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.Analysis.Normed.Group.SeparationQuotient | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 32
} | [
{
"pp": "case mpr.H\nM : Type u_1\ninst✝ : SeminormedAddCommGroup M\nh : ∀ (x : M), ‖x‖ = 0\nx : M\n⊢ normedMk x = 0 x",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real",
"NormedAddGroupHom",
"SeparationQuotient.instNormedAddCommGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 715,
"column": 8
} | {
"line": 716,
"column": 66
} | [
{
"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⟯\nthis✝ : Field E :=\n have this := inferInsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 48,
"column": 4
} | {
"line": 48,
"column": 33
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 59,
"column": 38
} | {
"line": 59,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k\nh✝¹ : Nonem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 736,
"column": 8
} | {
"line": 736,
"column": 58
} | [
{
"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⟯\nthis✝ : Field E :=\n have this := inferInsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bound k\nh✝¹ : Nonem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 59
} | [
{
"pp": "case h.refine_1\nα : Type u_1\nβ : Type u_2\nG : Type u_3\n𝓕 : Filter α\ninst✝¹ : NormedAddCommGroup G\ninst✝ : CompleteSpace G\nf : α → β → G\ng : β → G\nbound : β → ℝ\nh_sum : Summable bound\nhab : ∀ (k : β), Tendsto (fun x ↦ f x k) 𝓕 (𝓝 (g k))\nh_bound✝ : ∀ᶠ (n : α) in 𝓕, ∀ (k : β), ‖f n k‖ ≤ bo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 854,
"column": 22
} | {
"line": 854,
"column": 56
} | [
{
"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 : L\n⊢ spectralNorm K L (x - x) = 0",
"usedConstants": [
"Real",
"sub_self",
"Re... | simp [sub_self, spectralNorm_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 854,
"column": 22
} | {
"line": 854,
"column": 56
} | [
{
"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 : L\n⊢ spectralNorm K L (x - x) = 0",
"usedConstants": [
"Real",
"sub_self",
"Re... | simp [sub_self, spectralNorm_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 854,
"column": 22
} | {
"line": 854,
"column": 56
} | [
{
"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 : L\n⊢ spectralNorm K L (x - x) = 0",
"usedConstants": [
"Real",
"sub_self",
"Re... | simp [sub_self, spectralNorm_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 41,
"column": 31
} | {
"line": 41,
"column": 62
} | [
{
"pp": "E✝ : Type u_1\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedSpace ℝ E✝\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : 0 < r\nx : ↑{0}ᶜ\n⊢ 0 < ‖↑x‖",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 47,
"column": 25
} | {
"line": 47,
"column": 36
} | [
{
"pp": "E✝ : Type u_1\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedSpace ℝ E✝\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : 0 < r\nx : E\nhx : x ∈ {0}ᶜ\n⊢ 0 < ‖x‖",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Rea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 33
} | [
{
"pp": "R : Type u_7\nE : Type u_8\nF : Type u_9\ninst✝⁶ : NontriviallyNormedField R\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace R E\ninst✝³ : CompleteSpace E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace R F\ninst✝ : CompleteSpace F\nf : E →L[R] F\nhf : Injective ⇑f\nhf' : IsClosed (range ⇑f)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 291,
"column": 26
} | {
