module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 152,
"column": 8
} | {
"line": 152,
"column": 66
} | {
"line": 152,
"column": 67
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : NormedField L\ninst✝ : Algebra K L\nhd : DenseRange ⇑(algebraMap K L)\nf : L[X]\nhf : f.Monic\nε : ℝ\nhε : ε > 0\nh : ¬f.natDegree = 0\nc : ℕ → K\nhc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε\ni : ℕ\nhi : i ∈ Finset.Iio f.natDegree... | [
"K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : NormedField L\ninst✝ : Algebra K L\nhd : DenseRange ⇑(algebraMap K L)\nf : L[X]\nhf : f.Monic\nε : ℝ\nhε : ε > 0\nh : ¬f.natDegree = 0\nc : ℕ → K\nhc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε\ni : ℕ\nhi : i ∈ Finset.Iio f.natDegree\n⊢ i < f.na... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 24
} | {
"line": 116,
"column": 25
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nx : R\nhx : μ x = 0\nh0 : ∀ (n : ℕ), 1 ≤ n → μ (x ^ n) ^ (1 / ↑n) = 0\nhL0 : ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n) = 0\n⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 (smoothingFun μ x))",
"ppTerm": "?m.152",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nx : R\nhx : μ x = 0\nh0 : ∀ (n : ℕ), 1 ≤ n → μ (x ^ n) ^ (1 / ↑n) = 0\nhL0 : ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n) = 0\n⊢ Tendsto (smoothingSeminormSeq μ x) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Approximation | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 54
} | {
"line": 160,
"column": 55
} | [
{
"pp": "case h.refine_2.inl\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : NormedField L\ninst✝ : Algebra K L\nhd : DenseRange ⇑(algebraMap K L)\nf : L[X]\nhf : f.Monic\nε : ℝ\nhε : ε > 0\nh✝ : ¬f.natDegree = 0\nc : ℕ → K\nhc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε\nhdeg : (C 1 * X ... | [
"case h.refine_2.inl\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : NormedField L\ninst✝ : Algebra K L\nhd : DenseRange ⇑(algebraMap K L)\nf : L[X]\nhf : f.Monic\nε : ℝ\nhε : ε > 0\nh✝ : ¬f.natDegree = 0\nc : ℕ → K\nhc : ∀ (i : ℕ), dist (f.coeff i) ((algebraMap K L) (c i)) < ε\nhdeg : (C 1 * X ^ f.natDegre... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 22
} | {
"line": 232,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhx : ∀ (y : R), f (x * y) = f x * f y\ny : R\nhseq : seminormFromConst_seq c f (x * y) = fun n ↦ f x * seminormFromConst_seq c f y n\n⊢ Tendsto (seminormFromConst_seq c f (x * y)) atTop (... | [
"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhx : ∀ (y : R), f (x * y) = f x * f y\ny : R\nhseq : seminormFromConst_seq c f (x * y) = fun n ↦ f x * seminormFromConst_seq c f y n\n⊢ Tendsto (fun n ↦ f x * seminormFromConst_seq c f y n) atTop (𝓝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 211,
"column": 8
} | {
"line": 211,
"column": 36
} | {
"line": 211,
"column": 37
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x ≠ 0\nL : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)\nhL0 : 0 ≤ L\nε : ℝ\nhε : ε > 0\nm1 : ℕ+\nhm1 : μ (x ^ ↑m1) ^ (1 / ↑↑m1) < (⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)) + ε / 2\nm2 : ℕ\nhm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) ... | [
"case pos\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x ≠ 0\nL : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)\nhL0 : 0 ≤ L\nε : ℝ\nhε : ε > 0\nm1 : ℕ+\nhm1 : μ (x ^ ↑m1) ^ (1 / ↑↑m1) < (⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)) + ε / 2\nm2 : ℕ\nhm2 : ∀ n ≥ m2, (L + ε / 2) ^ (-(↑(n % ↑m1) / ↑n)) * (μ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 21
} | {
"line": 262,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhlim : Tendsto (fun n ↦ seminormFromConst_seq c f x (n + 1)) atTop (𝓝 (seminormFromConst' c f x))\nhterm : seminormFromConst_seq c f (c * x) = fun n ↦ f c * seminormFromConst_seq c f x (... | [
"R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\nhlim : Tendsto (fun n ↦ seminormFromConst_seq c f x (n + 1)) atTop (𝓝 (seminormFromConst' c f x))\nhterm : seminormFromConst_seq c f (c * x) = fun n ↦ f c * seminormFromConst_seq c f x (n + 1)\n⊢ Te... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsConjRoot | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 44
} | {
"line": 113,
"column": 45
} | [
{
"pp": "K : Type u_2\nS : Type u_4\ninst✝² : CommRing S\ninst✝¹ : Field K\ninst✝ : Algebra K S\nx y : S\nr : K\nh : IsConjRoot K x y\n⊢ IsConjRoot K (x - (algebraMap K S) r) (y - (algebraMap K S) r)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap"... | [
"K : Type u_2\nS : Type u_4\ninst✝² : CommRing S\ninst✝¹ : Field K\ninst✝ : Algebra K S\nx y : S\nr : K\nh : IsConjRoot K x y\n⊢ IsConjRoot K (x + -(algebraMap K S) r) (y + -(algebraMap K S) r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Minpoly.IsConjRoot | {
"line": 227,
"column": 53
