module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 467,
"column": 36
} | {
"line": 467,
"column": 50
} | {
"line": 467,
"column": 51
} | [
{
"pp": "case inr.inr\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 : 0 < r\nha : o.oangle x (r • (o.rotation ↑(π / 2)) x) = ↑(π / 2)\n⊢ ↑(Real.arctan (‖r‖ * ‖x‖ / ‖x‖)) = ↑(Real.arctan r)",
... | [
"case inr.inr\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 : 0 < r\nha : o.oangle x (r • (o.rotation ↑(π / 2)) x) = ↑(π / 2)\n⊢ ↑(Real.arctan (‖r‖ * (‖x‖ / ‖x‖))) = ↑(Real.arctan r)"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 504,
"column": 22
} | {
"line": 504,
"column": 25
} | {
"line": 504,
"column": 26
} | [
{
"pp": "case neg\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⊢ o.oangle (r • (o.rotation ↑(π / 2)) x) (r • (o.ro... | [
"case neg\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⊢ o.oangle (r • (o.rotation ↑(π / 2)) x)\n (r • (o.rotatio... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 144,
"column": 33
} | {
"line": 144,
"column": 46
} | {
"line": 144,
"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\nv w : V\np q : P\n⊢ ((signedDist v) p) (-w +ᵥ w +ᵥ q) = ((signedDist v) p) q",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"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\np q : P\n⊢ ((signedDist v) p) q = ((signedDist v) p) q"
] | neg_vadd_vadd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 953,
"column": 43
} | {
"line": 953,
"column": 72
} | {
"line": 953,
"column": 73
} | [
{
"pp": "case neg.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 : V\nh : ‖x‖ = ‖y‖\nhn : ¬x = y\nhs : o.oangle y (y - x) = ↑π\n⊢ |(o.oangle (y - x) y).toReal| < π / 2",
"ppTerm": "?neg.inl✝",
"assigned":... | [
"case neg.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 : V\nh : ‖x‖ = ‖y‖\nhn : ¬x = y\nhs : y ≠ 0 ∧ y - x ≠ 0 ∧ SameRay ℝ y (-(y - x))\n⊢ |(o.oangle (y - x) y).toReal| < π / 2"
] | oangle_eq_pi_iff_sameRay_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 251,
"column": 2
} | {
"line": 254,
"column": 64
} | {
"line": 256,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np q : P\nh : q ∈ s\n⊢ s.signedInfDist p = (signedDist (p -ᵥ ↑((orth... | [] | apply signedDist_left_congr
apply s.direction.inner_left_of_mem_orthogonal
· exact vsub_mem_direction (SetLike.coe_mem _) h
· exact vsub_orthogonalProjection_mem_direction_orthogonal s p | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 251,
"column": 2
} | {
"line": 254,
"column": 64
} | {
"line": 256,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace ℝ P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np q : P\nh : q ∈ s\n⊢ s.signedInfDist p = (signedDist (p -ᵥ ↑((orth... | [] | apply signedDist_left_congr
apply s.direction.inner_left_of_mem_orthogonal
· exact vsub_mem_direction (SetLike.coe_mem _) h
· exact vsub_orthogonalProjection_mem_direction_orthogonal s p | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 18
} | {
"line": 59,
"column": 19
} | [
{
"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\nc p₁ p₂ : P\n⊢ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0 ↔ ⟪((c -ᵥ p₁) +ᵥ c) -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0",
"ppTerm": "?m.47",
"assigned": true,
"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\nc p₁ p₂ : P\n⊢ ⟪⅟2 • (c -ᵥ p₁) + ⅟2 • (c -ᵥ p₂), p₂ -ᵥ p₁⟫ = 0 ↔ ⟪((c -ᵥ p₁) +ᵥ c) -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0"
] | vsub_midpoint, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Basic | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 32
} | {
"line": 68,
"column": 33
} | [
{
"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\nι₁ : Type u_3\ns₁ : Finset ι₁\nw₁ : ι₁ → ℝ\np₁ : ι₁ → P\nh₁ : ∑ i ∈ s₁, w₁ i = 0\nι₂ : Type u_4\ns₂ : Finset ι₂\nw₂ : ι₂ → ℝ\np₂ : ι₂ → P\nh₂ : ∑ i ∈ s₂, w₂ i ... | [
"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\nι₁ : Type u_3\ns₁ : Finset ι₁\nw₁ : ι₁ → ℝ\np₁ : ι₁ → P\nh₁ : ∑ i ∈ s₁, w₁ i = 0\nι₂ : Type u_4\ns₂ : Finset ι₂\nw₂ : ι₂ → ℝ\np₂ : ι₂ → P\nh₂ : ∑ i ∈ s₂, w₂ i = 0\n⊢ ⟪∑ i ... | Finset.weightedVSub_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Basic | {
"line": 68,
"column": 33
} | {
"line": 68,
"column": 59
} | {
"line": 69,
"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\nι₁ : Type u_3\ns₁ : Finset ι₁\nw₁ : ι₁ → ℝ\np₁ : ι₁ → P\nh₁ : ∑ i ∈ s₁, w₁ i = 0\nι₂ : Type u_4\ns₂ : Finset ι₂\nw₂ : ι₂ → ℝ\np₂ : ι₂ → P\nh₂ : ∑ i ∈ s₂, w₂ i ... | [
"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\nι₁ : Type u_3\ns₁ : Finset ι₁\nw₁ : ι₁ → ℝ\np₁ : ι₁ → P\nh₁ : ∑ i ∈ s₁, w₁ i = 0\nι₂ : Type u_4\ns₂ : Finset ι₂\nw₂ : ι₂ → ℝ\np₂ : ι₂ → P\nh₂ : ∑ i ∈ s₂, w₂ i = 0\n⊢ ⟪∑ i ... | Finset.weightedVSub_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 27
} | {
"line": 162,
"column": 4
} | [
{
"pp": "case refine_1\nP : Type u_2\ninst✝ : MetricSpace P\nps : Set P\nh : Cospherical ps\n⊢ ∃ s, ps ⊆ Metric.sphere s.center s.radius",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"EuclideanGeometry.Cospherical",
"Real",
"EuclideanGeometry.Sphere",
"Members... | [
"case refine_1\nP : Type u_2\ninst✝ : MetricSpace P\nps : Set P\nc : P\nr : ℝ\nh : ∀ p ∈ ps, dist p c = r\n⊢ ∃ s, ps ⊆ Metric.sphere s.center s.radius"
] | rcases h with ⟨c, r, h⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 15
} | {
"line": 84,
"column": 16
} | [
{
"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\nc p₁ p₂ : P\n⊢ c ∈ perpBisector p₁ p₂ ↔ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫",
"ppTerm": "?m.31",
"assigned": true,
"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\nc p₁ p₂ : P\n⊢ ⟪c -ᵥ p₁, p₂ -ᵥ p₁⟫ = ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ ↔ c ∈ perpBisector p₁ p₂"
] | Iff.comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 85,
"column": 44
} | {
"line": 85,
"column": 58
} | {
"line": 85,
"column": 59
} | [
{
"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\nc p₁ p₂ : P\n⊢ ⟪c -ᵥ p₁ + (c -ᵥ p₂), p₂ -ᵥ p₁⟫ = 0 ↔ ⟪c -ᵥ midpoint ℝ p₁ p₂, p₂ -ᵥ p₁⟫ = 0",
"ppTerm": "?m.73",
"assigned": true,
"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\nc p₁ p₂ : P\n⊢ ⟪c -ᵥ p₁ + (c -ᵥ p₂), p₂ -ᵥ p₁⟫ = 0 ↔ ⟪⅟2 • (c -ᵥ p₁) + ⅟2 • (c -ᵥ p₂), p₂ -ᵥ p₁⟫ = 0"
] | vsub_midpoint, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 75
} | {