"line": 291,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\nE : Type u_2\nF : Type u_4\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommMonoid E\ninst✝³ : Module R E\ninst✝² : TopologicalSpace F\ninst✝¹ : AddCommMonoid F\ninst✝ : Module R F\nf : E ≃L[R] F\ny : F\n⊢ f (⋯.rightInverse y) = f (↑f.symm y)",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 319,
"column": 34
} | {
"line": 319,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : Semiring R\nE : Type u_2\nF : Type u_4\nG : Type u_6\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : Module R E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\nf : E →L[R] F\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommMonoid G\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 80
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nβ : Type u_3\nL : SummationFilter β\nb : GeneralSchauderBasis β 𝕜 X L\nl : β →₀ 𝕜\nhl : (Finsupp.linearCombination 𝕜 ↑b) l = 0\n⊢ ∀ (a : β), l a = 0 a",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nβ : Type u_3\nL : SummationFilter β\nb : GeneralSchauderBasis β 𝕜 X L\nx : X\n⊢ Tendsto (fun A ↦ (b.proj A) x) L.filter (𝓝 x)",
"usedConstants": [
"Eq.mpr",
"Norm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 15
} | [
{
"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[instTopologicalSpaceSubtype] V\nx : ℝ\nhxU : x ∈ U\ny : { x // x ∈ sphere 0 r }\nhyV : y ∈ V\nhx₀ : 0 < -x\nthis : Neg.neg ⁻¹' (-(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 86,
"column": 41
} | {
"line": 86,
"column": 52
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nr : ℝ\nhr : r ≠ 0\nV : Set ↑(sphere 0 r)\nhV : IsOpen[instTopologicalSpaceSubtype] V\ny : { x // x ∈ sphere 0 r }\nhyV : y ∈ V\nU : Set ℝ\nhU : IsOpen U\nhU₀ : 0 ∉ U\nx : ℝ\nhxU : x ∈ U\nhx₀ : 0 < x\n⊢ 0 ≤ r",
"usedConstants": []... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 246,
"column": 27
} | {
"line": 246,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nβ : Type u_3\nb : UnconditionalSchauderBasis β 𝕜 X\ninst✝ : CompleteSpace X\nC : ℝ\nhC : ∀ (A : Finset β), ‖GeneralSchauderBasis.proj b A‖ ≤ C\n⊢ 0 ≤ C",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 335,
"column": 27
} | {
"line": 335,
"column": 50
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nb : SchauderBasis 𝕜 X\ninst✝ : CompleteSpace X\nC : ℝ\nhC : ∀ (n : ℕ), ‖b.proj n‖ ≤ C\n⊢ 0 ≤ C",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.DoubleDual | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\n⊢ ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 123,
"column": 12
} | {
"line": 123,
"column": 59
} | [
{
"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\nthis : ‖R • x‖ + 2 • ‖(1 - R) • P • R • x‖ ≤ ‖R • P • R • x‖ + ‖R • x - R • P • R • x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 195,
"column": 18
} | {
"line": 195,
"column": 50
} | [
{
"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 : { P // IsLprojection X P }\n⊢ ↑P = ↑(P ⊓ P)",
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"congrArg",
"id",
"Subtype",
"instOfNatNat",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 13
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf : ι → R\nhf : Summable fun i ↦ ‖f i‖\ni : ι\n⊢ ‖‖1 + f i‖ - 1‖ ≤ ‖f i‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"NormedCommRing.toC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 163,
"column": 38
} | {
"line": 163,
"column": 53
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf✝ : ι → R\nhf : Summable fun i ↦ ‖f✝ i‖\nε : ℝ\nhε : 0 < ε\nf : ℝ → ℝ := fun x ↦ Real.exp x - 1\n⊢ Set.Iio ε ∈ 𝓝 (f 0)",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Real",
"sub_self",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 213,
"column": 4
} | {
"line": 213,
"column": 15
} | [
{
