module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Manifold.VectorBundle.LocalFrame | {
"line": 471,
"column": 4
} | {
"line": 471,
"column": 15
} | {
"line": 471,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nF : Type u_5\ninst✝¹² : NormedA... | [
"𝕜 : Type u_1\ninst✝¹⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁷ : NormedAddCommGroup E\ninst✝¹⁶ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁵ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁴ : TopologicalSpace M\ninst✝¹³ : ChartedSpace H M\nF : Type u_5\ninst✝¹² : NormedAddCommGroup ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.LocalFrame | {
"line": 539,
"column": 4
} | {
"line": 539,
"column": 15
} | {
"line": 539,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H M\nF : Type u_5\ninst✝¹¹ : NormedA... | [
"𝕜 : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H M\nF : Type u_5\ninst✝¹¹ : NormedAddCommGroup ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.LocalFrame | {
"line": 529,
"column": 2
} | {
"line": 547,
"column": 22
} | {
"line": 549,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H M\nF : Type u_5\ninst✝¹¹ : NormedA... | [] | let aux := fun x ↦ b.repr (e ((T% s) x)).2 i
-- Since `e.baseSet` is open, this is sufficient.
suffices MDiffAt aux x by
apply this.congr_of_eventuallyEq
apply eventuallyEq_of_mem (s := e.baseSet) (by simp [e.open_baseSet.mem_nhds hxe])
intro y hy
simp [aux, e.localFrame_coeff_eq_coeff hy]
simp on... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.LocalFrame | {
"line": 529,
"column": 2
} | {
"line": 547,
"column": 22
} | {
"line": 549,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹³ : TopologicalSpace M\ninst✝¹² : ChartedSpace H M\nF : Type u_5\ninst✝¹¹ : NormedA... | [] | let aux := fun x ↦ b.repr (e ((T% s) x)).2 i
-- Since `e.baseSet` is open, this is sufficient.
suffices MDiffAt aux x by
apply this.congr_of_eventuallyEq
apply eventuallyEq_of_mem (s := e.baseSet) (by simp [e.open_baseSet.mem_nhds hxe])
intro y hy
simp [aux, e.localFrame_coeff_eq_coeff hy]
simp on... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 17
} | {
"line": 167,
"column": 18
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁰ : TopologicalSpace M\ninst✝⁹ : ChartedSpace H M\nF : Type u_5\ninst✝⁸ :... | [
"case pos\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹⁰ : TopologicalSpace M\ninst✝⁹ : ChartedSpace H M\nF : Type u_5\ninst✝⁸ : NormedAddCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 13
} | {
"line": 207,
"column": 14
} | [
{
"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\nF : Type u_5\ninst✝⁹ : NormedAd... | [
"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\nF : Type u_5\ninst✝⁹ : NormedAddCommGroup F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Basic | {
"line": 290,
"column": 4
} | {
"line": 290,
"column": 15
} | {
"line": 290,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : ChartedSpace H M\nF : Type u_5\ninst✝¹⁰ : NormedA... | [
"𝕜 : Type u_1\ninst✝¹⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁵ : NormedAddCommGroup E\ninst✝¹⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹² : TopologicalSpace M\ninst✝¹¹ : ChartedSpace H M\nF : Type u_5\ninst✝¹⁰ : NormedAddCommGroup ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorBundle.CovariantDerivative.Torsion | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 36
} | {
"line": 150,
"column": 37
} | [
{
"pp": "case mp\n𝕜 : 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\ninst✝³ : CompleteSpace 𝕜\ni... | [
"case mp\n𝕜 : 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\ninst✝³ : CompleteSpace 𝕜\ninst✝² : Comp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Polygon.Basic | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 13
} | {
"line": 87,
"column": 14
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : AddTorsor V P\ninst✝¹ : Nontrivial R\nm : ℕ\ninst✝ : NeZero m.succ\npoly : Polygon P m.succ\nh : HasNondegenerateVertices R poly\ni : Fin m.succ\n⊢ poly.vertices i ≠ poly.vertices ((finRota... | [
"R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : AddTorsor V P\ninst✝¹ : Nontrivial R\nm : ℕ\ninst✝ : NeZero m.succ\npoly : Polygon P m.succ\nh : HasNondegenerateVertices R poly\ni : Fin m.succ\n⊢ ¬poly.vertices i = poly.vertices (i + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Polygon.Basic | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 22
} | {
"line": 154,
"column": 23
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AffineSpace V P\nt : Triangle R P\nht : t.points = ![t.points 0, t.points 1, t.points 2]\n⊢ AffineIndependent R ![t.points 0, t.points 1, t.points 2]",
"ppTerm": "?m.126",
"assigned"... | [
"R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AffineSpace V P\nt : Triangle R P\nht : t.points = ![t.points 0, t.points 1, t.points 2]\n⊢ AffineIndependent R t.points"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.ClassEquation | {
"line": 72,
"column": 39
} | {
"line": 72,
"column": 50
} | {
"line": 72,
"column": 51
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\nval✝ : Fintype G\nx✝ : ConjClasses G\ng : G\nhg : (carrier (Quot.mk (⇑(IsConj.setoid G)) g)).Subsingleton\n⊢ ↑(carrier (Quot.mk (⇑(IsConj.setoid G)) g)).toFinset = ↑{g}",
"ppTerm": "?m.188",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\nval✝ : Fintype G\nx✝ : ConjClasses G\ng : G\nhg : (carrier (Quot.mk (⇑(IsConj.setoid G)) g)).Subsingleton\n⊢ carrier (Quot.mk (⇑(IsConj.setoid G)) g) = {g}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Commutator.Finite | {
"line": 41,
"column": 10
} | {
"line": 41,
"column": 21
} | {
"line": 41,
"column": 22
} | [
{
"pp": "case pos\nη : Type u_2\ninst✝¹ : Finite η\nGs : η → Type u_3\ninst✝ : (i : η) → Group (Gs i)\nH K : (i : η) → Subgroup (Gs i)\nhi : (i : η) → Gs i\nj : η\n_hj : j ∈ Set.univ\nx : Gs j\nhx : x ∈ ↑(H j)\n⊢ (MonoidHom.mulSingle Gs j) x ∈ comap (Pi.evalMonoidHom Gs j) (H j)",
"ppTerm": "?pos✝",
"as... | [
"case pos\nη : Type u_2\ninst✝¹ : Finite η\nGs : η → Type u_3\ninst✝ : (i : η) → Group (Gs i)\nH K : (i : η) → Subgroup (Gs i)\nhi : (i : η) → Gs i\nj : η\n_hj : j ∈ Set.univ\nx : Gs j\nhx : x ∈ ↑(H j)\n⊢ x ∈ H j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Commutator.Finite | {
"line": 41,
"column": 10
} | {
"line": 41,
"column": 21
} | {
"line": 41,
"column": 22
} | [
{