"line": 303,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₁ p₂ p₃ : P\nh₁₂ : s.IsDiameter p₁ p₂\nh₁₃ : s.IsDiameter p₁ p₃\n⊢ p₂ = p₃",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [] | rw [← h₁₂.pointReflection_center_left, ← h₁₃.pointReflection_center_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 75
} | {
"line": 303,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₁ p₂ p₃ : P\nh₁₂ : s.IsDiameter p₁ p₂\nh₁₃ : s.IsDiameter p₁ p₃\n⊢ p₂ = p₃",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [] | rw [← h₁₂.pointReflection_center_left, ← h₁₃.pointReflection_center_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 75
} | {
"line": 303,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₁ p₂ p₃ : P\nh₁₂ : s.IsDiameter p₁ p₂\nh₁₃ : s.IsDiameter p₁ p₃\n⊢ p₂ = p₃",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [] | rw [← h₁₂.pointReflection_center_left, ← h₁₃.pointReflection_center_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Basic | {
"line": 112,
"column": 65
} | {
"line": 112,
"column": 79
} | {
"line": 112,
"column": 80
} | [
{
"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\nhvi : ⟪v, v⟫ ≠ 0\nhd : discrim ⟪v, v⟫ (2 * ⟪v, p₁ -ᵥ p₂⟫) 0 = 2 * ⟪v, p₁ -ᵥ p₂⟫ * (2 * ⟪v, p₁ -ᵥ p₂⟫)\n⊢ r = 0 ∨ r = -2 * ... | [
"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\nhvi : ⟪v, v⟫ ≠ 0\nhd : discrim ⟪v, v⟫ (2 * ⟪v, p₁ -ᵥ p₂⟫) 0 = 2 * ⟪v, p₁ -ᵥ p₂⟫ * (2 * ⟪v, p₁ -ᵥ p₂⟫)\n⊢ r = 0 ∨ r = -2 * (2 * ⟪v, p₁ ... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 57,
"column": 6
} | {
"line": 57,
"column": 36
} | {
"line": 57,
"column": 37
} | [
{
"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 x : P\n⊢ x ∈ s.orthRadius p ↔ ⟪p -ᵥ s.center, x -ᵥ p⟫ = 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"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 x : P\n⊢ ⟪x -ᵥ p, p -ᵥ s.center⟫ = 0 ↔ ⟪p -ᵥ s.center, x -ᵥ p⟫ = 0"
] | mem_orthRadius_iff_inner_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 36
} | {
"line": 72,
"column": 37
} | [
{
"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 : P\n⊢ s.center ∈ s.orthRadius p ↔ p = s.center",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"In... | [
"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 : P\n⊢ ⟪s.center -ᵥ p, p -ᵥ s.center⟫ = 0 ↔ p = s.center"
] | mem_orthRadius_iff_inner_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 29
} | {
"line": 86,
"column": 4
} | [
{
"pp": "case refine_1\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 q : P\nh : s.orthRadius p ≤ s.orthRadius q\n⊢ p = q ∨ q = s.center",
"ppTerm": "?refine_1",
"assigned": true,
"usedC... | [
"case refine_1\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 q : P\nh : s.orthRadius p ≤ s.orthRadius q\nh' : (s.orthRadius p).direction ≤ (s.orthRadius q).direction\n⊢ p = q ∨ q = s.center"
] | have h' := direction_le h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 92,
"column": 8
} | {
"line": 92,
"column": 38
} | {
"line": 92,
"column": 39
} | [
{
"pp": "case refine_1\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 q : P\nh : s.orthRadius p ≤ s.orthRadius q\nh' : (ℝ ∙ (p -ᵥ s.center))ᗮ ≤ (ℝ ∙ (q -ᵥ s.center))ᗮ\nr : ℝ\nhr : r • (p -ᵥ s.center... | [
"case refine_1\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 q : P\nh : s.orthRadius p ≤ s.orthRadius q\nh' : (ℝ ∙ (p -ᵥ s.center))ᗮ ≤ (ℝ ∙ (q -ᵥ s.center))ᗮ\nr : ℝ\nhr : r • (p -ᵥ s.center) = q -ᵥ s.c... | mem_orthRadius_iff_inner_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 61
} | {
"line": 168,
"column": 4
} | [
{
"pp": "case 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\np q : P\nh : dist q s.center ^ 2 - dist p s.center ^ 2 = dist q p ^ 2\nh0 : s.radius < 0\n⊢ q ∈ s ↔ 0 ≤ s.radius ∧ dist q s.center ^ 2 ... | [
"case 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\np q : P\nh : dist q s.center ^ 2 - dist p s.center ^ 2 = dist q p ^ 2\nh0 : s.radius < 0\n⊢ q ∉ s"
] | simp only [h0.not_ge, sub_left_inj, false_and, iff_false] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 570,
"column": 12
} | {
"line": 570,
"column": 17
} | {
"line": 570,
"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\np₁ p₂ : P\ns : Sphere P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nhp₁p₂ : p₁ ≠ p₂\nhp₁' : ‖p₁ -ᵥ s.center‖ = s.radius\nhp₂' : ‖p₂ -ᵥ s.center‖ = s.radius\nhd : s.radius ^ 2... | [
"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\nhp₁' : ‖p₁ -ᵥ s.center‖ = s.radius\nhp₂' : ‖p₂ -ᵥ s.center‖ = s.radius\nhd : s.radius ^ 2 = ‖p₂ -ᵥ p₁... | hp₁', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 90,
"column": 12
} | {
"line": 90,
"column": 97
} | {
"line": 91,
"column": 10
} | [
{
"pp": "case h.left.right.inr\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\ns : AffineSubspace ℝ P\ninst✝ : s.direction.HasOrthogonalProjection\nps : Set P\nhnps : ps.Nonempty\np : P\nhps : ps ⊆ ↑s\nhp : p ∉ s\n... | [
"case h.left.right.inr\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\ns : AffineSubspace ℝ P\ninst✝ : s.direction.HasOrthogonalProjection\nps : Set P\nhnps : ps.Nonempty\np : P\nhps : ps ⊆ ↑s\nhp : p ∉ s\nthis : Nonem... | dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq _ (hps hp₁), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 283,
"column": 31
} | {
"line": 283,
"column": 75
} | {
"line": 283,
"column": 76
} | [
{
"pp": "V : Type u_3\nP : Type u_4\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)\ns : Sphere P\np₁ p₂ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nh : p₁ ≠ p₂\n⊢ ‖midpoint ℝ p₁ p₂ -ᵥ p₁‖ /\n ... | [
"V : Type u_3\nP : Type u_4\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)\ns : Sphere P\np₁ p₂ : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nh : p₁ ≠ p₂\n⊢ ‖midpoint ℝ p₁ p₂ -ᵥ p₁‖ / (↑(Real.arctan ... | o.oangle_add_right_smul_rotation_pi_div_two, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Incenter | {
"line": 204,
"column": 79
} | {
"line": 212,
"column": 17
} | {
"line": 214,
"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)\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nsigns : Finset (F... | [] | by
have hs := t.excenter_eq_incenter_or_excenter_singleton_of_ne signs h₁₂ h₁₃ h₂₃
rcases hs with hs | hs | hs | hs
· rw [hs, t.oangle_incenter_eq h₁₂ h₁₃ h₂₃] at h
simp [Real.Angle.pi_ne_zero] at h
· rw [hs, t.oangle_excenter_singleton_eq h₁₂ h₁₃ h₂₃] at h
simp [Real.Angle.pi_ne_zero] at h
