module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Euclidean.Triangle | {
"line": 492,
"column": 2
} | {
"line": 495,
"column": 51
} | {
"line": 497,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh₂₃₁ : ∠ p₂ p₃ p₁ ≤ ∠ p₁ p₂ p₃\nh₃₁₂ : ∠ p₃ p₁ p₂ ≤ ∠ p₁ p₂ p₃\n⊢ π / 3 ≤ ∠ p₁ p₂ p₃",
"ppTerm": "?m.47",
"assigned": true,
"usedCons... | [] | by_cases h : p₂ = p₁
· rw [h, angle_self_left]
linarith [Real.pi_pos]
· linarith [angle_add_angle_add_angle_eq_pi p₃ h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Triangle | {
"line": 492,
"column": 2
} | {
"line": 495,
"column": 51
} | {
"line": 497,
"column": 0
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh₂₃₁ : ∠ p₂ p₃ p₁ ≤ ∠ p₁ p₂ p₃\nh₃₁₂ : ∠ p₃ p₁ p₂ ≤ ∠ p₁ p₂ p₃\n⊢ π / 3 ≤ ∠ p₁ p₂ p₃",
"ppTerm": "?m.47",
"assigned": true,
"usedCons... | [] | by_cases h : p₂ = p₁
· rw [h, angle_self_left]
linarith [Real.pi_pos]
· linarith [angle_add_angle_add_angle_eq_pi p₃ h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1062,
"column": 8
} | {
"line": 1063,
"column": 51
} | {
"line": 1064,
"column": 4
} | [
{
"pp": "case refine_1.h\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)\nhne : i ≠ j\nr : ℝ\n... | [] | exact AffineMap.lineMap_mem _ h.excenter_mem_affineSpan_range
(s.touchpoint_mem_affineSpan_simplex _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1062,
"column": 8
} | {
"line": 1063,
"column": 51
} | {
"line": 1064,
"column": 4
} | [
{
"pp": "case refine_1.h\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)\nhne : i ≠ j\nr : ℝ\n... | [] | exact AffineMap.lineMap_mem _ h.excenter_mem_affineSpan_range
(s.touchpoint_mem_affineSpan_simplex _ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1062,
"column": 8
} | {
"line": 1063,
"column": 51
} | {
"line": 1064,
"column": 4
} | [
{
"pp": "case refine_1.h\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)\nhne : i ≠ j\nr : ℝ\n... | [] | exact AffineMap.lineMap_mem _ h.excenter_mem_affineSpan_range
(s.touchpoint_mem_affineSpan_simplex _ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Incenter | {
"line": 1113,
"column": 4
} | {
"line": 1114,
"column": 58
} | {
"line": 1115,
"column": 2
} | [
{
"pp": "case mp\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))\ni : Fin (n + 1)\nw : Fin (n + 1) → ℝ\nhw : ∑ j, w j = 1\n⊢ (Finset.affineCo... | [] | exact fun h ↦ (affineIndependent_iff_eq_of_fintype_affineCombination_eq ℝ s.points).1
s.independent _ _ hw (s.sum_touchpointWeights _ _) h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Euclidean.NinePointCircle | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 40
} | {
"line": 148,
"column": 2
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P n\ni : Fin (n + 1)\n⊢ s.points i -ᵥ s.eulerPoint i = ((↑n - 1) / ↑n) • (s.points i -ᵥ s.mongePoint)",
"ppTerm": "?m.67",
"assign... | [
"V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P n\ni : Fin (n + 1)\n⊢ s.points i -ᵥ s.mongePoint - (↑n)⁻¹ • (s.points i -ᵥ s.mongePoint) = ((↑n - 1) / ↑n) • (s.points i -ᵥ s.mongePoint)"
] | rw [eulerPoint, vsub_vadd_eq_vsub_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 39
} | {
"line": 166,
"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\nn : ℕ\ns : Simplex ℝ P (n + 2)\n⊢ (univ.weightedVSub s.pointsWithCircumcenter)\n ((↑(n + 2 + 1) / ↑(n + 2 - 1)) •\n (centroidWeightsWithCircumc... | [
"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)\n⊢ (affineCombination ℝ univ s.pointsWithCircumcenter)\n ((↑(n + 2 + 1) / ↑(n + 2 - 1)) •\n (centroidWeightsWithCircumcenter u... | weightedVSub_vadd_affineCombination | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Euclidean.Simplex | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 97
} | {
"line": 76,
"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\ni : Fin (n + 1)\n⊢ dist (s.points i) s.centroid = ↑n * dist s.centroid (s.faceOppositeCentroid i)",
"ppTerm": ... | [] | simp_rw [dist_eq_norm_vsub, s.point_vsub_centroid_eq_smul_vsub i, norm_smul, Real.norm_natCast] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Euclidean.Simplex | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 97
} | {
"line": 76,
"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\ni : Fin (n + 1)\n⊢ dist (s.points i) s.centroid = ↑n * dist s.centroid (s.faceOppositeCentroid i)",
"ppTerm": ... | [] | simp_rw [dist_eq_norm_vsub, s.point_vsub_centroid_eq_smul_vsub i, norm_smul, Real.norm_natCast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Simplex | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 97
} | {
"line": 76,
"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\ni : Fin (n + 1)\n⊢ dist (s.points i) s.centroid = ↑n * dist s.centroid (s.faceOppositeCentroid i)",
"ppTerm": ... | [] | simp_rw [dist_eq_norm_vsub, s.point_vsub_centroid_eq_smul_vsub i, norm_smul, Real.norm_natCast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.Power | {
"line": 62,
"column": 42
} | {
"line": 62,
"column": 49
} | {
"line": 63,
"column": 4
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y z : V\nh₃ : ‖z - y‖ = ‖z + y‖\nr : ℝ\nhr : x = r • y\nhzy : ⟪z, y⟫ = 0\nhzx : ⟪z, x⟫ = 0\n⊢ |(r - 1) * (r + 1) * ‖y‖ ^ 2| = |r ^ 2 * ‖y‖ ^ 2 - ‖y‖ ^ 2|",
"ppTerm": "?m.338",
"assigned": true,
"usedConstants": [
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Geometry.Euclidean.Sphere.Power | {
"line": 62,
"column": 42
} | {
"line": 62,
"column": 49
} | {
"line": 63,
"column": 4
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y z : V\nh₃ : ‖z - y‖ = ‖z + y‖\nr : ℝ\nhr : x = r • y\nhzy : ⟪z, y⟫ = 0\nhzx : ⟪z, x⟫ = 0\n⊢ |(r - 1) * (r + 1) * ‖y‖ ^ 2| = |r ^ 2 * ‖y‖ ^ 2 - ‖y‖ ^ 2|",
"ppTerm": "?m.338",
"assigned": true,
"usedConstants": [
... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.Sphere.Power | {
"line": 62,
"column": 42
} | {
"line": 62,
"column": 49
} | {
"line": 63,
"column": 4
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y z : V\nh₃ : ‖z - y‖ = ‖z + y‖\nr : ℝ\nhr : x = r • y\nhzy : ⟪z, y⟫ = 0\nhzx : ⟪z, x⟫ = 0\n⊢ |(r - 1) * (r + 1) * ‖y‖ ^ 2| = |r ^ 2 * ‖y‖ ^ 2 - ‖y‖ ^ 2|",
"ppTerm": "?m.338",
"assigned": true,
"usedConstants": [
... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Euclidean.Sphere.Power | {
"line": 129,
"column": 60
} | {
"line": 129,
"column": 90
} | {
"line": 130,
"column": 2
} | [
{
