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.Algebra.Lie.Weights.Killing | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 74
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁶ : LieRing L\ninst✝⁵ : Field K\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nthis : range (Weight.toLinear K (↥H) L) ⊆ insert 0 (Weight.toLine... | simpa only [Submodule.span_insert_zero] using Submodule.span_mono this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 74
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁶ : LieRing L\ninst✝⁵ : Field K\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nthis : range (Weight.toLinear K (↥H) L) ⊆ insert 0 (Weight.toLine... | simpa only [Submodule.span_insert_zero] using Submodule.span_mono this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 350,
"column": 6
} | {
"line": 350,
"column": 58
} | [
{
"pp": "case mem\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✝ : PerfectField K\nx : L\nhx : x ∈ H\nN S : End K... | induction hz using LieSubmodule.iSup_induction' with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 496,
"column": 60
} | {
"line": 496,
"column": 68
} | [
{
"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 : Weight K (↥H) L\nhα : ¬α.IsZero\n⊢ Weigh... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 496,
"column": 60
} | {
"line": 496,
"column": 68
} | [
{
"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 : Weight K (↥H) L\nhα : ¬α.IsZero\n⊢ Weigh... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 496,
"column": 60
} | {
"line": 496,
"column": 68
} | [
{
"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 : Weight K (↥H) L\nhα : ¬α.IsZero\n⊢ Weigh... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 548,
"column": 43
} | {
"line": 548,
"column": 81
} | [
{
"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 β : Weight K (↥H) L\nhyp : coroot ... | simpa using LinearMap.congr_fun this x | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Reflection | {
"line": 212,
"column": 32
} | {
"line": 212,
"column": 54
} | [
{
"pp": "case succ\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nf g : Dual R M\nhf : f x = 2\nhg : g y = 2\nz : M\nt : R\nht : t = f y * g x - 2\nm : ℕ\nS_eval_t_sub_two :\n ∀ (k : ℤ), Polynomial.eval t (S R (k - 2)) = t * Polynomial.eval t (S R (k - 1... | add_sub_assoc (2 * k), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 78
} | [
{
"pp": "R : 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 : AddCommGroup N'... | refine ⟨LinearMap.ker_eq_bot.mp <| eq_bot_iff.mpr fun m hm ↦ ?_, fun f ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 12
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : AddCommGroup M\ninst✝¹⁶ : Module R M\ninst✝¹⁵ : AddCommGroup N\ninst✝¹⁴ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ninst✝¹³ : p.IsPerfPair\nS : Type u_4\nM' : Type u_5\nN' : Type u_6\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Localization.NumDen | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 32
} | [
{
"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\n⊢ num A 0 = 0",
"usedConstants": [
"instHDiv",
"Algebra.algebraMap",
"CommSemiring.toSemiring",
"Is... | have := mk'_num_den' A (0 : K) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Localization.NumDen | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 10
} | [
{
"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\nthis : (algebraMap A K) (num A 0) = 0 ∨ (algebraMap A K) ↑(den A 0) = 0\n⊢ num A 0 = 0",
"usedConstants": [
"GroupWit... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Reflection | {
"line": 299,
"column": 69
} | {
"line": 299,
"column": 82
} | [
{
"pp": "case pred\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nf g : Dual R M\nhf : f x = 2\nhg : g y = 2\nt : R\nht : t = f y * g x - 2\nS_eval_t_sub_two :\n ∀ (k : ℤ),\n Polynomial.eval t (S R (k - 2)) = (f y * g x - 2) * Polynomial.eval t (S R (... | zpow_neg_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Reflection | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 85
} | [
{
"pp": "case e_a\nR : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : IsAddTorsionFree M\nr : ι ↪ M\nc : ι → Dual R M\nhfin : (range ⇑r).Finite\nh_two : ∀ (i : ι), (c i) (r i) = 2\nh_mapsTo : ∀ (i : ι), MapsTo (⇑(preReflection (r i) (c i))) (rang... | simpa using LinearMap.congr_fun hij ⟨r k, Submodule.subset_span (mem_range_self k)⟩ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 391,
"column": 15
} | {
"line": 391,
"column": 31
} | [
{
"pp": "case more\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nn : ℕ\nih1 : (U R ↑n).degree = ↑n\nih2 : (U R ↑(n + 1)).degree = ↑(n + 1)\n⊢ (U R (↑n + 2)).degree = ↑n + 2",