· exact .i... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 39,
"column": 63
} | {
"line": 39,
"column": 66
} | {
"line": 39,
"column": 67
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\n⊢ ‖x‖ * ‖y‖ ≤ 1",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"congrArg",
"AddGroupWit... | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\n⊢ 1 * ‖y‖ ≤ 1"
] | hx, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 71,
"column": 53
} | {
"line": 71,
"column": 56
} | {
"line": 71,
"column": 57
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\nh₁ : ¬x = y\nh₂ : ¬x = -y\nH1 : ‖x - ⟪y, x⟫ • y‖ ≠ 0\n⊢ (1 - ⟪x, y⟫ * ⟪x, y⟫) ^ 2 +\n (‖x‖ ^ 2 - ⟪x, y⟫ * ⟪x, y⟫ - (⟪x, y⟫ * ⟪x, y⟫ - ⟪x, y⟫ * (⟪x, y⟫ * ⟪y, y⟫))) * ⟪x, y⟫ ^ 2 =\n ‖x... | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\nh₁ : ¬x = y\nh₂ : ¬x = -y\nH1 : ‖x - ⟪y, x⟫ • y‖ ≠ 0\n⊢ (1 - ⟪x, y⟫ * ⟪x, y⟫) ^ 2 + (1 ^ 2 - ⟪x, y⟫ * ⟪x, y⟫ - (⟪x, y⟫ * ⟪x, y⟫ - ⟪x, y⟫ * (⟪x, y⟫ * ⟪y, y⟫))) * ⟪x, y⟫ ^ 2 =\n 1 ^ 2 - ⟪x, y⟫ * ⟪x, y⟫... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 118,
"column": 46
} | {
"line": 118,
"column": 60
} | {
"line": 118,
"column": 61
} | [
{
"pp": "case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhpi : angle x y ≠ π\nhxy : ¬x = y\nh : -(‖y‖ - ‖x‖) = (‖y‖ - ‖x‖) * ⟪x, y⟫ / (‖x‖ * ‖y‖)\nhx0 : ¬x = 0\nhy0 : ¬y = 0\n⊢ ‖x‖ = ‖y‖",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"N... | [
"case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhpi : angle x y ≠ π\nhxy : ¬x = y\nh : -(‖y‖ - ‖x‖) = (‖y‖ - ‖x‖) * (⟪x, y⟫ / (‖x‖ * ‖y‖))\nhx0 : ¬x = 0\nhy0 : ¬y = 0\n⊢ ‖x‖ = ‖y‖"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 601,
"column": 11
} | {
"line": 601,
"column": 25
} | {
"line": 601,
"column": 25
} | [
{
"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\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\nfs : Finset (Fin (n + 1))\nm : ℕ\nhfs : #fs = m + 1\nhne... | [
"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\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\nfs : Finset (Fin (n + 1))\nm : ℕ\nhfs : #fs = m + 1\nhne : m ≠ n\nhm... | fs.subset_univ | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 93,
"column": 32
} | {
"line": 93,
"column": 35
} | {
"line": 93,
"column": 36
} | [
{
"pp": "case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\nhxy : ¬x - ⟪x, y⟫ • y = 0\n⊢ x = ⟪x, y⟫ • y + (‖x - ⟪y, x⟫ • y‖⁻¹ * (‖x‖ ^ 2 - ⟪y, x⟫ * ⟪x, y⟫)) • ‖x - ⟪y, x⟫ • y‖⁻¹ • (x - ⟪y, x⟫ • y)",
"ppTerm": "?neg✝",
"assigned": tr... | [
"case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\nhxy : ¬x - ⟪x, y⟫ • y = 0\n⊢ x = ⟪x, y⟫ • y + (‖x - ⟪y, x⟫ • y‖⁻¹ * (1 ^ 2 - ⟪y, x⟫ * ⟪x, y⟫)) • ‖x - ⟪y, x⟫ • y‖⁻¹ • (x - ⟪y, x⟫ • y)"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 501,
"column": 2
} | {
"line": 501,
"column": 37
} | {
"line": 502,
"column": 2
} | [
{
"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\nn : ℕ\ns : Simplex ℝ P n\ni : Fin (n + 1)\n⊢ s.points i = (affineCombination ℝ univ s.pointsWithCircumcenter) (pointWeightsWithCircumcenter i)",
"ppTerm": ... | [
"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\nn : ℕ\ns : Simplex ℝ P n\ni : Fin (n + 1)\n⊢ s.pointsWithCircumcenter (pointIndex i) =\n (affineCombination ℝ univ s.pointsWithCircumcenter) (pointWeightsWithCircumcent... | rw [← pointsWithCircumcenter_point] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Congruence | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 31
} | {
"line": 72,
"column": 0
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | [] | simp [h, hd₁, hd₂, dist_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Congruence | {
"line": 90,
"column": 67
} | {
"line": 90,
"column": 86
} | {
"line": 90,
"column": 86
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | [
"V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\na b c : P₁\n... | angle_comm b' a' c' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1224,
"column": 2
} | {
"line": 1235,
"column": 68
} | {
"line": 1237,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\nn : ℕ\ninst✝¹ : NeZero n\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni j : Fin (n + 1)\n⊢ s.touchpoint... | [] | intro he
rw [eq_comm, ← Finset.univ.affineCombination_piSingle ℝ s.points (Finset.mem_univ _),
affineCombination_eq_touchpoint_iff (Fintype.sum_pi_single' _ _)] at he
have : 1 < n := Nat.AtLeastTwo.one_lt
obtain ⟨k, hki, hkj⟩ : ∃ k, k ≠ i ∧ k ≠ j := Fin.exists_ne_and_ne_of_two_lt i j (by lia)
have he' := co... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1224,
"column": 2
} | {
"line": 1235,
"column": 68
} | {
"line": 1237,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\nn : ℕ\ninst✝¹ : NeZero n\ns : Simplex ℝ P n\ninst✝ : n.AtLeastTwo\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni j : Fin (n + 1)\n⊢ s.touchpoint... | [] | intro he
rw [eq_comm, ← Finset.univ.affineCombination_piSingle ℝ s.points (Finset.mem_univ _),
affineCombination_eq_touchpoint_iff (Fintype.sum_pi_single' _ _)] at he
have : 1 < n := Nat.AtLeastTwo.one_lt
obtain ⟨k, hki, hkj⟩ : ∃ k, k ≠ i ∧ k ≠ j := Fin.exists_ne_and_ne_of_two_lt i j (by lia)
have he' := co... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1358,
"column": 6
} | {
"line": 1358,
"column": 83
} | {
"line": 1358,
"column": 83
} | [
{
"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\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nhw : ∑ j, Simplex.touchpointWeights t {i₁} i₂ j = 1\n⊢ Sbtw ℝ ((Finset.affineC... | [
"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\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nhw : ∑ j ∈ {i₁, i₂, i₃}, Simplex.touchpointWeights t {i₁} i₂ j = 1\n⊢ Sbtw ℝ ((Finset.affi... | (by clear hw; decide +revert : (Finset.univ : Finset (Fin 3)) = {i₁, i₂, i₃}) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 94,
"column": 67
} | {
"line": 94,
"column": 93
} | {
"line": 96,
"column": 0
} | [
{
"pp": "case e'_2.e'_5.e'_9\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\nn : ℕ\ns : Simplex ℝ P n\ne : Fin (n + 1) ≃ Fin (n + 1)\n⊢ univ = Finset.map e.toEmbedding univ",
"ppTerm": "?e'_2.e'_5.e'_9",
"as... | [] | simp [Function.comp_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 94,
"column": 67
} | {
"line": 94,
"column": 93
} | {
"line": 96,
"column": 0
} | [
{
