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.Topology.Algebra.Valued.ValuedField | {
"line": 390,
"column": 4
} | {
"line": 395,
"column": 18
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nγ : Γ₀ˣ\nx : hat K\nγ₀' : ValueGroup₀ v := extension x\nhγ₀'_def : γ₀' = extension x\nγ₀ : Γ₀ := extensionValuation x\nhγ₀_def : γ₀ = extensionValuation x\nheq : γ₀ = embedding γ₀'\nthis : γ₀ ≠ 0... | rcases eq_or_ne γ₀ 0 with h | h
· simp only [(Valuation.zero_iff _).mp h, mem_setOf_eq, Valuation.map_zero, Units.zero_lt,
iff_true]
apply subset_closure
exact ⟨0, by simp only [mem_setOf_eq, Valuation.map_zero, Units.zero_lt, true_and]; rfl⟩
· exact this h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AlgebraicIndependent.AlgebraicClosure | {
"line": 44,
"column": 2
} | {
"line": 45,
"column": 95
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nA : Type u_4\nx : ι → A\nS : Type u_5\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra S A\ninst✝¹ : IsScalarTower R S A\ninst✝ : NoZeroDivisors S\nhx : AlgebraicIndependent R x\nalg : Algebra.IsAlge... | let _ : Algebra Rt St :=
(Rt.inclusion (T := St.restrictScalars R) <| adjoin_le <| by exact subset_adjoin).toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 474,
"column": 2
} | {
"line": 524,
"column": 51
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\n⊢ ValueGroup₀ v ≃* ValueGroup₀ extensionValuation",
"usedConstants": [
"WithZero.instNontrivial",
"dite_cond_eq_true",
"Units.val",
"Eq.mpr",
"GroupWithZero.toMo... | refine MulEquiv.ofBijective (valueGroup₀_hom_extensionValuation (hv := hv)) ⟨?_, ?_⟩
· intro a b hab
set x := (restrict₀_surjective hv.v a).choose with hx_def
set hx := (restrict₀_surjective hv.v a).choose_spec
set y := (restrict₀_surjective hv.v b).choose with hy_def
set hy := (restrict₀_surjective h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 474,
"column": 2
} | {
"line": 524,
"column": 51
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\n⊢ ValueGroup₀ v ≃* ValueGroup₀ extensionValuation",
"usedConstants": [
"WithZero.instNontrivial",
"dite_cond_eq_true",
"Units.val",
"Eq.mpr",
"GroupWithZero.toMo... | refine MulEquiv.ofBijective (valueGroup₀_hom_extensionValuation (hv := hv)) ⟨?_, ?_⟩
· intro a b hab
set x := (restrict₀_surjective hv.v a).choose with hx_def
set hx := (restrict₀_surjective hv.v a).choose_spec
set y := (restrict₀_surjective hv.v b).choose with hy_def
set hy := (restrict₀_surjective h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 49
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type w\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : Algebra E K\ninst✝ : IsScalarTower F E K\nh : IsPurelyInseparable F K\nq : ℕ\nh✝ : ExpChar F q\nthis : ExpChar E q\n⊢ IsPurelyInseparable E K",
"usedConstan... | rw [isPurelyInseparable_iff_pow_mem _ q] at h ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Trace.Basic | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\nL : Type u_5\ninst✝⁶ : Field L\nF : Type u_6\ninst✝⁵ : Field F\ninst✝⁴ : Algebra R L\ninst✝³ : Algebra L F\ninst✝² : Algebra R F\ninst✝¹ : IsScalarTower R L F\ninst✝ : FiniteDimensional L F\nx : F\nhx : IsIntegral R x\nhx' : IsIntegral L x\ny : AlgebraicClosure F\nhy ... | rw [mem_roots_map (minpoly.ne_zero hx')] at hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 340,
"column": 6
} | {
"line": 343,
"column": 61
} | [
{
"pp": "F✝ : Type u\nE✝ : Type v\ninst✝¹¹ : Field F✝\ninst✝¹⁰ : Field E✝\ninst✝⁹ : Algebra F✝ E✝\nq✝ : ℕ\ninst✝⁸ : ExpChar F✝ q✝\ninst✝⁷ : IsPurelyInseparable F✝ E✝\ninst✝⁶ : FiniteDimensional F✝ E✝\nE : Type v\ninst✝⁵ : Field E\nq d : ℕ\nIH :\n ∀ m < d,\n ∀ (F : Type v) [inst : Field F] [inst_1 : Algebra ... | rw [← hd, ← finrank_mul_finrank F F⟮x⟯, Nat.lt_mul_iff_one_lt_left finrank_pos, this]
by_contra! H
refine hx (finrank_adjoin_simple_eq_one_iff.mp (le_antisymm (this ▸ H) ?_))
exact Nat.one_le_iff_ne_zero.mpr Module.finrank_pos.ne' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 340,
"column": 6
} | {
"line": 343,
"column": 61
} | [
{
"pp": "F✝ : Type u\nE✝ : Type v\ninst✝¹¹ : Field F✝\ninst✝¹⁰ : Field E✝\ninst✝⁹ : Algebra F✝ E✝\nq✝ : ℕ\ninst✝⁸ : ExpChar F✝ q✝\ninst✝⁷ : IsPurelyInseparable F✝ E✝\ninst✝⁶ : FiniteDimensional F✝ E✝\nE : Type v\ninst✝⁵ : Field E\nq d : ℕ\nIH :\n ∀ m < d,\n ∀ (F : Type v) [inst : Field F] [inst_1 : Algebra ... | rw [← hd, ← finrank_mul_finrank F F⟮x⟯, Nat.lt_mul_iff_one_lt_left finrank_pos, this]
by_contra! H
refine hx (finrank_adjoin_simple_eq_one_iff.mp (le_antisymm (this ▸ H) ?_))
exact Nat.one_le_iff_ne_zero.mpr Module.finrank_pos.ne' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 612,
"column": 4
} | {
"line": 612,
"column": 85
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁸ : Field F\ninst✝⁷ : Field E\ninst✝⁶ : Algebra F E\nK : Type w\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\ninst✝³ : Algebra E K\ninst✝² : IsScalarTower F E K\ninst✝¹ : Algebra.IsAlgebraic F E\ninst✝ : Algebra.IsSeparable E K\nx : K\nx✝² : x ∈ ⊤\nS : IntermediateField F K := s... | obtain ⟨y, rfl⟩ := IsPurelyInseparable.surjective_algebraMap_of_isSeparable L K x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.FieldTheory.SeparableClosure | {
"line": 366,
"column": 56
} | {
"line": 367,
"column": 64
} | [
{
"pp": "F : Type u\ninst✝ : Field F\n⊢ finInsepDegree F F = 1",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Field.finInsepDegree",
"Cardinal.instOne",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
"Cardinal.toN... | by
rw [finInsepDegree_def', insepDegree_self, Cardinal.one_toNat] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Oka | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 100
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nP : Ideal R → Prop\nhP : IsOka P\nI : Ideal R\nhI : Maximal (fun x ↦ ¬P x) I\na b : R\nhab : a * b ∈ I\nha : a ∉ I\nhb : b ∉ I\n⊢ False",
"usedConstants": [
"Iff.mpr",
"Submodule",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"Semi... | have h₁ : P (I ⊔ span {a}) := of_not_not <| hI.not_prop_of_gt (Submodule.lt_sup_iff_notMem.2 ha) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.SeparableDegree | {