"pp": "V : Type u_1\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace ℝ V\nP : Type u_2\ninst✝³ : MetricSpace P\ninst✝² : NormedAddTorsor V P\ninst✝¹ : Fact (finrank ℝ V = 2)\ninst✝ : Oriented ℝ V (Fin 2)\np₁ p₂ p₃ p₄ p : P\nh : dist p₁ p * dist p₂ p = dist p₃ p * dist p₄ p\nhp₁p₂ : ∠ p₁ p p₂ = π\nhp... | [] | by grind [Set.pair_comm p₄ p₃] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Sphere.SecondInter | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 21
} | {
"line": 174,
"column": 21
} | [
{
"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' : P\nhp : p ∈ s\nhp' : dist p' s.center ≤ s.radius\n⊢ Wbtw ℝ p p' (s.secondInter p (p' -ᵥ p))",
"ppTerm": "?m.37",
"assigned": true,... | [
"case pos\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' : P\nhp : p ∈ s\nhp' : dist p' s.center ≤ s.radius\nh : p' = p\n⊢ Wbtw ℝ p p' (s.secondInter p (p' -ᵥ p))",
"case neg\nV : Type u_1\nP : Typ... | by_cases h : p' = p | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Geometry.Group.Growth.LinearLowerBound | {
"line": 49,
"column": 34
} | {
"line": 51,
"column": 50
} | {
"line": 52,
"column": 4
} | [
{
"pp": "G✝ : Type u_1\ninst✝³ : Group G✝\ninst✝² : DecidableEq G✝\nX✝ : Finset G✝\nn : ℕ\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nX : Finset G\nhX₁ : 1 ∈ X\nhX : X.Nontrivial\nhn : True\nhXn : X = X ^ 2\nx y : G\nhx : x ∈ ↑X\nhy : y ∈ ↑X\n⊢ x * y ∈ ↑X",
"ppTerm": "?m.237",
"assigned": tr... | [] | by
norm_cast at *
simpa [← hXn, ← sq] using! mul_mem_mul hx hy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Sphere.Power | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 9
} | {
"line": 320,
"column": 0
} | [
{
"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\ns : Sphere P\nt p : P\nh_tangent : s.IsTangentAt t line[ℝ, p, t]\n⊢ s.radius ^ 2 + dist p t ^ 2 - s.radius ^ 2 = dist p t ^ 2",
"ppTerm": "?m.62",
"ass... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 515,
"column": 6
} | {
"line": 517,
"column": 63
} | {
"line": 518,
"column": 2
} | [
{
"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\nt₁ t₂ : Triangle ℝ P\ni₁ i₂ i₃ j₁ j₂ j₃ : Fin 3\nhi₁₂ : i₁ ≠ i₂\nhi₁₃ : i₁ ≠ i₃\nhi₂₃ : i₂ ≠ i₃\nhj₁₂ : j₁ ≠ j₂\nhj₁₃ : j₁ ≠ j₃\nhj₂₃ : j₂ ≠ j₃\... | [] | rw [direction_affineSpan, direction_affineSpan,
t₁.independent.finrank_vectorSpan (Fintype.card_fin _),
t₂.independent.finrank_vectorSpan (Fintype.card_fin _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 515,
"column": 6
} | {
"line": 517,
"column": 63
} | {
"line": 518,
"column": 2
} | [
{
"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\nt₁ t₂ : Triangle ℝ P\ni₁ i₂ i₃ j₁ j₂ j₃ : Fin 3\nhi₁₂ : i₁ ≠ i₂\nhi₁₃ : i₁ ≠ i₃\nhi₂₃ : i₂ ≠ i₃\nhj₁₂ : j₁ ≠ j₂\nhj₁₃ : j₁ ≠ j₃\nhj₂₃ : j₂ ≠ j₃\... | [] | rw [direction_affineSpan, direction_affineSpan,
t₁.independent.finrank_vectorSpan (Fintype.card_fin _),
t₂.independent.finrank_vectorSpan (Fintype.card_fin _)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Euclidean.MongePoint | {
"line": 515,
"column": 6
} | {
"line": 517,
"column": 63
} | {
"line": 518,
"column": 2
} | [
{
"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\nt₁ t₂ : Triangle ℝ P\ni₁ i₂ i₃ j₁ j₂ j₃ : Fin 3\nhi₁₂ : i₁ ≠ i₂\nhi₁₃ : i₁ ≠ i₃\nhi₂₃ : i₂ ≠ i₃\nhj₁₂ : j₁ ≠ j₂\nhj₁₃ : j₁ ≠ j₃\nhj₂₃ : j₂ ≠ j₃\... | [] | rw [direction_affineSpan, direction_affineSpan,
t₁.independent.finrank_vectorSpan (Fintype.card_fin _),
t₂.independent.finrank_vectorSpan (Fintype.card_fin _)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.Atlas | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 82
} | {
"line": 120,
"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... | refine ⟨(e.continuousOn_symm x hx).continuousAt (e.open_target.mem_nhds hx), ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.VectorBundle.Tangent | {
"line": 409,
"column": 6
} | {
"line": 412,
"column": 30
} | {
"line": 413,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝¹¹ : NormedAddCommGroup E'\ninst✝¹⁰ : NormedSpace 𝕜 E'\nH : Type u_4\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nH' : Type u_5\nins... | [
"𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nE : Type u_2\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nE' : Type u_3\ninst✝¹¹ : NormedAddCommGroup E'\ninst✝¹⁰ : NormedSpace 𝕜 E'\nH : Type u_4\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nH' : Type u_5\ninst✝⁸ : Topolo... | have : Continuous (chartAt (ModelProd H E) p).symm := by
rw [← continuousOn_univ]
convert! (chartAt (ModelProd H E) p).symm.continuousOn
simp only [mfld_simps] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.Algebra.LeftInvariantDerivation | {
"line": 63,
"column": 18
} | {
"line": 63,
"column": 25
} | {
"line": 63,
"column": 25
} | [
{
"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\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : ChartedSpace H G\ninst✝¹ : Monoid G\ninst✝ : ContMDiffM... | [
"case mk\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : ChartedSpace H G\ninst✝¹ : Monoid G\ninst✝ : ContMDiffMul ... | cases X | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 390,
"column": 4
} | {
"line": 390,
"column": 79
} | {
"line": 391,
"column": 4
} | [
{
"pp": "case refine_1\n𝕜 : 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... | [
"case refine_1\n𝕜 : 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 : NormedAdd... | refine (hf.mdifferentiableAt hn).isInteriorPoint_of_surjective_mfderiv ?_ h | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.IsManifold.InteriorBoundary | {
"line": 499,
"column": 2
} | {
"line": 500,
"column": 71
} | {
"line": 501,
"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 aux : (interior (range ↑I)) ×ˢ (interior (range J)) = interior (range (I.prod J)) := by
rw [← interior_prod_eq, ← range_prodMap, modelWithCorners_prod_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 411,
"column": 67
} | {
"line": 413,
"column": 42
} | {
"line": 415,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\nF₁ : Type u_17\ninst✝³ : NormedAddCom... | [] | by
rw [modelWithCornersSelf_prod, ← chartedSpaceSelf_prod]
exact mdifferentiableWithinAt_prod_iff f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 639,
"column": 6
} | {
"line": 639,
"column": 56
} | {
"line": 640,
"column": 6
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_21\ninst✝¹ : To... | [
"case inl\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_21\ninst✝¹ : TopologicalSpa... | let t := I.symm ⁻¹' (chartAt H x).target ∩ range I | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 652,
"column": 6
} | {
"line": 652,
"column": 56
} | {
"line": 653,
"column": 6
