module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 193,
"column": 2
} | {
"line": 199,
"column": 59
} | [
{
"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\nhx : IsIntegral A x\n⊢ 0 < (minpoly A x).natDegree",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"P... | rw [pos_iff_ne_zero]
intro ndeg_eq_zero
have eq_one : minpoly A x = 1 := by
rw [eq_C_of_natDegree_eq_zero ndeg_eq_zero]
convert! C_1 (R := A)
simpa only [ndeg_eq_zero.symm] using (monic hx).leadingCoeff
simpa only [eq_one, map_one, one_ne_zero] using aeval A x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Minpoly.Basic | {
"line": 193,
"column": 2
} | {
"line": 199,
"column": 59
} | [
{
"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\nhx : IsIntegral A x\n⊢ 0 < (minpoly A x).natDegree",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"P... | rw [pos_iff_ne_zero]
intro ndeg_eq_zero
have eq_one : minpoly A x = 1 := by
rw [eq_C_of_natDegree_eq_zero ndeg_eq_zero]
convert! C_1 (R := A)
simpa only [ndeg_eq_zero.symm] using (monic hx).leadingCoeff
simpa only [eq_one, map_one, one_ne_zero] using aeval A x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Abelian | {
"line": 295,
"column": 2
} | {
"line": 296,
"column": 92
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\n⊢ LieModule.ker R L L = center R L",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.instSetLike",
"NegZeroClass.toNeg",
"LieRing.toAddCommGroup",
"_private.Mathlib.Algebra... | ext y
simp only [LieModule.mem_maxTrivSubmodule, LieModule.mem_ker, ← lie_skew _ y, neg_eq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Abelian | {
"line": 295,
"column": 2
} | {
"line": 296,
"column": 92
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\n⊢ LieModule.ker R L L = center R L",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.instSetLike",
"NegZeroClass.toNeg",
"LieRing.toAddCommGroup",
"_private.Mathlib.Algebra... | ext y
simp only [LieModule.mem_maxTrivSubmodule, LieModule.mem_ker, ← lie_skew _ y, neg_eq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Squarefree.Basic | {
"line": 271,
"column": 43
} | {
"line": 271,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝² : CommMonoidWithZero R\ninst✝¹ : UniqueFactorizationMonoid R\ninst✝ : NormalizationMonoid R\nx : R\nx0 : x ≠ 0\n⊢ (∀ (x_1 : R), emultiplicity x_1 x ≤ 1 ∨ IsUnit x_1) ↔ (normalizedFactors x).Nodup",
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"C... | Multiset.nodup_iff_count_le_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Separable | {
"line": 281,
"column": 18
} | {
"line": 281,
"column": 83
} | [
{
"pp": "case neg\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\np : R[X]\nhsep : p.Separable\nx : R\nhp : ¬p = 0\n⊢ ↑(multiplicity (X - C x) p) ≤ 1",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"instCompleteLinearOrderENat",
"instAddMonoidWithOneENat",
"ChainComplete... | ← (finiteMultiplicity_X_sub_C x hp).emultiplicity_eq_multiplicity | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Perfect | {
"line": 88,
"column": 42
} | {
"line": 88,
"column": 57
} | [
{
"pp": "M : Type u_1\np : ℕ\ninst✝¹ : CommMonoid M\ninst✝ : PerfectRing M p\nn✝ n : ℕ\nih : powMulEquiv M (p ^ n) = powMulEquiv M p ^ n\n⊢ powMulEquiv M (p ^ n.succ) = powMulEquiv M (p ^ n) * powMulEquiv M p",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrA... | MulAut.mul_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Perfect | {
"line": 164,
"column": 91
} | {
"line": 165,
"column": 41
} | [
{
"pp": "R : Type u_1\np : ℕ\ninst✝² : CommSemiring R\ninst✝¹ : ExpChar R p\ninst✝ : PerfectRing R p\nx : R\n⊢ (iterateFrobeniusEquiv R p 1) x = x ^ p",
"usedConstants": [
"iterateFrobeniusEquiv",
"Eq.mpr",
"iterateFrobeniusEquiv_def",
"congrArg",
"CommSemiring.toSemiring",
... | by
rw [iterateFrobeniusEquiv_def, pow_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Perfect | {
"line": 310,
"column": 49
} | {
"line": 313,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Finite K\n⊢ PerfectField K",
"usedConstants": [
"PerfectField",
"NonAssocSemiring.toAddCommMonoidWithOne",
"IsDomain.to_noZeroDivisors",
"Nat.Prime",
"PerfectRing.toPerfectField",
"AddGroupWithOne.toAddMonoidWithOne",
... | by
obtain ⟨p, _instP⟩ := CharP.exists K
have : Fact p.Prime := ⟨CharP.char_is_prime K p⟩
exact PerfectRing.toPerfectField K p | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Perfect | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np n : ℕ\ninst✝ : ExpChar R p\nf : R[X]\n⊢ Multiset.map (⇑(iterateFrobenius R p n)) ((expand R (p ^ n)) f).roots ≤ p ^ n • f.roots",
"usedConstants": [
"Iff.mpr",
"Multiset.le_iff_count",
"instHSMul",
"Polynomial.roots",... | refine le_iff_count.2 fun r ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.Perfect | {
"line": 376,
"column": 2
} | {
"line": 377,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\np n : ℕ\ninst✝¹ : ExpChar R p\nf : R[X]\ninst✝ : DecidableEq R\n⊢ Finset.image (⇑(iterateFrobenius R p n)) ((expand R (p ^ n)) f).roots.toFinset ⊆ f.roots.toFinset",
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"instHSMu... | rw [Finset.image_toFinset, ← (roots f).toFinset_nsmul _ (expChar_pow_pos R p n).ne',