"pp": "case pos\nη : Type u_2\ninst✝¹ : Finite η\nGs : η → Type u_3\ninst✝ : (i : η) → Group (Gs i)\nH K : (i : η) → Subgroup (Gs i)\nhi : (i : η) → Gs i\nj : η\n_hj : j ∈ Set.univ\nx : Gs j\nhx : x ∈ ↑(K j)\n⊢ (MonoidHom.mulSingle Gs j) x ∈ comap (Pi.evalMonoidHom Gs j) (K j)",
"ppTerm": "?pos✝",
"as... | [
"case pos\nη : Type u_2\ninst✝¹ : Finite η\nGs : η → Type u_3\ninst✝ : (i : η) → Group (Gs i)\nH K : (i : η) → Subgroup (Gs i)\nhi : (i : η) → Gs i\nj : η\n_hj : j ∈ Set.univ\nx : Gs j\nhx : x ∈ ↑(K j)\n⊢ x ∈ K j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.VectorField.LieBracket | {
"line": 1043,
"column": 2
} | {
"line": 1044,
"column": 71
} | {
"line": 1045,
"column": 2
} | [
{
"pp": "case e_6.h'x\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\ninst✝¹ : IsManifold I (... | [
"case e_6.hx\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\ninst✝¹ : IsManifold I (minSmoothness... | · rw [inter_comm]
exact extChartAt_mem_closure_interior h's (mem_extChartAt_source x) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 98,
"column": 13
} | {
"line": 98,
"column": 24
} | {
"line": 98,
"column": 25
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsKleinFour G\nh : IsCyclic G\n⊢ False",
"ppTerm": "?m.4",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsKleinFour G\nh : IsCyclic G\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 119,
"column": 15
} | {
"line": 119,
"column": 26
} | {
"line": 119,
"column": 27
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\ninst✝² : IsKleinFour G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nx y : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\n⊢ x * y ∉ {x, y, 1}",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"CancelMonoid.toRightCan... | [
"G : Type u_1\ninst✝³ : Group G\ninst✝² : IsKleinFour G\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nx y : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\n⊢ ¬y = 1 ∧ ¬x = 1 ∧ ¬x * y = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 54
} | {
"line": 128,
"column": 55
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsKleinFour G\nx y z : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\nhz : z ≠ 1\nhzx : z ≠ x\nhzy : z ≠ y\nx✝ : Fintype G := ⋯\n⊢ z ∉ {x, y, 1}",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivIn... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsKleinFour G\nx y z : G\nhx : x ≠ 1\nhy : y ≠ 1\nhxy : x ≠ y\nhz : z ≠ 1\nhzx : z ≠ x\nhzy : z ≠ y\nx✝ : Fintype G := Fintype.ofFinite G\n⊢ ¬z = x ∧ ¬z = y ∧ ¬z = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 148,
"column": 8
} | {
"line": 149,
"column": 15
} | {
"line": 149,
"column": 16
} | [
{
"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\nh... | [
"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\nhx : ¬x = 1\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.KleinFour | {
"line": 154,
"column": 6
} | {
"line": 154,
"column": 17
} | {
"line": 154,
"column": 18
} | [
{
"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 := ⋯\n_inst₂ : Fintype G₂ := ⋯\nx y : G₁\nhx : e x ≠ 1\nhy : e y ≠ 1\n... | [
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 15
} | {
"line": 195,
"column": 16
} | [
{
"pp": "case inl\nn : ℕ\nhn : 0 < n\n⊢ ¬↑n = 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"AddMonoidWit... | [
"case inl\nn : ℕ\nhn : 0 < n\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 229,
"column": 22
} | {
"line": 229,
"column": 33
} | {
"line": 229,
"column": 34
} | [
{
"pp": "x✝¹ : 0 ≠ 1\nx✝ : 0 ≠ 2\nh' : IsMulCommutative (DihedralGroup 0)\n⊢ False",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x✝¹ : 0 ≠ 1\nx✝ : 0 ≠ 2\nh' : IsMulCommutative (DihedralGroup 0)\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 234,
"column": 4
} | {
"line": 234,
"column": 15
} | {
"line": 234,
"column": 16
} | [
{
"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": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nx✝¹ : n + 3 ≠ 1\nx✝ : n + 3 ≠ 2\nh' : IsMulCommutative (DihedralGroup (n + 3))\nthis : 2 % (n + 3) = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 242,
"column": 4
} | {
"line": 242,
"column": 36
} | {
"line": 242,
"column": 37
} | [
{
"pp": "case pos\nn : ℕ\nh1 : n ≠ 1\nh : IsCyclic (DihedralGroup n)\nh2 : n = 2\n⊢ False",
"ppTerm": "?pos✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case pos\nn : ℕ\nh1 : n ≠ 1\nh : IsCyclic (DihedralGroup n)\nh2 : n = 2\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 273,
"column": 8
} | {
"line": 274,
"column": 15
} | {
"line": 274,
"column": 16
} | [
{
"pp": "n : ℕ\nhn : Odd n\nu : (ZMod n)ˣ := ZMod.unitOfCoprime 2 ⋯\nhu : ∀ (a : ZMod n), a + a = 0 ↔ a = 0\ni j : ZMod n\nh : Commute (r i, sr j).1 (r i, sr j).2\n⊢ (fun x ↦\n match x with\n | Sum.inl i => ⟨(sr i, r 0), ⋯⟩\n | Sum.inr (Sum.inl j) => ⟨(r 0, sr j), ⋯⟩\n | Sum.inr (Sum... | [
"n : ℕ\nhn : Odd n\nu : (ZMod n)ˣ := ZMod.unitOfCoprime 2 ⋯\nhu : ∀ (a : ZMod n), a + a = 0 ↔ a = 0\ni j : ZMod n\nh : Commute (r i, sr j).1 (r i, sr j).2\n⊢ 0 = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Dihedral | {
"line": 276,
"column": 8
} | {
"line": 277,
"column": 15
} | {
"line": 277,
"column": 16
} | [
{
"pp": "n : ℕ\nhn : Odd n\nu : (ZMod n)ˣ := ZMod.unitOfCoprime 2 ⋯\nhu : ∀ (a : ZMod n), a + a = 0 ↔ a = 0\ni j : ZMod n\nh : Commute (sr i, r j).1 (sr i, r j).2\n⊢ (fun x ↦\n match x with\n | Sum.inl i => ⟨(sr i, r 0), ⋯⟩\n | Sum.inr (Sum.inl j) => ⟨(r 0, sr j), ⋯⟩\n | Sum.inr (Sum... | [
"n : ℕ\nhn : Odd n\nu : (ZMod n)ˣ := ZMod.unitOfCoprime 2 ⋯\nhu : ∀ (a : ZMod n), a + a = 0 ↔ a = 0\ni j : ZMod n\nh : Commute (sr i, r j).1 (sr i, r j).2\n⊢ 0 = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CommutingProbability | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 15
} | {
"line": 85,
"column": 16
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Mul M\ninst✝ : Finite M\nh : Nonempty M\nthis : Fintype M\n⊢ commProb M = 1 ↔ IsMulCommutative M",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"instHDiv",
"congrArg",
"Rat",
"Commute",
"i... | [
"M : Type u_1\ninst✝¹ : Mul M\ninst✝ : Finite M\nh : Nonempty M\nthis : Fintype M\n⊢ ↑(Nat.card { p // Commute p.1 p.2 }) / ↑(Nat.card M) ^ 2 = 1 ↔ IsMulCommutative M"
] | commProb, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CommutingProbability | {
"line": 94,
"column": 6
} | {
"line": 94,
"column": 15
} | {
"line": 94,
"column": 16
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\n⊢ commProb G = ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"Rat",
"Commute",
"ConjClasses",
"id",... | [
