module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 23
} | {
"line": 251,
"column": 24
} | [
{
"pp": "case mp\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nhx : x ≠ 0\nθ : Real.Angle\nh : (o.rotation θ) x = x\n⊢ 0 = θ",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"case mp\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nhx : x ≠ 0\nθ : Real.Angle\nh : (o.rotation θ) x = x\n⊢ 0 = θ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 249,
"column": 2
} | {
"line": 251,
"column": 55
} | {
"line": 252,
"column": 2
} | [
{
"pp": "case mp\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nhx : x ≠ 0\nθ : Real.Angle\n⊢ (o.rotation θ) x = x → θ = 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"LinearIsometryE... | [
"case mpr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nhx : x ≠ 0\nθ : Real.Angle\n⊢ θ = 0 → (o.rotation θ) x = x"
] | · intro h
rw [eq_comm]
simpa [hx, h] using o.oangle_rotation_right hx hx θ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 351,
"column": 10
} | {
"line": 351,
"column": 21
} | {
"line": 351,
"column": 22
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nhij : i ≠ j\nr : ℝ\nhr : r ≠ 0\nh : s.points j -ᵥ s.altitudeFoot j = r • (s.points i -ᵥ s.a... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\ni j : Fin (n + 1)\nhij : i ≠ j\nr : ℝ\nhr : r ≠ 0\nh : s.points j -ᵥ s.altitudeFoot j = r • (s.points i -ᵥ s.altitudeFoot ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Projection | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 13
} | {
"line": 211,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np : P\n⊢ s.direction.orthogona... | [
"𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np : P\n⊢ p -ᵥ ↑((orthogonalProjection s) p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Projection | {
"line": 210,
"column": 80
} | {
"line": 211,
"column": 68
} | {
"line": 213,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np : P\n⊢ s.direction.orthogona... | [] | by
simpa using vsub_orthogonalProjection_mem_direction_orthogonal _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Projection | {
"line": 226,
"column": 6
} | {
"line": 226,
"column": 27
} | {
"line": 226,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np q : P\nhqs : q ∈ s\nhpq : p ... | [
"𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np q : P\nhqs : q ∈ s\nhpq : p -ᵥ q ∈ s.dir... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Projection | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 13
} | {
"line": 237,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np : P\nq : ↥s\n⊢ (orthogonalPr... | [
"𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np : P\nq : ↥s\n⊢ (orthogonalProjection s) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 361,
"column": 45
} | {
"line": 362,
"column": 85
} | {
"line": 364,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nθ : Real.Angle\nf : V ≃ₗᵢ[ℝ] ℂ\nhf : (map (Fin 2) f.toLinearEquiv) o = Complex.orientation\nx : V\n⊢ f ((o.rotation θ) x) = ↑θ.toCircle * f x",
"ppTerm": "?m.76"... | [] | by
rw [← Complex.rotation, ← hf, o.rotation_map, LinearIsometryEquiv.symm_apply_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 32
} | {
"line": 108,
"column": 33
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nv₁ v₂ v₃ v : V\n⊢ ∠ (v₁ - v) (v₂ - v) (v₃ - v) = ∠ v₁ v₂ v₃",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nv₁ v₂ v₃ v : V\n⊢ ∠ (v₁ - v) (v₂ - v) (v₃ - v) = ∠ v₁ v₂ v₃"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 32
} | {
"line": 113,
"column": 33
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nv v₁ v₂ v₃ : V\n⊢ ∠ (v - v₁) (v - v₂) (v - v₃) = ∠ v₁ v₂ v₃",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nv v₁ v₂ v₃ : V\n⊢ ∠ (v - v₁) (v - v₂) (v - v₃) = ∠ v₁ v₂ v₃"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 29
} | {
"line": 118,
"column": 30
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nv₁ v₂ v₃ : V\n⊢ ∠ (-v₁) (-v₂) (-v₃) = ∠ v₁ v₂ v₃",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nv₁ v₂ v₃ : V\n⊢ ∠ (-v₁) (-v₂) (-v₃) = ∠ v₁ v₂ v₃"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Projection | {
"line": 362,
"column": 6
} | {
"line": 362,
"column": 44
} | {
"line": 363,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\np₁ p₂ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nr₁ r₂ : 𝕜\nv : V\nhv : v ∈ s.directionᗮ\n⊢ ‖p₁ -ᵥ p₂‖ * ‖p... | [
"𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\np₁ p₂ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nr₁ r₂ : 𝕜\nv : V\nhv : v ∈ s.directionᗮ\n⊢ ‖p₁ -ᵥ p₂‖ * ‖p₁ -ᵥ p₂‖ + ‖... | rw [norm_smul, dist_eq_norm_vsub V p₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Projection | {
"line": 463,
"column": 63
} | {
"line": 463,
"column": 98
} | {
"line": 464,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np : P\n⊢ s.direction.reflectio... | [
"𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np : P\n⊢ p -ᵥ ↑(Classical.arbitrary ↥s) ∈ ... | s.direction.reflection_eq_self_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Projection | {
"line": 481,
"column": 4
} | {
"line": 481,
"column": 15
} | {
"line": 481,
"column": 16
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace 𝕜 V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace 𝕜 P\ninst✝³ : Nonempty ↥s₁\ninst✝² : Nonempty ↥s₂\ninst✝¹ : s₁.direction.HasOrthogonalPro... | [
"case mp\n𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace 𝕜 V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\ns₁ s₂ : AffineSubspace 𝕜 P\ninst✝³ : Nonempty ↥s₁\ninst✝² : Nonempty ↥s₂\ninst✝¹ : s₁.direction.HasOrthogonalProjection\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 63
