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.FieldTheory.PerfectClosure | {
"line": 172,
"column": 10
} | {
"line": 172,
"column": 76
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\ne : PerfectClosure K p\nx✝ : ℕ × K\nn : ℕ\nx : K\n⊢ ((n, x).1 + (0, 1).1, (⇑(frobenius K p))^[(0, 1).1] (n, x).2 * (⇑(frobenius K p))^[(n, x).1] (0, 1).2) = (n, x)",
"usedConstants": [
"iterate_map_one",
... | simp only [iterate_map_one, iterate_zero_apply, mul_one, add_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Invariant.Basic | {
"line": 543,
"column": 2
} | {
"line": 543,
"column": 23
} | [
{
"pp": "G : Type u_1\nA : Type u_2\nB : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝¹⁸ : Group G\ninst✝¹⁷ : CommRing A\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : MulSemiringAction G B\ninst✝¹⁴ : Algebra A B\ninst✝¹³ : Field K\ninst✝¹² : Field L\ninst✝¹¹ : Algebra K L\ninst✝¹⁰ : Algebra A K\ninst✝⁹ : Algebra B L\ninst✝⁸ :... | refine ⟨fun x h ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.PerfectClosure | {
"line": 352,
"column": 14
} | {
"line": 353,
"column": 38
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℕ × K\nn : ℕ\nih : mk K p x ^ n = mk K p (x.1, x.2 ^ n)\n⊢ (⇑(frobenius K p))^[(x.1, x.2 ^ n * x.2).1 + 0]\n ((x.1, x.2 ^ n).1 + x.1,\n (⇑(frobenius K p))^[x.1] (x.1, x.2 ^ n).2 * (⇑(frobenius K p)... | by simp_rw [iterate_frobenius, add_zero, mul_pow, ← pow_mul,
← pow_add, mul_assoc, ← pow_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.IntegralClosure.IntegralRestrict | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 21
} | [
{
"pp": "case neg\nA : Type u_1\nB : Type u_6\ninst✝²⁴ : CommRing A\ninst✝²³ : CommRing B\ninst✝²² : Algebra A B\nAₘ : Type u_9\nBₘ : Type u_10\ninst✝²¹ : CommRing Aₘ\ninst✝²⁰ : CommRing Bₘ\ninst✝¹⁹ : Algebra Aₘ Bₘ\ninst✝¹⁸ : Algebra A Aₘ\ninst✝¹⁷ : Algebra B Bₘ\ninst✝¹⁶ : Algebra A Bₘ\ninst✝¹⁵ : IsScalarTower ... | letI := g.toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.RingTheory.IntegralClosure.IntegralRestrict | {
"line": 438,
"column": 47
} | {
"line": 438,
"column": 77
} | [
{
"pp": "case h.a\nA : Type u_1\nB : Type u_6\ninst✝¹⁰ : CommRing A\ninst✝⁹ : CommRing B\ninst✝⁸ : Algebra A B\ninst✝⁷ : IsIntegrallyClosed A\ninst✝⁶ : IsDomain A\ninst✝⁵ : IsDomain B\ninst✝⁴ : IsIntegrallyClosed B\ninst✝³ : Algebra.IsIntegral A B\ninst✝² : IsTorsionFree A B\ninst✝¹ : Free A B\ninst✝ : Module.F... | Algebra.norm_localization A A⁰ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IntegralClosure.IntegralRestrict | {
"line": 492,
"column": 2
} | {
"line": 492,
"column": 21
} | [
{
"pp": "case neg\nA : Type u_1\nB : Type u_6\ninst✝²⁵ : CommRing A\ninst✝²⁴ : CommRing B\ninst✝²³ : Algebra A B\nAₘ : Type u_9\nBₘ : Type u_10\ninst✝²² : CommRing Aₘ\ninst✝²¹ : CommRing Bₘ\ninst✝²⁰ : Algebra Aₘ Bₘ\ninst✝¹⁹ : Algebra A Aₘ\ninst✝¹⁸ : Algebra B Bₘ\ninst✝¹⁷ : Algebra A Bₘ\ninst✝¹⁶ : IsScalarTower ... | letI := g.toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.GroupTheory.CosetCover | {
"line": 239,
"column": 4
} | {
"line": 240,
"column": 91
} | [
{
"pp": "case hx\nG : Type u_1\ninst✝¹ : Group G\nι : Type u_2\nH : ι → Subgroup G\ng : ι → G\ns : Finset ι\ninst✝ : DecidablePred FiniteIndex\nhcovers : ⋃ i ∈ {i ∈ s | (H i).FiniteIndex} ∪ {a ∈ s | ¬(H a).FiniteIndex}, g i • ↑(H i) = Set.univ\nD : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k\nhD : D.Fi... | · by_cases hfi : (H i).FiniteIndex <;>
simp [← Set.smul_set_iUnion₂, Set.iUnion_subtype, ← leftCoset_assoc, f, K, ht, hfi] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.CosetCover | {
"line": 298,
"column": 8
} | {
"line": 298,
"column": 43
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nι : Type u_2\nH : ι → Subgroup G\ng : ι → G\ns : Finset ι\nhcovers : ⋃ i ∈ s, g i • ↑(H i) = Set.univ\ninst✝ : DecidablePred FiniteIndex\nD : Subgroup G := ⨅ k ∈ {i ∈ s | (H i).FiniteIndex}, H k\nhD : D.FiniteIndex\nhD_le : ∀ {i : ι}, i ∈ s → (H i).FiniteIndex → D ≤ H i\... | simpa [K, f, if_pos hi.2] using hxr | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.FieldTheory.PurelyInseparable.Exponent | {
"line": 152,
"column": 39
} | {
"line": 152,
"column": 53
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsPurelyInseparable K L\na : L\n⊢ (X ^ ringExpChar K ^ elemExponent K a).natDegree = ringExpChar K ^ elemExponent K a",
"usedConstants": [
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"HMul.hM... | natDegree_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PurelyInseparable.Exponent | {
"line": 189,
"column": 27
} | {
"line": 189,
"column": 41
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsPurelyInseparable K L\na : L\nn : ℕ\nh : a ^ ringExpChar K ^ n ∈ (algebraMap K L).range\np : ℕ\nh✝ : ExpChar K p\nhp : Nat.Prime p\nhchar✝ : CharP K p\ny : K\nhy : (algebraMap K L) y = a ^ ringExpChar K ^ n\... | natDegree_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PurelyInseparable.Exponent | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 65
