module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.BooleanGenerators | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 32
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nhT : ∅ ⊆ S\nha : IsAtom (sSup ∅)\nhaS : sSup ∅ ≤ sSup S\n⊢ sSup ∅ ∈ S",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"sS... | [
"case inl\nα : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nhT : ∅ ⊆ S\nhaS : sSup ∅ ≤ sSup S\nha : IsAtom ⊥\n⊢ sSup ∅ ∈ S"
] | simp only [sSup_empty] at ha | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.BooleanGenerators | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 46
} | {
"line": 127,
"column": 4
} | [
{
"pp": "case a.a\nα : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nT₁ T₂ : Set α\nhT₁ : T₁ ⊆ S\nhT₂ : T₂ ⊆ S\n⊢ sSup (T₁ ∩ T₂) ≤ sSup T₁",
"ppTerm": "?a.a✝",
"assigned": true,
"usedConstants": [
"Set.inter_subset_left",
"Set.... | [
"case a\nα : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nT₁ T₂ : Set α\nhT₁ : T₁ ⊆ S\nhT₂ : T₂ ⊆ S\n⊢ sSup (T₁ ∩ T₂) ≤ sSup T₂"
] | · apply sSup_le_sSup Set.inter_subset_left | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.BooleanGenerators | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 46
} | {
"line": 159,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nh : sSup S = ⊤\n⊢ ComplementedLattice α",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"DistribLattice",
"IsCompactlyGenerated.BooleanGenerators.distribLatti... | [
"α : Type u_1\ninst✝¹ : CompleteLattice α\nS : Set α\ninst✝ : IsCompactlyGenerated α\nhS : BooleanGenerators S\nh : sSup S = ⊤\n_i : DistribLattice α := hS.distribLattice_of_sSup_eq_top h\n⊢ ComplementedLattice α"
] | let _i := hS.distribLattice_of_sSup_eq_top h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 173,
"column": 6
} | {
"line": 173,
"column": 47
} | {
"line": 177,
"column": 4
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\nI : LieIdeal R L\nhI : IsAtom I\nJ : LieIdeal R ↥I\nJ' : LieIdeal R L :=\n let __spread.0 := Submodule.map ↑I.incl ↑J;\n { toSubmodule := __spread.0, lie_mem := ⋯ }\nhJ : ¬J = ⊤\nth... | [] | exact fun _ ↦ this (↑x) x.property hx rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 265,
"column": 14
} | {
"line": 265,
"column": 43
} | {
"line": 267,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\nI : LieIdeal R L\n_x : (s : Finset (LieIdeal R L)) ×' (_ : ↑s ⊆ {I | IsAtom I}) ×' I ≤ s.sup id\na✝⁵ :\n ∀ (y : (s : Finset (LieIdeal R L)) ×' (_ : ↑s ⊆ {I | IsAtom I}) ×' I ≤ s.sup ... | [] | exact Finset.card_lt_card hs' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 265,
"column": 14
} | {
"line": 265,
"column": 43
} | {
"line": 267,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\nI : LieIdeal R L\n_x : (s : Finset (LieIdeal R L)) ×' (_ : ↑s ⊆ {I | IsAtom I}) ×' I ≤ s.sup id\na✝⁵ :\n ∀ (y : (s : Finset (LieIdeal R L)) ×' (_ : ↑s ⊆ {I | IsAtom I}) ×' I ≤ s.sup ... | [] | exact Finset.card_lt_card hs' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 265,
"column": 14
} | {
"line": 265,
"column": 43
} | {
"line": 267,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\nI : LieIdeal R L\n_x : (s : Finset (LieIdeal R L)) ×' (_ : ↑s ⊆ {I | IsAtom I}) ×' I ≤ s.sup id\na✝⁵ :\n ∀ (y : (s : Finset (LieIdeal R L)) ×' (_ : ↑s ⊆ {I | IsAtom I}) ×' I ≤ s.sup ... | [] | exact Finset.card_lt_card hs' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanSubalgebra | {
"line": 89,
"column": 10
} | {
"line": 89,
"column": 15
} | {
"line": 90,
"column": 10
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nH : LieSubalgebra R L\nk : ℕ\nhk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = H.toLieSubmodule\n⊢ ∃ k, LieSubmodule.lcs k H.toLieSubmodule = ⊥",
"ppTerm": "?m.174",
"assigned": true,
"usedConstants": [
... | [
"case h\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nH : LieSubalgebra R L\nk : ℕ\nhk : ∀ (l : ℕ), k ≤ l → LieSubmodule.ucs l ⊥ = H.toLieSubmodule\n⊢ LieSubmodule.lcs k H.toLieSubmodule = ⊥"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Engel | {
"line": 98,
"column": 2
} | {
"line": 103,
"column": 59
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u₁\nL : Type u₂\nM : Type u₄\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule ... | [
"case refine_2\nR : Type u₁\nL : Type u₂\nM : Type u₄\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤\nN : Lie... | · rintro z ⟨y, n, hn : n ∈ N, rfl⟩
obtain ⟨t, z, hz, rfl⟩ := exists_smul_add_of_span_sup_eq_top hxI y
simp only [SetLike.mem_coe, Submodule.span_union, Submodule.mem_sup]
exact
⟨t • ⁅x, n⁆, Submodule.subset_span ⟨t • n, N.smul_mem' t hn, lie_smul t x n⟩, ⁅z, n⁆,
Submodule.subset_span ⟨z, hz, n... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 37
} | {
"line": 196,
"column": 37
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nh : lowerCentralSeries R L M 1 = ⊥\n⊢ IsTrivial L M",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants":... | [
"case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nh : ∀ m ∈ lowerCentralSeries R L M 1, m = 0\n⊢ IsTrivial L M"
] | rw [LieSubmodule.eq_bot_iff] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 198,
"column": 4
} | {
"line": 198,
"column": 90
} | {
"line": 199,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nh : ∀ m ∈ lowerCentralSeries R L M 1, m = 0\nx : L\nm : M\n⊢ ⁅x, m⁆ ∈ {x | ∃ x_1 n, ⁅↑x_1, ↑n⁆ = x}",
"ppTerm": "?... | [
"case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nh : ∀ m ∈ lowerCentralSeries R L M 1, m = 0\nx : L\nm : M\n⊢ ∃ a a_1, ⁅a, a_1⁆ = ⁅x, m⁆"
] | simp only [Subtype.exists, LieSubmodule.mem_top, exists_prop, true_and, Set.mem_setOf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 7
} | {
"line": 325,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : IsNilpotent L M\nk : ℕ\nhM : lowerCentralSeries R L M k = ⊥\n⊢ ∃ k, ∀ (x : L), (toEnd R L M) ... | [
"case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : IsNilpotent L M\nk : ℕ\nhM : lowerCentralSeries R L M k = ⊥\n⊢ ∀ (x : L), (toEnd R L M) x ^ k = 0... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 7
} | {
"line": 341,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : IsNilpotent L M\nx y : L\nk : ℕ\nhM : lowerCentralSeries R L M (2 * k) = ⊥\n⊢ _root_.IsNilpot... | [
"case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : IsNilpotent L M\nx y : L\nk : ℕ\nhM : lowerCentralSeries R L M (2 * k) = ⊥\n⊢ ((toEnd R L M) x ∘ₗ... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 394,
"column": 49
} | {
"line": 404,
"column": 16
} | {
"line": 406,
"column": 0
} | [
{
"pp": "L : Type v\nM : Type w\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : LieRingModule L M\ninst✝ : IsNilpotent L M\n⊢ nilpotencyLength L M = 0 ↔ Subsingleton M",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LieModule.lowerCentralSeries_zero",
"i... | [] | by