} | {
"line": 245,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\n⊢ o.oangle (-x) y + o.oangle (-y) x = 0",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | rw [oangle_neg_left_eq_neg_right, oangle_rev, neg_add_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 63
} | {
"line": 245,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\n⊢ o.oangle (-x) y + o.oangle (-y) x = 0",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | rw [oangle_neg_left_eq_neg_right, oangle_rev, neg_add_cancel] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 63
} | {
"line": 245,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\n⊢ o.oangle (-x) y + o.oangle (-y) x = 0",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | rw [oangle_neg_left_eq_neg_right, oangle_rev, neg_add_cancel] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 31
} | {
"line": 398,
"column": 32
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p p₂ p₃ : P\nh : Sbtw ℝ p₂ p₁ p\n⊢ ∠ p₁ p₂ p₃ = ∠ p p₂ p₃",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p p₂ p₃ : P\nh : Sbtw ℝ p₂ p₁ p\n⊢ ∠ p₁ p₂ p₃ = ∠ p p₂ p₃"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 31
} | {
"line": 411,
"column": 32
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p p₂ p₃ : P\nh : Wbtw ℝ p₂ p₁ p\nhp₁p₂ : p₁ ≠ p₂\n⊢ ∠ p₁ p₂ p₃ = ∠ p p₂ p₃",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"use... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p p₂ p₃ : P\nh : Wbtw ℝ p₂ p₁ p\nhp₁p₂ : p₁ ≠ p₂\n⊢ ∠ p₁ p₂ p₃ = ∠ p p₂ p₃"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 438,
"column": 6
} | {
"line": 438,
"column": 17
} | {
"line": 438,
"column": 18
} | [
{
"pp": "case refine_2.inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₃ : P\n⊢ Collinear ℝ {p₁, p₁, p₃}",
"ppTerm": "?refine_2.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"I... | [
"case refine_2.inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₃ : P\n⊢ Collinear ℝ {p₁, p₃}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine | {
"line": 439,
"column": 6
} | {
"line": 439,
"column": 17
} | {
"line": 439,
"column": 18
} | [
{
"pp": "case refine_2.inr.inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₃ : P\n⊢ Collinear ℝ {p₁, p₃, p₃}",
"ppTerm": "?refine_2.inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case refine_2.inr.inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₃ : P\n⊢ Collinear ℝ {p₁, p₃}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 15
} | {
"line": 323,
"column": 16
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nhn : p₂ ≠ p₃\nh : ‖p₁ -ᵥ p₂‖ = ‖p₁ -ᵥ p₃‖\n⊢ p₁ -ᵥ p₂ ≠ p₁ -ᵥ p₃",
"ppTerm": "?m... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nhn : p₂ ≠ p₃\nh : ‖p₁ -ᵥ p₂‖ = ‖p₁ -ᵥ p₃‖\n⊢ ¬p₂ = p₃"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 13
} | {
"line": 359,
"column": 14
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\n⊢ 0 ≤ (↑⟪x, y⟫ + (o.areaForm x) y • I).re ∧ (↑⟪x, y⟫ + (o.areaForm x) y • I).im = 0 ↔ SameRay ℝ x y",
"ppTerm": "?m.59",
"assigned": true,
"used... | [
"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\n⊢ 0 ≤ ⟪x, y⟫ ∧ (o.areaForm x) y = 0 ↔ SameRay ℝ x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 18
} | {
"line": 107,
"column": 19
} | [
{
"pp": "case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫ = 0\nh0 : x = 0 ∨ y ≠ 0\nhx : ¬x = 0\n⊢ ‖x‖ * ‖x‖ < ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.nor... | [
"case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : ⟪x, y⟫ = 0\nh0 : x = 0 ∨ y ≠ 0\nhx : ¬x = 0\n⊢ ¬y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 426,
"column": 25
} | {
"line": 426,
"column": 36
} | {
"line": 426,
"column": 37
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\ny : V\nhy : y ≠ 0\nr : ℝ\nhr : 0 ≤ r\nh₁ : ‖r • y‖ = ‖y‖\nh₂ : SameRay ℝ (r • y) y\n⊢ ‖y‖ ≠ 0",
"ppTerm": "?m.138",
"assigned": true,
"usedConstants": [
... | [
"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\ny : V\nhy : y ≠ 0\nr : ℝ\nhr : 0 ≤ r\nh₁ : ‖r • y‖ = ‖y‖\nh₂ : SameRay ℝ (r • y) y\n⊢ ¬y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 429,
"column": 6
} | {
"line": 429,
"column": 47
} | {
"line": 429,
"column": 48
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\ny : V\nhy : y ≠ 0\nr : ℝ\nhr : 0 ≤ r\nh₁ : ‖r • y‖ = ‖y‖\nh₂ : SameRay ℝ (r • y) y\nthis : ‖y‖ ≠ 0\n⊢ r * ‖y‖ = 1 * ‖y‖",
"ppTerm": "?m.163",
"assigned": tru... | [
"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\ny : V\nhy : y ≠ 0\nr : ℝ\nhr : 0 ≤ r\nh₁ : ‖r • y‖ = ‖y‖\nh₂ : SameRay ℝ (r • y) y\nthis : ‖y‖ ≠ 0\n⊢ r * ‖y‖ = ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 495,
"column": 2
} | {
"line": 495,
"column": 26
} | {
"line": 495,
"column": 26
} | [
{
"pp": "case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhp₂p₁ : ¬p₂ = p₁\n⊢ ∡ p₂ p₁ p₃ = 0",
"ppTerm": "?... | [
"case pos\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhp₂p₁ : ¬p₂ = p₁\nhp₃p₁ : p₃ = p₁\n⊢ ∡ p₂ p₁ p₃ = 0",
"case neg... | by_cases hp₃p₁ : p₃ = p₁ | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 333,
"column": 4
} | {
"line": 333,
"column": 34
} | {
"line": 333,
"column": 35
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\n⊢ ‖p₁ -ᵥ p₃‖ * ‖p₁ -ᵥ p₃‖ = ‖p₁ -ᵥ p₂‖ * ‖p₁ -ᵥ p₂‖ + ‖p₂ -ᵥ p₃‖ * ‖p₂ -ᵥ p₃‖ ↔\n ‖p₁ -ᵥ p₂ - (p₃ -ᵥ p₂)‖ * ‖p₁ -ᵥ p₂ - (p₃ -ᵥ p₂)‖ = ‖p₁ -ᵥ p... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\n⊢ ‖p₁ -ᵥ p₃‖ * ‖p₁ -ᵥ p₃‖ = ‖p₁ -ᵥ p₂‖ * ‖p₁ -ᵥ p₂‖ + ‖p₂ -ᵥ p₃‖ * ‖p₂ -ᵥ p₃‖ ↔\n ‖p₁ -ᵥ p₃‖ * ‖p₁ -ᵥ p₃‖ = ‖p₁ -ᵥ p₂‖ * ‖p₁ -ᵥ p₂‖ + ‖p₃ -ᵥ p₂‖ * ‖p₃ -ᵥ ... | vsub_sub_vsub_cancel_right p₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 340,
"column": 68