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_21\ninst✝¹ : To... | [
"case inr\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_21\ninst✝¹ : TopologicalSpa... | let t := I.symm ⁻¹' (chartAt H x).target ∩ range I | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 496,
"column": 2
} | {
"line": 496,
"column": 73
} | {
"line": 498,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace (TotalSpace F E)\ninst✝¹¹ : (x : B) → TopologicalSpace (E x)\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 F\ninst✝⁷ : FiberBundle F E\ni... | [] | · exact Set.compl_subset_iff_union.mp <| Set.compl_subset_compl.mpr ht' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable | {
"line": 517,
"column": 25
} | {
"line": 517,
"column": 37
} | {
"line": 517,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace (TotalSpace F E)\ninst✝¹¹ : (x : B) → TopologicalSpace (E x)\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 F\ninst✝⁷ : FiberBundle F E\ni... | [
"𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace (TotalSpace F E)\ninst✝¹¹ : (x : B) → TopologicalSpace (E x)\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedSpace 𝕜 F\ninst✝⁷ : FiberBundle F E\ninst✝⁶ : (x :... | tsum_eq_sum' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions | {
"line": 1108,
"column": 2
} | {
"line": 1108,
"column": 9
} | {
"line": 1110,
"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\ns : Set M\nz : M\nF' : Type u_21\nins... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Geometry.Manifold.MFDeriv.NormedSpace | {
"line": 509,
"column": 43
} | {
"line": 511,
"column": 5
} | {
"line": 513,
"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\nF : Type u_8\ninst✝¹ : NormedAddCommG... | [] | by
simp [mvfderivWithin, mfderivWithin_add hg hg' hs]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.MFDeriv.NormedSpace | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 32
} | {
"line": 556,
"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\nF : Type u_8\ninst✝¹ : NormedAddCommG... | [] | simp [mvfderiv, mfderiv_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.MFDeriv.NormedSpace | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 32
} | {
"line": 556,
"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\nF : Type u_8\ninst✝¹ : NormedAddCommG... | [] | simp [mvfderiv, mfderiv_const] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.NormedSpace | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 32
} | {
"line": 556,
"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\nF : Type u_8\ninst✝¹ : NormedAddCommG... | [] | simp [mvfderiv, mfderiv_const] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorField.Pullback | {
"line": 274,
"column": 2
} | {
"line": 277,
"column": 67
} | {
"line": 280,
"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... | have hv : MDifferentiableWithinAt I I'.tangent
(fun x ↦ (v x : TangentBundle I' M')) (s ∩ f ⁻¹' t) x₀ := by
apply hV.comp x₀ ((hf.mdifferentiableWithinAt (by positivity)).mono inter_subset_left)
exact MapsTo.mono_left (mapsTo_preimage _ _) inter_subset_right | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.VectorBundle.Hom | {
"line": 202,
"column": 4
} | {
"line": 204,
"column": 44
} | {
"line": 205,
"column": 4
} | [
{
"pp": "𝕜₁ : Type u_1\ninst✝²¹ : NontriviallyNormedField 𝕜₁\n𝕜₂ : Type u_2\ninst✝²⁰ : NontriviallyNormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nB : Type u_3\nF₁ : Type u_4\ninst✝¹⁹ : NormedAddCommGroup F₁\ninst✝¹⁸ : NormedSpace 𝕜₁ F₁\nE₁ : B → Type u_5\ninst✝¹⁷ : (x : B) → AddCommGroup (E₁ x)\ninst✝¹⁶ : (x : B) → Modu... | [
"𝕜₁ : Type u_1\ninst✝²¹ : NontriviallyNormedField 𝕜₁\n𝕜₂ : Type u_2\ninst✝²⁰ : NontriviallyNormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nB : Type u_3\nF₁ : Type u_4\ninst✝¹⁹ : NormedAddCommGroup F₁\ninst✝¹⁸ : NormedSpace 𝕜₁ F₁\nE₁ : B → Type u_5\ninst✝¹⁷ : (x : B) → AddCommGroup (E₁ x)\ninst✝¹⁶ : (x : B) → Module 𝕜₁ (E₁ x... | let L₂ : E₂ b ≃L[𝕜₂] F₂ :=
(trivializationAt F₂ E₂ b).continuousLinearEquivAt 𝕜₂ b
(mem_baseSet_trivializationAt _ _ _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 452,
"column": 2
} | {
"line": 452,
"column": 64
} | {
"line": 453,
"column": 2
} | [
{
"pp": "case hV₁\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\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ns : Set M\nx : M\nV W V₁ : ... | [
"case hs\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\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\ns : Set M\nx : M\nV W V₁ : (x : M) → Tan... | · exact hV₁.differentiableWithinAt_mpullbackWithin_vectorField | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.Instances.Icc | {
"line": 78,
"column": 4
} | {
"line": 82,
"column": 37
} | {
"line": 83,
"column": 4
} | [
{
"pp": "case pos\nx y : ℝ\nh : Fact (x < y)\nn : WithTop ℕ∞\nz : ↑(Icc x y)\nhz : ↑z < y\n⊢ ContDiffWithinAt ℝ n ((fun z ↦ ↑z) ∘ ↑(IccLeftChart x y).symm ∘ ↑(𝓡∂ 1).symm) (range ↑(𝓡∂ 1))\n (↑(𝓡∂ 1) (↑(IccLeftChart x y) z))",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"case pos\nx y : ℝ\nh : Fact (x < y)\nn : WithTop ℕ∞\nz : ↑(Icc x y)\nhz : ↑z < y\n⊢ ContDiffWithinAt ℝ n (fun x_1 ↦ min (max (x_1.ofLp 0) 0 + x) y) (range Subtype.val) (toLp 2 fun x_1 ↦ ↑z - x)"
] | simp? [IccLeftChart, Function.comp_def, modelWithCornersEuclideanHalfSpace] says
simp only [IccLeftChart, Fin.isValue, OpenPartialHomeomorph.coe_mk_symm,
PartialEquiv.coe_symm_mk, modelWithCornersEuclideanHalfSpace, ModelWithCorners.mk_symm,
Function.comp_def, Function.update_self, ModelWithCorner... | Mathlib.Tactic.Says._aux_Mathlib_Tactic_Says___elabRules_Mathlib_Tactic_Says_says_1 | Mathlib.Tactic.Says.says |
Mathlib.Geometry.Manifold.Instances.Icc | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 42
} | {
"line": 204,
"column": 2
} | [
{
"pp": "x y : ℝ\nh : Fact (x < y)\nz : ↑(Icc x y)\nA : (mfderiv[Icc x y] (Subtype.val ∘ projIcc x y ⋯) ↑z) 1 = (mfderiv[Icc x y] id ↑z) 1\n⊢ (mfderiv% Subtype.val z) 1 = 1",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"instOneTangentSpaceRealModelWithCornersSelf",
"Eq.mpr",
... | [
"x y : ℝ\nh : Fact (x < y)\nz : ↑(Icc x y)\nA : (mfderiv[Icc x y] (Subtype.val ∘ projIcc x y ⋯) ↑z) 1 = (mfderiv[Icc x y] id ↑z) 1\n⊢ (mfderiv[Icc x y] id ↑z) 1 = 1"
] | rw [← mfderivWithin_comp_projIcc_one, A] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.GroupAction.CardCommute | {
"line": 57,
"column": 19
} | {
"line": 57,
"column": 40
} | {