let s := {k | lowerCentralSeries ℤ L M k = ⊥}
have hs : s.Nonempty := by
obtain ⟨k, hk⟩ := IsNilpotent.nilpotent ℤ L M
exact ⟨k, hk⟩
change sInf s = 0 ↔ _
rw [← LieSubmodule.subsingleton_iff ℤ L M, ← subsingleton_iff_bot_eq_top, ←
lowerCentralSeries_zero, @eq_comm (LieSubmodule ℤ L M)]
refine ⟨... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Engel | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 68
} | {
"line": 232,
"column": 2
} | [
{
"pp": "R : Type u₁\nL : Type u₂\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := (toEnd R L M).range\nh : ∀ (y : ↥L'), IsNi... | [
"R : Type u₁\nL : Type u₂\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsNoetherian R L\nM : Type u_1\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L M\n_i4 : LieModule R L M\nL' : LieSubalgebra R (Module.End R M) := (toEnd R L M).range\nh : ∀ (y : ↥L'), IsNilpotent ↑y\n... | have hs : s.Nonempty := ⟨⊥, LieAlgebra.isEngelian_of_subsingleton⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 42
} | {
"line": 436,
"column": 0
} | [
{
"pp": "case inr\nL : Type v\nM : Type w\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : LieRingModule L M\ninst✝ : IsNilpotent L M\nh✝ : nilpotencyLength L M ≤ 1\na✝ : Nontrivial M\nh : nilpotencyLength L M = 1\n⊢ IsTrivial L M",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [] | · rwa [nilpotencyLength_eq_one_iff] at h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 518,
"column": 37
} | {
"line": 518,
"column": 42
} | {
"line": 518,
"column": 43
} | [
{
"pp": "case mp\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nhk : lowerCentralSeries R (↥(toEnd R L M).range) M k = ⊥\n⊢ ∃ k, lowerCentralSeries R ... | [
"case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nhk : lowerCentralSeries R (↥(toEnd R L M).range) M k = ⊥\n⊢ lowerCentralSeries R L M k = ⊥"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 518,
"column": 37
} | {
"line": 518,
"column": 42
} | {
"line": 518,
"column": 43
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nhk : lowerCentralSeries R L M k = ⊥\n⊢ ∃ k, lowerCentralSeries R (↥(toEnd R L M).rang... | [
"case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nhk : lowerCentralSeries R L M k = ⊥\n⊢ lowerCentralSeries R (↥(toEnd R L M).range) M k = ⊥"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 644,
"column": 2
} | {
"line": 644,
"column": 7
} | {
"line": 645,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGrou... | [
"case h\nR : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGroup M₂... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.FieldTheory.Fixed | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 24
} | {
"line": 223,
"column": 25
} | [
{
"pp": "G : Type u\ninst✝³ : Group G\nF : Type v\ninst✝² : Field F\ninst✝¹ : MulSemiringAction G F\ninst✝ : Fintype G\nx : F\nf : Polynomial ↥(subfield G F)\nhf : Polynomial.eval₂ (subfield G F).subtype x f = 0\ny : G ⧸ stabilizer G x\ng : G\n⊢ g • Polynomial.eval x (Polynomial.map (subfield G F).subtype f) = ... | [
"G : Type u\ninst✝³ : Group G\nF : Type v\ninst✝² : Field F\ninst✝¹ : MulSemiringAction G F\ninst✝ : Fintype G\nx : F\nf : Polynomial ↥(subfield G F)\nhf : Polynomial.eval₂ (subfield G F).subtype x f = 0\ny : G ⧸ stabilizer G x\ng : G\n⊢ g • Polynomial.eval₂ (subfield G F).subtype x f = 0"
] | Polynomial.eval_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 676,
"column": 2
} | {
"line": 676,
"column": 7
} | {
"line": 677,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGrou... | [
"case h\nR : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : AddCommGroup M₂... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 743,
"column": 2
} | {
"line": 743,
"column": 7
} | {
"line": 743,
"column": 7
} | [
{
"pp": "L : Type v\ninst✝ : LieRing L\nhL : LieModule.IsNilpotent L L\nk : ℕ\nh : LieModule.lowerCentralSeries ℤ L L k = ⊥\n⊢ IsSolvable L",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"LieAlgebra.IsSolvable.mk_int",
"LieSubmodule.instBot",
... | [
"case h\nL : Type v\ninst✝ : LieRing L\nhL : LieModule.IsNilpotent L L\nk : ℕ\nh : LieModule.lowerCentralSeries ℤ L L k = ⊥\n⊢ derivedSeries ℤ L k = ⊥"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 807,
"column": 50
} | {
"line": 812,
"column": 81
} | {
"line": 814,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nh₁ : I ≤ center R L\nh₂ : LieRing.IsNilpotent (L ⧸ I)\n⊢ LieRing.IsNilpotent L",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
... | [] | by
suffices LieModule.IsNilpotent L (L ⧸ I) by
exact LieModule.nilpotentOfNilpotentQuotient R L L h₁ this
simp only [LieRing.IsNilpotent, LieModule.isNilpotent_iff R] at h₂ ⊢
peel h₂ with k hk
simp [← LieSubmodule.toSubmodule_inj, coe_lowerCentralSeries_ideal_quot_eq, hk] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 830,
"column": 4
} | {
"line": 830,
"column": 45
} | {
"line": 831,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nk : ℕ\nf : L →ₗ⁅R⁆ L'\nh : Function.Surjective ⇑f\n⊢ f.idealRange = ⊤",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"LieHom.... | [] | exact f.idealRange_eq_top_of_surjective h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.SplittingField.IsSplittingField | {
"line": 69,
"column": 19
} | {
"line": 69,
"column": 26
} | {
"line": 69,
"column": 27
} | [
{
"pp": "F : Type u\nK : Type v\nL : Type w\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Field F\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra F K\ninst✝² : Algebra F L\ninst✝¹ : IsScalarTower F K L\nf : F[X]\ninst✝ : IsSplittingField F L f\n⊢ Subalgebra.restrictScalars F (Algebra.adjoin K ↑((Polynomial.map (algebra... | [
"F : Type u\nK : Type v\nL : Type w\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Field F\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra F K\ninst✝² : Algebra F L\ninst✝¹ : IsScalarTower F K L\nf : F[X]\ninst✝ : IsSplittingField F L f\n⊢ Subalgebra.restrictScalars F\n (Algebra.adjoin K ↑(Polynomial.map (algebraMap K ... | aroots, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.SplittingField.IsSplittingField | {
"line": 76,
"column": 51
} | {
"line": 76,
"column": 58
} | {
"line": 76,
"column": 59
} | [
{
"pp": "K : Type v\nL : Type w\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nf : K[X]\ninst✝ : IsSplittingField K L f\nh : f.Splits\n⊢ Algebra.adjoin K ↑(f.aroots L).toFinset ≤ ⊥",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
... | [
"K : Type v\nL : Type w\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nf : K[X]\ninst✝ : IsSplittingField K L f\nh : f.Splits\n⊢ Algebra.adjoin K ↑(Polynomial.map (algebraMap K L) f).roots.toFinset ≤ ⊥"
] | aroots, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.SplittingField.IsSplittingField | {
"line": 144,
"column": 2
} | {
"line": 145,
"column": 63
} | {
"line": 147,
"column": 0
} | [
{
