module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Fin.Tuple.Sort | {
"line": 203,
"column": 22
} | {
"line": 203,
"column": 33
} | {
"line": 203,
"column": 34
} | [
{
"pp": "n : ℕ\nσ : Equiv.Perm (Fin n)\ni✝ j✝ : Fin n\nhij : i✝ < j✝\nh : σ (σ⁻¹ i✝) = σ (σ⁻¹ j✝)\n⊢ i✝ = j✝",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nσ : Equiv.Perm (Fin n)\ni✝ j✝ : Fin n\nhij : i✝ < j✝\nh : σ (σ⁻¹ i✝) = σ (σ⁻¹ j✝)\n⊢ i✝ = j✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 44
} | {
"line": 189,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := T - μ • 1\nK : NNReal\nhK : AntilipschitzWith... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := T - μ • 1\nK : NNReal\nhK : AntilipschitzWith K ⇑S\nh₂ : ... | rw [iterate_succ', LinearMap.range_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 213,
"column": 55
} | {
"line": 213,
"column": 85
} | {
"line": 213,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := T - μ • 1\nK✝ : NNReal\nhK✝ : AntilipschitzWi... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := T - μ • 1\nK✝ : NNReal\nhK✝ : AntilipschitzWith K✝ ⇑S\nh₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.HermitianFunctionalCalculus | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 29
} | {
"line": 82,
"column": 30
} | [
{
"pp": "n : Type u_1\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.IsHermitian\nh0 : FiniteDimensional ℝ C(↑(spectrum ℝ A), ℝ)\nf : C(↑(spectrum ℝ A), ℝ)\nhf : ↑hA.eigenvectorUnitary * diagonal (RCLike.ofReal ∘ ⇑f ∘ fun i ↦ ⟨hA.eigenvalues i, ⋯⟩) = 0\n... | [
"n : Type u_1\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.IsHermitian\nh0 : FiniteDimensional ℝ C(↑(spectrum ℝ A), ℝ)\nf : C(↑(spectrum ℝ A), ℝ)\nhf : ↑hA.eigenvectorUnitary * diagonal (RCLike.ofReal ∘ ⇑f ∘ fun i ↦ ⟨hA.eigenvalues i, ⋯⟩) = 0\n⊢ (RCLike.of... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 59
} | {
"line": 98,
"column": 60
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ : 𝕜\nhμ : HasEigenvalue T μ\nv : E\nhv₁ : T v = μ • v\nhv₂ : v ≠ 0\n⊢ (starRingEnd 𝕜) μ = μ",
"ppTerm": "?m.66",
"assigned": false,
"usedCo... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ : 𝕜\nhμ : HasEigenvalue T μ\nv : E\nhv₁ : T v = μ • v\nhv₂ : v ≠ 0\n⊢ (starRingEnd 𝕜) μ = μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 14
} | {
"line": 106,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : v = 0\n⊢ ⟪((fun μ ↦ (eigenspace T μ).subtypeₗᵢ) μ) ⟨... | [] | simp [hv'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 14
} | {
"line": 106,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : v = 0\n⊢ ⟪((fun μ ↦ (eigenspace T μ).subtypeₗᵢ) μ) ⟨... | [] | simp [hv'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 14
} | {
"line": 106,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : v = 0\n⊢ ⟪((fun μ ↦ (eigenspace T μ).subtypeₗᵢ) μ) ⟨... | [] | simp [hv'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Matrix.PosDef | {
"line": 50,
"column": 40
} | {
"line": 50,
"column": 51
} | {
"line": 50,
"column": 52
} | [
{
"pp": "n : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype n\ninst✝¹ : RCLike 𝕜\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.PosSemidef\ni : n\nx✝ : i ∈ Finset.univ\n⊢ 0 ≤ ↑(⋯.eigenvalues i)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real"... | [
"n : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype n\ninst✝¹ : RCLike 𝕜\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.PosSemidef\ni : n\nx✝ : i ∈ Finset.univ\n⊢ 0 ≤ ⋯.eigenvalues i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 60
} | {
"line": 109,
"column": 61
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv✝ : v ∈ eigenspace T μ\nhv : T v = μ • v\nw : E\nhw✝ : w ∈ eigenspace T ν\nhw : T w = ν • w\nhv' : ¬v = 0\nH : (s... | [
"case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv✝ : v ∈ eigenspace T μ\nhv : T v = μ • v\nw : E\nhw✝ : w ∈ eigenspace T ν\nhw : T w = ν • w\nhv' : ¬v = 0\nH : (starRingEnd �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.PosDef | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 13
} | {
"line": 89,
"column": 14
} | [
{
"pp": "n : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype n\ninst✝¹ : RCLike 𝕜\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.PosDef\ni : n\na✝ : i ∈ Finset.univ\n⊢ 0 < ↑(⋯.eigenvalues i)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