} | [
{
"pp": "case zero\nF : Type u_1\nK : Type u_2\nL : Type u_3\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : IsPurelyInseparable K L\ninst✝¹ : FiniteDimensional K L\ninst✝ : CharZero K\nh✝ : ExpChar K 1\n⊢ HasExponent K L",
"usedConstants": [
"Subring.instSetLike",
"Algebra.a... | · exact ⟨0, fun a ↦ surjective_algebraMap_of_isSeparable K L _⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.LinearDisjoint | {
"line": 440,
"column": 24
} | {
"line": 440,
"column": 59
} | [
{
"pp": "case H\nF : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nA B : IntermediateField F E\ninst✝¹ : IsGalois F ↥A\ninst✝ : FiniteDimensional F E\nh₁ : A ⊔ B = ⊤\nh₂ : A ⊓ B = ⊥\n⊢ finrank F E = finrank F ↥A * finrank F ↥B",
"usedConstants": [
"Eq.mpr",
"Inter... | ← Module.finrank_mul_finrank F B E, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PurelyInseparable.Tower | {
"line": 80,
"column": 51
} | {
"line": 83,
"column": 60
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁷ : Field F\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nK : Type w\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower F E K\ninst✝ : IsPurelyInseparable F E\nι : Type u_1\nv : ι → K\nhsep : ∀ (i : ι), IsSeparable F (v i)\nh : LinearIndependen... | by
contrapose
refine fun hs ↦ (injective_iff_map_eq_zero _).mp (algebraMap F E).injective _ ?_
rw [hlF, Finsupp.notMem_support_iff.1 hs, zero_pow this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PurelyInseparable.PerfectClosure | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 9
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\nx : E\nn : ℕ\nhx : x ^ ringExpChar F ^ n ∈ (algebraMap F E).rangeS\n⊢ x⁻¹ ∈\n (have this := ⋯;\n Subalgebra.perfectClosure F E (ringExpChar F)).carrier",
"use... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.FieldTheory.RatFunc.IntermediateField | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 42
} | [
{
"pp": "case h.e'_3\nK : Type u_1\ninst✝ : Field K\nf : K⟮X⟯\nhf : ¬∃ c, f = C c\ne : K[X] ≃ₐ[K] ↥K[f] := Polynomial.algEquivOfTranscendental K f ⋯\nφ : K[X][X] :=\n Polynomial.map (algebraMap K K[X]) f.num - Polynomial.C Polynomial.X * Polynomial.map (algebraMap K K[X]) f.denom\nφ_map : (mapEquiv e.toRingEqu... | rw [add_comm, X_mul_C, map_neg, neg_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Relrank | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 52
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B : Subfield E\nh : A ≤ B\n⊢ A.relfinrank B * finrank (↥B) E = finrank (↥A) E",
"usedConstants": [
"Nat.instMulZeroOneClass",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Subfield.relrank",
"Semiring.toModule",
"HMul.hMul",
"Co... | simpa using congr(toNat $(relrank_mul_rank_top h)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.FieldTheory.Relrank | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 52
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B : Subfield E\nh : A ≤ B\n⊢ A.relfinrank B * finrank (↥B) E = finrank (↥A) E",
"usedConstants": [
"Nat.instMulZeroOneClass",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Subfield.relrank",
"Semiring.toModule",
"HMul.hMul",
"Co... | simpa using congr(toNat $(relrank_mul_rank_top h)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Relrank | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 52
} | [
{
"pp": "E : Type v\ninst✝ : Field E\nA B : Subfield E\nh : A ≤ B\n⊢ A.relfinrank B * finrank (↥B) E = finrank (↥A) E",
"usedConstants": [
"Nat.instMulZeroOneClass",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Subfield.relrank",
"Semiring.toModule",
"HMul.hMul",
"Co... | simpa using congr(toNat $(relrank_mul_rank_top h)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Relrank | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 52
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nA B : IntermediateField F E\nh : A ≤ B\n⊢ A.relfinrank B * finrank (↥B) E = finrank (↥A) E",
"usedConstants": [
"Nat.instMulZeroOneClass",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.to... | simpa using congr(toNat $(relrank_mul_rank_top h)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.FieldTheory.Relrank | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 52
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nA B : IntermediateField F E\nh : A ≤ B\n⊢ A.relfinrank B * finrank (↥B) E = finrank (↥A) E",
"usedConstants": [
"Nat.instMulZeroOneClass",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.to... | simpa using congr(toNat $(relrank_mul_rank_top h)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Relrank | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 52
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nA B : IntermediateField F E\nh : A ≤ B\n⊢ A.relfinrank B * finrank (↥B) E = finrank (↥A) E",
"usedConstants": [