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot",
"Polynomial.Chebyshev.U",
... | push_cast at ih2 | Lean.Elab.Tactic.NormCast.evalPushCast | Lean.Parser.Tactic.pushCast |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 438,
"column": 15
} | {
"line": 438,
"column": 31
} | [
{
"pp": "case more\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : NeZero 2\nthis : leadingCoeff 2 = 2\nn : ℕ\nih1 : (U R ↑n).leadingCoeff = 2 ^ n\nih2 : (U R ↑(n + 1)).leadingCoeff = 2 ^ (n + 1)\n⊢ (U R (↑n + 2)).leadingCoeff = 2 ^ (n + 2)",
"usedConstants": [
"Polynomial.Chebyshev.U... | push_cast at ih2 | Lean.Elab.Tactic.NormCast.evalPushCast | Lean.Parser.Tactic.pushCast |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 454,
"column": 17
} | {
"line": 454,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nx : R\nn : ℕ\nih1 : eval (-x) (U R ↑n) = ↑↑(↑n).negOnePow * eval x (U R ↑n)\nih2 : eval (-x) (U R ↑(n + 1)) = ↑↑(↑(n + 1)).negOnePow * eval x (U R ↑(n + 1))\n⊢ eval (-x) (U R (↑n + 2)) = ↑↑(↑n + 2).negOnePow * (2 * x * eval x (U R (↑n + 1)) - eval x (U R ↑n))",
"us... | push_cast at ih2 | Lean.Elab.Tactic.NormCast.evalPushCast | Lean.Parser.Tactic.pushCast |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 715,
"column": 4
} | {
"line": 720,
"column": 8
} | [
{
"pp": "case add_two\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nih1 : eval (-2) (S R (↑n + 1)) = ↑↑(↑n + 1).negOnePow * (↑(↑n + 1) + 1)\nih2 : eval (-2) (S R ↑n) = ↑↑(↑n).negOnePow * (↑↑n + 1)\n⊢ eval (-2) (S R (↑n + 2)) = ↑↑(↑n + 2).negOnePow * (↑(↑n + 2) + 1)",
"usedConstants": [
"Mathlib.Tactic.Rin... | simp only [S_add_two, eval_sub, eval_mul, eval_X, ih1,
Int.cast_add, Int.cast_natCast, Int.cast_one, neg_mul, ih2, Int.cast_ofNat, Int.negOnePow_add,
Int.negOnePow_def 2]
norm_cast
norm_num
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 715,
"column": 4
} | {
"line": 720,
"column": 8
} | [
{
"pp": "case add_two\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nih1 : eval (-2) (S R (↑n + 1)) = ↑↑(↑n + 1).negOnePow * (↑(↑n + 1) + 1)\nih2 : eval (-2) (S R ↑n) = ↑↑(↑n).negOnePow * (↑↑n + 1)\n⊢ eval (-2) (S R (↑n + 2)) = ↑↑(↑n + 2).negOnePow * (↑(↑n + 2) + 1)",
"usedConstants": [
"Mathlib.Tactic.Rin... | simp only [S_add_two, eval_sub, eval_mul, eval_X, ih1,
Int.cast_add, Int.cast_natCast, Int.cast_one, neg_mul, ih2, Int.cast_ofNat, Int.negOnePow_add,
Int.negOnePow_def 2]
norm_cast
norm_num
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Bezout | {
"line": 78,
"column": 64
} | {
"line": 78,
"column": 68
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : IsBezout R\ninst✝ : IsDomain R\ntfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R\ntfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R\ntfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R\nh : WellFounded DvdNotUnit\nf : { J // J.FG... | ← hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 926,
"column": 2
} | {
"line": 927,
"column": 87
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nh₁ :\n derivative ((1 - X ^ 2) * derivative (T R (n - 1 + 1))) =\n derivative ((↑(n - 1) + 1) * (T R (n - 1) - X * T R (n - 1 + 1)))\n⊢ (1 - X ^ 2) * (⇑derivative)^[2] (T R n) = X * derivative (T R n) - ↑n ^ 2 * T R n",
"usedConstants": [
"Polynomi... | simp only [derivative_sub, derivative_mul, derivative_X, derivative_one, derivative_X_pow,
C_eq_natCast, sub_add_cancel, Int.cast_sub, Int.cast_one, derivative_intCast] at h₁ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Valuation.Integers | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 17
} | [
{
"pp": "F : Type u\nΓ₀ : Type v\ninst✝³ : Field F\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation F Γ₀\nO : Type w\ninst✝¹ : CommRing O\ninst✝ : Algebra O F\nhv : v.Integers O\nx : O\nhx0 : x ≠ 0\nd : O\nhdu : v ((algebraMap O F) d) < 1\n⊢ v ((algebraMap O F) x) * v ((algebraMap O F) d) < v ((algebr... | contrapose! hdu | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.RingTheory.Valuation.Integers | {
"line": 228,
"column": 2
} | {
"line": 232,
"column": 94
} | [
{
"pp": "case mpr\nF : Type u\nΓ₀ : Type v\ninst✝³ : Field F\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation F Γ₀\nO : Type w\ninst✝¹ : CommRing O\ninst✝ : Algebra O F\nhv : v.Integers O\nI : Ideal O\nx : Γ₀\nhx : IsGreatest (⇑v ∘ ⇑(algebraMap O F) '' ↑I) x\n⊢ Submodule.IsPrincipal I",
"usedConst... | · obtain ⟨a, ha, rfl⟩ : ∃ a ∈ I, (v ∘ algebraMap O F) a = x := by simpa using hx.left
refine ⟨a, ?_⟩
ext b
simp only [Ideal.submodule_span_eq, Ideal.mem_span_singleton]
exact ⟨fun hb ↦ dvd_of_le hv (hx.2 <| mem_image_of_mem _ hb), fun hb ↦ I.mem_of_dvd hb ha⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 65