"G : Type u_2\ninst✝ : Group G\n⊢ ↑(Nat.card { p // Commute p.1 p.2 }) / ↑(Nat.card G) ^ 2 = ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)"
] | commProb, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CommutingProbability | {
"line": 94,
"column": 2
} | {
"line": 97,
"column": 33
} | {
"line": 99,
"column": 0
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\n⊢ commProb G = ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"mul_div_mul_right",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoi... | [] | rw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]
by_cases h : (Nat.card G : ℚ) = 0
· rw [h, zero_mul, div_zero, div_zero]
· exact mul_div_mul_right _ _ h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.CommutingProbability | {
"line": 94,
"column": 2
} | {
"line": 97,
"column": 33
} | {
"line": 99,
"column": 0
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\n⊢ commProb G = ↑(Nat.card (ConjClasses G)) / ↑(Nat.card G)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"mul_div_mul_right",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoi... | [] | rw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]
by_cases h : (Nat.card G : ℚ) = 0
· rw [h, zero_mul, div_zero, div_zero]
· exact mul_div_mul_right _ _ h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.CommutingProbability | {
"line": 108,
"column": 25
} | {
"line": 108,
"column": 73
} | {
"line": 108,
"column": 74
} | [
{
"pp": "G : Type u_2\ninst✝¹ : Group G\ninst✝ : Finite G\nH : Subgroup G\np q : { p // Commute p.1 p.2 }\nh : (fun p ↦ ⟨(↑(↑p).1, ↑(↑p).2), ⋯⟩) p = (fun p ↦ ⟨(↑(↑p).1, ↑(↑p).2), ⋯⟩) q\n⊢ p = q",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Subgro... | [
"G : Type u_2\ninst✝¹ : Group G\ninst✝ : Finite G\nH : Subgroup G\np q : { p // Commute p.1 p.2 }\nh : (fun p ↦ ⟨(↑(↑p).1, ↑(↑p).2), ⋯⟩) p = (fun p ↦ ⟨(↑(↑p).1, ↑(↑p).2), ⋯⟩) q\n⊢ ↑(↑p).1 = ↑(↑q).1 ∧ ↑(↑p).2 = ↑(↑q).2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.IsFreeGroup | {
"line": 149,
"column": 38
} | {
"line": 149,
"column": 86
} | {
"line": 150,
"column": 8
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nG✝ : Type u_3\nH : Type u_4\ninst✝² : Group G✝\ninst✝¹ : Group H\nG : Type u\ninst✝ : Group G\nX : Type u\nof : X → G\nlift : {H : Type u} → [inst : Group H] → (X → H) ≃ (G →* H)\nlift_of : ∀ {H : Type u} [inst : Group H] (f : X → H) (a : X), (lift f) (of a) = f a\n⊢ ∀ {H :... | [] | by intro H _ f a; simp [← lift_of (lift.symm f)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 205,
"column": 15
} | {
"line": 205,
"column": 26
} | {
"line": 205,
"column": 27
} | [
{
"pp": "case mul_of.inl\nM : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\nC : M ∗ N → Prop\none : C 1\ninl_mul : ∀ (m : M) (x : M ∗ N), C x → C (inl m * x)\ninr_mul : ∀ (n : N) (x : M ∗ N), C x → C (inr n * x)\nxs : FreeMonoid (M ⊕ N)\nih : C (mk xs)\nm : M\n⊢ C (mk (of (Sum.inl m) * ... | [
"case mul_of.inl\nM : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\nC : M ∗ N → Prop\none : C 1\ninl_mul : ∀ (m : M) (x : M ∗ N), C x → C (inl m * x)\ninr_mul : ∀ (n : N) (x : M ∗ N), C x → C (inr n * x)\nxs : FreeMonoid (M ⊕ N)\nih : C (mk xs)\nm : M\n⊢ C (inl m * mk xs)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 206,
"column": 15
} | {
"line": 206,
"column": 26
} | {
"line": 206,
"column": 27
} | [
{
"pp": "case mul_of.inr\nM : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\nC : M ∗ N → Prop\none : C 1\ninl_mul : ∀ (m : M) (x : M ∗ N), C x → C (inl m * x)\ninr_mul : ∀ (n : N) (x : M ∗ N), C x → C (inr n * x)\nxs : FreeMonoid (M ⊕ N)\nih : C (mk xs)\nn : N\n⊢ C (mk (of (Sum.inr n) * ... | [
"case mul_of.inr\nM : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\nC : M ∗ N → Prop\none : C 1\ninl_mul : ∀ (m : M) (x : M ∗ N), C x → C (inl m * x)\ninr_mul : ∀ (n : N) (x : M ∗ N), C x → C (inr n * x)\nxs : FreeMonoid (M ⊕ N)\nih : C (mk xs)\nn : N\n⊢ C (inr n * mk xs)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 211,
"column": 22
} | {
"line": 211,
"column": 33
} | {
"line": 211,
"column": 34
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\nC : M ∗ N → Prop\nm : M ∗ N\ninl : ∀ (m : M), C (Coprod.inl m)\ninr : ∀ (n : N), C (Coprod.inr n)\nmul : ∀ (x y : M ∗ N), C x → C y → C (x * y)\n⊢ C 1",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
... | [
"M : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\nC : M ∗ N → Prop\nm : M ∗ N\ninl : ∀ (m : M), C (Coprod.inl m)\ninr : ∀ (n : N), C (Coprod.inr n)\nmul : ∀ (x y : M ∗ N), C x → C y → C (x * y)\n⊢ C 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 207,
"column": 61
} | {
"line": 207,
"column": 72
} | {
"line": 207,
"column": 73
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nmotive : CoprodI M → Prop\none : motive 1\nmul : ∀ {i : ι} (m : M i) (x : CoprodI M), motive x → motive (of m * x)\nx : CoprodI M\nhx : x ∈ ⋃ i, range ⇑of\ny : CoprodI M\nihy : motive y\n⊢ ∃ i m, of m = x",
"ppTerm": "?m.52",
"assi... | [
"ι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nmotive : CoprodI M → Prop\none : motive 1\nmul : ∀ {i : ι} (m : M i) (x : CoprodI M), motive x → motive (of m * x)\nx : CoprodI M\nhx : x ∈ ⋃ i, range ⇑of\ny : CoprodI M\nihy : motive y\n⊢ ∃ i m, of m = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 137,
"column": 43
} | {
"line": 137,
"column": 54
} | {
"line": 137,
"column": 55
} | [
{
"pp": "B : Type u_1\nB' : Type u_2\nM : CoxeterMatrix B\ne : B ≃ B'\n⊢ Surjective ⇑↑(FreeGroup.freeGroupCongr e)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"MulEquiv.instEquivLike",
"MonoidHom.instFunLike",
"MonoidHom",
"Monoid.toMulOneClass",
"MulEquiv... | [
"B : Type u_1\nB' : Type u_2\nM : CoxeterMatrix B\ne : B ≃ B'\n⊢ Surjective ⇑(FreeGroup.freeGroupCongr e)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 244,
"column": 55
} | {
"line": 253,
"column": 25
} | {
"line": 255,
"column": 0
} | [
{
"pp": "ι : Type u_1\nG : ι → Type u_4\ninst✝¹ : (i : ι) → Group (G i)\nN : Type u_5\ninst✝ : Group N\nf : (i : ι) → G i →* N\ns : Subgroup N\nh : ∀ (i : ι), (f i).range ≤ s\n⊢ (lift f).range ≤ s",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Set.mem_range_self",
"Eq.mpr",
... | [] | by
rintro _ ⟨x, rfl⟩
induction x using CoprodI.induction_on with
| one => exact s.one_mem
| of i x =>
simp only [lift_of]
exact h i (Set.mem_range_self x)
| mul x y hx hy =>
simp only [map_mul]
exact s.mul_mem hx hy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.CoprodI | {
"line": 361,
"column": 6
} | {
"line": 362,
"column": 44
} | {
"line": 362,