"pp": "case adjoin_rootSet'\nF : Type u\nK : Type v\nL : Type w\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Field F\ninst✝² : Algebra K L\ninst✝¹ : Algebra K F\np : K[X]\nf : F ≃ₐ[K] L\ninst✝ : IsSplittingField K F p\n⊢ Algebra.adjoin K (p.rootSet L) = ⊤",
"ppTerm": "?adjoin_rootSet'",
"assigned": t... | [] | · rw [← (AlgHom.range_eq_top f.toAlgHom).mpr f.surjective,
(splits F p).adjoin_rootSet_eq_range, adjoin_rootSet F p] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 85,
"column": 46
} | {
"line": 85,
"column": 59
} | {
"line": 85,
"column": 60
} | [
{
"pp": "F : Type u_1\ninst✝⁷ : Field F\nE : Type u_2\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nS : Set E\nK : Type u_3\nL : Type u_4\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nE1 E2 : IntermediateField K L\ninst✝¹ : FiniteDimensional K ↥E1\ninst✝ : FiniteDimensional K ↥E2\ng : TensorProduct K ↥E... | [
"F : Type u_1\ninst✝⁷ : Field F\nE : Type u_2\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nS : Set E\nK : Type u_3\nL : Type u_4\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nE1 E2 : IntermediateField K L\ninst✝¹ : FiniteDimensional K ↥E1\ninst✝ : FiniteDimensional K ↥E2\ng : TensorProduct K ↥E1 ↥E2 →ₐ[K] ... | E1.range_val, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 269,
"column": 4
} | {
"line": 272,
"column": 100
} | {
"line": 274,
"column": 0
} | [
{
"pp": "case refine_2\nk : Type u\ninst✝³ : Field k\nK : Type v\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : Algebra k K\np q : k[X]\nh : ∀ (a : K), (aeval a) p ≠ 0 ∨ (aeval a) q ≠ 0\nx : k[X]\nhu : x ∈ nonunits k[X]\nh0 : x ≠ 0\n⊢ x ∣ p → ¬x ∣ q",
"ppTerm": "?refine_2",
"assigned": true,
"us... | [] | rintro ⟨_, rfl⟩ ⟨_, rfl⟩
obtain ⟨a, ha : _ = _⟩ := IsAlgClosed.exists_root (x.map <| algebraMap k K) <| by
simpa only [degree_map] using (ne_of_lt <| degree_pos_of_ne_zero_of_nonunit h0 hu).symm
exact not_and_or.mpr (h a) (by simp_rw [map_mul, ← eval_map_algebraMap, ha, zero_mul, true_and]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 269,
"column": 4
} | {
"line": 272,
"column": 100
} | {
"line": 274,
"column": 0
} | [
{
"pp": "case refine_2\nk : Type u\ninst✝³ : Field k\nK : Type v\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : Algebra k K\np q : k[X]\nh : ∀ (a : K), (aeval a) p ≠ 0 ∨ (aeval a) q ≠ 0\nx : k[X]\nhu : x ∈ nonunits k[X]\nh0 : x ≠ 0\n⊢ x ∣ p → ¬x ∣ q",
"ppTerm": "?refine_2",
"assigned": true,
"us... | [] | rintro ⟨_, rfl⟩ ⟨_, rfl⟩
obtain ⟨a, ha : _ = _⟩ := IsAlgClosed.exists_root (x.map <| algebraMap k K) <| by
simpa only [degree_map] using (ne_of_lt <| degree_pos_of_ne_zero_of_nonunit h0 hu).symm
exact not_and_or.mpr (h a) (by simp_rw [map_mul, ← eval_map_algebraMap, ha, zero_mul, true_and]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 61
} | {
"line": 335,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\n⊢ Module.rank F ↥F⟮α⟯ = 1 ↔ α ∈ ⊥",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Set.singleton_subset_iff",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"instSMulOfMul",
... | [] | rw [rank_adjoin_eq_one_iff]; exact Set.singleton_subset_iff | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 61
} | {
"line": 335,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\n⊢ Module.rank F ↥F⟮α⟯ = 1 ↔ α ∈ ⊥",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Set.singleton_subset_iff",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"instSMulOfMul",
... | [] | rw [rank_adjoin_eq_one_iff]; exact Set.singleton_subset_iff | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 568,
"column": 2
} | {
"line": 574,
"column": 47
} | {
"line": 575,
"column": 2
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : CharZero K\nf g : K[X]\nhf0 : f ≠ 0\nhg0 : ¬g = 0\nhdf0 : ¬derivative f = 0\nhdg : derivative f * g ≠ 0\na : K\nhaf : eval a f = 0\n⊢ (f.IsRoot a → g.IsRoot a) → rootMultiplicity a f ≤ rootMultiplicity a (derivative f) + rootMult... | [
"case neg\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : CharZero K\nf g : K[X]\nhf0 : f ≠ 0\nhg0 : ¬g = 0\nhdf0 : ¬derivative f = 0\nhdg : derivative f * g ≠ 0\na : K\nhaf : ¬eval a f = 0\n⊢ (f.IsRoot a → g.IsRoot a) → rootMultiplicity a f ≤ rootMultiplicity a (derivative f) + rootMultiplicity a ... | · have h0 : 0 < f.rootMultiplicity a := (rootMultiplicity_pos hf0).2 haf
rw [derivative_rootMultiplicity_of_root haf]
intro h
calc rootMultiplicity a f
= rootMultiplicity a f - 1 + 1 := (Nat.sub_add_cancel (Nat.succ_le_iff.1 h0)).symm
_ ≤ rootMultiplicity a f - 1 + rootMultiplicity a g := add_... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 9
} | {
"line": 113,
"column": 4
} | [
{
"pp": "case refine_2\nR : Type u_2\nL : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : LieRing L\ninst✝¹² : LieAlgebra R L\nM₁ : Type u_5\nM₂ : Type u_6\nM₃ : Type u_7\ninst✝¹¹ : AddCommGroup M₁\ninst✝¹⁰ : Module R M₁\ninst✝⁹ : LieRingModule L M₁\ninst✝⁸ : LieModule R L M₁\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Modul... | [
"case h\nR : Type u_2\nL : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : LieRing L\ninst✝¹² : LieAlgebra R L\nM₁ : Type u_5\nM₂ : Type u_6\nM₃ : Type u_7\ninst✝¹¹ : AddCommGroup M₁\ninst✝¹⁰ : Module R M₁\ninst✝⁹ : LieRingModule L M₁\ninst✝⁸ : LieModule R L M₁\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\ninst✝⁵ : Li... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 7
} | {
"line": 190,
"column": 2
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝ : LieRing.IsNilpotent ↥H\nx : L\nhx : x ∈ H\ny : ↥H\nk : ℕ\nhk : lowerCentralSeries R (↥H) (↥H) k = ⊥\n⊢ ∃ k, ((toEnd R (↥H) L) y ^ k) x = 0",
"ppTerm": "?m.60",
"assigned"... | [
"case h\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝ : LieRing.IsNilpotent ↥H\nx : L\nhx : x ∈ H\ny : ↥H\nk : ℕ\nhk : lowerCentralSeries R (↥H) (↥H) k = ⊥\n⊢ ((toEnd R (↥H) L) y ^ k) x = 0"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 7
} | {
"line": 338,
"column": 2
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : LieRing.IsNilpotent L\ninst✝ : IsNoetherian R M\nχ : L → R\nx : L\nthis :\n (toEnd R ... | [
"case h\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : LieRing.IsNilpotent L\ninst✝ : IsNoetherian R M\nχ : L → R\nx : L\nthis :\n (toEnd R L ↥(... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 468,
"column": 2
} | {
"line": 468,
"column": 7
} | {
"line": 469,
"column": 2
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : LieRingModule L M\ninst✝³ : LieModule R L M\ninst✝² : LieRing.IsNilpotent L\ninst✝¹ : IsNoetherian R M\ninst✝ : IsArtinian R M\nx : L\nk : ℕ... | [
"case h\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : LieRingModule L M\ninst✝³ : LieModule R L M\ninst✝² : LieRing.IsNilpotent L\ninst✝¹ : IsNoetherian R M\ninst✝ : IsArtinian R M\nx : L\nk : ℕ\nhk... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 528,
