module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 620,
"column": 4
} | {
"line": 620,
"column": 19
} | [
{
"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)\nhs : (∡ p₃ p₁ p₂).sign = 1\n⊢ dist p₁ p₂ / dist p₃ p₁ = d... | dist_comm p₁ p₃ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 624,
"column": 61
} | {
"line": 628,
"column": 51
} | [
{
"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⊢ (∡ p₂ p₃ p₁).sin = dist p₁ p₂ / dist p₁ p₃",
"usedC... | by
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
sin_angle_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)
(Or.inl (left_ne_of_oangle_eq_pi_div_two h))] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 720,
"column": 15
} | {
"line": 720,
"column": 17
} | [
{
"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 p₅ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₃p₄ : p₃ ≠ p₄\nhc : Collinear ℝ {p₁, p₂, p₃, p₄}\nhr : Same... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle | {
"line": 637,
"column": 4
} | {
"line": 637,
"column": 19
} | [
{
"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)\nhs : (∡ p₃ p₁ p₂).sign = 1\n⊢ dist p₃ p₂ / dist p₃ p₁ = d... | dist_comm p₁ p₃ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 674,
"column": 54
} | {
"line": 681,
"column": 43
} | [
{
"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\nh : (o.oangle x y).sign = -1\n⊢ o.oangle x y = -↑(InnerProductGeometry.angle x y)",
"usedConstants": [
"not_le",
"SignType.ctorIdx",
"... | by
by_cases hx : x = 0; · simp [hx] at h
by_cases hy : y = 0; · simp [hy] at h
refine (o.oangle_eq_angle_or_eq_neg_angle hx hy).resolve_left ?_
intro hxy
rw [hxy, ← SignType.neg_iff, ← not_le] at h
exact h (Real.Angle.sign_coe_nonneg_of_nonneg_of_le_pi (InnerProductGeometry.angle_nonneg _ _)
(InnerProdu... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 738,
"column": 15
} | {
"line": 738,
"column": 17
} | [
{
"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 p₅ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₃p₄ : p₃ ≠ p₄\nhc : Collinear ℝ {p₁, p₂, p₃, p₄}\nhr : Same... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Affine | {
"line": 744,
"column": 15
} | {
"line": 744,
"column": 17
} | [
{
"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 p₅ : P\nhp₁p₂ : p₁ ≠ p₂\nhp₃p₄ : p₃ ≠ p₄\nhc : Collinear ℝ {p₁, p₂, p₃, p₄}\nhr : Same... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 816,
"column": 15
} | {
"line": 816,
"column": 17
} | [
{
"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 ×... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 816,
"column": 15
} | {
"line": 816,
"column": 17
} | [
{
"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 ×... | s, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.PerpBisector | {
"line": 178,
"column": 8
} | {
"line": 178,
"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\na b p : P\nhab : a ≠ b\nhp : p ∈ perpBisector a b\ns : ℝ\nh_wbtw : Wbtw ℝ a b ((AffineMap.lineMap a b) s)\n⊢ ⟪p -ᵥ midpoint ℝ a b, s • (b -ᵥ a) + (a -ᵥ midpoin... | left_vsub_midpoint, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Basic | {
"line": 86,
"column": 55
} | {
"line": 86,
"column": 58
} | [
{
"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\nι : Type u_3\ns : Finset ι\nw₁ w₂ : ι → ℝ\np : ι → P\nh₁ : ∑ i ∈ s, w₁ i = 1\nh₂ : ∑ i ∈ s, w₂ i = 1\n⊢ 1 - ∑ x ∈ s, w₂ x = 0",
"usedConstants": [
"E... | h₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 320,
"column": 2
} | {
"line": 324,
"column": 12
} | [
{
"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\ni : Fin (n + 1)\nw : Fin (n + 1) → ℝ\nh : ∑ i, w i = 1\n⊢ (s.signedInfDist i) ((Finset.affineCombination ℝ Finset.... | rw [← ContinuousAffineMap.coe_toAffineMap, Finset.map_affineCombination _ _ _ h,
Finset.univ.affineCombination_apply_eq_lineMap_sum w
((s.signedInfDist i).toAffineMap ∘ s.points) 0
‖s.points i -ᵥ (s.faceOpposite i).orthogonalProjectionSpan (s.points i)‖
{i} h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 950,
"column": 8
} | {
"line": 950,
"column": 64
} | [
{
"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\nh : ‖x‖ = ‖y‖\nhn : ¬x = y\n⊢ (2 • o.oangle (y - x) y).sign = (o.oangle y x).sign",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Orienta... | o.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq hn h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 1004,