"pp": "case e'_3.e'_5\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\nn : ℕ\ns : Simplex ℝ P n\ne : Fin (n + 1) ≃ Fin (n + 1)\n⊢ s.points = (s.points ∘ ⇑e.symm) ∘ ⇑e.toEmbedding",
"ppTerm": "?e'_3.e'_5",
"... | [] | simp [Function.comp_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 129,
"column": 51
} | {
"line": 133,
"column": 49
} | {
"line": 135,
"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\nn : ℕ\ns : Simplex ℝ P n\nS : AffineSubspace ℝ P\nhS : affineSpan ℝ (Set.range s.points) ≤ S\n⊢ ↑(s.restrict S hS).mongePoint = s.mongePoint",
"ppTerm": "?... | [] | by
haveI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
simp_rw [mongePoint]
rw [← Simplex.centroid, ← Simplex.centroid]
simp [centroid_restrict, circumcenter_restrict] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.NinePointCircle | {
"line": 187,
"column": 37
} | {
"line": 187,
"column": 70
} | {
"line": 187,
"column": 71
} | [
{
"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\nn : ℕ\nhn : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\nhn1 : ¬n = 1\nhltn : 1 < n\nhnsub1 : ↑(n - 1) = ↑n - 1\n⊢ (↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (... | [
"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\nn : ℕ\nhn : NeZero n\ns : Simplex ℝ P n\ni : Fin (n + 1)\nhn1 : ¬n = 1\nhltn : 1 < n\nhnsub1 : ↑(n - 1) = ↑n - 1\n⊢ (↑n)⁻¹ • (s.centroid -ᵥ s.circumcenter) + (s.centroid -... | div_self (by simpa using hn.out), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Similarity | {
"line": 64,
"column": 15
} | {
"line": 64,
"column": 29
} | {
"line": 64,
"column": 30
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | [
"V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\na b c : P₁\n... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Similarity | {
"line": 64,
"column": 30
} | {
"line": 64,
"column": 44
} | {
"line": 64,
"column": 45
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | [
"V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\na b c : P₁\n... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Similarity | {
"line": 70,
"column": 15
} | {
"line": 70,
"column": 29
} | {
"line": 70,
"column": 30
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | [
"V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\na b c : P₁\n... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Similarity | {
"line": 70,
"column": 30
} | {
"line": 70,
"column": 44
} | {
"line": 70,
"column": 45
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | [
"V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\na b c : P₁\n... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 286,
"column": 4
} | {
"line": 286,
"column": 30
} | {
"line": 287,
"column": 2
} | [
{
"pp": "case e'_3.e'_5\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\nn : ℕ\ns : Simplex ℝ P (n + 2)\ne : Fin (n + 3) ≃ Fin (n + 3)\ni₁ i₂ : Fin (n + 3)\n⊢ s.points = (s.points ∘ ⇑e.symm) ∘ ⇑e.toEmbedding",
"p... | [] | simp [Function.comp_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 286,
"column": 4
} | {
"line": 286,
"column": 30
} | {
"line": 287,
"column": 2
} | [
{
"pp": "case e'_3.e'_5\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\nn : ℕ\ns : Simplex ℝ P (n + 2)\ne : Fin (n + 3) ≃ Fin (n + 3)\ni₁ i₂ : Fin (n + 3)\n⊢ s.points = (s.points ∘ ⇑e.symm) ∘ ⇑e.toEmbedding",
"p... | [] | simp [Function.comp_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 286,
"column": 4
} | {
"line": 286,
"column": 30
} | {
"line": 287,
"column": 2
} | [
{
"pp": "case e'_3.e'_5\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\nn : ℕ\ns : Simplex ℝ P (n + 2)\ne : Fin (n + 3) ≃ Fin (n + 3)\ni₁ i₂ : Fin (n + 3)\n⊢ s.points = (s.points ∘ ⇑e.symm) ∘ ⇑e.toEmbedding",
"p... | [] | simp [Function.comp_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 406,
"column": 2
} | {
"line": 410,
"column": 12
} | {
"line": 412,
"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\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\n⊢ altitude t i₁ = mongePlane t i₂ i₃",
"ppTerm": "?m.34",
"assigned": ... | [] | have hs : ({i₂, i₃}ᶜ : Finset (Fin 3)) = {i₁} := by decide +revert
have he : ({i₁}ᶜ : Set (Fin 3)) = {i₂, i₃} := by grind
rw [mongePlane_def, altitude_def, direction_affineSpan, hs, he, centroid_singleton,
vectorSpan_image_eq_span_vsub_set_left_ne ℝ _ (Set.mem_insert i₂ _)]
simp [h₂₃] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 406,
"column": 2
} | {
"line": 410,
"column": 12
} | {
"line": 412,
"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\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\n⊢ altitude t i₁ = mongePlane t i₂ i₃",
"ppTerm": "?m.34",
"assigned": ... | [] | have hs : ({i₂, i₃}ᶜ : Finset (Fin 3)) = {i₁} := by decide +revert
have he : ({i₁}ᶜ : Set (Fin 3)) = {i₂, i₃} := by grind
rw [mongePlane_def, altitude_def, direction_affineSpan, hs, he, centroid_singleton,
vectorSpan_image_eq_span_vsub_set_left_ne ℝ _ (Set.mem_insert i₂ _)]
simp [h₂₃] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.Tangent | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 78
} | {
"line": 106,
"column": 2
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝¹¹ : NormedAddCommGroup E'\ninst✝¹⁰ : NormedSpace 𝕜 E'\nH : Type u_4\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nH' ... | [] | exact mapsTo_iff_image_subset.2 (i.1.extend_image_source_inter j.1).subset | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 61
} | {
"line": 289,
"column": 2
} | [
{
"pp": "𝕜✝ : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜✝\nE✝ : Type u_2\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace 𝕜✝ E✝\nH✝ : Type u_3\ninst✝⁹ : TopologicalSpace H✝\nI✝ : ModelWithCorners 𝕜✝ E✝ H✝\nM✝ : Type u_4\ninst✝⁸ : TopologicalSpace M✝\ninst✝⁷ : ChartedSpace H✝ M✝\nn✝ : WithTop ℕ∞\ne... | [
"𝕜✝ : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜✝\nE✝ : Type u_2\ninst✝¹¹ : NormedAddCommGroup E✝\ninst✝¹⁰ : NormedSpace 𝕜✝ E✝\nH✝ : Type u_3\ninst✝⁹ : TopologicalSpace H✝\nI✝ : ModelWithCorners 𝕜✝ E✝ H✝\nM✝ : Type u_4\ninst✝⁸ : TopologicalSpace M✝\ninst✝⁷ : ChartedSpace H✝ M✝\nn✝ : WithTop ℕ∞\ne✝ e'✝ : Open... | rw [show e'.extend I x = φ (e.extend I x) by simp [φ, hex]] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.MFDeriv.Tangent | {
"line": 84,
"column": 2
} | {
"line": 85,
"column": 22
} | {
"line": 86,
"column": 2
} | [
{
"pp": "case e_g\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\ninst✝⁶ : IsManifold I 1 ... | [