"column": 45
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝¹ : (i : ι) → Monoid (M i)\ninst✝ : (i : ι) → DecidableEq (M i)\ni : ι\nm : M i\nw : Word M\nh : w.fstIdx ≠ some i\nm' : M i\nw' : Word M\nh' : w'.fstIdx ≠ some i\nhe : rcons { head := m, tail := w, fstIdx_ne := h } = rcons { head := m', tail := w', fstIdx_ne := h' ... | [
"ι : Type u_1\nM : ι → Type u_2\ninst✝¹ : (i : ι) → Monoid (M i)\ninst✝ : (i : ι) → DecidableEq (M i)\ni : ι\nm : M i\nw : Word M\nh : w.fstIdx ≠ some i\nm' : M i\nw' : Word M\nh' : w'.fstIdx ≠ some i\nhe : rcons { head := m, tail := w, fstIdx_ne := h } = rcons { head := m', tail := w', fstIdx_ne := h' }\nhm : ¬m =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 383,
"column": 20
} | {
"line": 383,
"column": 64
} | {
"line": 383,
"column": 65
} | [
{
"pp": "case cons\nB : Type u_1\nW : Type u_3\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nx : B\nω' : List B\nih : cs.wordProd ω'.reverse = (cs.wordProd ω')⁻¹\n⊢ cs.wordProd (x :: ω').reverse = (cs.wordProd (x :: ω'))⁻¹",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
... | [
"case cons\nB : Type u_1\nW : Type u_3\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nx : B\nω' : List B\nih : cs.wordProd ω'.reverse = (cs.wordProd ω')⁻¹\n⊢ cs.wordProd ω'.reverse = (cs.wordProd ω')⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 421,
"column": 17
} | {
"line": 421,
"column": 46
} | {
"line": 421,
"column": 47
} | [
{
"pp": "case succ\nB : Type u_1\nm : ℕ\nih : ∀ (i i' : B), (alternatingWord i i' m).length = m\ni i' : B\n⊢ (alternatingWord i i' (m + 1)).length = m + 1",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",
"Nat.instAddM... | [
"case succ\nB : Type u_1\nm : ℕ\nih : ∀ (i i' : B), (alternatingWord i i' m).length = m\ni i' : B\n⊢ (alternatingWord i' i m).length = m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 420,
"column": 32
} | {
"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... | [] | by simp [cons, fstIdx, Ne.symm ij] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 459,
"column": 35
} | {
"line": 459,
"column": 46
} | {
"line": 459,
"column": 47
} | [
{
"pp": "B : Type u_1\ni j : B\np k : ℕ\nh' : k < 2 * p → take k (alternatingWord i j (2 * p)) = if Even k then alternatingWord i j k else alternatingWord j i k\nh : k + 1 < 2 * p\nh_even : ¬Even k\nhk : take k (alternatingWord i j (2 * p)) = alternatingWord j i k\n⊢ Odd ?m.111",
"ppTerm": "?m.112",
"as... | [
"B : Type u_1\ni j : B\np k : ℕ\nh' : k < 2 * p → take k (alternatingWord i j (2 * p)) = if Even k then alternatingWord i j k else alternatingWord j i k\nh : k + 1 < 2 * p\nh_even : ¬Even k\nhk : take k (alternatingWord i j (2 * p)) = alternatingWord j i k\n⊢ Odd ?m.111"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 47
} | {
"line": 123,
"column": 48
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω₁ : List B\nhω₁ : cs.IsReduced ω₁\nω₂ : List B\nhω₂ : cs.IsReduced ω₂\nthis : cs.length (cs.wordProd (ω₁ ++ ω₂)) ≤ (ω₁ ++ ω₂).length\n⊢ cs.length (cs.wordProd ω₁ * cs.wordProd ω₂) ≤ cs.length (cs.wordProd ω₁) + c... | [
"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω₁ : List B\nhω₁ : cs.IsReduced ω₁\nω₂ : List B\nhω₂ : cs.IsReduced ω₂\nthis : cs.length (cs.wordProd (ω₁ ++ ω₂)) ≤ (ω₁ ++ ω₂).length\n⊢ cs.length (cs.wordProd ω₁ * cs.wordProd ω₂) ≤ ω₁.length + ω₂.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 24
} | {
"line": 126,
"column": 25
} | [
{
"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 w₂ ≤ cs.length (w₁ * w₂) + cs.length w₁",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg... | [
"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw₁ w₂ : W\n⊢ cs.length w₂ ≤ cs.length w₁ + cs.length (w₁ * w₂)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 13
} | {
"line": 129,
"column": 14
} | [
{
"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 ≤ cs.length (w₁ * w₂) + cs.length w₂",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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 ≤ cs.length (w₁ * w₂) + cs.length w₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 15
} | {
"line": 169,
"column": 16
} | [
{
"pp": "case h₁\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\n⊢ cs.length (cs.simple i) ≤ 1",
"ppTerm": "?h₁",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case h₁\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\n⊢ cs.length (cs.simple i) ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 13
} | {
"line": 194,
"column": 14
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.length (cs.simple i * w)⁻¹ ≠ cs.length w",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CoxeterSystem.inv_simple",
"DivInvMonoid.toInv",
... | [
"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ ¬cs.length (w⁻¹ * cs.simple i) = cs.length w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 15
} | {
"line": 199,
"column": 16
} | [
{
"pp": "case inl\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\nh : cs.length (w * cs.simple i) + 1 ≤ cs.length w\n⊢ cs.length w ≤ cs.length (w * cs.simple i) + 1",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"case inl\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\nh : cs.length (w * cs.simple i) + 1 ≤ cs.length w\n⊢ cs.length w ≤ cs.length (w * cs.simple i) + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 15
} | {
"line": 201,
"column": 16
} | [
{
"pp": "case inr\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\nh : cs.length w + 1 ≤ cs.length (w * cs.simple i)\n⊢ cs.length (w * cs.simple i) ≤ cs.length w + 1",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"case inr\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\nh : cs.length w + 1 ≤ cs.length (w * cs.simple i)\n⊢ cs.length (w * cs.simple i) ≤ cs.length w + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 205,
"column": 60
} | {
"line": 205,
"column": 70
} | {
"line": 205,
"column": 70
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\nthis : cs.length (cs.simple i * w) = cs.length w⁻¹ + 1 ∨ cs.length (cs.simple i * w) + 1 = cs.length w⁻¹\n⊢ cs.length (cs.simple i * w) = cs.length w + 1 ∨ cs.length (cs.simple i * w) + 1 = cs.length... | [
"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\nthis : cs.length (cs.simple i * w) = cs.length w + 1 ∨ cs.length (cs.simple i * w) + 1 = cs.length w\n⊢ cs.length (cs.simple i * w) = cs.length w + 1 ∨ cs.length (cs.simple i * w) + 1 = cs.length w"
] | length_inv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 261,