"line": 808,
"column": 2
} | {
"line": 808,
"column": 53
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁷ : Field F\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nK : Type w\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower F E K\ninst✝ : Algebra.IsSeparable F E\nx : K\nhsep : IsSeparable E x\nf : E[X] := minpoly E x\nhf : f = minpoly E x\n⊢ IsSe... | let E' : IntermediateField F E := adjoin F f.coeffs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 52
} | [
{
"pp": "case h\nΓ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nπ₁ π₂ : ↥v.valuationSubring\nh1 : v.IsUniformizer ↑π₁\nh2 : v.IsUniformizer ↑π₂\nhval : v ((↑π₁)⁻¹ * ↑π₂) = 1\np : ↥v.integer := ⟨(↑π₁)⁻¹ * ↑π₂, ⋯⟩\nhp : p = ⟨(↑π... | simp [hp, ← mul_assoc, mul_inv_cancel₀ h1.ne_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 304,
"column": 8
} | {
"line": 304,
"column": 16
} | [
{
"pp": "case hb\nΓ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nr : ↥v.valuationSubring\nhr : r ≠ 0\nπ : v.Uniformizer\nhr₀ : v ↑r ≠ 0\nvr : Γˣ := ⋯\nhvr_def : vr = Units.mk0 (v ↑r) hr₀\n⊢ ↑vr ∈ range ⇑v",
"usedConstants"... | hvr_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 55
} | [
{
"pp": "Γ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nr : ↥v.valuationSubring\nhr : r ≠ 0\nπ : v.Uniformizer\nhr₀ : v ↑r ≠ 0\nvr : Γˣ := Units.mk0 (v ↑r) hr₀\nhvr_def : vr = Units.mk0 (v ↑r) hr₀\nm : ℤ\nhm : Units.mk0 (v ↑π.... | rw [hn, zpow_natCast, pow_eq_zero_iff', not_and_or] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 148,
"column": 8
} | {
"line": 150,
"column": 86
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nx y : R\nhx : ¬x = 0\nhy : ¬y = 0\nhxy : ¬x + y = 0\n⊢ min ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors)\n ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y})... | set nmin :=
min ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors)
((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y})).factors) | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 154,
"column": 12
} | {
"line": 154,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nx y : R\nhx : ¬x = 0\nhy : ¬y = 0\nhxy : ¬x + y = 0\nnmin : ℕ :=\n min ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors)\n ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y}... | exact min_le_left _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 154,
"column": 12
} | {
"line": 154,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nx y : R\nhx : ¬x = 0\nhy : ¬y = 0\nhxy : ¬x + y = 0\nnmin : ℕ :=\n min ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors)\n ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y}... | exact min_le_left _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 154,
"column": 12
} | {
"line": 154,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nx y : R\nhx : ¬x = 0\nhy : ¬y = 0\nhxy : ¬x + y = 0\nnmin : ℕ :=\n min ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors)\n ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {y}... | exact min_le_left _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 349,
"column": 56
} | {
"line": 350,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nr : R\n⊢ (valuation K v) ↑r < 1 ↔ v.asIdeal ∣ Ideal.span {r}",
"usedConstants": [
"Algebra.cast",
"Eq.mpr",
"I... | by
rw [valuation_of_algebraMap]; exact v.intValuation_lt_one_iff_dvd r | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 265,
"column": 2
} | {
"line": 270,
"column": 32
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\na : R\nn : σ →₀ ℕ\n⊢ Commute φ ((monomial n) a) ↔ ∀ (m : σ →₀ ℕ), Commute ((coeff m) φ) a",
"usedConstants": [
"Eq.mpr",
"MvPowerSeries.coeff_mul_monomial",
"Nat.instCanonicallyOrderedAdd",
"Finsupp.instL... | rw [commute_iff_eq, MvPowerSeries.ext_iff]
refine ⟨fun h m => ?_, fun h m => ?_⟩
· have := h (m + n)
rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this
· rw [coeff_mul_monomial, coeff_monomial_mul]
split_ifs <;> [apply h; rfl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 265,
"column": 2
} | {
"line": 270,
"column": 32
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\na : R\nn : σ →₀ ℕ\n⊢ Commute φ ((monomial n) a) ↔ ∀ (m : σ →₀ ℕ), Commute ((coeff m) φ) a",
"usedConstants": [
"Eq.mpr",
"MvPowerSeries.coeff_mul_monomial",
"Nat.instCanonicallyOrderedAdd",
"Finsupp.instL... | rw [commute_iff_eq, MvPowerSeries.ext_iff]
refine ⟨fun h m => ?_, fun h m => ?_⟩
· have := h (m + n)
rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this
· rw [coeff_mul_monomial, coeff_monomial_mul]
split_ifs <;> [apply h; rfl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 482,
"column": 4
} | {
"line": 493,
"column": 25
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\ns t : σ\n⊢ X s = X t → s = t",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"Finsupp.single_eq_single_iff",
"NeZero.one",
"Semiring.toModule",
"Nat.ins... | classical
intro h
replace h := congr_arg (coeff (single s 1)) h
rw [coeff_X, if_pos rfl, coeff_X] at h
split_ifs at h with H
· rw [Finsupp.single_eq_single_iff] at H
rcases H with H | H
· exact H.1
· exfalso
exact one_ne_zero H.1
· exfalso
exact one_ne_zero h | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 482,
"column": 4
} | {
"line": 493,
"column": 25
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\ns t : σ\n⊢ X s = X t → s = t",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"Finsupp.single_eq_single_iff",
"NeZero.one",
"Semiring.toModule",
"Nat.ins... | classical
intro h
replace h := congr_arg (coeff (single s 1)) h
rw [coeff_X, if_pos rfl, coeff_X] at h
split_ifs at h with H
· rw [Finsupp.single_eq_single_iff] at H
rcases H with H | H
· exact H.1
· exfalso
exact one_ne_zero H.1
· exfalso
exact one_ne_zero h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 482,
"column": 4
} | {
"line": 493,
"column": 25
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\ns t : σ\n⊢ X s = X t → s = t",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"Finsupp.single_eq_single_iff",
"NeZero.one",
"Semiring.toModule",
"Nat.ins... | classical
intro h