} | [
{
"pp": "case refine_2\nR : 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✝ : AdicCompletion I M\nh : ∀ (n : ℕ), x✝ ≡ 0 [SMOD I ^ n • ⊤]\nn : ℕ\nx y : AdicCompletion ... | · simp only [val_add_apply, hx, val_zero_apply, hy, add_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Valuation.ValuationRing | {
"line": 243,
"column": 31
} | {
"line": 243,
"column": 74
} | [
{
"pp": "A : Type u\ninst✝⁵ : CommRing A\nK : Type v\ninst✝⁴ : Field K\ninst✝³ : Algebra A K\ninst✝² : IsDomain A\ninst✝¹ : ValuationRing A\ninst✝ : IsFractionRing A K\nx✝¹ x✝ : A\n⊢ ⟨(algebraMap A K) (x✝¹ + x✝), ⋯⟩ = ⟨(algebraMap A K) x✝¹, ⋯⟩ + ⟨(algebraMap A K) x✝, ⋯⟩",
"usedConstants": [
"Valuation... | by ext1; exact (algebraMap A K).map_add _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 1088,
"column": 11
} | {
"line": 1088,
"column": 38
} | [
{
"pp": "case one\nR : Type u_1\ninst✝ : CommRing R\nm : ℤ\n⊢ 2 * T R m * T R 1 = T R (m + 1) + T R (m - 1)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWit... | rw [T_add_one, T_one]; ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 1088,
"column": 11
} | {
"line": 1088,
"column": 38
} | [
{
"pp": "case one\nR : Type u_1\ninst✝ : CommRing R\nm : ℤ\n⊢ 2 * T R m * T R 1 = T R (m + 1) + T R (m - 1)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWit... | rw [T_add_one, T_one]; ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 536,
"column": 4
} | {
"line": 538,
"column": 31
} | [
{
"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\nf : AdicCauchySequence I M\n⊢ ∀ {m n : ℕ} (hmn : m ≤ n),\n (transitionMap I M hmn) ((fun n ↦ (I ^ n • ⊤).... | intro m n hmn
simp only [mkQ_apply]
exact (f.property hmn).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 536,
"column": 4
} | {
"line": 538,
"column": 31
} | [
{
"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\nf : AdicCauchySequence I M\n⊢ ∀ {m n : ℕ} (hmn : m ≤ n),\n (transitionMap I M hmn) ((fun n ↦ (I ^ n • ⊤).... | intro m n hmn
simp only [mkQ_apply]
exact (f.property hmn).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 879,
"column": 4
} | {
"line": 879,
"column": 36
} | [
{
"pp": "case a\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nx : R\nhx : x ∈ I\ny : R\nf : ℕ → R := fun n ↦ ∑ i ∈ range n, (x * y) ^ i\nm n✝ : ℕ\nh : m ≤ n✝\nn : ℕ\na✝ : n ∈ range (n✝ - m)\n⊢ (x * y) ^ (m + n) ∈ I ^ m",
"usedConstants": [
"Eq.mpr",
"Submodule",
... | rw [mul_pow, pow_add, mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 884,
"column": 2
} | {
"line": 884,
"column": 29
} | [
{
"pp": "case h\nR : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nx : R\nhx : x ∈ I\ny : R\nf : ℕ → R := ⋯\nhf : ∀ (m n : ℕ), m ≤ n → f m ≡ f n [SMOD I ^ m • ⊤]\nL : R\nhL : ∀ (n : ℕ), f n ≡ L [SMOD I ^ n • ⊤]\n⊢ (1 + -x * y) * L = 1",
"usedConstants": [
"AddGroup.toSubtract... | rw [← sub_eq_zero, neg_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.KrullDimension.Zero | {
"line": 56,
"column": 45
} | {
"line": 56,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\n⊢ Order.krullDim (PrimeSpectrum R) ≤ ↑0 ↔ ringKrullDim R = 0",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"instCompleteLinearOrderENat",
"ChainCompletePartialOrder.instOfCompleteLattice",... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.KrullDimension.Zero | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 53
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommSemiring R\ntfae_1_to_3 : KrullDimLE 0 R ∧ IsLocalRing R → ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x\nH : ∀ (x : R), IsNilpotent x ↔ ¬IsUnit x\ne : nilradical R = ⊤\n⊢ False",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toM... | obtain ⟨n, hn⟩ := (Ideal.eq_top_iff_one _).mp e | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 168,
"column": 2
} | {
"line": 184,
"column": 60
} | [
{
"pp": "case mp\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\n⊢ J.IsMaximal → (under R J).IsMaximal ∧ y ∉ under R J",
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Eq.mpr",
... | · refine fun h => ⟨?_, fun hy =>
h.ne_top (Ideal.eq_top_of_isUnit_mem _ hy (map_units _ ⟨y, Submonoid.mem_powers _⟩))⟩
have hJ : J.IsPrime := IsMaximal.isPrime h
rw [isPrime_iff_isPrime_disjoint (Submonoid.powers y)] at hJ
have : y ∉ (J.under R).1 := Set.disjoint_left.1 hJ.right (Submonoid.mem_powers ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 226,
"column": 10
} | {
"line": 226,
"column": 25
} | [
{
"pp": "R : 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\nP' : Ideal S\nhP'✝ : P'.IsPrime\nhP' : (under R P').IsPrime\nhPM : Disjoint ↑(powers y) ↑(under R P')\nhP : (under R P').jacobson = under R P'\nx : R\nhx : x ∈ sInf... | Ideal.jacobson, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 236,