"column": 8
} | {
"line": 1004,
"column": 16
} | [
{
"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 : V\nhx : x ≠ 0\nr : ℝ\nhr0 : 0 < r\nr' : ℝ\nhr'0 : 0 < r'\nhy : ¬r' • x = 0\nhe : InnerProductGeometry.angle x (r' • x) = 0\nhz : r • r' • x ≠ 0\nhr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 1008,
"column": 8
} | {
"line": 1008,
"column": 16
} | [
{
"pp": "case neg.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 : V\nhx : x ≠ 0\nr : ℝ\nhr0 : r < 0\nr' : ℝ\nhr'0 : r' < 0\nhy : ¬r' • x = 0\nhe : InnerProductGeometry.angle x (r' • x) = π\nhz : r • r' • x ≠ 0\nhr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 392,
"column": 8
} | {
"line": 392,
"column": 12
} | [
{
"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),... | ← hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Basic | {
"line": 166,
"column": 38
} | {
"line": 166,
"column": 52
} | [
{
"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\ns : AffineSubspace ℝ P\ninst✝ : FiniteDimensional ℝ ↥s.direction\nhd : finrank ℝ ↥s.direction = 2\nc₁ c₂ p₁ p₂ : P\nhc₁s : c₁ ∈ s\nhc₂s : c₂ ∈ s\nhp₁s : p₁ ∈ ... | simp [hp.symm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 45,
"column": 2
} | {
"line": 46,
"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\ns : Sphere P\np : P\n⊢ Nonempty ↥(s.orthRadius p)",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Real.i... | rw [orthRadius]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 45,
"column": 2
} | {
"line": 46,
"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\ns : Sphere P\np : P\n⊢ Nonempty ↥(s.orthRadius p)",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Real.i... | rw [orthRadius]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 102,
"column": 6
} | {
"line": 103,
"column": 21
} | [
{
"pp": "case refine_1.inl.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 : s.orthRadius p ≤ s.orthRadius q\nh' : (ℝ ∙ (p -ᵥ s.center))ᗮ ≤ (ℝ ∙ (q -ᵥ s.center))ᗮ\nhr : 0 • (p -ᵥ s.cente... | · rw [zero_smul, eq_comm, vsub_eq_zero_iff_eq] at hr
exact .inr hr | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 238,
"column": 6
} | {
"line": 238,
"column": 14
} | [
{
"pp": "case e_K.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\nas : AffineSubspace ℝ P\np : P\nhpar : as ∥ s.orthRadius p\nhp : p ∈ s\nq : P\nhqs : q ∈ s\nhqas : q ∈ as\nhqo : as ≤ s.orthRadius ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 238,
"column": 6
} | {
"line": 238,
"column": 14
} | [
{
"pp": "case e_K.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\nas : AffineSubspace ℝ P\np : P\nhpar : as ∥ s.orthRadius p\nhp : p ∈ s\nq : P\nhqs : q ∈ s\nhqas : q ∈ as\nhqo : as ≤ s.orthRadius ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 238,
"column": 6
} | {
"line": 238,
"column": 14
} | [
{
"pp": "case e_K.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\nas : AffineSubspace ℝ P\np : P\nhpar : as ∥ s.orthRadius p\nhp : p ∈ s\nq : P\nhqs : q ∈ s\nhqas : q ∈ as\nhqo : as ≤ s.orthRadius ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 12
} | [
{
"pp": "case 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\nhp : p ∈ s\nq : P\nhqs : q ∈ s\nhrad : s.radius ≠ 0\nhpar : s.orthRadius q ∥ s.orthRadius p\nhqas : q ∈ s.orthRadius q\nhqo ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 12
} | [
{
"pp": "case 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\nhp : p ∈ s\nq : P\nhqs : q ∈ s\nhrad : s.radius ≠ 0\nhpar : s.orthRadius q ∥ s.orthRadius p\nhqas : q ∈ s.orthRadius q\nhqo ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 12
} | [
{
"pp": "case 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\nhp : p ∈ s\nq : P\nhqs : q ∈ s\nhrad : s.radius ≠ 0\nhpar : s.orthRadius q ∥ s.orthRadius p\nhqas : q ∈ s.orthRadius q\nhqo ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 247,
"column": 6
} | {
"line": 247,
"column": 14
} | [
{
"pp": "case inr.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\nhp : p ∈ s\nq : P\nhqs : q ∈ s\nhrad : s.radius ≠ 0\nhpar : s.orthRadius q ∥ s.orthRadius p\nhqas : q ∈ s.orthRadius q\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 247,
"column": 6
} | {
"line": 247,
"column": 14
} | [
{