toFinset_subset] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.AdjoinRoot | {
"line": 177,
"column": 49
} | {
"line": 181,
"column": 9
} | [
{
"pp": "R : Type u_1\nT : Type u_3\ninst✝¹ : CommRing R\ninst✝ : Semiring T\np : R[X]\nf g : AdjoinRoot p →+* T\nhAlg : f.comp (of p) = g.comp (of p)\nhRoot : f (root p) = g (root p)\n⊢ f = g",
"usedConstants": [
"Polynomial.C",
"Ideal.Quotient.ringHom_ext",
"AdjoinRoot",
"congrArg"... | by
apply Ideal.Quotient.ringHom_ext
ext x
· simpa using congr($(hAlg) x)
· simpa | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AdjoinRoot | {
"line": 396,
"column": 23
} | {
"line": 397,
"column": 69
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝² : CommRing R\nf✝ g : R[X]\ninst✝¹ : CommRing S\ni : R →+* S\na : S\nh✝ : eval₂ i a f✝ = 0\ninst✝ : CommRing T\nf : R ≃+* S\np : R[X]\nq : S[X]\nh : Associated (Polynomial.map (↑f) p) q\n⊢ p ∣ Polynomial.map (↑f.symm) q",
"... | by
simpa [Polynomial.map_map] using map_dvd f.symm.toRingHom h.dvd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 83
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na : A\nk : ℕ\nh : a ^ k = 0\nh₁ :\n ∑ i ∈ range k, (↑i !)⁻¹ • a ^ i =\n ∑ i ∈ range (nilpotencyClass a), (↑i !)⁻¹ • a ^ i + ∑ i ∈ Ico (nilpotencyClass a) k, (↑i !)⁻¹ • a ^ i\nx✝ : ℕ\nh₂ : x✝ ∈ Ico (nilpotencyClass a) k\n⊢ (↑x✝!)⁻¹ • a ^ x✝ = 0",
... | rw [pow_eq_zero_of_le (mem_Ico.1 h₂).1 (pow_nilpotencyClass ⟨k, h⟩), smul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 83
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na : A\nk : ℕ\nh : a ^ k = 0\nh₁ :\n ∑ i ∈ range k, (↑i !)⁻¹ • a ^ i =\n ∑ i ∈ range (nilpotencyClass a), (↑i !)⁻¹ • a ^ i + ∑ i ∈ Ico (nilpotencyClass a) k, (↑i !)⁻¹ • a ^ i\nx✝ : ℕ\nh₂ : x✝ ∈ Ico (nilpotencyClass a) k\n⊢ (↑x✝!)⁻¹ • a ^ x✝ = 0",
... | rw [pow_eq_zero_of_le (mem_Ico.1 h₂).1 (pow_nilpotencyClass ⟨k, h⟩), smul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 83
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na : A\nk : ℕ\nh : a ^ k = 0\nh₁ :\n ∑ i ∈ range k, (↑i !)⁻¹ • a ^ i =\n ∑ i ∈ range (nilpotencyClass a), (↑i !)⁻¹ • a ^ i + ∑ i ∈ Ico (nilpotencyClass a) k, (↑i !)⁻¹ • a ^ i\nx✝ : ℕ\nh₂ : x✝ ∈ Ico (nilpotencyClass a) k\n⊢ (↑x✝!)⁻¹ • a ^ x✝ = 0",
... | rw [pow_eq_zero_of_le (mem_Ico.1 h₂).1 (pow_nilpotencyClass ⟨k, h⟩), smul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdjoinRoot | {
"line": 575,
"column": 90
} | {
"line": 579,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : R[X]\nhg : g.Monic\n⊢ Function.LeftInverse ⇑(mk g) ⇑(modByMonicHom hg)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"Dvd.dvd",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
... | by
intro f
induction f using AdjoinRoot.induction_on
rw [modByMonicHom_mk hg, mk_eq_mk, modByMonic_eq_sub_mul_div, sub_sub_cancel_left, dvd_neg]
apply dvd_mul_right | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AdjoinRoot | {
"line": 957,
"column": 16
} | {
"line": 957,
"column": 60
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\nK : Type u_5\ninst✝ : CommRing R\nI✝ : Ideal R\nf✝ f : R[X]\nI : Ideal R\nx : R\nthis : (algebraMap R (AdjoinRoot f ⧸ Ideal.map (of f) I)) x = (Ideal.Quotient.mk (Ideal.map (of f) I)) ((mk f) (C x))\n⊢ (quotAdjoinRootEquivQuotPolynomialQuot I f) (... | quotAdjoinRootEquivQuotPolynomialQuot_mk_of, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 1033,
"column": 2
} | {
"line": 1033,
"column": 22
} | [
{
"pp": "K : Type u_6\nL : Type u_7\ninst✝⁴ : CommRing K\ninst✝³ : IsDomain K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsIntegral K L\nf : L[X]\nhf : Irreducible f\nthis : Fact (Irreducible f)\n⊢ ∃ g, g.Monic ∧ Irreducible g ∧ f ∣ Polynomial.map (algebraMap K L) g",
"usedConstants": [
... | have h := hf.ne_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Dynamics.Newton | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 43
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : R[X]\nx : S\nh : IsNilpotent ((aeval x) P)\nh' : IsUnit ((aeval x) (derivative P))\nr₁ r₂ : S\nx✝¹ : IsNilpotent (r₁ - x) ∧ (aeval r₁) P = 0\nx✝ : IsNilpotent (r₂ - x) ∧ (aeval r₂) P = 0\nhr₁ :... | rw [← sub_sub_sub_cancel_right r₂ r₁ x] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Semisimple | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 82
} | [
{
"pp": "case inl\nM : Type u_2\ninst✝⁴ : AddCommGroup M\nK : Type u_3\ninst✝³ : Field K\ninst✝² : Module K M\nf g : End K M\ninst✝¹ : FiniteDimensional K M\ninst✝ : PerfectField K\ncomm : Commute f g\nhf : f.IsSemisimple\nhg : g.IsSemisimple\na : End K M\nha : a ∈ K[f, g]\nR : Type u_3 := K[X] ⧸ Ideal.span {mi... | · exact ⟨AdjoinRoot.of _ (AdjoinRoot.root _), (eval₂_C _ _).trans (aeval_X f)⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 231,
"column": 2
} | {
"line": 232,
"column": 82
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module ℚ M\ninst✝ : Module ℚ N\nfM : End R M\nfN : End R N\ng : M →ₗ[R] N\nh : fN ∘ₗ g = g ∘ₗ fM\nm : M\nk l : ℕ\nkl : ℕ := max k l... | have (i : ℕ) : (fN ^ i) (g m) = g ((fM ^ i) m) := by