} | {
"line": 230,
"column": 34
} | {
"line": 232,
"column": 0
} | [
{
"pp": "K : Type u_2\nS : Type u_4\ninst✝³ : CommRing S\ninst✝² : Field K\ninst✝¹ : Algebra K S\ninst✝ : IsDomain S\nx y : S\nh : IsIntegral K x\n⊢ IsConjRoot K x y ↔ y ∈ (minpoly K x).aroots S",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instIsTorsionFreeOfIsDom... | [] | by
rw [Polynomial.mem_aroots, isConjRoot_iff_aeval_eq_zero h]
simp only [iff_and_self]
exact fun _ => minpoly.ne_zero h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Minpoly.IsConjRoot | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 30
} | {
"line": 274,
"column": 31
} | [
{
"pp": "R : Type u_1\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : IsDomain S\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R S\nr : R\nx : S\nh : X - C r = minpoly R x\nhf : Function.Injective ⇑(algebraMap R S)\nthis : x ∈ (X - C r).aroots S\n⊢ x = (algebraMap R S) r",
... | [
"R : Type u_1\nS : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : IsDomain S\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R S\nr : R\nx : S\nh : X - C r = minpoly R x\nhf : Function.Injective ⇑(algebraMap R S)\nthis : x ∈ (X - C r).aroots S\n⊢ x = (algebraMap R S) r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 82,
"column": 25
} | {
"line": 82,
"column": 44
} | {
"line": 82,
"column": 45
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral... | [
"K : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral K y\nkr : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 98,
"column": 10
} | {
"line": 98,
"column": 48
} | {
"line": 98,
"column": 49
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral... | [
"K : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral K y\nkr : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 107,
"column": 12
} | {
"line": 107,
"column": 31
} | {
"line": 107,
"column": 32
} | [
{
"pp": "case a\nK : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : Is... | [
"case a\nK : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral K y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 108,
"column": 12
} | {
"line": 108,
"column": 46
} | {
"line": 108,
"column": 47
} | [
{
"pp": "case a\nK : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : Is... | [
"case a\nK : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral K y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 43
} | {
"line": 252,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\n⊢ ∀ᶠ (c : ℕ) in atTop, 0 ≤ smoothingSeminormSeq μ x c",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"Filter.Eventually",
"instArchimedeanN... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\n⊢ ∃ a, ∀ (b : ℕ), a ≤ b → 0 ≤ smoothingSeminormSeq μ x b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 18
} | {
"line": 294,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nn : ℕ\n⊢ mu μ hn n ≤ n",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Re... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nn : ℕ\n⊢ Classical.choose ⋯ ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Instances | {
"line": 27,
"column": 59
} | {
"line": 27,
"column": 70
} | {
"line": 27,
"column": 71
} | [
{
"pp": "F : Type u_1\ninst✝ : NormedField F\nf : Filter F\nhc : Cauchy f\nhn : nhds 0 ⊓ f = ⊥\nδ : ℝ\nδ_pos : δ > 0\nhδ : ∀ᶠ (y : F) in f, δ ≤ ‖y‖\ny : F\nhy : δ ≤ ‖y‖\n⊢ δ ≤ ((fun x ↦ ‖x⁻¹‖) y)⁻¹",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Re... | [
"F : Type u_1\ninst✝ : NormedField F\nf : Filter F\nhc : Cauchy f\nhn : nhds 0 ⊓ f = ⊥\nδ : ℝ\nδ_pos : δ > 0\nhδ : ∀ᶠ (y : F) in f, δ ≤ ‖y‖\ny : F\nhy : δ ≤ ‖y‖\n⊢ δ ≤ ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Instances | {
"line": 28,
"column": 56
} | {
"line": 28,
"column": 67
} | {
"line": 28,
"column": 68
} | [
{
"pp": "F : Type u_1\ninst✝ : NormedField F\nf : Filter F\nhc : Cauchy f\nhn : nhds 0 ⊓ f = ⊥\nδ : ℝ\nδ_pos : δ > 0\nhδ : ∀ᶠ (y : F) in f, δ ≤ ‖y‖\nf_bdd : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun x ↦ ‖x⁻¹‖\ny : F\nhy : δ ≤ ‖y‖\n⊢ y ≠ 0",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
... | [
"F : Type u_1\ninst✝ : NormedField F\nf : Filter F\nhc : Cauchy f\nhn : nhds 0 ⊓ f = ⊥\nδ : ℝ\nδ_pos : δ > 0\nhδ : ∀ᶠ (y : F) in f, δ ≤ ‖y‖\nf_bdd : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun x ↦ ‖x⁻¹‖\ny : F\nhy : δ ≤ ‖y‖\n⊢ ¬y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.ProperSpace | {
"line": 48,
"column": 26
} | {
"line": 48,
"column": 53
} | {
"line": 48,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : WeaklyLocallyCompactSpace 𝕜\nr : ℝ\nrpos : 0 < r\nhr : IsCompact (closedBall 0 r)\nc : 𝕜\nhc : 1 < ‖c‖\nn : ℕ\nthis : c ^ n ≠ 0\nx✝ : 𝕜\n⊢ 0 < ?m.159",
"ppTerm": "?m.160",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : WeaklyLocallyCompactSpace 𝕜\nr : ℝ\nrpos : 0 < r\nhr : IsCompact (closedBall 0 r)\nc : 𝕜\nhc : 1 < ‖c‖\nn : ℕ\nthis : c ^ n ≠ 0\nx✝ : 𝕜\n⊢ 0 < ?m.159"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 57,