"pp": "case inr.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\nhp : p ∈ s\nq : P\nhqs : q ∈ s\nhrad : s.radius ≠ 0\nhpar : s.orthRadius q ∥ s.orthRadius p\nhqas : q ∈ s.orthRadius q\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 247,
"column": 6
} | {
"line": 247,
"column": 14
} | [
{
"pp": "case inr.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\nhp : p ∈ s\nq : P\nhqs : q ∈ s\nhrad : s.radius ≠ 0\nhpar : s.orthRadius q ∥ s.orthRadius p\nhqas : q ∈ s.orthRadius q\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 262,
"column": 8
} | {
"line": 262,
"column": 76
} | [
{
"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... | simpa [hf] using inter_orthRadius_eq_empty_of_finrank_eq_one hp h.ne | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 451,
"column": 14
} | {
"line": 451,
"column": 40
} | [
{
"pp": "case neg.refine_3.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\ns₁ s₂ : Sphere P\nh : dist s₁.center s₂.center = s₁.radius + s₂.radius\nh₁ : 0 ≤ s₁.radius\nh₂ : 0 ≤ s₂.radius\nh0 : ¬s₁.radius + s... | div_le_one (by positivity) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Incenter | {
"line": 65,
"column": 6
} | {
"line": 65,
"column": 76
} | [
{
"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\np : P\nhp : p ∈ affineSpan ℝ (Set.range s.points)\ni₁ : Fin (n + 1)\nh :\n ∀ (i₂ : Fin (n + 1)),\n i₂ ≠ i₁ →\n... | ← s.exists_forall_dist_eq_iff_exists_excenterExists_and_eq_excenter hp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Incenter | {
"line": 97,
"column": 42
} | {
"line": 97,
"column": 50
} | [
{
"pp": "case «0».«0»\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)\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\np :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Incenter | {
"line": 97,
"column": 42
} | {
"line": 97,
"column": 50
} | [
{
"pp": "case «0».«1»\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)\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\np :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Incenter | {
"line": 97,
"column": 42
} | {
"line": 97,
"column": 50
} | [
{
"pp": "case «0».«2»\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)\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\np :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Incenter | {
"line": 97,
"column": 42
} | {
"line": 97,
"column": 50
} | [
{
"pp": "case «1».«0»\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)\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\np :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Incenter | {
"line": 97,
"column": 42
} | {
"line": 97,
"column": 50
} | [
{
"pp": "case «1».«1»\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)\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\np :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Incenter | {
"line": 97,
"column": 42
} | {
"line": 97,
"column": 50
} | [
{
"pp": "case «1».«2»\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)\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\np :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Incenter | {
"line": 97,
"column": 42
} | {
"line": 97,
"column": 50
} | [
{
"pp": "case «2».«0»\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)\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\np :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Incenter | {
"line": 97,
"column": 42
} | {
"line": 97,
"column": 50
} | [
{
"pp": "case «2».«1»\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)\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\np :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Incenter | {
"line": 97,
"column": 42
} | {
"line": 97,
"column": 50
} | [
{
"pp": "case «2».«2»\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)\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\np :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 372,
"column": 4
} | {
"line": 372,
"column": 39
} | [
{
"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\nhf2 : Fact (Module.finrank ℝ V = 2)\ns : Sphere P\np : P\nhp : dist p s.center < s.radius\nhpc : p ≠ s.center\nv : ↥(s.orthRadius p).direction\nhv : ∀ (w : ↥(s... | smul_eq_zero_iff_right two_ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 92,
"column": 18
} | {
"line": 92,