replace h := congr_arg (coeff (single s 1)) h
rw [coeff_X, if_pos rfl, coeff_X] at h
split_ifs at h with H
· rw [Finsupp.single_eq_single_iff] at H
rcases H with H | H
· exact H.1
· exfalso
exact one_ne_zero H.1
· exfalso
exact one_ne_zero h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 585,
"column": 2
} | {
"line": 585,
"column": 35
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\nS : Type u_3\ninst✝² : DivisionSemiring R\ninst✝¹ : Semiring S\ninst✝ : Nontrivial S\nφ : MvPowerSeries σ R\nf : R →+* S\n⊢ (map f) φ = 0 ↔ φ = 0",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.RingTheory.MvPowerSeries.Basic.0.MvPowerSeries.map_eq_zero._simp... | simp only [MvPowerSeries.ext_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 612,
"column": 8
} | {
"line": 613,
"column": 66
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nh : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0\nm : σ →₀ ℕ\nH : m - single s n + single s n = m\n⊢ (coeff m) φ =\n (coeff (single s n, m - single s n).1) (X s ^ n) *\n (coeff (single s n, m - single s ... | rw [coeff_X_pow, if_pos rfl, one_mul]
simpa using congr_arg (fun m : σ →₀ ℕ => coeff m φ) H.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 612,
"column": 8
} | {
"line": 613,
"column": 66
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nh : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0\nm : σ →₀ ℕ\nH : m - single s n + single s n = m\n⊢ (coeff m) φ =\n (coeff (single s n, m - single s n).1) (X s ^ n) *\n (coeff (single s n, m - single s ... | rw [coeff_X_pow, if_pos rfl, one_mul]
simpa using congr_arg (fun m : σ →₀ ℕ => coeff m φ) H.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 213,
"column": 56
} | {
"line": 213,
"column": 89
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf : MvPowerSeries σ R\nn : ℕ\nh : weightedOrder w f = ↑n\nd : σ →₀ ℕ\nhd : (coeff d) f ≠ 0 ∧ ↑((weight w) d) = weightedOrder w f\ne : σ →₀ ℕ\nhe : (weight w) e < n\n⊢ ↑((weight w) e) < weightedOrder w f",
"usedConstants": [
"Finsupp.i... | by simp only [h, Nat.cast_lt, he] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 783,
"column": 19
} | {
"line": 783,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\n⊢ ∀ (x : ↥(adicCompletionIntegers K v)),\n Valued.v ((algebraMap (↥(adicCompletionIntegers K v)) (adicCompletion K v)) x) ≤ 1",
... | simp [mem_adicCompletionIntegers] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 783,
"column": 19
} | {
"line": 783,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\n⊢ ∀ (x : ↥(adicCompletionIntegers K v)),\n Valued.v ((algebraMap (↥(adicCompletionIntegers K v)) (adicCompletion K v)) x) ≤ 1",
... | simp [mem_adicCompletionIntegers] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 783,
"column": 19
} | {
"line": 783,
"column": 52
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\n⊢ ∀ (x : ↥(adicCompletionIntegers K v)),\n Valued.v ((algebraMap (↥(adicCompletionIntegers K v)) (adicCompletion K v)) x) ≤ 1",
... | simp [mem_adicCompletionIntegers] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 784,
"column": 25
} | {
"line": 784,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\n⊢ ∀ ⦃r : adicCompletion K v⦄,\n Valued.v r ≤ 1 → ∃ x, (algebraMap (↥(adicCompletionIntegers K v)) (adicCompletion K v)) x = r",
... | simp [mem_adicCompletionIntegers] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 784,
"column": 25
} | {
"line": 784,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\n⊢ ∀ ⦃r : adicCompletion K v⦄,\n Valued.v r ≤ 1 → ∃ x, (algebraMap (↥(adicCompletionIntegers K v)) (adicCompletion K v)) x = r",
... | simp [mem_adicCompletionIntegers] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 784,
"column": 25
} | {
"line": 784,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\n⊢ ∀ ⦃r : adicCompletion K v⦄,\n Valued.v r ≤ 1 → ∃ x, (algebraMap (↥(adicCompletionIntegers K v)) (adicCompletion K v)) x = r",
... | simp [mem_adicCompletionIntegers] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Trunc | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nn : ℕ\nr : R\nf : R⟦X⟧\n⊢ (trunc n) (C r * f) = Polynomial.C r * (trunc n) f",
"usedConstants": [
"Polynomial.C",
"PowerSeries.coeff_trunc",
"Semiring.toModule",
"HMul.hMul",
"Polynomial.ext",
"congrArg",
"LinearMap.instFun... | ext i; simp [coeff_trunc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Trunc | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nn : ℕ\nr : R\nf : R⟦X⟧\n⊢ (trunc n) (C r * f) = Polynomial.C r * (trunc n) f",
"usedConstants": [
"Polynomial.C",
"PowerSeries.coeff_trunc",
"Semiring.toModule",
"HMul.hMul",
"Polynomial.ext",
"congrArg",
"LinearMap.instFun... | ext i; simp [coeff_trunc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Trunc | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nn : ℕ\nf : R⟦X⟧\nr : R\n⊢ (trunc n) (f * C r) = (trunc n) f * Polynomial.C r",
"usedConstants": [
"PowerSeries.coeff_mul_C",
"Polynomial.C",
"PowerSeries.coeff_trunc",
"Semiring.toModule",
"HMul.hMul",
"Polynomial.ext",
"co... | ext i; simp [coeff_trunc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Trunc | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nn : ℕ\nf : R⟦X⟧\nr : R\n⊢ (trunc n) (f * C r) = (trunc n) f * Polynomial.C r",
"usedConstants": [
"PowerSeries.coeff_mul_C",
"Polynomial.C",
"PowerSeries.coeff_trunc",
"Semiring.toModule",
"HMul.hMul",
"Polynomial.ext",
"co... | ext i; simp [coeff_trunc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.LinearTopology | {
"line": 184,
"column": 2
} | {
"line": 229,
"column": 61
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nM : Type u_3\ninst✝⁸ : Ring R\ninst✝⁷ : Ring R'\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : Module R' M\ninst✝³ : SMulCommClass R R' M\ninst✝² : TopologicalSpace M\ninst✝¹ : IsLinearTopology R M\ninst✝ : IsLinearTopology R' M\n⊢ (𝓝 0).HasBasis (fun I ↦ ↑I ∈ 𝓝 ... | refine IsLinearTopology.hasBasis_submodule R |>.to_hasBasis (fun I hI ↦ ?_)
(fun I hI ↦ ⟨{I with smul_mem' := fun r x hx ↦ hI.2.1 r x hx}, hI.1, subset_rfl⟩)
-- `I` itself is a neighborhood of zero, so it contains some open sub-`R'`-module `J`.
