module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Normed.Unbundled.FiniteExtension | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 51
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : NormedField K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nhfd : FiniteDimensional K L\nhna : IsNonarchimedean norm\nh1 : LinearIndepOn K id {1}\nι : Type u_2 := { x // x ∈ h1.extend ⋯ }\nB : Basis ι K L := Basis.extend h1\nhfin : Fintype ι := FiniteDimensional.fintypeBa... | have hg_neg : ∀ a : L, g (-a) = g a := B.norm_neg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Field.ProperSpace | {
"line": 40,
"column": 2
} | {
"line": 51,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : WeaklyLocallyCompactSpace 𝕜\n⊢ ProperSpace 𝕜",
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Set.ext",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"E... | rcases exists_isCompact_closedBall (0 : 𝕜) with ⟨r, rpos, hr⟩
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
have hC n : IsCompact (closedBall (0 : 𝕜) (‖c‖ ^ n * r)) := by
have : c ^ n ≠ 0 := pow_ne_zero _ <| fun h ↦ by simp [h, zero_le_one.not_gt] at hc
convert hr.smul (c ^ n)
ext
simp onl... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.ProperSpace | {
"line": 40,
"column": 2
} | {
"line": 51,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : WeaklyLocallyCompactSpace 𝕜\n⊢ ProperSpace 𝕜",
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Set.ext",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"E... | rcases exists_isCompact_closedBall (0 : 𝕜) with ⟨r, rpos, hr⟩
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩
have hC n : IsCompact (closedBall (0 : 𝕜) (‖c‖ ^ n * r)) := by
have : c ^ n ≠ 0 := pow_ne_zero _ <| fun h ↦ by simp [h, zero_le_one.not_gt] at hc
convert hr.smul (c ^ n)
ext
simp onl... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 72
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : r ≠ 0\nU : Set ℝ\nV : Set ↑(sphere 0 r)\nhU : IsOpen U\nhU₀ : 0 ∉ U\nhV : IsOpen V\nx : ℝ\nhxU : x ∈ U\ny : { x // x ∈ sphere 0 r }\nhyV : y ∈ V\nhx₀ : 0 < -x\nthis : -U • Subtype.val '' V ∈ 𝓝 ((fun x1 x2 ↦ x1 ... | simp only [neg_smul, nhds_neg, Set.neg_smul, Filter.mem_neg] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 76,
"column": 80
} | {
"line": 99,
"column": 54
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : r ≠ 0\nU : Set ℝ\nV : Set ↑(sphere 0 r)\nhU : IsOpen U\nhU₀ : 0 ∉ U\nhV : IsOpen V\n⊢ IsOpen (U • Subtype.val '' V)",
"usedConstants": [
"le_iff_eq_or_lt",
"norm_eq_of_mem_sphere",
"Filter.instMember... | by
rw [isOpen_iff_mem_nhds]
rintro _ ⟨x, hxU, _, ⟨y, hyV, rfl⟩, rfl⟩
wlog hx₀ : 0 < x generalizing x U
· replace hx₀ : 0 < -x := by
rw [not_lt, le_iff_eq_or_lt, ← neg_pos] at hx₀
exact hx₀.resolve_left <| ne_of_mem_of_not_mem hxU hU₀
specialize this hU.neg (by simpa) (-x) (by simpa) hx₀
simp... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 119,
"column": 63
} | {
"line": 123,
"column": 64
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nR : M\nh₃ : IsLprojection X R\nx : X\n⊢ ‖R • P • R • x‖ + ‖R • x - R • P • R • x‖ + 2 • ‖(1 - R) • P • R • x‖ ≥ ‖R • x‖ + 2... | by
rw [ge_iff_le]
have :=
add_le_add_left (norm_le_insert' (R • x) (R • P • R • x)) (2 • ‖(1 - R) • P • R • x‖)
simpa only [mul_smul, sub_smul, one_smul] using this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 207,
"column": 8
} | {
"line": 207,
"column": 70
} | [
{
"pp": "X : Type u_1\ninst✝² : NormedAddCommGroup X\nM : Type u_2\ninst✝¹ : Ring M\ninst✝ : Module M X\nx : X\n⊢ ‖x‖ = ‖0 • x‖ + ‖(1 - 0) • x‖",
"usedConstants": [
"Norm.norm",
"Real",
"instHSMul",
"Ring.toNonAssocRing",
"Real.instZero",
"Real.instAddMonoid",
"AddG... | simp only [zero_smul, norm_zero, sub_zero, one_smul, zero_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 207,
"column": 8
} | {
"line": 207,
"column": 70
} | [
{
"pp": "X : Type u_1\ninst✝² : NormedAddCommGroup X\nM : Type u_2\ninst✝¹ : Ring M\ninst✝ : Module M X\nx : X\n⊢ ‖x‖ = ‖0 • x‖ + ‖(1 - 0) • x‖",
"usedConstants": [
"Norm.norm",
"Real",
"instHSMul",
"Ring.toNonAssocRing",
"Real.instZero",
"Real.instAddMonoid",
"AddG... | simp only [zero_smul, norm_zero, sub_zero, one_smul, zero_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 207,
"column": 8
} | {
"line": 207,
"column": 70
} | [
{
"pp": "X : Type u_1\ninst✝² : NormedAddCommGroup X\nM : Type u_2\ninst✝¹ : Ring M\ninst✝ : Module M X\nx : X\n⊢ ‖x‖ = ‖0 • x‖ + ‖(1 - 0) • x‖",
"usedConstants": [
"Norm.norm",
"Real",
"instHSMul",
"Ring.toNonAssocRing",
"Real.instZero",
"Real.instAddMonoid",
"AddG... | simp only [zero_smul, norm_zero, sub_zero, one_smul, zero_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 245,
"column": 15
} | {
"line": 245,
"column": 23
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n⊢ ↑P * (↑P + ↑Q * ↑R * ↑Pᶜ) + ↑Pᶜ * ↑R * (↑P + ↑Q * ↑R * ↑Pᶜ) = ↑P + ↑Q * ↑R * ↑Pᶜ",
"usedConstants": [
"Distrib.leftDistribClass",
... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 245,
"column": 24
} | {
"line": 245,
"column": 32
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n⊢ ↑P * ↑P + ↑P * (↑Q * ↑R * ↑Pᶜ) + ↑Pᶜ * ↑R * (↑P + ↑Q * ↑R * ↑Pᶜ) = ↑P + ↑Q * ↑R * ↑Pᶜ",
"usedConstants": [
"Distrib.leftDistribClass... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 249,
"column": 14
} | {
"line": 249,
"column": 23
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n⊢ ↑P * ↑P + 0 + (↑R * 0 + ↑Pᶜ * (↑Q * (↑R * ↑R) * ↑Pᶜ)) = ↑P + ↑Pᶜ * (↑Q * ↑R)",