"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\np₁ p₂ p₃ : P\ns : Sphere P\nhd : s.IsDiameter p₁ p₃\no : P := s.center\nh_center : o = midpoint ℝ p₁ p₃\n⊢ p₁ -ᵥ midpoint ℝ p₁ p₃ = -(p₃ -ᵥ midpoint ℝ p₁ p₃)",... | left_vsub_midpoint, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 412,
"column": 67
} | {
"line": 412,
"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)\nt : Triangle ℝ P\np : P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh : 2 • ∡ ... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 299,
"column": 2
} | {
"line": 299,
"column": 60
} | [
{
"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\ni : Fin (n + 1)\n⊢ (s.height i)⁻¹ < ∑ i_1 ∈ {i}ᶜ, s.excenterWeightsUnnorm {i} i_1",
"us... | convert! s.inv_height_lt_sum_inv_height i using 2 with j h | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 115,
"column": 8
} | {
"line": 116,
"column": 30
} | [
{
"pp": "case pos\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhpi : angle x y ≠ π\nhxy : ¬x = y\nh : ‖x‖ - ‖y‖ = ⟪x, y⟫ * (‖x‖⁻¹ - ‖y‖⁻¹)\nhx0 : ¬x = 0\nhy0 : y = 0\n⊢ ‖x‖ = ‖y‖",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Sem... | rw [hy0, norm_zero, inner_zero_right, zero_mul, sub_zero] at h
rw [hy0, norm_zero, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 115,
"column": 8
} | {
"line": 116,
"column": 30
} | [
{
"pp": "case pos\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhpi : angle x y ≠ π\nhxy : ¬x = y\nh : ‖x‖ - ‖y‖ = ⟪x, y⟫ * (‖x‖⁻¹ - ‖y‖⁻¹)\nhx0 : ¬x = 0\nhy0 : y = 0\n⊢ ‖x‖ = ‖y‖",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Sem... | rw [hy0, norm_zero, inner_zero_right, zero_mul, sub_zero] at h
rw [hy0, norm_zero, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 510,
"column": 4
} | {
"line": 510,
"column": 42
} | [
{
"pp": "case pointIndex\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\ni a✝ : Fin (n + 1)\nhi : pointIndex a✝ ∈ univ\nhn : pointIndex a✝ ≠ pointIndex i\nh : a✝ ≠ i\n⊢ pointWeightsWithCirc... | simp [pointWeightsWithCircumcenter, h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 607,
"column": 51
} | {
"line": 607,
"column": 60
} | [
{
"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\ni₁ i₂ : Fin (n + 1)\nh : i₁ ≠ i₂\nhc : #{i₁, i₂} = 2\nW : AffineSubspace ℝ P := affineSpan ℝ (s.points '' {i₁, i₂})\nh_faces : ↑((ort... | ite_smul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 22
} | [
{
"pp": "case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\nh₁ : ¬x = y\n⊢ (‖x - ⟪y, x⟫ • y‖⁻¹ * (⟪x, x⟫ - ⟪y, x⟫ * ⟪x, y⟫)) ^ 2 + ⟪x, y⟫ ^ 2 = 1",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"InnerP... | by_cases h₂ : x = -y | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 570,
"column": 2
} | {
"line": 570,
"column": 20
} | [
{
"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\ni : Fin (n + 1)\n⊢ SignType.sign (s.excenterWeights sign... | convert! mul_one _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 77,
"column": 4
} | {
"line": 78,
"column": 60
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\n⊢ ⟪x, NormedSpace.normalize (ortho y x)⟫ ^ 2 = Real.sin (angle x y) ^ 2",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"Inner.inner",
"Real.cos",... | simp [Real.sin_sq, ← inner_eq_cos_angle_of_norm_eq_one hx hy,
← inner_normalized_ortho_sq_add_inner_sq_eq_one hx hy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 643,
"column": 68
} | {
"line": 643,
"column": 76
} | [
{
"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)\nhij : ¬i = j\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 643,
"column": 68
} | {
"line": 643,
"column": 76
} | [
{
"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)\nhij : ¬i = j\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 643,
"column": 68
} | {
"line": 643,
"column": 76
} | [
{
"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)\nhij : ¬i = j\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 668,
"column": 26
} | {
"line": 668,
"column": 34
} | [
{
"pp": "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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns₁ signs₂ : Finset (Fin (n + 1))\nh₁ : s.ExcenterExists signs₁\nh₂ : s.ExcenterExists signs₂\nh : s.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 668,
"column": 26
} | {
"line": 668,
"column": 34
} | [
{