rcases (hasBasis_submodule R').mem_iff.mp hI with ⟨J, hJ, J_sub_I⟩
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.LinearTopology | {
"line": 184,
"column": 2
} | {
"line": 229,
"column": 61
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nM : Type u_3\ninst✝⁸ : Ring R\ninst✝⁷ : Ring R'\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : Module R' M\ninst✝³ : SMulCommClass R R' M\ninst✝² : TopologicalSpace M\ninst✝¹ : IsLinearTopology R M\ninst✝ : IsLinearTopology R' M\n⊢ (𝓝 0).HasBasis (fun I ↦ ↑I ∈ 𝓝 ... | refine IsLinearTopology.hasBasis_submodule R |>.to_hasBasis (fun I hI ↦ ?_)
(fun I hI ↦ ⟨{I with smul_mem' := fun r x hx ↦ hI.2.1 r x hx}, hI.1, subset_rfl⟩)
-- `I` itself is a neighborhood of zero, so it contains some open sub-`R'`-module `J`.
rcases (hasBasis_submodule R').mem_iff.mp hI with ⟨J, hJ, J_sub_I⟩
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 258,
"column": 2
} | {
"line": 265,
"column": 5
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nn : ℕ\nh : ↑n < ψ.order\n⊢ (coeff n) (φ * ψ) = 0",
"usedConstants": [
"Eq.mpr",
"lt_of_le_of_lt",
"PowerSeries.coeff_of_lt_order",
"Semiring.toModule",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"HMul.hMul"... | suffices coeff n (φ * ψ) = ∑ p ∈ antidiagonal n, 0 by rw [this, Finset.sum_const_zero]
rw [coeff_mul]
apply Finset.sum_congr rfl
intro x hx
refine mul_eq_zero_of_right (coeff x.fst φ) (coeff_of_lt_order x.snd (lt_of_le_of_lt ?_ h))
rw [mem_antidiagonal] at hx
norm_cast
lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 258,
"column": 2
} | {
"line": 265,
"column": 5
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nn : ℕ\nh : ↑n < ψ.order\n⊢ (coeff n) (φ * ψ) = 0",
"usedConstants": [
"Eq.mpr",
"lt_of_le_of_lt",
"PowerSeries.coeff_of_lt_order",
"Semiring.toModule",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"HMul.hMul"... | suffices coeff n (φ * ψ) = ∑ p ∈ antidiagonal n, 0 by rw [this, Finset.sum_const_zero]
rw [coeff_mul]
apply Finset.sum_congr rfl
intro x hx
refine mul_eq_zero_of_right (coeff x.fst φ) (coeff_of_lt_order x.snd (lt_of_le_of_lt ?_ h))
rw [mem_antidiagonal] at hx
norm_cast
lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 338,
"column": 4
} | {
"line": 339,
"column": 27
} | [
{
"pp": "case inr.coe\nR : Type u_2\ninst✝ : Semiring R\nφ : R⟦X⟧\nhφ : φ ≠ 0\nn : ℕ\nho : φ.order = ↑n\nhn : φ.order.toNat = n\n⊢ ↑φ.order.toNat ≤ emultiplicity X φ",
"usedConstants": [
"le_emultiplicity_of_pow_dvd",
"PowerSeries.X",
"MvPowerSeries.instSemiring",
"Unit",
"Powe... | · apply le_emultiplicity_of_pow_dvd
apply X_pow_order_dvd | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 346,
"column": 22
} | {
"line": 346,
"column": 36
} | [
{
"pp": "case h\nR : Type u_2\ninst✝ : Semiring R\nφ : R⟦X⟧\nhφ : φ ≠ 0\nn : ℕ\nho : φ.order = ↑n\nhn : φ.order.toNat = n\nψ : R⟦X⟧\nH : φ = X ^ (φ.order.toNat + 1) * ψ\nthis : (coeff n) φ = (coeff n) (ψ * X ^ (φ.order.toNat + 1))\n⊢ ↑n < ((monomial (φ.order.toNat + 1)) 1).order",
"usedConstants": [
"... | order_monomial | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.LinearTopology | {
"line": 151,
"column": 10
} | {
"line": 151,
"column": 11
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsLinearTopology R R\nf : MvPowerSeries σ R\nhf : IsTopologicallyNilpotent (constantCoeff f)\nd : σ →₀ ℕ\n⊢ ∀ (i : Ideal R), ↑i ∈ 𝓝 0 → ∀ᶠ (x : ℕ) in atTop, (coeff d) (f ^ x) ∈ ↑i",
"usedConstants": [
"Idea... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.MvPowerSeries.LinearTopology | {
"line": 168,
"column": 2
} | {
"line": 171,
"column": 30
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsLinearTopology R R\nf : MvPowerSeries σ R\n⊢ Tendsto (fun n ↦ f ^ n) atTop (𝓝 0) ↔ IsTopologicallyNilpotent (constantCoeff f)",
"usedConstants": [
"RingHom.instRingHomClass",
"MvPowerSeries.instZero... | refine ⟨fun H ↦ ?_, isTopologicallyNilpotent_of_constantCoeff⟩
replace H : Tendsto (fun n ↦ constantCoeff (f ^ n)) atTop (nhds 0) :=
continuous_constantCoeff R |>.tendsto' 0 0 constantCoeff_zero |>.comp H
simpa only [map_pow] using H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.LinearTopology | {
"line": 168,
"column": 2
} | {
"line": 171,
"column": 30
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsLinearTopology R R\nf : MvPowerSeries σ R\n⊢ Tendsto (fun n ↦ f ^ n) atTop (𝓝 0) ↔ IsTopologicallyNilpotent (constantCoeff f)",
"usedConstants": [
"RingHom.instRingHomClass",
"MvPowerSeries.instZero... | refine ⟨fun H ↦ ?_, isTopologicallyNilpotent_of_constantCoeff⟩
replace H : Tendsto (fun n ↦ constantCoeff (f ^ n)) atTop (nhds 0) :=
continuous_constantCoeff R |>.tendsto' 0 0 constantCoeff_zero |>.comp H
simpa only [map_pow] using H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Evaluation | {
"line": 168,
"column": 2
} | {
"line": 171,
"column": 35
} | [
{
"pp": "case hf\nσ : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : UniformSpace R\nS : Type u_3\ninst✝⁴ : CommRing S\ninst✝³ : UniformSpace S\nφ : R →+* S\na : σ → S\ninst✝² : IsUniformAddGroup R\ninst✝¹ : IsUniformAddGroup S\ninst✝ : IsLinearTopology S S\nhφ : Continuous[inst✝⁵.toTopologicalSpace, ins... | have hn_ne : ∀ s, Set.Nonempty {n : ℕ | (a s) ^ n.succ ∈ I} := fun s ↦ by
