module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Square | {
"line": 123,
"column": 54
} | {
"line": 123,
"column": 67
} | {
"line": 123,
"column": 67
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nsq : Square C\nh : sq.IsPullback\ninst✝ : Mono sq.f₂₄\n⊢ MorphismProperty.monomorphisms C sq.f₂₄",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"inst✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Square | {
"line": 141,
"column": 52
} | {
"line": 141,
"column": 65
} | {
"line": 141,
"column": 65
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nsq : Square C\nh : sq.IsPushout\ninst✝ : Epi sq.f₁₃\n⊢ MorphismProperty.epimorphisms C sq.f₁₃",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"inst✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.SpectralObject.Page | {
"line": 846,
"column": 2
} | {
"line": 848,
"column": 60
} | {
"line": 849,
"column": 2
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\ni j : ι\nf : i ⟶ j\nhn₂ : n₁ + 1 = n₁ + 1\nh : (X.shortComplex (𝟙 i) f (𝟙 j) n₀ n₁ (n₁ + 1) hn₁ hn₂).HomologyData := X.homologyDataIdId ... | [
"C : Type u_1\nι : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} ι\ninst✝ : Abelian C\nX : SpectralObject C ι\nn₀ n₁ : ℤ\nhn₁ : n₀ + 1 = n₁\ni j : ι\nf : i ⟶ j\nhn₂ : n₁ + 1 = n₁ + 1\nh : (X.shortComplex (𝟙 i) f (𝟙 j) n₀ n₁ (n₁ + 1) hn₁ hn₂).HomologyData := X.homologyDataIdId f n₀ n₁ (n₁ ... | rw [Category.assoc, ← this,
h.left_homologyIso_eq_right_homologyIso_trans_iso_symm,
← ShortComplex.RightHomologyData.homologyIso_hom_comp_ι] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Artinian.Module | {
"line": 504,
"column": 59
} | {
"line": 504,
"column": 90
} | {
"line": 506,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : IsArtinianRing R\nx✝ : R\n⊢ x✝ ∈ IsUnit.submonoid R ↔ IsRightRegular x✝",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"IsArtinianRing.isUnit_iff_isRightRegular",
"Ring.toSemiring"
],
"usedFVars": [
"R",
"inst... | [] | exact isUnit_iff_isRightRegular | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 523,
"column": 2
} | {
"line": 524,
"column": 15
} | {
"line": 526,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nK : LieSubalgebra R L\n⊢ K = ⊥ ↔ ∀ x ∈ K, x = 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LieSubalgebra.instPartialOrder_1",
"LieRing.toAddCommGroup",
... | [] | rw [_root_.eq_bot_iff]
exact Iff.rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Subalgebra | {
"line": 523,
"column": 2
} | {
"line": 524,
"column": 15
} | {
"line": 526,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nK : LieSubalgebra R L\n⊢ K = ⊥ ↔ ∀ x ∈ K, x = 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LieSubalgebra.instPartialOrder_1",
"LieRing.toAddCommGroup",
... | [] | rw [_root_.eq_bot_iff]
exact Iff.rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Ideal | {
"line": 127,
"column": 24
} | {
"line": 127,
"column": 53
} | {
"line": 128,
"column": 2
} | [
{
"pp": "case mp\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nK K' : LieSubalgebra R L\nh : K ≤ K'\nh' : ∀ (x y : ↥K'), ↑y ∈ K → ⁅↑x, ↑y⁆ ∈ K\nx y : L\nhx : x ∈ K'\nhy : y ∈ K\n⊢ ⁅x, y⁆ ∈ K",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Li... | [] | exact h' ⟨x, hx⟩ ⟨y, h hy⟩ hy | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Ideal | {
"line": 424,
"column": 21
} | {
"line": 424,
"column": 64
} | {
"line": 425,
"column": 2
} | [
{
"pp": "case a\nR : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nJ : LieIdeal R L'\nh : f.IsIdealMorphism\n⊢ map f (comap f J) ≤ f.idealRange ∧ map f (comap f J) ≤ J",
"ppTerm": "?a✝",
"... | [] | exact ⟨f.map_le_idealRange _, map_comap_le⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 30
} | {
"line": 109,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : Ring B\ninst✝³ : Algebra A B\nx : B\ninst✝² : Nontrivial B\nR : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nf : A →+* R\n⊢ map f (minpoly A x) ≠ 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Polynomial.i... | [
"case pos\nA : Type u_1\nB : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : Ring B\ninst✝³ : Algebra A B\nx : B\ninst✝² : Nontrivial B\nR : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nf : A →+* R\nhx : IsIntegral A x\n⊢ map f (minpoly A x) ≠ 1",
"case neg\nA : Type u_1\nB : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴... | by_cases hx : IsIntegral A x | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 30
} | {
"line": 117,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nthis : Nontrivial A\n⊢ ¬IsUnit (minpoly A x)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"IsUnit",
"Classical.propD... | [
"case pos\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nontrivial B\nthis : Nontrivial A\nhx : IsIntegral A x\n⊢ ¬IsUnit (minpoly A x)",
"case neg\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Ring B\ninst✝¹ : Algebra A B\nx : B\ninst✝ : Nont... | by_cases hx : IsIntegral A x | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 30
} | {
"line": 172,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\na : A\n⊢ minpoly A (x + (algebraMap A B) a) = (minpoly A x).comp (X - C a)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Algebra.algebraMap",
"CommSe... | [
"case pos\nA : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\na : A\nhx : IsIntegral A x\n⊢ minpoly A (x + (algebraMap A B) a) = (minpoly A x).comp (X - C a)",
"case neg\nA : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nx : B\na : ... | by_cases hx : IsIntegral A x | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 30
} | {
"line": 193,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\n⊢ minpoly A (-x) = (-1) ^ (minpoly A x).natDegree * (minpoly A x).comp (-X)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"Polynomial.instOne",
"Pol... | [
"case pos\nA : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\n⊢ minpoly A (-x) = (-1) ^ (minpoly A x).natDegree * (minpoly A x).comp (-X)",
"case neg\nA : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\nhx : ¬IsInt... | by_cases hx : IsIntegral A x | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 10
} | {
"line": 140,
"column": 0
} | [
{
"pp": "A : Type u_3\ninst✝ : CommRing A\nI : Ideal A\nx y : A\nn : ℕ\nhxy : x ≡ y [SMOD I]\n⊢ x ^ n ≡ y ^ n [SMOD I]",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHom.instRingHomClass",
"Semiring.toModule",
"congrArg",
"Co... | [] | simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_pow] at hxy ⊢
rw [hxy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SModEq.Basic | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 10
} | {
"line": 140,
"column": 0
} | [
{