"pp": "case left\nι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf : ι → R\nhu : Summable fun n ↦ ‖f n‖\ni : ι\n⊢ -‖f i‖ ≤ ‖1 + f i‖ - 1",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 26
} | [
{
"pp": "case right\nι : Type u_1\nR : Type u_2\ninst✝¹ : NormedCommRing R\ninst✝ : NormOneClass R\nf : ι → R\nhu : Summable fun n ↦ ‖f n‖\ni : ι\n⊢ ‖1 + f i‖ - 1 ≤ ‖f i‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toCommRing",
"Re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 19
} | [
{
"pp": "ι : Type uι\ninst✝² : Fintype ι\n𝕜 : Type u𝕜\nE : ι → Type uE\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝ : NormedField 𝕜\np : FreeAddMonoid (𝕜 × ((i : ι) → E i))\na : ℝ\nx : 𝕜\nm : (i : ι) → E i\na✝ : (x, m) ∈ FreeAddMonoid.toList p\nh : ‖x‖ * ∏ x, ‖m x‖ = a\n⊢ 0 ≤ a",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 42
} | [
{
"pp": "ι : Type uι\ninst✝³ : Fintype ι\n𝕜 : Type u𝕜\nE : ι → Type uE\ninst✝² : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝¹ : NormedField 𝕜\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\na : 𝕜\nx : ⨂[𝕜] (i : ι), E i\np : ↑x.lifts\n⊢ ⨅ p, projectiveSeminormAux ↑p ≤ ‖a‖ * projectiveSeminormAux ↑p",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 54
} | [
{
"pp": "ι : Type uι\ninst✝⁵ : Fintype ι\n𝕜 : Type u𝕜\nE : ι → Type uE\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nG : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ContinuousMultilinearMap 𝕜 E G\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn | {
"line": 44,
"column": 75
} | {
"line": 44,
"column": 86
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nK : Set α\nu : ι → ℝ\nf : ι → α → ℂ\nhu : Summable u\nh : ∀ᶠ (i : ι) in cofinite, ∀ x ∈ K, ‖f i x‖ ≤ u i\ni : ι\nhi : ∀ x ∈ K, ‖f i x‖ ≤ u i\nhi' : u i ≤ 1 / 2\nx : α\nhx : x ∈ K\n⊢ 3 / 2 * ‖f i x‖ ≤ 3 / 2 * u i",
"usedConstants": [
"Real.instIsOrderedRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 46
} | [
{
"pp": "case refine_1\nα : Type u_1\nι : Type u_2\ns : Set α\nf : ι → α → ℂ\nhf : SummableUniformlyOn (fun i x ↦ log (f i x)) s\nhfn : ∀ x ∈ s, ∀ (i : ι), f i x ≠ 0\nhg : BddAbove ((fun x ↦ (∑' (i : ι), log (f i x)).re) '' s)\nr : α → ℂ\nhr : HasSumUniformlyOn (fun i x ↦ log (f i x)) r s\n⊢ BddAbove ((fun x ↦ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn | {
"line": 101,
"column": 74
} | {
"line": 101,
"column": 85
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nK : Set α\nu : ι → ℝ\nR : Type u_3\ninst✝³ : NormedCommRing R\ninst✝² : NormOneClass R\ninst✝¹ : CompleteSpace R\ninst✝ : TopologicalSpace α\nf : ι → α → R\nhK : IsCompact K\nhu : Summable u\nh : ∀ᶠ (i : ι) in cofinite, ∀ x ∈ K, ‖f i x‖ ≤ u i\nhcts : ∀ (i : ι), ContinuousOn ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.UpperLower | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 30
} | [
{
"pp": "ι : Type u_2\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ MonotoneOn (dist x) (Ici x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.UpperLower | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 30
} | [
{
"pp": "ι : Type u_2\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ AntitoneOn (dist x) (Iic x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Int | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 55
} | [
{
"pp": "n : ℤ\n⊢ ↑n.toNat + ↑(-n).toNat = ‖n‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 103,
"column": 16
} | {
"line": 103,
"column": 38
} | [
{
"pp": "ι : Type uι\ninst✝⁵ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type uF\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nx : ⨂[𝕜] (i : ι), E i\nx✝ : Continuo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 18