"case e_f.e_f\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\ninst✝⁶ : IsManifold I 1 M\nE' : ... | · have : MDiffAt (extChartAt I' (g x₀)) (g x) := mdifferentiableAt_extChartAt hy
simp_all [mfderiv] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 864,
"column": 35
} | {
"line": 864,
"column": 47
} | {
"line": 864,
"column": 47
} | [
{
"pp": "case e'_21\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nE' : Type u_5\ninst✝¹ : N... | [] | rw [neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 864,
"column": 35
} | {
"line": 864,
"column": 47
} | {
"line": 864,
"column": 47
} | [
{
"pp": "case e'_23\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nE' : Type u_5\ninst✝¹ : N... | [] | rw [neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 876,
"column": 31
} | {
"line": 876,
"column": 43
} | {
"line": 876,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nE' : Type u_5\ninst✝¹ : NormedAddComm... | [] | rw [neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 879,
"column": 32
} | {
"line": 879,
"column": 44
} | {
"line": 879,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nE' : Type u_5\ninst✝¹ : NormedAddComm... | [] | rw [neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.VectorBundle.Hom | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 56
} | {
"line": 155,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF₁ : Type u_3\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nn : WithTop ℕ∞\nEB : Type u_4\ninst✝¹⁶ : NormedAddCommGroup EB\ninst✝¹⁵ : NormedSpace 𝕜 EB\nHB : Type u_5\ninst✝¹⁴ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : ChartedSpa... | [
"𝕜 : Type u_1\nB : Type u_2\nF₁ : Type u_3\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nn : WithTop ℕ∞\nEB : Type u_4\ninst✝¹⁶ : NormedAddCommGroup EB\ninst✝¹⁵ : NormedSpace 𝕜 EB\nHB : Type u_5\ninst✝¹⁴ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : ChartedSpace HB B\nE₁ ... | refine contMDiffAt_totalSpace.mpr ⟨contMDiffAt_id, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 201,
"column": 2
} | {
"line": 202,
"column": 47
} | {
"line": 204,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁹ : TopologicalSpace H\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nH' : Type u_5\ninst✝⁴ : TopologicalS... | [] | ext x
simp [mpullback_apply, mpullbackWithin_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 201,
"column": 2
} | {
"line": 202,
"column": 47
} | {
"line": 204,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁹ : TopologicalSpace H\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nH' : Type u_5\ninst✝⁴ : TopologicalS... | [] | ext x
simp [mpullback_apply, mpullbackWithin_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.VectorBundle.Hom | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 45
} | {
"line": 162,
"column": 0
} | [
{
"pp": "case hb\n𝕜₁ : Type u_1\ninst✝²⁰ : NontriviallyNormedField 𝕜₁\n𝕜₂ : Type u_2\ninst✝¹⁹ : NontriviallyNormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nB : Type u_3\nF₁ : Type u_4\ninst✝¹⁸ : NormedAddCommGroup F₁\ninst✝¹⁷ : NormedSpace 𝕜₁ F₁\nE₁ : B → Type u_5\ninst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)\ninst✝¹⁵ : (x : ... | [] | exacts [⟨hb.2.1, hb.1.1⟩, ⟨hb.1.2, hb.2.2⟩] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 71
} | {
"line": 398,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nx : M\nV W : (x : M) → TangentSpace I... | [] | exact mlieBracketWithin_smul_right hf hW (uniqueMDiffWithinAt_univ I) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Homeomorph.TransferInstance | {
"line": 38,
"column": 25
} | {
"line": 42,
"column": 9
} | {
"line": 42,
"column": 10
} | [
{
"pp": "R : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝ : TopologicalSpace β\ne : α ≃ β\nthis : TopologicalSpace α := e.topologicalSpace\n⊢ Continuous e.invFun",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"Equiv.instEquivLike",
"congrArg"... | [] | by
simp only [Equiv.invFun_as_coe]
convert! continuous_coinduced_rng
rw [e.coinduced_symm]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 190,
"column": 19
} | {
"line": 194,
"column": 44
} | {
"line": 195,
"column": 2
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ if h : IsRCLikeNormedField ℝ then Convex ℝ (range Subtype.val) else range Subtype.val = univ",
"ppTerm": "?m.158",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Convex.convex_isRCLik... | [] | by
simp only [instIsRCLikeNormedField, ↓reduceDIte]
apply Convex.convex_isRCLikeNormedField
rw [range_euclideanHalfSpace n]
exact EuclideanHalfSpace.convex (n := n) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 421,
"column": 2
} | {
"line": 421,
"column": 43
} | {
"line": 423,
"column": 0
} | [
{
"pp": "x y : ℝ\nhxy : Fact (x < y)\np : ↑(Icc x y)\nhp : x < ↑p ∧ ↑p < y\n⊢ ↑((IccLeftChart x y).extend (𝓡∂ 1)) p ∈ {y | 0 < y.ofLp 0}",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"IccLeftChart_extend_interior_pos"
],
"usedFVars": [
"x",
"y",
"hxy",
... | [] | exact IccLeftChart_extend_interior_pos hp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 31
} | {
"line": 316,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nn : WithTop ℕ∞\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝... | [
"𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nn : WithTop ℕ∞\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAd... | rw [← contMDiffOn_univ] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.Instances.Icc | {
"line": 120,
"column": 4
} | {
"line": 123,
"column": 54
} | {
"line": 124,
"column": 4
} | [
{
"pp": "case pos\nx y : ℝ\nh : Fact (x < y)\nn : WithTop ℕ∞\nz : ℝ\nhz : z ∈ Icc x y\nh'z : ↑(projIcc x y ⋯ z) < y\n⊢ ContDiffWithinAt ℝ n ((↑(𝓡∂ 1) ∘ ↑(IccLeftChart x y)) ∘ projIcc x y ⋯) (Icc x y) z",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"ContDiff.sub",
"Iff.mpr",
... | [
"case pos\nx y : ℝ\nh : Fact (x < y)\nn : WithTop ℕ∞\nz : ℝ\nhz : z ∈ Icc x y\nh'z : ↑(projIcc x y ⋯ z) < y\nthis :\n ContDiff ℝ n fun w ↦\n have this := toLp 2 fun x_1 ↦ w - x;\n this\n⊢ ContDiffWithinAt ℝ n ((↑(𝓡∂ 1) ∘ ↑(IccLeftChart x y)) ∘ projIcc x y ⋯) (Icc x y) z"
] | have : ContDiff ℝ n (fun (w : ℝ) ↦
(show EuclideanSpace ℝ (Fin 1) from toLp 2 fun (_ : Fin 1) ↦ w - x)) := by
dsimp