"column": 4
} | {
"line": 261,
"column": 15
} | {
"line": 261,
"column": 16
} | [
{
"pp": "case step\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni i' : B\nm✝¹ : ℕ\nhM : M.M i i' ≠ 0\nm✝ : ℕ\nm : (M.M i i').succ.le m✝\nih : cs.IsReduced (drop 1 ((if Even m✝ then i' else i) :: alternatingWord i i' m✝))\n⊢ cs.IsReduced (alternatingWord i i' m✝)",
... | [
"case step\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni i' : B\nm✝¹ : ℕ\nhM : M.M i i' ≠ 0\nm✝ : ℕ\nm : (M.M i i').succ.le m✝\nih : cs.IsReduced (drop 1 ((if Even m✝ then i' else i) :: alternatingWord i i' m✝))\n⊢ cs.IsReduced (alternatingWord i i' m✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 13
} | {
"line": 283,
"column": 14
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.IsRightDescent w⁻¹ i ↔ cs.IsLeftDescent w i",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.IsRightDescent w⁻¹ i ↔ cs.IsLeftDescent w i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 318,
"column": 2
} | {
"line": 322,
"column": 7
} | {
"line": 324,
"column": 0
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.IsRightDescent w i ↔ cs.length (w * cs.simple i) + 1 = cs.length w",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"_private.Mathlib.GroupTheory.Coxeter.Length.0.Coxe... | [] | unfold IsRightDescent
constructor
· intro _
exact (cs.length_mul_simple w i).resolve_left (by lia)
· lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coxeter.Length | {
"line": 318,
"column": 2
} | {
"line": 322,
"column": 7
} | {
"line": 324,
"column": 0
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw : W\ni : B\n⊢ cs.IsRightDescent w i ↔ cs.length (w * cs.simple i) + 1 = cs.length w",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"_private.Mathlib.GroupTheory.Coxeter.Length.0.Coxe... | [] | unfold IsRightDescent
constructor
· intro _
exact (cs.length_mul_simple w i).resolve_left (by lia)
· lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 29
} | {
"line": 122,
"column": 30
} | [
{
"pp": "case mp\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw t : W\nh : cs.IsReflection (w * t * w⁻¹)\n⊢ cs.IsReflection t",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nw t : W\nh : cs.IsReflection (w * t * w⁻¹)\n⊢ cs.IsReflection t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 231,
"column": 8
} | {
"line": 231,
"column": 19
} | {
"line": 231,
"column": 20
} | [
{
"pp": "case cons\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\nω : List B\nih : cs.leftInvSeq ω = (cs.rightInvSeq ω.reverse).reverse\n⊢ cs.leftInvSeq (i :: ω) = (cs.rightInvSeq (i :: ω).reverse).reverse",
"ppTerm": "?cons",
"assigned": true,
"use... | [
"case cons\nB : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni : B\nω : List B\nih : cs.leftInvSeq ω = (cs.rightInvSeq ω.reverse).reverse\n⊢ cs.simple i :: List.map (⇑(MulAut.conj (cs.simple i))) (cs.leftInvSeq ω) = (cs.rightInvSeq (i :: ω).reverse).reverse"
] | leftInvSeq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.CoprodI | {
"line": 698,
"column": 42
} | {
"line": 698,
"column": 59
} | {
"line": 698,
"column": 60
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nx y : (i : ι) × M i\nl : List ((i : ι) × M i)\nhnot1✝ : ∀ l_1 ∈ x :: y :: l, l_1.snd ≠ 1\nhnot1 : x.snd ≠ 1 ∧ ∀ x ∈ y :: l, x.snd ≠ 1\nhchain✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) (x :: y :: l)\nhchain : x.fst ≠ y.fst ∧ List.IsChain ... | [
"ι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nx y : (i : ι) × M i\nl : List ((i : ι) × M i)\nhnot1✝ : ∀ l_1 ∈ x :: y :: l, l_1.snd ≠ 1\nhnot1 : x.snd ≠ 1 ∧ ∀ x ∈ y :: l, x.snd ≠ 1\nhchain✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) (x :: y :: l)\nhchain : x.fst ≠ y.fst ∧ List.IsChain (fun l l' ↦ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 700,
"column": 6
} | {
"line": 700,
"column": 26
} | {
"line": 700,
"column": 27
} | [
{
"pp": "case cons.cons\nι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nx : (i : ι) × M i\nl : List ((i : ι) × M i)\ni j : ι\nw' : NeWord M i j\nhnot1✝ : ∀ l_1 ∈ x :: ⟨i, w'.head⟩ :: l, l_1.snd ≠ 1\nhnot1 : x.snd ≠ 1 ∧ ∀ x ∈ ⟨i, w'.head⟩ :: l, x.snd ≠ 1\nhchain✝ : List.IsChain (fun l l' ↦ l.fst... | [
"case cons.cons\nι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nx : (i : ι) × M i\nl : List ((i : ι) × M i)\ni j : ι\nw' : NeWord M i j\nhnot1✝ : ∀ l_1 ∈ x :: ⟨i, w'.head⟩ :: l, l_1.snd ≠ 1\nhnot1 : x.snd ≠ 1 ∧ ∀ x ∈ ⟨i, w'.head⟩ :: l, x.snd ≠ 1\nhchain✝ : List.IsChain (fun l l' ↦ l.fst ≠ l'.fst) (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 817,
"column": 27
} | {
"line": 817,
"column": 38
} | {
"line": 817,
"column": 39
} | [
{
"pp": "case singleton\nι : Type u_1\nG : Type u_4\ninst✝² : Group G\nH : ι → Type u_5\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝ : MulAction G α\nX : ι → Set α\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i\ni j i✝ : ι\nx : H i✝\nhne_one : x ≠ 1\nk : ι\nhk... | [
"case singleton\nι : Type u_1\nG : Type u_4\ninst✝² : Group G\nH : ι → Type u_5\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝ : MulAction G α\nX : ι → Set α\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i\ni j i✝ : ι\nx : H i✝\nhne_one : x ≠ 1\nk : ι\nhk : i✝ ≠ k\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 830,
"column": 25
} | {
"line": 830,
"column": 43
} | {
"line": 830,
"column": 44
} | [
{
"pp": "ι : Type u_1\nG : Type u_4\ninst✝² : Group G\nH : ι → Type u_5\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • ... | [
"ι : Type u_1\nG : Type u_4\ninst✝² : Group G\nH : ι → Type u_5\ninst✝¹ : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X j ⊆ X i\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 420,
"column": 50
} | {
"line": 420,
"column": 61
} | {
"line": 420,
"column": 62
} | [
{
"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' < (cs.rightInvSeq ω).length\ndup : (cs.rightInvSeq ω)[j]? = (cs.rightInvSeq ω)[j']?\n⊢ j' < ω.length",
"ppTerm": "?m.50",
"ass... | [
"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' < (cs.rightInvSeq ω).length\ndup : (cs.rightInvSeq ω)[j]? = (cs.rightInvSeq ω)[j']?\n⊢ j' < ω.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 870,
"column": 8
} | {
"line": 870,
"column": 94
} | {
"line": 871,
"column": 6
} | [
{
"pp": "case neg\nι : Type u_1\nG : Type u_4\ninst✝³ : Group G\nH : ι → Type u_5\ninst✝² : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝¹ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 →... | [] | exact lift_word_prod_nontrivial_of_head_card f X hXnonempty hXdisj hpp w hcard hl.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.CoprodI | {