"Nat.instMulZeroOneClass",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.to... | simpa using congr(toNat $(relrank_mul_rank_top h)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.Luroth | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 97
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nE : IntermediateField K K⟮X⟯\nh : E ≠ ⊥\n⊢ (algebraMap K[X] K⟮X⟯) (c E).num * generator E ≠ 0",
"usedConstants": [
"Iff.mpr",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"_private.Mathlib.FieldTheory.RatFunc.Luroth.0.RatFunc.Luroth.c",
"Al... | exact mul_ne_zero_iff.mpr ⟨algebraMap_ne_zero (num_ne_zero (c_ne_zero h)), generator_ne_zero h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 178,
"column": 94
} | {
"line": 181,
"column": 77
} | [
{
"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\nm n : ℕ\ninst✝¹ : NeZero m\ninst✝ : NeZero n\ns : Simplex ℝ P n\ne : Fin (n + 1) ≃ Fin (m + 1)\n⊢ (s.reindex e).altitudeFoot = s.altitudeFoot ∘ ⇑e.symm",
... | by
ext i
simp only [altitudeFoot, reindex_points, Function.comp_apply]
exact orthogonalProjectionSpan_congr (s.range_faceOpposite_reindex e i) rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Altitude | {
"line": 290,
"column": 2
} | {
"line": 295,
"column": 6
} | [
{
"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 j : Fin (n + 1)\nh : i ≠ j\nthis : NeZero n\n⊢ have this := ⋯;\n ⟪s.points j -ᵥ s.altitudeFoot i, s.points i -ᵥ s.altitudeFoot i⟫ ... | refine Submodule.inner_right_of_mem_orthogonal
(K := vectorSpan ℝ (s.points '' {i}ᶜ))
(vsub_mem_vectorSpan_of_mem_affineSpan_of_mem_affineSpan
(s.mem_affineSpan_image_iff.2 h.symm)
(Affine.Simplex.altitudeFoot_mem_affineSpan_image_compl _ _))
?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 44
} | [
{
"pp": "case «0»\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)\nθ : Real.Angle\nx : V\nhx : x ≠ 0\n⊢ ↑(o.rotation θ).toLinearEquiv ((o.basisRightAngleRotation x hx) ((fun i ↦ i) ⟨0, ⋯⟩)) =\n ((Matrix.toLin (o.basisRi... | · rw [Matrix.toLin_self]
simp [rotation_apply, Fin.sum_univ_succ] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Projection | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np : P\nx : ↑↑s\n⊢ dist p ↑((or... | simp [mul_self_nonneg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Projection | {
"line": 429,
"column": 24
} | {
"line": 429,
"column": 52
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ns : AffineSubspace 𝕜 P\ninst✝¹ : Nonempty ↥s\ninst✝ : s.direction.HasOrthogonalProjection\np : P\n⊢ s.direction.reflectio... | orthogonalProjection_apply', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 303,
"column": 61
} | {
"line": 304,
"column": 46
} | [
{
"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\nr : ℝ\nhr : r ≠ 0\n⊢ 2 • o.oangle (r • x) y = 2 • o.oangle x y",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NegZeroClass.toNeg",... | by
rcases hr.lt_or_gt with (h | h) <;> simp [h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 310,
"column": 61
} | {
"line": 311,
"column": 46
} | [
{
"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\nr : ℝ\nhr : r ≠ 0\n⊢ 2 • o.oangle x (r • y) = 2 • o.oangle x y",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NegZeroClass.toNeg",... | by
rcases hr.lt_or_gt with (h | h) <;> simp [h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 331,
"column": 51
} | {
"line": 331,
"column": 77
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\n⊢ ‖p₁ -ᵥ p₃‖ * ‖p₁ -ᵥ p₃‖ = dist p₁ p₂ * dist p₁ p₂ + dist p₂ p₃ * dist p₂ p₃ ↔ ∠ p₁ p₂ p₃ = π / 2",
"usedConstants": [
"Norm.norm",
... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 349,
"column": 13
} | {
"line": 349,
"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\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃) = Real.arcsin (dist p₁ p₂ / di... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 358,
"column": 13
} | {
"line": 358,
"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\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0\n⊢ InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃) = Real.arctan (dist p₁ p₂ / dist p₃ p₂)",
... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 401,
"column": 13
} | {
"line": 401,
"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\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ Real.sin (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) = dist p₁ p₂ / dist... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 409,
"column": 13
} | {
"line": 409,
"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\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ Real.tan (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) = dist p₁ p₂ / dist p₃ p₂",
"usedConstants": [
... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 427,
"column": 13
} | {
"line": 427,
"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\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\n⊢ Real.sin (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) * dist p₁ p₃ = dist p₁ p₂",
