z = π
hxz₂ : angle x z = 0
⊢ angle x y = 0

Tactic:
by

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

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ¬Collinear ℝ {a, b, c}
ha₁ : ∠ a b c = ∠ a' b' c'
hd : dist b c = dist b' c'
ha₂ : ∠ b c a = ∠ b' c' a'
h' : ¬Collinear ℝ {a', b', c'}
ha₃' : ∠ c' a' b' + ∠ a' b' c' + ∠ b' c' a' = Real.pi
ha₃ : ∠ c a b = ∠ c' a' b'
h_bac : ¬Collinear ℝ {b, a, c}
⊢ ¬Collinear ℝ {b', a', c'}

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x z : V
hxz₁ : ¬angle x z = π
hxz₂ : x ≠ 0 ∧ ∃ r, 0 < r ∧ z = r • x
left✝ : x ≠ 0
r2 : ℝ≥0
hr2 : 0 < ↑r2
hy : ↑r2 • x ≠ 0
h : angle x z = angle x (↑r2 • x) + angle (↑r2 • x) z
⊢ x ∈ {x, z}

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
h : angle x z = angle x y + angle y z
hxz₁ : ¬angle x z = π
hxz₂ : ¬angle x z = 0
⊢ Real.sin (angle x z) ≠ 0

Tactic:
by

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

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
n : ℕ
inst✝¹ : NeZero n
s : Simplex ℝ P n
inst✝ : n.AtLeastTwo
signs : Finset (Fin (n + 1))
h : s.ExcenterExists signs
i j : Fin (n + 1)
⊢ s.touchpoint signs i ≠ s.points j

Tactic:
by

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

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
t : Triangle ℝ P
signs : Finset (Fin 3)
⊢ signs = ∅ ∨ signs = ∅ᶜ ∨ ∃ i, signs = {i} ∨ signs = {i}ᶜ

Tactic:
by

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

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ¬Collinear ℝ {a, b, c}
ha₁ : ∠ a b c = ∠ a' b' c'
hd : dist b c = dist b' c'
ha₂ : ∠ b c a = ∠ b' c' a'
h' : ¬Collinear ℝ {a', b', c'}
ha₃' : ∠ c' a' b' + ∠ a' b' c' + ∠ b' c' a' = Real.pi
ha₃ : ∠ c a b = ∠ c' a' b'
h_bac : ¬Collinear ℝ {b, a, c}
h_bac' : ¬Collinear ℝ {b', a', c'}
⊢ dist a b = dist a' b'

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x : V
hx : ¬x = 0
hxz₁ : ¬angle x 0 = π
hxz₂ : ¬angle x 0 = 0
h_sin_xz : Real.sin (angle x 0) ≠ 0
left✝ : x ≠ 0
r : ℝ≥0
hr : 0 < ↑r
hy : ↑r • x ≠ 0
⊢ x ∈ {x, 0}

Tactic:
by

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

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ¬Collinear ℝ {a, b, c}
ha₁ : ∠ a b c = ∠ a' b' c'
hd : dist b c = dist b' c'
ha₂ : ∠ b c a = ∠ b' c' a'
⊢ ![a, b, c] ≅ ![a', b', c']

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
h : angle x z = angle x y + angle y z
hxz₁ : ¬angle x z = π
hxz₂ : ¬angle x z = 0
h_sin_xz : Real.sin (angle x z) ≠ 0
hx : ¬x = 0
hz : ¬z = 0
⊢ 0 ≤ Real.sin (angle y z) / ‖x‖

Tactic:
by

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

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ¬Collinear ℝ {a, b, c}
ha₁ : ∠ a b c = ∠ a' b' c'
ha₂ : ∠ b c a = ∠ b' c' a'
hd : dist c a = dist c' a'
ha₃ : ∠ c a b + ∠ a b c + ∠ b c a = Real.pi
⊢ ¬Collinear ℝ {a', b', c'}

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
h : angle x z = angle x y + angle y z
hxz₁ : ¬angle x z = π
hxz₂ : ¬angle x z = 0
h_sin_xz : Real.sin (angle x z) ≠ 0
hx : ¬x = 0
hz : ¬z = 0
hnneg₁ : 0 ≤ Real.sin (angle y z) / ‖x‖
⊢ 0 ≤ Real.sin (angle x y) / ‖z‖

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
h : angle x z = angle x y + angle y z
hxz₁ : ¬angle x z = π
hxz₂ : ¬angle x z = 0
h_sin_xz : Real.sin (angle x z) ≠ 0
hx : ¬x = 0
hz : ¬z = 0
hnneg₁ : 0 ≤ Real.sin (angle y z) / ‖x‖
hnneg₂ : 0 ≤ Real.sin (angle x y) / ‖z‖
⊢ 0 ≤ Real.sin (angle x z) / ‖y‖

Tactic:
by

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

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
t : Triangle ℝ P
signs : Finset (Fin 3)
⊢ Simplex.ExcenterExists t signs

Tactic:
by

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

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
t : Triangle ℝ P
signs : Finset (Fin 3)
⊢ signs = ∅ ∨ signs = Finset.univ ∨ ∃ i, signs = {i} ∨ signs = {i}ᶜ

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
h : angle x z = angle x y + angle y z
hxz₁ : ¬angle x z = π
hxz₂ : ¬angle x z = 0
h_sin_xz : Real.sin (angle x z) ≠ 0
hx : ¬x = 0
hz : ¬z = 0
hnneg₁ : 0 ≤ Real.sin (angle y z) / ‖x‖
hnneg₂ : 0 ≤ Real.sin (angle x y) / ‖z‖
hnneg₃ : 0 ≤ Real.sin (angle x z) / ‖y‖
⊢ ⟨Real.sin (angle x z) / ‖y‖, hnneg₃⟩ ≠ 0