"column": 4
} | {
"line": 528,
"column": 9
} | {
"line": 529,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra R L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : LieRingModule L M\ninst✝⁵ : LieModule R L M\ninst✝⁴ : LieRing.IsNilpotent L\nM₂ : Type u_5\ninst✝³ : AddCommGroup M₂\ninst✝... | [
"case h\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra R L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : LieRingModule L M\ninst✝⁵ : LieModule R L M\ninst✝⁴ : LieRing.IsNilpotent L\nM₂ : Type u_5\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\nin... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Killing | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 65
} | {
"line": 65,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_3\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsKilling R L\n⊢ LinearMap.ker (killingForm R L) = ⊥",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"Submodule",
"Semiring.toModule",
... | [] | simp [← LieIdeal.coe_killingCompl_top, killingCompl_top_eq_bot] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Killing | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 65
} | {
"line": 65,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_3\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsKilling R L\n⊢ LinearMap.ker (killingForm R L) = ⊥",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"Submodule",
"Semiring.toModule",
... | [] | simp [← LieIdeal.coe_killingCompl_top, killingCompl_top_eq_bot] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Killing | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 65
} | {
"line": 65,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_3\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsKilling R L\n⊢ LinearMap.ker (killingForm R L) = ⊥",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"Submodule",
"Semiring.toModule",
... | [] | simp [← LieIdeal.coe_killingCompl_top, killingCompl_top_eq_bot] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 620,
"column": 4
} | {
"line": 621,
"column": 29
} | {
"line": 622,
"column": 4
} | [
{
"pp": "case inr.h\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : LieRingModule L M\ninst✝³ : LieModule R L M\ninst✝² : LieRing.IsNilpotent L\ninst✝¹ : IsNoetherian R M\ninst✝ : IsArtinian R M\n... | [
"case inr.hcomp\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : LieRingModule L M\ninst✝³ : LieModule R L M\ninst✝² : LieRing.IsNilpotent L\ninst✝¹ : IsNoetherian R M\ninst✝ : IsArtinian R M\nh : ∀ N ... | · rw [disjoint_iff, ← LieSubmodule.map_inf M₀ₓ.injective_incl, h₂.inf_eq_bot,
LieSubmodule.map_bot] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Sl2 | {
"line": 267,
"column": 10
} | {
"line": 267,
"column": 40
} | {
"line": 267,
"column": 41
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\ninst✝⁶ : LieModule R L M\nh e f : L\ninst✝⁵ : IsDomain R\ninst✝⁴ : CharZero R\ninst✝³ : Nontrivial M\ninst✝² : IsTorsi... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\ninst✝⁶ : LieModule R L M\nh e f : L\ninst✝⁵ : IsDomain R\ninst✝⁴ : CharZero R\ninst✝³ : Nontrivial M\ninst✝² : IsTorsionFree R M\n... | Module.End.mem_eigenspace_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.TraceForm | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 44
} | {
"line": 130,
"column": 45
} | [
{
"pp": "case mem\nR : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh : traceForm R L M = 0\ny : End R M\nhy : ∀ z ∈ φ.range, ⁅y, z⁆ ∈ φ.range\nx a b c : ... | [
"case mem\nR : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nh : traceForm R L M = 0\ny : End R M\nhy : ∀ z ∈ φ.range, ⁅y, z⁆ ∈ φ.range\nx a b c : L\nhbc : ⁅φ ... | ← LieRing.of_associative_ring_bracket, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 370,
"column": 22
} | {
"line": 370,
"column": 73
} | {
"line": 370,
"column": 73
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra R L\nM : Type u_3\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : LieRingModule L M\ninst✝⁵ : LieModule R L M\ninst✝⁴ : LieRing.IsNilpotent L\ninst✝³ : IsAddTorsionFree R\ninst✝² : IsDomain R\ninst✝¹ : Is... | [
"R : Type u_1\nL : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra R L\nM : Type u_3\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : LieRingModule L M\ninst✝⁵ : LieModule R L M\ninst✝⁴ : LieRing.IsNilpotent L\ninst✝³ : IsAddTorsionFree R\ninst✝² : IsDomain R\ninst✝¹ : IsTorsionFree ... | genWeightSpace_add_chainTop _ _ (by simpa using hα) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.BaseChange | {
"line": 46,
"column": 54
} | {
"line": 46,
"column": 70
} | {
"line": 47,
"column": 2
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nL : Type u_3\ninst✝³ : Finite m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ninst✝ : Field L\ne : m ≃ n\nK : Subfield L\nA : Matrix m n L\nB : Matrix n m L\nhAB : A * B = 1\nh_mem : ∀ (i : m) (j : n), A i j ∈ K\ni : n\nj : m\nval✝ : Fintype m\nA' : Matrix m m ↥K := of fun i ... | [] | by simp [← this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 48
} | {
"line": 207,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nhn : Odd n\n⊢ eval 0 (T R n) = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Units.val",
"Polynomial.eval",
"Int.instDiv",
"False",
"instHDiv",
... | [] | simp [T_eval_zero, ← Int.not_odd_iff_even, hn] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 48
} | {
"line": 207,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nhn : Odd n\n⊢ eval 0 (T R n) = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Units.val",
"Polynomial.eval",
"Int.instDiv",
"False",
"instHDiv",
... | [] | simp [T_eval_zero, ← Int.not_odd_iff_even, hn] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 48
} | {
"line": 207,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nhn : Odd n\n⊢ eval 0 (T R n) = 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Units.val",
"Polynomial.eval",
"Int.instDiv",
"False",
"instHDiv",
... | [] | simp [T_eval_zero, ← Int.not_odd_iff_even, hn] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Reflection | {
"line": 255,
"column": 44
} | {
"line": 255,
"column": 58
} | {
"line": 255,
"column": 58
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nf g : Dual R M\nhf : f x = 2\nhg : g y = 2\nz : M\nt : R\nht : t = f y * g x - 2\na✝ :\n ∀ (n : ℕ),\n ((reflection hf * reflection hg) ^ ↑n) z =\n z +\n (Polynomial.eval t (S R ((↑n ... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nf g : Dual R M\nhf : f x = 2\nhg : g y = 2\nz : M\nt : R\nht : t = f y * g x - 2\na✝ :\n ∀ (n : ℕ),\n ((reflection hf * reflection hg) ^ ↑n) z =\n z +\n (Polynomial.eval t (S R ((↑n - 2) / 2)) *... | mul_comm (g x) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.RationalRoot | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 35
} | {
"line": 99,
"column": 2