} | {
"line": 340,
"column": 100
} | {
"line": 341,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃) = Real.arccos (‖p₂ -ᵥ p₃‖ / ‖p₁ -ᵥ p₃‖)",
"ppTerm": "?m.112"... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃)) = Real.arccos (‖p₂ -ᵥ p₃‖ / ‖p₁ -ᵥ p₂ + (p₂ -ᵥ p₃)‖)"
] | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 349,
"column": 67
} | {
"line": 349,
"column": 99
} | {
"line": 350,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃) = Real.arcsin (‖p₁ -ᵥ p₂‖ / ‖p... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃)) = Real.arcsin (‖p₁ -ᵥ p₂‖ / ‖... | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 358,
"column": 68
} | {
"line": 358,
"column": 100
} | {
"line": 359,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃) = Real.arctan (‖p₁ -ᵥ p₂‖ / ‖p₂ -ᵥ p₃‖)",
... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃)) = Real.arctan (‖p₁ -ᵥ p₂‖ / ‖p₂ -ᵥ p₃‖)"
] | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 367,
"column": 13
} | {
"line": 367,
"column": 45
} | {
"line": 367,
"column": 46
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ 0 < InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)",
"ppTerm": "?m.148",
... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ 0 < InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃))"
] | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 375,
"column": 13
} | {
"line": 375,
"column": 45
} | {
"line": 375,
"column": 46
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃) ≤ π / 2",
"ppTerm": "?m.101",
"assigned": true,
"use... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃)) ≤ π / 2"
] | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 384,
"column": 13
} | {
"line": 384,
"column": 45
} | {
"line": 384,
"column": 46
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃) < π / 2",
"ppTerm": "?m.136",
"assign... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃)) < π / 2"
] | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 392,
"column": 68
} | {
"line": 392,
"column": 100
} | {
"line": 393,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ Real.cos (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) = ‖p₂ -ᵥ p₃‖ / ‖p₁ -ᵥ p₃‖",
"ppTerm": "?m.112",
... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ Real.cos (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃))) = ‖p₂ -ᵥ p₃‖ / ‖p₁ -ᵥ p₂ + (p₂ -ᵥ p₃)‖"
] | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 401,
"column": 67
} | {
"line": 401,
"column": 99
} | {
"line": 402,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ Real.sin (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) = ‖p₁ -ᵥ p₂‖ / ‖p₁ ... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ Real.sin (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃))) = ‖p₁ -ᵥ p₂‖ / ‖p₁... | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 409,
"column": 68
} | {
"line": 409,
"column": 100
} | {
"line": 410,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ Real.tan (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) = ‖p₁ -ᵥ p₂‖ / ‖p₂ -ᵥ p₃‖",
"ppTerm": "?m.112",
... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ Real.tan (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃))) = ‖p₁ -ᵥ p₂‖ / ‖p₂ -ᵥ p₃‖"
] | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 418,
"column": 68
} | {
"line": 418,
"column": 100
} | {
"line": 419,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ Real.cos (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) * ‖p₁ -ᵥ p₃‖ = ‖p₂ -ᵥ p₃‖",
"ppTerm": "?m.112",
... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ Real.cos (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃))) * ‖p₁ -ᵥ p₂ + (p₂ -ᵥ p₃)‖ = ‖p₂ -ᵥ p₃‖"
] | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 436,
"column": 2
} | {
"line": 439,
"column": 58
} | {
"line": 441,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nh : o.oangle x y = ↑(π / 2)\nhs : (o.oangle y (y - x)).sign = 1\n⊢ ‖x‖ / (o.oangle y (y - x)).tan = ‖y‖",
"ppTerm": "?m.59",
"assigned": true,
"use... | [] | rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.tan_coe,
InnerProductGeometry.norm_div_tan_angle_sub_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)
(Or.inr (o.left_ne_zero_of_oangle_eq_pi_div_two h))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 601,
"column": 8
} | {
"line": 601,
"column": 19
} | {
"line": 601,
"column": 20
} | [
{
"pp": "case inl.inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nh : InnerProductGeometry.angle 0 x = InnerProductGeometry.angle y z\nhs : (o.oangle 0 x).sign = (o.oangle y z).sign\n⊢ (o.oangle 0 x).sign =... | [
"case inl.inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nh : InnerProductGeometry.angle 0 x = InnerProductGeometry.angle y z\nhs : (o.oangle 0 x).sign = (o.oangle y z).sign\n⊢ (o.oangle y z).sign = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 427,
"column": 67
} | {
"line": 427,
"column": 99
} | {
"line": 428,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ Real.sin (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) * ‖p₁ -ᵥ p₃‖ = ‖p₁ -ᵥ p₂‖",
"ppTerm": "?m.112",
... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ Real.sin (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃))) * ‖p₁ -ᵥ p₂ + (p₂ -ᵥ p₃)‖ = ‖p₁ -ᵥ p₂‖"
] | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 602,
"column": 8
} | {
"line": 602,
"column": 19
} | {
"line": 602,
"column": 20
} | [
{
"pp": "case inl.inr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw y z : V\nh : InnerProductGeometry.angle w 0 = InnerProductGeometry.angle y z\nhs : (o.oangle w 0).sign = (o.oangle y z).sign\n⊢ (o.oangle w 0).sign =... | [