rcases ha.hpow s |>.eventually_mem hI |>.exists_forall_of_atTop with ⟨n, hn⟩
use n
simpa using hn n.succ n.le_succ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.SetTheory.Cardinal.Continuum | {
"line": 46,
"column": 6
} | {
"line": 46,
"column": 30
} | [
{
"pp": "c : Cardinal.{u}\n⊢ 𝔠 ≤ lift.{v, u} c ↔ 𝔠 ≤ c",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Cardinal.lift",
"id",
"LE.le",
"Cardinal.instLE",
"Cardinal.continuum",
"Iff",
"Eq.symm",
"Eq",
"Cardinal.lift_continuum"
... | ← lift_continuum.{v, u}, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Continuum | {
"line": 50,
"column": 6
} | {
"line": 50,
"column": 30
} | [
{
"pp": "c : Cardinal.{u}\n⊢ lift.{v, u} c ≤ 𝔠 ↔ c ≤ 𝔠",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Cardinal.lift",
"id",
"LE.le",
"Cardinal.instLE",
"Cardinal.continuum",
"Iff",
"Eq.symm",
"Eq",
"Cardinal.lift_continuum"
... | ← lift_continuum.{v, u}, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Continuum | {
"line": 54,
"column": 6
} | {
"line": 54,
"column": 30
} | [
{
"pp": "c : Cardinal.{u}\n⊢ 𝔠 < lift.{v, u} c ↔ 𝔠 < c",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Cardinal.lift",
"id",
"Cardinal.continuum",
"Cardinal.partialOrder",
"Iff",
"LT.lt",
... | ← lift_continuum.{v, u}, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Continuum | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 30
} | [
{
"pp": "c : Cardinal.{u}\n⊢ lift.{v, u} c < 𝔠 ↔ c < 𝔠",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Cardinal.lift",
"id",
"Cardinal.continuum",
"Cardinal.partialOrder",
"Iff",
"LT.lt",
... | ← lift_continuum.{v, u}, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 516,
"column": 4
} | {
"line": 518,
"column": 23
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝²¹ : CommSemiring A\nR✝ : Type u_3\ninst✝²⁰ : CommRing R✝\ninst✝¹⁹ : Algebra A R✝\nτ : Type u_4\nS : Type u_5\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra A S\ninst✝¹⁶ : Algebra R✝ S\ninst✝¹⁵ : IsScalarTower A R✝ S\na✝¹ a✝ : σ → MvPowerSeries τ S\nT✝ : Type u_6\ninst✝¹⁴ : C... | intros
ext
exact mul_add _ _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 516,
"column": 4
} | {
"line": 518,
"column": 23
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝²¹ : CommSemiring A\nR✝ : Type u_3\ninst✝²⁰ : CommRing R✝\ninst✝¹⁹ : Algebra A R✝\nτ : Type u_4\nS : Type u_5\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra A S\ninst✝¹⁶ : Algebra R✝ S\ninst✝¹⁵ : IsScalarTower A R✝ S\na✝¹ a✝ : σ → MvPowerSeries τ S\nT✝ : Type u_6\ninst✝¹⁴ : C... | intros
ext
exact mul_add _ _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 251,
"column": 2
} | {
"line": 253,
"column": 61
} | [
{
"pp": "case h\nι : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nthis : UniformSpace R := ⋯\n⊢ IsClosed (Set.range DFunLike.coe)",
"usedConstants": [
"Set.ext",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"CommSemiring.toSemiring",
"setOf"... | have : Set.range ((↑) : ArithmeticFunction R → (ℕ → R)) = {f | f 0 = 0} := by
ext f
exact ⟨by rintro ⟨f, rfl⟩; simp, fun hf ↦ ⟨⟨f, hf⟩, rfl⟩⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LSeries.Basic | {
"line": 188,
"column": 8
} | {
"line": 188,
"column": 35
} | [
{
"pp": "f g : ℕ → ℂ\ns : ℂ\nh : f =ᶠ[cofinite] g\nhf : LSeriesSummable f s\n⊢ term f s =ᶠ[cofinite] term g s",
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Membership.mem",
"Exists",
"Eq.mp",
"Filter.EventuallyEq",
"Set.EqOn",
"And",
"Nat",
... | eventuallyEq_iff_exists_mem | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.Basic | {
"line": 358,
"column": 4
} | {
"line": 358,
"column": 57
} | [
{
"pp": "case inl\nf : ℕ → ℂ\nx : ℝ\ns : ℂ\nhs : x < s.re\nC : ℝ\nhC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)\nhC₀ : 0 ≤ C\nhsum : Summable fun n ↦ ‖↑C / ↑n ^ (s + (1 - ↑x))‖\n⊢ ‖term f s 0‖ ≤ ‖↑C / ↑0 ^ (s + (1 - ↑x))‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSem... | simpa only [term_zero, norm_zero] using norm_nonneg _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.LSeries.Basic | {
"line": 358,
"column": 4
} | {
"line": 358,
"column": 57
} | [
{
"pp": "case inl\nf : ℕ → ℂ\nx : ℝ\ns : ℂ\nhs : x < s.re\nC : ℝ\nhC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)\nhC₀ : 0 ≤ C\nhsum : Summable fun n ↦ ‖↑C / ↑n ^ (s + (1 - ↑x))‖\n⊢ ‖term f s 0‖ ≤ ‖↑C / ↑0 ^ (s + (1 - ↑x))‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSem... | simpa only [term_zero, norm_zero] using norm_nonneg _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.Basic | {
"line": 358,
"column": 4
} | {
"line": 358,
"column": 57
} | [
{
"pp": "case inl\nf : ℕ → ℂ\nx : ℝ\ns : ℂ\nhs : x < s.re\nC : ℝ\nhC : ∀ (n : ℕ), n ≠ 0 → ‖f n‖ ≤ C * ↑n ^ (x - 1)\nhC₀ : 0 ≤ C\nhsum : Summable fun n ↦ ‖↑C / ↑n ^ (s + (1 - ↑x))‖\n⊢ ‖term f s 0‖ ≤ ‖↑C / ↑0 ^ (s + (1 - ↑x))‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSem... | simpa only [term_zero, norm_zero] using norm_nonneg _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.EGauge | {
"line": 160,
"column": 84
} | {
"line": 173,
"column": 82
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\nE : Type u_2\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\nx : E\ninst✝ : (𝓝[≠] 0).NeBot\nr : ℝ≥0∞\nhs₀ : 0 ∈ s\nh : ∀ (c : 𝕜), c ≠ 0 → x ∈ c • s → r ≤ ‖c‖ₑ\n⊢ r ≤ egauge 𝕜 s x",