simpa using LinearMap.congr_fun (Module.End.commute_pow_left_of_commute h i) m | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.BilinearForm.Properties | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 51
} | [
{
"pp": "V : Type u_5\nK : Type u_6\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nB : BilinForm K V\nhB : B.Nondegenerate\nf : Dual K V\nv : V\n⊢ (B ((B.toDual hB).symm f)) v = f v",
"usedConstants": [
"LinearEquiv.symm",
"Algebra.to_smulCommClas... | change B.toDual hB ((B.toDual hB).symm f) v = f v | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Order.BooleanGenerators | {
"line": 123,
"column": 46
} | {
"line": 139,
"column": 65
} | [
{
"pp": "α : 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 ⊓ sSup T₂",
"usedConstants": [
"Eq.mpr",
"IsCompactlyGenerated.BooleanGenerators.mono",
"Eq.ge",
... | by
apply le_antisymm
· apply le_inf
· apply sSup_le_sSup Set.inter_subset_left
· apply sSup_le_sSup Set.inter_subset_right
obtain ⟨X, hX, hX'⟩ := hS.atomistic (sSup T₁ ⊓ sSup T₂) (inf_le_left.trans (sSup_le_sSup hT₁))
rw [hX']
apply _root_.sSup_le
intro I hI
apply _root_.le_sSup
constructor
· ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 74,
"column": 27
} | {
"line": 74,
"column": 28
} | [
{
"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 = ⊥\n⊢ ∀ (I : LieIdeal R L), IsLieAbelian ↥I → I = ⊥",
"usedConstants": [
"LieIdeal"
]
}
] | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 74,
"column": 27
} | {
"line": 74,
"column": 28
} | [
{
"pp": "case mpr\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), IsLieAbelian ↥I → I = ⊥\n⊢ ∀ (I : LieIdeal R L), IsSolvable ↥I → I = ⊥",
"usedConstants": [
"LieIdeal"
]
}
] | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 206,
"column": 13
} | {
"line": 206,
"column": 14
} | [
{
"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\ns : Finset (LieIdeal R L)\nhs : ↑s ⊆ {I | IsAtom I}\n⊢ ∀ I ≤ s.sup id, ∃ t ⊆ s, I = t.sup id",
"usedConstants": [
"LieIdeal"
]
}
] | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Lie.Normalizer | {
"line": 73,
"column": 73
} | {
"line": 76,
"column": 24
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\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\nN₁ N₂ : LieSubmodule R L M\nh : N₁ ≤ N₂\n⊢ N₁.normalizer ≤ N₂.normalizer",
"usedConstants": ... | by
intro m hm
rw [mem_normalizer] at hm ⊢
exact fun x ↦ h (hm x) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 319,
"column": 2
} | {
"line": 323,
"column": 36
} | [
{
"pp": "R : 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⊢ IsLieAbelian ↥(radical R L) ↔ ∀ (I : LieIdeal R L), IsSolvable ↥I → IsLieAbelian ↥I",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.instSetLike",
"in... | constructor
· rintro h₁ I h₂
rw [LieIdeal.solvable_iff_le_radical] at h₂
exact (LieIdeal.inclusion_injective h₂).isLieAbelian h₁
· intro h; apply h; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 319,
"column": 2
} | {
"line": 323,
"column": 36
} | [
{
"pp": "R : 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⊢ IsLieAbelian ↥(radical R L) ↔ ∀ (I : LieIdeal R L), IsSolvable ↥I → IsLieAbelian ↥I",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.instSetLike",
"in... | constructor
· rintro h₁ I h₂
rw [LieIdeal.solvable_iff_le_radical] at h₂
exact (LieIdeal.inclusion_injective h₂).isLieAbelian h₁
· intro h; apply h; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.Divisibility.Lemmas | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 58
} | [
{
"pp": "case h\nR : Type u_1\nx y : R\nn m p : ℕ\ninst✝ : Semiring R\nhp : n + m ≤ p + 1\nh_comm : Commute x y\nhy : y ^ n = 0\nx✝ : ℕ × ℕ\ni j : ℕ\nhij : i + j = p\n⊢ x ^ m ∣ x ^ (i, j).1 * y ^ (i, j).2",
"usedConstants": [
"le_or_gt",
"Nat",
"Nat.instLinearOrder"
]
}
] | rcases le_or_gt m i with (hi : m ≤ i) | (hi : i + 1 ≤ m) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.Ring.Divisibility.Lemmas | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 27
} | [
{
"pp": "R : Type u_1\nx y : R\nn m p : ℕ\ninst✝ : Ring R\nhp : n + m ≤ p + 1\nh_comm : Commute x y\nh : (x + y) ^ n = 0\n⊢ x ^ m ∣ (-y) ^ p * (-1) ^ p",
"usedConstants": [
"dvd_mul_of_dvd_left",
"MulOne.toOne",
"Ring.toNonAssocRing",
"Monoid.toMulOneClass",
"SemigroupWithZero.... | apply dvd_mul_of_dvd_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Ring.Divisibility.Lemmas | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 27
} | [
{
"pp": "R : Type u_1\nx y : R\nn m p : ℕ\ninst✝ : Ring R\nhp : n + m ≤ p + 1\nh_comm : Commute x y\nhx : x ^ n = 0\n⊢ y ^ m ∣ (y - x) ^ p * (-1) ^ p",
"usedConstants": [
"dvd_mul_of_dvd_left",
"MulOne.toOne",
"Ring.toNonAssocRing",
"Monoid.toMulOneClass",
"HSub.hSub",
"S... | apply dvd_mul_of_dvd_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Lie.Engel | {
"line": 243,
"column": 6
} | {
"line": 243,
"column": 30
} | [
{
"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... | refine hK₁ _ fun x => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 775,
"column": 75
} | {
"line": 789,
"column": 55
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nk : ℕ\n⊢ ↑(lowerCentralSeries R L (L ⧸ I) k) = ↑(lowerCentralSeries R (L ⧸ I) (L ⧸ I) k)",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.instSetLike",