} | [
{
"pp": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : (aeval r) p = 0\n⊢ ↑(den A r) ∣ p.leadingCoeff * num A r ^ p.natDegree",
"ppTerm": "?m.58",
"assi... | [
"A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : (aeval r) p = 0\n⊢ ↑(den A r) ∣ (p.scaleRoots ?s).coeff p.natDegree * num A r ^ p.natDegree",
"case s\nA : Type u_1... | rw [← coeff_scaleRoots_natDegree] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 426,
"column": 17
} | {
"line": 426,
"column": 25
} | {
"line": 426,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℤ\nhn : n = -1\n⊢ (U R n).natDegree = (n + 1).natAbs - 1",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Polynomial.Chebyshev.U",
"CommSemiring.toSemiring",
"HSub.hSub",
"Int.ins... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\n⊢ (U R (-1)).natDegree = (-1 + 1).natAbs - 1"
] | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.RingTheory.FiniteLength | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 16
} | {
"line": 60,
"column": 16
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : CompositionSeries (Submodule R M)\ns_head : RelSeries.head s = ⊥\ns_last : RelSeries.last s = ⊤\n⊢ IsFiniteLength R ↥⊤",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"R : Type u_1\ninst✝² : Ring R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ns : CompositionSeries (Submodule R M)\ns_head : RelSeries.head s = ⊥\ns_last : RelSeries.last s = ⊤\n⊢ IsFiniteLength R ↥(RelSeries.last s)"
] | ← s_last | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 367,
"column": 6
} | {
"line": 368,
"column": 53
} | {
"line": 369,
"column": 2
} | [
{
"pp": "case succ\nR : Type u_1\nM : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nx : M\nΦ : Set M\nhΦ₁ : Φ.Finite\nhΦ₂ : span R Φ = ⊤\nf g : Dual R M\nhf₁ : f x = 2\nhf₂ : MapsTo (⇑(preReflection x f)) Φ Φ\nhg... | [] | simp_rw [Module.End.mul_eq_comp, LinearMap.comp_id, LinearMap.id_comp, this, add_zero,
add_assoc, Nat.cast_succ, add_smul, one_smul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Length | {
"line": 282,
"column": 47
} | {
"line": 282,
"column": 65
} | {
"line": 282,
"column": 65
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ IsSimpleOrder (Submodule R M) ↔ IsSimpleModule R M",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"AddCommGroup.toAddCommMonoid",... | [
"R : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ IsSimpleOrder (Submodule R M) ↔ IsSimpleOrder (Submodule R M)"
] | isSimpleModule_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.Integers | {
"line": 417,
"column": 2
} | {
"line": 422,
"column": 14
} | {
"line": 423,
"column": 2
} | [
{
"pp": "case mp\nΓ₀ : Type v\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nK : Type u_1\ninst✝ : Field K\nv : Valuation K Γ₀\nγ : Γ₀\nx : K\n⊢ (∃ y, v ↑y ≤ γ ∧ (algebraMap (↥v.integer) K) y = x) → v x ≤ min 1 γ",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearOrde... | [
"case mpr\nΓ₀ : Type v\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nK : Type u_1\ninst✝ : Field K\nv : Valuation K Γ₀\nγ : Γ₀\nx : K\n⊢ v x ≤ min 1 γ → ∃ y, v ↑y ≤ γ ∧ (algebraMap (↥v.integer) K) y = x"
] | · rintro ⟨y, hy, rfl⟩
rcases min_cases 1 γ with ⟨h, _⟩ | ⟨h, _⟩
· rw [h]
exact y.prop
· rw [h]
exact hy | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 691,
"column": 2
} | {
"line": 691,
"column": 67
} | {
"line": 693,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nm n : ℕ\nx : AdicCompletion I M\nm_ge : n ≤ m\nh : ↑x n = 0\n⊢ ↑x m ∈ I ^ n • ⊤",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"LinearMap.id",
... | [] | simpa [mapQ, h, ← LinearMap.mem_ker, ker_liftQ] using x.prop m_ge | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 883,
"column": 2
} | {
"line": 883,
"column": 7
} | {
"line": 884,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nx : R\nhx : x ∈ I\ny : R\nf : ℕ → R := fun n ↦ ∑ i ∈ range n, (x * y) ^ i\nhf : ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]\nL : R\nhL : ∀ (n : ℕ), f n ≡ L [SMOD I ^ n • ⊤]\n⊢ ∃ b, (1 + -x * y) * b = 1",
"ppTerm": "?m.2... | [
"case h\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nx : R\nhx : x ∈ I\ny : R\nf : ℕ → R := ⋯\nhf : ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]\nL : R\nhL : ∀ (n : ℕ), f n ≡ L [SMOD I ^ n • ⊤]\n⊢ (1 + -x * y) * L = 1"
] | use L | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 1119,
"column": 2
} | {
"line": 1119,
"column": 52
} | {
"line": 1120,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm n : ℤ\n⊢ T R (m * n) = (T R m).comp (T R n)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocS... | [] | induction m using Polynomial.Chebyshev.induct with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 91
} | {
"line": 316,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ PreValuationRing R ↔ Std.Total fun x1 x2 ↦ x1 ≤ x2",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Semigroup.toMul",
"PreValuationRing.iff_dvd_total",
"Dvd.dvd",
"Semiring.toModule",
"PreValuationRi... | [
"R : Type u_1\ninst✝ : CommRing R\nH : Std.Total fun x1 x2 ↦ x1 ≤ x2\na b : R\n⊢ a ∣ b ∨ b ∣ a"
] | refine ⟨fun _ => ⟨le_total⟩, fun H => PreValuationRing.iff_dvd_total.mpr ⟨fun a b => ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 1135,
"column": 2
} | {
"line": 1135,
"column": 52
} | {
"line": 1136,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm n : ℤ\n⊢ C R (m * n) = (C R m).comp (C R n)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocS... | [] | induction m using Polynomial.Chebyshev.induct with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 195,
"column": 44
} | {
"line": 195,
"column": 97
} | {
"line": 195,
"column": 97
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ny : R\ninst✝² : Algebra R S\ninst✝¹ : Away y S\ninst✝ : IsJacobsonRing R\nI : Ideal R\nhI : I.IsMaximal\nhy : y ∉ I\n⊢ (under R (Ideal.map (algebraMap R S) I)).IsMaximal ∧ y ∉ under R (Ideal.map (algebraMap R S) I)",
"ppTerm": "?... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ny : R\ninst✝² : Algebra R S\ninst✝¹ : Away y S\ninst✝ : IsJacobsonRing R\nI : Ideal R\nhI : I.IsMaximal\nhy : y ∉ I\n⊢ I.IsMaximal ∧ y ∉ I",
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ny : R\ninst✝² : Algebra R S\nin... | under_map_of_isPrime_disjoint (powers y) S hI.isPrime | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 24
} | {
"line": 257,
"column": 4
} | [
{
"pp": "case h.a\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^... | [
"case pos\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ... | by_cases hr0 : r = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 260,
"column": 4
} | {
"line": 260,
"column": 21
} | {
"line": 261,
"column": 4
} | [
{
"pp": "case neg\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^... | [