"usedConstants": [
"Eq.mpr",
"IsLprojection",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 260,
"column": 45
} | {
"line": 260,
"column": 53
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : { P // IsLprojection X P }\n⊢ ↑P = ↑P * (↑P + (↑Q - ↑P * ↑Q))",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"IsLprojection",
"HMul.hM... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 263,
"column": 45
} | {
"line": 263,
"column": 53
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : { P // IsLprojection X P }\n⊢ ↑Q = ↑Q * (↑P + (↑Q - ↑P * ↑Q))",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"IsLprojection",
"HMul.hM... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 589,
"column": 2
} | {
"line": 590,
"column": 41
} | [
{
"pp": "K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\n⊢ spectralNorm K L 1 = 1",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real",
"NormedRing.toRing",
"congrArg... | have h1 : (1 : L) = algebraMap K L 1 := by rw [map_one]
rw [h1, spectralNorm_extends, norm_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 589,
"column": 2
} | {
"line": 590,
"column": 41
} | [
{
"pp": "K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\n⊢ spectralNorm K L 1 = 1",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real",
"NormedRing.toRing",
"congrArg... | have h1 : (1 : L) = algebraMap K L 1 := by rw [map_one]
rw [h1, spectralNorm_extends, norm_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 284,
"column": 8
} | {
"line": 284,
"column": 16
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n⊢ ↑P * (↑P + ↑Pᶜ * ↑R) + ↑Pᶜ * ↑Q * (↑P + ↑Pᶜ * ↑R) = ↑P + ↑Q * ↑R * ↑Pᶜ",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mp... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 284,
"column": 46
} | {
"line": 284,
"column": 54
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n⊢ ↑P * ↑P + ↑P * (↑Pᶜ * ↑R) + ↑Q * ↑Pᶜ * (↑P + ↑Pᶜ * ↑R) = ↑P + ↑Q * ↑R * ↑Pᶜ",
"usedConstants": [
"Distrib.leftDistribClass",
"... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 285,
"column": 63
} | {
"line": 285,
"column": 72
} | [
{
"pp": "X : Type u_1\ninst✝³ : NormedAddCommGroup X\nM : Type u_2\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q R : { P // IsLprojection X P }\n⊢ ↑P * ↑P + 0 + (↑Q * 0 + ↑Q * ↑Pᶜ * (↑Pᶜ * ↑R)) = ↑P + ↑Q * ↑R * ↑Pᶜ",
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"Sem... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 734,
"column": 29
} | {
"line": 734,
"column": 44
} | [
{
"pp": "K : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nx : L\nE : Type v := id ↥K⟮x⟯\nhE : Field E :=\n { add := fun a b ↦ ⟨↑a + ↑... | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 734,
"column": 29
} | {
"line": 734,
"column": 44
} | [
{
"pp": "K : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nx : L\nE : Type v := id ↥K⟮x⟯\nhE : Field E :=\n { add := fun a b ↦ ⟨↑a + ↑... | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 734,
"column": 29
} | {
"line": 734,
"column": 44
} | [
{
"pp": "K : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nx : L\nE : Type v := id ↥K⟮x⟯\nhE : Field E :=\n { add := fun a b ↦ ⟨↑a + ↑... | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 754,
"column": 29
} | {
"line": 754,
"column": 44
} | [
{
"pp": "K : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nx : L\nE : Type v := id ↥K⟮x⟯\nhE : Field E :=\n { add := fun a b ↦ ⟨↑a + ↑... | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 754,
"column": 29
} | {
"line": 754,
"column": 44
} | [
{
"pp": "K : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nx : L\nE : Type v := id ↥K⟮x⟯\nhE : Field E :=\n { add := fun a b ↦ ⟨↑a + ↑... | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 754,
"column": 29
} | {
"line": 754,
"column": 44
} | [
{
"pp": "K : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nx : L\nE : Type v := id ↥K⟮x⟯\nhE : Field E :=\n { add := fun a b ↦ ⟨↑a + ↑... | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 874,
"column": 6
} | {
"line": 874,
"column": 69
} | [
{
"pp": "R : Type u_1\nK : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nx✝ y✝ : L\nhxy : spectralNorm K L (x✝ - y✝) = 0\n⊢ x✝ - y✝ = 0",
"usedConstants": [
"MulAlgebr... | exact (map_eq_zero_iff_eq_zero (spectralMulAlgNorm K L)).mp hxy | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 991,
"column": 6
} | {
"line": 991,
"column": 67
} | [
{
"pp": "K : Type u\ninst✝⁹ : NontriviallyNormedField K\nL : Type v\ninst✝⁸ : Field L\ninst✝⁷ : Algebra K L\nhu : IsUltrametricDist K\ninst✝⁶ : CompleteSpace K\nx : L\nE : Type u_2\ninst✝⁵ : Field E\ninst✝⁴ : Algebra K E\ninst✝³ : Algebra L E\ninst✝² : IsScalarTower K L E\ninst✝¹ : IsSplittingField L E ((mapAlg... | rw [← hr, ← has, spectralMulAlgNorm_eq_of_mem_roots K L x ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 178,
"column": 35
} | {
"line": 178,
"column": 82
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | simp [FreeAddMonoid.lift_eval_of, C0 r' f i hf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 178,
"column": 35
} | {
"line": 178,
"column": 82
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | simp [FreeAddMonoid.lift_eval_of, C0 r' f i hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 178,
"column": 35
} | {
"line": 178,
"column": 82
} | [
{