"column": 10
} | {
"line": 57,
"column": 21
} | {
"line": 57,
"column": 22
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDegree ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 61,
"column": 10
} | {
"line": 61,
"column": 21
} | {
"line": 61,
"column": 22
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDegree ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 388,
"column": 2
} | {
"line": 406,
"column": 55
} | {
"line": 407,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\ns : ℕ → ℕ\nhs_le : ∀ (n : ℕ), s n ≤ n\nx : R\na : ℝ\na_in : a ∈ Set.Icc 0 1\nψ : ℕ → ℕ\nhψ_mono : StrictMono ψ\nhψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 a)\nha : a = 0\n⊢ limsup (fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ... | [
"case neg\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\ns : ℕ → ℕ\nhs_le : ∀ (n : ℕ), s n ≤ n\nx : R\na : ℝ\na_in : a ∈ Set.Icc 0 1\nψ : ℕ → ℕ\nhψ_mono : StrictMono ψ\nhψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 a)\nha : ¬a = 0\n⊢ limsup (fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n))) at... | · rw [ha] at hψ_lim
calc limsup (fun n : ℕ => μ (x ^ s (ψ n)) ^ (1 / (ψ n : ℝ))) atTop ≤
limsup (fun n : ℕ => μ x ^ ((s (ψ n) : ℝ) * (1 / (ψ n : ℝ)))) atTop := by
apply csInf_le_csInf _ (μ_nonempty μ hs_le ψ)
· intro b hb
simp only [eventually_map, eventually_atTop, Set.mem... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 98,
"column": 10
} | {
"line": 98,
"column": 42
} | {
"line": 98,
"column": 43
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDegree ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 102,
"column": 6
} | {
"line": 102,
"column": 86
} | {
"line": 102,
"column": 87
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDegree ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 68
} | {
"line": 117,
"column": 8
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDegree ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Field.Dense | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 69
} | {
"line": 119,
"column": 6
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.nat... | [
"K : Type u_1\nL : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : NontriviallyNormedField L\ninst✝⁴ : CompleteSpace L\ninst✝³ : CharZero L\ninst✝² : IsUltrametricDist L\ninst✝¹ : Algebra K L\nhi : DenseRange ⇑(algebraMap K L)\ninst✝ : IsAlgClosed K\nf : L[X]\nfmon : f.Monic\nfirr : Irreducible f\nfnatdeg0 : f.natDegree ≠ 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 82
} | {
"line": 515,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nhna : IsNonarchimedean ⇑μ\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nmu : ℕ → ℕ := ⋯\nnu : ℕ → ℕ := ⋯\nhnu : nu = fun n ↦ n - mu n\nhmu_le : ∀ (n : ℕ), mu n ≤ n\nhmu_b... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nhna : IsNonarchimedean ⇑μ\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nmu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n\nnu : ℕ → ℕ := fun n ↦ n - mu n\nhnu : nu = fun n ↦ n - mu n\nhmu_le : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 147,
"column": 4
} | {
"line": 147,
"column": 15
} | {
"line": 147,
"column": 16
} | [
{
"pp": "G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nx : G\n⊢ ‖f x‖ ≤ ‖f.completion‖ * ‖x‖",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nx : G\n⊢ ‖f x‖ ≤ ‖f.completion‖ * ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 67
} | {
"line": 163,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\nhatg : Completion G\nhatg_in : f.completion hatg = 0\nε : ℝ\nε_pos : 0 < ε\nC' : ℝ\nC'_pos : C' > 0\nhC' : f.SurjectiveOnWith f.range C'\n... | [
"G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\nhatg : Completion G\nhatg_in : f.completion hatg = 0\nε : ℝ\nε_pos : 0 < ε\nC' : ℝ\nC'_pos : C' > 0\nhC' : f.SurjectiveOnWith f.range C'\nδ : ℝ\nδ_pos... | rcases exists_pos_mul_lt ε_pos (1 + C' * ‖f‖) with ⟨δ, δ_pos, hδ⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 36
} | {
"line": 170,
"column": 37
} | [
{
"pp": "G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\nhatg : Completion G\nhatg_in : f.completion hatg = 0\nε : ℝ\nε_pos : 0 < ε\nC' : ℝ\nC'_pos : C' > 0\nhC' : f.SurjectiveOnWith f.range C'\n... | [
"G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\nhatg : Completion G\nhatg_in : f.completion hatg = 0\nε : ℝ\nε_pos : 0 < ε\nC' : ℝ\nC'_pos : C' > 0\nhC' : f.SurjectiveOnWith f.range C'\nδ : ℝ\nδ_pos... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 571,
"column": 2
} | {
"line": 571,
"column": 20
} | {