"column": 6
} | {
"line": 236,
"column": 21
} | [
{
"pp": "R : 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\nP' : Ideal S\nhP'✝ : P'.IsPrime\nhP' : (under R P').IsPrime\nhPM : Disjoint ↑(powers y) ↑(under R P')\nhP : (under R P').jacobson = under R P'\nthis : sInf {I | Ide... | Ideal.jacobson, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 313,
"column": 38
} | {
"line": 313,
"column": 61
} | [
{
"pp": "A : Type u_2\nB : Type u_3\nE : Type u_4\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDomain A\ninst✝⁴ : CommRing B\ninst✝³ : IsDomain B\ninst✝² : IsDiscreteValuationRing A\ninst✝¹ : EquivLike E A B\ninst✝ : RingEquivClass E A B\ne : E\na : ↥(maximalIdeal A)\nha : a ≠ 0\n⊢ e ↑a ∈ nonunits B",
"usedConstants": ... | map_mem_nonunits_iff e, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 354,
"column": 65
} | {
"line": 354,
"column": 73
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nϖ : R\nhϖ : Irreducible ϖ\nx y : R\nhy : y ∈ nonZeroDivisors R\nhx : (algebraMap R K) x / (algebraMap R K) y ≠ 0\n⊢ x ≠ 0",
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 354,
"column": 65
} | {
"line": 354,
"column": 73
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nϖ : R\nhϖ : Irreducible ϖ\nx y : R\nhy : y ∈ nonZeroDivisors R\nhx : (algebraMap R K) x / (algebraMap R K) y ≠ 0\n⊢ x ≠ 0",
"... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 354,
"column": 65
} | {
"line": 354,
"column": 73
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : IsDiscreteValuationRing R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nϖ : R\nhϖ : Irreducible ϖ\nx y : R\nhy : y ∈ nonZeroDivisors R\nhx : (algebraMap R K) x / (algebraMap R K) y ≠ 0\n⊢ x ≠ 0",
"... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 55
} | [
{
"pp": "case h\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\ns : Ideal R\nhs : s ≠ ⊥\nϖ : R\nhirr : Irreducible ϖ\ngen_ne_zero : generator s ≠ 0\nn : ℕ\nu : Rˣ\nhnu : generator s * ↑u = ϖ ^ n\n⊢ s = span {ϖ ^ n}",
"usedConstants": [
"Semiring.toModule",
... | have : span _ = _ := Ideal.span_singleton_generator s | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 416,
"column": 4
} | {
"line": 416,
"column": 64
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nr : R\nu : Rˣ\nϖ : R\nhϖ : Irreducible ϖ\nn : ℕ\nhr : r = ↑u * ϖ ^ n\n⊢ emultiplicity ϖ (ϖ ^ n) = ↑n",
"usedConstants": [
"Eq.mpr",
"ENat.instNatCast",
"congrArg",
"CommSemiring.toSemi... | emultiplicity_pow_self_of_prime (irreducible_iff_prime.1 hϖ) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 510,
"column": 4
} | {
"line": 510,
"column": 12
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nx : R\nu : Rˣ\n⊢ (addVal R) x = (addVal R) (x * ↑u)",
"usedConstants": [
"Units.val",
"instAddMonoidWithOneENat",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 328,
"column": 6
} | {
"line": 328,
"column": 31
} | [
{
"pp": "R : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nx y : R\nh : v (y - x) < v x\nthis : v (y - x + x) = max (v (y - x)) (v x)\n⊢ v y = v x",
"usedConstants": [
"AddGroupWithOne.toAddGroup",
"congrArg",
"PartialOrder.toPreo... | max_eq_right (le_of_lt h) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.Basic | {
"line": 374,
"column": 20
} | {
"line": 374,
"column": 28
} | [
{
"pp": "case pos\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁷ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝⁴ : LinearOrderedCommMonoidWithZero Γ'₀\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\nv : Valuation R Γ₀\nx✝ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 374,
"column": 20
} | {
"line": 374,
"column": 28
} | [
{
"pp": "case neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁷ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝⁴ : LinearOrderedCommMonoidWithZero Γ'₀\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\nv : Valuation R Γ₀\nx✝ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 374,
"column": 20
} | {
"line": 374,
"column": 28
} | [
{
"pp": "case pos\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁷ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝⁴ : LinearOrderedCommMonoidWithZero Γ'₀\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\nv : Valuation R Γ₀\nx✝ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 374,
"column": 20
} | {
"line": 374,
"column": 28
} | [
{
"pp": "case neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁷ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝⁴ : LinearOrderedCommMonoidWithZero Γ'₀\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\nv : Valuation R Γ₀\nx✝ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 374,