"pp": "ι : Type u_1\nι₂ : Type u_2\nι₃ : Type u_3\nR : Type u_4\ninst✝⁷ : CommSemiring R\nR₁ : Type u_5\nR₂ : Type u_6\ns : ι → Type u_7\ninst✝⁶ : (i : ι) → AddCommMonoid (s i)\ninst✝⁵ : (i : ι) → Module R (s i)\nM : Type u_8\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nE : Type u_9\ninst✝² : AddCommMonoid... | simp [FreeAddMonoid.lift_eval_of, C0 r' f i hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.PiTensorProduct | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 51
} | [
{
"pp": "ι : Type u_1\nR : Type u_4\ninst✝² : CommSemiring R\ns : ι → Type u_7\ninst✝¹ : (i : ι) → AddCommMonoid (s i)\ninst✝ : (i : ι) → Module R (s i)\nx : ⨂[R] (i : ι), s i\np : FreeAddMonoid (R × ((i : ι) → s i))\nh : (List.map (fun x ↦ x.1 • ⨂ₜ[R] (i : ι), x.2 i) (FreeAddMonoid.toList p)).sum = x\na : R\n⊢... | simp [Function.comp_def, mul_smul, List.smul_sum] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.ODE.Gronwall | {
"line": 54,
"column": 74
} | {
"line": 64,
"column": 8
} | [
{
"pp": "δ K ε x : ℝ\n⊢ HasDerivAt (gronwallBound δ K ε) (K * gronwallBound δ K ε x + ε) x",
"usedConstants": [
"Semigroup",
"NormedCommRing.toNormedRing",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Distrib.leftDistribClass",
"Eq.mpr",
"NormedCommRing.t... | by
by_cases hK : K = 0
· subst K
simp only [gronwallBound_K0, zero_mul, zero_add]
convert ((hasDerivAt_id x).const_mul ε).const_add δ
rw [mul_one]
· simp only [gronwallBound_of_K_ne_0 hK]
convert (((hasDerivAt_id x).const_mul K).exp.const_mul δ).add
((((hasDerivAt_id x).const_mul K).exp.sub_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.ODE.Gronwall | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 62
} | [
{
"pp": "case h₁\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nv : ℝ → E → E\ns : ℝ → Set E\nK : ℝ≥0\nf g : ℝ → E\na b t₀ : ℝ\nhv : ∀ t ∈ Ioo a b, LipschitzOnWith K (v t) (s t)\nht : t₀ ∈ Ioo a b\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ t ∈ Ioo a b, HasDerivAt f (v t (f t)) t\nhfs : ∀ t ... | have hss : Ioc a t₀ ⊆ Ioo a b := Ioc_subset_Ioo_right ht.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 294,
"column": 6
} | {
"line": 295,
"column": 51
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x y : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nα : FunSpace t₀ x₀ r L\nt₁ t₂ : ↑(Icc tmin tmax)\n⊢ ‖(∫ (τ : ℝ) in ↑t₀..↑t₁, f τ (α.com... | integral_interval_sub_left (intervalIntegrable_comp_compProj hf _ t₁)
(intervalIntegrable_comp_compProj hf _ t₂), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Polynomial.Basic | {
"line": 92,
"column": 73
} | {
"line": 97,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP : 𝕜[X]\ninst✝ : OrderTopology 𝕜\n⊢ (IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) atTop fun x ↦ |eval x P|) ↔ P.degree ≤ 0",
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
"W... | by
refine ⟨fun h => ?_, fun h => ⟨|P.coeff 0|, eventually_map.mpr (Eventually.of_forall
(forall_imp (fun _ => le_of_eq) fun x => congr_arg abs <| _root_.trans (congr_arg (eval x)
(eq_C_of_degree_le_zero h)) eval_C))⟩⟩
contrapose! h
exact not_isBoundedUnder_of_tendsto_atTop (abs_tendsto_atTop P h) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 422,
"column": 30
} | {
"line": 422,
"column": 61
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x y : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nhy : y ∈ closedBall x₀ ↑r\nα : FunSpace t₀ x₀ r L\nthis : Nonempty ↑(Icc tmin tmax)\n⊢ ... | ContinuousMap.norm_eq_iSup_norm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Polynomial.Basic | {
"line": 180,
"column": 38
} | {
"line": 180,
"column": 49
} | [
{
"pp": "case pos.inl\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhQ : Q ≠ 0\nh : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)\nhP0 : P.leadingCoeff = 0\n⊢ degree 0 < Q.degree",
"usedConstants": [
"With... | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 15
} | [
{
"pp": "S : Type u_2\nT : Type u_3\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nf : S →+* T\nhf : Function.Injective ⇑f\n⊢ Function.Injective ⇑(map f)",
"usedConstants": [
"PowerSeries"
]
}
] | intro u v huv | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 512,
"column": 2
} | {
"line": 512,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : R⟦X⟧\nT : Subring R\nhp : ∀ (n : ℕ), (coeff n) p ∈ T\nn : ℕ\n⊢ ↑((coeff n) (p.toSubring T hp)) = (coeff n) p",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Subring.instSetLike",
"Ring.toNonAssocRing",
"congrArg",
"LinearMap... | rw [toSubring, coeff_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 512,
"column": 2
} | {
"line": 512,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : R⟦X⟧\nT : Subring R\nhp : ∀ (n : ℕ), (coeff n) p ∈ T\nn : ℕ\n⊢ ↑((coeff n) (p.toSubring T hp)) = (coeff n) p",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Subring.instSetLike",
"Ring.toNonAssocRing",
"congrArg",
"LinearMap... | rw [toSubring, coeff_mk] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 512,
"column": 2
} | {
"line": 512,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : R⟦X⟧\nT : Subring R\nhp : ∀ (n : ℕ), (coeff n) p ∈ T\nn : ℕ\n⊢ ↑((coeff n) (p.toSubring T hp)) = (coeff n) p",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Subring.instSetLike",
"Ring.toNonAssocRing",
"congrArg",
"LinearMap... | rw [toSubring, coeff_mk] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 84,
"column": 2
} | {
"line": 89,
"column": 77