"Eq.mpr",
... | by
induction k with
| zero =>
simp only [LieModule.lowerCentralSeries_zero, LieSubmodule.top_toSubmodule]
| succ k ih =>
simp only [LieModule.lowerCentralSeries_succ, LieSubmodule.lieIdeal_oper_eq_linear_span]
congr
ext x
constructor
· rintro ⟨⟨y, -⟩, ⟨z, hz⟩, rfl : ⁅y, z⁆ = x⟩
rw [←... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.SplittingField.IsSplittingField | {
"line": 128,
"column": 2
} | {
"line": 131,
"column": 66
} | [
{
"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\n⊢ FiniteDimensional K L",
"usedConstants": [
"Multiset.toFinset",
"Submodule",
"False",
"Lattice.toSemilatticeSup",
"Algebra.algebraMap",
"... | exact ⟨@Algebra.top_toSubmodule K L _ _ _ ▸
adjoin_rootSet L f ▸ fg_adjoin_of_finite (Finset.finite_toSet _) fun y hy ↦
if hf : f = 0 then by rw [hf, rootSet_zero] at hy; cases hy
else IsAlgebraic.isIntegral ⟨f, hf, (mem_rootSet'.mp hy).2⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.SplittingField.IsSplittingField | {
"line": 128,
"column": 2
} | {
"line": 131,
"column": 66
} | [
{
"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\n⊢ FiniteDimensional K L",
"usedConstants": [
"Multiset.toFinset",
"Submodule",
"False",
"Lattice.toSemilatticeSup",
"Algebra.algebraMap",
"... | exact ⟨@Algebra.top_toSubmodule K L _ _ _ ▸
adjoin_rootSet L f ▸ fg_adjoin_of_finite (Finset.finite_toSet _) fun y hy ↦
if hf : f = 0 then by rw [hf, rootSet_zero] at hy; cases hy
else IsAlgebraic.isIntegral ⟨f, hf, (mem_rootSet'.mp hy).2⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SplittingField.IsSplittingField | {
"line": 128,
"column": 2
} | {
"line": 131,
"column": 66
} | [
{
"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\n⊢ FiniteDimensional K L",
"usedConstants": [
"Multiset.toFinset",
"Submodule",
"False",
"Lattice.toSemilatticeSup",
"Algebra.algebraMap",
"... | exact ⟨@Algebra.top_toSubmodule K L _ _ _ ▸
adjoin_rootSet L f ▸ fg_adjoin_of_finite (Finset.finite_toSet _) fun y hy ↦
if hf : f = 0 then by rw [hf, rootSet_zero] at hy; cases hy
else IsAlgebraic.isIntegral ⟨f, hf, (mem_rootSet'.mp hy).2⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Adjoin.Field | {
"line": 74,
"column": 10
} | {
"line": 74,
"column": 20
} | [
{
"pp": "case refine_1\nF : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝⁴ : Field F\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra F K\ninst✝ : Algebra F L\ns : Finset K\nx✝ : ∀ x ∈ ∅, IsIntegral F x ∧ (map (algebraMap F L) (minpoly F x)).Splits\n⊢ Nonempty (↥(Algebra.adjoin F ↑∅) →ₐ[F] L)",
"usedCons... | coe_empty, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 446,
"column": 2
} | {
"line": 447,
"column": 42
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nK : IntermediateField F E\nS : Set E\nh : K ≤ adjoin F S\n⊢ adjoin F S = adjoin F (↑K ∪ S)",
"usedConstants": [
"Iff.mpr",
"IntermediateField.instPartialOrder",
"IntermediateField.adjoin.mono",
... | exact le_antisymm (adjoin.mono F S _ Set.subset_union_right) <| adjoin_le_iff.2 <|
Set.union_subset h (subset_adjoin F S) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 49
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\np : 𝕜[X]\nk : 𝕜\nhk : k ∈ σ a\nq : 𝕜[X] := ⋯\nhroot : (C k - X) * -(q / (X - C k)) = q\naeval_q_eq : ↑ₐ (eval k p) - (aeval a) p = (aeval a) q\nhcomm : Commute ((aeval a) (C k - X)) ((aeval a) (-(q / (X - C k))... | simpa only [aeval_X, aeval_C, map_sub] using hk | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Eigenspace.Triangularizable | {
"line": 83,
"column": 2
} | {
"line": 84,
"column": 86
} | [
{
"pp": "case h.zero\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : IsAlgClosed K\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : End K V\nih :\n ∀ m < 0,\n ∀ {V : Type u_2} [inst : AddCommGroup V] [inst_1 : Module K V] [FiniteDimensional K V] (f : End K V),\n ... | · rw [← top_le_iff]
simp only [Submodule.finrank_eq_zero.1 (Eq.trans (finrank_top _ _) h_dim), bot_le] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 12
} | [
{
"pp": "case h\nk : Type u\ninst✝¹⁸ : Field k\nK✝ : Type u\ninst✝¹⁷ : Field K✝\nL : Type v\nM : Type w\ninst✝¹⁶ : Field L\ninst✝¹⁵ : Algebra K✝ L\ninst✝¹⁴ : Field M\ninst✝¹³ : Algebra K✝ M\ninst✝¹² : IsAlgClosed M\ninst✝¹¹ : Algebra.IsAlgebraic K✝ L\nR : Type u\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\nS : T... | rw [hroot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 561,
"column": 8
} | {
"line": 561,
"column": 39
} | [
{
"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\n⊢ (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ derivative f * g",
"usedConstants": [
"Polynomial.eq_C_of_derivative_eq_zero",
"Polynomial.d... | eq_C_of_derivative_eq_zero hdf0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 73
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ι → End R M\nμ : ι → R\ni : ι\nh✝ : ∀ (j : ι), MapsTo ⇑(f j) ↑((f i).maxGenEigenspace (μ i)) ↑((f i).maxGenEigenspace (μ i))\nthis✝ : Nonempty ι\np : Submodule R M := (f i).maxGenEigenspace (... | rw [Submodule.map_iInf _ p.injective_subtype, this, Submodule.inf_iInf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Extension | {