"column": 20
} | {
"line": 374,
"column": 28
} | [
{
"pp": "case pos\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁷ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝⁴ : LinearOrderedCommMonoidWithZero Γ'₀\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\nv : Valuation R Γ₀\nx✝ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 374,
"column": 20
} | {
"line": 374,
"column": 28
} | [
{
"pp": "case neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁷ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝⁴ : LinearOrderedCommMonoidWithZero Γ'₀\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\nv : Valuation R Γ₀\nx✝ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 374,
"column": 20
} | {
"line": 374,
"column": 28
} | [
{
"pp": "case pos\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁷ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝⁴ : LinearOrderedCommMonoidWithZero Γ'₀\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\nv : Valuation R Γ₀\nx✝ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 374,
"column": 20
} | {
"line": 374,
"column": 28
} | [
{
"pp": "case neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁷ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝⁴ : LinearOrderedCommMonoidWithZero Γ'₀\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\nv : Valuation R Γ₀\nx✝ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 388,
"column": 16
} | {
"line": 388,
"column": 24
} | [
{
"pp": "case pos\nR : Type u_3\nΓ₀ : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝² : Nontrivial R\ninst✝¹ : NoZeroDivisors R\ninst✝ : DecidablePred fun x ↦ x = 0\nx : R\nh✝ : x = 0\n⊢ 0 ≤ 1",
"usedConstants": [
"MulOne.toOne",
"LinearOrderedCommMonoidWithZero.to... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 388,
"column": 16
} | {
"line": 388,
"column": 24
} | [
{
"pp": "case neg\nR : Type u_3\nΓ₀ : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝² : Nontrivial R\ninst✝¹ : NoZeroDivisors R\ninst✝ : DecidablePred fun x ↦ x = 0\nx : R\nh✝ : ¬x = 0\n⊢ 1 ≤ 1",
"usedConstants": [
"MulOne.toOne",
"instReflLe",
"PartialOrder.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 393,
"column": 16
} | {
"line": 393,
"column": 24
} | [
{
"pp": "case pos\nR : Type u_3\nΓ₀ : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝³ : Nontrivial R\ninst✝² : NoZeroDivisors R\ninst✝¹ : DecidablePred fun x ↦ x = 0\ninst✝ : Nontrivial Γ₀\nx : R\nh✝ : x = 0\n⊢ 0 < 1 ↔ x = 0",
"usedConstants": [
"MulOne.toOne",
"Pr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 393,
"column": 16
} | {
"line": 393,
"column": 24
} | [
{
"pp": "case neg\nR : Type u_3\nΓ₀ : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝³ : Nontrivial R\ninst✝² : NoZeroDivisors R\ninst✝¹ : DecidablePred fun x ↦ x = 0\ninst✝ : Nontrivial Γ₀\nx : R\nh✝ : ¬x = 0\n⊢ 1 < 1 ↔ x = 0",
"usedConstants": [
"MulOne.toOne",
"F... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 37
} | {
"line": 469,
"column": 45
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v x ≠ 0\nH : ¬v (x + y) = 0\nhy : v y = 0\n⊢ v x ≤ v x",
"usedConstants": [
"Li... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 37
} | {
"line": 469,
"column": 45
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v x ≠ 0\nH : ¬v (x + y) = 0\nhy : v y = 0\n⊢ v x ≤ v x",
"usedConstants": [
"Li... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 37
} | {
"line": 469,
"column": 45
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v x ≠ 0\nH : ¬v (x + y) = 0\nhy : v y = 0\n⊢ v x ≤ v x",
"usedConstants": [
"Li... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 51
} | {
"line": 469,
"column": 59
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v x ≠ 0\nH : ¬v (x + y) = 0\nhy : v y = 0\n⊢ v y ≤ v x",
"usedConstants": [
"Gr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 51
} | {
"line": 469,
"column": 59
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v x ≠ 0\nH : ¬v (x + y) = 0\nhy : v y = 0\n⊢ v y ≤ v x",
"usedConstants": [
"Gr... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 51
} | {
"line": 469,
"column": 59
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v x ≠ 0\nH : ¬v (x + y) = 0\nhy : v y = 0\n⊢ v y ≤ v x",
"usedConstants": [
"Gr... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 37
} | {
"line": 469,
"column": 45
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v y ≠ 0\nH : ¬v (x + y) = 0\nhy : v x = 0\n⊢ v x ≤ v y",
"usedConstants": [
"Gr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 37
} | {
"line": 469,
"column": 45
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v y ≠ 0\nH : ¬v (x + y) = 0\nhy : v x = 0\n⊢ v x ≤ v y",
"usedConstants": [