apply contDiff_euclidean.2 (fun i ↦ by fun_prop) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.IntegralCurve.Basic | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 34
} | {
"line": 195,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nγ : ℝ → M\nv : (x : M) → TangentSpace I x\nt₀ : ℝ\ninst✝ : IsManifold I 1 M\nhγ : IsMIntegr... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type u_2\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nγ : ℝ → M\nv : (x : M) → TangentSpace I x\nt₀ : ℝ\ninst✝ : IsManifold I 1 M\nhγ : IsMIntegralCurveAt γ ... | have hsrc := mem_of_mem_nhds ht1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 40
} | {
"line": 114,
"column": 40
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : IsManifold I 1 M\nv : (x : M) → TangentSpace I x\nt₀ : ℝ\nx₀ : M\ninst✝ : Complete... | [
"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : IsManifold I 1 M\nv : (x : M) → TangentSpace I x\nt₀ : ℝ\nx₀ : M\ninst✝ : CompleteSpace E\nhx ... | ← (extChartAt I x₀).right_inv hf3' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 155,
"column": 6
} | {
"line": 155,
"column": 20
} | {
"line": 156,
"column": 4
} | [
{
"pp": "case hf\nE : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : (x : M) → ENorm (TangentSpace I x)\na b : ℝ\nγ : ℝ → M\ninst✝ : ∀ (x : M)... | [] | exact h'f _ ht | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 187,
"column": 6
} | {
"line": 187,
"column": 20
} | {
"line": 188,
"column": 4
} | [
{
"pp": "case hf\nE : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : (x : M) → ENorm (TangentSpace I x)\na b : ℝ\nγ : ℝ → M\ninst✝ : ∀ (x : M)... | [] | exact h'f _ ht | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 313,
"column": 2
} | {
"line": 314,
"column": 56
} | {
"line": 316,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : (x : M) → ENorm (TangentSpace I x)\ninst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSp... | [] | exact (riemannianEDist_le_pathELength (γ := fun (t : ℝ) ↦ x) (a := 0) (b := 0)
contMDiffOn_const rfl rfl le_rfl).trans_eq (by simp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace | {
"line": 211,
"column": 6
} | {
"line": 212,
"column": 82
} | {
"line": 213,
"column": 6
} | [
{
"pp": "case refine_2\n𝕜 : Type u\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nEM : Type u_1\ninst✝¹⁴ : NormedAddCommGroup EM\ninst✝¹³ : NormedSpace 𝕜 EM\nHM : Type u_2\ninst✝¹² : TopologicalSpace HM\nIM : ModelWithCorners 𝕜 EM HM\nM : Type u\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace HM M\nEN : Type u... | [
"case refine_2\n𝕜 : Type u\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nEM : Type u_1\ninst✝¹⁴ : NormedAddCommGroup EM\ninst✝¹³ : NormedSpace 𝕜 EM\nHM : Type u_2\ninst✝¹² : TopologicalSpace HM\nIM : ModelWithCorners 𝕜 EM HM\nM : Type u\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace HM M\nEN : Type u_3\ninst✝⁹ :... | let b : V' ≃ₜ ha.isOpenMap.functor.obj V :=
U.isOpenEmbedding'.homeomorphOfSubsetRange <| Set.image_subset_range _ V.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Geometry.Manifold.VectorBundle.Tensoriality | {
"line": 122,
"column": 6
} | {
"line": 122,
"column": 48
} | {
"line": 123,
"column": 6
} | [
{
"pp": "case insert\n𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nF : Type u_5\ninst... | [
"case insert\n𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nF : Type u_5\ninst✝⁹ : NormedA... | simp only [Finset.sum_insert ha, ← h hσ.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Torsion | {
"line": 55,
"column": 16
} | {
"line": 57,
"column": 10
} | {
"line": 58,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : IsManifold I 2 M\ninst✝ : Co... | [] | by
simp [torsionAux, hcov.leibniz hX hf, VectorField.mlieBracket_smul_left hf hX]
module | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Commensurable | {
"line": 62,
"column": 63
} | {
"line": 62,
"column": 86
} | {
"line": 64,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ H.Commensurable H",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"and_self",
"Subgroup.relIndex_self",
"one_ne_zero._simp_1",
... | [] | by simp [Commensurable] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.PresentedGroup | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 91
} | {
"line": 200,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nrels₁ : Set (FreeGroup α)\nrels₂ : Set (FreeGroup β)\nr : FreeGroup (α ⊕ β)\nhr : r ∈ ⇑(FreeGroup.map Sum.inl) '' rels₁ ∪ ⇑(FreeGroup.map Sum.inr) '' rels₂\n⊢ (FreeGroup.lift (toCoprod rels₁ rels₂)) r = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨r, hr, rfl⟩ | ⟨r, hr, rfl⟩ := hr <;> simp [← MonoidHom.comp_apply, one_of_mem hr] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.GroupTheory.PresentedGroup | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 91
} | {
"line": 200,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nrels₁ : Set (FreeGroup α)\nrels₂ : Set (FreeGroup β)\nr : FreeGroup (α ⊕ β)\nhr : r ∈ ⇑(FreeGroup.map Sum.inl) '' rels₁ ∪ ⇑(FreeGroup.map Sum.inr) '' rels₂\n⊢ (FreeGroup.lift (toCoprod rels₁ rels₂)) r = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨r, hr, rfl⟩ | ⟨r, hr, rfl⟩ := hr <;> simp [← MonoidHom.comp_apply, one_of_mem hr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.PresentedGroup | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 91
} | {
"line": 200,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nrels₁ : Set (FreeGroup α)\nrels₂ : Set (FreeGroup β)\nr : FreeGroup (α ⊕ β)\nhr : r ∈ ⇑(FreeGroup.map Sum.inl) '' rels₁ ∪ ⇑(FreeGroup.map Sum.inr) '' rels₂\n⊢ (FreeGroup.lift (toCoprod rels₁ rels₂)) r = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨r, hr, rfl⟩ | ⟨r, hr, rfl⟩ := hr <;> simp [← MonoidHom.comp_apply, one_of_mem hr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.CoprodI | {
"line": 252,
"column": 4
} | {
"line": 253,
"column": 25
} | {
"line": 255,
"column": 0
} | [
{
"pp": "case mul\nι : Type u_1\nG : ι → Type u_4\ninst✝¹ : (i : ι) → Group (G i)\nN : Type u_5\ninst✝ : Group N\nf : (i : ι) → G i →* N\ns : Subgroup N\nh : ∀ (i : ι), (f i).range ≤ s\nx y : CoprodI G\nhx : (lift f) x ∈ s\nhy : (lift f) y ∈ s\n⊢ (lift f) (x * y) ∈ s",
"ppTerm": "?mul",
"assigned": true... | [] | simp only [map_mul]
exact s.mul_mem hx hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.CoprodI | {
"line": 252,
"column": 4
} | {
"line": 253,
"column": 25
} | {