"line": 128,
"column": 4
} | {
"line": 130,
"column": 62
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nc : Set (Lifts F E K)\nhc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c\nσ : Lifts F E K\nhσ✝ : σ ∈ c\nhσ : σ ∈ insert ⊥ c\nt : ↑(insert ⊥ c) → IntermediateField F E := fun i ↦... | dsimp only [union, AlgHom.comp_apply]
exact Subalgebra.iSupLift_inclusion (K := (toSubalgebra <| t ·))
(i := ⟨σ, hσ⟩) x (le_iSup (toSubalgebra <| t ·) ⟨σ, hσ⟩) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Extension | {
"line": 128,
"column": 4
} | {
"line": 130,
"column": 62
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nc : Set (Lifts F E K)\nhc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c\nσ : Lifts F E K\nhσ✝ : σ ∈ c\nhσ : σ ∈ insert ⊥ c\nt : ↑(insert ⊥ c) → IntermediateField F E := fun i ↦... | dsimp only [union, AlgHom.comp_apply]
exact Subalgebra.iSupLift_inclusion (K := (toSubalgebra <| t ·))
(i := ⟨σ, hσ⟩) x (le_iSup (toSubalgebra <| t ·) ⟨σ, hσ⟩) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 646,
"column": 8
} | {
"line": 646,
"column": 23
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nf g : K[X]\nhfm : f.Monic\nhgm : g.Monic\nhf : Irreducible f\nhg :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))\nhf' : f.natDegree ≠ 0\nhg' : g.natDeg... | adjoin.finrank, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 742,
"column": 28
} | {
"line": 742,
"column": 39
} | [
{
"pp": "F : Type u\ninst✝² : Field F\nE : Type v\ninst✝¹ : Field E\ninst✝ : Algebra F E\ns : Set E\n⊢ Cardinal.lift.{u, v} #↑(Set.range ⇑(algebraMap F E) ∪ s) ≤\n max (max (Cardinal.lift.{v, u} #F) (Cardinal.lift.{u, v} #↑s)) ℵ₀ ∧\n Cardinal.lift.{u, v} ℵ₀ ≤ max (max (Cardinal.lift.{v, u} #F) (Cardinal... | lift_aleph0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 744,
"column": 21
} | {
"line": 744,
"column": 32
} | [
{
"pp": "F : Type u\ninst✝² : Field F\nE : Type v\ninst✝¹ : Field E\ninst✝ : Algebra F E\ns : Set E\n⊢ max (max (Cardinal.lift.{u, v} #↑(Set.range ⇑(algebraMap F E))) (Cardinal.lift.{u, v} #↑s))\n (Cardinal.lift.{u, v} ℵ₀) ≤\n max (max (Cardinal.lift.{v, u} #F) (Cardinal.lift.{u, v} #↑s)) ℵ₀",
"used... | lift_aleph0 | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Multiset.Fintype | {
"line": 189,
"column": 6
} | {
"line": 189,
"column": 31
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\nthis : map Prod.fst m.toEnumFinset.val = m\n⊢ map (fun x ↦ x.fst) Finset.univ.val = m",
"usedConstants": [
"Finset.univ",
"Multiset.map",
"congrArg",
"Multiset.coeEmbedding",
"Finset",
"Multiset.fintypeCoe",
... | ← m.map_univ_coeEmbedding | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 333,
"column": 4
} | {
"line": 334,
"column": 23
} | [
{
"pp": "case pos\nL : Type u_2\ninst✝⁵ : LieRing L\nK : Type u_4\ninst✝⁴ : Field K\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nI : LieIdeal K L\nα : Weight K (↥H) L\nhα : α.IsZero\n⊢ LieSubmodule.restr I H ... | · rw [show genWeightSpace L (α : H → K) = H.toLieSubmodule from by ext; simp [hα.eq]]
exact le_sup_left | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 342,
"column": 4
} | {
"line": 342,
"column": 13
} | [
{
"pp": "case pos\nL : Type u_2\ninst✝⁵ : LieRing L\nK : Type u_4\ninst✝⁴ : Field K\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα : Weight K (↥H) L\nhα : α.IsZero\n⊢ genWeightSpace L ⇑α ≤ H.toLieSubmodule ⊔ ... | simp [hα] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 342,
"column": 4
} | {
"line": 342,
"column": 13
} | [
{
"pp": "case pos\nL : Type u_2\ninst✝⁵ : LieRing L\nK : Type u_4\ninst✝⁴ : Field K\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα : Weight K (↥H) L\nhα : α.IsZero\n⊢ genWeightSpace L ⇑α ≤ H.toLieSubmodule ⊔ ... | simp [hα] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 342,
"column": 4
} | {
"line": 342,
"column": 13
} | [
{
"pp": "case pos\nL : Type u_2\ninst✝⁵ : LieRing L\nK : Type u_4\ninst✝⁴ : Field K\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα : Weight K (↥H) L\nhα : α.IsZero\n⊢ genWeightSpace L ⇑α ≤ H.toLieSubmodule ⊔ ... | simp [hα] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Killing | {
"line": 116,
"column": 58
} | {
"line": 118,
"column": 26
} | [
{
"pp": "R : Type u_1\nL : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nL' : Type u_4\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\ne : L ≃ₗ⁅R⁆ L'\nx y : L\n⊢ ((killingForm R L') (e x)) (e y) = ((killingForm R L) x) y",
"usedConstants": [
"LieHom",
"LieAlgebra.toM... | by
simp_rw [killingForm_apply_apply, ← LieAlgebra.conj_ad_apply, ← LinearEquiv.conj_comp,
LinearMap.trace_conj'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 740,
"column": 4
} | {
"line": 741,
"column": 29
} | [
{
"pp": "R : 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\ninst✝³ : IsDomain R\ninst✝² : IsPrincipalIdealRing R\ninst✝¹ ... | rwa [map_sub, map_smul, trace_id, sub_eq_zero, smul_eq_mul, mul_comm,
← nsmul_eq_mul] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 782,
"column": 2
} | {
"line": 782,
"column": 96
} | [
{