"Gr... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 37
} | {
"line": 469,
"column": 45
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v y ≠ 0\nH : ¬v (x + y) = 0\nhy : v x = 0\n⊢ v x ≤ v y",
"usedConstants": [
"Gr... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 51
} | {
"line": 469,
"column": 59
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v y ≠ 0\nH : ¬v (x + y) = 0\nhy : v x = 0\n⊢ v y ≤ v y",
"usedConstants": [
"Li... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 51
} | {
"line": 469,
"column": 59
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v y ≠ 0\nH : ¬v (x + y) = 0\nhy : v x = 0\n⊢ v y ≤ v y",
"usedConstants": [
"Li... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.Basic | {
"line": 469,
"column": 51
} | {
"line": 469,
"column": 59
} | [
{
"pp": "K : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝² : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx✝ y✝ x y : R\nh : v y ≠ 0\nH : ¬v (x + y) = 0\nhy : v x = 0\n⊢ v y ≤ v y",
"usedConstants": [
"Li... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 498,
"column": 47
} | {
"line": 509,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R[X]\nhP : P.IsMaximal\ninst✝ : Nontrivial R\nhR : IsJacobsonRing R\nhP' : ∀ (x : R), C x ∈ P → x = 0\nP' : Ideal R := comap C P\npX : R[X]\nhpX : pX ∈ P\nhp0 : map (Ideal.Quotient.mk (comap C P)) pX ≠ 0\na : R ⧸ P' := (map (Ideal.Quotient.mk P') pX).leading... | by
rw [this]
refine RingHom.IsIntegral.trans (algebraMap (R ⧸ P') (Localization M))
(IsLocalization.map (Localization M') φ M.le_comap_map) ?_ ?_
· exact (algebraMap (R ⧸ P') (Localization M)).isIntegral_of_surjective
(IsField.localization_map_bijective hM ((Quotient.maximal_ideal_if... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 580,
"column": 2
} | {
"line": 580,
"column": 55
} | [
{
"pp": "case intro\nR : Type u_1\ninst✝² : CommRing R\nι : Type u_2\ninst✝¹ : _root_.Finite ι\ninst✝ : IsJacobsonRing R\nval✝ : Fintype ι\ne : ι ≃ Fin (Fintype.card ι) := Fintype.equivFin ι\n⊢ IsJacobsonRing (MvPolynomial ι R)",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMon... | rw [isJacobsonRing_iso (renameEquiv R e).toRingEquiv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Valuation.Basic | {
"line": 872,
"column": 55
} | {
"line": 872,
"column": 99
} | [
{
"pp": "case neg\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : LinearOrderedCommGroupWithZero Γ'₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ''₀\ninst✝ : Ring R\nv : Valuation R Γ₀\nw : Val... | ← Units.mk0_mul _ _ (mul_ne_zero hx20 hy10), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 39
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\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\nS : Type u_6\ninst✝¹ : CommRing S\ninst✝ : Algebra S R\n⊢ P.IsValuedIn S ↔ ∀ (i j : ι), P.pairing i j ∈ range... | simp only [isValuedIn_iff, mem_range] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 39
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\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\nS : Type u_6\ninst✝¹ : CommRing S\ninst✝ : Algebra S R\n⊢ P.IsValuedIn S ↔ ∀ (i j : ι), P.pairing i j ∈ range... | simp only [isValuedIn_iff, mem_range] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.IsValuedIn | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 39
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\nN : Type u_5\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\nS : Type u_6\ninst✝¹ : CommRing S\ninst✝ : Algebra S R\n⊢ P.IsValuedIn S ↔ ∀ (i j : ι), P.pairing i j ∈ range... | simp only [isValuedIn_iff, mem_range] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 110,
"column": 40
} | {
"line": 112,
"column": 28
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nS : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁹ : CommRing S\ninst✝⁸ : LinearOrder S\ninst✝⁷ : CommRing R\ninst✝⁶ : Algebra S 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\ninst✝¹ : P.IsValuedIn S\nB :... | by
obtain ⟨s, hs, hs'⟩ := B.exists_pos_eq i
simpa [← hs'] using hs.ne' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 124,
"column": 8
} | {
"line": 124,
"column": 11
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Fintype ι\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : CommRing R\ninst✝⁴ : Module R M\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsAnisotropic\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R N\ni j k : ι\nm n ... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.RootPositive | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 33