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : NormedSpace 𝕜 W\ninst✝¹ : SeparatingDual 𝕜 V\ninst✝ : SeparatingDual 𝕜 W\nf : (V →L[𝕜] V) ≃A[𝕜] W →L[�... | set TL : V ≃L[𝕜] W := { Tₗ with
continuous_toFun := T.continuous
continuous_invFun := by
change Continuous Tₗ.symm.toLinearMap
suffices T'.toLinearMap = Tₗ.symm from this ▸ T'.continuous
simp [LinearMap.ext_iff, ← Tₗ.injective.eq_iff, T', this, hT, hd, Tₗ] } | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 22
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nn : ℕ\nhn : ↑n < φ.order + ψ.order\ni j : ℕ\nhij : (i, j) ∈ antidiagonal n\nhi : φ.order ≤ ↑i\nhj : ψ.order ≤ ↑j\n⊢ (coeff (i, j).1) φ * (coeff (i, j).2) ψ = 0",
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
"Fin... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 630,
"column": 2
} | {
"line": 630,
"column": 85
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf : MvPowerSeries σ R\np : ℕ\nhf : ↑p = weightedOrder w f\nd : σ →₀ ℕ\nhd : ¬(coeff d) f = 0 ∧ (weight w) d = p\nhf' : (coeff d) ((weightedHomogeneousComponent w p) f) = (coeff d) 0\n⊢ (coeff d) f = 0",
"usedConstants": [
"Finsupp.ins... | simp only [coeff_weightedHomogeneousComponent, coeff_zero, ite_eq_right_iff] at hf' | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 239,
"column": 2
} | {
"line": 247,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\nn : ℕ\na : R\ninst✝ : Decidable (a = 0)\n⊢ ((monomial n) a).order = if a = 0 then ⊤ else ↑n",
"usedConstants": [
"Eq.mpr",
"MvPowerSeries.instZero",
"Semiring.toModule",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"Semilinea... | split_ifs with h
· rw [h, order_eq_top, map_zero]
· rw [order_eq]
constructor <;> intro i hi
· simp only [Nat.cast_inj] at hi
rwa [hi, coeff_monomial_same]
· simp only [Nat.cast_lt] at hi
rw [coeff_monomial, if_neg]
exact ne_of_lt hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 239,
"column": 2
} | {
"line": 247,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\nn : ℕ\na : R\ninst✝ : Decidable (a = 0)\n⊢ ((monomial n) a).order = if a = 0 then ⊤ else ↑n",
"usedConstants": [
"Eq.mpr",
"MvPowerSeries.instZero",
"Semiring.toModule",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"Semilinea... | split_ifs with h
· rw [h, order_eq_top, map_zero]
· rw [order_eq]
constructor <;> intro i hi
· simp only [Nat.cast_inj] at hi
rwa [hi, coeff_monomial_same]
· simp only [Nat.cast_lt] at hi
rw [coeff_monomial, if_neg]
exact ne_of_lt hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 263,
"column": 6
} | {
"line": 263,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nn : ℕ\nh : ↑n < ψ.order\nx : ℕ × ℕ\nhx : x ∈ antidiagonal n\n⊢ ↑x.2 ≤ ↑n",
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
"Finset",
"Nat.instAddMonoid",
"Membership.mem",
"Eq.mp",
"Prod.fst",
"... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 594,
"column": 8
} | {
"line": 594,
"column": 24
} | [
{
"pp": "case mp.hnc\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nm j : σ →₀ ℕ\nhij : (single s n, j) ∈ antidiagonal m\n⊢ n ≤ m s",
"usedConstants": [
"Finsupp.instHasAntidiagonal",
"Nat.instMulZeroClass",
"AddMonoid.toAddSemigroup",
"congrArg... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 595,
"column": 15
} | {
"line": 595,
"column": 33
} | [
{
"pp": "case mp.hnc\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nm j : σ →₀ ℕ\nhij : (single s n, j).1 + (single s n, j).2 = m\n⊢ n ≤ ((single s n, j).1 + (single s n, j).2) s",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroCl... | Finsupp.add_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 340,
"column": 6
} | {
"line": 350,
"column": 14
} | [
{
"pp": "R : Type u_2\ninst✝ : Semiring R\nφ : R⟦X⟧\nhφ : φ ≠ 0\nn : ℕ\nho : φ.order = ↑n\nhn : φ.order.toNat = n\n⊢ emultiplicity X φ ≤ ↑φ.order.toNat",
"usedConstants": [
"not_le",
"PowerSeries.coeff_mul_of_lt_order",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemiri... | apply Order.le_of_lt_add_one
rw [← not_le, ← Nat.cast_one, ← Nat.cast_add, ← pow_dvd_iff_le_emultiplicity]
rintro ⟨ψ, H⟩
have := congr_arg (coeff n) H
rw [X_pow_mul, coeff_mul_of_lt_order, ← hn] at this
· exact coeff_order hφ this
· rw [X_pow_eq, order_monomial]
split_ifs
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 340,
"column": 6
} | {
"line": 350,
"column": 14
} | [
{
"pp": "R : Type u_2\ninst✝ : Semiring R\nφ : R⟦X⟧\nhφ : φ ≠ 0\nn : ℕ\nho : φ.order = ↑n\nhn : φ.order.toNat = n\n⊢ emultiplicity X φ ≤ ↑φ.order.toNat",
"usedConstants": [
"not_le",
"PowerSeries.coeff_mul_of_lt_order",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemiri... | apply Order.le_of_lt_add_one
rw [← not_le, ← Nat.cast_one, ← Nat.cast_add, ← pow_dvd_iff_le_emultiplicity]
rintro ⟨ψ, H⟩
have := congr_arg (coeff n) H
rw [X_pow_mul, coeff_mul_of_lt_order, ← hn] at this
· exact coeff_order hφ this
· rw [X_pow_eq, order_monomial]
split_ifs
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 605,
"column": 12
} | {
"line": 605,
"column": 28
} | [
{
"pp": "case pos.h₀\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\ni j : σ →₀ ℕ\nhij : (i, j) ∈ antidiagonal m\nhne : (i, j) ≠ (single s n, m - single s n)\n⊢ ((coeff (i, j).1)... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 621,
"column": 12
} | {
"line": 621,
"column": 28
} | [
{