"pp": "A : Type u_3\ninst✝ : CommRing A\nI : Ideal A\nx y : A\nn : ℕ\nhxy : x ≡ y [SMOD I]\n⊢ x ^ n ≡ y ^ n [SMOD I]",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHom.instRingHomClass",
"Semiring.toModule",
"congrArg",
"Co... | [] | simp only [SModEq.def, Ideal.Quotient.mk_eq_mk, map_pow] at hxy ⊢
rw [hxy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerBasis | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 35
} | {
"line": 263,
"column": 4
} | [
{
"pp": "S : Type u_2\ninst✝⁴ : Ring S\nA : Type u_4\ninst✝³ : CommRing A\ninst✝² : Algebra A S\nS' : Type u_7\ninst✝¹ : Ring S'\ninst✝ : Algebra A S'\npb : PowerBasis A S\ny : S'\nhy : (aeval y) (minpoly A pb.gen) = 0\nf : A[X]\nh✝ : Nontrivial A\nhf : ¬f %ₘ minpoly A pb.gen = 0\n⊢ (f %ₘ minpoly A pb.gen).natD... | [
"S : Type u_2\ninst✝⁴ : Ring S\nA : Type u_4\ninst✝³ : CommRing A\ninst✝² : Algebra A S\nS' : Type u_7\ninst✝¹ : Ring S'\ninst✝ : Algebra A S'\npb : PowerBasis A S\ny : S'\nhy : (aeval y) (minpoly A pb.gen) = 0\nf : A[X]\nh✝ : Nontrivial A\nhf : ¬f %ₘ minpoly A pb.gen = 0\n⊢ (f %ₘ minpoly A pb.gen).degree < (minpol... | apply natDegree_lt_natDegree hf | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 114,
"column": 21
} | {
"line": 114,
"column": 60
} | {
"line": 115,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nf g : (R ⧸ I)[X]\n⊢ (eval₂RingHom (Quotient.lift I ((Quotient.mk (map C I)).comp C) ⋯) ((Quotient.mk (map C I)) X)) (f * g) =\n (eval₂RingHom (Quotient.lift I ((Quotient.mk (map C I)).comp C) ⋯) ((Quotient.mk (map C I)) X)) f *\n (eval₂RingHom (Quo... | [] | simp only [coe_eval₂RingHom, eval₂_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 114,
"column": 21
} | {
"line": 114,
"column": 60
} | {
"line": 115,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nf g : (R ⧸ I)[X]\n⊢ (eval₂RingHom (Quotient.lift I ((Quotient.mk (map C I)).comp C) ⋯) ((Quotient.mk (map C I)) X)) (f * g) =\n (eval₂RingHom (Quotient.lift I ((Quotient.mk (map C I)).comp C) ⋯) ((Quotient.mk (map C I)) X)) f *\n (eval₂RingHom (Quo... | [] | simp only [coe_eval₂RingHom, eval₂_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 114,
"column": 21
} | {
"line": 114,
"column": 60
} | {
"line": 115,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nf g : (R ⧸ I)[X]\n⊢ (eval₂RingHom (Quotient.lift I ((Quotient.mk (map C I)).comp C) ⋯) ((Quotient.mk (map C I)) X)) (f * g) =\n (eval₂RingHom (Quotient.lift I ((Quotient.mk (map C I)).comp C) ⋯) ((Quotient.mk (map C I)) X)) f *\n (eval₂RingHom (Quo... | [] | simp only [coe_eval₂RingHom, eval₂_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerBasis | {
"line": 484,
"column": 74
} | {
"line": 487,
"column": 11
} | {
"line": 489,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nB : PowerBasis R S\nx : S\nhx : B.gen ∈ R[x]\n⊢ R[x] = ⊤",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"L... | [] | by
rw [_root_.eq_top_iff, ← B.adjoin_gen_eq_top]
refine adjoin_le ?_
simp [hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Squarefree.Basic | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 63
} | {
"line": 200,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\ninst✝ : IsCancelMulZero R\nx y p : R\nk : ℕ\nhy : Squarefree y\nhp : Prime p\nh : p ^ (k + 1) ∣ y * x\n⊢ p ^ k ∣ x",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Squarefree.pow_dvd_of_squarefree_of_pow_succ_dvd_mul_right"
],... | [] | exact pow_dvd_of_squarefree_of_pow_succ_dvd_mul_right hy hp h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Separable | {
"line": 242,
"column": 4
} | {
"line": 242,
"column": 14
} | {
"line": 243,
"column": 2
} | [
{
"pp": "case inl\nR : Type u\ninst✝ : CommRing R\nu : Rˣ\na✝ : Nontrivial R\nhn : IsUnit ↑0\n⊢ (X ^ 0 - C ↑u).Separable",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Units.val",
"Polynomial.C",
"MulOne.toOne",
"False",
"NeZero.one",
"congrArg",
... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.Separable | {
"line": 242,
"column": 4
} | {
"line": 242,
"column": 14
} | {
"line": 243,
"column": 2
} | [
{
"pp": "case inl\nR : Type u\ninst✝ : CommRing R\nu : Rˣ\na✝ : Nontrivial R\nhn : IsUnit ↑0\n⊢ (X ^ 0 - C ↑u).Separable",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Units.val",
"Polynomial.C",
"MulOne.toOne",
"False",
"NeZero.one",
"congrArg",
... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Separable | {
"line": 242,
"column": 4
} | {
"line": 242,
"column": 14
} | {
"line": 243,
"column": 2
} | [
{
"pp": "case inl\nR : Type u\ninst✝ : CommRing R\nu : Rˣ\na✝ : Nontrivial R\nhn : IsUnit ↑0\n⊢ (X ^ 0 - C ↑u).Separable",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Units.val",
"Polynomial.C",
"MulOne.toOne",
"False",
"NeZero.one",
"congrArg",
... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Perfect | {
"line": 366,
"column": 2
} | {
"line": 369,
"column": 87
} | {
"line": 370,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np n : ℕ\ninst✝ : ExpChar R p\nf : R[X]\nr : R\nh : ∃ s, r = s ^ p ^ n\n⊢ count r (Multiset.map (⇑(iterateFrobenius R p n)) ((expand R (p ^ n)) f).roots) ≤ count r (p ^ n • f.roots)",
"ppTerm": "?pos✝",
"assigned": true,
"used... | [
"case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np n : ℕ\ninst✝ : ExpChar R p\nf : R[X]\nr : R\nh : ¬∃ s, r = s ^ p ^ n\n⊢ count r (Multiset.map (⇑(iterateFrobenius R p n)) ((expand R (p ^ n)) f).roots) ≤ count r (p ^ n • f.roots)"
] | · obtain ⟨s, rfl⟩ := h
simp_rw [count_nsmul, count_roots, ← rootMultiplicity_expand_pow, ← count_roots, count_map,
count_eq_card_filter_eq]
exact card_le_card (monotone_filter_right _ fun _ h ↦ iterateFrobenius_inj R p n h) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.Separable | {
"line": 406,
"column": 23
} | {
"line": 406,
"column": 34
} | {
"line": 406,
"column": 35
} | [
{
"pp": "F✝ : Type u\ninst✝¹ : Field F✝\np✝ : ℕ\nF : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn₁ : ℕ\ng₁ : F[X]\nhg₁ : g₁.Separable\nhgf₁ : (expand F (p ^ n₁)) g₁ = f\ng₂ : F[X]\nhg₂ : g₂.Separable\nk : ℕ\nhgf₂ : (expand F (p ^ n₁ * p ^ k)) g₂ = (expand F (p ^ n₁... | [
"F✝ : Type u\ninst✝¹ : Field F✝\np✝ : ℕ\nF : Type u\ninst✝ : Field F\np : ℕ\nHF : CharP F p\nf : F[X]\nhf : Irreducible f\nhp : 0 < p\nn₁ : ℕ\ng₁ : F[X]\nhg₁ : g₁.Separable\nhgf₁ : (expand F (p ^ n₁)) g₁ = f\ng₂ : F[X]\nhg₂ : g₂.Separable\nk : ℕ\nhgf₂ : (expand F (p ^ n₁)) ((expand F (p ^ k)) g₂) = (expand F (p ^ n... | expand_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.AdjointAction.Derivation | {
"line": 99,
"column": 45
} | {
"line": 99,
"column": 65
} | {
"line": 99,
"column": 65