"case neg\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : UniqueFactorizationMonoid R\nh₁ : ∃ p, Irreducible p\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nI : Ideal R\nI0 : ¬I = ⊥\nx : R\nhxI : x ∈ I\nhx0 : x ≠ 0\np : R\nleft✝ : Irreducible p\nH : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (p ^ n) x\nex : ... | apply pow_dvd_pow | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 62
} | {
"line": 357,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nϖ : R\nhϖ : Irreducible ϖ\nn : ℕ\nu : Rˣ\nm : ℕ\nv : Rˣ\nhy : ↑v * ϖ ^ m ∈ nonZeroDivisors R\nhx : (algebraMap R K) (↑u * ϖ ^ n) ... | [
"R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nϖ : R\nhϖ : Irreducible ϖ\nn : ℕ\nu : Rˣ\nm : ℕ\nv : Rˣ\nhy : ↑v * ϖ ^ m ∈ nonZeroDivisors R\nhx : (algebraMap R K) (↑u * ϖ ^ n) / (algebraMa... | have hϖ' : algebraMap R K ϖ ≠ 0 := by simpa using hϖ.ne_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.KrullDimension.Zero | {
"line": 125,
"column": 2
} | {
"line": 126,
"column": 59
} | {
"line": 128,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : KrullDimLE 0 R\ninst✝ : IsLocalRing R\nI : Ideal R\nhI : I ≠ ⊤\n⊢ ∀ b ∈ {J | I ≤ J ∧ J.IsPrime}, IsLocalRing.maximalIdeal R ≤ b",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Ring.KrullDimLE.eq_maximalIdeal... | [] | · rintro J ⟨h₁, h₂⟩
exact (Ring.KrullDimLE.eq_maximalIdeal_of_isPrime J).ge | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Ideal.IsPrincipal | {
"line": 123,
"column": 17
} | {
"line": 124,
"column": 75
} | {
"line": 124,
"column": 75
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nx : Associates R\n⊢ x ∈ (Associates R)⁰ ↔ ↑((associatesEquivIsPrincipal R) x) ∈ (Ideal R)⁰",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Associates.mk",
"IsDomain.to_noZeroDivisors",
"Semi... | [] | by rw [← quot_out x, mk_mem_nonZeroDivisors_associates,
associatesEquivIsPrincipal_apply, span_singleton_nonZeroDivisors] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Flat.TorsionFree | {
"line": 112,
"column": 61
} | {
"line": 112,
"column": 85
} | {
"line": 112,
"column": 86
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\nhtors : torsion R M = ⊥\nI : Ideal R\nhFG : I.FG\nh : I ≠ ⊥\nhprinc : Submodule.IsPrincipal I\nthis : IsPrincipal.generator I ≠ 0\n⊢ Function.Injective\n ⇑(lift (ls... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\nhtors : torsion R M = ⊥\nI : Ideal R\nhFG : I.FG\nh : I ≠ ⊥\nhprinc : Submodule.IsPrincipal I\nthis : IsPrincipal.generator I ≠ 0\n⊢ Function.Injective ⇑(lift (lsmul R M ∘ₗ Submod... | LinearEquiv.coe_rTensor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Flat.TorsionFree | {
"line": 144,
"column": 2
} | {
"line": 145,
"column": 16
} | {
"line": 147,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDedekindDomain R\ninst✝ : IsTorsionFree R M\n⊢ Flat R M",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"IsDedekindDomain.toIsDomain",... | [] | rw [IsDedekindDomain.flat_iff_torsion_eq_bot, ← Submodule.isTorsionFree_iff_torsion_eq_bot]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Flat.TorsionFree | {
"line": 144,
"column": 2
} | {
"line": 145,
"column": 16
} | {
"line": 147,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDedekindDomain R\ninst✝ : IsTorsionFree R M\n⊢ Flat R M",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"IsDedekindDomain.toIsDomain",... | [] | rw [IsDedekindDomain.flat_iff_torsion_eq_bot, ← Submodule.isTorsionFree_iff_torsion_eq_bot]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 65
} | {
"line": 95,
"column": 4
} | [
{
"pp": "case inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : CharZero R\ninst✝² : IsAddTorsionFree M\ninst✝¹ : P.IsReduced\nn : ℕ\ninst✝ : n.AtLeastTwo... | [
"case inr\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : CharZero R\ninst✝² : IsAddTorsionFree M\ninst✝¹ : P.IsReduced\nn : ℕ\ninst✝ : n.AtLeastTwo\ni j : ι\nh... | replace this : (n : ℤ) • P.root i = -1 • P.root i := by simpa | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 38
} | {
"line": 151,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : CommRing R\ninst✝¹² : Module R M\ninst✝¹¹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹⁰ : CommRing S\ninst✝⁹ : IsDomain R\ninst✝⁸ : IsDomain S\ninst✝... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : AddCommGroup M\ninst✝¹⁴ : AddCommGroup N\ninst✝¹³ : CommRing R\ninst✝¹² : Module R M\ninst✝¹¹ : Module R N\nP : RootPairing ι R M N\nS : Type u_5\ninst✝¹⁰ : CommRing S\ninst✝⁹ : IsDomain R\ninst✝⁸ : IsDomain S\ninst✝⁷ : Algebra ... | rw [algebraMap_rootFormIn] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 68,
"column": 46
} | {
"line": 68,
"column": 71
} | {
"line": 69,
"column": 6
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup M\ninst✝ : Module K M\np : ι → Submodule K M\nh₁ : ∀ (i : ι), p i ≠ ⊤\nj : ι\ns : Finset ι\nhj : j ∉ s\nhs : s.Nonempty\nh₂ : ↑s.card + 1 < ENat.card K\nhj' : ↑(p j) ∪ ⋃ i ∈ s, ↑(p i) = univ\nx : M\nhx : x ∈ p j\nhx₀ : x ... | [] | by specialize hf z; aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Basic | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 96
} | {
"line": 270,
"column": 0
} | [
{
"pp": "case h\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R N\nP : RootPairing ι R M N\ninst✝⁶ : Finite ι\ninst✝⁵ : P.IsRootSystem\nk : Type u_5\ninst✝⁴ : Field k\ninst✝³ : CharZero k\nin... | [] | rw [equiv_of_mapsTo_apply, (exist_eq_reflection_of_mapsTo p root coroot i j hs).choose_spec] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 158,
"column": 8
} | {
"line": 158,
"column": 67
} | {
"line": 159,
"column": 6
} | [
{
"pp": "case neg.a\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : ¬α.I... | [
"case neg.a\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : ¬α.IsZero\n⊢ cha... | ← Nat.le_sub_iff_add_le (chainTopCoeff_le_chainLength α β), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Basic | {
"line": 754,
"column": 4
} | {
"line": 754,
"column": 19
} | {
"line": 754,
"column": 20
} | [
{
"pp": "case single.single.single\nk : Type u_1\nG : Type u_2\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\nα✝ : Type u_3\nA : Type u_4\nB : Type u_5\ninst✝³ : AddCommMonoid A\ninst✝² : Module k A\nρ : Representation k G A\ninst✝¹ : AddCommMonoid B\ninst✝ : Module k B\nτ : Representation k G B\nα : Type u_6\ng ... | [] | | single i b => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.RootSystem.Finite.Lemmas | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 62
} | {
"line": 89,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : P.IsCrystallographic\ni j : ι\nthis✝ : Fintype ι\nn : ℕ\nhcn ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : P.IsCrystallographic\ni j : ι\nthis✝ : Fintype ι\nn : ℕ\nhcn : P.coxeterW... | simp only [hcn, mem_insert_iff, mem_singleton_iff] at this ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 304,