"pp": "case neg\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\ni j : σ →₀ ℕ\nhij : (i, j) ∈ antidiagonal m\n⊢ ((coeff (i, j).1) (X s ^ n) * (coeff (i, j).2) fun m ↦ (coeff (m... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Polynomial.Norm | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 58
} | [
{
"pp": "A : Type u_1\ninst✝¹ : SeminormedRing A\ninst✝ : NormOneClass A\n⊢ X.supNorm = 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"SeminormedRing.toNorm",
"Real",
"Semiring.toModule",
"Polynomial.monomial_one_one_eq_X",... | rw [← monomial_one_one_eq_X, supNorm_monomial, norm_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Polynomial.Norm | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 58
} | [
{
"pp": "A : Type u_1\ninst✝¹ : SeminormedRing A\ninst✝ : NormOneClass A\n⊢ X.supNorm = 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"SeminormedRing.toNorm",
"Real",
"Semiring.toModule",
"Polynomial.monomial_one_one_eq_X",... | rw [← monomial_one_one_eq_X, supNorm_monomial, norm_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Polynomial.Norm | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 58
} | [
{
"pp": "A : Type u_1\ninst✝¹ : SeminormedRing A\ninst✝ : NormOneClass A\n⊢ X.supNorm = 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"SeminormedRing.toNorm",
"Real",
"Semiring.toModule",
"Polynomial.monomial_one_one_eq_X",... | rw [← monomial_one_one_eq_X, supNorm_monomial, norm_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 234,
"column": 4
} | {
"line": 235,
"column": 74
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedAddCommGroup V\ninst✝⁸ : InnerProductSpace 𝕜 V\ninst✝⁷ : CompleteSpace V\ninst✝⁶ : NormedAddCommGroup W\ninst✝⁵ : InnerProductSpace 𝕜 W\ninst✝⁴ : CompleteSpace W\nF : Type u_4\ninst✝³ : EquivLike F (V →L[𝕜] V) (W →L[𝕜] W... | obtain ⟨U, hU⟩ := StarAlgEquiv.eq_linearIsometryEquivConjStarAlgEquiv
(StarAlgEquivClass.toStarAlgEquiv f : _ ≃⋆ₐ[𝕜] _) (map_continuous f) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 707,
"column": 4
} | {
"line": 707,
"column": 30
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\nm : σ →₀ ℕ\na : R\nn : ℕ\n⊢ (monomial (n • m)) (∏ i ∈ range n, a) = (monomial (n • m)) (a ^ n)",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"instHSMul",
"Semiring.toModule",
"congrArg",
"CommSemiring.t... | ← Finset.pow_eq_prod_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Polynomial.Order | {
"line": 88,
"column": 79
} | {
"line": 88,
"column": 91
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\ny : ℝ\nhy : eval (-y) P = 0\n⊢ P.IsRoot (-y)",
"usedConstants": [
"Polynomial.eval",
"NegZeroClass.toNeg",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.IsRoot",
... | ← IsRoot.def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.OfDigits | {
"line": 120,
"column": 31
} | {
"line": 120,
"column": 39
} | [
{
"pp": "case succ\nx : ℝ\nb : ℕ\ninst✝ : NeZero b\nhx : x ∈ Set.Ico 0 1\nthis : b ≠ 0\nn : ℕ\nih : ↑b ^ n * ∑ i ∈ Finset.range n, ofDigitsTerm (x.digits b) i = ↑⌊↑b ^ n * x⌋₊\n⊢ ↑b ^ (n + 1) * (∑ x_1 ∈ Finset.range n, ofDigitsTerm (x.digits b) x_1 + ofDigitsTerm (x.digits b) n) =\n ↑⌊↑b ^ (n + 1) * x⌋₊",
... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 386,
"column": 6
} | {
"line": 392,
"column": 60
} | [
{
"pp": "n : ℕ\np : ℂ[X]\nhp : ¬p = 0\nhn : n ≤ p.natDegree\nS : Multiset (Multiset ℂ) := powersetCard (p.natDegree - n) p.roots\nthis : ∀ x ∈ S.toFinset, ∏ x_1 ∈ x.toFinset, ‖x_1‖ ^ count x_1 x ≤ ∏ m ∈ p.roots.toFinset, max 1 ‖m‖ ^ count m p.roots\n⊢ ∑ x ∈ S.toFinset, ↑(count x S) * (Multiset.map (fun a ↦ max ... | rw [← Finset.sum_mul]
congr
norm_cast
simp only [mem_powersetCard, mem_toFinset, imp_self, implies_true, sum_count_eq_card,
card_powersetCard, S, ← Nat.choose_symm hn]
congr
exact splits_iff_card_roots.mp <| IsAlgClosed.splits p | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 386,
"column": 6
} | {
"line": 392,
"column": 60
} | [
{
"pp": "n : ℕ\np : ℂ[X]\nhp : ¬p = 0\nhn : n ≤ p.natDegree\nS : Multiset (Multiset ℂ) := powersetCard (p.natDegree - n) p.roots\nthis : ∀ x ∈ S.toFinset, ∏ x_1 ∈ x.toFinset, ‖x_1‖ ^ count x_1 x ≤ ∏ m ∈ p.roots.toFinset, max 1 ‖m‖ ^ count m p.roots\n⊢ ∑ x ∈ S.toFinset, ↑(count x S) * (Multiset.map (fun a ↦ max ... | rw [← Finset.sum_mul]
congr
norm_cast
simp only [mem_powersetCard, mem_toFinset, imp_self, implies_true, sum_count_eq_card,
card_powersetCard, S, ← Nat.choose_symm hn]
congr
exact splits_iff_card_roots.mp <| IsAlgClosed.splits p | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Real.Hyperreal | {
"line": 1031,
"column": 2
} | {
"line": 1031,
"column": 39
} | [
{
"pp": "x : ℝ*\nh0 : x ≠ 0\nhi : x⁻¹.Infinitesimal\n⊢ x.Infinite",
"usedConstants": [
"Hyperreal.instField",
"lt_or_gt_of_ne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"Hyperreal.instLine... | rcases lt_or_gt_of_ne h0 with hn | hp | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Real.Hyperreal | {
"line": 1111,
"column": 2
} | {
"line": 1112,
"column": 60
} | [
{
"pp": "x y : ℝ*\n⊢ x.InfinitePos → ¬y.Infinitesimal → y < 0 → (x * y).InfiniteNeg",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Hyperreal.instField",
"Eq.mpr",
"NegZeroClass.toNeg",
"Hyperreal.infinitePos_mul_of_infinitePos_not_infinitesimal_pos",
"Preorder.toLT... | rw [← infinitePos_neg, ← neg_pos, neg_mul_eq_mul_neg, ← infinitesimal_neg]
exact infinitePos_mul_of_infinitePos_not_infinitesimal_pos | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Real.Hyperreal | {
"line": 1111,