"case inl.inr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw y z : V\nh : InnerProductGeometry.angle w 0 = InnerProductGeometry.angle y z\nhs : (o.oangle w 0).sign = (o.oangle y z).sign\n⊢ (o.oangle y z).sign = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 603,
"column": 8
} | {
"line": 603,
"column": 19
} | {
"line": 603,
"column": 20
} | [
{
"pp": "case inr.inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x z : V\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle 0 z\nhs : (o.oangle w x).sign = (o.oangle 0 z).sign\n⊢ (o.oangle w x).sign =... | [
"case inr.inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x z : V\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle 0 z\nhs : (o.oangle w x).sign = (o.oangle 0 z).sign\n⊢ (o.oangle w x).sign = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 604,
"column": 8
} | {
"line": 604,
"column": 19
} | {
"line": 604,
"column": 20
} | [
{
"pp": "case inr.inr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x y : V\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle y 0\nhs : (o.oangle w x).sign = (o.oangle y 0).sign\n⊢ (o.oangle w x).sign =... | [
"case inr.inr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x y : V\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle y 0\nhs : (o.oangle w x).sign = (o.oangle y 0).sign\n⊢ (o.oangle w x).sign = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 608,
"column": 8
} | {
"line": 608,
"column": 19
} | {
"line": 608,
"column": 20
} | [
{
"pp": "case inl.inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhsyz : (o.oangle y z).sign = 0\nh : InnerProductGeometry.angle 0 x = InnerProductGeometry.angle y z\nhs : (o.oangle 0 x).sign = (o.oangle y ... | [
"case inl.inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhsyz : (o.oangle y z).sign = 0\nh : InnerProductGeometry.angle 0 x = InnerProductGeometry.angle y z\nhs : (o.oangle 0 x).sign = (o.oangle y z).sign\nhsw... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 609,
"column": 8
} | {
"line": 609,
"column": 19
} | {
"line": 609,
"column": 20
} | [
{
"pp": "case inl.inr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw y z : V\nhsyz : (o.oangle y z).sign = 0\nh : InnerProductGeometry.angle w 0 = InnerProductGeometry.angle y z\nhs : (o.oangle w 0).sign = (o.oangle y ... | [
"case inl.inr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw y z : V\nhsyz : (o.oangle y z).sign = 0\nh : InnerProductGeometry.angle w 0 = InnerProductGeometry.angle y z\nhs : (o.oangle w 0).sign = (o.oangle y z).sign\nhsw... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 437,
"column": 68
} | {
"line": 437,
"column": 100
} | {
"line": 438,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ = 0\n⊢ Real.tan (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) * ‖p₂ -ᵥ p₃‖ = ‖p₁ ... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ = 0\n⊢ Real.tan (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃))) * ‖p₂ -ᵥ p₃‖ = ‖p₁... | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 610,
"column": 8
} | {
"line": 610,
"column": 19
} | {
"line": 610,
"column": 20
} | [
{
"pp": "case inr.inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x z : V\nhswx : (o.oangle w x).sign = 0\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle 0 z\nhs : (o.oangle w x).sign = (o.oangle 0 ... | [
"case inr.inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x z : V\nhswx : (o.oangle w x).sign = 0\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle 0 z\nhs : (o.oangle w x).sign = (o.oangle 0 z).sign\nhsy... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 457,
"column": 6
} | {
"line": 457,
"column": 17
} | {
"line": 457,
"column": 18
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\nhr : r < 0\n⊢ (o.rotation ↑(π / 2)) x ≠ 0",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"LinearIsometryEquiv.ins... | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\nhr : r < 0\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 611,
"column": 8
} | {
"line": 611,
"column": 19
} | {
"line": 611,
"column": 20
} | [
{
"pp": "case inr.inr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x y : V\nhswx : (o.oangle w x).sign = 0\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle y 0\nhs : (o.oangle w x).sign = (o.oangle y ... | [
"case inr.inr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x y : V\nhswx : (o.oangle w x).sign = 0\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle y 0\nhs : (o.oangle w x).sign = (o.oangle y 0).sign\nhsy... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 447,
"column": 68
} | {
"line": 447,
"column": 100
} | {
"line": 448,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ = 0\n⊢ ‖p₂ -ᵥ p₃‖ / Real.cos (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) = ‖p₁ ... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ = 0\n⊢ ‖p₂ -ᵥ p₃‖ / Real.cos (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃))) = ‖p₁... | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 620,
"column": 6
} | {
"line": 620,
"column": 49
} | {
"line": 620,
"column": 50
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x y z : V\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z\nhs : (o.oangle w x).sign = (o.oangle y z).sign\nh0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = ... | [
"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x y z : V\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z\nhs : (o.oangle w x).sign = (o.oangle y z).sign\nh0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0\nhswx : (o... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 623,
"column": 6
} | {
"line": 623,
"column": 49
} | {