"line": 255,
"column": 0
} | [
{
"pp": "case mul\nι : Type u_1\nG : ι → Type u_4\ninst✝¹ : (i : ι) → Group (G i)\nN : Type u_5\ninst✝ : Group N\nf : (i : ι) → G i →* N\ns : Subgroup N\nh : ∀ (i : ι), (f i).range ≤ s\nx y : CoprodI G\nhx : (lift f) x ∈ s\nhy : (lift f) y ∈ s\n⊢ (lift f) (x * y) ∈ s",
"ppTerm": "?mul",
"assigned": true... | [] | simp only [map_mul]
exact s.mul_mem hx hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 312,
"column": 2
} | {
"line": 313,
"column": 59
} | {
"line": 314,
"column": 2
} | [
{
"pp": "case mp\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ ¬cs.length (cs.simple i * w) < cs.length w → cs.length (cs.simple i * w) = cs.length w + 1",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Group... | [
"case mpr\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.length (cs.simple i * w) = cs.length w + 1 → ¬cs.length (cs.simple i * w) < cs.length w"
] | · intro _
exact (cs.length_simple_mul w i).resolve_right (by lia) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 50
} | {
"line": 100,
"column": 2
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nt : W\nht : cs.IsReflection t\nw : W\nthis : cs.length (w * t) = cs.length w\n⊢ cs.lengthParity (w * t) = cs.lengthParity w",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MonoidHom.in... | [] | simp only [lengthParity_eq_ofAdd_length, this] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 50
} | {
"line": 107,
"column": 2
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nt : W\nht : cs.IsReflection t\nw : W\nthis : cs.length (t * w) = cs.length w\n⊢ cs.lengthParity (t * w) = cs.lengthParity w",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MonoidHom.in... | [] | simp only [lengthParity_eq_ofAdd_length, this] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.CoprodI | {
"line": 589,
"column": 18
} | {
"line": 589,
"column": 21
} | {
"line": 589,
"column": 22
} | [
{
"pp": "case mul\nι : Type u_1\nM : ι → Type u_2\ninst✝² : (i : ι) → Monoid (M i)\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (M i)\nx y : CoprodI M\nhx : ∀ (w : Word M), (x • w).prod = x * w.prod\nhy : ∀ (w : Word M), (y • w).prod = y * w.prod\nw : Word M\n⊢ (x • y • w).prod = x * y * w.prod",
... | [
"case mul\nι : Type u_1\nM : ι → Type u_2\ninst✝² : (i : ι) → Monoid (M i)\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (M i)\nx y : CoprodI M\nhx : ∀ (w : Word M), (x • w).prod = x * w.prod\nhy : ∀ (w : Word M), (y • w).prod = y * w.prod\nw : Word M\n⊢ x * (y • w).prod = x * y * w.prod"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Descent | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 78
} | {
"line": 217,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : CommGroup G\nh : G → ℝ\nC : ℝ\nH : ∀ (x y : G), |h (x * y) + h (x / y) - 2 * (h x + h y)| ≤ C\ninst✝ : Northcott h\n⊢ Finite ↥(torsion G)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"InvOneClass.toOne",
"HMu... | [
"G : Type u_1\ninst✝¹ : CommGroup G\nh : G → ℝ\nC : ℝ\nH : ∀ (x y : G), |h (x * y) + h (x / y) - 2 * (h x + h y)| ≤ C\ninst✝ : Northcott h\nH' : ∀ (x : G), 4 * h x - (h 1 + C) ≤ h (x ^ 2)\n⊢ Finite ↥(torsion G)"
] | have H' x : 4 * h x - (h 1 + C) ≤ h (x ^ 2) := by grind [pow_two, div_self'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.VectorBundle.Riemannian | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 62
} | {
"line": 236,
"column": 2
} | [
{
"pp": "case e\nB : Type u_1\ninst✝⁷ : TopologicalSpace B\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nE : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (x : B) → NormedAddCommGroup (E x)\ninst✝² : (x : B) → InnerProductSpace ℝ (E x)\ninst✝¹ : FiberBundle F E\nin... | [] | simp [Trivialization.coe_continuousLinearEquivAt_eq _ h'x] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.DoubleCoset | {
"line": 106,
"column": 47
} | {
"line": 106,
"column": 67
} | {
"line": 106,
"column": 67
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\na b : G\nh : b * a⁻¹ ∈ H\n⊢ b = b * a⁻¹ * a",
"ppTerm": "?m.139",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"co... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\na b : G\nh : b * a⁻¹ ∈ H\n⊢ b = b"
] | inv_mul_cancel_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Transfer | {
"line": 98,
"column": 30
} | {
"line": 98,
"column": 44
} | {
"line": 98,
"column": 45
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\nq : G ⧸ H\n⊢ ↑(g ^ ((H.quotientEquivSigmaZMod g) q).snd.cast * Quotient.out (Quotient.out ((H.quotientEquivSigmaZMod g) q).fst)) =\n q",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\nq : G ⧸ H\n⊢ ↑(g ^ ((H.quotientEquivSigmaZMod g) q).snd.cast • Quotient.out (Quotient.out ((H.quotientEquivSigmaZMod g) q).fst)) =\n q"
] | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 43
} | {
"line": 135,
"column": 43
} | [
{
"pp": "G : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nhM : HasEnoughRootsOfUnity M (Monoid.exponent G)\nH : Subgroup G\nthis : HasEnoughRootsOfUnity M (Monoid.exponent (G ⧸ H))\n⊢ Nat.card (G ⧸ H →* Mˣ) = Nat.card (G ⧸ H)",
"ppTerm": "?m.43",
"assigned": tru... | [
"G : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nhM : HasEnoughRootsOfUnity M (Monoid.exponent G)\nH : Subgroup G\nthis : HasEnoughRootsOfUnity M (Monoid.exponent (G ⧸ H))\n⊢ Nat.card (G ⧸ H) = Nat.card (G ⧸ H)"
] | card_monoidHom_of_hasEnoughRootsOfUnity | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 155,
"column": 53
} | {
"line": 155,
"column": 92
} | {
"line": 155,
"column": 92
} | [
{
"pp": "case refine_2\nG : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nhM : HasEnoughRootsOfUnity M (Monoid.exponent G)\nthis : HasEnoughRootsOfUnity M (Monoid.exponent (G →* Mˣ))\n⊢ Nat.card G = Nat.card (G →* Mˣ)",
"ppTerm": "?refine_2",
"assigned": true,
... | [
"case refine_2\nG : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nhM : HasEnoughRootsOfUnity M (Monoid.exponent G)\nthis : HasEnoughRootsOfUnity M (Monoid.exponent (G →* Mˣ))\n⊢ Nat.card G = Nat.card G"
] | card_monoidHom_of_hasEnoughRootsOfUnity | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 131,
"column": 10
} | {
"line": 131,
"column": 27
} | {
"line": 132,
"column": 10
} | [
{