"column": 2
} | {
"line": 1112,
"column": 60
} | [
{
"pp": "x y : ℝ*\n⊢ x.InfinitePos → ¬y.Infinitesimal → y < 0 → (x * y).InfiniteNeg",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Hyperreal.instField",
"Eq.mpr",
"NegZeroClass.toNeg",
"Hyperreal.infinitePos_mul_of_infinitePos_not_infinitesimal_pos",
"Preorder.toLT... | rw [← infinitePos_neg, ← neg_pos, neg_mul_eq_mul_neg, ← infinitesimal_neg]
exact infinitePos_mul_of_infinitePos_not_infinitesimal_pos | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Real.Irrational | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 16
} | [
{
"pp": "n : ℕ\nm : ℤ\nhnpos : 0 < n\nN : ℤ\nD : ℕ\nP : D ≠ 0\nC : N.natAbs.Coprime D\nhxr : ↑{ num := N, den := D, den_nz := P, reduced := C } ^ n = ↑m\nhv : ¬∃ y, ↑{ num := N, den := D, den_nz := P, reduced := C } = ↑y\n⊢ False",
"usedConstants": [
"Int.cast",
"Real",
"DivisionRing.toRat... | ← cast_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 24
} | [
{
"pp": "n : ℕ\nm : ℤ\nhnpos : 0 < n\nN : ℤ\nD : ℕ\nP : D ≠ 0\nC : N.natAbs.Coprime D\nhxr : ↑{ num := N, den := D, den_nz := P, reduced := C } ^ n = ↑m\nhv : ¬∃ y, ↑{ num := N, den := D, den_nz := P, reduced := C } = ↑y\n⊢ False",
"usedConstants": [
"Int.cast",
"Real",
"DivisionRing.toRat... | rw [← cast_pow] at hxr | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Real.Irrational | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 71
} | [
{
"pp": "case inl\nq : ℚ\nhq : 0 ≤ q\n⊢ Irrational √↑q ↔ ¬IsSquare q ∧ 0 ≤ q",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"Rat.instMul",
"Real",
"congrArg",
"Real.instRatCast",
"Rat",
"PartialOrder.toPreorder",
"and_iff_left",
"Rat.linearOrder",... | simp_rw [irrational_sqrt_ratCast_iff_of_nonneg hq, and_iff_left hq] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.Real.Irrational | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 71
} | [
{
"pp": "case inl\nq : ℚ\nhq : 0 ≤ q\n⊢ Irrational √↑q ↔ ¬IsSquare q ∧ 0 ≤ q",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"Rat.instMul",
"Real",
"congrArg",
"Real.instRatCast",
"Rat",
"PartialOrder.toPreorder",
"and_iff_left",
"Rat.linearOrder",... | simp_rw [irrational_sqrt_ratCast_iff_of_nonneg hq, and_iff_left hq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Real.Irrational | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 71
} | [
{
"pp": "case inl\nq : ℚ\nhq : 0 ≤ q\n⊢ Irrational √↑q ↔ ¬IsSquare q ∧ 0 ≤ q",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"Rat.instMul",
"Real",
"congrArg",
"Real.instRatCast",
"Rat",
"PartialOrder.toPreorder",
"and_iff_left",
"Rat.linearOrder",... | simp_rw [irrational_sqrt_ratCast_iff_of_nonneg hq, and_iff_left hq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Real.Irrational | {
"line": 352,
"column": 84
} | {
"line": 354,
"column": 38
} | [
{
"pp": "x : ℝ\nh : Irrational x\nq : ℚ\nhq : q ≠ 0\n⊢ Irrational (x / ↑q)",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Divi... | by
rw [div_eq_mul_inv, ← cast_inv]
exact h.mul_ratCast (inv_ne_zero hq) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Real.Irrational | {
"line": 510,
"column": 22
} | {
"line": 510,
"column": 49
} | [
{
"pp": "m : ℤ\nx : ℝ\n⊢ Irrational (x / ↑↑m) ↔ m ≠ 0 ∧ Irrational x",
"usedConstants": [
"Rat.instOfNat",
"Int.cast",
"Eq.mpr",
"Real",
"instHDiv",
"DivisionRing.toRatCast",
"congrArg",
"Real.instDivInvMonoid",
"Real.instRatCast",
"Rat",
"Ra... | irrational_div_ratCast_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 516,
"column": 22
} | {
"line": 516,
"column": 49
} | [
{
"pp": "n : ℕ\nx : ℝ\n⊢ Irrational (x / ↑↑n) ↔ n ≠ 0 ∧ Irrational x",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"Real",
"instHDiv",
"DivisionRing.toRatCast",
"congrArg",
"Real.instDivInvMonoid",
"Real.instRatCast",
"Rat",
"id",
"HDiv.hDi... | irrational_div_ratCast_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.Pi.Wallis | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 44
} | [
{
"pp": "case succ.refine_1\nn : ℕ\nIH :\n ∏ i ∈ range n, (2 * ↑i + 2) / (2 * ↑i + 1) * ((2 * ↑i + 2) / (2 * ↑i + 3)) =\n 2 ^ (4 * n) * ↑n ! ^ 4 / (↑(2 * n)! ^ 2 * (2 * ↑n + 1))\n⊢ ↑(2 * n)! ^ 2 * (2 * ↑n + 1) * ((2 * ↑n + 1) * (2 * ↑n + 3)) ≠ 0",
"usedConstants": [
"Iff.mpr",
"Real.instIsOr... | any_goals exact ne_of_gt (by positivity) | Lean.Elab.Tactic.evalAnyGoals | Lean.Parser.Tactic.anyGoals |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 132,
"column": 40
} | {
"line": 132,
"column": 49
} | [
{
"pp": "case h.e'_6\nz : ℂ\nhz : ‖z‖ < 1\nthis :\n HasSum (fun x ↦ -I / 2 * (((-1) ^ (2 * x.1 + ↑x.2 + 1) + 1) * (z * I) ^ (2 * x.1 + ↑x.2) / ↑(2 * x.1 + ↑x.2)))\n (-I / 2 * log ((1 + z * I) / (1 - z * I)))\nk : ℕ\n⊢ (-1) ^ k * z ^ (2 * k + 1) / ↑(2 * k + 1) =\n -I / 2 * 0 + -I / 2 * (((-1) ^ (2 * k + 1... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.Pi.Leibniz | {
"line": 30,
"column": 6
} | {
"line": 30,
"column": 37
} | [
{
"pp": "case hf0\n⊢ Tendsto (fun i ↦ (2 * ↑i + 1)⁻¹) atTop (𝓝 0)",
"usedConstants": [
"Real",
"HMul.hMul",
"Nat.instAtLeastTwoHAddOfNat",
"PseudoMetricSpace.toUniformSpace",
"instOfNatNat",
"instOrderTopologyReal",
"Nat.cast",
"Field.toSemifield",
"Rea... | apply Tendsto.inv_tendsto_atTop | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.Arcosh | {
"line": 95,
"column": 69
} | {
"line": 95,
"column": 95
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\n⊢ √(sinh x ^ 2) ^ 2 = sinh x ^ 2",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"IsOrderedRing.toPosMulMono",