"line": 623,
"column": 50
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x y z : V\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z\nhs : (o.oangle w x).sign = (o.oangle y z).sign\nh0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = ... | [
"V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x y z : V\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z\nhs : (o.oangle w x).sign = (o.oangle y z).sign\nh0 : (w = 0 ∨ x = 0) ∨ y = 0 ∨ z = 0\nhswx : (o... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 457,
"column": 67
} | {
"line": 457,
"column": 99
} | {
"line": 458,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ ‖p₁ -ᵥ p₂‖ / Real.sin (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) = ‖p₁ ... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ ‖p₁ -ᵥ p₂‖ / Real.sin (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃))) = ‖p₁... | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 467,
"column": 68
} | {
"line": 467,
"column": 100
} | {
"line": 468,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ ‖p₁ -ᵥ p₂‖ / Real.tan (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) = ‖p₂ ... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ ‖p₁ -ᵥ p₂‖ / Real.tan (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₂ + (p₂ -ᵥ p₃))) = ‖p₂... | ← vsub_add_vsub_cancel p₁ p₂ p₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 505,
"column": 2
} | {
"line": 505,
"column": 18
} | {
"line": 505,
"column": 19
} | [
{
"pp": "case neg.h\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\nhr : ¬r = 0\nhx : -x = r⁻¹ • (o.rotation ↑(π / 2)) (r • (o.rotation ↑(π / 2)) x)\n⊢ r • (o.rotation ↑(π / 2)) x ≠ 0",
"ppTerm":... | [
"case neg.h\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\nhr : ¬r = 0\nhx : -x = r⁻¹ • (o.rotation ↑(π / 2)) (r • (o.rotation ↑(π / 2)) x)\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 691,
"column": 4
} | {
"line": 691,
"column": 15
} | {
"line": 691,
"column": 16
} | [
{
"pp": "case refine_2\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh : InnerProductGeometry.angle x y = 0\nha : o.oangle x y = ↑0 ∨ o.oangle x y = -↑0\n⊢ o.oangle x y = 0",
"ppTerm... | [
"case refine_2\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : x ≠ 0\nhy : y ≠ 0\nh : InnerProductGeometry.angle x y = 0\nha : o.oangle x y = ↑0 ∨ o.oangle x y = -↑0\n⊢ o.oangle x y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 707,
"column": 4
} | {
"line": 707,
"column": 15
} | {
"line": 707,
"column": 16
} | [
{
"pp": "case neg.refine_2\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : ¬x = 0\nhy : ¬y = 0\nh : InnerProductGeometry.angle x y = π\nha : o.oangle x y = ↑π ∨ o.oangle x y = -↑π\n⊢ o.oangle x y = ↑π",
... | [
"case neg.refine_2\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : ¬x = 0\nhy : ¬y = 0\nh : InnerProductGeometry.angle x y = π\nha : o.oangle x y = ↑π ∨ o.oangle x y = -↑π\n⊢ o.oangle x y = ↑π"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 518,
"column": 2
} | {
"line": 518,
"column": 18
} | {
"line": 518,
"column": 19
} | [
{
"pp": "case neg.h\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\nhr : ¬r = 0\nhx : x = r⁻¹ • (o.rotation ↑(π / 2)) (-(r • (o.rotation ↑(π / 2)) x))\n⊢ -(r • (o.rotation ↑(π / 2)) x) ≠ 0",
"ppT... | [
"case neg.h\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nhd2 : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx : V\nh : x ≠ 0\nr : ℝ\nhr : ¬r = 0\nhx : x = r⁻¹ • (o.rotation ↑(π / 2)) (-(r • (o.rotation ↑(π / 2)) x))\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 755,
"column": 2
} | {
"line": 755,
"column": 55
} | {
"line": 757,
"column": 0
} | [
{
"pp": "case neg\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nhx : ¬x = 0\nhy : ¬y = 0\n⊢ (o.oangle x (-y)).sign = -(o.oangle x y).sign",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | rw [o.oangle_neg_right hx hy, Real.Angle.sign_add_pi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 820,
"column": 6
} | {
"line": 820,
"column": 22
} | {
"line": 820,
"column": 23
} | [
{
"pp": "case refine_1\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V ×... | [
"case refine_1\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 821,
"column": 6
} | {
"line": 821,
"column": 22
} | {
"line": 821,
"column": 23
} | [
{
"pp": "case refine_2\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V ×... | [
"case refine_2\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\nr : ℝ\nh : ¬(o.oangle x y = 0 ∨ o.oangle x y = ↑π)\nh' : ∀ (r' : ℝ), o.oangle x (r' • x + y) ≠ 0 ∧ o.oangle x (r' • x + y) ≠ ↑π\ns : Set (V × V) := (fun ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 757,
"column": 58
} | {
"line": 757,
"column": 69
} | {
"line": 757,
"column": 70
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ : P\nh : Sbtw ℝ p₁ p₂ p₃\n⊢ Collinear ℝ {p₁, p₂, p₂, p₃}",
"ppTerm": "?m.60",
... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ : P\nh : Sbtw ℝ p₁ p₂ p₃\n⊢ Collinear ℝ {p₁, p₂, p₃}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 766,
"column": 4
} | {
"line": 766,
"column": 36
} | {
"line": 766,
"column": 37
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhne : p₁ ≠ p₂\n⊢ Collinear ℝ {p₁, p₂, p₁, p₃}",
"ppTerm"... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ : P\nh : Wbtw ℝ p₁ p₂ p₃\nhne : p₁ ≠ p₂\n⊢ Collinear ℝ {p₁, p₂, p₃}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 733,
"column": 91
} | {
"line": 737,
"column": 52
} | {
"line": 739,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ : P\nh : ∡ p₁ p₂ p₃ = ↑(π / 2)\n⊢ dist p₃ p₂ / (∡ p₃ p₁ p₂).sin = dist p₁ p₃",
"ppTer... | [] | by