"line": 571,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nm : ℕ\nhm : 1 ≤ m\nhlim : Tendsto (fun n ↦ smoothingSeminormSeq μ x (m * n)) atTop (𝓝 (smoothingFun μ x))\nh_eq : ∀ (n : ℕ), smoothingSeminormSeq μ x (m * n) ^ m = smoothingSeminormSeq μ (x ^ m) n\n⊢ Tendsto (fun x_1 ↦ smoothi... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nm : ℕ\nhm : 1 ≤ m\nhlim : Tendsto (fun n ↦ smoothingSeminormSeq μ x (m * n)) atTop (𝓝 (smoothingFun μ x))\nh_eq : ∀ (n : ℕ), smoothingSeminormSeq μ x (m * n) ^ m = smoothingSeminormSeq μ (x ^ m) n\n⊢ Tendsto (fun x_1 ↦ smoothingSeminormSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.HomCompletion | {
"line": 177,
"column": 16
} | {
"line": 177,
"column": 62
} | {
"line": 177,
"column": 63
} | [
{
"pp": "G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\nhatg : Completion G\nhatg_in : f.completion hatg = 0\nε : ℝ\nε_pos : 0 < ε\nC' : ℝ\nC'_pos : C' > 0\nhC' : f.SurjectiveOnWith f.range C'\n... | [
"G : Type u_1\ninst✝¹ : SeminormedAddCommGroup G\nH : Type u_2\ninst✝ : SeminormedAddCommGroup H\nf : NormedAddGroupHom G H\nC : ℝ\nh : f.SurjectiveOnWith f.range C\nhatg : Completion G\nhatg_in : f.completion hatg = 0\nε : ℝ\nε_pos : 0 < ε\nC' : ℝ\nC'_pos : C' > 0\nhC' : f.SurjectiveOnWith f.range C'\nδ : ℝ\nδ_pos... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.SemiNormedGrp.Kernels | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 74
} | {
"line": 102,
"column": 75
} | [
{
"pp": "V W : SemiNormedGrp\nf g : V ⟶ W\nv : { carrier := ↥(Hom.hom (f - g)).ker, str := AddSubgroup.seminormedAddCommGroup }.carrier\nthis : ↑v ∈ (Hom.hom (f - g)).ker\n⊢ (Hom.hom (ofHom (NormedAddGroupHom.incl (Hom.hom (f - g)).ker) ≫ f)) v =\n (Hom.hom (ofHom (NormedAddGroupHom.incl (Hom.hom (f - g)).ke... | [
"V W : SemiNormedGrp\nf g : V ⟶ W\nv : { carrier := ↥(Hom.hom (f - g)).ker, str := AddSubgroup.seminormedAddCommGroup }.carrier\nthis : ↑v ∈ (Hom.hom (f - g)).ker\n⊢ (Hom.hom f) ↑v = (Hom.hom g) ↑v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 334,
"column": 43
} | {
"line": 337,
"column": 60
} | {
"line": 338,
"column": 8
} | [
{
"pp": "K : Type u_2\ninst✝³ : NormedField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : DecidableEq L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\nhf1 : f 1 = 1\np : K[X]\ns : Multiset L\nhp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod\nh_le : 0 ≤ ⨆ ... | [] | by
have hs0 : 0 < s.card := hps ▸ hm.pos
obtain ⟨x, hx⟩ := card_pos_iff_exists_mem.mp hs0
exact Finset.card_pos.mpr ⟨x, mem_toFinset.mpr hx⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 360,
"column": 6
} | {
"line": 361,
"column": 61
} | {
"line": 362,
"column": 4
} | [
{
"pp": "case pos\nK : Type u_2\ninst✝³ : NormedField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : DecidableEq L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\nhf1 : f 1 = 1\np : K[X]\ns : Multiset L\nhp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod\nh_l... | [] | exact le_trans this (pow_le_pow_left₀ (apply_nonneg _ _)
(le_trans (by rw [if_pos hyx]) (le_ciSup h_bdd y)) _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 429,
"column": 2
} | {
"line": 429,
"column": 46
} | {
"line": 430,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\ny : L\nhy : y ≠ 0\nhy_alg : IsAlgebraic K y\n⊢ 0 < spectralNorm K L y",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"Real.instZero",
"s... | [
"K : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\ny : L\nhy : y ≠ 0\nhy_alg : IsAlgebraic K y\n⊢ 0 ≠ spectralNorm K L y"
] | apply lt_of_le_of_ne (spectralNorm_nonneg _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Group.SeparationQuotient | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 32
} | {
"line": 141,
"column": 33
} | [
{
"pp": "case mpr\nM : Type u_1\ninst✝ : SeminormedAddCommGroup M\nh : ∀ (x : M), ‖x‖ = 0\nx : M\n⊢ normedMk x = 0 x",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real",
"NormedAddGroupHom",
"Separ... | [
"case mpr\nM : Type u_1\ninst✝ : SeminormedAddCommGroup M\nh : ∀ (x : M), ‖x‖ = 0\nx : M\n⊢ ‖‖x‖‖ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 48,
"column": 4
} | {
"line": 48,
"column": 33
} | {
"line": 48,
"column": 34
} | [
{
"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... | [
"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✝ : IsEmpty ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 59,
"column": 38
} | {
"line": 59,
"column": 66
} | {
"line": 59,
"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... | [
"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 : Nonempty β\nh✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 67
} | {
"line": 71,
"column": 68
} | [
{
"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... | [
"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 : Nonempty β\nh✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Tannery | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 59
} | {
"line": 79,
"column": 60
} | [
{