"congrArg",
"pow_two_nonneg",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | sq_sqrt (pow_two_nonneg _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 155,
"column": 2
} | {
"line": 158,
"column": 56
} | [
{
"pp": "x y : ℝ≥0\n⊢ Tendsto (fun t ↦ dist x y / 2 ^ (t + 1)) atTop (𝓝 0)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"DivisionCommMonoid.toDivisionMonoid",
... | conv =>
rw [← zero_mul (dist x y / 2)]
enter [1, n]
rw [pow_succ', ← div_div, div_eq_inv_mul, ← inv_pow] | Lean.Elab.Tactic.Conv.evalConv | Lean.Parser.Tactic.Conv.conv |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 29
} | [
{
"pp": "case h.e'_2\nθ : ℝ\nn : ℕ\nf : ℝ → ℝ := fun x ↦ 1 - x ^ 2\nu₁ : ℝ → ℝ := fun x ↦ f x ^ (n + 1)\nu₁' : ℝ → ℝ := fun x ↦ -(2 * (↑n + 1) * x * f x ^ n)\nv₁ : ℝ → ℝ := fun x ↦ sin (x * θ)\nv₁' : ℝ → ℝ := fun x ↦ cos (x * θ) * θ\nu₂ : ℝ → ℝ := fun x ↦ x * f x ^ n\nu₂' : ℝ → ℝ := fun x ↦ f x ^ n - 2 * ↑n * x... | rw [← integral_const_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 222,
"column": 15
} | {
"line": 222,
"column": 39
} | [
{
"pp": "p : ℤ[X]\na b : ℤ\nk : ℕ\nhp : p.natDegree ≤ k\nhb : b ≠ 0\n| eval₂ (Int.castRingHom ℝ) (↑a / ↑b) p * ↑b ^ k",
"usedConstants": [
"Int.cast",
"Real",
"instHDiv",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"Real.instDivInvMonoid",
"Polynomial.sum",
... | rw [← sum_monomial_eq p] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 51
} | [
{
"pp": "case inr\np : ℤ[X]\na b : ℤ\nk : ℕ\nhp : p.natDegree ≤ k\nhb : b ≠ 0\n⊢ eval₂ (Int.castRingHom ℝ) (↑a / ↑b) (p.sum fun n a ↦ (monomial n) a) * ↑b ^ k =\n ↑(∑ i ∈ p.support, p.coeff i * a ^ i * b ^ (k - i))",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.instAddCommMonoid",
... | rw [eval₂_sum, sum, Finset.sum_mul, Int.cast_sum] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 242,
"column": 4
} | {
"line": 242,
"column": 53
} | [
{
"pp": "case refine_1\nn : ℕ\nx : ℝ\nhx : x ∈ Ioc (-1) 1\n⊢ 0 ≤ 1 - x ^ 2",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"AddGroupWithOne.toAddGroup",
"abs",
"congrArg",
"abs_le",
"Real.instSub",
"cova... | rw [sub_nonneg, sq_le_one_iff_abs_le_one, abs_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 300,
"column": 38
} | {
"line": 300,
"column": 47
} | [
{
"pp": "h' : ¬Irrational (π / 2)\na : ℤ\nb : ℕ\nhb : 0 < b\nh : π / 2 = ↑a / ↑b\nha : 0 < ↑a\nk : ∀ (n : ℕ), 0 < ↑a ^ (2 * n + 1) / ↑n !\nj : ∀ᶠ (n : ℕ) in atTop, ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1\nn : ℕ\nhn : ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1\nhn' : 0 < ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2)\nz... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Artanh | {
"line": 116,
"column": 78
} | {
"line": 117,
"column": 45
} | [
{
"pp": "x : ℝ\n⊢ artanh x = 0 ↔ x ≤ -1 ∨ x = 0 ∨ 1 ≤ x",
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Artanh.0.Real.artanh_eq_zero_iff._proof_1_2"
]
}
] | by
grind [artanh, log_eq_zero, div_nonpos_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 192,
"column": 2
} | {
"line": 197,
"column": 44
} | [
{
"pp": "case pos\np : ℝ\nhp : p ≠ 0 ∧ p ≠ 1\n⊢ deriv binEntropy p = log (1 - p) - log p",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.deriv_negMulLog",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWi... | · obtain ⟨hp₀, hp₁⟩ := hp
rw [ne_comm, ← sub_ne_zero] at hp₁
rw [binEntropy_eq_negMulLog_add_negMulLog_one_sub', deriv_fun_add, deriv_comp_const_sub,
deriv_negMulLog hp₀, deriv_negMulLog hp₁]
· ring
all_goals fun_prop (disch := assumption) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Log.RpowTendsto | {
"line": 31,
"column": 2
} | {
"line": 44,
"column": 37
} | [
{
"pp": "p x : ℝ\np_pos : 0 < p\nx_pos : 0 < x\nhx : ‖p * log x‖ ≤ 1\n⊢ ‖p⁻¹ * (x ^ p - 1) - log x‖ ≤ p * ‖log x‖ ^ 2",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real.p... | have pinv_nonneg : 0 ≤ p⁻¹ := by grind [_root_.inv_nonneg]
calc
_ = ‖p⁻¹ * ((x ^ p - 1) - p * log x)‖ := by grind
_ = p⁻¹ * ‖(rexp (p * log x) - 1) - p * log x‖ := by
simp only [norm_mul, Real.norm_of_nonneg (r := p⁻¹) pinv_nonneg]
congr
rw [mul_comm, Real.exp_mul, Real.exp_log (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.RpowTendsto | {
"line": 31,
"column": 2
} | {
"line": 44,
"column": 37
} | [
{
"pp": "p x : ℝ\np_pos : 0 < p\nx_pos : 0 < x\nhx : ‖p * log x‖ ≤ 1\n⊢ ‖p⁻¹ * (x ^ p - 1) - log x‖ ≤ p * ‖log x‖ ^ 2",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real.p... | have pinv_nonneg : 0 ≤ p⁻¹ := by grind [_root_.inv_nonneg]
calc
_ = ‖p⁻¹ * ((x ^ p - 1) - p * log x)‖ := by grind
_ = p⁻¹ * ‖(rexp (p * log x) - 1) - p * log x‖ := by
simp only [norm_mul, Real.norm_of_nonneg (r := p⁻¹) pinv_nonneg]
congr
rw [mul_comm, Real.exp_mul, Real.exp_log (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Digamma | {
"line": 50,
"column": 13
} | {
"line": 50,
"column": 23
} | [
{
"pp": "⊢ -(↑√Real.pi * (2 * log 2 + ↑Real.eulerMascheroniConstant)) / ↑Real.pi ^ (1 / 2) =\n -2 * log 2 - ↑Real.eulerMascheroniConstant",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Complex.log",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCo... | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 304,
"column": 10
} | {
"line": 305,
"column": 69
} | [
{
"pp": "case refine_1\np : ℝ\nhp : p ∈ Ioo 0 1\n⊢ 0 ≤ᶠ[ae (volume.restrict (Ioi 0))] fun t ↦ p.rpowIntegrand₀₁ t 1",