"usedConstants": [
... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 437,
"column": 13
} | {
"line": 437,
"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\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ ≠ 0 ∨ p₁ -ᵥ p₂ = 0\n⊢ Real.tan (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) * dist p₃ p₂ = dist... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 444,
"column": 2
} | {
"line": 448,
"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\np₁ p₂ p₃ : P\nh : ∠ p₁ p₂ p₃ = π / 2\nh0 : p₁ = p₂ ∨ p₃ ≠ p₂\n⊢ dist p₃ p₂ / Real.cos (∠ p₂ p₃ p₁) = dist p₁ p₃",
"usedConstants": [
"NormedCommRing.... | rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←
inner_neg_left, neg_vsub_eq_vsub_rev] at h
rw [ne_comm, ← @vsub_ne_zero V, ← @vsub_eq_zero_iff_eq V, or_comm] at h0
rw [angle, dist_eq_norm_vsub' V p₃ p₂, dist_eq_norm_vsub V p₁ p₃, ← vsub_add_vsub_cancel p₁ p₂ p₃,
add_com... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 444,
"column": 2
} | {
"line": 448,
"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\np₁ p₂ p₃ : P\nh : ∠ p₁ p₂ p₃ = π / 2\nh0 : p₁ = p₂ ∨ p₃ ≠ p₂\n⊢ dist p₃ p₂ / Real.cos (∠ p₂ p₃ p₁) = dist p₁ p₃",
"usedConstants": [
"NormedCommRing.... | rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←
inner_neg_left, neg_vsub_eq_vsub_rev] at h
rw [ne_comm, ← @vsub_ne_zero V, ← @vsub_eq_zero_iff_eq V, or_comm] at h0
rw [angle, dist_eq_norm_vsub' V p₃ p₂, dist_eq_norm_vsub V p₁ p₃, ← vsub_add_vsub_cancel p₁ p₂ p₃,
add_com... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 457,
"column": 13
} | {
"line": 457,
"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\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ dist p₁ p₂ / Real.sin (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) = dist... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle | {
"line": 467,
"column": 13
} | {
"line": 467,
"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\np₁ p₂ p₃ : P\nh : ⟪p₂ -ᵥ p₃, p₁ -ᵥ p₂⟫ = 0\nh0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ dist p₁ p₂ / Real.tan (InnerProductGeometry.angle (p₂ -ᵥ p₃) (p₁ -ᵥ p₃)) = dist... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 562,
"column": 53
} | {
"line": 562,
"column": 83
} | [
{
"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\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ ⟪x, y⟫ / (‖x‖ * ‖y‖) = Real.cos (InnerProductGeometry.angle x y)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
... | InnerProductGeometry.cos_angle | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 613,
"column": 4
} | {
"line": 617,
"column": 25
} | [
{
"pp": "case pos\nV : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nw x y z : V\nh : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z\nhs : (o.oangle w x).sign = (o.oangle y z).sign\nh0 : (w = 0 ∨ x = 0) ∨ y ... | have hpi : π / 2 ≠ π := by
intro hpi
rw [div_eq_iff, eq_comm, ← sub_eq_zero, mul_two, add_sub_cancel_right] at hpi
· exact Real.pi_pos.ne.symm hpi
· exact two_ne_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 783,
"column": 42
} | {
"line": 783,
"column": 52
} | [
{
"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\nm : Fin (Nat.succ 0).succ → ℝ\nhm : m 0 ≠ 0 ∨ m 1 ≠ 0\nh : m 0 • x + (m 1 * r) • x + m 1 • y = 0\n⊢ (m 0 + m 1 * r) • x + m 1 • y = 0 ... | ← add_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 898,
"column": 6
} | {
"line": 898,
"column": 42
} | [
{
"pp": "V : Type u_1\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : Fact (finrank ℝ V = 2)\no : Orientation ℝ V (Fin 2)\nx y : V\n⊢ (o.oangle (y - x) y).sign = -(o.oangle x y).sign",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"in... | ← o.oangle_sign_smul_sub_left x y 1, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.SignedDist | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 77
} | [
{
"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\nv : V\np q : P\nh : ¬v = 0\n⊢ |⟪v, q -ᵥ p⟫| = ‖v‖ * ‖q -ᵥ p‖ ↔ ∃ a, a • v = q -ᵥ p",
"usedConstants": [
"Norm.norm",
"SeminormedAddGr... | rw [← Real.norm_eq_abs, ((norm_inner_eq_norm_tfae ℝ v (q -ᵥ p)).out 0 2 :)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Basic | {
"line": 105,
"column": 36
} | {
"line": 105,
"column": 62
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv : V\np₁ p₂ : P\nhv : v ≠ 0\nr : ℝ\n| ⟪v, v⟫ * (r * r) + 2 * ⟪v, p₁ -ᵥ p₂⟫ * r + (⟪p₁ -ᵥ p₂, p₁ -ᵥ p₂⟫ - dist p₁ p₂ * dist p₁ p₂) = 0",
"usedConstants": [... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Geometry.Euclidean.Angle.Oriented.Basic | {
"line": 959,
"column": 8
} | {
"line": 959,
"column": 18
} | [
{
"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\nhy : y ≠ 0\nr : ℝ\nhr0 : 0 ≤ r\nhr : r • y + 1 • y = x\n⊢ |(o.oangle (y - x) y).toReal| < π / 2",
"usedConstan... | ← add_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 13
} | [
{
"pp": "P : Type u_2\ninst✝ : MetricSpace P\ns : Sphere P\n⊢ { center := s.center, radius := s.radius } = s",
"usedConstants": [
"Real",
"EuclideanGeometry.Sphere.ext",
"EuclideanGeometry.Sphere.mk",
"EuclideanGeometry.Sphere.center",
"Eq.refl",
"EuclideanGeometry.Sphere... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 13
} | [
{
"pp": "P : Type u_2\ninst✝ : MetricSpace P\ns : Sphere P\n⊢ { center := s.center, radius := s.radius } = s",
"usedConstants": [
"Real",
"EuclideanGeometry.Sphere.ext",
"EuclideanGeometry.Sphere.mk",
"EuclideanGeometry.Sphere.center",
"Eq.refl",
"EuclideanGeometry.Sphere... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 13
} | [
{
"pp": "P : Type u_2\ninst✝ : MetricSpace P\ns : Sphere P\n⊢ { center := s.center, radius := s.radius } = s",
"usedConstants": [
"Real",
"EuclideanGeometry.Sphere.ext",
"EuclideanGeometry.Sphere.mk",
"EuclideanGeometry.Sphere.center",
"Eq.refl",
"EuclideanGeometry.Sphere... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.Basic | {
"line": 587,
"column": 4
} | {
"line": 587,
"column": 31
} | [
{
"pp": "case h₂\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np₁ p₂ : P\nx✝ : p₁ ∈ s ∧ p₂ ∈ s ∧ dist p₁ p₂ = 2 * s.radius\nh₁ : p₁ ∈ s\nh₂ : p₂ ∈ s\nhr : dist p₁ p₂ = 2 * s.radius\n⊢ dist p₂ p₁ = s.... | rw [dist_comm, hr, two_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 112,
"column": 2
} | {
"line": 113,
"column": 37
} | [
{
"pp": "case refine_2\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np q : P\nas : AffineSubspace ℝ P\nh : s.IsTangentAt p as\n⊢ q = p → q ∈ s ∧ q ∈ as",
"usedConstants": [
"InnerProductSpa... | · rintro rfl
exact ⟨h.mem_sphere, h.mem_space⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 132,
"column": 46
} | {
"line": 133,
"column": 84
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\nas : AffineSubspace ℝ P\np q : P\nh : s.IsTangentAt p as\nhq : q ∈ as\nhqp : q ≠ p\nthis : s.radius ^ 2 < dist q s.center ^ 2\n⊢ s.radius < dist ... | by
simpa [sq_lt_sq, abs_of_nonneg (s.radius_nonneg_of_mem h.mem_sphere)] using this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Sphere.OrthRadius | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 63
} | [
{
"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... | have hp : p ∈ s.orthRadius q := h (s.self_mem_orthRadius p) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 166,
"column": 58
} | {
"line": 170,
"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\ns : Sphere P\nas : AffineSubspace ℝ P\nh : s.IsTangent as\np : P\nhp : p ∈ as\n⊢ s.radius ≤ dist p s.center",
"usedConstants": [
"Real.instIsOrderedR... | by
obtain ⟨x, h⟩ := h
refine le_of_sq_le_sq ?_ dist_nonneg
rw [h.dist_sq_eq_of_mem hp, le_add_iff_nonneg_right]
exact sq_nonneg _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 446,
"column": 10
} | {
"line": 446,
"column": 42
} | [
{
"pp": "case neg.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 + s₂.radius ... | abs_of_nonneg (add_nonneg h₁ h₂) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 440,
"column": 6
} | {
"line": 452,
"column": 18
} | [
{
"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\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 + s₂.radius = 0\n⊢ s₁... | refine ⟨?_, ?_, ?_⟩
· simp only [mem_sphere, dist_lineMap_left, norm_div, Real.norm_eq_abs, h, abs_of_nonneg h₁,
abs_of_nonneg (add_nonneg h₁ h₂)]
field
· simp only [mem_sphere, dist_lineMap_right, Real.norm_eq_abs, h]
rw [one_sub_div h0, add_sub_cancel_left, abs_div, abs_of_nonneg... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Tangent | {
"line": 440,
"column": 6
} | {
"line": 452,
"column": 18
} | [
{
"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\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 + s₂.radius = 0\n⊢ s₁... | refine ⟨?_, ?_, ?_⟩
· simp only [mem_sphere, dist_lineMap_left, norm_div, Real.norm_eq_abs, h, abs_of_nonneg h₁,
abs_of_nonneg (add_nonneg h₁ h₂)]
field
· simp only [mem_sphere, dist_lineMap_right, Real.norm_eq_abs, h]
rw [one_sub_div h0, add_sub_cancel_left, abs_div, abs_of_nonneg... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Angle.Sphere | {
"line": 398,
"column": 4
} | {
"line": 398,
"column": 41
} | [
{
"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)\nt₁ t₂ : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nh₁ : t₁.point... | Real.Angle.tan_eq_of_two_zsmul_eq h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 155,
"column": 8
} | {
"line": 155,
"column": 54
} | [
{
"pp": "case h.right\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\nι : Type u_3\nhne : Nonempty ι\ninst✝ : Finite ι\np : ι → P\nha : AffineIndependent ℝ p\nval✝ : Fintype ι\nhm :\n ∀ {ι : Type u_3} [hne : Nonem... | replace hdist : 0 = cr := by simpa using hdist | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 34
} | [
{
"pp": "case inr.inr\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhy : y ≠ 0\nhx : x ≠ 0\n⊢ Real.sin (angle x y) * ‖x‖ = Real.sin (angle y (x - y)) * ‖x - y‖",
"usedConstants": [
"eq_or_ne"
]
}
] | obtain rfl | hxy := eq_or_ne x y | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 301,
"column": 2
} | {
"line": 315,
"column": 35
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Simplex ℝ P 1\n⊢ s.circumcenter = Finset.centroid ℝ univ s.points",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.... | have hr :
Set.Pairwise Set.univ fun i j : Fin 2 =>
dist (s.points i) (Finset.univ.centroid ℝ s.points) =
dist (s.points j) (Finset.univ.centroid ℝ s.points) := by
intro i hi j hj hij
rw [Finset.centroid_pair_fin, dist_eq_norm_vsub V (s.points i),
dist_eq_norm_vsub V (s.points j), vsub_va... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 301,
"column": 2
} | {
"line": 315,
"column": 35
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Simplex ℝ P 1\n⊢ s.circumcenter = Finset.centroid ℝ univ s.points",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.... | have hr :
Set.Pairwise Set.univ fun i j : Fin 2 =>
dist (s.points i) (Finset.univ.centroid ℝ s.points) =
dist (s.points j) (Finset.univ.centroid ℝ s.points) := by
intro i hi j hj hij
rw [Finset.centroid_pair_fin, dist_eq_norm_vsub V (s.points i),
dist_eq_norm_vsub V (s.points j), vsub_va... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 245,
"column": 33
} | {
"line": 245,
"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\np₁ p₂ p₃ : P\n⊢ ‖p₁ -ᵥ p₃‖ * ‖p₁ -ᵥ p₃‖ =\n dist p₁ p₂ * dist p₁ p₂ + dist p₃ p₂ * dist p₃ p₂ - 2 * dist p₁ p₂ * dist p₃ p₂ * Real.cos (∠ p₁ p₂ p₃)",
"u... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 45,
"column": 37
} | {
"line": 45,
"column": 75
} | [
{
"pp": "case h\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ ⟪x - (ℝ ∙ y).starProjection x, y⟫ = 0",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Inner.inner",
"Real.instRCLike",
"Submodule.starProject... | Submodule.starProjection_inner_eq_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 287,
"column": 6
} | {
"line": 287,
"column": 32
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : dist p₁ p₂ = dist p₁ p₃\n⊢ ∠ p₁ p₂ p₃ = ∠ p₁ p₃ p₂",
"usedConstants": [
"Norm.norm",
"Real",
"congrArg",
"AddComm... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 297,
"column": 6
} | {
"line": 297,
"column": 32
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : InnerProductGeometry.angle (p₁ -ᵥ p₂) (p₃ -ᵥ p₂) = InnerProductGeometry.angle (p₁ -ᵥ p₃) (p₂ -ᵥ p₃)\nhpi : InnerProductGeometry.angle (p₂ -ᵥ ... | dist_eq_norm_vsub V p₁ p₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 906,
"column": 2
} | {
"line": 906,
"column": 34
} | [
{
"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⊢ (s.exsphere signs).IsTangentAt (s.tou... | exact h.isTangentAt_touchpoint i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 251,
"column": 10
} | {
"line": 251,
"column": 28
} | [
{
"pp": "case pos\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\ny : V\nhy left✝ : y ≠ 0\nr : ℝ≥0\nhr : 0 < ↑r\nhxz₁ : ¬angle 0 (↑r • y) = π\nhxz₂ : ¬angle 0 (↑r • y) = 0\nh_sin_xz : Real.sin (angle 0 (↑r • y)) ≠ 0\n⊢ y ∈ Submodule.span ℝ≥0 {0, ↑r • y}",
"usedConstants": [
... | ← NNReal.smul_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1358,
"column": 7
} | {
"line": 1358,
"column": 34
} | [
{
"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⊢ Finset.univ = {i₁, i₂, ... | by clear hw; decide +revert | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.NinePointCircle | {
"line": 214,
"column": 6
} | {
"line": 214,
"column": 32
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Triangle ℝ P\ni : Fin 3\n⊢ s.points i -ᵥ Simplex.eulerPoint s i = s.points i -ᵥ midpoint ℝ s.orthocenter (s.points i)",
"usedConstants": [
"Eq.mp... | orthocenter_eq_mongePoint, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Sphere.Power | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 10
} | [
{
"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 c d p : P\nh : Cospherical {a, b, c, d}\nhapb : p ∈ affineSpan ℝ {a, b}\nhcpd : p ∈ affineSpan ℝ {c, d}\nq : P\nr : ℝ\nh' : ∀ p ∈ {a, b, c, d}, dist p q = ... | ← hd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 42
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P (n + 2)\ni₁ i₂ : Fin (n + 3)\n⊢ (∀ u ∈ ℝ ∙ (s.points i₁ -ᵥ s.points i₂), ⟪s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points, u⟫ = 0) ... | refine ⟨?_, s.mongePoint_mem_affineSpan⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 383,
"column": 6
} | {
"line": 383,
"column": 32
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nt : Triangle ℝ P\n⊢ t.orthocenter = 3 • (Finset.centroid ℝ univ t.points -ᵥ circumcenter t) +ᵥ circumcenter t",
"usedConstants": [
"Eq.mpr",
"I... | orthocenter_eq_mongePoint, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 417,
"column": 6
} | {
"line": 417,
"column": 32
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₂₃ : i₂ ≠ i₃\nh₁₃ : i₁ ≠ i₃\n⊢ t.orthocenter ∈ altitude t i₁",
"usedConstants": [
"Eq.mpr",
... | orthocenter_eq_mongePoint, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 704,
"column": 6
} | {
"line": 704,
"column": 45
} | [
{
"pp": "case inr.refine_2\nX : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nC r : ℝ≥0\nf : X → Y\ns : Set X\nh : HolderOnWith C r f s\nhr : 0 < r\nd : ℝ\nhd : 0 ≤ d\nhC0 : 0 < C\nhC... | refine ENNReal.tsum_le_tsum fun n => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 842,
"column": 2
} | {
"line": 842,
"column": 98
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\ne : X ≃ᵢ Y\nd : ℝ\n⊢ Measure.map ⇑e μH[d] = μH[d]",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real"... | rw [e.isometry.map_hausdorffMeasure (Or.inr e.surjective), e.surjective.range_eq, restrict_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 842,
"column": 2
} | {
"line": 842,
"column": 98
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\ne : X ≃ᵢ Y\nd : ℝ\n⊢ Measure.map ⇑e μH[d] = μH[d]",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real"... | rw [e.isometry.map_hausdorffMeasure (Or.inr e.surjective), e.surjective.range_eq, restrict_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 842,
"column": 2
} | {
"line": 842,
"column": 98
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\ne : X ≃ᵢ Y\nd : ℝ\n⊢ Measure.map ⇑e μH[d] = μH[d]",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real"... | rw [e.isometry.map_hausdorffMeasure (Or.inr e.surjective), e.surjective.range_eq, restrict_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 945,
"column": 8
} | {
"line": 945,
"column": 40
} | [
{
"pp": "ι : Type u_4\ninst✝ : Fintype ι\na b : ι → ℚ\nH : ∀ (i : ι), a i < b i\nI : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)\nγ : ℕ → Type u_4 := ⋯\nt : (n : ℕ) → γ n → Set (ι → ℝ) := ⋯\nA : Tendsto (fun n ↦ 1 / ↑n) atTop (𝓝 0)\nB : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), ediam (t n i) ≤ 1 / ↑n\nC : ∀ᶠ (n : ℕ) in atTop, (uni... | simp only [ENNReal.rpow_natCast] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 962,
"column": 8
} | {
"line": 963,
"column": 33
} | [
{
"pp": "ι : Type u_4\ninst✝ : Fintype ι\na b : ι → ℚ\nH : ∀ (i : ι), a i < b i\nI : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)\nγ : ℕ → Type u_4 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊\nt : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)\nA : Te... | simp only [ENNReal.ofReal_div_of_pos (Nat.cast_pos.mpr hn), comp_apply,
ENNReal.ofReal_natCast] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.LocalDiffeomorph | {
"line": 354,
"column": 6
} | {
"line": 354,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nH : Type u_4\ninst✝⁵ : TopologicalSpace H\nG : Type u_5\ninst✝⁴ : TopologicalSpace G\nI : ModelWithCorners... | obtain ⟨hx, hfx⟩ := hyp x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Geometry.Manifold.MFDeriv.Tangent | {
"line": 53,
"column": 2
} | {
"line": 59,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : IsManifold I 1 M\np : Tangent... | dsimp only [tangentMap]
rw [MDifferentiableAt.mfderiv (mdifferentiableAt_atlas_symm (chart_mem_atlas _ _) h)]
simp only [TangentBundle.chartAt, tangentBundleCore,
mfld_simps, (· ∘ ·)]
-- `simp` fails to apply `PartialEquiv.prod_symm` with `ModelProd`
congr
exact ((chartAt H (TotalSpace.proj p)).right_inv ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.Tangent | {
"line": 53,
"column": 2
} | {
"line": 59,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : IsManifold I 1 M\np : Tangent... | dsimp only [tangentMap]
rw [MDifferentiableAt.mfderiv (mdifferentiableAt_atlas_symm (chart_mem_atlas _ _) h)]
simp only [TangentBundle.chartAt, tangentBundleCore,
mfld_simps, (· ∘ ·)]
-- `simp` fails to apply `PartialEquiv.prod_symm` with `ModelProd`
congr
exact ((chartAt H (TotalSpace.proj p)).right_inv ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 167,
"column": 2
} | {
"line": 170,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\ns : Set M\np : TangentBundle I M\nhs :... | simp only [tangentMapWithin, id]
rw [mfderivWithin_id]
· rcases p with ⟨⟩; rfl
· exact hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 167,
"column": 2
} | {
"line": 170,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\ns : Set M\np : TangentBundle I M\nhs :... | simp only [tangentMapWithin, id]
rw [mfderivWithin_id]
· rcases p with ⟨⟩; rfl
· exact hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 701,
"column": 2
} | {
"line": 701,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁸ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : ChartedSpace H M\nF : Type u_5\ninst✝⁵ : NormedAddCom... | let w : F := (t ⟨x, σ₀⟩).2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Geometry.Manifold.GroupLieAlgebra | {
"line": 181,
"column": 2
} | {
"line": 183,
"column": 45