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, Real.Angle.sin_coe, dist_comm p₁ p₃,
dist_div_sin_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(Or.inl (right_ne_of_o... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 54,
"column": 20
} | {
"line": 54,
"column": 50
} | {
"line": 55,
"column": 4
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv x y : V\n⊢ ContinuousAffineMap.const ℝ P ⟪-NormedSpace.normalize v, x + y⟫ =\n ContinuousAffineMap.const ℝ P ⟪-NormedSpace.normalize v, x⟫ +\n Contin... | [] | by ext; simp [inner_add_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 13
} | {
"line": 97,
"column": 14
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np q : P\n⊢ ((signedDist 0) p) q = 0",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np q : P\n⊢ ((signedDist 0) p) q = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 13
} | {
"line": 100,
"column": 14
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\n⊢ ((signedDist (-v)) p) q = -((signedDist v) p) q",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\n⊢ ((signedDist (-v)) p) q = -((signedDist v) p) q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 17
} | {
"line": 105,
"column": 18
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv w : V\nr : ℝ\nleft✝ : r > 0\nright✝ : r • v = w\np q : P\n⊢ ((signedDist v) p) q = ((signedDist w) p) q",
"ppTerm": "?m.78",
"assigned": false,
"... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv w : V\nr : ℝ\nleft✝ : r > 0\nright✝ : r • v = w\np q : P\n⊢ ((signedDist v) p) q = ((signedDist w) p) q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 62
} | {
"line": 149,
"column": 63
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nh : ⟪v, p -ᵥ q⟫ = 0\nr : P\n⊢ ((signedDist v) p) r = ((signedDist v) q) r",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants": ... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nh : ⟪v, p -ᵥ q⟫ = 0\nr : P\n⊢ ((signedDist v) p) r = ((signedDist v) q) r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 62
} | {
"line": 153,
"column": 63
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q r : P\nh : ⟪v, q -ᵥ r⟫ = 0\n⊢ ((signedDist v) p) q = ((signedDist v) p) r",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q r : P\nh : ⟪v, q -ᵥ r⟫ = 0\n⊢ ((signedDist v) p) q = ((signedDist v) p) r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 13
} | {
"line": 157,
"column": 14
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nh : ⟪v, q -ᵥ p⟫ = 0\n⊢ ((signedDist v) p) q = 0",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nh : ⟪v, q -ᵥ p⟫ = 0\n⊢ ((signedDist v) p) q = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 13
} | {
"line": 217,
"column": 14
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nc : ℝ\n⊢ ((signedDist v) ((AffineMap.lineMap p q) c)) p = -c * ((signedDist v) p) q",
"ppTerm": "?m.36",
"assigned": true,
"usedCon... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nc : ℝ\n⊢ (AffineMap.lineMap 0 (((signedDist v) q) p)) c = -(c * ((signedDist v) p) q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 13
} | {
"line": 221,
"column": 14
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nc : ℝ\n⊢ ((signedDist v) p) ((AffineMap.lineMap p q) c) = c * ((signedDist v) p) q",
"ppTerm": "?m.34",
"assigned": true,
"usedCons... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nc : ℝ\n⊢ (AffineMap.lineMap 0 (((signedDist v) p) q)) c = c * ((signedDist v) p) q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 13
} | {
"line": 225,
"column": 14
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nc : ℝ\n⊢ ((signedDist v) ((AffineMap.lineMap p q) c)) q = (1 - c) * ((signedDist v) p) q",
"ppTerm": "?m.40",
"assigned": true,
"us... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nc : ℝ\n⊢ (AffineMap.lineMap (((signedDist v) p) q) 0) c = (1 - c) * ((signedDist v) p) q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 13
} | {
"line": 229,
"column": 14
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nc : ℝ\n⊢ ((signedDist v) q) ((AffineMap.lineMap p q) c) = (c - 1) * ((signedDist v) p) q",
"ppTerm": "?m.40",
"assigned": true,
"us... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np q : P\nc : ℝ\n⊢ (AffineMap.lineMap (((signedDist v) q) p) 0) c = (c - 1) * ((signedDist v) p) q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 900,
"column": 4
} | {
"line": 900,
"column": 15
} | {
"line": 900,
"column": 16
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ : P\nh₁₂ : p₁ ≠ p₂\nha : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄\nhs : line[ℝ, p₁, p₂].SOppSide p₃ p₄\... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nhd2 : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ : P\nh₁₂ : p₁ ≠ p₂\nha : ∡ p₁ p₂ p₃ = ∡ p₃ p₂ p₄\nhs : line[ℝ, p₁, p₂].SOppSide p₃ p₄\nh₃₂ : p₃ ≠ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 13
} | {
"line": 284,
"column": 14
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np q : P\n⊢ (affineSpan ℝ {q}).signedInfDist p = (signedDist (p -ᵥ q)) q",
"ppTerm": "?m.50",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np q : P\n⊢ (affineSpan ℝ {q}).signedInfDist p = (signedDist (p -ᵥ q)) q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 999,
"column": 14
} | {
"line": 999,
"column": 29
} | {
"line": 999,
"column": 30
} | [
{
"pp": "case inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhx : x ≠ 0\nhz : z ≠ 0\nhy : ¬y = 0\nhr : ¬SameRay ℝ x z\nhs : ¬(o.oangle x y).sign = (o.oangle y z).sign\nhn : o.oangle x y ≠ o.oangle y z\nhe ... | [
"case inl\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhx : x ≠ 0\nhz : z ≠ 0\nhy : ¬y = 0\nhr : ¬SameRay ℝ x z\nhs : ¬(o.oangle x y).sign = (o.oangle y z).sign\nhn : o.oangle x y ≠ o.oangle y z\nhe : InnerProdu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 999,