"pp": "case 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‖ ≤ boun... | [
"case 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‖ ≤ bound k\nh✝¹ : N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 41,
"column": 31
} | {
"line": 41,
"column": 62
} | {
"line": 41,
"column": 63
} | [
{
"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‖",
"ppTerm": "?m.177",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"No... | [
"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⊢ ¬↑x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 47,
"column": 25
} | {
"line": 47,
"column": 36
} | {
"line": 47,
"column": 37
} | [
{
"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‖",
"ppTerm": "?m.204",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",... | [
"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⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 80
} | {
"line": 147,
"column": 4
} | [
{
"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",
"ppTerm": "?m.29",
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 15
} | {
"line": 85,
"column": 16
} | [
{
"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' (-(... | [
"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' (-(U • Subtype.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 246,
"column": 27
} | {
"line": 246,
"column": 72
} | {
"line": 246,
"column": 73
} | [
{
"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",
"ppTerm": "?m.42",
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.RadialEquiv | {
"line": 86,
"column": 41
} | {
"line": 86,
"column": 52
} | {
"line": 86,
"column": 53
} | [
{
"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",
"ppTerm": "?m.207",... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 335,
"column": 27
} | {
"line": 335,
"column": 50
} | {
"line": 335,
"column": 51
} | [
{
"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",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"usedF... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 855,
"column": 22
} | {
"line": 855,
"column": 56
} | {
"line": 856,
"column": 4
} | [
{
"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",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants"... | [] | simp [sub_self, spectralNorm_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 855,
"column": 22
} | {
"line": 855,
"column": 56
} | {
"line": 856,
"column": 4
} | [
{
"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",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants"... | [] | simp [sub_self, spectralNorm_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 855,
"column": 22
} | {
"line": 855,
"column": 56
} | {
"line": 856,
"column": 4
} | [
{
"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",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants"... | [] | simp [sub_self, spectralNorm_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 291,
"column": 26
} | {
"line": 291,
"column": 37
} | {
"line": 291,
"column": 38
} | [
{
"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)",
"ppTerm": "?m.109",
... | [
"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) = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.DoubleDual | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 13
} | {
"line": 74,
"column": 14
} | [
{
"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‖",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 319,
"column": 34
} | {
"line": 319,
"column": 45
} | {
"line": 319,
"column": 46
} | [
{
"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... | [
"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✝ : Module R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 123,
"column": 12
} | {
"line": 123,
"column": 59
} | {
"line": 123,
"column": 60
} | [
{
"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... | [
"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‖ + 2 • ‖(1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 29
} | {
"line": 131,
"column": 30
} | [
{
"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\nPR_eq_RPR : ∀ (R : M), IsLprojection X R → P * R = R * P * R\ne1 : Q * P - Q * P * Q = 0\n⊢ Q * P = Q * P * Q",
"ppTerm... | [
"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\nPR_eq_RPR : ∀ (R : M), IsLprojection X R → P * R = R * P * R\ne1 : Q * P - Q * P * Q = 0\n⊢ Q * P = Q * P * Q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MStructure | {
"line": 191,
"column": 18
} | {
"line": 191,
"column": 50
} | {
"line": 191,
"column": 51
} | [
{
"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)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsLprojection",
"congrArg",
... | [
"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 ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn | {
"line": 44,
"column": 75
} | {
"line": 44,
"column": 86
} | {
"line": 44,
"column": 87
} | [
{