"line": 877,
"column": 8
} | {
"line": 877,
"column": 19
} | {
"line": 877,
"column": 20
} | [
{
"pp": "case pos\nι : Type u_1\nG : Type u_4\ninst✝³ : Group G\nH : ι → Type u_5\ninst✝² : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝¹ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 →... | [
"case pos\nι : Type u_1\nG : Type u_4\ninst✝³ : Group G\nH : ι → Type u_5\ninst✝² : (i : ι) → Group (H i)\nf : (i : ι) → H i →* G\nα : Type u_6\ninst✝¹ : MulAction G α\nX : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhpp : Pairwise fun i j ↦ ∀ (h : H i), h ≠ 1 → (f i) h • X... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 15
} | {
"line": 204,
"column": 16
} | [
{
"pp": "case inl.e_m.e_a\nM : Type u_2\ninst✝ : AddCommGroup M\na : ℚ\nm : M\ns : ℕ+\nh : 0 ≤ a\n⊢ ↑(have this := ⟨a, h⟩;\n this).num =\n a.num",
"ppTerm": "?inl.e_m.e_a✝",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Rat.instOfNat",
"Int.cast",
"E... | [
"case inl.e_m.e_a\nM : Type u_2\ninst✝ : AddCommGroup M\na : ℚ\nm : M\ns : ℕ+\nh : 0 ≤ a\n⊢ 0 ≤ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.VectorBundle.Riemannian | {
"line": 269,
"column": 6
} | {
"line": 269,
"column": 17
} | {
"line": 269,
"column": 18
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 17
} | {
"line": 209,
"column": 18
} | [
{
"pp": "case e_a\nM : Type u_2\ninst✝ : AddCommGroup M\na : ℚ\nm : M\ns : ℕ+\nh : a ≤ 0\n⊢ ↑a.num.natAbs = -a.num",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"abs_eq_neg_self._simp_1",
"Rat.instOfNat",
"Int.cast",
"Eq.mpr",
"... | [
"case e_a\nM : Type u_2\ninst✝ : AddCommGroup M\na : ℚ\nm : M\ns : ℕ+\nh : a ≤ 0\n⊢ a ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 452,
"column": 51
} | {
"line": 452,
"column": 62
} | {
"line": 452,
"column": 63
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni j : B\np k : ℕ\nh : k + 1 < 2 * p\n⊢ k + 1 < (cs.leftInvSeq (alternatingWord i j (2 * p))).length",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"c... | [
"B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\ni j : B\np k : ℕ\nh : k + 1 < 2 * p\n⊢ k + 1 < 2 * p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DoubleCoset | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 23
} | {
"line": 153,
"column": 24
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\na b : Quotient ↑H ↑K\nh : ¬Disjoint (doubleCoset (Quotient.out a) ↑H ↑K) (doubleCoset (Quotient.out b) ↑H ↑K)\n⊢ a = b",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\na b : Quotient ↑H ↑K\nh : ¬Disjoint (doubleCoset (Quotient.out a) ↑H ↑K) (doubleCoset (Quotient.out b) ↑H ↑K)\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DoubleCoset | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 45
} | {
"line": 175,
"column": 4
} | [
{
"pp": "case mp\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\na x : G\ny : ↥K\nh_h : x * ((↑y)⁻¹ * a⁻¹) ∈ H\n⊢ ∃ x_1 ∈ H, ∃ y ∈ K, x = x_1 * a * y",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
... | [
"case mp\nG : Type u_1\ninst✝ : Group G\nH K : Subgroup G\na x : G\ny : ↥K\nh_h : x * ((↑y)⁻¹ * a⁻¹) ∈ H\n⊢ x = x * (↑y⁻¹ * a⁻¹) * a * ↑y"
] | refine ⟨x * (y⁻¹ * a⁻¹), h_h, y, y.2, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.DivisibleHull | {
"line": 322,
"column": 38
} | {
"line": 322,
"column": 49
} | {
"line": 322,
"column": 50
} | [
{
"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 : ℚ≥0\nha : 0 < a\nmb : M\nsb : ℕ+\nmc : M\nsc : ℕ+\nh : ↑sc • mb < ↑sb • mc\n⊢ ↑⟨a.den, ⋯⟩ * a.num ≠ 0",
"ppTerm": "?m.77",
"assigned": true,
"use... | [
"M✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\na : ℚ≥0\nha : 0 < a\nmb : M\nsb : ℕ+\nmc : M\nsc : ℕ+\nh : ↑sc • mb < ↑sb • mc\n⊢ ¬a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 1033,
"column": 8
} | {
"line": 1033,
"column": 19
} | {
"line": 1033,
"column": 20
} | [
{
"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)... | [
"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), a i • (Y i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 329,
"column": 6
} | {
"line": 329,
"column": 17
} | {
"line": 329,
"column": 18
} | [
{
"pp": "case mk.refine_2\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 : ↑s • 0 < ↑1 • m\n⊢ 0 < m",
"ppTerm": "?mk.refine_2",
"assigned": false,
"usedConstants": ... | [
"case mk.refine_2\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 : ↑s • 0 < ↑1 • m\n⊢ 0 < m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 351,
"column": 24
} | {
"line": 351,
"column": 35
} | {
"line": 351,
"column": 36
} | [
{
"pp": "M✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\na b : M\nh : a ≤ b\n⊢ (↑(coeAddMonoidHom M)).toFun a ≤ (↑(coeAddMonoidHom M)).toFun b",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"PNat.... | [
"M✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\na b : M\nh : a ≤ b\n⊢ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DoubleCoset | {
"line": 278,
"column": 46
} | {
"line": 278,
"column": 57
} | {
"line": 278,
"column": 58
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nt : Finset (Quotient ↑H ↑K)\nht : ⋃ q ∈ t, doubleCoset (out q) ↑H ↑K ≠ Set.univ\nx : G\ny : Quotient ↑H ↑K\nhy : y ∈ t\nq : G\nhq : q ∈ doubleCoset (out y) ↑H ↑K\nhx : Quot.mk (⇑(rightRel H)) q = Quot.mk (⇑(rightRel H)) x\na : ↥H\nha : x = ↑a * q\n⊢ x = ... | [
"G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nt : Finset (Quotient ↑H ↑K)\nht : ⋃ q ∈ t, doubleCoset (out q) ↑H ↑K ≠ Set.univ\nx : G\ny : Quotient ↑H ↑K\nhy : y ∈ t\nq : G\nhq : q ∈ doubleCoset (out y) ↑H ↑K\nhx : Quot.mk (⇑(rightRel H)) q = Quot.mk (⇑(rightRel H)) x\na : ↥H\nha : x = ↑a * q\n⊢ x = ↑a * q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 391,
"column": 6
} | {
"line": 391,
"column": 45
} | {
"line": 391,
"column": 46
} | [
{
"pp": "M✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\nx : DivisibleHull M\nx✝³ x✝² : M\nx✝¹ x✝ : ℕ+\nh : ArchimedeanClass.mk (mk x✝³ x✝¹) = ArchimedeanClass.mk (mk x✝² x✝)\n⊢ (archimedeanClassOrderHom M) (ArchimedeanClass.... | [
"M✝ : Type u_1\ninst✝³ : AddCommMonoid M✝\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedAddMonoid M\nx : DivisibleHull M\nx✝³ x✝² : M\nx✝¹ x✝ : ℕ+\nh : ArchimedeanClass.mk (mk x✝³ x✝¹) = ArchimedeanClass.mk (mk x✝² x✝)\n⊢ (archimedeanClassOrderHom M) (ArchimedeanClass.mk x✝³) = (a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 37,
"column": 4
} | {
"line": 38,
"column": 11