"column": 2
} | {
"line": 305,
"column": 100
} | {
"line": 306,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\... | [
"case pos\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\nh : LinearI... | have : P.chainBotCoeff i j = 0 ↔ Iic (P.chainBotCoeff i j) = {0} := by
simpa [Set.ext_iff, mem_Iic, mem_singleton_iff] using ⟨fun h ↦ by simp [h], fun h ↦ by rw [← h]⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 437,
"column": 2
} | {
"line": 439,
"column": 67
} | {
"line": 440,
"column": 2
} | [
{
"pp": "case pos\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\... | [
"case pos\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Finite ι\ninst✝⁷ : CommRing R\ninst✝⁶ : CharZero R\ninst✝⁵ : IsDomain R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\ni j : ι\naux : Linea... | have hS₂ : S₂ = Icc (-P.chainBotCoeff i (P.chainTopIdx i j) : ℤ)
(P.chainTopCoeff i (P.chainTopIdx i j)) := by
ext; rw [S₂_def, mem_setOf_eq, root_add_zsmul_mem_range_iff h'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Real.Basic | {
"line": 283,
"column": 25
} | {
"line": 283,
"column": 31
} | {
"line": 283,
"column": 31
} | [
{
"pp": "f g : CauSeq ℚ abs\n⊢ { cauchy := ⟦f⟧ }.lt { cauchy := ⟦g⟧ } ↔ f < g",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CauSeq.instLTAbs",
"Real",
"Real.equivCauchy._proof_1",
"abs",
"congrArg",
"IsAbsoluteValue.abs_isAbsoluteValue... | [
"f g : CauSeq ℚ abs\n⊢ (match { cauchy := ⟦f⟧ }, { cauchy := ⟦g⟧ } with\n | { cauchy := x }, { cauchy := y } => Quotient.liftOn₂ x y (fun x1 x2 ↦ x1 < x2) definition._proof_1✝) ↔\n f < g"
] | lt_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.Basic | {
"line": 507,
"column": 14
} | {
"line": 507,
"column": 55
} | {
"line": 508,
"column": 2
} | [
{
"pp": "x : ℝ\n⊢ 0⁻¹ = 0",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"CauSeq.Completion.instInvCauchy",
"GroupWithZero.toMonoidWithZero",
"Real",
"DivInvMonoid.toInv",
"abs",
"congrArg",
"Real.instDivInvMonoid",
"inv_zero",
"IsAbso... | [] | by simp [← ofCauchy_zero, ← ofCauchy_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Eigenspace.Matrix | {
"line": 64,
"column": 16
} | {
"line": 64,
"column": 28
} | {
"line": 64,
"column": 28
} | [
{
"pp": "case mp\nR : Type u_1\nn : Type u_2\nM : Type u_3\ninst✝⁷ : DecidableEq n\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : Nontrivial R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nd : n → R\nμ : R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nb : Basis n R M\nthis✝ : ∀ (i : n), HasEigenvalue ((... | [
"case mp\nR : Type u_1\nn : Type u_2\nM : Type u_3\ninst✝⁷ : DecidableEq n\ninst✝⁶ : Fintype n\ninst✝⁵ : CommRing R\ninst✝⁴ : Nontrivial R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nd : n → R\nμ : R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nb : Basis n R M\nthis✝ : ∀ (i : n), HasEigenvalue ((toLin b b) (... | disjoint_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.ZPow | {
"line": 226,
"column": 63
} | {
"line": 226,
"column": 78
} | {
"line": 226,
"column": 78
} | [
{
"pp": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : IsUnit A.det\nm n : ℕ\n⊢ A ^ (↑m * ↑n) = A ^ ↑(m * n)",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Matrix",
... | [
"n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nh : IsUnit A.det\nm n : ℕ\n⊢ A ^ (↑m * ↑n) = A ^ (↑m * ↑n)"
] | Int.natCast_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 205,
"column": 17
} | {
"line": 205,
"column": 51
} | {
"line": 205,
"column": 51
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : DecidableEq n\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nd : n → R\nh : (diagonal d).PosDef\ni : n\n⊢ 0 < d i",
"ppTerm": "?m.26",
"assigned": true,
"usedCo... | [] | simpa using h.2 (x := .single i 1) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 205,
"column": 17
} | {
"line": 205,
"column": 51
} | {
"line": 205,
"column": 51
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : DecidableEq n\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nd : n → R\nh : (diagonal d).PosDef\ni : n\n⊢ 0 < d i",
"ppTerm": "?m.26",
"assigned": true,
"usedCo... | [] | simpa using h.2 (x := .single i 1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 205,
"column": 17
} | {
"line": 205,
"column": 51
} | {
"line": 205,
"column": 51
} | [
{
"pp": "n : Type u_2\nR : Type u_3\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : DecidableEq n\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nd : n → R\nh : (diagonal d).PosDef\ni : n\n⊢ 0 < d i",
"ppTerm": "?m.26",
"assigned": true,
"usedCo... | [] | simpa using h.2 (x := .single i 1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.PosDef | {
"line": 447,
"column": 27
} | {
"line": 450,
"column": 96
} | {
"line": 452,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : StarRing R\ninst✝¹ : Fintype n\ninst✝ : Fintype m\nA : Matrix n n R\nB : Matrix n m R\nhA : A.PosDef\nhB : Function.Injective B.mulVec\n⊢ (Bᴴ * A * B).PosDef",
"ppTerm": "?m.32",
"assigned": true,
"... | [] | by
refine of_dotProduct_mulVec_pos (isHermitian_conjTranspose_mul_mul _ hA.1) fun x hx => ?_
have : B *ᵥ x ≠ 0 := fun h => hx <| hB.eq_iff' (mulVec_zero _) |>.1 h
simpa only [star_mulVec, dotProduct_mulVec, vecMul_vecMul] using hA.dotProduct_mulVec_pos this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Eigenspace.Minpoly | {
"line": 100,
"column": 59
} | {
"line": 105,
"column": 79
} | {
"line": 107,
"column": 0
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nf : End R M\ninst✝² : IsDomain R\ninst✝¹ : Module.Finite R M\ninst✝ : IsTorsionFree R M\n⊢ Set.Finite f.HasEigenvalue",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Module.End.instRi... | [] | by
have h : minpoly R f ≠ 0 := minpoly.ne_zero (Algebra.IsIntegral.isIntegral (R := R) f)
convert! (minpoly R f).rootSet_finite R
ext μ
change f.HasEigenvalue μ ↔ _
rw [hasEigenvalue_iff_isRoot, mem_rootSet_of_ne h, IsRoot, coe_aeval_eq_eval] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 410,
"column": 10
} | {
"line": 410,
"column": 23
} | {
"line": 410,
"column": 24
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : Ring R\nn k : ℕ\nih : eval (↑n) (descPochhammer R k) = ↑(n.descFactorial k)\nh : k ≤ n\n⊢ (↑n - ↑k) * ↑(n.descFactorial k) = ↑((n - k) * n.descFactorial k)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCo... | [
"case neg\nR : Type u\ninst✝ : Ring R\nn k : ℕ\nih : eval (↑n) (descPochhammer R k) = ↑(n.descFactorial k)\nh : k ≤ n\n⊢ (↑n - ↑k) * ↑(n.descFactorial k) = ↑(n - k) * ↑(n.descFactorial k)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Basis | {
"line": 117,
"column": 32
} | {
"line": 117,
"column": 41
} | {
"line": 117,
"column": 41
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\n⊢ lieSpan R L (range b.e ∪ range b.f) = ⊤",