"line": 58,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : Group α\ninst✝⁴ : MulAction α β\ninst✝³ : Fintype α\ninst✝² : Fintype β\ninst✝¹ : Fintype (Quotient (orbitRel α β))\ninst✝ : (b : β) → Fintype ↥(stabilizer α b)\nφ : Quotient (orbitRel α β) → β\nhφ : LeftInverse Quotient.mk'' φ\nthis :\n ∀ (ω : Quotient (orbitRel α... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁵ : Group α\ninst✝⁴ : MulAction α β\ninst✝³ : Fintype α\ninst✝² : Fintype β\ninst✝¹ : Fintype (Quotient (orbitRel α β))\ninst✝ : (b : β) → Fintype ↥(stabilizer α b)\nφ : Quotient (orbitRel α β) → β\nhφ : LeftInverse Quotient.mk'' φ\nthis :\n ∀ (ω : Quotient (orbitRel α β)),\n F... | ← Fintype.card_sigma, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 75,
"column": 82
} | {
"line": 75,
"column": 89
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case r.r.r\nn : ℕ\na b c : ZMod n\n⊢ r (a + b + c) = r (a + (b + c))",
"ppTerm": "?r.r.r",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.RingNF.add_assoc_rev",
"HMul.hMul",
"Nat.rawCast",
"ZMod.c... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 75,
"column": 82
} | {
"line": 75,
"column": 89
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case r.r.sr\nn : ℕ\na b c : ZMod n\n⊢ sr (c - (a + b)) = sr (c - b - a)",
"ppTerm": "?r.r.sr",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 75,
"column": 82
} | {
"line": 75,
"column": 89
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case r.sr.r\nn : ℕ\na b c : ZMod n\n⊢ sr (b - a + c) = sr (b + c - a)",
"ppTerm": "?r.sr.r",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.R... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 75,
"column": 82
} | {
"line": 75,
"column": 89
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case r.sr.sr\nn : ℕ\na b c : ZMod n\n⊢ r (c - (b - a)) = r (a + (c - b))",
"ppTerm": "?r.sr.sr",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tact... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 75,
"column": 82
} | {
"line": 75,
"column": 89
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case sr.r.r\nn : ℕ\na b c : ZMod n\n⊢ sr (a + b + c) = sr (a + (b + c))",
"ppTerm": "?sr.r.r",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.RingNF.add_assoc_rev",
"HMul.hMul",
"Nat.rawCast",
"ZM... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 75,
"column": 82
} | {
"line": 75,
"column": 89
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case sr.r.sr\nn : ℕ\na b c : ZMod n\n⊢ r (c - (a + b)) = r (c - b - a)",
"ppTerm": "?sr.r.sr",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 75,
"column": 82
} | {
"line": 75,
"column": 89
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case sr.sr.r\nn : ℕ\na b c : ZMod n\n⊢ r (b - a + c) = r (b + c - a)",
"ppTerm": "?sr.sr.r",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.R... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 75,
"column": 82
} | {
"line": 75,
"column": 89
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case sr.sr.sr\nn : ℕ\na b c : ZMod n\n⊢ sr (c - (b - a)) = sr (a + (c - b))",
"ppTerm": "?sr.sr.sr",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 234,
"column": 4
} | {
"line": 234,
"column": 76
} | {
"line": 236,
"column": 0
} | [
{
"pp": "n : ℕ\nx✝¹ : n + 3 ≠ 1\nx✝ : n + 3 ≠ 2\nh' : IsMulCommutative (DihedralGroup (n + 3))\nthis : 2 % (n + 3) = 0\n⊢ False",
"ppTerm": "?m.190",
"assigned": true,
"usedConstants": [
"False",
"of_decide_eq_true",
"Nat.Simproc.add_le_gt",
"False.elim",
"Eq.mp",
... | [] | simpa using Nat.le_of_dvd Nat.zero_lt_two <| Nat.dvd_of_mod_eq_zero this | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 237,
"column": 8
} | {
"line": 237,
"column": 68
} | {
"line": 238,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ IsMulCommutative (DihedralGroup n) → n = 1 ∨ n = 2",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"DihedralGroup.instGroup",
"id",
"MulOne.toMul",
"DivInvMonoid.toMonoid",
"Ne",
"instOfNatN... | [] | by contrapose!; rintro ⟨h1, h2⟩; exact not_commutative h1 h2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.CommutingProbability | {
"line": 48,
"column": 2
} | {
"line": 49,
"column": 100
} | {
"line": 51,
"column": 0
} | [
{
"pp": "case e_a\nM : Type u_1\ninst✝¹ : Mul M\nM' : Type u_2\ninst✝ : Mul M'\n⊢ Nat.card { p // (p.1 * p.2).1 = (p.2 * p.1).1 ∧ (p.1 * p.2).2 = (p.2 * p.1).2 } =\n Nat.card ({ p // p.1 * p.2 = p.2 * p.1 } × { p // p.1 * p.2 = p.2 * p.1 })",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": ... | [] | exact Nat.card_congr ⟨fun x => ⟨⟨⟨x.1.1.1, x.1.2.1⟩, x.2.1⟩, ⟨⟨x.1.1.2, x.1.2.2⟩, x.2.2⟩⟩,
fun x => ⟨⟨⟨x.1.1.1, x.2.1.1⟩, ⟨x.1.1.2, x.2.1.2⟩⟩, ⟨x.1.2, x.2.2⟩⟩, fun x => rfl, fun x => rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.CommutingProbability | {
"line": 85,
"column": 59
} | {
"line": 85,
"column": 83
} | {
"line": 85,
"column": 83
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Mul M\ninst✝ : Finite M\nh : Nonempty M\nthis : Fintype M\n⊢ ↑(card ↑{x | Commute x.1 x.2}) / ↑(Nat.card M) ^ 2 = 1 ↔ IsMulCommutative M",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"instHDiv",
"congrArg"... | [
"M : Type u_1\ninst✝¹ : Mul M\ninst✝ : Finite M\nh : Nonempty M\nthis : Fintype M\n⊢ ↑(card ↑{x | Commute x.1 x.2}) / ↑(card M) ^ 2 = 1 ↔ IsMulCommutative M"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 272,
"column": 31
} | {
"line": 272,
"column": 59
} | {
"line": 272,
"column": 60
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_5\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nf : M ∗ N →* P\n⊢ Submonoid.map f ⊤ = MonoidHom.mrange (f.comp inl) ⊔ MonoidHom.mrange (f.comp inr)",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"M : Type u_1\nN : Type u_2\nP : Type u_5\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nf : M ∗ N →* P\n⊢ Submonoid.map f (MonoidHom.mrange inl ⊔ MonoidHom.mrange inr) =\n MonoidHom.mrange (f.comp inl) ⊔ MonoidHom.mrange (f.comp inr)"
] | ← mrange_inl_sup_mrange_inr, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CoprodI | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 11
} | {
"line": 184,
"column": 12
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝¹ : (i : ι) → Monoid (M i)\nN : Type u_4\ninst✝ : Monoid N\nf : (i : ι) → M i →* N\n⊢ MonoidHom.mrange (lift f) = ⨆ i, MonoidHom.mrange (f i)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"FreeMonoid.lift",
"Eq.mpr",
"Monoid... | [