"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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns₁ signs₂ : Finset (Fin (n + 1))\nh₁ : s.ExcenterExists signs₁\nh₂ : s.ExcenterExists signs₂\nh : s.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 668,
"column": 26
} | {
"line": 668,
"column": 34
} | [
{
"pp": "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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns₁ signs₂ : Finset (Fin (n + 1))\nh₁ : s.ExcenterExists signs₁\nh₂ : s.ExcenterExists signs₂\nh : s.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 668,
"column": 26
} | {
"line": 668,
"column": 34
} | [
{
"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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns₁ signs₂ : Finset (Fin (n + 1))\nh₁ : s.ExcenterExists signs₁\nh₂ : s.ExcenterExists signs₂\nh : s.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 675,
"column": 26
} | {
"line": 675,
"column": 34
} | [
{
"pp": "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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns₁ signs₂ : Finset (Fin (n + 1))\nh₁ : s.ExcenterExists signs₁\nh₂ : s.ExcenterExists signs₂\nh : s.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 675,
"column": 26
} | {
"line": 675,
"column": 34
} | [
{
"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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns₁ signs₂ : Finset (Fin (n + 1))\nh₁ : s.ExcenterExists signs₁\nh₂ : s.ExcenterExists signs₂\nh : s.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 675,
"column": 26
} | {
"line": 675,
"column": 34
} | [
{
"pp": "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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns₁ signs₂ : Finset (Fin (n + 1))\nh₁ : s.ExcenterExists signs₁\nh₂ : s.ExcenterExists signs₂\nh : s.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 675,
"column": 26
} | {
"line": 675,
"column": 34
} | [
{
"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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns₁ signs₂ : Finset (Fin (n + 1))\nh₁ : s.ExcenterExists signs₁\nh₂ : s.ExcenterExists signs₂\nh : s.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 712,
"column": 4
} | {
"line": 712,
"column": 12
} | [
{
"pp": "case inr\nV : 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\ni j : Fin (n + 1)\nthis : 2 ≤ n\nk : Fin (n + 1)\nhki : k ≠ i\nhkj : k ≠ j\nhij :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 219,
"column": 28
} | {
"line": 219,
"column": 36
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx z : V\nkx kz : ℝ≥0\nhy : kx • x + kz • z ≠ 0\nhkx : 0 < kx\nhkz : 0 < kz\nhz : z = 0\n⊢ x ≠ 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"False",
"Real",
"instHSMul",
"eq_false",... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 277,
"column": 4
} | {
"line": 279,
"column": 61
} | [
{
"pp": "case mpr\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y z : V\nhy : y ≠ 0\nh : angle x z = π ∨ y ∈ Submodule.span ℝ≥0 {x, z}\n⊢ angle x z = angle x y + angle y z",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Submodule",
"Real",
"R... | obtain h | h := h
· grind [angle_add_angle_eq_pi_of_angle_eq_pi, angle_comm]
· exact angle_eq_angle_add_add_angle_add_of_mem_span hy h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 277,
"column": 4
} | {
"line": 279,
"column": 61
} | [
{
"pp": "case mpr\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y z : V\nhy : y ≠ 0\nh : angle x z = π ∨ y ∈ Submodule.span ℝ≥0 {x, z}\n⊢ angle x z = angle x y + angle y z",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Submodule",
"Real",
"R... | obtain h | h := h
· grind [angle_add_angle_eq_pi_of_angle_eq_pi, angle_comm]
· exact angle_eq_angle_add_add_angle_add_of_mem_span hy h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Congruence | {
"line": 105,
"column": 49
} | {
"line": 105,
"column": 65
} | [
{
"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... | Set.insert_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1261,
"column": 2
} | {
"line": 1267,
"column": 25
} | [
{
"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\nsigns : Finset (Fin 3)\n⊢ Simplex.excenter t signs = Simplex.incenter t ∨ ∃ i, Simplex.excenter t signs = Simplex.excenter t {i}",