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nS : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝¹² : CommRing S\ninst✝¹¹ : LinearOrder S\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Algebra S 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\ninst✝⁴ : P.IsValuedIn S\n... | B.zero_lt_apply_root_root_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Finite.Nondegenerate | {
"line": 329,
"column": 35
} | {
"line": 329,
"column": 43
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : Fintype ι\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Field R\ninst✝² : Module R M\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsAnisotropic\nx✝ : M\n⊢ x✝ ∈ P.RootForm.orthogonal ⊤ → x✝ ∈ P.RootForm.orthogonal... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.RootSystem.Basic | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 12
} | [
{
"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\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\np₁ : M →ₗ[R] N →ₗ[R] R\nisPerfPair_toLinearMap... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 12
} | [
{
"pp": "case refine_1\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\ni j : ι\nhij : i ≠ j\nhLin : ¬LinearIndependent R ![P.root i, P.root j]\nh : P.root i = P.root ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 10
} | [
{
"pp": "ι : 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\ni j : ι\ninst✝ : P.IsReduced\nh : i ≠ j\nh' : P.root i ≠ -P.root j\nthis : ¬LinearIndependent R ![P.root i, P... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 158,
"column": 2
} | {
"line": 161,
"column": 16
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ni j : ι\ninst✝² : NeZero 2\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\ns t : R\nh₁ : s • P.root i + t • ... | have h₃ : t ≠ 0 := by
rcases eq_or_ne t 0 with rfl | ht
· exact False.elim <| h₂ <| (smul_eq_zero_iff_left <| P.ne_zero i).mp <| by simpa using h₁
· assumption | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 195,
"column": 48
} | {
"line": 195,
"column": 51
} | [
{
"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\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nthis : IsAdd... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Reduced | {
"line": 241,
"column": 8
} | {
"line": 241,
"column": 11
} | [
{
"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... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Submodule.Union | {
"line": 72,
"column": 59
} | {
"line": 72,
"column": 67
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\ninst✝² : Field K\ninst✝¹ : AddCommGroup M\ninst✝ : Module K M\np : ι → Submodule K M\nh₁ : ∀ (i : ι), p i ≠ ⊤\nj : ι\ns : Finset ι\nhj : j ∉ s\nhs : s.Nonempty\nh₂ : ↑s.card + 1 < ENat.card K\nhj' : ↑(p j) ∪ ⋃ i ∈ s, ↑(p i) = univ\nx : M\nhx : x ∈ p j\nhx₀ : x ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.QuadraticForm.Prod | {
"line": 308,
"column": 46
} | {
"line": 308,
"column": 59
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nP : Type u_7\nMᵢ : ι → Type u_8\ninst✝⁵ : CommSemiring R\ninst✝⁴ : (i : ι) → AddCommMonoid (Mᵢ i)\ninst✝³ : AddCommMonoid P\ninst✝² : (i : ι) → Module R (Mᵢ i)\ninst✝¹ : Module R P\ninst✝ : Fintype ι\nQ : (i : ι) → QuadraticMap R (Mᵢ i) P\nh : ∀ (x : (i : ι) → Mᵢ i), ∑ i, (Q... | Pi.zero_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 62,
"column": 17
} | {
"line": 62,
"column": 50
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\nx :... | ← lie_eq_smul_of_mem_rootSpace hx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 132,
"column": 2
} | {
"line": 137,
"column": 8
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\nn :... | have H₂ : ((1 + n + chainTopCoeff (-α) W) • α + chainTop (-α) W : H → K) =
(chainTopCoeff α β + 1) • α + β := by
simp only [Weight.coe_neg, ← Nat.cast_smul_eq_nsmul ℤ, Nat.cast_add, Nat.cast_one, coe_chainTop,
smul_neg, ← neg_smul, hW, ← add_assoc, ← add_smul, ← sub_eq_add_neg]
congr 2
ring | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 245,
"column": 50
} | {
"line": 248,
"column": 77
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : IsKilling K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα β : Weight K (↥H) L\nhα : α.IsNonZero\nβ' ... | by
have : (β' : H → K) = -n • (-α) + β := by rwa [neg_smul, smul_neg, neg_neg]
rw [chainBotCoeff, chainBotCoeff, ← Weight.coe_neg,
chainTopCoeff_of_eq_zsmul_add (-α) β hα.neg β' (-n) this, sub_neg_eq_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 262,
"column": 2
} | {
"line": 289,
"column": 59
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα : Weight K (↥H) L\ninst✝ : Nontrivial L\n... | symm
apply eq_of_le_of_not_lt
· rw [Nat.one_le_iff_ne_zero]
intro e
exact α.2 (by simpa [e, Weight.coe_zero] using
genWeightSpace_chainTopCoeff_add_one_nsmul_add α (0 : Weight K H L) hα)