"ppTerm": "?m.132",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LieAlgebra.Basis.e",
... | [
"ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\n⊢ ⊤ = ⊤"
] | b.span_ef | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Lagrange | {
"line": 154,
"column": 55
} | {
"line": 154,
"column": 64
} | {
"line": 154,
"column": 65
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\nx y : F\nhxy : C (x - y)⁻¹ = 0 ∨ False\n⊢ x = y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"False",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"AddGroupWithOne.toAddGroup... | [
"F : Type u_1\ninst✝ : Field F\nx y : F\nhxy : C (x - y)⁻¹ = 0\n⊢ x = y"
] | or_false, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Vandermonde | {
"line": 189,
"column": 2
} | {
"line": 195,
"column": 92
} | {
"line": 196,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝ : Field K\nn✝ n : ℕ\nih : ∀ (v w : Fin n → K), (projVandermonde v w).det = ∏ i, ∏ j ∈ Ioi i, (v j * w i - v i * w j)\nv w : Fin (n + 1) → K\nh0 : w 0 ≠ 0\nr : K := v 0 / w 0\nhr : r = v 0 / w 0\nW : Matrix (Fin (n + 1)) (Fin (n + 1)) K :=\n of fun i ↦\n Fin.cons (projVandermonde... | [
"K : Type u_2\ninst✝ : Field K\nn✝ n : ℕ\nih : ∀ (v w : Fin n → K), (projVandermonde v w).det = ∏ i, ∏ j ∈ Ioi i, (v j * w i - v i * w j)\nv w : Fin (n + 1) → K\nh0 : w 0 ≠ 0\nr : K := v 0 / w 0\nhr : r = v 0 / w 0\nW : Matrix (Fin (n + 1)) (Fin (n + 1)) K :=\n of fun i ↦\n Fin.cons (projVandermonde v w i 0) fu... | · rw [succAbove_zero, hW_eq, det_mul_column, ih]
simp only [Nat.succ_eq_add_one, coe_ofNat_eq_mod, Nat.zero_mod,
pow_zero, show W 0 0 = w 0 ^ n by simp [W, projVandermonde_apply], one_mul, hr]
field_simp
simp only [Finset.prod_div_distrib, Finset.prod_const, Finset.card_fin, Function.comp_apply]
f... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Lagrange | {
"line": 321,
"column": 2
} | {
"line": 324,
"column": 55
} | {
"line": 326,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\ni : ι\nv r : ι → F\nhvs : Set.InjOn v ↑s\nhi : i ∈ s\n⊢ eval (v i) ((interpolate s v) r) = r i",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Poly... | [] | rw [interpolate_apply, eval_finsetSum, ← add_sum_erase _ _ hi]
simp_rw [eval_mul, eval_C, eval_basis_self hvs hi, mul_one, add_eq_left]
refine sum_eq_zero fun j H => ?_
rw [eval_basis_of_ne (mem_erase.mp H).1 hi, mul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Lagrange | {
"line": 321,
"column": 2
} | {
"line": 324,
"column": 55
} | {
"line": 326,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\ni : ι\nv r : ι → F\nhvs : Set.InjOn v ↑s\nhi : i ∈ s\n⊢ eval (v i) ((interpolate s v) r) = r i",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Poly... | [] | rw [interpolate_apply, eval_finsetSum, ← add_sum_erase _ _ hi]
simp_rw [eval_mul, eval_C, eval_basis_self hvs hi, mul_one, add_eq_left]
refine sum_eq_zero fun j H => ?_
rw [eval_basis_of_ne (mem_erase.mp H).1 hi, mul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Vandermonde | {
"line": 230,
"column": 4
} | {
"line": 230,
"column": 49
} | {
"line": 232,
"column": 0
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝¹ : CommRing R\nn : ℕ\ninst✝ : IsDomain R\nv : Fin n → R\ni j : Fin n\nh₁ : v i = v j\nh₂ : ¬i = j\nk : Fin n\n⊢ vandermonde v i k = vandermonde v j k",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.... | [] | rw [vandermonde_apply, vandermonde_apply, h₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.EngelSubalgebra | {
"line": 120,
"column": 2
} | {
"line": 121,
"column": 69
} | {
"line": 122,
"column": 2
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsArtinian R L\nH : LieSubalgebra R L\nx : L\nh : engel R x ≤ H\nN : LieSubalgebra R L := H.normalizer\naux₁ : ∀ n ∈ N, ⁅x, n⁆ ∈ H\naux₂ : ∀ n ∈ N, ⁅x, n⁆ ∈ N\ndx : ↥N →ₗ[R] ↥N := LinearMap.restrict ((... | [
"R : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsArtinian R L\nH : LieSubalgebra R L\nx : L\nh : engel R x ≤ H\nN : LieSubalgebra R L := H.normalizer\naux₁ : ∀ n ∈ N, ⁅x, n⁆ ∈ H\naux₂ : ∀ n ∈ N, ⁅x, n⁆ ∈ N\ndx : ↥N →ₗ[R] ↥N := LinearMap.restrict ((ad R L) x) a... | obtain ⟨k, hk⟩ : ∃ a, ∀ b ≥ a, Codisjoint (LinearMap.ker (dx ^ b)) (LinearMap.range (dx ^ b)) :=
eventually_atTop.mp <| dx.eventually_codisjoint_ker_pow_range_pow | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Lie.EngelSubalgebra | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 40
} | {
"line": 171,
"column": 0
} | [
{
"pp": "case h.inr\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsNoetherian R L\nH : LieSubalgebra R L\nx : ↥H\nK : ℕ →o Submodule R ↥H := { toFun := fun n ↦ LinearMap.ker ((ad R ↥H) x ^ n), monotone' := ⋯ }\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → K n = K m\ny... | [] | rwa [this, ← hn m hmn, ← this] at hm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.Lagrange | {
"line": 599,
"column": 4
} | {
"line": 603,
"column": 54
} | {
"line": 605,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ns : Finset ι\nv : ι → R\ninst✝ : DecidableEq ι\ni : ι\nt : Finset ι\nhit : i ∉ t\nIH : derivative (nodal t v) = ∑ i ∈ t, nodal (t.erase i) v\n⊢ derivative (nodal (insert i t) v) = ∑ i_1 ∈ insert i t, nodal ((insert i t).erase i_1) v",
... | [] | rw [nodal_insert_eq_nodal hit, derivative_mul, IH, derivative_sub, derivative_X, derivative_C,
sub_zero, one_mul, sum_insert hit, mul_sum, erase_insert hit, add_right_inj]
refine sum_congr rfl fun j hjt => ?_
rw [t.erase_insert_of_ne (ne_of_mem_of_not_mem hjt hit).symm,
nodal_insert_eq_nodal (mem_of... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Lagrange | {
"line": 599,
"column": 4
} | {
"line": 603,
"column": 54
} | {
"line": 605,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ns : Finset ι\nv : ι → R\ninst✝ : DecidableEq ι\ni : ι\nt : Finset ι\nhit : i ∉ t\nIH : derivative (nodal t v) = ∑ i ∈ t, nodal (t.erase i) v\n⊢ derivative (nodal (insert i t) v) = ∑ i_1 ∈ insert i t, nodal ((insert i t).erase i_1) v",
... | [] | rw [nodal_insert_eq_nodal hit, derivative_mul, IH, derivative_sub, derivative_X, derivative_C,
sub_zero, one_mul, sum_insert hit, mul_sum, erase_insert hit, add_right_inj]
refine sum_congr rfl fun j hjt => ?_
rw [t.erase_insert_of_ne (ne_of_mem_of_not_mem hjt hit).symm,
nodal_insert_eq_nodal (mem_of... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Monad | {
"line": 214,
"column": 52
} | {
"line": 217,
"column": 40
} | {
"line": 219,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring T\nf : R →+* MvPolynomial σ S\ng : S →+* T\nφ : MvPolynomial σ R\n⊢ (map g) ((bind₂ f) φ) = (bind₂ ((map g).comp f)) φ",
"ppTerm": "?m.38",
"assigned": true,
"usedC... | [] | by
simp only [bind₂, eval₂_comp_right, coe_eval₂Hom, eval₂_map]
congr 1 with : 1
simp only [Function.comp_apply, map_X] | [anonymous] | Lean.Parser.Term.byTactic |
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