"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",
"ppTerm": "?m.133",
"assigned": true,
"usedConsta... | [
"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⊢ ‖f i x‖ ≤ u i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 46
} | {
"line": 69,
"column": 47
} | [
{
"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 ↦ ... | [
"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 a ↦ (∑' (b : ι),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn | {
"line": 101,
"column": 74
} | {
"line": 101,
"column": 85
} | {
"line": 101,
"column": 86
} | [
{
"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 ... | [
"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 (f i) K\nhKe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 13
} | {
"line": 138,
"column": 14
} | [
{
"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‖",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"NormedCommRing.toS... | [
"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‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 163,
"column": 38
} | {
"line": 163,
"column": 53
} | {
"line": 163,
"column": 54
} | [
{
"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)",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
... | [
"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 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 50,
"column": 20
} | {
"line": 50,
"column": 50
} | {
"line": 50,
"column": 51
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\nE : α → Type u_3\nF : α → Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\ninst✝³ : (i : α) → NormedSpace 𝕜 (E i)\ninst✝² : (i : α) → NormedAddCommGroup (F i)\ninst✝¹ : (i : α) → NormedSpace 𝕜 (F i)\np✝ q r p : ℝ≥0∞\ninst✝ : Fact... | [
"α : Type u_1\n𝕜 : Type u_2\nE : α → Type u_3\nF : α → Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\ninst✝³ : (i : α) → NormedSpace 𝕜 (E i)\ninst✝² : (i : α) → NormedAddCommGroup (F i)\ninst✝¹ : (i : α) → NormedSpace 𝕜 (F i)\np✝ q r p : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\nT ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 36
} | {
"line": 58,
"column": 37
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\nE : α → Type u_3\nF : α → Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\ninst✝³ : (i : α) → NormedSpace 𝕜 (E i)\ninst✝² : (i : α) → NormedAddCommGroup (F i)\ninst✝¹ : (i : α) → NormedSpace 𝕜 (F i)\np✝ q r p : ℝ≥0∞\ninst✝ : Fact... | [
"α : Type u_1\n𝕜 : Type u_2\nE : α → Type u_3\nF : α → Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : α) → NormedAddCommGroup (E i)\ninst✝³ : (i : α) → NormedSpace 𝕜 (E i)\ninst✝² : (i : α) → NormedAddCommGroup (F i)\ninst✝¹ : (i : α) → NormedSpace 𝕜 (F i)\np✝ q r p : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\nT ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 81
} | {
"line": 92,
"column": 82
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : (i : ι) → NormedAddCommGroup (F i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝¹ : (i : ι) → NormedAdd... | [
"ι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : (i : ι) → NormedAddCommGroup (F i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝¹ : (i : ι) → NormedAddCommGroup (G... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 15
} | {
"line": 212,
"column": 16
} | [
{
"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",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",... | [
"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⊢ 1 ≤ ‖1 + f i‖ + ‖f i‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 213,
"column": 4
} | {
"line": 213,
"column": 26
} | {
"line": 213,
"column": 27
} | [
{
"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‖",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr"... | [
"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⊢ ‖f i + 1‖ ≤ ‖f i‖ + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 103,
"column": 48
} | {
"line": 103,
"column": 59
} | {
"line": 103,
"column": 60
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : (i : ι) → NormedAddCommGroup (F i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝¹ : (i : ι) → NormedAdd... | [
"ι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : (i : ι) → NormedAddCommGroup (F i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝¹ : (i : ι) → NormedAddCommGroup (G... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 59,
"column": 16
} | {
"line": 59,
"column": 38
} | {
"line": 59,
"column": 39
} | [
{
"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 u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nx : ⨂[𝕜] (i : ι), E i\nf : Continuo... | [
"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 u_1\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nx : ⨂[𝕜] (i : ι), E i\nf : ContinuousMultilinea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 101,