"ι : Type u_1\nM : ι → Type u_2\ninst✝¹ : (i : ι) → Monoid (M i)\nN : Type u_4\ninst✝ : Monoid N\nf : (i : ι) → M i →* N\n⊢ MonoidHom.mrange\n ({ toFun := fun fi ↦ (conGen (Rel M)).lift (FreeMonoid.lift fun p ↦ (fi p.fst) p.snd) ⋯,\n invFun := fun f x ↦ f.comp of, left_inv := ⋯, right_inv := ⋯ }\n ... | lift, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 51
} | {
"line": 121,
"column": 2
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw₁ w₂ : W\n⊢ cs.length (w₁ * w₂) ≤ cs.length w₁ + cs.length w₂",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"CoxeterSystem.exists_isR... | [
"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw₂ : W\nω₁ : List B\nhω₁ : cs.IsReduced ω₁\n⊢ cs.length (cs.wordProd ω₁ * w₂) ≤ cs.length (cs.wordProd ω₁) + cs.length w₂"
] | rcases cs.exists_isReduced w₁ with ⟨ω₁, hω₁, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 66
} | {
"line": 139,
"column": 0
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw t : W\nht : cs.IsReflection t\n⊢ cs.length (w⁻¹ * t) < cs.length w⁻¹ ↔ cs.length (t * w) < cs.length w",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.to... | [] | rw [← length_inv, mul_inv_rev, inv_inv, ht.inv, cs.length_inv w] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 163,
"column": 62
} | {
"line": 165,
"column": 37
} | {
"line": 167,
"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\nw : W\n⊢ cs.IsLeftInversion (t * w) t ↔ ¬cs.IsLeftInversion w t",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CoxeterSystem.IsLeftInver... | [] | by
rw [← isRightInversion_inv_iff, ← isRightInversion_inv_iff, mul_inv_rev, ht.inv,
ht.isRightInversion_mul_left_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 318,
"column": 2
} | {
"line": 323,
"column": 75
} | {
"line": 325,
"column": 0
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nj : ℕ\n⊢ cs.rightInvSeq (drop j ω) = drop j (cs.rightInvSeq ω)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.recAux",
"HMul.hMul",
"DivInv... | [] | induction j generalizing ω with
| zero => simp
| succ j ih₁ =>
induction ω with
| nil => simp
| cons k ω _ => rw [drop_succ_cons, ih₁ ω, rightInvSeq, drop_succ_cons] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 318,
"column": 2
} | {
"line": 323,
"column": 75
} | {
"line": 325,
"column": 0
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nj : ℕ\n⊢ cs.rightInvSeq (drop j ω) = drop j (cs.rightInvSeq ω)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.recAux",
"HMul.hMul",
"DivInv... | [] | induction j generalizing ω with
| zero => simp
| succ j ih₁ =>
induction ω with
| nil => simp
| cons k ω _ => rw [drop_succ_cons, ih₁ ω, rightInvSeq, drop_succ_cons] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 318,
"column": 2
} | {
"line": 323,
"column": 75
} | {
"line": 325,
"column": 0
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nj : ℕ\n⊢ cs.rightInvSeq (drop j ω) = drop j (cs.rightInvSeq ω)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.recAux",
"HMul.hMul",
"DivInv... | [] | induction j generalizing ω with
| zero => simp
| succ j ih₁ =>
induction ω with
| nil => simp
| cons k ω _ => rw [drop_succ_cons, ih₁ ω, rightInvSeq, drop_succ_cons] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.DivisibleHull | {
"line": 239,
"column": 6
} | {
"line": 239,
"column": 13
} | {
"line": 240,
"column": 4
} | [
{
"pp": "case e'_2\nM✝ : Type u_1\ninst✝¹ : AddCommMonoid M✝\nM : Type u_2\ninst✝ : AddCommGroup M\na b : ℚ\nm : M\ns : ℕ+\nthis : ((a * b).num * ↑a.den * ↑b.den * ↑↑s) • m = (a.num * b.num * ↑(a * b).den * ↑↑s) • m\n⊢ (↑a.den * (↑b.den * ↑↑s) * (a * b).num) • m = ((a * b).num * ↑a.den * ↑b.den * ↑↑s) • m",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.DivisibleHull | {
"line": 239,
"column": 6
} | {
"line": 239,
"column": 13
} | {
"line": 240,
"column": 4
} | [
{
"pp": "case e'_3\nM✝ : Type u_1\ninst✝¹ : AddCommMonoid M✝\nM : Type u_2\ninst✝ : AddCommGroup M\na b : ℚ\nm : M\ns : ℕ+\nthis : ((a * b).num * ↑a.den * ↑b.den * ↑↑s) • m = (a.num * b.num * ↑(a * b).den * ↑↑s) • m\n⊢ (↑(a * b).den * ↑↑s * (a.num * b.num)) • m = (a.num * b.num * ↑(a * b).den * ↑↑s) • m",
"... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.DivisibleHull | {
"line": 229,
"column": 6
} | {
"line": 229,
"column": 13
} | {
"line": 230,
"column": 4
} | [
{
"pp": "case e'_2\nM✝ : Type u_1\ninst✝¹ : AddCommMonoid M✝\nM : Type u_2\ninst✝ : AddCommGroup M\na b : ℚ\nm : M\ns : ℕ+\nthis :\n ((a + b).num * ↑a.den * ↑b.den * (↑↑s * ↑↑s)) • m =\n ((a.num * ↑b.den + b.num * ↑a.den) * ↑(a + b).den * (↑↑s * ↑↑s)) • m\n⊢ (↑a.den * ↑↑s * (↑b.den * ↑↑s) * (a + b).num) • m... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.DivisibleHull | {
"line": 229,
"column": 6
} | {
"line": 229,
"column": 13
} | {
"line": 230,
"column": 4
} | [
{
"pp": "case e'_3\nM✝ : Type u_1\ninst✝¹ : AddCommMonoid M✝\nM : Type u_2\ninst✝ : AddCommGroup M\na b : ℚ\nm : M\ns : ℕ+\nthis :\n ((a + b).num * ↑a.den * ↑b.den * (↑↑s * ↑↑s)) • m =\n ((a.num * ↑b.den + b.num * ↑a.den) * ↑(a + b).den * (↑↑s * ↑↑s)) • m\n⊢ (↑(a + b).den * ↑↑s * (↑b.den * ↑↑s * a.num + ↑a.... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.DoubleCoset | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 100
} | {
"line": 66,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\na b l : G\nhl : l ∈ ↑H\nr : G\nhr : r ∈ ↑K\ny : G\nhy : y ∈ ↑H\nr' : G\nhr' : r' ∈ ↑K\nhrx : y * b * r' = l * a * r\n⊢ b = y⁻¹ * l * a * (r * r'⁻¹)",
"ppTerm": "?m.141",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.to... | [] | rwa [mul_assoc, mul_assoc, eq_inv_mul_iff_mul_eq, ← mul_assoc, ← mul_assoc, eq_mul_inv_iff_mul_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.GroupTheory.DivisibleHull | {
"line": 258,
"column": 4
} | {
"line": 258,
"column": 11
} | {
"line": 259,
"column": 2
} | [
{
"pp": "case e'_1.e'_3\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nm n m' n' : M\ns t s' t' : ℕ+\nh : ↑s' • m = ↑s • m'\nh' : ↑t' • n = ↑t • n'\n⊢ (↑t' * (↑t * ↑s)) • m' = (↑s * (↑t * ↑t')) • m'",
"ppTerm": "?e'_1.e'_3",
"assigned": true,
"us... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.DivisibleHull | {
"line": 260,
"column": 4
} | {
"line": 260,
"column": 11
} | {
"line": 262,
"column": 0
} | [
{
"pp": "case e'_1.e'_4\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nm n m' n' : M\ns t s' t' : ℕ+\nh : ↑s' • m = ↑s • m'\nh' : ↑t' • n = ↑t • n'\n⊢ (↑s' * (↑t' * ↑s)) • n = (↑s' * (↑s * ↑t')) • n",
"ppTerm": "?e'_1.e'_4",
"assigned": true,