} | [
{
"pp": "case h\nι : Type uι\ninst✝³ : Fintype ι\n𝕜 : Type u𝕜\ninst✝² : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\np : Seminorm 𝕜 (⨂[𝕜] (i : ι), E i)\nG : Type (max uι u𝕜 uE)\nx✝¹ : SeminormedAddCommGroup G\nx✝ : Nor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 138,
"column": 2
} | {
"line": 139,
"column": 9
} | [
{
"pp": "ι : Type uι\ninst✝³ : Fintype ι\n𝕜 : Type u𝕜\ninst✝² : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E i)\nx : ⨂[𝕜] (i : ι), E i\n⊢ injectiveSeminorm x = ⨆ p, ↑p x",
"usedConstants": [
"PiTensorProduct.instMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Basic | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 92
} | [
{
"pp": "case a\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\nt₀ ε y : ℝ\nhy : y ∈ Metric.ball t₀ ε\n⊢ HasDerivAt γ (v y (γ y)) y ↔ HasDerivWithinAt γ (v y (γ y)) (Metric.ball t₀ ε) y",
"usedConstants": [
"IsIntegralCurveOn._proof_1",
"HasDerivA... | exact ⟨HasDerivAt.hasDerivWithinAt, fun h ↦ h.hasDerivAt (Metric.isOpen_ball.mem_nhds hy)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 48
} | [
{
"pp": "α : Type u_1\ninst✝ : EMetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\nx y : α\nh : edist x y ≠ ∞\nhy : IsFixedPt f y\n⊢ edist x y ≤ edist x (f x) / (1 - ↑K)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 66
} | [
{
"pp": "α : Type u_1\ninst✝ : EMetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\nx y : α\nhx : IsFixedPt f x\nhy : IsFixedPt f y\nh : ¬edist x y = ∞\n⊢ edist x y ≤ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.DiscreteGronwall | {
"line": 56,
"column": 10
} | {
"line": 56,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nu b c : ℕ → R\nn₀ : ℕ\nhu : ∀ n ≥ n₀, u (n + 1) ≤ c n * u n + b n\nhc : ∀ n ≥ n₀, 0 ≤ c n\nn k : ℕ\nhk : n₀ ≤ k\nih : u k ≤ u n₀ * ∏ i ∈ Ico n₀ k, c i + ∑ k_1 ∈ Ico n₀ k, b k_1 * ∏ i ∈ Ico (k_1 + 1) k, c i\nhck : 0... | prod_Ico_succ_top (by have := mem_Ico.mp hj; omega) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.ODE.DiscreteGronwall | {
"line": 87,
"column": 8
} | {
"line": 87,
"column": 29
} | [
{
"pp": "case hbc\nu b c : ℕ → ℝ\nn₀ : ℕ\nhun₀ : 0 ≤ u n₀\nhu : ∀ n ≥ n₀, u (n + 1) ≤ (1 + c n) * u n + b n\nhc : ∀ n ≥ n₀, 0 ≤ c n\nhb : ∀ n ≥ n₀, 0 ≤ b n\nn : ℕ\nhn : n₀ ≤ n\n⊢ ∏ i ∈ Ico n₀ n, (1 + c i) ≤ rexp (∑ i ∈ Ico n₀ n, c i)",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝ : MetricSpace α\nK : ℝ≥0\nf : α → α\nhf : ContractingWith K f\nx y : α\nhy : IsFixedPt f y\n⊢ dist x y ≤ dist x (f x) / (1 - ↑K)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝² : MetricSpace α\nK : ℝ≥0\nf : α → α\ninst✝¹ : Nonempty α\ninst✝ : CompleteSpace α\nn : ℕ\nhf : ContractingWith K f^[n]\nx : α := fixedPoint f^[n] hf\nhx : f^[n] x = x\nthis : ¬IsFixedPt f x\n⊢ ↑K * dist x (f x) < dist x (f x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Transform | {
"line": 40,
"column": 2
} | {
"line": 41,
"column": 94
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\ns : Set ℝ\nhγ : IsIntegralCurveOn γ v s\ndt t : ℝ\nht : t ∈ -dt +ᵥ s\n⊢ HasDerivWithinAt (γ ∘ fun x ↦ x + dt) ((v ∘ fun x ↦ x + dt) t ((γ ∘ fun x ↦ x + dt) t)) (-dt +ᵥ s) t",
"usedConstants": [
"No... | rw [comp_apply, hasDerivWithinAt_iff_hasFDerivWithinAt, Function.comp_def,