"usedConstants": [
"Eq.mpr",
"ne_or_eq",
... | by
rw [le_egauge_iff]
intro c hc
rcases ne_or_eq c 0 with hc₀ | rfl
· exact h c hc₀ hc
obtain rfl : x = 0 := by
grw [zero_smul_set_subset, Set.mem_zero] at hc
exact hc
apply le_of_forall_gt
intro b hb
rcases Filter.nonempty_of_mem <|
inter_mem_nhdsWithin {(0 : 𝕜)}ᶜ (Metric.eball_mem_nhds 0 ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.EGauge | {
"line": 276,
"column": 8
} | {
"line": 276,
"column": 85
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\nI : Set ι\nhI : I.Finite\nU : (i : ι) → Set (E i)\nhU : ∀ i ∈ I, Balanced 𝕜 (U i)\nx : (i : ι) → E i\nhI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨... | have : (𝓝[≠] (0 : 𝕜)).NeBot := (hI₀.resolve_left H.2).resolve_left (by simpa) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 386,
"column": 2
} | {
"line": 386,
"column": 56
} | [
{
"pp": "case h\nR : Type u_2\ninst✝ : CommRing R\nr : R\nf : R⟦X⟧\nn : ℕ\n⊢ (coeff n) ((rescale r) f) = (coeff n) ((MvPowerSeries.rescale fun x ↦ r) f)",
"usedConstants": [
"Eq.mpr",
"Unit.unit",
"PowerSeries.coeff_rescale",
"Nat.instMulZeroClass",
"MvPowerSeries.rescale",
... | rw [coeff_rescale, coeff, MvPowerSeries.coeff_rescale] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 486,
"column": 12
} | {
"line": 486,
"column": 77
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommRing R\nP : R⟦X⟧\nhP : constantCoeff P = 0\ninst✝ : Invertible ((coeff 1) P)\nn : ℕ\nB : R⟦X⟧\nhB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B\nhB' : constantCoeff B = 0\nk : R\nhk : ⅟((coeff 1) P) * (coeff (n + 1 + 1)) (subst B P) = k\n⊢ (∑ᶠ (i : ℕ), (coeff i) P * (coeff (n + 1 ... | simp only [↓reduceIte, ← hk, mul_invOf_cancel_left', sub_eq_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 604,
"column": 2
} | {
"line": 604,
"column": 44
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\nf : R⟦X⟧\ne : Fin 2 →₀ ℕ\n⊢ MvPolynomial.coeff e ((MvPolynomial.X 0 + MvPolynomial.X 1) ^ (single () (e 0 + e 1)) PUnit.unit) *\n (MvPowerSeries.coeff (single () (e 0 + e 1))) f =\n ↑((e 0 + e 1).choose (e 0)) * (coeff (e 0 + e 1)) f",
"usedConstants": [
... | · simp [MvPolynomial.coeff_add_pow, coeff] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.FDeriv.Congr | {
"line": 48,
"column": 6
} | {
"line": 48,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\nF : Type u_3\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf : E → F\nf' : E →L[𝕜] F\nx : E\ns t : Set E\nh : s =ᶠ[𝓝[≠] x] t\n⊢ H... | suffices 𝓝[s \ {x}] x = 𝓝[t \ {x}] x by simp only [HasFDerivWithinAt, this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Analytic.ChangeOrigin | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 71
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nx_mem_b... | refine .of_nnnorm_bounded (p.changeOriginSeries_summable_aux₁ h) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Analytic.CPolynomialDef | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 42
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nr : ℝ≥0∞\nhf : ∀ y ∈ Metric.eball x r, f y = 0\nr_pos ... | refine ⟨⟨?_, r_pos, ?_⟩, fun n _ ↦ hp n⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Module.Multilinear.Curry | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 9
} | [
{
"pp": "𝕜 : Type u\nn : ℕ\nEi : Fin n.succ → Type wEi\nG : Type wG\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)\ninst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : Ei 0 →L[𝕜] ContinuousMultilinearMap �... | ext m x | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 211,
"column": 6
} | {
"line": 212,
"column": 47
} | [
{
"pp": "case neg.hs.refine_1.refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing E\ninst✝ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\nr : ℝ≥0\nhr : r ≠ 0\nhc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)\nr' : ℝ≥0\nhr' : r' * r < 1\nhrz : ¬r' = 0\n⊢ Tendsto (fun n ↦ ‖‖‖... | simp_rw [norm_norm]
exact tendsto_succ_norm_div_norm c hrz hc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 211,
"column": 6
} | {
"line": 212,
"column": 47
} | [
{
"pp": "case neg.hs.refine_1.refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing E\ninst✝ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\nr : ℝ≥0\nhr : r ≠ 0\nhc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)\nr' : ℝ≥0\nhr' : r' * r < 1\nhrz : ¬r' = 0\n⊢ Tendsto (fun n ↦ ‖‖‖... | simp_rw [norm_norm]
exact tendsto_succ_norm_div_norm c hrz hc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Multilinear.Curry | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 9
} | [
{
"pp": "𝕜 : Type u\nn : ℕ\nEi : Fin n.succ → Type wEi\nG : Type wG\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)\ninst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ContinuousMultilinearMap 𝕜 (fun i ↦ E... | ext m x | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Analysis.Normed.Module.Multilinear.Curry | {
"line": 235,
"column": 2
} | {
"line": 237,
"column": 25
} | [
{
"pp": "𝕜 : Type u\nn : ℕ\nEi : Fin n.succ → Type wEi\nG : Type wG\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)\ninst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ContinuousMultilinearMap 𝕜 (fun i ↦ E... | ext m x