"pp": "K : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\ninst✝⁶ : LieRing.IsNilpotent L\ninst✝⁵ : Field K\ninst✝⁴ : LieAlgebra K L\ninst✝³ : Module K M\ninst✝² : LieModule K L M\ninst✝¹ : FiniteDimensional K M\ninst✝ : IsTriangularizable K L M\n... | have hN := (LieSubmodule.map_mono (le_top : genWeightSpace N χ_N ≤ ⊤)).trans N.map_incl_top.le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.PerfectPairing.Basic | {
"line": 171,
"column": 36
} | {
"line": 173,
"column": 20
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ninst✝ : p.IsPerfPair\nU : Submodule R M\n⊢ p.IsPerfectCompl U ⊤ ↔ U = ⊤",
"usedConstants": [
"Eq.mpr",
"Submo... | by
rw [← IsPerfectCompl.flip_iff]
exact left_top_iff | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 520,
"column": 38
} | {
"line": 520,
"column": 47
} | [
{
"pp": "case h\nK : 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 : Weight K (↥H) L\nhα : α.IsZero\n... | simp [hα] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 69,
"column": 50
} | {
"line": 69,
"column": 71
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ninst✝⁴ : p.IsPerfPair\nM' : Type u_4\nN' : Type u_5\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : ... | LinearEquiv.symm_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 70
} | [
{
"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... | obtain ⟨w, hw₁, hw₂, hw₃⟩ := exists_linearIndependent L (N' : Set N) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 745,
"column": 2
} | {
"line": 745,
"column": 38
} | [
{
"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\nχ : Weight K (↥H) L\nhχ : χ.IsNonZero\n⊢ fin... | simp [finrank_rootSpace_eq_one _ hχ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 214,
"column": 6
} | {
"line": 214,
"column": 86
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℕ\nih1 : (T R (↑n + 1)).degree = ↑(↑n + 1).natAbs\nih2 : (T R ↑n).degree = ↑(↑n).natAbs\n⊢ (2 * X * T R (↑n + 1)).degree = ↑(n + 2)",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
... | rw [mul_assoc, ← C_ofNat, degree_C_mul two_ne_zero, mul_comm, degree_mul_X, ih1] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Reflection | {
"line": 438,
"column": 2
} | {
"line": 438,
"column": 23
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nx : M\nf : Dual R M\ny : M\ng : Dual R M\ninst✝ : IsAddTorsionFree M\nΦ : Set M\nhfx : f x = 2\nhgy : g y = 2\nhgx : g x = 2\nhfy : f y = 2\nhxfΦ : MapsTo (⇑(preReflection x f)) Φ Φ\nhygΦ : MapsTo (⇑(preRefle... | rw [range_subset_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Length | {
"line": 183,
"column": 31
} | {
"line": 183,
"column": 65
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u_3\nP : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : N →ₗ[R] M\ng : M →ₗ[R] P\nhf : Function.Injective ⇑f\nhg : Function.Surjective ⇑g\nH : Fu... | simpa [s', t', hs₁, ht₂] using hfg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Length | {
"line": 183,
"column": 31
} | {
"line": 183,
"column": 65
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u_3\nP : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : N →ₗ[R] M\ng : M →ₗ[R] P\nhf : Function.Injective ⇑f\nhg : Function.Surjective ⇑g\nH : Fu... | simpa [s', t', hs₁, ht₂] using hfg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Length | {
"line": 183,
"column": 31
} | {
"line": 183,
"column": 65
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u_3\nP : Type u_4\ninst✝³ : AddCommGroup N\ninst✝² : AddCommGroup P\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : N →ₗ[R] M\ng : M →ₗ[R] P\nhf : Function.Injective ⇑f\nhg : Function.Surjective ⇑g\nH : Fu... | simpa [s', t', hs₁, ht₂] using hfg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.Integers | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 68
} | [
{
"pp": "R : Type u\nΓ₀ : Type v\ninst✝³ : CommRing R\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nO : Type w\ninst✝¹ : CommRing O\ninst✝ : Algebra O R\nhv : v.Integers O\nx z : O\n⊢ v ((algebraMap O R) (x * z)) ≤ v ((algebraMap O R) x)",
"usedConstants": [
"CommMonoidWithZero.toCo... | grw [← mul_one (v (algebraMap O R x)), map_mul, v.map_mul, hv.2 z] | Mathlib.Tactic._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_grwSeq_1 | Mathlib.Tactic.grwSeq |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 217,
"column": 18
} | {
"line": 217,
"column": 29
} | [
{
"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\nx✝ : Hausdorffification I M\nx : M\nhx : ∀ (n : ℕ), Quotient.mk'' x ≡ 0 [SMOD I ^ n • ⊤]\nn : ℕ\nthis : Subm... | map_smul'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 559,
"column": 20
} | {
"line": 559,
"column": 25
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : AdicCompletion I M\na : ℕ → M\nha : ∀ (n : ℕ), Submodule.Quotient.mk (a n) = ↑x n\nm n : ℕ\nhmn : m ≤ n\n⊢ Submodule.Quotient.mk (a m) = Submodule.Quotient.mk (a n)",
"usedC... | ha m, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 38
} | [
{
"pp": "case inr.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 ^ 1) p\nthis : Irreducible (ϖ ^ 1)\n⊢ Associated p ϖ",
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
... | simpa only [pow_one] using hn.symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 38
} | [
{