"usedC... | have h : signs = ∅ ∨ signs = Finset.univ ∨ ∃ i, signs = {i} ∨ signs = {i}ᶜ := by decide +revert
rcases h with rfl | rfl | ⟨i, rfl | rfl⟩
· exact .inl rfl
· exact .inl t.excenter_univ
· exact .inr ⟨i, rfl⟩
· refine .inr ⟨i, ?_⟩
rw [t.excenter_compl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1261,
"column": 2
} | {
"line": 1267,
"column": 25
} | [
{
"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\nsigns : Finset (Fin 3)\n⊢ Simplex.excenter t signs = Simplex.incenter t ∨ ∃ i, Simplex.excenter t signs = Simplex.excenter t {i}",
"usedC... | have h : signs = ∅ ∨ signs = Finset.univ ∨ ∃ i, signs = {i} ∨ signs = {i}ᶜ := by decide +revert
rcases h with rfl | rfl | ⟨i, rfl | rfl⟩
· exact .inl rfl
· exact .inl t.excenter_univ
· exact .inr ⟨i, rfl⟩
· refine .inr ⟨i, ?_⟩
rw [t.excenter_compl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1342,
"column": 6
} | {
"line": 1342,
"column": 33
} | [
{
"pp": "case h.e'_4.h.e'_10\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\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\n⊢ Simplex.face t ⋯ = Simplex.faceOpposite t i₂",
"use... | Affine.Simplex.faceOpposite | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1350,
"column": 6
} | {
"line": 1350,
"column": 33
} | [
{
"pp": "case h.e'_4.h.e'_10\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\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\n⊢ Simplex.face t ⋯ = Simplex.faceOpposite t i₂",
"use... | Affine.Simplex.faceOpposite | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Similarity | {
"line": 251,
"column": 36
} | {
"line": 254,
"column": 7
} | [
{
"pp": "ι : Type u_1\nP₁ : Type u_3\nP₂ : Type u_4\nv₁ : ι → P₁\nv₂ : ι → P₂\ninst✝¹ : PseudoMetricSpace P₁\ninst✝ : PseudoMetricSpace P₂\n⊢ Similar v₁ v₂ ↔ ∃ r, 0 < r ∧ Pairwise fun i₁ i₂ ↦ dist (v₁ i₁) (v₁ i₂) = r * dist (v₂ i₁) (v₂ i₂)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"LinearO... | by
simp_rw [similar_iff_exists_pairwise_dist_eq]
simp_rw [← pos_iff_ne_zero, NNReal.exists, ← NNReal.coe_pos, NNReal.coe_mk]
grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Similarity | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 46
} | [
{
"pp": "P₁ : Type u_3\nP₂ : Type u_4\ninst✝¹ : PseudoMetricSpace P₁\ninst✝ : PseudoMetricSpace P₂\na b c : P₁\na' b' c' : P₂\nh_ne : dist a b ≠ 0\nh_ne' : dist a' b' ≠ 0\nheq1 : dist a b * dist b' c' = dist b c * dist a' b'\nheq2 : dist a b * dist c' a' = dist c a * dist a' b'\nr : ℝ := dist a b / dist a' b'\n... | apply Similar.of_exists_pos_pairwise_dist_eq | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Euclidean.NinePointCircle | {
"line": 82,
"column": 2
} | {
"line": 84,
"column": 33
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\nV₂ : Type u_3\nP₂ : Type u_4\ninst✝³ : NormedAddCommGroup V₂\ninst✝² : InnerProductSpace ℝ V₂\ninst✝¹ : MetricSpace P₂\ninst✝ : NormedAddTorsor V₂ P₂\nn : ℕ\n... | ext
· simp [ninePointCircle_center, centroid_map]
· simp [ninePointCircle_radius] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.NinePointCircle | {
"line": 82,
"column": 2
} | {
"line": 84,
"column": 33
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MetricSpace P\ninst✝⁴ : NormedAddTorsor V P\nV₂ : Type u_3\nP₂ : Type u_4\ninst✝³ : NormedAddCommGroup V₂\ninst✝² : InnerProductSpace ℝ V₂\ninst✝¹ : MetricSpace P₂\ninst✝ : NormedAddTorsor V₂ P₂\nn : ℕ\n... | ext
· simp [ninePointCircle_center, centroid_map]
· simp [ninePointCircle_radius] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.Power | {
"line": 53,
"column": 2
} | {
"line": 55,
"column": 14
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y z : V\nh₃ : ‖z - y‖ = ‖z + y‖\nr : ℝ\nhr : x = r • y\n⊢ ‖x - y‖ * ‖x + y‖ = |‖z + y‖ ^ 2 - ‖z - x‖ ^ 2|",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"Real.pi",
"Inn... | have hzy : ⟪z, y⟫ = 0 := by
rwa [inner_eq_zero_iff_angle_eq_pi_div_two, ← norm_add_eq_norm_sub_iff_angle_eq_pi_div_two,
eq_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Sphere.SecondInter | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 31
} | [
{
"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\ns : Sphere P\nn : ℕ\ninst✝ : n.AtLeastTwo\nsx : Affine.Simplex ℝ P n\ni : Fin (n + 1)\nhi : sx.points i ∈ s\nhsx : ∀ (j : Fin (n + 1)), dist (sx.points j) s.c... | Affine.Simplex.faceOpposite | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 249,