"column": 14
} | {
"line": 999,
"column": 29
} | {
"line": 999,
"column": 30
} | [
{
"pp": "case inr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhx : x ≠ 0\nhz : z ≠ 0\nhy : ¬y = 0\nhr : ¬SameRay ℝ x z\nhs : ¬(o.oangle x y).sign = (o.oangle y z).sign\nhn : o.oangle x y ≠ o.oangle y z\nhe ... | [
"case inr\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y z : V\nhx : x ≠ 0\nhz : z ≠ 0\nhy : ¬y = 0\nhr : ¬SameRay ℝ x z\nhs : ¬(o.oangle x y).sign = (o.oangle y z).sign\nhn : o.oangle x y ≠ o.oangle y z\nhe : InnerProdu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 26
} | {
"line": 224,
"column": 27
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : Nontrivial V\ns : Sphere P\nh : 0 ≤ s.radius\nv : V\nhv : v ∈ Metric.sphere 0 s.radius\n⊢ v +ᵥ s.center ∈ Metric.sphere s.center s.radius",
"ppTerm": "?... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : Nontrivial V\ns : Sphere P\nh : 0 ≤ s.radius\nv : V\nhv : v ∈ Metric.sphere 0 s.radius\n⊢ ‖v‖ = s.radius"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 51
} | {
"line": 106,
"column": 52
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ : P\n⊢ p₂ ∈ perpBisector p₁ p₂ ↔ p₁ = p₂",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ : P\n⊢ p₂ = p₁ ↔ p₁ = p₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 136,
"column": 40
} | {
"line": 136,
"column": 83
} | {
"line": 136,
"column": 84
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c p : P\nh_sbtw : Sbtw ℝ a b c\nh_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0\nt : ℝ\nhb_eq : (AffineMap.lineMap a c) t = b\nht0 : 0 < t\nht1 : t < 1\nhb : b -ᵥ a = t • (... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c p : P\nh_sbtw : Sbtw ℝ a b c\nh_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0\nt : ℝ\nhb_eq : (AffineMap.lineMap a c) t = b\nht0 : 0 < t\nht1 : t < 1\nhb : b -ᵥ a = t • (c -ᵥ a)\n⊢ ⟪... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Basic | {
"line": 107,
"column": 30
} | {
"line": 107,
"column": 41
} | {
"line": 107,
"column": 42
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np₁ p₂ : P\nhv : v ≠ 0\nr : ℝ\n⊢ ⟪v, v⟫ ≠ 0",
"ppTerm": "?m.113",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np₁ p₂ : P\nhv : v ≠ 0\nr : ℝ\n⊢ ¬v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 45
} | {
"line": 142,
"column": 46
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c p : P\nh_sbtw : Sbtw ℝ a b c\nh_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0\nt : ℝ\nhb_eq : (AffineMap.lineMap a c) t = b\nht0 : 0 < t\nht1 : t < 1\nhb : b -ᵥ a = t • (... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c p : P\nh_sbtw : Sbtw ℝ a b c\nh_inner : ⟪p -ᵥ a, b -ᵥ a⟫ = 0\nt : ℝ\nhb_eq : (AffineMap.lineMap a c) t = b\nht0 : 0 < t\nht1 : t < 1\nhb : b -ᵥ a = t • (c -ᵥ a)\nhpc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 45
} | {
"line": 155,
"column": 46
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c p : P\nh_inner : ⟪p -ᵥ a, c -ᵥ a⟫ = 0\nt : ℝ\nhb_eq : (AffineMap.lineMap a c) t = b\nht0 : 0 ≤ t\nht1 : t ≤ 1\nh_sq_ineq : dist p b ^ 2 ≤ dist p c ^ 2\n⊢... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c p : P\nh_inner : ⟪p -ᵥ a, c -ᵥ a⟫ = 0\nt : ℝ\nhb_eq : (AffineMap.lineMap a c) t = b\nht0 : 0 ≤ t\nht1 : t ≤ 1\nh_sq_ineq : dist p b ^ 2 ≤ dist p c ^ 2\n⊢ dist p b ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 396,
"column": 30
} | {
"line": 396,
"column": 45
} | {
"line": 396,
"column": 46
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nhs : Cospherical s\np : Fin 3 → P\nhps : Set.range p ⊆ s\nhpi : Function.Injective p\nv : V\nhv : ∀ (i : Fin 3), ∃ r, p i = r • v +ᵥ p 0\nh : v = 0\... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nhs : Cospherical s\np : Fin 3 → P\nhps : Set.range p ⊆ s\nhpi : Function.Injective p\nv : V\nhv : ∀ (i : Fin 3), ∃ r, p i = r • v +ᵥ p 0\nh : v = 0\n⊢ p 1 = p 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 408,
"column": 4
} | {
"line": 408,
"column": 21
} | {
"line": 408,
"column": 22
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\np : Fin 3 → P\nhps : Set.range p ⊆ s\nhpi : Function.Injective p\nv : V\nhv0 : v ≠ 0\nc : P\nr : ℝ\nhs : ∀ p ∈ s, dist p c = r\nhs' : ∀ (i : Fin 3),... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\np : Fin 3 → P\nhps : Set.range p ⊆ s\nhpi : Function.Injective p\nv : V\nhv0 : v ≠ 0\nc : P\nr : ℝ\nhs : ∀ p ∈ s, dist p c = r\nhs' : ∀ (i : Fin 3), dist (p i) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 419,
"column": 4
} | {
"line": 419,
"column": 26
} | {
"line": 419,
"column": 27
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\np : Fin 3 → P\nhps : Set.range p ⊆ s\nhpi : Function.Injective p\nv : V\nhv0 : v ≠ 0\nc : P\nr : ℝ\nhs : ∀ p ∈ s, dist p c = r\nhs' : ∀ (i : Fin 3),... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\np : Fin 3 → P\nhps : Set.range p ⊆ s\nhpi : Function.Injective p\nv : V\nhv0 : v ≠ 0\nc : P\nr : ℝ\nhs : ∀ p ∈ s, dist p c = r\nhs' : ∀ (i : Fin 3), dist (p i) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 420,
"column": 34
} | {
"line": 420,
"column": 54
} | {
"line": 420,
"column": 55
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\np : Fin 3 → P\nhps : Set.range p ⊆ s\nhpi : Function.Injective p\nv : V\nhv0 : v ≠ 0\nc : P\nr : ℝ\nhs : ∀ p ∈ s, dist p c = r\nhs' : ∀ (i : Fin 3),... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\np : Fin 3 → P\nhps : Set.range p ⊆ s\nhpi : Function.Injective p\nv : V\nhv0 : v ≠ 0\nc : P\nr : ℝ\nhs : ∀ p ∈ s, dist p c = r\nhs' : ∀ (i : Fin 3), dist (p i) ... | hfn0' 1 (by decide), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 15