"pp": "case inr.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 ^ 1) p\nthis : Irreducible (ϖ ^ 1)\n⊢ Associated p ϖ",
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
... | simpa only [pow_one] using hn.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 38
} | [
{
"pp": "case inr.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 ^ 1) p\nthis : Irreducible (ϖ ^ 1)\n⊢ Associated p ϖ",
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
... | simpa only [pow_one] using hn.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 60
} | [
{
"pp": "case neg\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ny : R\ninst✝¹ : Algebra R S\ninst✝ : Away y S\nH : IsJacobsonRing R\nJ : Ideal S\nh : J.IsMaximal\nhJ✝ : (under R J).IsPrime ∧ Disjoint ↑(powers y) ↑(under R J)\nI : Ideal R\nhI' : y ∉ I\nhJI : under R J ≤ I\nhIm : I.IsMaxi... | exact absurd (h.1.2 _ (lt_of_le_of_ne this hJ)) hI_p.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 280,
"column": 2
} | {
"line": 281,
"column": 89
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nP : Ideal R[X]\npX : R[X]\nhpX : pX ∈ P\ninst✝³ : Algebra (R ⧸ comap C P) Rₘ\ninst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ\ninst✝¹ : Algebra (R[X] ⧸ P) S... | let M : Submonoid (R ⧸ P') :=
Submonoid.powers (pX.map (Ideal.Quotient.mk (P.comap (C : R →+* R[X])))).leadingCoeff | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nP : Ideal R[X]\npX : R[X]\nhpX : pX ∈ P\ninst✝³ : Algebra (R ⧸ comap C P) Rₘ\ninst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ\ninst✝¹ : Algebra (R[X] ⧸ P) S... | dsimp at hp | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.RingTheory.Valuation.Basic | {
"line": 342,
"column": 67
} | {
"line": 343,
"column": 45
} | [
{
"pp": "R : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nx y : R\nhy : v y = 0\n⊢ v (x + y) = v x",
"usedConstants": [
"Eq.mpr",
"Ring.toNonAssocRing",
"congrArg",
"id",
"Distrib.toAdd",
"NonUnitalNonAssocRing.... | by
rw [add_comm, map_add_of_left_eq_zero v hy] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 445,
"column": 2
} | {
"line": 450,
"column": 81
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nP : Ideal R[X]\nhP : P.IsMaximal\ninst✝¹ : IsJacobsonRing R\ninst✝ : Nontrivial R\nhP' : ∀ (x : R), C x ∈ P → x = 0\nP' : Ideal R := comap C P\nhP'_prime : P'.IsPrime\nm : R[X]\nhmem_P : m ∈ P\nhm : ⟨m, hmem_P⟩ ≠ 0\nhm' : m ≠ 0\nφ : R ⧸ P' →+* R[X] ⧸ P := quotientMap ... | have hp0 : a ≠ 0 := fun hp0' =>
hm' <| map_injective (Ideal.Quotient.mk (P.comap (C : R →+* R[X]) : Ideal R))
((injective_iff_map_eq_zero (Ideal.Quotient.mk (P.comap (C : R →+* R[X]) : Ideal R))).2
fun x hx => by
rwa [Quotient.eq_zero_iff_mem, (by rwa [eq_bot_iff] : (P.comap C : Ideal R) = ⊥... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Valuation.Basic | {
"line": 521,
"column": 6
} | {
"line": 521,
"column": 35
} | [
{
"pp": "R : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx : R\n⊢ v.restrict x ≤ 1 ↔ v x ≤ 1",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
... | restrict_le_iff_le_embedding, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 527,
"column": 4
} | {
"line": 527,
"column": 80
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R[X]\nhP : P.IsMaximal\ninst✝ : IsJacobsonRing R\nP' : Ideal R := comap C P\nthis : P'.IsPrime\nf : R[X] →+* (R ⧸ P')[X] := mapRingHom (Ideal.Quotient.mk P')\nhf : Function.Surjective ⇑f\np : R[X]\nhp : p ∈ comap f ⊥\nn : ℕ\n⊢ (Ideal.Quotient.mk (comap C P))... | simpa only [f, coeff_map, coe_mapRingHom] using (Polynomial.ext_iff.mp hp) n | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 145,
"column": 45
} | {
"line": 146,
"column": 38
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\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 : Nontrivial R\ninst✝¹ : P.IsCrystallographic\ninst✝ : Algebra ℚ R\ni j : ι\n⊢ P.pairingIn ℚ i j = ↑(P... | by
simp [← P.algebraMap_pairingIn' ℚ ℤ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 88
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\n⊢ ⨅ i, LinearMap.ker (P.root' i) = (span R (range P.root')).dualCoannihilator",
"usedConstants": [
"... | rw [← rootSpan_dualAnnihilator_map_eq, rootSpan_dualAnnihilator_map_eq_iInf_ker_root'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 88
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\n⊢ ⨅ i, LinearMap.ker (P.root' i) = (span R (range P.root')).dualCoannihilator",
"usedConstants": [
"... | rw [← rootSpan_dualAnnihilator_map_eq, rootSpan_dualAnnihilator_map_eq_iInf_ker_root'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 88
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\n⊢ ⨅ i, LinearMap.ker (P.root' i) = (span R (range P.root')).dualCoannihilator",
"usedConstants": [
"... | rw [← rootSpan_dualAnnihilator_map_eq, rootSpan_dualAnnihilator_map_eq_iInf_ker_root'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 164,
"column": 61
} | {
"line": 166,
"column": 55
} | [
{
"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✝... | by
rw [← finrank_range_polarization_eq_finrank_span_coroot]
exact LinearMap.finrank_range_le (P.PolarizationIn S) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 60,