"column": 4
} | {
"line": 253,
"column": 45
} | [
{
"pp": "case neg\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)\ni₁ i₂ : Fin (n + 3)\nh : ¬i₁ = i₂\nhs : ∑ i, (pointWeightsWithCircumcenter i₁ - pointWeightsWithCircumcenter i₂) i = ... | · simp_rw [fs, sum_insert (notMem_singleton.2 h), sum_singleton]
repeat rw [← sum_subset fs.subset_univ _]
· simp_rw [fs, sum_insert (notMem_singleton.2 h), sum_singleton]
simp [h, Ne.symm h, dist_comm (s.points i₁)]
all_goals intro i _ hi; simp [hfs i hi] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Sphere.Power | {
"line": 306,
"column": 42
} | {
"line": 311,
"column": 6
} | [
{
"pp": "V : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_2\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b t p : P\ns : Sphere P\nha : a ∈ s\nhb : b ∈ s\nhp : p ∈ affineSpan ℝ {a, b}\nh_tangent : s.IsTangentAt t (affineSpan ℝ {p, t})\n⊢ dist p t ^ 2 = dist p a *... | by
have hr := radius_nonneg_of_mem ha
have radius_le_dist := h_tangent.isTangent.radius_le_dist_center (left_mem_affineSpan_pair ℝ p t)
rw [mul_dist_eq_power_of_radius_le_dist_center hr hp ha hb radius_le_dist,
Sphere.power, h_tangent.dist_sq_eq_of_mem (left_mem_affineSpan_pair ℝ p t)]
ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 511,
"column": 8
} | {
"line": 511,
"column": 11
} | [
{
"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\nt₁ t₂ : Triangle ℝ P\ni₁ i₂ i₃ j₁ j₂ j₃ : Fin 3\nhi₁₂ : i₁ ≠ i₂\nhi₁₃ : i₁ ≠ i₃\nhi₂₃ : i₂ ≠ i₃\nhj₁₂ : j₁ ≠ j₂\nhj₁₃ : j₁ ≠ j₃\nhj₂₃ : j₂ ≠ j₃\... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Group.Growth.QuotientInter | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\ninst✝² : DecidableEq G\nH : Subgroup G\ninst✝¹ : DecidablePred fun x ↦ x ∈ H\ninst✝ : H.Normal\nA : Finset G\nm n : ℕ\nπ : G →* G ⧸ H := QuotientGroup.mk' H\nφ : G ⧸ H → G := invFunOn (⇑π) (↑A ^ m)\nhφ : Set.InjOn φ (⇑π '' ↑A ^ m)\nhφA : ∀ {a : G ⧸ H}, a ∈ ⇑π '' ↑A ^ m →... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 204,
"column": 61
} | {
"line": 204,
"column": 79
} | [
{
"pp": "case neg\nX : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n ... | ENNReal.add_ne_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Group.Growth.QuotientInter | {
"line": 78,
"column": 32
} | {
"line": 78,
"column": 52
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\ninst✝² : DecidableEq G\nH : Subgroup G\ninst✝¹ : DecidablePred fun x ↦ x ∈ H\ninst✝ : H.Normal\nA : Finset G\nhAsymm : A⁻¹ = A\nπ : G →* G ⧸ H := QuotientGroup.mk' H\na : G\nha : a ∈ A\n⊢ #({x ∈ A⁻¹ * A | x ∈ H}) = #({x ∈ A ^ 2 | x ∈ H})",
"usedConstants": [
"H... | by simp [hAsymm, sq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorBundle.Tangent | {
"line": 454,
"column": 2
} | {
"line": 460,
"column": 53
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_4\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\n⊢ ContMDiff I.tangent 𝓘(𝕜, E) n fun p ↦ p.snd",
"usedConstants": [
"ContMDiff.comp",
... | change ContMDiff I.tangent 𝓘(𝕜, E) n
((id Prod.snd : ModelProd H E → E) ∘ (tangentBundleModelSpaceHomeomorph I))
apply ContMDiff.comp (I' := I.prod 𝓘(𝕜, E))
· convert! contMDiff_snd
rw [chartedSpaceSelf_prod]
rfl
· exact contMDiff_tangentBundleModelSpaceHomeomorph | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.Tangent | {
"line": 454,
"column": 2
} | {
"line": 460,
"column": 53
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_4\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\n⊢ ContMDiff I.tangent 𝓘(𝕜, E) n fun p ↦ p.snd",
"usedConstants": [
"ContMDiff.comp",
... | change ContMDiff I.tangent 𝓘(𝕜, E) n
((id Prod.snd : ModelProd H E → E) ∘ (tangentBundleModelSpaceHomeomorph I))
apply ContMDiff.comp (I' := I.prod 𝓘(𝕜, E))
· convert! contMDiff_snd
rw [chartedSpaceSelf_prod]
rfl
· exact contMDiff_tangentBundleModelSpaceHomeomorph | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Diffeomorph | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 33
} | [
{
"pp": "case h.e'_12\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nH : Type u_5\ninst✝⁵ : TopologicalSpace H\nG : Type u_7\ninst✝⁴ : TopologicalSpace G\nI : Mo... | simp [h.image_eq_preimage_symm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.Complex | {
"line": 59,
"column": 34