"pp": "case left\nG A : Type u\ninst✝³ : Group G\ninst✝² : IsFreeGroup G\ninst✝¹ : MulAction G A\nX : Type u\ninst✝ : Group X\nf : Labelling (Generators (ActionCategory G A)) X\nf' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G := fun e ↦ ⟨fun b ↦ f ⟨e, ⋯⟩, IsFreeGroup.of e⟩\nF' : G →* (A → X) ⋊[mulAut... | [
"case e'_2\nG A : Type u\ninst✝³ : Group G\ninst✝² : IsFreeGroup G\ninst✝¹ : MulAction G A\nX : Type u\ninst✝ : Group X\nf : Labelling (Generators (ActionCategory G A)) X\nf' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G := ⋯\nF' : G →* (A → X) ⋊[mulAutArrow] G\nhF' : ∀ (a : IsFreeGroup.Generators G), F' (I... | convert! hE _ _ _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 193,
"column": 10
} | {
"line": 193,
"column": 18
} | {
"line": 193,
"column": 18
} | [
{
"pp": "α : Type u\nL : List (α × Bool)\ninst✝ : DecidableEq α\nh : IsReduced L\nn : ℕ\nih :\n reduce (replicate (n + 1) L).flatten =\n conjugator L ++ (replicate (n + 1) (reduceCyclically L)).flatten ++ invRev (conjugator L)\nL₁ L₂ L₃ L₄ L₅ : List (α × Bool)\n⊢ mk L₁ * mk L₂ * mk (invRev L₃) * (mk L₃ * mk... | [
"α : Type u\nL : List (α × Bool)\ninst✝ : DecidableEq α\nh : IsReduced L\nn : ℕ\nih :\n reduce (replicate (n + 1) L).flatten =\n conjugator L ++ (replicate (n + 1) (reduceCyclically L)).flatten ++ invRev (conjugator L)\nL₁ L₂ L₃ L₄ L₅ : List (α × Bool)\n⊢ mk L₁ * mk L₂ * (mk L₃)⁻¹ * (mk L₃ * mk L₄ * mk L₅) = mk... | ← inv_mk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.VectorBundle.Riemannian | {
"line": 296,
"column": 6
} | {
"line": 296,
"column": 64
} | {
"line": 297,
"column": 4
} | [
{
"pp": "case e'_1\nB : Type u_1\ninst✝⁷ : TopologicalSpace B\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nE : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (x : B) → NormedAddCommGroup (E x)\ninst✝² : (x : B) → InnerProductSpace ℝ (E x)\ninst✝¹ : FiberBundle F E\... | [] | simp [Trivialization.coe_continuousLinearEquivAt_eq _ h'x] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Nilpotent | {
"line": 895,
"column": 2
} | {
"line": 895,
"column": 38
} | {
"line": 896,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nhH : IsNilpotent G\ninst✝ : Nontrivial G\n⊢ nilpotencyClass G = nilpotencyClass (G ⧸ center G) + 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub",
"Subgroup.instNormalCenter",
"id",
... | [
"G : Type u_1\ninst✝¹ : Group G\nhH : IsNilpotent G\ninst✝ : Nontrivial G\n⊢ nilpotencyClass G = nilpotencyClass G - 1 + 1"
] | rw [nilpotencyClass_quotient_center] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Goursat | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 70
} | {
"line": 81,
"column": 2
} | [
{
"pp": "case mp\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\nhI₁ : Surjective (Prod.fst ∘ ⇑I.subtype)\nhI₂ : Surjective (Prod.snd ∘ ⇑I.subtype)\nx y : G × H\nhx : x ∈ I\nhy : y ∈ I\nthis✝ : I.goursatFst.Normal\nthis : I.goursatSnd.Normal\nh : (y.1 / x.1, 1) ∈ I\n⊢ (1, x... | [
"case mpr\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\nhI₁ : Surjective (Prod.fst ∘ ⇑I.subtype)\nhI₂ : Surjective (Prod.snd ∘ ⇑I.subtype)\nx y : G × H\nhx : x ∈ I\nhy : y ∈ I\nthis✝ : I.goursatFst.Normal\nthis : I.goursatSnd.Normal\nh : (1, x.2 / y.2) ∈ I\n⊢ (y.1 / x.1, 1) ∈... | · simpa [Prod.mul_def, Prod.div_def] using div_mem (mul_mem h hx) hy | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Nilpotent | {
"line": 1110,
"column": 2
} | {
"line": 1113,
"column": 75
} | {
"line": 1115,
"column": 0
} | [
{
"pp": "η : Type u_2\nGs : η → Type u_3\ninst✝¹ : (i : η) → Group (Gs i)\ninst✝ : ∀ (i : η), IsNilpotent (Gs i)\nn : ℕ\nh : ∀ (i : η), nilpotencyClass (Gs i) ≤ n\n⊢ IsNilpotent ((i : η) → Gs i)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"le_bot_iff",
"Iff.mpr",
"Eq.... | [] | rw [nilpotent_iff_lowerCentralSeries]
refine ⟨n, eq_bot_iff.mpr <| (top_lowerCentralSeries_pi_le _).trans ?_⟩
rw [le_bot_iff, pi_eq_bot_iff]
exact fun i => lowerCentralSeries_eq_bot_iff_nilpotencyClass_le.mpr (h i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Nilpotent | {
"line": 1110,
"column": 2
} | {
"line": 1113,
"column": 75
} | {
"line": 1115,
"column": 0
} | [
{
"pp": "η : Type u_2\nGs : η → Type u_3\ninst✝¹ : (i : η) → Group (Gs i)\ninst✝ : ∀ (i : η), IsNilpotent (Gs i)\nn : ℕ\nh : ∀ (i : η), nilpotencyClass (Gs i) ≤ n\n⊢ IsNilpotent ((i : η) → Gs i)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"le_bot_iff",
"Iff.mpr",
"Eq.... | [] | rw [nilpotent_iff_lowerCentralSeries]
refine ⟨n, eq_bot_iff.mpr <| (top_lowerCentralSeries_pi_le _).trans ?_⟩
rw [le_bot_iff, pi_eq_bot_iff]
exact fun i => lowerCentralSeries_eq_bot_iff_nilpotencyClass_le.mpr (h i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Nilpotent | {
"line": 1187,
"column": 46
} | {
"line": 1187,
"column": 61
} | {
"line": 1188,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsNilpotent G\nih : ∀ (H : Subgroup (G ⧸ center G)), normalizer ↑H = H → H = ⊤\nH : Subgroup G\nhH : normalizer ↑H = H\nhch : center G ≤ H\n⊢ (mk' (center G)).ker ≤ H",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mon... | [] | simpa using hch | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.Goursat | {
"line": 170,
"column": 11
} | {
"line": 170,
"column": 16
} | {
"line": 170,
"column": 17
} | [
{
"pp": "case mpr\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\nG' : Subgroup G := map (MonoidHom.fst G H) I\nH' : Subgroup H := map (MonoidHom.snd G H) I\nP : ↥I →* ↥G' := (MonoidHom.fst G H).subgroupMap I\nQ : ↥I →* ↥H' := (MonoidHom.snd G H).subgroupMap I\nI' : Subgrou... | [
"case mpr\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\nG' : Subgroup G := map (MonoidHom.fst G H) I\nH' : Subgroup H := map (MonoidHom.snd G H) I\nP : ↥I →* ↥G' := (MonoidHom.fst G H).subgroupMap I\nQ : ↥I →* ↥H' := (MonoidHom.snd G H).subgroupMap I\nI' : Subgroup (↥G' × ↥H'... | ← hP, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 81,
"column": 2
} | {
"line": 85,
"column": 50
} | {
"line": 87,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\n⊢ Setoid.IsPartition (range fun a ↦ orbit G a)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"MulAction.nonempty_orbit",
"False",
"Set.Nonempty.ne_empty",
"Set.PairwiseDisjoint.isPartitio... | [] | apply orbit.pairwiseDisjoint.isPartition_of_exists_of_ne_empty
· intro x
exact ⟨_, ⟨x, rfl⟩, mem_orbit_self x⟩
· rintro ⟨a, ha : orbit G a = ∅⟩
exact (MulAction.nonempty_orbit a).ne_empty ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.