} | {
"line": 38,
"column": 12
} | [
{
"pp": "ι : Type u_1\nG : Type u_2\ninst✝ : Monoid G\nn : ι → ℕ\ne : G ≃* ((i : ι) → Multiplicative (ZMod (n i)))\ni : ι\n⊢ n i = orderOf (e.symm (Pi.mulSingle i (Multiplicative.ofAdd 1)))",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulEquiv.instEquivLike",
... | [
"ι : Type u_1\nG : Type u_2\ninst✝ : Monoid G\nn : ι → ℕ\ne : G ≃* ((i : ι) → Multiplicative (ZMod (n i)))\ni : ι\n⊢ n i = addOrderOf 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.DivisibleHull | {
"line": 396,
"column": 6
} | {
"line": 396,
"column": 17
} | {
"line": 396,
"column": 18
} | [
{
"pp": "case 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\nnum✝¹ : M\nden✝¹ : ℕ+\nnum✝ : M\nden✝ : ℕ+\nh : (archimedeanClassOrderHom M) (ArchimedeanClass.mk num✝¹) ≤ (archimedeanClassOrderHom M) (ArchimedeanClass.mk... | [
"case 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\nnum✝¹ : M\nden✝¹ : ℕ+\nnum✝ : M\nden✝ : ℕ+\nh : (archimedeanClassOrderHom M) (ArchimedeanClass.mk num✝¹) ≤ (archimedeanClassOrderHom M) (ArchimedeanClass.mk num✝)\n⊢ Ar... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 60
} | {
"line": 76,
"column": 4
} | [
{
"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)\ng g' : G\nh : ∀ (φ : G →* Mˣ), φ g = φ g'\n⊢ g = g'",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"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)\ng g' : G\nh : ∀ (φ : G →* Mˣ), φ g = φ g'\n⊢ g = g'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 89,
"column": 6
} | {
"line": 90,
"column": 13
} | {
"line": 90,
"column": 14
} | [
{
"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)\nι : Type\nw✝ : Fintype ι\nn : ι → ℕ\nh₁ : ∀ (i : ι), 1 < n i\nh₂ : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))\ne : G ≃* ((i : ι) → Multiplicative (ZM... | [
"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)\nι : Type\nw✝ : Fintype ι\nn : ι → ℕ\nh₁ : ∀ (i : ι), 1 < n i\nh₂ : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))\ne : G ≃* ((i : ι) → Multiplicative (ZMod (n i))) :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteIndexNormalSubgroup | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 17
} | {
"line": 133,
"column": 18
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Type u_2\nN : Type u_3\ninst✝¹ : Group H\ninst✝ : Group N\nf : G →* H\nK : FiniteIndexNormalSubgroup H\ng : G →* H ⧸ K.toSubgroup := (QuotientGroup.mk' K.toSubgroup).comp f\n⊢ Subgroup.comap f K.toSubgroup = g.ker",
"ppTerm": "?m.51",
"assigned": false,
"... | [
"G : Type u_1\ninst✝² : Group G\nH : Type u_2\nN : Type u_3\ninst✝¹ : Group H\ninst✝ : Group N\nf : G →* H\nK : FiniteIndexNormalSubgroup H\ng : G →* H ⧸ K.toSubgroup := (QuotientGroup.mk' K.toSubgroup).comp f\n⊢ Subgroup.comap f K.toSubgroup = g.ker"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteIndexNormalSubgroup | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 22
} | {
"line": 134,
"column": 23
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Type u_2\nN : Type u_3\ninst✝¹ : Group H\ninst✝ : Group N\nf : G →* H\nK : FiniteIndexNormalSubgroup H\ng : G →* H ⧸ K.toSubgroup := (QuotientGroup.mk' K.toSubgroup).comp f\nhker : Subgroup.comap f K.toSubgroup = g.ker\n⊢ (Subgroup.comap f K.toSubgroup).FiniteIndex",... | [
"G : Type u_1\ninst✝² : Group G\nH : Type u_2\nN : Type u_3\ninst✝¹ : Group H\ninst✝ : Group N\nf : G →* H\nK : FiniteIndexNormalSubgroup H\ng : G →* H ⧸ K.toSubgroup := (QuotientGroup.mk' K.toSubgroup).comp f\nhker : Subgroup.comap f K.toSubgroup = g.ker\n⊢ g.ker.FiniteIndex"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Transfer | {
"line": 193,
"column": 2
} | {
"line": 204,
"column": 89
} | {
"line": 206,
"column": 0
} | [
{
"pp": "case neg\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\nkey : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k\nhH : ¬H.index = 0\nthis : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH\n⊢ g ^ H.index ∈ H",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | classical
replace key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H := fun k g₀ hk =>
(congr_arg (· ∈ H) (key k g₀ hk)).mp hk
replace key : ∀ q : G ⧸ H, g ^ Function.minimalPeriod (g • ·) q ∈ H := fun q =>
key (Function.minimalPeriod (g • ·) q) q.out
(QuotientGroup.out_conj_pow_min... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.GroupTheory.FixedPointFree | {
"line": 59,
"column": 31
} | {
"line": 59,
"column": 60
} | {
"line": 59,
"column": 61
} | [
{
"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 : (⇑φ)^[2] = _root_.id\ng : G\n⊢ g * φ g = 1",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"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 : (⇑φ)^[2] = _root_.id\ng : G\n⊢ g * φ g = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Radical | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 15
} | {
"line": 50,
"column": 16
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : CompleteLattice α\ninst✝ : IsCoatomic α\na : α\nh : a ⊔ radical α = ⊤\nm : α\nc : IsCoatom m\nle : a ≤ m\nq : m = ⊤\n⊢ a = ⊤",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nα : Type u_1\ninst✝¹ : CompleteLattice α\ninst✝ : IsCoatomic α\na : α\nh : a ⊔ radical α = ⊤\nm : α\nc : IsCoatom m\nle : a ≤ m\nq : m = ⊤\n⊢ a = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Focal | {
"line": 100,
"column": 27
} | {
"line": 100,
"column": 38
} | {
"line": 100,
"column": 39
} | [
{
"pp": "case mul\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx✝ y✝ : G\nhx✝ : x✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nhy✝ : y✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nIHa : g * x✝ * g⁻¹ ∈ H.focalSubgroup\nIHb : g * y✝ * g⁻¹ ∈ H.focalSubgroup\n⊢ g * (x✝ * y✝) * g⁻... | [
"case mul\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx✝ y✝ : G\nhx✝ : x✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nhy✝ : y✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nIHa : g * x✝ * g⁻¹ ∈ H.focalSubgroup\nIHb : g * y✝ * g⁻¹ ∈ H.focalSubgroup\n⊢ g * (x✝ * y✝) * g⁻¹ ∈ H.focalS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Focal | {
"line": 101,
"column": 18
} | {
"line": 101,
"column": 41
} | {
"line": 101,
"column": 42
} | [
{
"pp": "case inv\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx✝ : G\nhx✝ : x✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nIH : g * x✝ * g⁻¹ ∈ H.focalSubgroup\n⊢ g * x✝⁻¹ * g⁻¹ ∈ H.focalSubgroup",