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh : LieAlgebra.center R L = ⊥\nD : LieDerivation R L L\nhD : D ∈ LieModule.maxTrivSubmodule R L (LieDerivation R L L)\nx : L\nthis : D x ∈ ⊥\n⊢ D x = 0 x",
"ppTerm": "?m.111",
"assigned": true,
"use... | [
"R : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh : LieAlgebra.center R L = ⊥\nD : LieDerivation R L L\nhD : D ∈ LieModule.maxTrivSubmodule R L (LieDerivation R L L)\nx : L\nthis : D x = 0\n⊢ D x = 0 x"
] | LieSubmodule.mem_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 117,
"column": 8
} | {
"line": 120,
"column": 11
} | {
"line": 121,
"column": 6
} | [
{
"pp": "case calc_3.i_inj\nA : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := ⋯\nh₄ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\nR2N : Finset ℕ := ⋯\nhR2N : R2N = range (2 * N + 1)\nRN : Finset ℕ := ⋯\nhRN : RN = range (N + 1)\... | [] | simp only [mem_sigma, mem_range, Prod.mk.injEq, and_imp]
rintro ⟨x₁, y₁⟩ - h₁ ⟨x₂, y₂⟩ - h₂ h₃ h₄
simp_all
lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 117,
"column": 8
} | {
"line": 120,
"column": 11
} | {
"line": 121,
"column": 6
} | [
{
"pp": "case calc_3.i_inj\nA : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := ⋯\nh₄ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\nR2N : Finset ℕ := ⋯\nhR2N : R2N = range (2 * N + 1)\nRN : Finset ℕ := ⋯\nhRN : RN = range (N + 1)\... | [] | simp only [mem_sigma, mem_range, Prod.mk.injEq, and_imp]
rintro ⟨x₁, y₁⟩ - h₁ ⟨x₂, y₂⟩ - h₂ h₃ h₄
simp_all
lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.BilinearForm.Basic | {
"line": 86,
"column": 43
} | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 54
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nS : Type u_3\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\nB : BilinForm R M\nr : S\nx y : M\n⊢ (B ((algebraMap S R) r • x)) y = r • (B x) y",
"ppT... | [
"R : Type u_1\nM : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nS : Type u_3\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\nB : BilinForm R M\nr : S\nx y : M\n⊢ (algebraMap S R) r * (B x) y = r • (B x) y"
] | smul_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.BilinearForm.Hom | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 77
} | {
"line": 310,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : BilinForm R M\nι : Type u_9\nb : Basis ι R M\nx y : M\n⊢ ((b.repr x).sum fun i xi ↦ (b.repr y).sum fun j yj ↦ xi • yj • (B (b i)) (b j)) = (B x) y",
"ppTerm": "?m.67",
"assigned": true,
"u... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : BilinForm R M\nι : Type u_9\nb : Basis ι R M\nx y : M\n⊢ ((b.repr x).sum fun i xi ↦ (b.repr y).sum fun j yj ↦ xi • yj • (B (b i)) (b j)) =\n (B ((Finsupp.linearCombination R ⇑b) (b.repr x))) ((Finsupp.linearCo... | conv_rhs => rw [← b.linearCombination_repr x, ← b.linearCombination_repr y] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.LinearAlgebra.BilinearForm.Hom | {
"line": 310,
"column": 78
} | {
"line": 310,
"column": 88
} | {
"line": 310,
"column": 89
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : BilinForm R M\nι : Type u_9\nb : Basis ι R M\nx y : M\n⊢ ∑ x_1 ∈ (b.repr x).support, ∑ x_2 ∈ (b.repr y).support, (b.repr x) x_1 • (b.repr y) x_2 • (B (b x_1)) (b x_2) =\n ∑ x_1 ∈ (b.repr x).support... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : BilinForm R M\nι : Type u_9\nb : Basis ι R M\nx y : M\n⊢ ∑ x_1 ∈ (b.repr x).support, ∑ x_2 ∈ (b.repr y).support, (b.repr x) x_1 • (b.repr y) x_2 • (B (b x_1)) (b x_2) =\n ∑ x_1 ∈ (b.repr x).support, ∑ x_2 ∈ (b... | smul_left, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Lie.BaseChange | {
"line": 81,
"column": 17
} | {
"line": 81,
"column": 19
} | {
"line": 81,
"column": 20
} | [
{
"pp": "case refine_3.refine_2.refine_3\nR : Type u_1\nA : Type u_2\nL : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nx z₁ z₂ : A ⊗[R] L\nh₁ : ((bracket' R A L L) z₁) z₁ = 0\nh₂ : ((bracket' R A L L) z₂) z₂ = 0\na₁ : A\nl₁ : L\ny₁ : A ⊗[R... | [
"case refine_3.refine_2.refine_3\nR : Type u_1\nA : Type u_2\nL : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nx z₁ z₂ : A ⊗[R] L\nh₁ : ((bracket' R A L L) z₁) z₁ = 0\nh₂ : ((bracket' R A L L) z₂) z₂ = 0\na₁ : A\nl₁ : L\ny₁ y₂ : A ⊗[R] L\n⊢ ((... | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Lie.BaseChange | {
"line": 83,
"column": 15
} | {
"line": 83,
"column": 17
} | {
"line": 83,
"column": 18
} | [
{
"pp": "case refine_3.refine_3\nR : Type u_1\nA : Type u_2\nL : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nx z₁ z₂ : A ⊗[R] L\nh₁ : ((bracket' R A L L) z₁) z₁ = 0\nh₂ : ((bracket' R A L L) z₂) z₂ = 0\ny₁ : A ⊗[R] L\n⊢ ∀ (y : A ⊗[R] L),\... | [
"case refine_3.refine_3\nR : Type u_1\nA : Type u_2\nL : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nx z₁ z₂ : A ⊗[R] L\nh₁ : ((bracket' R A L L) z₁) z₁ = 0\nh₂ : ((bracket' R A L L) z₂) z₂ = 0\ny₁ y₂ : A ⊗[R] L\n⊢ ((bracket' R A L L) y₁) z₂ ... | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Lie.Solvable | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 37
} | {
"line": 228,
"column": 2
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nL : Type u_3\ninst✝⁴ : CommRing R₁\ninst✝³ : CommRing R₂\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R₁ L\ninst✝ : LieAlgebra R₂ L\nk : ℕ\nih : ∀ (x : L), x ∈ derivedSeriesOfIdeal R₁ L k ⊤ ↔ x ∈ derivedSeriesOfIdeal R₂ L k ⊤\nI : LieIdeal R₂ L := derivedSeriesOfIdeal R₂ L k ⊤... | [
"R₁ : Type u_1\nR₂ : Type u_2\nL : Type u_3\ninst✝⁴ : CommRing R₁\ninst✝³ : CommRing R₂\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R₁ L\ninst✝ : LieAlgebra R₂ L\nk : ℕ\nih : ∀ (x : L), x ∈ derivedSeriesOfIdeal R₁ L k ⊤ ↔ x ∈ derivedSeriesOfIdeal R₂ L k ⊤\nI : LieIdeal R₂ L := derivedSeriesOfIdeal R₂ L k ⊤\nS : Set L ... | simp only [SetLike.mem_coe] at hx ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Solvable | {
"line": 307,
"column": 6
} | {
"line": 307,
"column": 30
} | {
"line": 307,
"column": 30
} | [
{
"pp": "case h\nR : Type u\nL : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nA : Type u_1\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Module.FaithfullyFlat R A\nk : ℕ\nh : derivedSeries A (A ⊗[R] L) k ≤ ⊥\nx : L\nhx : x ∈ derivedSeries R L k\n⊢ x ∈ ⊥",
"ppTerm": "?h",
... | [
"case h\nR : Type u\nL : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nA : Type u_1\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Module.FaithfullyFlat R A\nk : ℕ\nh : LieSubmodule.baseChange A (derivedSeries R L k) ≤ ⊥\nx : L\nhx : x ∈ derivedSeries R L k\n⊢ x ∈ ⊥"
] | derivedSeries_baseChange | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Solvable | {
"line": 309,
"column": 6
} | {
"line": 309,
"column": 26
} | {
"line": 309,
"column": 26
} | [
{