"us... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.GroupTheory.DoubleCoset | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 34
} | {
"line": 201,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\na✝ b✝ : G\n⊢ (setoid ↑⊥ ↑H) a✝ b✝ ↔ (leftRel H) a✝ b✝",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"congrArg",
"QuotientGroup.leftRel",
"Subgroup",
"Bot.bot",
"iff_self",
"Iff",
"SetLike.co... | [] | simp_rw [← bot_rel_eq_leftRel H] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 182,
"column": 17
} | {
"line": 187,
"column": 8
} | {
"line": 189,
"column": 0
} | [
{
"pp": "G : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nhM : HasEnoughRootsOfUnity M (Monoid.exponent G)\nΦ : (Subgroup (G →* Mˣ))ᵒᵈ\n⊢ (fun H ↦ OrderDual.toDual (restrictHom H Mˣ).ker)\n ((fun Φ ↦ (monoidHomMonoidHomEquiv G M).mapSubgroup (restrictHom (OrderDual... | [] | by
have : HasEnoughRootsOfUnity M (Monoid.exponent (G →* Mˣ)) := by
rwa [Monoid.exponent_eq_of_mulEquiv (monoidHom_mulEquiv_of_hasEnoughRootsOfUnity G M).some]
ext φ
rw [OrderDual.ofDual_toDual, mem_ker, restrictHom_apply, restrict_eq_one_iff]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Transfer | {
"line": 135,
"column": 30
} | {
"line": 135,
"column": 42
} | {
"line": 135,
"column": 42
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)\nk : ZMod (minimalPeriod (fun x ↦ g • x) (Quotient.out q))\nhk : k = 0\n⊢ g ^ ↑(minimalPeriod (fun x ↦ g • x) (Quotient.out ⟨q, k - 1⟩.fst)) * Quotient.out (Quotient.out ⟨q, k - 1⟩.fst) =\n g... | [
"case pos\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)\nk : ZMod (minimalPeriod (fun x ↦ g • x) (Quotient.out q))\nhk : k = 0\n⊢ g ^ minimalPeriod (fun x ↦ g • x) (Quotient.out ⟨q, k - 1⟩.fst) * Quotient.out (Quotient.out ⟨q, k - 1⟩.fst) =\n g ^ minimalPerio... | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Transfer | {
"line": 136,
"column": 8
} | {
"line": 136,
"column": 18
} | {
"line": 136,
"column": 19
} | [
{
"pp": "case neg\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)\nk : ZMod (minimalPeriod (fun x ↦ g • x) (Quotient.out q))\nhk : ¬k = 0\n⊢ (g ^ if k = 0 then ↑(minimalPeriod (fun x ↦ g • x) (Quotient.out ⟨q, k - 1⟩.fst)) else k.cast) *\n Quotient.out (Q... | [
"case neg\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\nq : orbitRel.Quotient (↥(zpowers g)) (G ⧸ H)\nk : ZMod (minimalPeriod (fun x ↦ g • x) (Quotient.out q))\nhk : ¬k = 0\n⊢ g ^ k.cast * Quotient.out (Quotient.out ⟨q, k - 1⟩.fst) =\n if k = 0 then g ^ minimalPeriod (fun x ↦ g • x) (Quotient.out q) * Q... | if_neg hk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Transfer | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 10
} | {
"line": 314,
"column": 2
} | [
{
"pp": "G : Type u_3\ninst✝¹ : Group G\ninst✝ : Finite G\np : ℕ\nhp : (Nat.card G).minFac = p\nP : Sylow p G\nhP : IsCyclic ↥↑P\n⊢ normalizer ↑P ≤ centralizer ↑P",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Sylow.toSubgroup",
"Sylow.instSetLike",
"Sylow",
"Subg... | [
"G : Type u_3\ninst✝¹ : Group G\ninst✝ : Finite G\nP : Sylow (Nat.card G).minFac G\nhP : IsCyclic ↥↑P\n⊢ normalizer ↑P ≤ centralizer ↑P"
] | subst hp | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.GroupTheory.FixedPointFree | {
"line": 90,
"column": 46
} | {
"line": 90,
"column": 70
} | {
"line": 90,
"column": 70
} | [
{
"pp": "F : Type u_1\nG : Type u_2\ninst✝³ : Group G\ninst✝² : FunLike F G G\ninst✝¹ : MonoidHomClass F G G\nφ : F\ninst✝ : Finite G\nhφ : FixedPointFree ⇑φ\nh2 : Function.Involutive ⇑φ\nthis : Fintype G\nh : 2 ∣ Nat.card G\n⊢ False",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"N... | [
"F : Type u_1\nG : Type u_2\ninst✝³ : Group G\ninst✝² : FunLike F G G\ninst✝¹ : MonoidHomClass F G G\nφ : F\ninst✝ : Finite G\nhφ : FixedPointFree ⇑φ\nh2 : Function.Involutive ⇑φ\nthis : Fintype G\nh : 2 ∣ Fintype.card G\n⊢ False"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Transfer | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 10
} | {
"line": 345,
"column": 2
} | [
{
"pp": "G : Type u_3\ninst✝¹ : Group G\ninst✝ : Finite G\np : ℕ\nhp : (Nat.card G).minFac = p\nP : Sylow p G\nhP : IsCyclic ↥↑P\n⊢ (MonoidHom.transferSylow P ⋯).ker.IsComplement' ↑P",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Sylow.toSubgroup",
"Sylow",
"Monoid.toMu... | [
"G : Type u_3\ninst✝¹ : Group G\ninst✝ : Finite G\nP : Sylow (Nat.card G).minFac G\nhP : IsCyclic ↥↑P\n⊢ (MonoidHom.transferSylow P ⋯).ker.IsComplement' ↑P"
] | subst hp | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.GroupTheory.Schreier | {
"line": 160,
"column": 41
} | {
"line": 160,
"column": 65
} | {
"line": 160,
"column": 65
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.FiniteIndex\nS : Finset G\nhS : closure ↑S = ⊤\nthis✝¹ : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex\nthis✝ : DecidableEq G\nR₀ : Set G\nthis : Fintype ↑R₀\nR : Finset G := R₀.toFinset\nhR : IsComplement ↑H ↑R\nhR1 : 1 ∈ R\n⊢ Fintype.card (G... | [
"G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.FiniteIndex\nS : Finset G\nhS : closure ↑S = ⊤\nthis✝¹ : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex\nthis✝ : DecidableEq G\nR₀ : Set G\nthis : Fintype ↑R₀\nR : Finset G := R₀.toFinset\nhR : IsComplement ↑H ↑R\nhR1 : 1 ∈ R\n⊢ Fintype.card (G ⧸ H) = Fint... | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 232,
"column": 9
} | {
"line": 232,
"column": 62
} | {
"line": 233,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nH : Type u_2\ninst✝ : Group H\ne : H ≃* G\n⊢ comap (↑e) (upperCentralSeries G 0) = upperCentralSeries H 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulEquiv.instEquivLike",
"_private.Mathlib.GroupTheory.Nilpotent.0... | [] | by simpa [MonoidHom.ker_eq_bot_iff] using e.injective | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Nilpotent | {
"line": 444,
"column": 4
} | {
"line": 445,
"column": 40
} | {
"line": 447,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS : Subgroup G\nn : ℕ\n⊢ S ≤ normalizer ↑(S.lowerCentralSeries (n + 1))",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.normalizer_commutator_ge_right",
"congrArg",
"PartialOrder.toPreorder",
"Bracket... | [] | rw [lowerCentralSeries_succ]
apply normalizer_commutator_ge_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Nilpotent | {
"line": 444,
"column": 4
} | {
"line": 445,
"column": 40
} | {
"line": 447,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS : Subgroup G\nn : ℕ\n⊢ S ≤ normalizer ↑(S.lowerCentralSeries (n + 1))",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.normalizer_commutator_ge_right",