rw [ContinuousMultilinearMap.curryRight_apply, ContinuousMultilinearMap.uncurryRight_apply,
snoc_last, init_snoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Multilinear.Curry | {
"line": 235,
"column": 2
} | {
"line": 237,
"column": 25
} | [
{
"pp": "𝕜 : Type u\nn : ℕ\nEi : Fin n.succ → Type wEi\nG : Type wG\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)\ninst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ContinuousMultilinearMap 𝕜 (fun i ↦ E... | ext m x
rw [ContinuousMultilinearMap.curryRight_apply, ContinuousMultilinearMap.uncurryRight_apply,
snoc_last, init_snoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Composition | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝¹⁵ : CommRing 𝕜\ninst✝¹⁴ : AddCommGroup E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : AddCommGroup G\ninst✝¹¹ : Module 𝕜 E\ninst✝¹⁰ : Module 𝕜 F\ninst✝⁹ : Module 𝕜 G\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : Topologica... | refine q.congr (by simp) fun i hi1 hi2 => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Analytic.Composition | {
"line": 279,
"column": 41
} | {
"line": 279,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝¹⁵ : CommRing 𝕜\ninst✝¹⁴ : AddCommGroup E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : AddCommGroup G\ninst✝¹¹ : Module 𝕜 E\ninst✝¹⁰ : Module 𝕜 F\ninst✝⁹ : Module 𝕜 G\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : Topologica... | ext n; simp [FormalMultilinearSeries.comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Composition | {
"line": 279,
"column": 41
} | {
"line": 279,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝¹⁵ : CommRing 𝕜\ninst✝¹⁴ : AddCommGroup E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : AddCommGroup G\ninst✝¹¹ : Module 𝕜 E\ninst✝¹⁰ : Module 𝕜 F\ninst✝⁹ : Module 𝕜 G\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : Topologica... | ext n; simp [FormalMultilinearSeries.comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Composition | {
"line": 768,
"column": 8
} | {
"line": 768,
"column": 49
} | [
{
"pp": "case hy\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ng : F → G\nf : E → F\nq : F... | simpa [edist_eq_enorm_sub] using fy_mem.2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Analytic.Composition | {
"line": 768,
"column": 8
} | {
"line": 768,
"column": 49
} | [
{
"pp": "case hy\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ng : F → G\nf : E → F\nq : F... | simpa [edist_eq_enorm_sub] using fy_mem.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Composition | {
"line": 768,
"column": 8
} | {
"line": 768,
"column": 49
} | [
{
"pp": "case hy\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ng : F → G\nf : E → F\nq : F... | simpa [edist_eq_enorm_sub] using fy_mem.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Basic | {
"line": 904,
"column": 2
} | {
"line": 905,
"column": 100
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nr' : ℝ≥0\nhf : HasFPowerSeriesWit... | obtain ⟨a, ha, C, -, hp⟩ : ∃ a ∈ Ioo (0 : ℝ) 1, ∃ C > 0, ∀ y ∈ Metric.ball (0 : E) r', ∀ n,
x + y ∈ insert x s → ‖f (x + y) - p.partialSum n y‖ ≤ C * a ^ n := hf.uniform_geometric_approx h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Analytic.Inverse | {
"line": 413,
"column": 8
} | {
"line": 415,
"column": 59
} | [
{
"pp": "n : ℕ\np : ℕ → ℝ\nhp : ∀ (k : ℕ), 0 ≤ p k\nr a : ℝ\nhr : 0 ≤ r\nha : 0 ≤ a\n⊢ ∑ e ∈ (Ico 2 (n + 1)).sigma fun n_1 ↦ Fintype.piFinset fun _i ↦ Ico 1 n, ∏ j, r * (a ^ e.snd j * p (e.snd j)) =\n ∑ j ∈ Ico 2 (n + 1), r ^ j * (∑ k ∈ Ico 1 n, a ^ k * p k) ^ j",
"usedConstants": [
"Eq.mpr",
... | ← sum_sigma' (Ico 2 (n + 1))
(fun k : ℕ => (Fintype.piFinset fun _ : Fin k => Ico 1 n : Finset (Fin k → ℕ)))
(fun n e => ∏ j : Fin n, r * (a ^ e j * p (e j))) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Within | {
"line": 190,
"column": 6
} | {
"line": 191,
"column": 43
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\ns : Set E\nx : E\ng : E → F\nhf : f =ᶠ[𝓝[insert x s] x] g\nh... | · have : y ∈ u ∩ insert x s := ⟨h'y, hy⟩
simpa [g', h'y, this] using hu this | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Analytic.Inverse | {
"line": 689,
"column": 2
} | {
"line": 692,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE✝ : Type u_2\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedSpace 𝕜 E✝\nF✝ : Type u_3\ninst✝¹ : NormedAddCommGroup F✝\ninst✝ : NormedSpace 𝕜 F✝\nf : OpenPartialHomeomorph E✝ F✝\na : E✝\ni : E✝ ≃L[𝕜] F✝\nh0 : a ∈ f.source\np : FormalMultilinearSer... | have : y + f a ∈ Metric.eball (f a) r := by
simp only [Metric.mem_eball, edist_eq_enorm_sub, sub_zero, lt_min_iff,
add_sub_cancel_right] at hy ⊢
exact hy.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 132,
"column": 8
} | {
"line": 132,
"column": 58
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\niso : E ≃L[𝕜] F\nf : G → ... | fderivWithin_zero_of_not_differentiableWithinAt h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 223,
"column": 8
} | {
"line": 223,
"column": 58
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\niso : E ≃L[𝕜] F\nf : F → ... | fderivWithin_zero_of_not_differentiableWithinAt h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 40