"column": 2
} | {
"line": 102,
"column": 9
} | {
"line": 102,
"column": 10
} | [
{
"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",
"ppTerm": "?m.77",
"assigned": true,
"us... | [
"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⊢ (sSup\n {p |\n ∃ G x x_1,\n p = (normSeminorm 𝕜 (ContinuousMultilinear... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 156,
"column": 4
} | {
"line": 162,
"column": 19
} | {
"line": 164,
"column": 0
} | [] | [] | _ ≤ (∑ i ∈ s, ‖e i‖ ^ p.toReal) ^ (r.toReal / p.toReal) *
(∑ i ∈ s, ‖f i‖ ^ q.toReal) ^ (r.toReal / q.toReal) := by
apply Real.Lr_rpow_le_Lp_mul_Lq_of_nonneg s hpqr <;> (intros; positivity)
_ ≤ _ := by
gcongr
· exact hCe s
· exact hDf s | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 19
} | {
"line": 80,
"column": 20
} | [
{
"pp": "ι : Type u_1\ninst✝² : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\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",
"ppTerm":... | [
"ι : Type u_1\ninst✝² : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\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 ≤ ‖x‖ * ∏ x, ‖m x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 42
} | {
"line": 117,
"column": 4
} | [
{
"pp": "ι : Type u_1\ninst✝³ : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\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",
"ppTe... | [
"ι : Type u_1\ninst✝³ : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.PiTensorProduct.ProjectiveSeminorm | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 54
} | {
"line": 157,
"column": 4
} | [
{
"pp": "ι : Type u_1\ninst✝⁵ : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nG : Type u_4\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ContinuousMultilinearMap 𝕜 E ... | [
"ι : Type u_1\ninst✝⁵ : Fintype ι\n𝕜 : Type u_2\nE : ι → Type u_3\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nG : Type u_4\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ContinuousMultilinearMap 𝕜 E G\nx : ⨂[𝕜]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.UpperLower | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 30
} | {
"line": 136,
"column": 31
} | [
{
"pp": "ι : Type u_2\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ MonotoneOn (dist x) (Ici x)",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_2\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ MonotoneOn (dist x) (Ici x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Order.UpperLower | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 30
} | {
"line": 145,
"column": 31
} | [
{
"pp": "ι : Type u_2\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ AntitoneOn (dist x) (Iic x)",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_2\ninst✝ : Fintype ι\nx : ι → ℝ\n⊢ AntitoneOn (dist x) (Iic x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Basic | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 92
} | {
"line": 87,
"column": 0
} | [
{
"pp": "E : 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",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"IsIntegr... | [] | exact ⟨HasDerivAt.hasDerivWithinAt, fun h ↦ h.hasDerivAt (Metric.isOpen_ball.mem_nhds hy)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.ODE.DiscreteGronwall | {
"line": 56,
"column": 10
} | {
"line": 56,
"column": 61
} | {
"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... | [
"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 ≤ c k\nj : ... | prod_Ico_succ_top (by have := mem_Ico.mp hj; omega) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 48
} | {
"line": 72,
"column": 49
} | [
{
"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)",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 66
} | {
"line": 77,
"column": 67
} | [
{
"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",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.DiscreteGronwall | {
"line": 87,
"column": 8
} | {
"line": 87,
"column": 29
} | {
"line": 87,
"column": 30
} | [
{
"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)",
"ppTerm": "?hbc",
"assigned": true,
"usedConstants": [
"Eq... | [
"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) ≤ ∏ x ∈ Ico n₀ n, rexp (c x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Gronwall | {
"line": 151,
"column": 77
} | {
"line": 151,
"column": 88
} | {
"line": 151,
"column": 89
} | [
{
"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‖ +... | [
"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 : ℝ), a ≤ x → x < b → ‖f' x‖ ≤ K * ‖f x‖"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 47
} | {
"line": 255,
"column": 48
} | [
{
"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)",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Contracting | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 13
} | {
"line": 323,
"column": 14
} | [
{
"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)",