} | {
"line": 193,
"column": 16
} | [
{
"pp": "case mp\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np : P\nhp : dist p s.center = s.radius\np' : P\nhp's : p' ∈ s\nhp'i : p' ∈ s.orthRadius p\nh' : s.radius ^ 2 = s.radius ^ 2 + dist p' p ... | [
"case mp\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np : P\nhp : dist p s.center = s.radius\np' : P\nhp's : p' ∈ s\nhp'i : p' ∈ s.orthRadius p\nh' : s.radius ^ 2 = s.radius ^ 2 + dist p' p ^ 2\n⊢ p' = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 15
} | {
"line": 195,
"column": 16
} | [
{
"pp": "case mpr\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np' : P\nhp : dist p' s.center = s.radius\n⊢ p' ∈ s ∧ p' ∈ s.orthRadius p'",
"ppTerm": "?mpr",
"assigned": true,
"usedConsta... | [
"case mpr\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np' : P\nhp : dist p' s.center = s.radius\n⊢ p' ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 533,
"column": 8
} | {
"line": 533,
"column": 19
} | {
"line": 533,
"column": 20
} | [
{
"pp": "case neg.refine_2.inl.refine_2\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₂ : P\nhp₂ : dist p₂ s.center ≤ s.radius\nhp₂' : dist p₂ s.center < dist s.center s.center\nhp₁ : dist s.center ... | [
"case neg.refine_2.inl.refine_2\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₂ : P\nhp₂ : dist p₂ s.center ≤ s.radius\nhp₂' : dist p₂ s.center < dist s.center s.center\nhp₁ : dist s.center s.center = s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 208,
"column": 6
} | {
"line": 208,
"column": 41
} | {
"line": 208,
"column": 42
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np q : P\nh : Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p) = {q}\nhq : q ∈ Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p)\nhr : 0 ≤ s.... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np q : P\nh : Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p) = {q}\nhq : q ∈ Metric.sphere s.center s.radius ∩ ↑(s.orthRadius p)\nhr : 0 ≤ s.radius\nh' :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 541,
"column": 6
} | {
"line": 541,
"column": 17
} | {
"line": 541,
"column": 18
} | [
{
"pp": "case neg.refine_2.inr\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₂ : P\nhp₂ : dist p₂ s.center ≤ s.radius\nhp₁ : dist s.center s.center = s.radius\nh : ¬s.center = p₂\nhp₂' : ‖p₂ -ᵥ s.ce... | [
"case neg.refine_2.inr\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₂ : P\nhp₂ : dist p₂ s.center ≤ s.radius\nhp₁ : dist s.center s.center = s.radius\nh : ¬s.center = p₂\nhp₂' : ‖p₂ -ᵥ s.center‖ = ‖s.c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 565,
"column": 2
} | {
"line": 571,
"column": 76
} | {
"line": 573,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ : P\ns : Sphere P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₁p₂ : p₁ ≠ p₂\n⊢ 0 < ⟪p₂ -ᵥ p₁, s.center -ᵥ p₁⟫",
"ppTerm": "?m.27",
"assigned": true,
"used... | [] | have hp₁' : ‖p₁ -ᵥ s.center‖ = s.radius := norm_vsub_center_eq_radius hp₁
have hp₂' : ‖p₂ -ᵥ s.center‖ = s.radius := norm_vsub_center_eq_radius hp₂
have hd : ‖p₂ -ᵥ s.center‖ ^ 2 =
‖p₂ -ᵥ p₁‖ ^ 2 + 2 * ⟪p₂ -ᵥ p₁, p₁ -ᵥ s.center⟫ + ‖p₁ -ᵥ s.center‖ ^ 2 := by
rw [← vsub_add_vsub_cancel p₂ p₁ s.center, norm_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 565,
"column": 2
} | {
"line": 571,
"column": 76
} | {
"line": 573,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ : P\ns : Sphere P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₁p₂ : p₁ ≠ p₂\n⊢ 0 < ⟪p₂ -ᵥ p₁, s.center -ᵥ p₁⟫",
"ppTerm": "?m.27",
"assigned": true,
"used... | [] | have hp₁' : ‖p₁ -ᵥ s.center‖ = s.radius := norm_vsub_center_eq_radius hp₁
have hp₂' : ‖p₂ -ᵥ s.center‖ = s.radius := norm_vsub_center_eq_radius hp₂
have hd : ‖p₂ -ᵥ s.center‖ ^ 2 =
‖p₂ -ᵥ p₁‖ ^ 2 + 2 * ⟪p₂ -ᵥ p₁, p₁ -ᵥ s.center⟫ + ‖p₁ -ᵥ s.center‖ ^ 2 := by
rw [← vsub_add_vsub_cancel p₂ p₁ s.center, norm_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 262,
"column": 8
} | {
"line": 262,
"column": 24
} | {
"line": 262,
"column": 25
} | [
{
"pp": "case inl.inr.inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np : P\nh : dist p s.center < s.radius\nhp : p ≠ s.center\nhb : ℝ ∙ (p -ᵥ s.center) = ⊤\nhb' : ∀ (v : V), ∃ r, r • (p -ᵥ s.cente... | [
"case inl.inr.inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np : P\nh : dist p s.center < s.radius\nhp : p ≠ s.center\nhb : ℝ ∙ (p -ᵥ s.center) = ⊤\nhb' : ∀ (v : V), ∃ r, r • (p -ᵥ s.center) = v\nhf :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 13
} | {
"line": 87,
"column": 14
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np q : P\nas : AffineSubspace ℝ P\nhp : s.IsTangentAt p as\nhq : s.IsTangentAt q as\nhqp : ⟪p -ᵥ q, p -ᵥ q⟫ = 0\nhpq : ⟪p -ᵥ q, q -ᵥ s.center⟫ = 0... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np q : P\nas : AffineSubspace ℝ P\nhp : s.IsTangentAt p as\nhq : s.IsTangentAt q as\nhqp : ⟪p -ᵥ q, p -ᵥ q⟫ = 0\nhpq : ⟪p -ᵥ q, q -ᵥ s.center⟫ = 0\n⊢ p = q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 616,
"column": 4
} | {
"line": 616,
"column": 15
} | {
"line": 616,
"column": 16
} | [
{
"pp": "case refine_2\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₁ p₂ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nh : s.center ∈ line[ℝ, p₁, p₂]\n⊢ dist s.center s.center ≤ s.radius",
... | [
"case refine_2\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₁ p₂ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nh : s.center ∈ line[ℝ, p₁, p₂]\n⊢ 0 ≤ s.radius"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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