"usedConstants": [
"instClosedIicTopology",
"Real.partialOrder",
"Real.instLE",
"Real",
"Set.Ioi",
"Preorder.toLT",
"Real.lattice",
"Real.inst... | refine ae_restrict_of_forall_mem measurableSet_Ioi fun t ht => ?_
exact rpowIntegrand₀₁_nonneg hp.1 (le_of_lt ht) zero_le_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 304,
"column": 10
} | {
"line": 305,
"column": 69
} | [
{
"pp": "case refine_1\np : ℝ\nhp : p ∈ Ioo 0 1\n⊢ 0 ≤ᶠ[ae (volume.restrict (Ioi 0))] fun t ↦ p.rpowIntegrand₀₁ t 1",
"usedConstants": [
"instClosedIicTopology",
"Real.partialOrder",
"Real.instLE",
"Real",
"Set.Ioi",
"Preorder.toLT",
"Real.lattice",
"Real.inst... | refine ae_restrict_of_forall_mem measurableSet_Ioi fun t ht => ?_
exact rpowIntegrand₀₁_nonneg hp.1 (le_of_lt ht) zero_le_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 33,
"column": 53
} | {
"line": 84,
"column": 19
} | [
{
"pp": "n : ℕ\n⊢ deriv Gamma (↑n + 1) = ↑n ! * (-γ + ↑(harmonic n))",
"usedConstants": [
"CharP.cast_eq_zero",
"Iff.mpr",
"NormedCommRing.toNormedRing",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instIsOrderedRing",
"Not.intro",
"Eq.mpr",
... | by
/- This follows from two properties of the function `f n = log (Gamma n)`:
firstly, the elementary computation that `deriv f (n + 1) = deriv f n + 1 / n`, so that
`deriv f n = deriv f 1 + harmonic n`; secondly, the convexity of `f` (the Bohr-Mollerup theorem),
which shows that `deriv f n` is `log n + o(1)` a... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 113,
"column": 60
} | {
"line": 113,
"column": 69
} | [
{
"pp": "h_diff : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ Gamma s\nh_diff' : ∀ {s : ℝ}, 0 < s → DifferentiableAt ℝ (fun s ↦ Gamma (2 * s)) s\n⊢ deriv Gamma (1 / 2) = deriv Gamma (1 / 2) + √π * 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 129,
"column": 76
} | {
"line": 132,
"column": 90
} | [
{
"pp": "a : ℝ\nha : 0 < a\n⊢ ∑' (n : ℤ), rexp (-π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), rexp (-π / a * ↑n ^ 2)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnit... | by
simpa only [← ofReal_inj, ofReal_tsum, ofReal_exp, ofReal_mul, ofReal_neg, ofReal_pow,
ofReal_intCast, ofReal_div, ofReal_one, ofReal_cpow ha.le, ofReal_ofNat, mul_zero, zero_mul,
add_zero] using Complex.tsum_exp_neg_quadratic (by rwa [ofReal_re] : 0 < (a : ℂ).re) 0 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Int.Log | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 27
} | [
{
"pp": "case inr.inr\nR : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\nhcri : 1 < r⁻¹\nthis : 1 ≤ Nat.clog b ⌈r⁻¹⌉₊\n⊢ ↑(b ^ (Nat.clog b ⌈r⁻¹⌉₊ - 1)) < r⁻¹",
"usedConstants": [
"NonA... | refine Nat.lt_ceil.1 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Int.Log | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 69
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\n⊢ r ≤ ↑b ^ clog b r",
"usedConstants": [
"le_inv_comm₀",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | rw [← neg_log_inv_eq_clog, zpow_neg, le_inv_comm₀ hr (zpow_pos ..)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 35,
"column": 4
} | {
"line": 35,
"column": 90
} | [
{
"pp": "⊢ ∀ᶠ (x : ℝ) in nhdsWithin 0 (Set.Ioo (-1) 0), 0 < log x * x",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"HMul.hMul",
"Ring.toNonAssocRing",
"IsStrictO... | refine eventually_nhdsWithin_of_forall fun x ⟨hx₁, hx₂⟩ ↦ mul_pos_of_neg_of_neg ?_ hx₂ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 373,
"column": 6
} | {
"line": 373,
"column": 64
} | [
{
"pp": "case pos.a\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\nhb1' : 1 < ↑b\n⊢ ↑b ^ ↑⌊logb (↑b) r⌋ ≤ ↑b ^ logb (↑b) r",
"usedConstants": [
"Int.cast",
"Real",
"Int.floor",
"LT.lt.le",
"Real.instFloorRing",
"Real.instRing",
"Nat.cast",
"Real.instOne",... | exact rpow_le_rpow_of_exponent_le hb1'.le (Int.floor_le _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 374,
"column": 6
} | {
"line": 375,
"column": 38
} | [
{
"pp": "case pos.a\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\nhb1' : 1 < ↑b\n⊢ Int.log b r ≤ ⌊logb (↑b) r⌋",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"instConditionallyCompleteLinearOrder",
"Int.floor",
"FloorRi... | rw [Int.le_floor, le_logb_iff_rpow_le hb1' hr, rpow_intCast]
exact Int.zpow_log_le_self hb hr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 374,
"column": 6
} | {
"line": 375,
"column": 38
} | [
{
"pp": "case pos.a\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\nhb1' : 1 < ↑b\n⊢ Int.log b r ≤ ⌊logb (↑b) r⌋",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"instConditionallyCompleteLinearOrder",
"Int.floor",
"FloorRi... | rw [Int.le_floor, le_logb_iff_rpow_le hb1' hr, rpow_intCast]
exact Int.zpow_log_le_self hb hr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 401,
"column": 2
} | {
"line": 408,
"column": 81
} | [
{
"pp": "b n : ℕ\n⊢ ⌊logb ↑b ↑n⌋₊ = Nat.log b n",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"CharP.cast_eq_zero",
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOn... | obtain _ | _ | b := b
· simp [Real.logb]
· simp [Real.logb]
obtain rfl | hn := eq_or_ne n 0
· simp
rw [← Nat.cast_inj (R := ℤ), Int.natCast_floor_eq_floor, floor_logb_natCast (by simp),
Int.log_natCast]
exact logb_nonneg (by simp [Nat.cast_add_one_pos]) (Nat.one_le_cast.2 (by lia)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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