hasFDerivWithinAt_comp_add_right, ← hasDerivWithinAt_iff_hasFDerivWithinAt, vadd_neg_vadd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.ODE.Transform | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\ns : Set ℝ\ndt : ℝ\n⊢ IsIntegralCurveOn (γ ∘ fun x ↦ x - dt) (v ∘ fun x ↦ x - dt) (dt +ᵥ s) ↔ IsIntegralCurveOn γ v s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Transform | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\nt₀ dt : ℝ\n⊢ IsIntegralCurveAt (γ ∘ fun x ↦ x - dt) (v ∘ fun x ↦ x - dt) (t₀ + dt) ↔ IsIntegralCurveAt γ v t₀",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Transform | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\nhγ : IsIntegralCurveOn γ v univ\ndt : ℝ\n⊢ IsIntegralCurveOn (γ ∘ fun x ↦ x + dt) (v ∘ fun x ↦ x + dt) univ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Transform | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nγ : ℝ → E\nv : ℝ → E → E\ndt : ℝ\n⊢ IsIntegralCurve (γ ∘ fun x ↦ x - dt) (v ∘ fun x ↦ x - dt) ↔ IsIntegralCurve γ v",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Gronwall | {
"line": 151,
"column": 77
} | {
"line": 151,
"column": 88
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → E\nK a b : ℝ\nhf : ContinuousOn f (Icc a b)\nhf' : ∀ x ∈ Ico a b, HasDerivWithinAt f (f' x) (Ici x) x\nha : f a = 0\nbound : ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖\nx : ℝ\nhx : x ∈ Icc a b\n⊢ ∀ x ∈ Ico a b, ‖f' x‖ ≤ K * ‖f x‖ +... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 43
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\nα : ℝ → E\ns : Set ℝ\nu : Set E\nn : WithTop ℕ∞\nhf : ContDiffOn ℝ n (uncurry f) (s ×ˢ u)\nhα : ContDiffOn ℝ n α s\nhmem : ∀ t ∈ s, α t ∈ u\n⊢ ContDiffOn ℝ n (fun t ↦ f t (α t)) s",
"usedConstants": [
"InnerP... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 176,
"column": 63
} | {
"line": 176,
"column": 74
} | [
{
"pp": "case mk.mk\nE : Type u_1\ninst✝ : NormedAddCommGroup E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L : ℝ≥0\ntoFun✝¹ : ↑(Icc tmin tmax) → E\nlipschitzWith✝¹ : LipschitzWith L toFun✝¹\nmem_closedBall₀✝¹ : toFun✝¹ t₀ ∈ closedBall x₀ ↑r\ntoFun✝ : ↑(Icc tmin tmax) → E\nlipschitzWith✝ : LipschitzWith ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\nL : ℝ≥0\nα : FunSpace t₀ x₀ 0 L\n⊢ α.toFun t₀ = x₀",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 339,
"column": 12
} | {
"line": 339,
"column": 23
} | [
{
"pp": "case zero\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ x : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nhx : x ∈ closedBall x₀ ↑r\nα β : FunSpace t₀ x₀ r L\nt : ↑(Icc tmin tmax)\n⊢ dist (((next hf hx)^[0] α).t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Basic | {
"line": 180,
"column": 10
} | {
"line": 180,
"column": 37
} | [
{
"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⊢ P.degree < Q.degree",
"usedConstants": [
"With... | leadingCoeff_eq_zero.1 hP0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 15
} | [
{
"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[�... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Basic | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 29
} | [
{
"pp": "𝕜 : 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) atBot (𝓝 0)\n⊢ Q.comp (-X) ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | rw [Ne, comp_eq_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Polynomial.Basic | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : (Q.comp (-X)).degree < (P.comp (-X)).degree\nhQ : Q ≠ 0\n⊢ Q.comp (-X) ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | rw [Ne, comp_eq_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 45
} | [
{
"pp": "𝕜 : 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[𝕜] W\nhV :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Basic | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 20
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : P.degree ≤ Q.degree\nhp : P = 0\n⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 115,
"column": 15
} | {
"line": 115,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\ne : V ≃L[𝕜] W\nα α' : 𝕜\nhα : α ≠ 0\nhα2 : α' * α' = α⁻¹\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 118,
"column": 15
} | {
"line": 118,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\ne : V ≃L[𝕜] W\nα α' : 𝕜\nhα : α ≠ 0\nhα2 : α' * α' = α⁻¹\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Fourier | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 13
} | [
{
"pp": "p : ℂ[X]\n⊢ Integrable (⇑(toAddCircle p)) haarAddCircle",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 15
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Fourier | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 17
} | [
{
"pp": "case inl\np : ℂ[X]\nthis :\n ∑' (i : ℤ), ‖if 0 ≤ i then p.coeff i.natAbs else 0‖ ^ 2 =\n ∫ (t : AddCircle (2 * π)), ‖↑↑((ContinuousMap.toLp 2 haarAddCircle ℂ) (toAddCircle p)) t‖ ^ 2 ∂haarAddCircle\nb : ℕ\nhb : ↑b ∉ Finset.map { toFun := Nat.cast, inj' := ⋯ } p.support\n⊢ ‖if 0 ≤ ↑b then p.coeff (↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 188,
"column": 23
} | {
"line": 188,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuous ⇑f\nh✝ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 195,
"column": 15
} | {
"line": 195,
"column": 74
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuous ⇑f\nh✝ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.GaussNorm | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 23
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : Semiring R\nv : R → ℝ\nc : ℝ\nf : R⟦X⟧\nh : HasGaussNorm v c f\nx✝ : ℝ\ny : ℕ\nhy : v ((coeff y) f) * c ^ y = x✝\n⊢ v ((MvPowerSeries.coeff ((Finsupp.uniqueEquiv ()).symm y)) f) * c ^ ((Finsupp.uniqueEquiv ()).symm y) PUnit.unit = x✝",
"usedConstants": [
"Finsupp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 15
} | [
{
"pp": "case h\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuous ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 456,
"column": 2
} | {
"line": 472,
"column": 64
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L K : ℝ≥0\ninst✝ : CompleteSpace E\nhf : IsPicardLindelof f t₀ x₀ a r L K\n⊢ ∃ L',\n ∀ (x y : E) (hx : x ∈ closedBall x₀ ↑r) (hy : y ∈ closedBall x₀ ↑r) (α β : FunS... | obtain ⟨m, C, h⟩ := exists_contractingWith_iterate_next hf
let L' := (∑ i ∈ Finset.range m, (K * max (tmax - t₀) (t₀ - tmin)) ^ i / i !) * (1 - C)⁻¹
have hL' : 0 ≤ L' := by
have : 0 ≤ max (tmax - t₀) (t₀ - tmin) := le_max_of_le_left <| sub_nonneg_of_le t₀.2.2
positivity
refine ⟨.mk L' hL', fun x y hx hy α... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 456,
"column": 2
} | {
"line": 472,
"column": 64
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L K : ℝ≥0\ninst✝ : CompleteSpace E\nhf : IsPicardLindelof f t₀ x₀ a r L K\n⊢ ∃ L',\n ∀ (x y : E) (hx : x ∈ closedBall x₀ ↑r) (hy : y ∈ closedBall x₀ ↑r) (α β : FunS... | obtain ⟨m, C, h⟩ := exists_contractingWith_iterate_next hf
let L' := (∑ i ∈ Finset.range m, (K * max (tmax - t₀) (t₀ - tmin)) ^ i / i !) * (1 - C)⁻¹
have hL' : 0 ≤ L' := by
have : 0 ≤ max (tmax - t₀) (t₀ - tmin) := le_max_of_le_left <| sub_nonneg_of_le t₀.2.2
positivity
refine ⟨.mk L' hL', fun x y hx hy α... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 236,
"column": 51
} | {
"line": 236,
"column": 62
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 561,
"column": 2
} | {
"line": 561,
"column": 12
} | [
{
"pp": "case coe\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E → E\nα : ℝ → E\nu : Set E\nt₀ tmin tmax : ℝ\nht₀ : t₀ ∈ Icc tmin tmax\nhα : ContinuousOn α (Icc tmin tmax)\nhmem : ∀ t ∈ Icc tmin tmax, α t ∈ u\nx₀ : E\nheqon : ∀ t ∈ Icc tmin tmax, α t =... | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Analysis.Polynomial.Norm | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedRing A\np : A[X]\ni : ℕ\n⊢ ‖p.coeff i‖ ≤ p.supNorm",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Norm | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedRing A\np : A[X]\n⊢ ∃ i, p.supNorm = ‖p.coeff i‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Norm | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 27
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedRing A\np : A[X]\n⊢ p.supNorm ∈ Set.range fun x ↦ ‖p.coeff x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real",
"SeminormedRing.toRing",
"Membership.mem",
"Exists",
"id",
"Polynomial.coe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Norm | {
"line": 90,
"column": 52
} | {
"line": 90,
"column": 81
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedRing A\np : A[X]\n⊢ p.supNorm ∈ upperBounds (Set.range fun x ↦ ‖p.coeff x‖)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real.instLE",
"Real",
"Preorder.toLE",
"SeminormedRing.toRing",
"Membersh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 116,
"column": 8
} | {
"line": 116,
"column": 23
} | [
{
"pp": "case inl.h\nR : Type u_1\nσ : Type u_2\ninst✝ : Semiring R\nv : R → ℝ\nc : σ → ℝ\nf g : MvPowerSeries σ R\nhc : 0 ≤ c\nvNonneg : ∀ (a : R), v a ≥ 0\nhv : ∀ (x y : R), v (x + y) ≤ max (v x) (v y)\nhbfd : HasGaussNorm v c f\nhbgd : HasGaussNorm v c g\nH : ∀ (t : σ →₀ ℕ), 0 ≤ ∏ i ∈ t.support, c i ^ t i\nF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 23
} | [
{
"pp": "case inr.h\nR : Type u_1\nσ : Type u_2\ninst✝ : Semiring R\nv : R → ℝ\nc : σ → ℝ\nf g : MvPowerSeries σ R\nhc : 0 ≤ c\nvNonneg : ∀ (a : R), v a ≥ 0\nhv : ∀ (x y : R), v (x + y) ≤ max (v x) (v y)\nhbfd : HasGaussNorm v c f\nhbgd : HasGaussNorm v c g\nH : ∀ (t : σ →₀ ℕ), 0 ≤ ∏ i ∈ t.support, c i ^ t i\nF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Order | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 17
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\nh : Even P.natDegree\nhlc' : 0 ≤ (P.comp (-X)).leadingCoeff\nthis : 0 < eval (-x) (P.comp (-X))\n⊢ 0 < eval x P",
"usedConstants": [
"Eq.mpr",
"Polyno... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 797,
"column": 2
} | {
"line": 797,
"column": 29
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L K : ℝ≥0\nhf : IsPicardLindelof f t₀ x₀ a r L K\nα : E → ℝ → E\nhα1 : ∀ x ∈ closedBall x₀ ↑r, α x ↑t₀ = x ∧ ∀ t ∈ Icc tmin tmax, HasDerivWith... | refine ⟨uncurry α, hα1, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Polynomial.Order | {
"line": 105,
"column": 28
} | {
"line": 105,
"column": 39
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\nh : Odd P.natDegree\nhlc' : 0 ≤ -(P.comp (-X)).leadingCoeff\nthis : eval (-x) (P.comp (-X)) < 0\n⊢ eval x P < 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.ContinuousMap | {
"line": 39,
"column": 28
} | {
"line": 39,
"column": 39
} | [
{
"pp": "case h\nX : Type u_1\n𝕜 : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : RCLike 𝕜\nf g : C(X, ℝ)\nhfg : realToRCLike 𝕜 f = realToRCLike 𝕜 g\nx : X\n⊢ f x = g x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 130,
"column": 10
} | {
"line": 130,
"column": 48
} | [
{
"pp": "R : Type u_1\nF : Type u_2\ninst✝³ : Semiring R\ninst✝² : FunLike F R ℝ\nv : F\nc : ℝ\np : R[X]\ninst✝¹ : ZeroHomClass F R ℝ\ninst✝ : NonnegHomClass F R ℝ\nh_eq_zero : ∀ (x : R), v x = 0 → x = 0\nhc : 0 < c\n⊢ PowerSeries.HasGaussNorm (⇑v) c ↑p",
"usedConstants": [
"Eq.mpr",
"Real.instL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 40
} | [
{
"pp": "case hbd\nR : Type u_1\nF : Type u_2\ninst✝³ : Semiring R\ninst✝² : FunLike F R ℝ\nv : F\nc : ℝ\np : R[X]\ninst✝¹ : ZeroHomClass F R ℝ\ninst✝ : NonnegHomClass F R ℝ\nhc : 0 ≤ c\ni : ℕ\n⊢ PowerSeries.HasGaussNorm (⇑v) c ↑p",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 183,
"column": 53
} | {
"line": 186,
"column": 37
} | [
{
"pp": "R : Type u_1\nF : Type u_2\ninst✝³ : Semiring R\ninst✝² : FunLike F R ℝ\nv : F\ninst✝¹ : ZeroHomClass F R ℝ\ninst✝ : NonnegHomClass F R ℝ\nhna : IsNonarchimedean ⇑v\nc : ℝ\nhc : 0 ≤ c\np q : R[X]\nh✝¹ : p ≠ 0\nh✝ : q ≠ 0\nhpq : p + q ≠ 0\ni : ℕ\na✝ : i ∈ (p + q).support\n⊢ v ((p + q).coeff i) * c ^ i ≤... | by
rw [coeff_add]
gcongr
exact hna (p.coeff i) (q.coeff i) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 41
} | [
{
"pp": "case pos\np q : ℂ[X]\nhpq : p * q = 0\n⊢ (p * q).mahlerMeasure = p.mahlerMeasure * q.mahlerMeasure",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Real",
"HMul.hMul",
"Polynomial.mahlerMeasure_eq_zero_iff._simp_1",
"Real.instZero",
"congrArg"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 340,
"column": 8
} | {
"line": 340,
"column": 20
} | [
{
"pp": "p : ℂ[X]\nthis✝¹ : IsFiniteMeasure (volume.restrict (uIoc 0 (2 * π)))\nthis✝ : NeZero (volume (uIoc 0 (2 * π)))\nhp : p ≠ 0\nthis : ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), 0 < ‖eval (circleMap 0 1 θ) p‖\nhlogAe :\n ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), rexp (log ‖eval (circleMap 0 1 θ) p... | rw [sqrt_sq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Real.Hyperreal | {
"line": 211,
"column": 13
} | {
"line": 211,
"column": 24
} | [
{
"pp": "n : ℕ\n⊢ n • |ArchimedeanOrder.val (ArchimedeanOrder.of 1)| < |ArchimedeanOrder.val (ArchimedeanOrder.of ω)|",
"usedConstants": [
"ArchimedeanOrder.of",
"Hyperreal.instField",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Preorder.toLT",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 13
} | [
{
"pp": "⊢ Irrational √2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 192,
"column": 48
} | {
"line": 192,
"column": 79
} | [
{
"pp": "x : ℝ\nh : Irrational x\n⊢ x ≠ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 45
} | [
{
"pp": "q : ℚ\nx : ℝ\nh : Irrational x\n⊢ Irrational (x - ↑q)",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instSub",
"AddMonoid.toAddZeroClass",
"Real.instRatCast",
"sub_eq_add_neg",
"HSub.hSub",
"AddZeroClass.toAddZero",
"id",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 35
} | [
{
"pp": "q : ℚ\nx : ℝ\nh : Irrational x\n⊢ Irrational (↑q - x)",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instSub",
"AddMonoid.toAddZeroClass",
"Real.instRatCast",
"sub_eq_add_neg",
"HSub.hSub",
"AddZeroClass.toAddZero",
"id",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 272,
"column": 28
} | {
"line": 272,
"column": 71
} | [
{
"pp": "q : ℚ\nx : ℝ\nh : Irrational (x - ↑q)\n⊢ Irrational (x + ↑(-q))",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"DivisionRing.toRatCast",
"congrArg",
"Real.instRatCast",
"Rat",
"id",
"Rat.cast",
"DivisionRing.toRing",
"S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 274,
"column": 31
} | {
"line": 274,
"column": 64
} | [
{
"pp": "q : ℚ\nx : ℝ\nh : Irrational (↑q - x)\n⊢ Irrational (↑q + -x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 37
} | [
{
"pp": "x : ℝ\nh : Irrational x\nm : ℤ\n⊢ Irrational (x - ↑m)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 37
} | [
{
"pp": "x : ℝ\nh : Irrational x\nm : ℤ\n⊢ Irrational (↑m - x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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