"column": 2
} | {
"line": 62,
"column": 34
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\nh : ∀ (i j : ι), i ≠ j → ¬LinearIndependent R ![P.root i, P.root j] → P.root i = -P.root j\ni j... | · rcases eq_or_ne i j with rfl | h'
· tauto
· exact Or.inr (h i j h' hLin) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 25
} | [
{
"pp": "case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝¹ : NeZero 2\ninst✝ : IsAddTorsionFree M\nhl : LinearIndependent R ![P.root i, P... | rw [range_subset_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 241,
"column": 12
} | {
"line": 241,
"column": 45
} | [
{
"pp": "case refine_1\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝⁴ : Finite ι\ninst✝³ : CharZero R\ninst✝² : IsDomain R\ninst✝¹ : IsTorsionFree R... | show (4 : R) = 2 * 2 by norm_num, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Dual | {
"line": 60,
"column": 37
} | {
"line": 60,
"column": 47
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ne : (Module.Dual R M × Module.Dual R (Module.Dual R M)) ≃ₗ[R] Module.Dual R (Module.Dual R M × M) :=\n LinearEquiv.prodComm R (Module.Dual R M) (Module.Dual R (Module.Dual R M)) ≪≫ₗ\n Module.dualProdDualEq... | ← coe_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.WeylGroup | {
"line": 82,
"column": 70
} | {
"line": 82,
"column": 99
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\npred : (g : P.Aut) → g ∈ P.weylGroup → Prop\nmem : ∀ (i : ι), pred (Equiv.reflection P i) ⋯\none : pred 1 ⋯\nm... | rwa [← weylGroup_toSubmonoid] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.RootSystem.WeylGroup | {
"line": 207,
"column": 4
} | {
"line": 208,
"column": 76
} | [
{
"pp": "case mul\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\nB : P.InvariantForm\ng g₁ g₂ : P.Aut\nhg₁ : g₁ ∈ P.weylGroup\nhg₂ : g₂ ∈ P.weylGroup\nhg₁' : ∀ (x y ... | intro x y
rw [← Submonoid.mk_mul_mk _ _ _ hg₁ hg₂, mul_smul, mul_smul, hg₁', hg₂'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.WeylGroup | {
"line": 207,
"column": 4
} | {
"line": 208,
"column": 76
} | [
{
"pp": "case mul\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nP : RootPairing ι R M N\nB : P.InvariantForm\ng g₁ g₂ : P.Aut\nhg₁ : g₁ ∈ P.weylGroup\nhg₂ : g₂ ∈ P.weylGroup\nhg₁' : ∀ (x y ... | intro x y
rw [← Submonoid.mk_mul_mk _ _ _ hg₁ hg₂, mul_smul, mul_smul, hg₁', hg₂'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Irreducible | {
"line": 140,
"column": 13
} | {
"line": 140,
"column": 39
} | [
{
"pp": "case refine_1.one\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing 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✝ : Nontrivial M\nh : IsSimpleOrder ↥P.weylGroupRootRep.invtSubmodule\nq : Submodule R... | simp [← Submonoid.one_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.Irreducible | {
"line": 140,
"column": 13
} | {
"line": 140,
"column": 39
} | [
{
"pp": "case refine_1.one\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing 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✝ : Nontrivial M\nh : IsSimpleOrder ↥P.weylGroupRootRep.invtSubmodule\nq : Submodule R... | simp [← Submonoid.one_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Irreducible | {
"line": 140,
"column": 13
} | {
"line": 140,
"column": 39
} | [
{
"pp": "case refine_1.one\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing 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✝ : Nontrivial M\nh : IsSimpleOrder ↥P.weylGroupRootRep.invtSubmodule\nq : Submodule R... | simp [← Submonoid.one_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Basic | {
"line": 633,
"column": 89
} | {
"line": 638,
"column": 5
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\ninst✝¹ : AddCommMonoid W\ninst✝ : Module k W\nρV : Representation k G V\nr : k[G]\nx : V\ny : W\n⊢ (r •\n have this := x ⊗ₜ[k] y;\n this) =\n (... | by
change asAlgebraHom (ρV ⊗ 1) _ _ = asAlgebraHom ρV _ _ ⊗ₜ _
simp only [asAlgebraHom_def, MonoidAlgebra.lift_apply, tprod_apply, MonoidHom.one_apply,
LinearMap.finsupp_sum_apply, LinearMap.smul_apply, TensorProduct.map_tmul, Module.End.one_apply]
simp only [Finsupp.sum, TensorProduct.sum_tmul]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Set.OrdConnectedLinear | {
"line": 77,
"column": 88
} | {
"line": 78,
"column": 99
} | [
{
"pp": "I : Set ℕ\nh₀ : I.Nonempty\nh₂ : BddAbove I\n⊢ I = Icc (sInf I) (sSup I) ↔ ∀ x ∈ I, ∀ y ∈ I, Disjoint (Ioo x y) I → y ≤ x + 1",
"usedConstants": [
"Order.Ioo_eq_empty_iff_le_succ._simp_2",
"Order.succ",
"Order.succ_eq_add_one",
"Nat.instOne",
"CompleteBooleanAlgebra.to... | by
simp [← h₀.ordConnected_iff_of_bdd (OrderBot.bddBelow I) h₂, ordConnected_iff_disjoint_Ioo_empty] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 43
} | [
{
"pp": "case refine_2\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... | · simpa using this (-P.chainTopCoeff i j) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.Chain | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 43
} | [
{
"pp": "case refine_2\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... | · simpa using this (-P.chainTopCoeff i j) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.