} | {
"line": 78,
"column": 86
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℂ F\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℂ E H\ninst✝³ : I.Boundaryless\nM : Type u_4\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace ... | by
set e := extChartAt I c
have hI : range I = univ := ModelWithCorners.Boundaryless.range_eq_univ
have H₁ : 𝓝[range I] (e c) = 𝓝 (e c) := by rw [hI, nhdsWithin_univ]
have H₂ : map e.symm (𝓝 (e c)) = 𝓝 c := by
rw [← map_extChartAt_symm_nhdsWithin_range (I := I) c, H₁]
rw [← H₂, eventually_map]
repla... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation | {
"line": 207,
"column": 75
} | {
"line": 207,
"column": 94
} | [
{
"pp": "case H\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\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : ChartedSpace H G\ninst✝¹ : Monoid G\ninst✝ : Co... | fdifferential_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.MFDeriv.Tangent | {
"line": 89,
"column": 6
} | {
"line": 91,
"column": 20
} | [
{
"pp": "case e_f.e_f.hc\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 : IsManifo... | apply mdifferentiableWithinAt_extChartAt_symm
apply (extChartAt I (f x₀)).map_source
simpa using hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.Tangent | {
"line": 89,
"column": 6
} | {
"line": 91,
"column": 20
} | [
{
"pp": "case e_f.e_f.hc\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 : IsManifo... | apply mdifferentiableWithinAt_extChartAt_symm
apply (extChartAt I (f x₀)).map_source
simpa using hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 10
} | [
{
"pp": "case h\n𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝²¹ : NontriviallyNormedField 𝕜\ninst✝²⁰ : NormedAddCommGroup F\ninst✝¹⁹ : NormedSpace 𝕜 F\ninst✝¹⁸ : TopologicalSpace (TotalSpace F E)\ninst✝¹⁷ : (x : B) → TopologicalSpace (E x)\nEB : Type u_7\ninst✝¹⁶ : NormedAd... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 496,
"column": 15
} | {
"line": 496,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\ninst✝²⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁹ : NormedAddCommGroup E\ninst✝¹⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁶ : TopologicalSpace M\ninst✝¹⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝¹⁴ : Normed... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 496,
"column": 52
} | {
"line": 496,
"column": 60
} | [
{
"pp": "𝕜 : Type u_1\ninst✝²⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁹ : NormedAddCommGroup E\ninst✝¹⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁶ : TopologicalSpace M\ninst✝¹⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝¹⁴ : Normed... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 497,
"column": 2
} | {
"line": 497,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\ninst✝²⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁹ : NormedAddCommGroup E\ninst✝¹⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁶ : TopologicalSpace M\ninst✝¹⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝¹⁴ : Normed... | rw [writtenInExtChartAt_prod] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 560,
"column": 2
} | {
"line": 564,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace (TotalSpace F E)\ninst✝¹¹ : (x : B) → TopologicalSpace (E x)\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 F\ninst✝⁷ : FiberBundle F E\ni... | have : {x | t x y ≠ 0} ⊆ {i | ((fun i ↦ {x | t i x ≠ 0}) i ∩ U).Nonempty} := by
intro x hx
rw [Set.mem_setOf] at hx ⊢
use y
simpa using ⟨hx, mem_of_mem_nhds hu⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 143,
"column": 55
} | {
"line": 143,
"column": 68
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nV : (x : M) → TangentSpace I x... | Pi.zero_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nV : (x : M) → TangentSpace I x\n⊢ mlie... | ext x; simp_rw [mlieBracket, mlieBracketWithin_self, Pi.zero_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nV : (x : M) → TangentSpace I x\n⊢ mlie... | ext x; simp_rw [mlieBracket, mlieBracketWithin_self, Pi.zero_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 10
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\ns t : Set M\nx : M\nV W : (x :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
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