"pp": "case h\nR : Type u\nL : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nA : Type u_1\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Module.FaithfullyFlat R A\nk : ℕ\nx : L\nh : 1 ⊗ₜ[R] x ∈ ⊥\nhx : x ∈ derivedSeries R L k\n⊢ x ∈ ⊥",
"ppTerm": "?h",
"assigned": true... | [
"case h\nR : Type u\nL : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nA : Type u_1\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Module.FaithfullyFlat R A\nk : ℕ\nx : L\nh : 1 ⊗ₜ[R] x = 0\nhx : x ∈ derivedSeries R L k\n⊢ x = 0"
] | LieSubmodule.mem_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Solvable | {
"line": 454,
"column": 8
} | {
"line": 454,
"column": 75
} | {
"line": 454,
"column": 75
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nI : LieIdeal R L\ninst✝ : IsSolvable ↥I\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nk : ℕ\nhk : derivedSeries R (↥I) k = ⊥\n⊢ k ∈ s",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
... | [] | by rwa [derivedSeries_def, LieIdeal.derivedSeries_eq_bot_iff] at hk | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 119,
"column": 23
} | {
"line": 119,
"column": 33
} | {
"line": 119,
"column": 34
} | [
{
"pp": "V : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : B.iIsOrtho v\nhv₂ : ∀ (i : n), ¬B.IsOrtho (v i) (v i)\ns : Finset n\nw : n → K\nhs : ∑ i ∈ s, w i • v i = 0\ni : n\nhi : i ∈ s\nhsum : ∑ j ∈ s, w j * (B (v j)) (v i... | [
"V : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nn : Type w\nB : BilinForm K V\nv : n → V\nhv₁ : B.iIsOrtho v\nhv₂ : ∀ (i : n), ¬B.IsOrtho (v i) (v i)\ns : Finset n\nw : n → K\nhs : ∑ i ∈ s, w i • v i = 0\ni : n\nhi : i ∈ s\nhsum : ∑ j ∈ s, w j * (B (v j)) (v i) = w i * (B... | smul_left, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 222,
"column": 9
} | {
"line": 222,
"column": 61
} | {
"line": 223,
"column": 2
} | [
{
"pp": "V : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nB : BilinForm K V\nW : Subspace K V\nx : V\n⊢ x ∈ (domRestrict B W).range.dualCoannihilator ↔ x ∈ B.orthogonal W",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submod... | [
"case mp\nV : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nB : BilinForm K V\nW : Subspace K V\nx : V\nhx : x ∈ (domRestrict B W).range.dualCoannihilator\n⊢ ∀ n ∈ W, B.IsOrtho n x",
"case mpr\nV : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : ... | constructor <;> rw [mem_orthogonal_iff] <;> intro hx | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 50
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh₁ : ∀ (I : LieIdeal R L), IsSolvable ↥I → I = ⊥\nI : LieIdeal R L\nh₂ : IsLieAbelian ↥I\n⊢ I = ⊥",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
... | [] | exact h₁ _ <| LieAlgebra.ofAbelianIsSolvable I | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 50
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh₁ : ∀ (I : LieIdeal R L), IsSolvable ↥I → I = ⊥\nI : LieIdeal R L\nh₂ : IsLieAbelian ↥I\n⊢ I = ⊥",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
... | [] | exact h₁ _ <| LieAlgebra.ofAbelianIsSolvable I | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 50
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case mp\nR : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh₁ : ∀ (I : LieIdeal R L), IsSolvable ↥I → I = ⊥\nI : LieIdeal R L\nh₂ : IsLieAbelian ↥I\n⊢ I = ⊥",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
... | [] | exact h₁ _ <| LieAlgebra.ofAbelianIsSolvable I | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 208,
"column": 31
} | {
"line": 208,
"column": 79
} | {
"line": 209,
"column": 2
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nΦ : LinearMap.BilinForm K L\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nhΦ_refl : Φ.IsRefl\nhL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥I\nI : ... | [] | simp only [this, map_zero, LinearMap.zero_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 208,
"column": 31
} | {
"line": 208,
"column": 79
} | {
"line": 209,
"column": 2
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nΦ : LinearMap.BilinForm K L\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nhΦ_refl : Φ.IsRefl\nhL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥I\nI : ... | [] | simp only [this, map_zero, LinearMap.zero_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 208,
"column": 31
} | {
"line": 208,
"column": 79
} | {
"line": 209,
"column": 2
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nΦ : LinearMap.BilinForm K L\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nhΦ_refl : Φ.IsRefl\nhL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥I\nI : ... | [] | simp only [this, map_zero, LinearMap.zero_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Quotient | {
"line": 118,
"column": 19
} | {
"line": 118,
"column": 21
} | {
"line": 118,
"column": 22
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝¹ : LieAlgebra R L\ninst✝ : LieModule R L M\nI J : LieIdeal R L\nx y : L ⧸ I\nx₁ x₂ y₁ : L\n⊢ ∀ (b₂ : L), (↑I).quotientR... | [
"R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝¹ : LieAlgebra R L\ninst✝ : LieModule R L M\nI J : LieIdeal R L\nx y : L ⧸ I\nx₁ x₂ y₁ y₂ : L\n⊢ (↑I).quotientRel x₁ y₁ → (↑I).quoti... | y₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 36
} | {
"line": 325,
"column": 0
} | [
{
"pp": "case mpr\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\n⊢ (∀ (I : LieIdeal R L), IsSolvable ↥I → IsLieAbelian ↥I) → IsLieAbelian ↥(radical R L)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"LieAlgebr... | [] | intro h; apply h; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 36
} | {
"line": 325,
"column": 0
} | [
{
"pp": "case mpr\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\n⊢ (∀ (I : LieIdeal R L), IsSolvable ↥I → IsLieAbelian ↥I) → IsLieAbelian ↥(radical R L)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"LieAlgebr... | [] | intro h; apply h; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanSubalgebra | {
"line": 68,
"column": 17
} | {
"line": 68,
"column": 26
} | {
"line": 70,
"column": 0
} | [
{
"pp": "case succ\nR : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝ : H.IsCartanSubalgebra\nk : ℕ\nih : LieSubmodule.ucs k H.toLieSubmodule = H.toLieSubmodule\n⊢ LieSubmodule.ucs (k + 1) H.toLieSubmodule = H.toLieSubmodule",
"ppTerm": "?... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.CartanSubalgebra | {
"line": 68,
"column": 17
} | {
"line": 68,
"column": 26
} | {
"line": 70,
"column": 0
} | [
{
"pp": "case succ\nR : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝ : H.IsCartanSubalgebra\nk : ℕ\nih : LieSubmodule.ucs k H.toLieSubmodule = H.toLieSubmodule\n⊢ LieSubmodule.ucs (k + 1) H.toLieSubmodule = H.toLieSubmodule",
"ppTerm": "?... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.CartanSubalgebra | {
"line": 68,
"column": 17
} | {
"line": 68,
"column": 26
} | {
"line": 70,
"column": 0
} | [
{
"pp": "case succ\nR : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝ : H.IsCartanSubalgebra\nk : ℕ\nih : LieSubmodule.ucs k H.toLieSubmodule = H.toLieSubmodule\n⊢ LieSubmodule.ucs (k + 1) H.toLieSubmodule = H.toLieSubmodule",
"ppTerm": "?... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 37
} | {
"line": 103,
"column": 2
} | [
{
"pp": "R₁ : Type u_1\nR₂ : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁹ : CommRing R₁\ninst✝⁸ : CommRing R₂\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R₁ L\ninst✝⁴ : LieAlgebra R₂ L\ninst✝³ : Module R₁ M\ninst✝² : Module R₂ M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R₁ L M\nk : ℕ\... | [
"R₁ : Type u_1\nR₂ : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁹ : CommRing R₁\ninst✝⁸ : CommRing R₂\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R₁ L\ninst✝⁴ : LieAlgebra R₂ L\ninst✝³ : Module R₁ M\ninst✝² : Module R₂ M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R₁ L M\nk : ℕ\nI : LieSubm... | simp only [SetLike.mem_coe] at hx ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 159,
"column": 2
} | {
"line": 160,
"column": 9
} | {
"line": 161,
"column": 2
} | [
{
"pp": "case refine_1\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\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : LieModule R L M\nh : lowerCentralSeries R L (↥N) k = ⊥\n⊢ lcs k N = ⊥",
... | [
"case refine_2\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\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : LieModule R L M\nh : lcs k N = ⊥\n⊢ lowerCentralSeries R L (↥N) k = ⊥"
] | · rw [← N.lowerCentralSeries_map_eq_lcs, ← LieModuleHom.le_ker_iff_map]
simpa | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 49
} | {
"line": 398,
"column": 4
} | [
{
"pp": "L : Type v\nM : Type w\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : LieRingModule L M\ninst✝ : IsNilpotent L M\ns : Set ℕ := {k | lowerCentralSeries ℤ L M k = ⊥}\n⊢ s.Nonempty",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"instLieModuleInt",
"LieSubmodule.i... | [
"L : Type v\nM : Type w\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : LieRingModule L M\ninst✝ : IsNilpotent L M\ns : Set ℕ := {k | lowerCentralSeries ℤ L M k = ⊥}\nk : ℕ\nhk : lowerCentralSeries ℤ L M k = ⊥\n⊢ s.Nonempty"
] | obtain ⟨k, hk⟩ := IsNilpotent.nilpotent ℤ L M | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.FieldTheory.Normal.Defs | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 11
} | {
"line": 97,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝⁵ : Field F\nE : Type u_3\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nM : Type u_5\nN : Type u_6\ninst✝² : Field N\ninst✝¹ : Field M\ninst✝ : Algebra M N\nh : Normal F E\nf : F ≃+* M\ng : E ≃+* N\nhcomp : (algebraMap M N).comp ↑f = (↑g).comp (algebraMap F E)\n⊢ Normal M N",
"ppTerm"... | [
"F : Type u_1\ninst✝⁵ : Field F\nE : Type u_3\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nM : Type u_5\nN : Type u_6\ninst✝² : Field N\ninst✝¹ : Field M\ninst✝ : Algebra M N\nh : Normal F E\nf : F ≃+* M\ng : E ≃+* N\nhcomp : (algebraMap M N).comp ↑f = (↑g).comp (algebraMap F E)\nthis : Normal F E\n⊢ Normal M N"
] | have := h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Adjoin.Singleton | {
"line": 46,
"column": 74
} | {
"line": 49,
"column": 56
} | {
"line": 51,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝⁶ : CommSemiring A\ninst✝⁵ : CommSemiring B\ninst✝⁴ : CommSemiring C\ninst✝³ : Algebra A B\ninst✝² : Algebra B C\ninst✝¹ : Algebra A C\ninst✝ : IsScalarTower A B C\nb : B\n⊢ Function.Surjective ⇑(adjoinAlgebraMap b)",
"ppTerm": "?m.21",
"assigned":... | [] | by
intro c
obtain ⟨p, hp⟩ := adjoin_eq_exists_aeval A (algebraMap B C b) c
aesop (add safe ((aeval_algebraMap_apply C b p).symm)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 920,
"column": 2
} | {
"line": 923,
"column": 54
} | {
"line": 925,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\nI : LieIdeal R L\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nk : ℕ\ninst✝ : LieModule.IsNilpotent L ↥I\n⊢ LieRing.IsNilpotent ↥I",
"ppTerm": "?m.14",
"assig... | [] | let f : I →ₗ⁅R⁆ L := I.incl
let g : I →ₗ⁅R⁆ I := LieHom.id
have hfg : ∀ x m, ⁅f x, g m⁆ = g ⁅x, m⁆ := by aesop
exact Function.injective_id.lieModuleIsNilpotent hfg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 920,
"column": 2
} | {
"line": 923,
"column": 54
} | {
"line": 925,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\nI : LieIdeal R L\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nk : ℕ\ninst✝ : LieModule.IsNilpotent L ↥I\n⊢ LieRing.IsNilpotent ↥I",
"ppTerm": "?m.14",
"assig... | [] | let f : I →ₗ⁅R⁆ L := I.incl
let g : I →ₗ⁅R⁆ I := LieHom.id
have hfg : ∀ x m, ⁅f x, g m⁆ = g ⁅x, m⁆ := by aesop
exact Function.injective_id.lieModuleIsNilpotent hfg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 33
} | {
"line": 114,
"column": 2
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < p.degree\nk : 𝕜\nhprod : C k - p = C (C k - p).leadingCoeff * (Multiset.map (fun x ↦ X - C x) (C k - p).roots).prod\nh_ne : C k - p ≠ 0\nlead_ne : (C k - p).leadingC... | [
"𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < p.degree\nk : 𝕜\nhprod : C k - p = C (C k - p).leadingCoeff * (Multiset.map (fun x ↦ X - C x) (C k - p).roots).prod\nh_ne : C k - p ≠ 0\nlead_ne : (C k - p).leadingCoeff ≠ 0\nle... | rcases hk with ⟨r, r_mem, r_ev⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 410,
"column": 10
} | {
"line": 415,
"column": 18
} | {
"line": 416,
"column": 8
} | [
{
"pp": "case right.refine_1\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\nα : E\nK : Type u\ninst✝¹ : Field K\ninst✝ : Algebra F K\nh : IsIntegral F α\nf : AdjoinRoot (minpoly F α) →+* ↥F⟮α⟯ := AdjoinRoot.lift (↑(Algebra.ofId F ↥F⟮α⟯)) (AdjoinSimple.gen F α) ⋯\nthis : F... | [] | obtain ⟨y, hy⟩ := hx
refine ⟨y, ?_⟩
rw [RingHom.comp_apply]
dsimp only [coe_type_toSubfield]
rw [AdjoinRoot.lift_of (aeval_gen_minpoly F α)]
exact hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 410,
"column": 10
} | {
"line": 415,
"column": 18
} | {
"line": 416,
"column": 8
} | [
{
"pp": "case right.refine_1\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\nα : E\nK : Type u\ninst✝¹ : Field K\ninst✝ : Algebra F K\nh : IsIntegral F α\nf : AdjoinRoot (minpoly F α) →+* ↥F⟮α⟯ := AdjoinRoot.lift (↑(Algebra.ofId F ↥F⟮α⟯)) (AdjoinSimple.gen F α) ⋯\nthis : F... | [] | obtain ⟨y, hy⟩ := hx
refine ⟨y, ?_⟩
rw [RingHom.comp_apply]
dsimp only [coe_type_toSubfield]
rw [AdjoinRoot.lift_of (aeval_gen_minpoly F α)]
exact hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Eigenspace.Triangularizable | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 64
} | {
"line": 151,
"column": 2
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\nf : End K V\ninst✝ : FiniteDimensional K V\nh : ∀ x ∈ p, f x ∈ p\nk : ℕ∞\nm : V\nhm : m ∈ p ⊓ ⨆ μ, (f.genEigenspace μ) k\n⊢ m ∈ ⨆ μ, p ⊓ (f.genEigenspace μ) k",
"ppTerm": "?m.56",
"ass... | [
"K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\nf : End K V\ninst✝ : FiniteDimensional K V\nh : ∀ x ∈ p, f x ∈ p\nk : ℕ∞\nm : V\nhm₀ : m ∈ p\nhm₁ : m ∈ ⨆ μ, (f.genEigenspace μ) k\n⊢ m ∈ ⨆ μ, p ⊓ (f.genEigenspace μ) k"
] | obtain ⟨hm₀ : m ∈ p, hm₁ : m ∈ ⨆ μ, f.genEigenspace μ k⟩ := hm | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.Eigenspace.Triangularizable | {
"line": 158,
"column": 11
} | {
"line": 158,
"column": 39
} | {
"line": 158,
"column": 39
} | [
{
"pp": "case pos\nK : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\nf : End K V\ninst✝ : FiniteDimensional K V\nh : ∀ x ∈ p, f x ∈ p\nk : ℕ∞\nm : K →₀ V\nhm₂ : ∀ (i : K), m i ∈ (f.genEigenspace i) k\nhm₀ : (m.sum fun _i xi ↦ xi) ∈ p\nhm₁ : (m.sum fun... | [
"case pos\nK : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\nf : End K V\ninst✝ : FiniteDimensional K V\nh : ∀ x ∈ p, f x ∈ p\nk : ℕ∞\nm : K →₀ V\nhm₂ : ∀ (i : K), m i ∈ (f.genEigenspace i) k\nhm₀ : (m.sum fun _i xi ↦ xi) ∈ p\nhm₁ : (m.sum fun _i xi ↦ xi)... | Module.End.mem_genEigenspace | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Eigenspace.Triangularizable | {
"line": 238,
"column": 70
} | {
"line": 243,
"column": 17
} | {
"line": 244,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\nf : End K V\ninst✝ : FiniteDimensional K V\nk : ℕ∞\nh : ∀ x ∈ p, f x ∈ p\nh' : ⨆ μ, (f.genEigenspace μ) k = ⊤\n⊢ ⨆ μ, (genEigenspace (LinearMap.restrict f h) μ) k = ⊤",
"ppTerm": "?m.61",
... | [] | by
have := congr_arg (Submodule.comap p.subtype) (Submodule.eq_iSup_inf_genEigenspace k h h')
have h_inj : Function.Injective p.subtype := Subtype.coe_injective
simp_rw [Submodule.inf_genEigenspace f p h, Submodule.comap_subtype_self,
← Submodule.map_iSup, Submodule.comap_map_eq_of_injective h_inj] at this
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 693,
"column": 4
} | {
"line": 693,
"column": 52
} | {
"line": 693,
"column": 52
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : L\nhx : IsAlgebraic K x\nh_mp : minpoly K x = minpoly K y\nhy : IsAlgebraic K y\n⊢ (adjoinRootEquivAdjoin K ⋯) (AdjoinRoot.root (minpoly K y)) = AdjoinSimple.gen K y",
"ppTerm": "?m.138",
"assigned": true... | [
"K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : L\nhx : IsAlgebraic K x\nh_mp : minpoly K x = minpoly K y\nhy : IsAlgebraic K y\n⊢ AdjoinSimple.gen K y = AdjoinSimple.gen K y"
] | adjoinRootEquivAdjoin_apply_root K hy.isIntegral | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 762,
"column": 2
} | {
"line": 762,
"column": 37
} | {
"line": 763,
"column": 2
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : L\nhx : IsIntegral K x\nhy : IsIntegral K y\n⊢ FiniteDimensional K ↥K⟮x, y⟯",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"instSMulOfMul",
"IntermediateField",
"Intermedi... | [
"K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : L\nhx : IsIntegral K x\nhy : IsIntegral K y\nthis : FiniteDimensional K ↥K⟮x⟯\n⊢ FiniteDimensional K ↥K⟮x, y⟯"
] | have := adjoin.finiteDimensional hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 196,
"column": 10
} | {
"line": 196,
"column": 30
} | {
"line": 196,
"column": 30
} | [
{
"pp": "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 = ⊥\nf : Module.End R ↥H := (toEnd R ↥H ↥H) y\ng : Module.End R L := (toEn... | [
"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 = ⊥\nf : Module.End R ↥H := (toEnd R ↥H ↥H) y\ng : Module.End R L := (toEnd R (↥H) L) ... | LieSubmodule.mem_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 232,
"column": 39
} | {
"line": 233,
"column": 63
} | {
"line": 235,
"column": 0
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\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✝ : LieRing.IsNilpotent L\nχ : Weight R L M\n⊢ ∃ x ∈ genWeightSpace M ⇑χ, x ≠ 0",
"ppTe... | [] | by
simpa [LieSubmodule.eq_bot_iff] using χ.genWeightSpace_ne_bot | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure | {
"line": 183,
"column": 10
} | {
"line": 183,
"column": 22
} | {
"line": 184,
"column": 10
} | [
{
"pp": "case mul_X\nk : Type u\ninst✝ : Field k\np : MvPolynomial (Vars k) k\nfi : Vars k\nih :\n ∀ (z : AlgebraicClosure k), (Ideal.Quotient.mk (maxIdeal k)) p = z → IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) p)\nz : AlgebraicClosure k\nhp : (Ideal.Quotient.mk (maxIdeal k)) (p * MvPolynomial.X fi) = z\n⊢... | [
"case mul_X\nk : Type u\ninst✝ : Field k\np : MvPolynomial (Vars k) k\nfi : Vars k\nih :\n ∀ (z : AlgebraicClosure k), (Ideal.Quotient.mk (maxIdeal k)) p = z → IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) p)\nz : AlgebraicClosure k\nhp : (Ideal.Quotient.mk (maxIdeal k)) (p * MvPolynomial.X fi) = z\n⊢ IsIntegral ... | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 585,
"column": 2
} | {
"line": 588,
"column": 55
} | {
"line": 590,
"column": 0
} | [
{
"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\n⊢ IsC... | [] | simpa only [isCompl_iff, codisjoint_iff, disjoint_iff, ← LieSubmodule.toSubmodule_inj,
LieSubmodule.sup_toSubmodule, LieSubmodule.inf_toSubmodule,
LieSubmodule.top_toSubmodule, LieSubmodule.bot_toSubmodule, coe_genWeightSpaceOf_zero] using!
(toEnd R L M x).isCompl_iSup_ker_pow_iInf_range_pow | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 585,
"column": 2
} | {
"line": 588,
"column": 55
} | {
"line": 590,
"column": 0
} | [
{
"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\n⊢ IsC... | [] | simpa only [isCompl_iff, codisjoint_iff, disjoint_iff, ← LieSubmodule.toSubmodule_inj,
LieSubmodule.sup_toSubmodule, LieSubmodule.inf_toSubmodule,
LieSubmodule.top_toSubmodule, LieSubmodule.bot_toSubmodule, coe_genWeightSpaceOf_zero] using!
(toEnd R L M x).isCompl_iSup_ker_pow_iInf_range_pow | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 585,
"column": 2
} | {
"line": 588,
"column": 55
} | {
"line": 590,
"column": 0
} | [
{
"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\n⊢ IsC... | [] | simpa only [isCompl_iff, codisjoint_iff, disjoint_iff, ← LieSubmodule.toSubmodule_inj,
LieSubmodule.sup_toSubmodule, LieSubmodule.inf_toSubmodule,
LieSubmodule.top_toSubmodule, LieSubmodule.bot_toSubmodule, coe_genWeightSpaceOf_zero] using!
(toEnd R L M x).isCompl_iSup_ker_pow_iInf_range_pow | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Derivation.Killing | {
"line": 85,
"column": 69
} | {
"line": 85,
"column": 89
} | {
"line": 85,
"column": 89
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁴ : Field R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : Module.Finite R L\ninst✝ : LieAlgebra.IsKilling R L\nx : L\nh : x ∈ ⊥\n⊢ x = 0",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.instS... | [
"R : Type u_1\nL : Type u_2\ninst✝⁴ : Field R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : Module.Finite R L\ninst✝ : LieAlgebra.IsKilling R L\nx : L\nh : x = 0\n⊢ x = 0"
] | LieSubmodule.mem_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Sl2 | {
"line": 275,
"column": 18
} | {
"line": 275,
"column": 43
} | {
"line": 275,
"column": 44
} | [
{
"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... | [] | rwa [lie_e_pow_toEnd_e n] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Lie.Sl2 | {
"line": 275,
"column": 18
} | {
"line": 275,
"column": 43
} | {
"line": 275,
"column": 44
} | [
{
"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... | [] | rwa [lie_e_pow_toEnd_e n] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Sl2 | {
"line": 275,
"column": 18
} | {
"line": 275,
"column": 43
} | {
"line": 275,
"column": 44
} | [
{
"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... | [] | rwa [lie_e_pow_toEnd_e n] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 409,
"column": 13
} | {
"line": 409,
"column": 33
} | {
"line": 409,
"column": 33
} | [
{
"pp": "case h\nL : Type u_2\ninst✝¹⁰ : LieRing L\nM : Type u_3\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : LieRingModule L M\nK : Type u_4\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieAlgebra K L\nH : LieSubalgebra K L\ninst✝⁴ : LieRing.IsNilpotent ↥H\ninst✝³ : Module K M\ninst✝² : LieModule K L M\ninst✝¹ : IsT... | [
"case h\nL : Type u_2\ninst✝¹⁰ : LieRing L\nM : Type u_3\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : LieRingModule L M\nK : Type u_4\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieAlgebra K L\nH : LieSubalgebra K L\ninst✝⁴ : LieRing.IsNilpotent ↥H\ninst✝³ : Module K M\ninst✝² : LieModule K L M\ninst✝¹ : IsTriangulariza... | LieSubmodule.mem_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.TraceForm | {
"line": 181,
"column": 2
} | {
"line": 183,
"column": 18
} | {
"line": 186,
"column": 0
} | [
{
"pp": "R : 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\nx y : L\nhx : x ∈ lowerCentralSeries R L L 1\nhy : y ∈ LieAlgebra.center R L\n⊢ ((traceForm R L ... | [] | apply traceForm_eq_zero_if_mem_lcs_of_mem_ucs R L M 1
· simpa using hx
· simpa using hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.TraceForm | {
"line": 181,
"column": 2
} | {
"line": 183,
"column": 18
} | {
"line": 186,
"column": 0
} | [
{
"pp": "R : 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\nx y : L\nhx : x ∈ lowerCentralSeries R L L 1\nhy : y ∈ LieAlgebra.center R L\n⊢ ((traceForm R L ... | [] | apply traceForm_eq_zero_if_mem_lcs_of_mem_ucs R L M 1
· simpa using hx
· simpa using hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Defs | {
"line": 665,
"column": 4
} | {
"line": 665,
"column": 38
} | {
"line": 666,
"column": 4
} | [
{
"pp": "ι : 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\ni j : ι\nι₂ : Type u_5\nM₂ : Type u_6\nN₂ : Type u_7\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ ... | [
"ι : 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\ni j : ι\nι₂ : Type u_5\nM₂ : Type u_6\nN₂ : Type u_7\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommGro... | have : IsReflexive R N₂ := equiv g | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 547,
"column": 4
} | {
"line": 548,
"column": 80
} | {
"line": 549,
"column": 4
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα β : Weight K (↥H) L\nhyp : coroot α = coro... | [
"K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : CharZero K\nα β : Weight K (↥H) L\nhyp : coroot α = coroot β\nhα : ¬... | have : α.ker = β.ker := by
rw [← orthogonal_span_coroot_eq_ker α, hyp, orthogonal_span_coroot_eq_ker] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 168,
"column": 55
} | {
"line": 168,
"column": 81
} | {
"line": 168,
"column": 82
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module L M\ninst✝⁴ : Module L N\ninst✝³ : Module K M\ninst✝² : Module K N\ninst✝¹ : IsScalarTower K L M\np : M →ₗ[L] N →ₗ[L] L\ni... | [
"K : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module L M\ninst✝⁴ : Module L N\ninst✝³ : Module K M\ninst✝² : Module K N\ninst✝¹ : IsScalarTower K L M\np : M →ₗ[L] N →ₗ[L] L\ninst✝ : p.IsP... | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 188,
"column": 56
} | {
"line": 188,
"column": 82
} | {
"line": 188,
"column": 83
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module L M\ninst✝⁴ : Module L N\ninst✝³ : Module K M\ninst✝² : Module K N\ninst✝¹ : IsScalarTower K L M\np : M →ₗ[L] N →ₗ[L] L\ni... | [
"K : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module L M\ninst✝⁴ : Module L N\ninst✝³ : Module K M\ninst✝² : Module K N\ninst✝¹ : IsScalarTower K L M\np : M →ₗ[L] N →ₗ[L] L\ninst✝ : p.IsP... | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 219,
"column": 16
} | {
"line": 219,
"column": 25
} | {
"line": 221,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℤ\nih : (T R n).degree = ↑n.natAbs\n⊢ (T R (-n)).degree = ↑(-n).natAbs",
"ppTerm": "?neg",
"assigned": true,
"usedConstants": [
"WithBot",
"Polynomial.Chebyshev.T",
"congrArg",
"C... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 219,
"column": 16
} | {
"line": 219,
"column": 25
} | {
"line": 221,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℤ\nih : (T R n).degree = ↑n.natAbs\n⊢ (T R (-n)).degree = ↑(-n).natAbs",
"ppTerm": "?neg",
"assigned": true,
"usedConstants": [
"WithBot",
"Polynomial.Chebyshev.T",
"congrArg",
"C... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 219,
"column": 16
} | {
"line": 219,
"column": 25
} | {
"line": 221,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℤ\nih : (T R n).degree = ↑n.natAbs\n⊢ (T R (-n)).degree = ↑(-n).natAbs",
"ppTerm": "?neg",
"assigned": true,
"usedConstants": [
"WithBot",
"Polynomial.Chebyshev.T",
"congrArg",
"C... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.NumDen | {
"line": 135,
"column": 2
} | {
"line": 136,
"column": 54
} | {
"line": 137,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nhx : x ≠ 0\n⊢ Associated (num A x) ↑(den A x⁻¹)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
... | [
"A : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nhx : x ≠ 0\nthis : Associated (num A x⁻¹⁻¹) ↑(den A x⁻¹)\n⊢ Associated (num A x) ↑(den A x⁻¹)"
] | have : Associated (num A x⁻¹⁻¹ : A) (den A x⁻¹) :=
(associated_den_num_inv x⁻¹ (inv_ne_zero hx)).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 241,
"column": 16
} | {
"line": 241,
"column": 25
} | {
"line": 243,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℤ\nih : (T R n).leadingCoeff = 2 ^ (n.natAbs - 1)\n⊢ (T R (-n)).leadingCoeff = 2 ^ ((-n).natAbs - 1)",
"ppTerm": "?neg",
"assigned": true,
"usedConstants": [
"Polynomial.Chebyshev.T",
"congrA... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 241,
"column": 16
} | {
"line": 241,
"column": 25
} | {
"line": 243,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℤ\nih : (T R n).leadingCoeff = 2 ^ (n.natAbs - 1)\n⊢ (T R (-n)).leadingCoeff = 2 ^ ((-n).natAbs - 1)",
"ppTerm": "?neg",
"assigned": true,
"usedConstants": [
"Polynomial.Chebyshev.T",
"congrA... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 241,
"column": 16
} | {
"line": 241,
"column": 25
} | {
"line": 243,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℤ\nih : (T R n).leadingCoeff = 2 ^ (n.natAbs - 1)\n⊢ (T R (-n)).leadingCoeff = 2 ^ ((-n).natAbs - 1)",
"ppTerm": "?neg",
"assigned": true,
"usedConstants": [
"Polynomial.Chebyshev.T",
"congrA... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 254,
"column": 16
} | {
"line": 254,
"column": 25
} | {
"line": 256,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nx : R\nn : ℤ\nih : eval (-x) (T R n) = ↑↑n.negOnePow * eval x (T R n)\n⊢ eval (-x) (T R (-n)) = ↑↑(-n).negOnePow * eval x (T R (-n))",
"ppTerm": "?neg",
"assigned": true,
"usedConstants": [
"Int.cast",
"Units.val",
"Polynomial.ev... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 254,
"column": 16
} | {
"line": 254,
"column": 25
} | {
"line": 256,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nx : R\nn : ℤ\nih : eval (-x) (T R n) = ↑↑n.negOnePow * eval x (T R n)\n⊢ eval (-x) (T R (-n)) = ↑↑(-n).negOnePow * eval x (T R (-n))",
"ppTerm": "?neg",
"assigned": true,
"usedConstants": [
"Int.cast",
"Units.val",
"Polynomial.ev... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 254,
"column": 16
} | {
"line": 254,
"column": 25
} | {
"line": 256,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nx : R\nn : ℤ\nih : eval (-x) (T R n) = ↑↑n.negOnePow * eval x (T R n)\n⊢ eval (-x) (T R (-n)) = ↑↑(-n).negOnePow * eval x (T R (-n))",
"ppTerm": "?neg",
"assigned": true,
"usedConstants": [
"Int.cast",
"Units.val",
"Polynomial.ev... | [] | simp [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Quotient.PowTransition | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 54
} | {
"line": 61,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : ℕ → Submodule R M\nhp : Antitone p\nx : (n : ℕ) → M ⧸ p n\nh : ∀ (m : ℕ), x m = (factor ⋯) (x (m + 1))\nm n : ℕ\ng : m ≤ n\n⊢ x m = (factor ⋯) (x n)",
"ppTerm": "?m.46",
"assigned": true,
"usedCons... | [
"R : Type u_1\ninst✝² : Ring R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : ℕ → Submodule R M\nhp : Antitone p\nx : (n : ℕ) → M ⧸ p n\nh : ∀ (m : ℕ), x m = (factor ⋯) (x (m + 1))\nm n : ℕ\ng : m ≤ n\nthis : n = m + (n - m)\n⊢ x m = (factor ⋯) (x n)"
] | have : n = m + (n - m) := (Nat.add_sub_of_le g).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 953,
"column": 2
} | {
"line": 953,
"column": 9
} | {
"line": 953,
"column": 10
} | [
{
"pp": "case a\nR : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nh :\n (⇑derivative)^[k + 2] (T R n) -\n ((⇑derivative)^[k + 2] (T R n) * X ^ 2 + (2 * k) • (⇑derivative)^[k + 1] (T R n) * X +\n (k * (k - 1)) • (⇑derivative)^[k] (T R n)) =\n (⇑derivative)^[k + 1] (T R n) * X + k • (⇑derivative)^[... | [
"case a.zero\nR : Type u_1\ninst✝ : CommRing R\nn : ℤ\nh :\n (⇑derivative)^[0 + 2] (T R n) -\n ((⇑derivative)^[0 + 2] (T R n) * X ^ 2 + (2 * 0) • (⇑derivative)^[0 + 1] (T R n) * X +\n (0 * (0 - 1)) • (⇑derivative)^[0] (T R n)) =\n (⇑derivative)^[0 + 1] (T R n) * X + 0 • (⇑derivative)^[0] (T R n) - ↑... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 967,
"column": 2
} | {
"line": 967,
"column": 9
} | {
"line": 967,
"column": 10
} | [
{
"pp": "case a\nR : Type u_1\ninst✝ : CommRing R\nn : ℤ\nk : ℕ\nh :\n (⇑derivative)^[k + 2] (U R n) -\n ((⇑derivative)^[k + 2] (U R n) * X ^ 2 + (2 * k) • (⇑derivative)^[k + 1] (U R n) * X +\n (k * (k - 1)) • (⇑derivative)^[k] (U R n)) =\n ↑3 * ((⇑derivative)^[k + 1] (U R n) * X + k • (⇑derivat... | [
"case a.zero\nR : Type u_1\ninst✝ : CommRing R\nn : ℤ\nh :\n (⇑derivative)^[0 + 2] (U R n) -\n ((⇑derivative)^[0 + 2] (U R n) * X ^ 2 + (2 * 0) • (⇑derivative)^[0 + 1] (U R n) * X +\n (0 * (0 - 1)) • (⇑derivative)^[0] (U R n)) =\n ↑3 * ((⇑derivative)^[0 + 1] (U R n) * X + 0 • (⇑derivative)^[0] (U R ... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 445,
"column": 4
} | {
"line": 445,
"column": 32
} | {
"line": 447,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nr : R\nf : ℕ → M\nhf : f ∈ {f | IsAdicCauchy I M f}\nm n : ℕ\nhmn : m ≤ n\n⊢ (r • f) m ≡ (r • f) n [SMOD I ^... | [] | exact SModEq.smul (hf hmn) r | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 1085,
"column": 77
} | {
"line": 1098,
"column": 94
} | {
"line": 1100,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm k : ℤ\n⊢ 2 * T R m * T R k = T R (m + k) + T R (m - k)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | [] | by
induction k using Polynomial.Chebyshev.induct with
| zero => simp [two_mul]
| one => rw [T_add_one, T_one]; ring
| add_two k ih1 ih2 =>
have h₁ := T_add_two R (m + k)
have h₂ := T_sub_two R (m - k)
have h₃ := T_add_two R k
linear_combination (norm := ring_nf) 2 * T R m * h₃ - h₂ - h₁ - ih2 + ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 167,
"column": 4
} | {
"line": 170,
"column": 12
} | {
"line": 171,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : CommRing R\nϖ : R\nhϖ : Irreducible ϖ\nhR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x\np : R\nhp : Irreducible p\nn : ℕ\nhn : Associated (ϖ ^ n) p\nthis : Irreducible (ϖ ^ n)\nH : n < 1\n⊢ Associated p ϖ",
"ppTerm": "?inl",
"assigned": true,
"usedConstants... | [
"case inl\nR : Type u_1\ninst✝ : CommRing R\nϖ : R\nhϖ : Irreducible ϖ\nhR : ∀ {x : R}, x ≠ 0 → ∃ n, Associated (ϖ ^ n) x\np : R\nhp : Irreducible p\nhn : Associated (ϖ ^ 0) p\nthis : Irreducible (ϖ ^ 0)\nH : 0 < 1\n⊢ Associated p ϖ"
] | obtain rfl : n = 0 := by
clear hn this
revert H n
decide | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 811,
"column": 2
} | {
"line": 812,
"column": 74
} | {
"line": 814,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\na : ℕ → ℕ\nha : StrictMono a\nf : (n : ℕ) → M →ₗ[R] N ⧸ I ^ a n • ⊤\nhf : ∀ {m : ℕ}, Submodule.factorPow I N ⋯ ∘ₗ f (m + 1) = f m\nm n :... | [] | ext
simp [extend, ← factorPow_comp_eq_of_factorPow_comp_succ_eq ha f hf hle] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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