Tactic:
by

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

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ¬Collinear ℝ {a, b, c}
ha₁ : ∠ a b c = ∠ a' b' c'
ha₂ : ∠ b c a = ∠ b' c' a'
hd : dist c a = dist c' a'
h' : ¬Collinear ℝ {a', b', c'}
ha₃' : ∠ c' a' b' + ∠ a' b' c' + ∠ b' c' a' = Real.pi
ha₃ : ∠ c a b = ∠ c' a' b'
⊢ ¬Collinear ℝ {b, c, a}

Tactic:
by

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

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
t : Triangle ℝ P
signs : Finset (Fin 3)
⊢ Simplex.excenter t signs = Simplex.incenter t ∨ ∃ i, Simplex.excenter t signs = Simplex.excenter t {i}

Tactic:
by

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

Goal state before:
pp: V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
a b c : P₁
a' b' c' : P₂
h : ¬Collinear ℝ {a, b, c}
ha₁ : ∠ a b c = ∠ a' b' c'
ha₂ : ∠ b c a = ∠ b' c' a'
hd : dist c a = dist c' a'
⊢ ![a, b, c] ≅ ![a', b', c']

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
h : angle x z = angle x y + angle y z
hxz₁ : ¬angle x z = π
hxz₂ : ¬angle x z = 0
h_sin_xz : Real.sin (angle x z) ≠ 0
hx : ¬x = 0
hz : ¬z = 0
hnneg₁ : 0 ≤ Real.sin (angle y z) / ‖x‖
hnneg₂ : 0 ≤ Real.sin (angle x y) / ‖z‖
hnneg₃ : 0 ≤ Real.sin (angle x z) / ‖y‖
hnonzero : ⟨Real.sin (angle x z) / ‖y‖, hnneg₃⟩ ≠ 0
h₁ :
  angle x z ≠ π →
    angle x z = angle x y + angle y z →
      Real.sin (angle x z) • NormedSpace.normalize y =
        Real.sin (angle y z) • NormedSpace.normalize x + Real.sin (angle x y) • NormedSpace.normalize z
h₂ :
  Real.sin (angle x z) • NormedSpace.normalize y =
    Real.sin (angle y z) • NormedSpace.normalize x + Real.sin (angle x y) • NormedSpace.normalize z
⊢ ⟨Real.sin (angle x z) / ‖y‖, hnneg₃⟩ • y =
    ⟨Real.sin (angle y z) / ‖x‖, hnneg₁⟩ • x + ⟨Real.sin (angle x y) / ‖z‖, hnneg₂⟩ • z

Tactic:
by

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

Goal state before:
pp: ι : Type u_1
V₁ : Type u_2
V₂ : Type u_3
P₁ : Type u_4
P₂ : Type u_5
inst✝⁷ : NormedAddCommGroup V₁
inst✝⁶ : NormedAddCommGroup V₂
inst✝⁵ : InnerProductSpace ℝ V₁
inst✝⁴ : InnerProductSpace ℝ V₂
inst✝³ : MetricSpace P₁
inst✝² : MetricSpace P₂
inst✝¹ : NormedAddTorsor V₁ P₁
inst✝ : NormedAddTorsor V₂ P₂
v₁ : ι → P₁
v₂ : ι → P₂
h : v₁ ≅ v₂
i j k : ι
⊢ ∠ (v₁ i) (v₁ j) (v₁ k) = ∠ (v₂ i) (v₂ j) (v₂ k)

Tactic:
by

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

Goal state before:
pp: V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y z : V
hy : y ≠ 0
⊢ angle x z = angle x y + angle y z ↔ angle x z = π ∨ y ∈ Submodule.span ℝ≥0 {x, z}

Tactic:
by

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

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
t : Triangle ℝ P
signs : Finset (Fin 3)
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
i : Fin (2 + 1)
h : Simplex.excenter t signs = Simplex.excenter t {i}
⊢ i = i₁ ∨ i = i₂ ∨ i = i₃

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Inversion.Calculus

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
c x : E → F
R : E → ℝ
a : E
hc : DifferentiableAt ℝ c a
hR : DifferentiableAt ℝ R a
hx : DifferentiableAt ℝ x a
hne : x a ≠ c a
⊢ DifferentiableAt ℝ (fun a ↦ inversion (c a) (R a) (x a)) a

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Inversion.Calculus

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
c : F
R : ℝ
x : F
hx : (fun x ↦ c + x) x ≠ c
A : HasFDerivAt (fun x ↦ _root_.id x - c) (ContinuousLinearMap.id ℝ F) (c + x)
⊢ ‖_root_.id (c + x) - c‖ ^ 2 ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Euclidean.Inversion.Calculus

Goal state before:
pp: F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
c : F
R : ℝ
x : F
hx : (fun x ↦ c + x) x ≠ c
⊢ HasFDerivAt (inversion c R) ?refine_1 (c + x)

Tactic:
by

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

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
t : Triangle ℝ P
signs : Finset (Fin 3)
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
⊢ Simplex.excenter t signs = Simplex.incenter t ∨
    Simplex.excenter t signs = Simplex.excenter t {i₁} ∨
      Simplex.excenter t signs = Simplex.excenter t {i₂} ∨ Simplex.excenter t signs = Simplex.excenter t {i₃}

Tactic:
by

Goal state after: