module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 576,
"column": 86
} | {
"line": 581,
"column": 62
} | {
"line": 583,
"column": 0
} | [
{
"pp": "n : ℕ\ni₁ i₂ : Fin (n + 1)\nh : i₁ ≠ i₂\n⊢ ∑ i, reflectionCircumcenterWeightsWithCircumcenter i₁ i₂ i = 1",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"M... | [] | by
simp_rw [sum_pointsWithCircumcenter, reflectionCircumcenterWeightsWithCircumcenter, sum_ite,
sum_const, filter_or, filter_eq']
rw [card_union_of_disjoint]
· norm_num
· simpa only [if_true, mem_univ, disjoint_singleton] using h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 799,
"column": 67
} | {
"line": 802,
"column": 45
} | {
"line": 804,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\nS : AffineSubspace ℝ P\nhS : affineSpan ℝ (Set.range s.p... | [] | by
rw [← s.excenterExists_restrict S hS] at h
haveI := Nonempty.map (AffineSubspace.inclusion hS) inferInstance
exact (h.touchpoint_map S.subtypeₐᵢ i).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 971,
"column": 2
} | {
"line": 971,
"column": 68
} | {
"line": 972,
"column": 2
} | [
{
"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)\n⊢ (∃ r, ∀ (i : Fin (n + 1)), dist p ↑((s.faceOpposite i).orthog... | [
"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)\n⊢ (∃ r, ∀ (i : Fin (n + 1)), |(s.signedInfDist i) p| = r) ↔ ∃ signs, s.Exce... | simp_rw [← abs_signedInfDist_eq_dist_of_mem_affineSpan_range _ hp] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Euclidean.Angle.Unoriented.TriangleInequality | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 89
} | {
"line": 169,
"column": 2
} | [
{
"pp": "case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y z : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\nhz : ‖z‖ = 1\nH : angle x z ≠ π\nH0 : angle x z = angle x y + angle y z\nH1 : ¬angle x z = 0\nHxz : Real.sin (angle x z) ≠ 0\nH2 : ¬angle x y = 0\nH3 : ¬angle y z = 0\nH4 : ¬angl... | [
"case neg\nV : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y z : V\nhx : ‖x‖ = 1\nhy : ‖y‖ = 1\nhz : ‖z‖ = 1\nH : angle x z ≠ π\nH0 : angle x z = angle x y + angle y z\nH1 : ¬angle x z = 0\nHxz : Real.sin (angle x z) ≠ 0\nH2 : ¬angle x y = 0\nH3 : ¬angle y z = 0\nH4 : ¬angle x y = π\nH... | nth_rw 2 [angle_le_angle_add_angle_aux hx hy, angle_le_angle_add_angle_aux hz hy] at H6 | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 970,
"column": 64
} | {
"line": 984,
"column": 21
} | {
"line": 986,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\np : P\nhp : p ∈ affineSpan ℝ (Set.range s.points)\n⊢ (∃ r, ∀ (i : Fin (n + 1)), dist p ↑((s.faceOpposite i).orthog... | [] | by
simp_rw [← abs_signedInfDist_eq_dist_of_mem_affineSpan_range _ hp]
refine ⟨?_, ?_⟩
· rintro ⟨r, h⟩
have h' : ∀ i, s.signedInfDist i p = r ∨ s.signedInfDist i p = -r :=
fun i ↦ eq_or_eq_neg_of_abs_eq (h i)
refine ⟨{i ∈ (Finset.univ : Finset (Fin (n + 1))) | s.signedInfDist i p = -r}, ?_⟩
apply... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 446,
"column": 2
} | {
"line": 457,
"column": 14
} | {
"line": 459,
"column": 0
} | [
{
"pp": "case mpr\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c : P\nh : ¬Collinear ℝ {a, b, c}\n⊢ dist a b < dist a c → ∠ a c b < ∠ a b c",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [... | [] | case mpr =>
intro h1
by_contra! w
rcases w.eq_or_lt with h2 | h3
· have h4 : dist a b = dist a c := by
apply dist_eq_of_angle_eq_angle_of_angle_ne_pi h2
rw [show ({a, b, c} : Set P) = {b, a, c} by exact Set.insert_comm a b {c}] at h
linarith [angle_lt_pi_of_not_collinear h]
... | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1020,
"column": 4
} | {
"line": 1021,
"column": 66
} | {
"line": 1023,
"column": 0
} | [
{
"pp": "case neg.refine_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\nn : ℕ\ninst✝ : NeZero n\ns : Simplex ℝ P n\nsigns : Finset (Fin (n + 1))\nh : s.ExcenterExists signs\ni j : Fin (n + 1)\nhij : s.touchpoint... | [] | · rw [hij, ← direction_affineSpan, ← range_faceOpposite_points]
exact vsub_orthogonalProjection_mem_direction_orthogonal _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Congruence | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 85
} | {
"line": 106,
"column": 2
} | [
{
"pp": "V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\n... | [
"V₁ : Type u_2\nV₂ : Type u_3\nP₁ : Type u_4\nP₂ : Type u_5\ninst✝⁷ : NormedAddCommGroup V₁\ninst✝⁶ : NormedAddCommGroup V₂\ninst✝⁵ : InnerProductSpace ℝ V₁\ninst✝⁴ : InnerProductSpace ℝ V₂\ninst✝³ : MetricSpace P₁\ninst✝² : MetricSpace P₂\ninst✝¹ : NormedAddTorsor V₁ P₁\ninst✝ : NormedAddTorsor V₂ P₂\na b c : P₁\n... | have h_bca : ¬Collinear ℝ {b, c, a} := by rwa [Set.insert_comm, Set.pair_comm] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1286,
"column": 2
} | {
"line": 1288,
"column": 7
} | {
"line": 1290,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\n⊢ line[ℝ, t.points i₂, t.points i₃].SSameSide (Simplex.incenter t) (t.points i... | [] | convert! t.sSameSide_incenter_point i₁
simp
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1286,
"column": 2
} | {
"line": 1288,
"column": 7
} | {
"line": 1290,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nt : Triangle ℝ P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\n⊢ line[ℝ, t.points i₂, t.points i₃].SSameSide (Simplex.incenter t) (t.points i... | [] | convert! t.sSameSide_incenter_point i₁
simp
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Circumcenter | {
"line": 821,
"column": 4
} | {
"line": 823,
"column": 17
} | {
"line": 825,
"column": 0
} | [
{
"pp": "case neg.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\nn : ℕ\ns : Simplex ℝ P n\np p₁ p₂ : P\nr : ℝ\nh₁ : ∀ (i : Fin (n + 1)), dist (s.points i) p₁ = r\nh₂ : ∀ (i : Fin (n + 1)), dist (s.points i) p₂ ... | [] | · right
rw [hd₁, reflection_vadd_smul_vsub_orthogonalProjection p r₂ s.circumcenter_mem_affineSpan,
neg_smul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Euclidean.Simplex | {
"line": 41,
"column": 2
} | {
"line": 52,
"column": 7
} | {
"line": 54,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P n\nhe : s.Equilateral\ni₁ i₂ i₃ : Fin (n + 1)\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\n⊢ ∠ (s.points i₁) (s.points i₂) (s.points i₃... | [] | rcases he with ⟨r, hr⟩
rw [angle, InnerProductGeometry.angle,
real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two]
refine Real.arccos_eq_of_eq_cos (by linarith [Real.pi_nonneg]) (by linarith [Real.pi_nonneg]) ?_
simp only [vsub_sub_vsub_cancel_right, ← dist_eq_norm_vsub, hr _ _ h₁₂, hr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Simplex | {
"line": 41,
"column": 2
} | {
"line": 52,
"column": 7
} | {
"line": 54,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P n\nhe : s.Equilateral\ni₁ i₂ i₃ : Fin (n + 1)\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\n⊢ ∠ (s.points i₁) (s.points i₂) (s.points i₃... | [] | rcases he with ⟨r, hr⟩
rw [angle, InnerProductGeometry.angle,
real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two]
refine Real.arccos_eq_of_eq_cos (by linarith [Real.pi_nonneg]) (by linarith [Real.pi_nonneg]) ?_
simp only [vsub_sub_vsub_cancel_right, ← dist_eq_norm_vsub, hr _ _ h₁₂, hr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 287,
"column": 30
} | {
"line": 287,
"column": 50
} | {
"line": 287,
"column": 51
} | [
{
"pp": "case e'_3.e'_6\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P (n + 2)\ne : Fin (n + 3) ≃ Fin (n + 3)\ni₁ i₂ : Fin (n + 3)\n⊢ Function.const (Fin (n + 3)) (↑(#{e.symm i₁, e.symm i₂}ᶜ)... | [
"case e'_3.e'_6\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P (n + 2)\ne : Fin (n + 3) ≃ Fin (n + 3)\ni₁ i₂ : Fin (n + 3)\n⊢ Function.const (Fin (n + 3)) (↑(#{e.symm i₁, e.symm i₂}ᶜ))⁻¹ = Functi... | Function.const_comp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Sphere.SecondInter | {
"line": 80,
"column": 54
} | {
"line": 87,
"column": 73
} | {
"line": 89,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Sphere P\np : P\nv : V\n⊢ s.secondInter p v = p ↔ ⟪v, p -ᵥ s.center⟫ = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Norme... | [] | by
refine ⟨fun hp => ?_, fun hp => ?_⟩
· by_cases hv : v = 0
· simp [hv]
rwa [Sphere.secondInter, eq_comm, eq_vadd_iff_vsub_eq, vsub_self, eq_comm, smul_eq_zero,
or_iff_left hv, div_eq_zero_iff, inner_self_eq_zero, or_iff_left hv, mul_eq_zero,
or_iff_right (by simp : (-2 : ℝ) ≠ 0)] at hp
· rw ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Sphere.SecondInter | {
"line": 140,
"column": 2
} | {
"line": 141,
"column": 26
} | {
"line": 142,
"column": 2
} | [
{
"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\ns : Sphere P\np : P\nv : V\nhv : ¬v = 0\nhv' : ⟪v, v⟫ ≠ 0\n⊢ s.secondInter (s.secondInter p v) v = p",
"ppTerm": "?neg✝",
"assigned": true,
... | [
"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\ns : Sphere P\np : P\nv : V\nhv : ¬v = 0\nhv' : ⟪v, v⟫ ≠ 0\n⊢ ((-2 * (-2 * ⟪v, p -ᵥ s.center⟫ + ⟪v, p -ᵥ s.center⟫) / ⟪v, v⟫) • v + (-2 * ⟪v, p -ᵥ s.center⟫ / ⟪v,... | simp only [Sphere.secondInter, vadd_vsub_assoc, vadd_vadd, inner_add_right, inner_smul_right,
div_mul_cancel₀ _ hv'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 159,
"column": 2
} | {
"line": 161,
"column": 6
} | {
"line": 163,
"column": 0
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_4\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : NormSMulClass 𝕜 E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nd : ℕ\nr : 𝕜\nhr : r ≠ 0\ns : Set E\n⊢ μHE[d] (r • s) = ‖r‖₊ ^ d • μHE[d] s",
"ppTerm": "?m.40",
"ass... | [] | rw [euclideanHausdorffMeasure_def, Measure.smul_apply, hausdorffMeasure_smul₀ (by simp) hr,
Measure.smul_apply, smul_comm]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 159,
"column": 2
} | {
"line": 161,
"column": 6
} | {
"line": 163,
"column": 0
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_4\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : NormSMulClass 𝕜 E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nd : ℕ\nr : 𝕜\nhr : r ≠ 0\ns : Set E\n⊢ μHE[d] (r • s) = ‖r‖₊ ^ d • μHE[d] s",
"ppTerm": "?m.40",
"ass... | [] | rw [euclideanHausdorffMeasure_def, Measure.smul_apply, hausdorffMeasure_smul₀ (by simp) hr,
Measure.smul_apply, smul_comm]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 659,
"column": 2
} | {
"line": 673,
"column": 49
} | {
"line": 675,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nho : OrthocentricSystem s\n⊢ ∃ r, ∀ (t : Triangle ℝ P), Set.range t.points ⊆ s → Simplex.circumradius t = r",
"ppTerm": "?m.37",
"assigned":... | [] | rcases ho with ⟨t, hto, hts⟩
use t.circumradius
intro t₂ ht₂
have ht₂s := ht₂
rw [hts] at ht₂
rcases exists_dist_eq_circumradius_of_subset_insert_orthocenter hto ht₂
t₂.independent.injective with
⟨c, hc, h⟩
rw [Set.forall_mem_range] at h
have hs : Set.range t.points ⊆ s := by
rw [hts]
ex... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 659,
"column": 2
} | {
"line": 673,
"column": 49
} | {
"line": 675,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nho : OrthocentricSystem s\n⊢ ∃ r, ∀ (t : Triangle ℝ P), Set.range t.points ⊆ s → Simplex.circumradius t = r",
"ppTerm": "?m.37",
"assigned":... | [] | rcases ho with ⟨t, hto, hts⟩
use t.circumradius
intro t₂ ht₂
have ht₂s := ht₂
rw [hts] at ht₂
rcases exists_dist_eq_circumradius_of_subset_insert_orthocenter hto ht₂
t₂.independent.injective with
⟨c, hc, h⟩
rw [Set.forall_mem_range] at h
have hs : Set.range t.points ⊆ s := by
rw [hts]
ex... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 928,
"column": 10
} | {
"line": 929,
"column": 76
} | {
"line": 930,
"column": 8
} | [
{
"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... | [] | gcongr
exact Nat.floor_le (mul_nonneg (sub_nonneg.2 (hx i).1.le) npos.le) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 928,
"column": 10
} | {
"line": 929,
"column": 76
} | {
"line": 930,
"column": 8
} | [
{
"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... | [] | gcongr
exact Nat.floor_le (mul_nonneg (sub_nonneg.2 (hx i).1.le) npos.le) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.Tangent | {
"line": 271,
"column": 74
} | {
"line": 272,
"column": 11
} | {
"line": 274,
"column": 0
} | [
{
"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_4\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_6\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : IsManifold I 1 M\nb₀ b : M\nh... | [] | by
simp [hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorBundle.Tangent | {
"line": 282,
"column": 74
} | {
"line": 283,
"column": 11
} | {
"line": 285,
"column": 0
} | [
{
"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_4\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_6\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : IsManifold I 1 M\nb₀ b : M\nh... | [] | by
simp [hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorBundle.Tangent | {
"line": 466,
"column": 2
} | {
"line": 472,
"column": 8
} | {
"line": 474,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nV : (x : E) → TangentSpace 𝓘(𝕜, E) x\ns : Set E\nx : E\n⊢ ContMDiffWithinAt 𝓘(𝕜, E) (𝓘(𝕜, E).prod 𝓘(𝕜, E)) n (fun x ↦ ⟨x, V x⟩) s x ↔ ContDiffWithinAt 𝕜 n V s x",... | [] | refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· exact ContMDiffWithinAt.contDiffWithinAt <|
(contMDiff_snd_tangentBundle_modelSpace E 𝓘(𝕜, E)).contMDiffAt.comp_contMDiffWithinAt _ h
· apply Bundle.contMDiffWithinAt_totalSpace.2
refine ⟨contMDiffWithinAt_id, ?_⟩
convert! h.contMDiffWithinAt with y
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.Tangent | {
"line": 466,
"column": 2
} | {
"line": 472,
"column": 8
} | {
"line": 474,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nV : (x : E) → TangentSpace 𝓘(𝕜, E) x\ns : Set E\nx : E\n⊢ ContMDiffWithinAt 𝓘(𝕜, E) (𝓘(𝕜, E).prod 𝓘(𝕜, E)) n (fun x ↦ ⟨x, V x⟩) s x ↔ ContDiffWithinAt 𝕜 n V s x",... | [] | refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· exact ContMDiffWithinAt.contDiffWithinAt <|
(contMDiff_snd_tangentBundle_modelSpace E 𝓘(𝕜, E)).contMDiffAt.comp_contMDiffWithinAt _ h
· apply Bundle.contMDiffWithinAt_totalSpace.2
refine ⟨contMDiffWithinAt_id, ?_⟩
convert! h.contMDiffWithinAt with y
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.Atlas | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 52
} | {
"line": 129,
"column": 2
} | [
{
"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\ne : OpenPar... | [
"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\ne : OpenPartialHomeomor... | have B := A.differentiableOn one_ne_zero (I x) mem | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.MFDeriv.Atlas | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 82
} | {
"line": 167,
"column": 2
} | [
{
"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... | [
"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 : NormedAddCommGroup E'\ni... | have : mfderiv% (_root_.id : M → M) x = ContinuousLinearMap.id _ _ := mfderiv_id | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.MFDeriv.Atlas | {
"line": 190,
"column": 6
} | {
"line": 192,
"column": 11
} | {
"line": 193,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedA... | [
"𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedAddCommGroup ... | have :
(ContinuousLinearMap.id 𝕜 _ : TangentSpace I' (e x) →L[𝕜] TangentSpace I' (e x)) y = y :=
rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.MFDeriv.Atlas | {
"line": 190,
"column": 6
} | {
"line": 195,
"column": 9
} | {
"line": 195,
"column": 10
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedA... | [] | have :
(ContinuousLinearMap.id 𝕜 _ : TangentSpace I' (e x) →L[𝕜] TangentSpace I' (e x)) y = y :=
rfl
conv_rhs => rw [← this, ← he.comp_symm_deriv (e.map_source hx)]
rw [e.left_inv hx]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.Atlas | {
"line": 190,
"column": 6
} | {
"line": 195,
"column": 9
} | {
"line": 195,
"column": 10
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedA... | [] | have :
(ContinuousLinearMap.id 𝕜 _ : TangentSpace I' (e x) →L[𝕜] TangentSpace I' (e x)) y = y :=
rfl
conv_rhs => rw [← this, ← he.comp_symm_deriv (e.map_source hx)]
rw [e.left_inv hx]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Diffeomorph | {
"line": 644,
"column": 4
} | {
"line": 644,
"column": 44
} | {
"line": 645,
"column": 4
} | [
{
"pp": "case hf\n𝕜 : Type u_1\ninst✝²⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝²⁵ : NormedAddCommGroup E\ninst✝²⁴ : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝²³ : NormedAddCommGroup E'\ninst✝²² : NormedSpace 𝕜 E'\nF : Type u_4\ninst✝²¹ : NormedAddCommGroup F\ninst✝²⁰ : NormedSpace 𝕜 F\nH : Type u_5\nin... | [
"case hg\n𝕜 : Type u_1\ninst✝²⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝²⁵ : NormedAddCommGroup E\ninst✝²⁴ : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝²³ : NormedAddCommGroup E'\ninst✝²² : NormedSpace 𝕜 E'\nF : Type u_4\ninst✝²¹ : NormedAddCommGroup F\ninst✝²⁰ : NormedSpace 𝕜 F\nH : Type u_5\ninst✝¹⁹ : Topo... | · exact ContMDiff.inl.comp ContMDiff.inl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 48
} | {
"line": 317,
"column": 49
} | [
{
"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\nn : WithTop ℕ∞\ninst✝ : IsManifold I ... | [
"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\nn : WithTop ℕ∞\ninst✝ : IsManifold I n M\ne : Ope... | I.isInteriorPoint_iff_of_mem_atlas hn he hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.MFDeriv.Tangent | {
"line": 86,
"column": 2
} | {
"line": 91,
"column": 20
} | {
"line": 93,
"column": 0
} | [
{
"pp": "case e_f.e_f\n𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\ninst✝⁶ : IsManifold ... | [] | · simp only [mfderivWithin, writtenInExtChartAt, modelWithCornersSelf_coe, range_id, inter_univ]
rw [if_pos]
· simp [Function.comp_def, OpenPartialHomeomorph.left_inv (chartAt H (f x₀)) hx]
· apply mdifferentiableWithinAt_extChartAt_symm
apply (extChartAt I (f x₀)).map_source
simpa using hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 496,
"column": 32
} | {
"line": 496,
"column": 45
} | {
"line": 496,
"column": 45
} | [
{
"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... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 496,
"column": 69
} | {
"line": 496,
"column": 82
} | {
"line": 496,
"column": 82
} | [
{
"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... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 213,
"column": 44
} | {
"line": 217,
"column": 10
} | {
"line": 219,
"column": 0
} | [
{
"pp": "𝕜 : 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 : NormedAddCommGro... | [] | by
rw [Trivialization.mem_source] at he he'
refine (hf.coordChange he'f he he').congr_of_eventuallyEq ?_ (by simp [he])
filter_upwards [hf.continuousWithinAt (e.open_baseSet.mem_nhds he)] with y hy
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 277,
"column": 4
} | {
"line": 277,
"column": 56
} | {
"line": 278,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : NormedAddCommGroup F\ninst✝¹³ : NormedSpace 𝕜 F\ninst✝¹² : TopologicalSpace (TotalSpace F E)\ninst✝¹¹ : (x : B) → TopologicalSpace (E x)\nEB : Type u_7\ninst✝¹⁰ : NormedAddCommGroup EB\ninst✝⁹ ... | [
"𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : NormedAddCommGroup F\ninst✝¹³ : NormedSpace 𝕜 F\ninst✝¹² : TopologicalSpace (TotalSpace F E)\ninst✝¹¹ : (x : B) → TopologicalSpace (E x)\nEB : Type u_7\ninst✝¹⁰ : NormedAddCommGroup EB\ninst✝⁹ : NormedSpac... | change MDifferentiableWithinAt IB IB id u b₀ ∧ _ ↔ _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 609,
"column": 4
} | {
"line": 609,
"column": 75
} | {
"line": 610,
"column": 4
} | [
{
"pp": "case e_a.hg\n𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\nins... | [
"case e_a.hf\n𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : Normed... | · exact hf.comp _ (mdifferentiableAt_id.prodMk mdifferentiableAt_const) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.MFDeriv.NormedSpace | {
"line": 354,
"column": 95
} | {
"line": 356,
"column": 5
} | {
"line": 359,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nx : M\nV : Type u_18\ninst✝¹ : Normed... | [] | by
rw [mfderiv_smul hf hg]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 74
} | {
"line": 196,
"column": 0
} | [
{
"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\ns : Set M\nx : M\nV W : (x : M) → Tang... | [] | rw [← mlieBracketWithin_univ, ← univ_inter s, mlieBracketWithin_inter h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 74
} | {
"line": 196,
"column": 0
} | [
{
"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\ns : Set M\nx : M\nV W : (x : M) → Tang... | [] | rw [← mlieBracketWithin_univ, ← univ_inter s, mlieBracketWithin_inter h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 74
} | {
"line": 196,
"column": 0
} | [
{
"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\ns : Set M\nx : M\nV W : (x : M) → Tang... | [] | rw [← mlieBracketWithin_univ, ← univ_inter s, mlieBracketWithin_inter h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.VectorBundle.Hom | {
"line": 110,
"column": 43
} | {
"line": 115,
"column": 19
} | {
"line": 116,
"column": 2
} | [
{
"pp": "𝕜₁ : Type u_1\ninst✝²⁰ : NontriviallyNormedField 𝕜₁\n𝕜₂ : Type u_2\ninst✝¹⁹ : NontriviallyNormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nB : Type u_3\nF₁ : Type u_4\ninst✝¹⁸ : NormedAddCommGroup F₁\ninst✝¹⁷ : NormedSpace 𝕜₁ F₁\nE₁ : B → Type u_5\ninst✝¹⁶ : (x : B) → AddCommGroup (E₁ x)\ninst✝¹⁵ : (x : B) → Modu... | [] | by
simp only [Prod.mk_right_inj]
ext v
dsimp only [comp_apply]
rw [Trivialization.continuousLinearMapAt_symmL, Trivialization.continuousLinearMapAt_symmL]
exacts [h₁, h₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 55
} | {
"line": 316,
"column": 4
} | [
{
"pp": "𝕜 : 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 : Topologic... | [
"𝕜 : 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 : TopologicalSpace H'\n... | apply hf.continuousWithinAt.preimage_mem_nhdsWithin | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 88
} | {
"line": 322,
"column": 2
} | [
{
"pp": "𝕜 : 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 : Topologic... | [
"𝕜 : 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 : TopologicalSpace H'\n... | simp only [inverse_equiv_comp, inverse_comp_equiv, ContinuousLinearEquiv.symm_symm, ϕ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 446,
"column": 4
} | {
"line": 446,
"column": 55
} | {
"line": 447,
"column": 4
} | [
{
"pp": "𝕜 : 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 : Topologic... | [
"𝕜 : 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 : TopologicalSpace H'\n... | apply hf.continuousWithinAt.preimage_mem_nhdsWithin | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 452,
"column": 2
} | {
"line": 452,
"column": 88
} | {
"line": 453,
"column": 2
} | [
{
"pp": "𝕜 : 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 : Topologic... | [
"𝕜 : 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 : TopologicalSpace H'\n... | simp only [inverse_equiv_comp, inverse_comp_equiv, ContinuousLinearEquiv.symm_symm, ϕ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 406,
"column": 72
} | {
"line": 409,
"column": 12
} | {
"line": 411,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ns : Set M\nx : M\nV W : (x : M) → Tan... | [] | by
rw [mlieBracketWithin_swap, Pi.neg_apply, mlieBracketWithin_smul_right hf hV (V := W) hs,
mlieBracketWithin_swap]
simp; abel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 457,
"column": 2
} | {
"line": 458,
"column": 70
} | {
"line": 460,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nx : M\nV W V₁ : (x : M) → TangentSpac... | [] | simp only [← mlieBracketWithin_univ] at hV hV₁ ⊢
exact mlieBracketWithin_add_left hV hV₁ (uniqueMDiffWithinAt_univ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 457,
"column": 2
} | {
"line": 458,
"column": 70
} | {
"line": 460,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nx : M\nV W V₁ : (x : M) → TangentSpac... | [] | simp only [← mlieBracketWithin_univ] at hV hV₁ ⊢
exact mlieBracketWithin_add_left hV hV₁ (uniqueMDiffWithinAt_univ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.GroupLieAlgebra | {
"line": 128,
"column": 35
} | {
"line": 128,
"column": 40
} | {
"line": 129,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝⁶ : TopologicalSpace H\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : ChartedSpace H G\ninst✝¹ : Group G\ninst✝ : LieGroup I ... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.Geometry.Manifold.LocalSourceTargetProperty | {
"line": 201,
"column": 6
} | {
"line": 201,
"column": 58
} | {
"line": 202,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_4\nH : Type u_6\nG : Type u_8\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace G\nI : ModelWithCorners... | [] | rw [hfg.inter_preimage_eq]; exact inter_subset_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.LocalSourceTargetProperty | {
"line": 201,
"column": 6
} | {
"line": 201,
"column": 58
} | {
"line": 202,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_4\nH : Type u_6\nG : Type u_8\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace G\nI : ModelWithCorners... | [] | rw [hfg.inter_preimage_eq]; exact inter_subset_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Instances.Real | {
"line": 294,
"column": 17
} | {
"line": 298,
"column": 46
} | {
"line": 299,
"column": 2
} | [
{
"pp": "x y : ℝ\nh : Fact (x < y)\n⊢ IsOpen {z | (↑z).ofLp 0 < y - x}",
"ppTerm": "?m.196",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Lattice.toSemilatticeSup",
"Real.lattice",
"continuous_subtype_val",
"Real.instZero",
"instHasSolidNormR... | [] | by
have : IsOpen { z : ℝ | z < y - x } := isOpen_Iio
have : IsOpen { z : EuclideanSpace ℝ (Fin 1) | z 0 < y - x } :=
this.preimage (@PiLp.continuous_apply 2 (Fin 1) (fun _ => ℝ) _ 0)
exact this.preimage continuous_subtype_val | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 359,
"column": 6
} | {
"line": 359,
"column": 35
} | {
"line": 360,
"column": 4
} | [
{
"pp": "case hf₁\n𝕜 : 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\nIs : IsManifold I 1 M\nx : M... | [] | exact differentiableAt_fun_id | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 359,
"column": 6
} | {
"line": 359,
"column": 35
} | {
"line": 360,
"column": 4
} | [
{
"pp": "case hf₁\n𝕜 : 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\nIs : IsManifold I 1 M\nx : M... | [] | exact differentiableAt_fun_id | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiffMFDeriv | {
"line": 359,
"column": 6
} | {
"line": 359,
"column": 35
} | {
"line": 360,
"column": 4
} | [
{
"pp": "case hf₁\n𝕜 : 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\nIs : IsManifold I 1 M\nx : M... | [] | exact differentiableAt_fun_id | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 677,
"column": 75
} | {
"line": 677,
"column": 88
} | {
"line": 677,
"column": 88
} | [
{
"pp": "𝕜 : 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 : Topolo... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 13
} | {
"line": 281,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : IsManifold I 1 M\nγ : ℝ → M\nv : (x : M) → TangentSpace I x\ninst✝¹... | [] | exact hne | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.Riemannian.PathELength | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 40
} | {
"line": 281,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : (x : M) → ENorm (TangentSpace I x)\ninst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSp... | [
"case ha\nE : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nH : Type u_2\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_3\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ninst✝¹ : (x : M) → ENorm (TangentSpace I x)\ninst✝ : ∀ (x : M), ENormSMulClass ℝ (TangentSpace... | apply mul_nonpos_of_nonneg_of_nonpos | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.VectorBundle.Riemannian | {
"line": 162,
"column": 6
} | {
"line": 162,
"column": 70
} | {
"line": 163,
"column": 4
} | [
{
"pp": "B : Type u_1\ninst✝⁷ : TopologicalSpace B\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nE : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (x : B) → NormedAddCommGroup (E x)\ninst✝² : (x : B) → InnerProductSpace ℝ (E x)\ninst✝¹ : FiberBundle F E\ninst✝ : Ve... | [] | exact tendsto_const_nhds.sub (tendsto_id.mul tendsto_const_nhds) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.VectorBundle.Riemannian | {
"line": 163,
"column": 4
} | {
"line": 165,
"column": 54
} | {
"line": 166,
"column": 4
} | [
{
"pp": "B✝ : Type u_1\ninst✝⁷ : TopologicalSpace B✝\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nE : B✝ → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (x : B✝) → NormedAddCommGroup (E x)\ninst✝² : (x : B✝) → InnerProductSpace ℝ (E x)\ninst✝¹ : FiberBundle F E\ninst✝... | [
"B✝ : Type u_1\ninst✝⁷ : TopologicalSpace B✝\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nE : B✝ → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (x : B✝) → NormedAddCommGroup (E x)\ninst✝² : (x : B✝) → InnerProductSpace ℝ (E x)\ninst✝¹ : FiberBundle F E\ninst✝ : VectorBun... | have B' : ∀ᶠ δ in 𝓝[>] 0, (r' ^ 2)⁻¹ < 1 - δ * C := by
apply (tendsto_order.1 B).1
simpa using inv_lt_one_of_one_lt₀ (by nlinarith) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.Submersion | {
"line": 367,
"column": 64
} | {
"line": 371,
"column": 59
} | {
"line": 373,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE'' : Type u_3\nF : Type u_5\nH : Type u_7\nG : Type u_9\nE : Type u\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\ninst✝⁹ : NormedAddCommGroup E''\ninst✝⁸ : NormedSpace 𝕜 E''\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\n... | [] | by
have aux : IsSubmersionAtOfComplement F I J n f x := by
apply IsSubmersionAtOfComplement.mk_of_charts <;> assumption
use aux.smallComplement, by infer_instance, by infer_instance
rwa [← IsSubmersionAtOfComplement.congr_F aux.smallEquiv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 23
} | {
"line": 142,
"column": 24
} | [
{
"pp": "G : Type u_1\ninst✝⁴ : Group G\ninst✝³ : IsKleinFour G\nG₁ : Type u_2\nG₂ : Type u_3\ninst✝² : Group G₁\ninst✝¹ : Group G₂\ninst✝ : IsKleinFour G₁\ne : G₁ ≃ G₂\nhe : e 1 = 1\nh : Monoid.exponent G₂ = 2\n_inst₁ : Fintype G₁ := Fintype.ofFinite G₁\n_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e\nx y : G₁\n⊢... | [
"case pos\nG : Type u_1\ninst✝⁴ : Group G\ninst✝³ : IsKleinFour G\nG₁ : Type u_2\nG₂ : Type u_3\ninst✝² : Group G₁\ninst✝¹ : Group G₂\ninst✝ : IsKleinFour G₁\ne : G₁ ≃ G₂\nhe : e 1 = 1\nh : Monoid.exponent G₂ = 2\n_inst₁ : Fintype G₁ := Fintype.ofFinite G₁\n_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e\nx y : G₁\nhx ... | by_cases hx : x = 1 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 147,
"column": 6
} | {
"line": 149,
"column": 75
} | {
"line": 150,
"column": 6
} | [
{
"pp": "case neg\nG : Type u_1\ninst✝⁴ : Group G\ninst✝³ : IsKleinFour G\nG₁ : Type u_2\nG₂ : Type u_3\ninst✝² : Group G₁\ninst✝¹ : Group G₂\ninst✝ : IsKleinFour G₁\ne : G₁ ≃ G₂\nhe : e 1 = 1\nh : Monoid.exponent G₂ = 2\n_inst₁ : Fintype G₁ := Fintype.ofFinite G₁\n_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e\nx... | [
"case neg\nG : Type u_1\ninst✝⁴ : Group G\ninst✝³ : IsKleinFour G\nG₁ : Type u_2\nG₂ : Type u_3\ninst✝² : Group G₁\ninst✝¹ : Group G₂\ninst✝ : IsKleinFour G₁\ne : G₁ ≃ G₂\nhe : e 1 = 1\nh : Monoid.exponent G₂ = 2\n_inst₁ : Fintype G₁ := Fintype.ofFinite G₁\n_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e\nx y : G₁\nhx ... | have univ₂ : {e (x * y), e x, e y, (1 : G₂)} = Finset.univ := by
simpa [map_univ_equiv e, map_insert, he]
using congr(Finset.map e.toEmbedding $(eq_finset_univ hx hy hxy)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 9
} | {
"line": 131,
"column": 10
} | [
{
"pp": "n : ℕ\ni : ZMod n\nk : ℤ\n⊢ r i ^ k = r (i * ↑k)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Int.cast",
"HMul.hMul",
"ZMod.commRing",
"CommSemiring.toSemiring",
"DivInvMonoid.toZPow",
"Int.casesOn",
"DihedralGroup.instGroup",
"I... | [
"case ofNat\nn : ℕ\ni : ZMod n\na✝ : ℕ\n⊢ r i ^ Int.ofNat a✝ = r (i * ↑(Int.ofNat a✝))",
"case negSucc\nn : ℕ\ni : ZMod n\na✝ : ℕ\n⊢ r i ^ Int.negSucc a✝ = r (i * ↑(Int.negSucc a✝))"
] | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.GroupTheory.PresentedGroup | {
"line": 108,
"column": 40
} | {
"line": 108,
"column": 53
} | {
"line": 108,
"column": 53
} | [
{
"pp": "α : Type u_1\nrels : Set (FreeGroup α)\nH : Subgroup (PresentedGroup rels)\nh : ∀ (j : α), of j ∈ H\nx : PresentedGroup rels\nx✝ : α\na✝ : Quot.mk (⇑(QuotientGroup.leftRel (Subgroup.normalClosure rels))) (FreeGroup.of x✝) ∈ H\n⊢ ?m.50 ∈ H",
"ppTerm": "?m.53",
"assigned": true,
"usedConstant... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.PresentedGroup | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 10
} | {
"line": 154,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_3\ninst✝ : Group G\nrels : Set (FreeGroup α)\nφ ψ : PresentedGroup rels →* G\nhx : ∀ (x : α), φ (of x) = ψ (of x)\na✝ : α\n⊢ (φ.comp (QuotientGroup.mk' (Subgroup.normalClosure rels))) (FreeGroup.of a✝) =\n (ψ.comp (QuotientGroup.mk' (Subgroup.normalClosure rels))) (FreeGroup... | [] | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.PresentedGroup | {
"line": 150,
"column": 2
} | {
"line": 152,
"column": 10
} | {
"line": 154,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_3\ninst✝ : Group G\nrels : Set (FreeGroup α)\nφ ψ : PresentedGroup rels →* G\nhx : ∀ (x : α), φ (of x) = ψ (of x)\n⊢ φ = ψ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"PresentedGroup",
"MonoidHom",
"Monoid.toMulOneClass",
"Subgr... | [] | unfold PresentedGroup
ext
apply hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.PresentedGroup | {
"line": 150,
"column": 2
} | {
"line": 152,
"column": 10
} | {
"line": 154,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_3\ninst✝ : Group G\nrels : Set (FreeGroup α)\nφ ψ : PresentedGroup rels →* G\nhx : ∀ (x : α), φ (of x) = ψ (of x)\n⊢ φ = ψ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"PresentedGroup",
"MonoidHom",
"Monoid.toMulOneClass",
"Subgr... | [] | unfold PresentedGroup
ext
apply hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.CoprodI | {
"line": 203,
"column": 74
} | {
"line": 208,
"column": 21
} | {
"line": 210,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nmotive : CoprodI M → Prop\nm : CoprodI M\none : motive 1\nmul : ∀ {i : ι} (m : M i) (x : CoprodI M), motive x → motive (of m * x)\n⊢ motive m",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Monoid",
"Mono... | [] | by
induction m using Submonoid.induction_of_closure_eq_top_left mclosure_iUnion_range_of with
| one => exact one
| mul_left x hx y ihy =>
obtain ⟨i, m, rfl⟩ : ∃ (i : ι) (m : M i), of m = x := by simpa using hx
exact mul m y ihy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 8
} | {
"line": 261,
"column": 0
} | [
{
"pp": "B : Type u_1\nW : Type u_3\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\np : W → Prop\nw : W\nsimple : ∀ (i : B), p (cs.simple i)\none : p 1\nmul : ∀ (w w' : W), p w → p w' → p (w * w')\n⊢ p w",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"_private.Mathlib... | [] | have := cs.submonoid_closure_range_simple.symm ▸ Submonoid.mem_top w
exact Submonoid.closure_induction (fun x ⟨i, hi⟩ ↦ hi ▸ simple i) one (fun _ _ _ _ ↦ mul _ _)
this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 8
} | {
"line": 261,
"column": 0
} | [
{
"pp": "B : Type u_1\nW : Type u_3\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\np : W → Prop\nw : W\nsimple : ∀ (i : B), p (cs.simple i)\none : p 1\nmul : ∀ (w w' : W), p w → p w' → p (w * w')\n⊢ p w",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"_private.Mathlib... | [] | have := cs.submonoid_closure_range_simple.symm ▸ Submonoid.mem_top w
exact Submonoid.closure_induction (fun x ⟨i, hi⟩ ↦ hi ▸ simple i) one (fun _ _ _ _ ↦ mul _ _)
this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.CoprodI | {
"line": 389,
"column": 6
} | {
"line": 389,
"column": 14
} | {
"line": 389,
"column": 15
} | [
{
"pp": "case cons.refine_1\nι : Type u_1\nM : ι → Type u_2\ninst✝¹ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝ : Monoid N\nmotive : Word M → Sort u_4\nempty : motive Word.empty\ncons :\n (i : ι) → (m : M i) → (w : Word M) → (h1 : w.fstIdx ≠ some i) → (h2 : m ≠ 1) → motive w → motive (Word.cons m w h1 h2)\nm... | [
"case cons.refine_1\nι : Type u_1\nM : ι → Type u_2\ninst✝¹ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝ : Monoid N\nmotive : Word M → Sort u_4\nempty : motive Word.empty\ncons :\n (i : ι) → (m : M i) → (w : Word M) → (h1 : w.fstIdx ≠ some i) → (h2 : m ≠ 1) → motive w → motive (Word.cons m w h1 h2)\nm : (i : ι) ×... | intro m' | Lean.Elab.Tactic.evalIntro | null |
Mathlib.GroupTheory.CoprodI | {
"line": 420,
"column": 35
} | {
"line": 420,
"column": 66
} | {
"line": 420,
"column": 66
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝² : Monoid N\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (M i)\ni : ι\nw✝ : Word M\nj : ι\nm : M j\nw : Word M\nh1 : w.fstIdx ≠ some j\nh2 : m ≠ 1\nx✝ : { p // rcons p = w }\nij : ¬i = j\n⊢ (cons m w h1 h2).f... | [] | simp [cons, fstIdx, Ne.symm ij] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.CoprodI | {
"line": 420,
"column": 35
} | {
"line": 420,
"column": 66
} | {
"line": 420,
"column": 66
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝² : Monoid N\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (M i)\ni : ι\nw✝ : Word M\nj : ι\nm : M j\nw : Word M\nh1 : w.fstIdx ≠ some j\nh2 : m ≠ 1\nx✝ : { p // rcons p = w }\nij : ¬i = j\n⊢ (cons m w h1 h2).f... | [] | simp [cons, fstIdx, Ne.symm ij] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.CoprodI | {
"line": 420,
"column": 35
} | {
"line": 420,
"column": 66
} | {
"line": 420,
"column": 66
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝³ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝² : Monoid N\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (M i)\ni : ι\nw✝ : Word M\nj : ι\nm : M j\nw : Word M\nh1 : w.fstIdx ≠ some j\nh2 : m ≠ 1\nx✝ : { p // rcons p = w }\nij : ¬i = j\n⊢ (cons m w h1 h2).f... | [] | simp [cons, fstIdx, Ne.symm ij] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 6
} | {
"line": 80,
"column": 0
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nt : W\nht : cs.IsReflection t\n⊢ t ^ 2 = 1",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneC... | [] | rcases ht with ⟨w, i, rfl⟩
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 6
} | {
"line": 80,
"column": 0
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nt : W\nht : cs.IsReflection t\n⊢ t ^ 2 = 1",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneC... | [] | rcases ht with ⟨w, i, rfl⟩
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 6
} | {
"line": 84,
"column": 0
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nt : W\nht : cs.IsReflection t\n⊢ t * t = 1",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"conj_mul",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvO... | [] | rcases ht with ⟨w, i, rfl⟩
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 6
} | {
"line": 84,
"column": 0
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nt : W\nht : cs.IsReflection t\n⊢ t * t = 1",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"conj_mul",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvO... | [] | rcases ht with ⟨w, i, rfl⟩
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 85
} | {
"line": 148,
"column": 2
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\n⊢ M.IsLiftable fun x ↦ Multiplicative.ofAdd 1",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Multiplicative.monoid",
"Equiv.instEquivLike",... | [
"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\n⊢ ∀ (i i' : B), Multiplicative.ofAdd 0 ^ M.M i i' = 1"
] | simp_rw [CoxeterMatrix.IsLiftable, ← ofAdd_add, (by decide : (1 + 1 : ZMod 2) = 0)] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 7
} | {
"line": 115,
"column": 0
} | [
{
"pp": "case h\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw u : W\ni : B\n⊢ w * (u * cs.simple i * u⁻¹) * w⁻¹ = w * u * cs.simple i * (w * u)⁻¹",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"DivInvMonoid.toInv",... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 9
} | {
"line": 226,
"column": 0
} | [
{
"pp": "case cons\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni j : B\nω : List B\nih : cs.rightInvSeq (ω.concat i) = (List.map (⇑(MulAut.conj (cs.simple i))) (cs.rightInvSeq ω)).concat (cs.simple i)\n⊢ cs.simple i * (cs.wordProd ω)⁻¹ * cs.simple j * (cs.wordProd ... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.CoprodI | {
"line": 658,
"column": 52
} | {
"line": 658,
"column": 65
} | {
"line": 658,
"column": 65
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\ni j i✝ j✝ k✝ l✝ : ι\n_w₁✝ : NeWord M i✝ j✝\n_hne✝ : j✝ ≠ k✝\n_w₂✝ : NeWord M k✝ l✝\n_w₁_ih✝ : ⟨j✝, _w₁✝.last⟩ ∈ _w₁✝.toList.getLast?\n_w₂_ih✝ : ⟨l✝, _w₂✝.last⟩ ∈ _w₂✝.toList.getLast?\n⊢ ⟨l✝, (_w₁✝.append _hne✝ _w₂✝).last⟩ ∈ _w₂✝.toList.get... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.CoprodI | {
"line": 672,
"column": 34
} | {
"line": 672,
"column": 47
} | {
"line": 672,
"column": 47
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝¹ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝ : Monoid N\ni j i✝ j✝ k✝ l✝ : ι\n_w₁✝ : NeWord M i✝ j✝\n_hne✝ : j✝ ≠ k✝\n_w₂✝ : NeWord M k✝ l✝\n_w₁_ih✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) _w₁✝.toList\n_w₂_ih✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) _w₂✝.to... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.CoprodI | {
"line": 672,
"column": 50
} | {
"line": 672,
"column": 63
} | {
"line": 672,
"column": 63
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝¹ : (i : ι) → Monoid (M i)\nN : Type u_3\ninst✝ : Monoid N\ni j i✝ j✝ k✝ l✝ : ι\n_w₁✝ : NeWord M i✝ j✝\n_hne✝ : j✝ ≠ k✝\n_w₂✝ : NeWord M k✝ l✝\n_w₁_ih✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) _w₁✝.toList\n_w₂_ih✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) _w₂✝.to... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 341,
"column": 6
} | {
"line": 341,
"column": 11
} | {
"line": 342,
"column": 4
} | [
{
"pp": "case h\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\nω : List B\nih :\n (cs.wordProd ω)⁻¹ * cs.simple i * cs.wordProd ω ∈ cs.rightInvSeq ω →\n cs.IsReflection ((cs.wordProd ω)⁻¹ * cs.simple i * cs.wordProd ω)\n⊢ (cs.wordProd ω)⁻¹ * cs.simple i * c... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.Descent | {
"line": 83,
"column": 46
} | {
"line": 83,
"column": 81
} | {
"line": 83,
"column": 81
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nf : G →* G\nhf : ∀ (U : Subgroup G), map f U ≤ U\ns : Set G\nh : G → ℝ\na b c : ℝ\nha : 0 ≤ a\nH₀ : a < b\nhs : s.Finite\nH₁ : s * ↑f.range = Set.univ\nH₂ : ∀ g ∈ s, ∀ (x : G), h x ≤ a * h (g * x) + c\nH₃ : ∀ (x : G), b * h x - c ≤ h (f x)\ninst✝ : Northcott h\nq : G → G... | [
"G : Type u_1\ninst✝¹ : Group G\nf : G →* G\nhf : ∀ (U : Subgroup G), map f U ≤ U\ns : Set G\nh : G → ℝ\na b c : ℝ\nha : 0 ≤ a\nH₀ : a < b\nhs : s.Finite\nH₁ : s * ↑f.range = Set.univ\nH₂ : ∀ g ∈ s, ∀ (x : G), h x ≤ a * h (g * x) + c\nH₃ : ∀ (x : G), b * h x - c ≤ h (f x)\ninst✝ : Northcott h\nq : G → G ⧸ map f ⊤ :... | field_simp [sub_pos.mpr H₀] at this | Mathlib.Tactic.FieldSimp._aux_Mathlib_Tactic_FieldSimp___elabRules_Mathlib_Tactic_FieldSimp_fieldSimp_1 | Mathlib.Tactic.FieldSimp.fieldSimp |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 434,
"column": 49
} | {
"line": 434,
"column": 54
} | {
"line": 435,
"column": 4
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nrω : cs.IsReduced ω\nj j' : ℕ\nj_lt_j' : j < j'\nj'_lt_length : j' < ω.length\ndup : (cs.rightInvSeq ω).getD j 1 = (cs.rightInvSeq ω).getD j' 1\nt : W := (cs.rightInvSeq ω).getD j 1\nh₁ : t = (cs.right... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.DivisibleHull | {
"line": 273,
"column": 4
} | {
"line": 273,
"column": 29
} | {
"line": 274,
"column": 4
} | [
{
"pp": "M✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\na : DivisibleHull M\n⊢ a ≤ a",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"DivisibleHull",
"LE.le",
"DivisibleHull.ins... | [
"case mk\nM✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nm : M\ns : ℕ+\n⊢ mk m s ≤ mk m s"
] | induction a with | mk m s | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.GroupTheory.DivisibleHull | {
"line": 324,
"column": 4
} | {
"line": 324,
"column": 29
} | {
"line": 325,
"column": 4
} | [
{
"pp": "M✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\na : DivisibleHull M\nha : 0 < a\nb c : ℚ≥0\nh : b < c\n⊢ b • a < c • a",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"instHSMul",
... | [
"case mk\nM✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nb c : ℚ≥0\nh : b < c\nm : M\ns : ℕ+\nha : 0 < mk m s\n⊢ b • mk m s < c • mk m s"
] | induction a with | mk m s | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.GroupTheory.DivisibleHull | {
"line": 408,
"column": 4
} | {
"line": 408,
"column": 29
} | {
"line": 409,
"column": 4
} | [
{
"pp": "case h₁.mk\nM✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\na : DivisibleHull M\n⊢ ((archimedeanClassOrderHom M).comp (archimedeanClassOrderHomInv M)) (ArchimedeanClass.mk a) =\n OrderHom.id (ArchimedeanClass.mk a... | [
"case h₁.mk.mk\nM✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\nm : M\ns : ℕ+\n⊢ ((archimedeanClassOrderHom M).comp (archimedeanClassOrderHomInv M)) (ArchimedeanClass.mk (mk m s)) =\n OrderHom.id (ArchimedeanClass.mk (mk m s))... | induction a with | mk m s | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.GroupTheory.CoprodI | {
"line": 972,
"column": 96
} | {
"line": 1055,
"column": 25
} | {
"line": 1057,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝² : Nontrivial ι\nG : Type u_1\ninst✝¹ : Group G\na : ι → G\nα : Type u_4\ninst✝ : MulAction G α\nX Y : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhYdisj : Pairwise (Disjoint on Y)\nhXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)\nhX : ∀ (i : ι)... | [] | by
-- Step one: express the free group lift via the free product lift
have : FreeGroup.lift a =
(CoprodI.lift fun i => FreeGroup.lift fun _ => a i).comp
(@freeGroupEquivCoprodI ι).toMonoidHom := by
ext i
simp
rw [this, MonoidHom.coe_comp]
clear this
refine Function.Injective.comp ?_ (Mul... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Focal | {
"line": 98,
"column": 75
} | {
"line": 98,
"column": 80
} | {
"line": 98,
"column": 80
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx : G\nhxP : x ∈ H\nu : G\nhzH : ⁅x, u⁆ ∈ H\n⊢ g * ⁅x, u⁆ * g⁻¹ = ⁅g * x * g⁻¹, g * u * g⁻¹⁆",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semigroup.toMul",
"DivInvMonoid.toIn... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.Focal | {
"line": 98,
"column": 75
} | {
"line": 98,
"column": 80
} | {
"line": 98,
"column": 80
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx : G\nhxP : x ∈ H\nu : G\nhzH : ⁅x, u⁆ ∈ H\n⊢ g * ⁅x, u⁆ * g⁻¹ = ⁅g * x * g⁻¹, g * u * g⁻¹⁆",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semigroup.toMul",
"DivInvMonoid.toIn... | [] | group | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Focal | {
"line": 98,
"column": 75
} | {
"line": 98,
"column": 80
} | {
"line": 98,
"column": 80
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx : G\nhxP : x ∈ H\nu : G\nhzH : ⁅x, u⁆ ∈ H\n⊢ g * ⁅x, u⁆ * g⁻¹ = ⁅g * x * g⁻¹, g * u * g⁻¹⁆",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semigroup.toMul",
"DivInvMonoid.toIn... | [] | group | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Focal | {
"line": 124,
"column": 67
} | {
"line": 124,
"column": 72
} | {
"line": 124,
"column": 72
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : G\nhh : h ∈ H\ng : G\nhconj : g⁻¹ * h * g ∈ H\n⊢ (g⁻¹ * h * g)⁻¹ * h = ⁅(g⁻¹ * h * g)⁻¹, g⁆",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semigroup.toMul",
"DivInvMonoid.toInv",
"NonUnita... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.GroupTheory.Focal | {
"line": 124,
"column": 67
} | {
"line": 124,
"column": 72
} | {
"line": 124,
"column": 72
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nh : G\nhh : h ∈ H\ng : G\nhconj : g⁻¹ * h * g ∈ H\n⊢ (g⁻¹ * h * g)⁻¹ * h = ⁅(g⁻¹ * h * g)⁻¹, g⁆",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semigroup.toMul",
"DivInvMonoid.toInv",
"NonUnita... | [] | group | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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