"ppTerm": "?inv",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case inv\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx✝ : G\nhx✝ : x✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nIH : g * x✝ * g⁻¹ ∈ H.focalSubgroup\n⊢ g * (x✝⁻¹ * g⁻¹) ∈ H.focalSubgroup"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Focal | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 13
} | {
"line": 198,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : (↑P).FiniteIndex\ng : ↥P\nhQ : IsPGroup p (↥↑P ⧸ (↑P).focalSubgroupOf)\n⊢ ↑g ^ (↑P).index = 1 ↔ g ∈ (↑P).focalSubgroupOf",
"ppTerm": "?m.68",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"G : Type u_1\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : (↑P).FiniteIndex\ng : ↥P\nhQ : IsPGroup p (↥↑P ⧸ (↑P).focalSubgroupOf)\n⊢ ↑g ^ (↑P).index = 1 ↔ g ∈ (↑P).focalSubgroupOf"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 76,
"column": 16
} | {
"line": 76,
"column": 39
} | {
"line": 76,
"column": 40
} | [
{
"pp": "α : Type u\nL : List (α × Bool)\nn✝ n : ℕ\nhead : α × Bool\ntail : List (α × Bool)\nh : IsCyclicallyReduced (head :: tail)\n⊢ ∀ l ∈ replicate (n + 1) (head :: tail), IsChain (fun a b ↦ a.1 = b.1 → a.2 = b.2) l",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"α : Type u\nL : List (α × Bool)\nn✝ n : ℕ\nhead : α × Bool\ntail : List (α × Bool)\nh : IsCyclicallyReduced (head :: tail)\n⊢ IsChain (fun a b ↦ a.1 = b.1 → a.2 = b.2) (head :: tail)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 142,
"column": 19
} | {
"line": 142,
"column": 30
} | {
"line": 142,
"column": 31
} | [
{
"pp": "α : Type u\nL : List (α × Bool)\ninst✝ : DecidableEq α\na : α × Bool\nl : List (α × Bool)\nb : α × Bool\nih : IsReduced l → IsCyclicallyReduced (reduceCyclically l)\nh : IsReduced (a :: (l ++ [b]))\nh' : ¬(b.1 = a.1 ∧ (!b.2) = a.2)\n⊢ b.1 = a.1 → b.2 = a.2",
"ppTerm": "?m.71",
"assigned": false... | [
"α : Type u\nL : List (α × Bool)\ninst✝ : DecidableEq α\na : α × Bool\nl : List (α × Bool)\nb : α × Bool\nih : IsReduced l → IsCyclicallyReduced (reduceCyclically l)\nh : IsReduced (a :: (l ++ [b]))\nh' : ¬(b.1 = a.1 ∧ (!b.2) = a.2)\n⊢ b.1 = a.1 → b.2 = a.2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 232,
"column": 12
} | {
"line": 232,
"column": 50
} | {
"line": 232,
"column": 51
} | [
{
"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... | [
"G : Type u_1\ninst✝¹ : Group G\nH : Type u_2\ninst✝ : Group H\ne : H ≃* G\n⊢ Function.Injective ⇑e"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 33
} | {
"line": 175,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝² : Groupoid G\ninst✝¹ : IsFreeGroupoid G\nT : WideSubquiver (Symmetrify (Generators G))\ninst✝ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)\na : G\np : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) a\n⊢ treeHom T a = homOfPath T p",
"ppTer... | [] | rw [treeHom, Unique.default_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 33
} | {
"line": 175,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝² : Groupoid G\ninst✝¹ : IsFreeGroupoid G\nT : WideSubquiver (Symmetrify (Generators G))\ninst✝ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)\na : G\np : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) a\n⊢ treeHom T a = homOfPath T p",
"ppTer... | [] | rw [treeHom, Unique.default_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 33
} | {
"line": 175,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝² : Groupoid G\ninst✝¹ : IsFreeGroupoid G\nT : WideSubquiver (Symmetrify (Generators G))\ninst✝ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)\na : G\np : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) a\n⊢ treeHom T a = homOfPath T p",
"ppTer... | [] | rw [treeHom, Unique.default_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 244,
"column": 8
} | {
"line": 244,
"column": 17
} | {
"line": 245,
"column": 8
} | [
{
"pp": "case refine_1\nG : Type u\ninst✝³ : Groupoid G\ninst✝² : IsFreeGroupoid G\nT : WideSubquiver (Symmetrify (Generators G))\ninst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)\nX : Type u\ninst✝ : Group X\nf : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X\nf' : Lab... | [
"case refine_1\nG : Type u\ninst✝³ : Groupoid G\ninst✝² : IsFreeGroupoid G\nT : WideSubquiver (Symmetrify (Generators G))\ninst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)\nX : Type u\ninst✝ : Group X\nf : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X\nf' : Labelling (Gene... | intro a p | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 232,
"column": 6
} | {
"line": 232,
"column": 78
} | {
"line": 232,
"column": 79
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nheq : (fun a ↦ a ^ n) x = (fun a ↦ a ^ n) y\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bo... | [
"α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nheq : (fun a ↦ a ^ n) x = (fun a ↦ a ^ n) y\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 234,
"column": 6
} | {
"line": 234,
"column": 78
} | {
"line": 234,
"column": 79
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((... | [
"α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((replicate k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 235,
"column": 35
} | {
"line": 235,
"column": 50
} | {
"line": 235,
"column": 51
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((... | [
"α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((replicate k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 236,
"column": 48
} | {
"line": 236,
"column": 63
} | {
"line": 236,
"column": 64
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((... | [
"α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((replicate k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 651,
"column": 2
} | {
"line": 651,
"column": 13
} | {
"line": 651,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS : Subgroup G\nn : ℕ\n⊢ S.lowerCentralSeries n ≤ S",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nS : Subgroup G\nn : ℕ\n⊢ S.lowerCentralSeries n ≤ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Orbit | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 13
} | {
"line": 46,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\na b : α × Bool\nh : ∀ (x : FreeGroup α), x.toWord[0]? = some a ↔ x.toWord[0]? = some b\n⊢ a = b",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : DecidableEq α\na b : α × Bool\nh : ∀ (x : FreeGroup α), x.toWord[0]? = some a ↔ x.toWord[0]? = some b\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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