"ppTerm": "?m.74",
"assigned": false,
"use... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 15
} | {
"line": 58,
"column": 16
} | [
{
"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[�... | [
"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[𝕜] W\nhV✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 45
} | {
"line": 80,
"column": 46
} | [
{
"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 :... | [
"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 : Nontrivial ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Transform | {
"line": 40,
"column": 2
} | {
"line": 41,
"column": 94
} | {
"line": 42,
"column": 2
} | [
{
"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",
"ppTerm": "?m.32",
"assi... | [
"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 (v (t + dt) ((fun x ↦ γ (x + dt)) t)) s (t + dt)"
] | 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.PicardLindelof | {
"line": 172,
"column": 63
} | {
"line": 172,
"column": 74
} | {
"line": 172,
"column": 75
} | [
{
"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 ... | [
"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 L toFun✝\nme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 13
} | {
"line": 180,
"column": 14
} | [
{
"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",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.Transform | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 13
} | {
"line": 85,
"column": 14
} | [
{
"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",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 115,
"column": 15
} | {
"line": 115,
"column": 26
} | {
"line": 115,
"column": 27
} | [
{
"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... | [
"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\nhe : adjoint ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 118,
"column": 15
} | {
"line": 118,
"column": 26
} | {
"line": 118,
"column": 27
} | [
{
"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... | [
"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\nhe : adjoint ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 15
} | {
"line": 167,
"column": 16
} | [
{
"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... | [
"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 : Continuous ⇑f\nh✝¹ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.ExistUnique | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 29
} | {
"line": 121,
"column": 2
} | [
{
"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... | [
"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, HasDerivWithinAt (α x) (... | refine ⟨uncurry α, hα1, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 188,
"column": 23
} | {
"line": 188,
"column": 34
} | {
"line": 188,
"column": 35
} | [
{
"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✝ :... | [
"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✝ : Nontrivial ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 195,
"column": 15
} | {
"line": 195,
"column": 74
} | {
"line": 196,
"column": 6
} | [
{
"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✝ :... | [
"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✝ : Nontrivial ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 15
} | {
"line": 212,
"column": 16
} | [
{
"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✝ :... | [
"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✝ : Nontrivial ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Fourier | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 13
} | {
"line": 52,
"column": 14
} | [
{
"pp": "p : ℂ[X]\n⊢ Integrable (⇑(toAddCircle p)) haarAddCircle",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℂ[X]\n⊢ Integrable (⇑(toAddCircle p)) haarAddCircle"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Fourier | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 17
} | {
"line": 99,
"column": 18
} | [
{
"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 (↑... | [
"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⊢ p.coeff b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Basic | {
"line": 180,
"column": 10
} | {
"line": 180,
"column": 37
} | {
"line": 180,
"column": 38
} | [
{
"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",
"ppTerm": "?pos.inl✝",
"as... | [
"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"
] | leadingCoeff_eq_zero.1 hP0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.GaussNorm | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 23
} | {
"line": 77,
"column": 24
} | [
{
"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✝",
"ppTerm": "?h",
"assigned": t... | [
"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 ((coeff y) f) * c ^ y = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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