} | [
{
"pp": "case h.e'_22.h.h.e'_5.h\n𝕜 : 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\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : ChartedSpace H G\ninst✝¹ : Gro... | · simp only [comp_apply, tangentMap, F₃, F₂, F₁, fg, fv]
rw [mfderiv_prod_eq_add_apply ((contMDiff_mul I (minSmoothness 𝕜 3)).mdifferentiableAt M)]
simp +instances [mulInvariantVectorField] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 408,
"column": 2
} | {
"line": 408,
"column": 52
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\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 : NormedAddCom... | simp only [tangentMap_prodFst, tangentMap_prodSnd] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 482,
"column": 2
} | {
"line": 483,
"column": 81
} | [
{
"pp": "case h.inr.inr\nx✝ y✝ : ℝ\nhxy : Fact (x✝ < y✝)\nE : 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✝\nx y : ℝ\ninst✝ : Fact (x < y)\nn : ℕ∞ω\... | · -- `e = right chart`, `e' = right chart`
exact (mem_groupoid_of_pregroupoid.mpr (symm_trans_mem_contDiffGroupoid _)).1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 484,
"column": 4
} | {
"line": 484,
"column": 71
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nn : WithTop ℕ∞\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\nE' : Type ... | change fderivWithin 𝕜 (φ ∘ Prod.fst) _ _ _ = fderivWithin 𝕜 φ _ _ _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Geometry.Manifold.IntegralCurve.Basic | {
"line": 188,
"column": 63
} | {
"line": 201,
"column": 5
} | [
{
"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... | by
apply eventually_mem_nhds_iff.mpr
(hγ.continuousAt.preimage_mem_nhds (extChartAt_source_mem_nhds (I := I) _)) |>.and hγ |>.mono
rintro t ⟨ht1, ht2⟩
have hsrc := mem_of_mem_nhds ht1
rw [mem_preimage, extChartAt_source I (γ t₀)] at hsrc
rw [hasDerivAt_iff_hasFDerivAt, ← hasMFDerivAt_iff_hasFDerivAt]
ap... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 698,
"column": 2
} | {
"line": 698,
"column": 26
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹² : TopologicalSpace H\nE : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nH' : Type u_5\ninst✝⁷ :... | set s' := s ∩ u with hs' | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 73,
"column": 64
} | {
"line": 74,
"column": 73
} | [
{
"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\ninst✝ : (x : M) → ENorm (TangentSpace I x)\na b : ℝ\nγ : ℝ → M\n⊢ pathELength I γ a b = ∫⁻ ... | by
rw [pathELength_eq_lintegral_mfderiv_Icc, restrict_Ioo_eq_restrict_Icc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.Riemannian.Basic | {
"line": 112,
"column": 6
} | {
"line": 113,
"column": 44
} | [
{
"pp": "case h\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\nn : ℕ∞ω\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F... | simp only [Metric.mem_ball, dist_zero_right, norm_eq_sqrt_re_inner (𝕜 := ℝ),
RCLike.re_to_real, Set.mem_setOf_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Rank | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 37
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝³ : Group G\ninst✝² : Group H\ninst✝¹ : FG G\ninst✝ : FG H\nf : G →* H\nhf : Surjective ⇑f\n⊢ rank H ≤ rank G",
"usedConstants": [
"Group.rank_spec"
]
}
] | obtain ⟨S, hS1, hS2⟩ := rank_spec G | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic | {
"line": 266,
"column": 19
} | {
"line": 270,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁰ : TopologicalSpace M\ninst✝⁹ : ChartedSpace H M\nF : Type u_5\ninst✝⁸ : NormedAdd... | by
rw [← Finset.sum_add_distrib]
congr
ext i
rw [← smul_add, (h i).add hσ hσ' hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 36
} | [
{
"pp": "case r\nn : ℕ\na : ZMod n\n⊢ r a * 1 = r a",
"usedConstants": [
"ZMod.commRing",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
"Distrib.toAdd",
"ZMod",
"instDistribOfSemiring",
"congr_arg",
"Dihedra... | · exact congr_arg r (add_zero a) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 33
} | [
{
"pp": "case r\nn : ℕ\nhodd : Odd n\nhne1 : n ≠ 1\ni : ZMod n\nh : ∀ (g : DihedralGroup n), g * r i = r i * g\n⊢ r i = 1",
"usedConstants": [
"ZMod.commRing",
"AddGroupWithOne.toAddGroup",
"CommSemiring.toSemiring",
"HSub.hSub",
"Distrib.toAdd",
"DihedralGroup.sr.inj",
... | have heq := sr.inj (h (sr i)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.CommutingProbability | {
"line": 73,
"column": 44
} | {
"line": 77,
"column": 30
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Mul M\ninst✝ : Finite M\n⊢ commProb M ≤ 1",
"usedConstants": [
"div_le_one_of_le₀",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"Finite.card_subtype_le",
"MulZeroClass.toMul",
"IsStrictOrderedRing.toMulPosStrictM... | by
refine div_le_one_of_le₀ ?_ (sq_nonneg (Nat.card M : ℚ))
norm_cast
rw [sq, ← Nat.card_prod]
apply Finite.card_subtype_le | [anonymous] | Lean.Parser.Term.byTactic |
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