"congrArg",
"PartialOrder.toPreorder",
"Bracket... | [] | rw [lowerCentralSeries_succ]
apply normalizer_commutator_ge_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Nilpotent | {
"line": 676,
"column": 39
} | {
"line": 679,
"column": 60
} | {
"line": 681,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nN : Subgroup G\ninst✝¹ : N.Normal\nS : Subgroup G\ninst✝ : S.Normal\nn : ℕ\n⊢ (S.lowerCentralSeries n).Normal",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.recAux",
"congrArg",
"Bracket.bracket",
"infer... | [] | by
induction n with
| zero => simpa
| succ n _ => rw [lowerCentralSeries_succ]; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 278,
"column": 64
} | {
"line": 290,
"column": 46
} | {
"line": 292,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝¹ : Groupoid G\ninst✝ : IsFreeGroupoid G\na b : G\n⊢ Nonempty (a ⟶ b) → Nonempty (Path (symgen a) (symgen b))",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"FreeGroup.of",
"Eq.mpr",
"Unit.unit",
"MulOne.toOne",
"Cat... | [] | by
rintro ⟨p⟩
rw [← @WeaklyConnectedComponent.eq (Generators G), eq_comm, ← FreeGroup.of_injective.eq_iff, ←
mul_inv_eq_one]
let X := FreeGroup (WeaklyConnectedComponent <| Generators G)
let f : G → X := fun g => FreeGroup.of (WeaklyConnectedComponent.mk g)
let F : G ⥤ CategoryTheory.SingleObj.{u} (X : Ty... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Nilpotent | {
"line": 747,
"column": 30
} | {
"line": 747,
"column": 42
} | {
"line": 747,
"column": 42
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Type u_2\ninst✝¹ : Group H\nf : G →* H\nhf1 : f.ker ≤ center G\ninst✝ : Group.IsNilpotent H\nn : ℕ\nhn : ⊤.lowerCentralSeries n = ⊥\n⊢ (map f ⊤).lowerCentralSeries n = ⊥",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"le_bot_iff",
"... | [
"G : Type u_1\ninst✝² : Group G\nH : Type u_2\ninst✝¹ : Group H\nf : G →* H\nhf1 : f.ker ≤ center G\ninst✝ : Group.IsNilpotent H\nn : ℕ\nhn : ⊤.lowerCentralSeries n = ⊥\n⊢ (map f ⊤).lowerCentralSeries n ≤ ⊥"
] | ← le_bot_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfStabilizer | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 28
} | {
"line": 96,
"column": 28
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\na : α\nthis : Fintype α := Fintype.ofFinite α\n⊢ Fintype.card α = Nat.card α",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Fintype.card",
"id",
... | [
"G : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\na : α\nthis : Fintype α := Fintype.ofFinite α\n⊢ Fintype.card α = Fintype.card α",
"G : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\na : α\nthis : Fintype α := Fintype.ofFinite α\n⊢ ∀ (x... | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Embedding | {
"line": 56,
"column": 9
} | {
"line": 56,
"column": 26
} | {
"line": 56,
"column": 27
} | [
{
"pp": "case inl\nα : Type u_1\nn : ℕ\ns : Set α\ninst✝ : Finite ↑s\nhs : ↑s.ncard + ↑n ≤ ENat.card α\nhα : Finite α\nx✝ : Fintype α := ⋯\n⊢ Fintype.card (Fin n) ≤ Fintype.card ↑sᶜ",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.card_fin",
"congrArg",
... | [
"case inl\nα : Type u_1\nn : ℕ\ns : Set α\ninst✝ : Finite ↑s\nhs : ↑s.ncard + ↑n ≤ ENat.card α\nhα : Finite α\nx✝ : Fintype α := Fintype.ofFinite α\n⊢ n ≤ Fintype.card ↑sᶜ"
] | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 377,
"column": 8
} | {
"line": 377,
"column": 39
} | {
"line": 377,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα4 : 4 ≤ Nat.card α\ng : Perm α\nhg : g ∈ alternatingGroup α\nhg' : ⟨g, hg⟩ ∈ center ↥(alternatingGroup α)\na : α\nha : a ∈ g.support\nhab : g a ≠ a\nc d : α\nhcd : c ≠ d\nhc : c ≠ a ∧ c ≠ g a\nhd : d ≠ a ∧ d ≠ g a\nk : Perm α := swap (g a) d * ... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα4 : 4 ≤ Nat.card α\ng : Perm α\nhg : g ∈ alternatingGroup α\nhg' : ⟨g, hg⟩ ∈ center ↥(alternatingGroup α)\na : α\nha : a ∈ g.support\nhab : g a ≠ a\nc d : α\nhcd : c ≠ d\nhc : c ≠ a ∧ c ≠ g a\nhd : d ≠ a ∧ d ≠ g a\nk : Perm α := swap (g a) d * swap (g a) c... | swap_apply_of_ne_of_ne hc.2 hcd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 83,
"column": 27
} | {
"line": 83,
"column": 45
} | {
"line": 85,
"column": 0
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\nH : Type u_3\nβ : Type u_4\ninst✝¹ : Group H\ninst✝ : MulAction H β\nσ : G → H\nf : α →ₑ[σ] β\nι : Type u_5\nhf : Injective ⇑f\nm : G\nx : ι ↪ α\nx✝ : ι\n⊢ { toFun := f.toFun ∘ (m • x).toFun, inj' := ⋯ } x✝ = (σ m • { toFun := f.toFu... | [] | simp [f.map_smul'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 17
} | {
"line": 112,
"column": 0
} | [
{
"pp": "case h\nG : Type u_1\nα : Type u_2\ninst✝³ : Group G\ninst✝² : MulAction G α\nH : Type u_3\nβ : Type u_4\ninst✝¹ : Group H\ninst✝ : MulAction H β\nσ : G → H\nf : α →ₑ[σ] β\nι : Type u_5\nhf : Bijective ⇑f\ny : ι ↪ β\ng : β → α\nleft✝ : LeftInverse g ⇑f\nhfg : RightInverse g ⇑f\nx✝ : ι\n⊢ f (g (y x✝)) =... | [] | exact hfg (y _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 477,
"column": 10
} | {
"line": 477,
"column": 25
} | {
"line": 477,
"column": 26
} | [
{
"pp": "k : ℕ\nhrec :\n ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],\n IsMultiplyPretransitive G α k →\n ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!\nG : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulActio... | [
"k : ℕ\nhrec :\n ∀ {G : Type u_1} [inst : Group G] {α : Type u_2} [inst_1 : MulAction G α] [Finite α],\n IsMultiplyPretransitive G α k →\n ∀ {s : Set α}, s.ncard = k → (fixingSubgroup G s).index * (Nat.card α - k)! = (Nat.card α)!\nG : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝... | ← Nat.succ_inj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 18
} | {
"line": 570,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : Finite α\nG : Subgroup (Perm α)\nthis : Fintype α\nhmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 1)\nj : Fin (Fintype.card α - 1) ↪ Fin (Fintype.card α) := castLEEmb ⋯\nk : Perm α\na✝ : k ∈ ⊤\nx : Fin (Fintype.card α) ↪ α := (Fintype.equivFinOfCardEq ⋯).symm.toEmbedding\n... | [
"α : Type u_1\ninst✝ : Finite α\nG : Subgroup (Perm α)\nthis : Fintype α\nhmt : IsMultiplyPretransitive (↥G) α (Fintype.card α - 1)\nj : Fin (Fintype.card α - 1) ↪ Fin (Fintype.card α) := castLEEmb ⋯\nk : Perm α\na✝ : k ∈ ⊤\nx : Fin (Fintype.card α) ↪ α := (Fintype.equivFinOfCardEq ⋯).symm.toEmbedding\nx' : Fin (Fi... | specialize hgk i | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 595,