obtain ⟨x, hx, x_ne0⟩ := (chainTop α (0 : Weight K H L)).exists_ne_zero
obtain ⟨h, e, f, isSl2, he, hf⟩ := exis... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.RootSystem | {
"line": 262,
"column": 2
} | {
"line": 289,
"column": 59
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁸ : Field K\ninst✝⁷ : CharZero K\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : IsKilling K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsTriangularizable K (↥H) L\nα : Weight K (↥H) L\ninst✝ : Nontrivial L\n... | symm
apply eq_of_le_of_not_lt
· rw [Nat.one_le_iff_ne_zero]
intro e
exact α.2 (by simpa [e, Weight.coe_zero] using
genWeightSpace_chainTopCoeff_add_one_nsmul_add α (0 : Weight K H L) hα)
obtain ⟨x, hx, x_ne0⟩ := (chainTop α (0 : Weight K H L)).exists_ne_zero
obtain ⟨h, e, f, isSl2, he, hf⟩ := exis... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Dual | {
"line": 148,
"column": 4
} | {
"line": 149,
"column": 29
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\ninst✝ : Invertible 2\nx : M × M\n⊢ ⅟2 • (2 * (Q x.1 - Q x.2)) = Q x.1 - Q x.2",
"usedConstants": [
"Module.End.instRing",
"NonAssocSemiring.toAddCommMonoid... | simp only [Module.End.smul_def, half_moduleEnd_apply_eq_half_smul, smul_eq_mul,
invOf_mul_cancel_left'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.WeylGroup | {
"line": 109,
"column": 4
} | {
"line": 113,
"column": 74
} | [
{
"pp": "case refine_1\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\n⊢ range P.reflection ⊆ ↑((Equiv.weightHom P).restrict P.weylGroup).range",
"usedConstants":... | rintro - ⟨i, rfl⟩
simp only [MonoidHom.restrict_range, Subgroup.coe_map, Equiv.weightHom_apply, mem_image,
SetLike.mem_coe]
use Equiv.reflection P i
exact ⟨reflection_mem_weylGroup P i, Equiv.reflection_weightEquiv P i⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.WeylGroup | {
"line": 109,
"column": 4
} | {
"line": 113,
"column": 74
} | [
{
"pp": "case refine_1\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\n⊢ range P.reflection ⊆ ↑((Equiv.weightHom P).restrict P.weylGroup).range",
"usedConstants":... | rintro - ⟨i, rfl⟩
simp only [MonoidHom.restrict_range, Subgroup.coe_map, Equiv.weightHom_apply, mem_image,
SetLike.mem_coe]
use Equiv.reflection P i
exact ⟨reflection_mem_weylGroup P i, Equiv.reflection_weightEquiv P i⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.WeylGroup | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 12
} | [
{
"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\nB : P.InvariantForm\ng : ↥P.weylGroup\nx y : M\n⊢ (B.form (g • x)) (g • y) = (B.form x) y",
"usedConstants... | revert x y | Lean.Elab.Tactic.evalRevert | Lean.Parser.Tactic.revert |
Mathlib.LinearAlgebra.RootSystem.Irreducible | {
"line": 141,
"column": 31
} | {
"line": 141,
"column": 55
} | [
{
"pp": "case refine_1.mul\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... | apply invtSubmodule.comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RepresentationTheory.Basic | {
"line": 358,
"column": 49
} | {
"line": 358,
"column": 57
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝⁵ : Semiring k\ninst✝⁴ : Group G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝¹ : S.Normal\ninst✝ : IsTrivial (MonoidHom.comp ρ S.subtype)\ny : G\nz : Gᵐᵒᵖ\nhz : z ∈ S.op\nw : V\n⊢ ↑(y * MulOpposite.unop z)... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RepresentationTheory.Basic | {
"line": 358,
"column": 49
} | {
"line": 358,
"column": 57
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝⁵ : Semiring k\ninst✝⁴ : Group G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝¹ : S.Normal\ninst✝ : IsTrivial (MonoidHom.comp ρ S.subtype)\ny : G\nz : Gᵐᵒᵖ\nhz : z ∈ S.op\nw : V\n⊢ ↑(y * MulOpposite.unop z)... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Basic | {
"line": 358,
"column": 49
} | {
"line": 358,
"column": 57
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝⁵ : Semiring k\ninst✝⁴ : Group G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝¹ : S.Normal\ninst✝ : IsTrivial (MonoidHom.comp ρ S.subtype)\ny : G\nz : Gᵐᵒᵖ\nhz : z ∈ S.op\nw : V\n⊢ ↑(y * MulOpposite.unop z)... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Basic | {
"line": 358,
"column": 49
} | {
"line": 358,
"column": 57
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\ninst✝⁵ : Semiring k\ninst✝⁴ : Group G\ninst✝³ : AddCommMonoid V\ninst✝² : Module k V\nρ : Representation k G V\nS : Subgroup G\ninst✝¹ : S.Normal\ninst✝ : IsTrivial (MonoidHom.comp ρ S.subtype)\ny : G\nz : Gᵐᵒᵖ\nhz : z ∈ S.op\nw : V\n⊢ ↑(y * MulOpposite.unop z)... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
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