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nf' : E →L[ℝ] F\nx : E\nL : Filter E\n⊢ Tendsto (fun x' ↦ ‖x' - x‖⁻¹ • (f x' - f x - f' (x' - x))) L (𝓝 0) ↔\n Tendsto (fun x' ↦ ‖x' - x‖⁻¹ * ‖f x' ... | rw [tendsto_iff_norm_sub_tendsto_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 585,
"column": 8
} | {
"line": 585,
"column": 58
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nx : E\ns : Set E\nhxs : UniqueDiffWithinAt 𝕜 s x\nh : ¬DifferentiableWithinAt 𝕜 f s x... | fderivWithin_zero_of_not_differentiableWithinAt h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 729,
"column": 2
} | {
"line": 729,
"column": 66
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nm n : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpTo n f p\nhmn : m ≤ n\n⊢ Has... | rw [← hasFTaylorSeriesUpToOn_univ_iff] at h ⊢; exact h.of_le hmn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 729,
"column": 2
} | {
"line": 729,
"column": 66
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nm n : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpTo n f p\nhmn : m ≤ n\n⊢ Has... | rw [← hasFTaylorSeriesUpToOn_univ_iff] at h ⊢; exact h.of_le hmn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 988,
"column": 6
} | {
"line": 988,
"column": 24
} | [
{
"pp": "case succ\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\ns : Set 𝕜\nn : ℕ\nih : ∀ (a : 𝕜), iteratedFDerivWithin 𝕜 n (fun x ↦ f (-x)) s a = (-1) ^ n • iteratedFDerivWithin 𝕜 n f (-s) (-a)\na : 𝕜\nih' : iteratedFDe... | ← mul_smul _ (-1), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Alternating.Basic | {
"line": 522,
"column": 11
} | {
"line": 522,
"column": 24
} | [
{
"pp": "case h\n𝕜 : Type u\nn : ℕ\nE : Type wE\nF : Type wF\nG : Type wG\nι : Type v\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : SeminormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ni... | dg.le_opNorm, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 832,
"column": 4
} | {
"line": 834,
"column": 7
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nx : E\nr : ℝ≥0∞\ny : E\nn : ℕ\nih :\n ∀ {F : Type (max u v)} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace 𝕜 F] [CompleteSpace F]\n {p : FormalMultilinearSerie... | rw [factorial_succ, mul_comm, mul_smul, ← derivSeries_apply_diag,
← ContinuousLinearMap.smul_apply, ih h.fderiv, iteratedFDeriv_succ_apply_right]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 832,
"column": 4
} | {
"line": 834,
"column": 7
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nx : E\nr : ℝ≥0∞\ny : E\nn : ℕ\nih :\n ∀ {F : Type (max u v)} [inst : NormedAddCommGroup F] [inst_1 : NormedSpace 𝕜 F] [CompleteSpace F]\n {p : FormalMultilinearSerie... | rw [factorial_succ, mul_comm, mul_smul, ← derivSeries_apply_diag,
← ContinuousLinearMap.smul_apply, ih h.fderiv, iteratedFDeriv_succ_apply_right]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 73,
"column": 6
} | {
"line": 74,
"column": 42
} | [
{
"pp": "case h.refine_1\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ m <... | have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 73,
"column": 6
} | {
"line": 74,
"column": 42
} | [
{
"pp": "case h.refine_1\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nh : HasFPowerSeriesAt 0 p x\nn k : ℕ\nhk : ∀ m <... | have := Finset.mem_range_succ_iff.mp hb
simp only [hk b (this.lt_of_ne hnb)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 36
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\ns : Set k\nhf : MonotoneOn f s\nhx : x ∈ s\nhy : y ∈ s... | rcases le_total x y with hxy | hxy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 34
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nc d : 𝕜 → 𝔸\nc' d' : 𝔸\nhc : HasDerivWithinAt c c' univ x\nhd : HasDerivWithinAt d d' univ x\n⊢ HasDerivWithinAt (c * d) (c' * d x + c x * d') univ x",
"usedConstants": [... | exact HasDerivWithinAt.mul hc hd | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.LogDeriv | {
"line": 123,
"column": 4
} | {
"line": 138,
"column": 68
} | [
{
"pp": "case inr.mp\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NontriviallyNormedField 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : IsRCLikeNormedField 𝕜\nf g : 𝕜 → 𝕜'\ns : Set 𝕜\nhf : DifferentiableOn 𝕜 f s\nhg : DifferentiableOn 𝕜 g s\nhs2 : IsOpen[PseudoMetricSpace.to... | · refine fun h ↦ ⟨f t * (g t)⁻¹, by grind, fun y hy ↦ ?_⟩
have hderiv : s.EqOn (deriv (f * g⁻¹)) (deriv f * g⁻¹ - f * deriv g / g ^ 2) := by
intro z hz
rw [deriv_mul (hf.differentiableAt (hs2.mem_nhds hz)) ((hg.differentiableAt
(hs2.mem_nhds hz)).inv (hgn z hz))]
simp only [Pi.in... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 97,
"column": 2
} | {
"line": 107,
"column": 35
} | [
{
"pp": "case inr\nf f' : ℝ → ℝ\na b : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r\nB B' : ℝ → ℝ\nha : f a ≤ B a\nhB : ContinuousOn B (Icc a b)\nhB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x\nbound : ∀ x ∈ Ico a b, f x = B x → f'... | · rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩
specialize hf' x xab r hfr
have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z :=
(hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici
(Ioi_mem_nhds hrB)
obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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