"column": 36
} | {
"line": 595,
"column": 53
} | {
"line": 595,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh2 : 2 ≤ Nat.card α\nh2le : Nat.card α - 2 ≤ Nat.card α\nthis✝ : IsMultiplyPretransitive (Equiv.Perm α) α (Nat.card α)\nthis : IsMultiplyPretransitive (Equiv.Perm α) α (Nat.card α - 2)\nx y : Fin (Nat.card α - 2) ↪ α\ng : Equiv.Perm α\nhg : g • x... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh2 : 2 ≤ Nat.card α\nh2le : Nat.card α - 2 ≤ Nat.card α\nthis✝ : IsMultiplyPretransitive (Equiv.Perm α) α (Nat.card α)\nthis : IsMultiplyPretransitive (Equiv.Perm α) α (Nat.card α - 2)\nx y : Fin (Nat.card α - 2) ↪ α\ng : Equiv.Perm α\nhg : g • x = y\nh : Eq... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Period | {
"line": 79,
"column": 75
} | {
"line": 79,
"column": 87
} | {
"line": 79,
"column": 87
} | [
{
"pp": "α : Type v\nG : Type u\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng : G\na : α\nn : ℕ\n⊢ g ^ ↑n • a = a ↔ g ^ n • a = a",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"instHSMul",
"congrArg",
"DivInvMonoid.toZPow",
"Gro... | [
"α : Type v\nG : Type u\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng : G\na : α\nn : ℕ\n⊢ g ^ n • a = a ↔ g ^ n • a = a"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 242,
"column": 65
} | {
"line": 248,
"column": 24
} | {
"line": 249,
"column": 2
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns B : Set α\nhB_ss_sc : B ⊂ s\nhB : IsBlock M B\nhG : Function.Surjective toPerm\nthis : IsPreprimitive ↥(stabilizer M s) ↑s\n⊢ IsTrivialBlock (Subtype.val ⁻¹' B)",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
... | [] | by
apply this.isTrivialBlock_of_isBlock
let φ' : stabilizer M (s : Set α) → M := Subtype.val
let f' : (s : Set α) →ₑ[φ'] α := {
toFun := Subtype.val
map_smul' _ _ := rfl }
exact hB.preimage f' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 89
} | {
"line": 108,
"column": 4
} | [
{
"pp": "case property\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ns : Set α\nhs : sᶜ.Nontrivial\nk : Perm α\nhk_swap : k.IsSwap\nhk_support : _root_.Disjoint s ↑k.support\nhks : k • s = s\ng : Perm ↑s\nhsg : sign g = -1\n⊢ ⟨Perm.ofSubtype g * k, ⋯⟩ ∈ stabilizer (↥(alternatingGroup α)) s",
"pp... | [
"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ns : Set α\nhs : sᶜ.Nontrivial\nk : Perm α\nhk_swap : k.IsSwap\nhk_support : _root_.Disjoint s ↑k.support\nhks : k • s = s\ng : Perm ↑s\nhsg : sign g = -1\n⊢ toPerm ⟨⟨Perm.ofSubtype g * k, ⋯⟩, ⋯⟩ = g"
] | · rw [mem_stabilizer_iff, Submonoid.mk_smul, mul_smul, hks, ofSubtype_mem_stabilizer] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 137,
"column": 45
} | {
"line": 143,
"column": 7
} | {
"line": 145,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ns : Set α\nhs : s.Nonempty\nhsc : sᶜ.Nontrivial\n⊢ stabilizer (↥(alternatingGroup α)) s ≠ ⊤",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Equiv.Perm.applyMulAction",
... | [] | by
obtain ⟨a, ha⟩ := hs
obtain ⟨b, hb, c, hc, hbc⟩ := hsc
suffices ∃ g, g ∉ stabilizer (alternatingGroup α) s by contrapose! this; simp [this]
use ⟨Equiv.swap a b * Equiv.swap a c, by aesop⟩
simp_rw [mem_stabilizer_set, Subgroup.mk_smul, mul_smul, Perm.smul_def]
grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 325,
"column": 72
} | {
"line": 325,
"column": 89
} | {
"line": 326,
"column": 8
} | [
{
"pp": "case neg\nα : Type u_1\nK : Type u_2\ninst✝¹ : Group K\ninst✝ : MulAction K α\nhα : Nat.card α = 2\nhK : fixedPoints K α ≠ _root_.Set.univ\nn : ℕ\nthis✝ : Finite α\nthis : Fintype α\nh2 : IsMultiplyPretransitive K α 2\nhn : ¬n ≤ 2\n⊢ ¬Fintype.card (Fin n) ≤ Fintype.card α",
"ppTerm": "?neg✝",
"... | [
"case neg\nα : Type u_1\nK : Type u_2\ninst✝¹ : Group K\ninst✝ : MulAction K α\nhα : Nat.card α = 2\nhK : fixedPoints K α ≠ _root_.Set.univ\nn : ℕ\nthis✝ : Finite α\nthis : Fintype α\nh2 : IsMultiplyPretransitive K α 2\nhn : ¬n ≤ 2\n⊢ ¬n ≤ Fintype.card α"
] | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 157,
"column": 2
} | {
"line": 160,
"column": 27
} | {
"line": 162,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulAction G α\nn : ℕ\ninst✝ : DecidableEq α\n⊢ Function.Surjective ⇑(mulActionHom_of_embedding G α n)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.card_fin",
"Function.Embedding.exists_o... | [] | intro ⟨s, hs⟩
obtain ⟨f : Fin n ↪ α, hf⟩ :=
Function.Embedding.exists_of_card_eq_finset (by rw [hs, Fintype.card_fin])
exact ⟨f, Subtype.ext hf⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 157,
"column": 2
} | {
"line": 160,
"column": 27
} | {
"line": 162,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulAction G α\nn : ℕ\ninst✝ : DecidableEq α\n⊢ Function.Surjective ⇑(mulActionHom_of_embedding G α n)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.card_fin",
"Function.Embedding.exists_o... | [] | intro ⟨s, hs⟩
obtain ⟨f : Fin n ↪ α, hf⟩ :=
Function.Embedding.exists_of_card_eq_finset (by rw [hs, Fintype.card_fin])
exact ⟨f, Subtype.ext hf⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 260,
"column": 6
} | {
"line": 260,
"column": 48
} | {
"line": 260,
"column": 48
} | [
{
"pp": "α : Type u_2\ninst✝ : DecidableEq α\nn : ℕ\nh_one_le : 1 ≤ n\nhn : n < Nat.card α\nhα : Nat.card α ≠ 2 * n\nthis✝² : Finite α\nthis✝¹ : Fintype α\nthis✝ : IsPretransitive (Perm α) ↑(powersetCard α n)\nthis : Nontrivial ↑(powersetCard α n)\ns : ↑(powersetCard α n)\n⊢ IsPreprimitive (Perm α) ↑(powersetCa... | [
"α : Type u_2\ninst✝ : DecidableEq α\nn : ℕ\nh_one_le : 1 ≤ n\nhn : n < Nat.card α\nhα : Nat.card α ≠ 2 * n\nthis✝² : Finite α\nthis✝¹ : Fintype α\nthis✝ : IsPretransitive (Perm α) ↑(powersetCard α n)\nthis : Nontrivial ↑(powersetCard α n)\ns : ↑(powersetCard α n)\n⊢ IsCoatom (stabilizer (Perm α) s)"
] | ← isCoatom_stabilizer_iff_preprimitive _ s | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 202,
"column": 21
} | {
"line": 202,
"column": 57
} | {
"line": 202,
"column": 57
} | [
{
"pp": "N : Type u_1\nG : Type u_2\ninst✝² : Group N\ninst✝¹ : Group G\nE : Type u_3\ninst✝ : Group E\nS : GroupExtension N E G\ns₁ s₂ s₃ : S.Splitting\nn₁ : N\nhn₁ : ⇑s₁ = fun g ↦ S.inl n₁ * s₂ g * (S.inl n₁)⁻¹\nn₂ : N\nhn₂ : ⇑s₂ = fun g ↦ S.inl n₂ * s₃ g * (S.inl n₂)⁻¹\n⊢ ⇑s₁ = fun g ↦ S.inl (n₁ * n₂) * s₃ g... | [] | simp only [hn₁, hn₂, map_mul]; group | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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