... | [
"n : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype n\ninst✝¹ : RCLike 𝕜\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.PosDef\ni : n\na✝ : i ∈ Finset.univ\n⊢ 0 < ⋯.eigenvalues i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 97
} | {
"line": 198,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhT : T.IsSymmetric\nv : E\nμ : Eigenvalues T\nthis :\n ∀ (w : PiLp 2 fun μ ↦ ↥(eigenspace T (↑T 1 μ))),\n T (hT.diagonalization.symm w) = ... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhT : T.IsSymmetric\nv : E\nμ : Eigenvalues T\nthis :\n ∀ (w : PiLp 2 fun μ ↦ ↥(eigenspace T (↑T 1 μ))),\n T (hT.diagonalization.symm w) = hT.diagonali... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.PosDef | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 50
} | {
"line": 124,
"column": 51
} | [
{
"pp": "m : Type u_1\nn : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : RCLike 𝕜\nA M : Matrix n n 𝕜\nhM : M.PosDef\nx : n → 𝕜\nhx : M *ᵥ x ⬝ᵥ star x = 0\nh : x ≠ 0\n⊢ False",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"m : Type u_1\nn : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : RCLike 𝕜\nA M : Matrix n n 𝕜\nhM : M.PosDef\nx : n → 𝕜\nhx : M *ᵥ x ⬝ᵥ star x = 0\nh : x ≠ 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 268,
"column": 26
} | {
"line": 268,
"column": 37
} | {
"line": 268,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\ni : Fin n\nv : E := (hT.unsortedEigenvectorBasis hn) i\nμ : 𝕜 := ↑T (DirectSum.IsInternal.s... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\ni : Fin n\nv : E := (hT.unsortedEigenvectorBasis hn) i\nμ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 20
} | {
"line": 273,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\ni : Fin n\nv : E := (hT.unsortedEigenvectorBasis hn) i\nμ : 𝕜 := ↑T (DirectSum.IsInternal.s... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\ni : Fin n\nv : E := (hT.unsortedEigenvectorBasis hn) i\nμ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 48
} | {
"line": 341,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\nv : E\ni : Fin n\nthis :\n ∀ (w : EuclideanSpace 𝕜 (Fin n)),\n T ((hT.eigenvectorBasis ... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\nv : E\ni : Fin n\nthis :\n ∀ (w : EuclideanSpace 𝕜 (Fin n)),\n T ((hT.eigenvectorBasis hn).repr.sym... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 412,
"column": 38
} | {
"line": 412,
"column": 85
} | {
"line": 412,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nμ : ℝ\nT : E →ₗ[𝕜] E\nhμ : HasEigenvalue T ↑μ\nhnn : ∀ (x : E), 0 ≤ RCLike.re ⟪x, T x⟫\nv : E\nhv₁ : v ∈ (genEigenspace T ↑μ) 1\nhv₂ : v ≠ 0\n⊢ 0 < ‖v‖ ^ 2",
"ppTerm": "?m.73",
"assi... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nμ : ℝ\nT : E →ₗ[𝕜] E\nhμ : HasEigenvalue T ↑μ\nhnn : ∀ (x : E), 0 ≤ RCLike.re ⟪x, T x⟫\nv : E\nhv₁ : v ∈ (genEigenspace T ↑μ) 1\nhv₂ : v ≠ 0\n⊢ v ≠ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 421,
"column": 38
} | {
"line": 421,
"column": 85
} | {
"line": 421,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nμ : ℝ\nT : E →ₗ[𝕜] E\nhμ : HasEigenvalue T ↑μ\nhnn : ∀ (x : E), 0 < RCLike.re ⟪x, T x⟫\nv : E\nhv₁ : v ∈ (genEigenspace T ↑μ) 1\nhv₂ : v ≠ 0\n⊢ 0 < ‖v‖ ^ 2",
"ppTerm": "?m.73",
"assi... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nμ : ℝ\nT : E →ₗ[𝕜] E\nhμ : HasEigenvalue T ↑μ\nhnn : ∀ (x : E), 0 < RCLike.re ⟪x, T x⟫\nv : E\nhv₁ : v ∈ (genEigenspace T ↑μ) 1\nhv₂ : v ≠ 0\n⊢ v ≠ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 520,
"column": 2
} | {
"line": 521,
"column": 68
} | {
"line": 521,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn : ℕ∞\nK : Compacts E\nf g : 𝓓^{n}_{K}(E, F)\nhfg ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn : ℕ∞\nK : Compacts E\nf g : 𝓓^{n}_{K}(E, F)\nhfg : (structure... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 459,
"column": 2
} | {
"line": 459,
"column": 17
} | {
"line": 459,
"column": 18
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsCompactOperator ⇑T\nhT' : (↑T).IsSymmetric\nS : ↥(⨆ μ, eigenspace (↑T) μ)ᗮ →L[𝕜] ↥(⨆ μ, eigenspace (↑T) μ)ᗮ := T.restrict ⋯\nhS_compact : IsC... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsCompactOperator ⇑T\nhT' : (↑T).IsSymmetric\nS : ↥(⨆ μ, eigenspace (↑T) μ)ᗮ →L[𝕜] ↥(⨆ μ, eigenspace (↑T) μ)ᗮ := T.restrict ⋯\nhS_compact : IsCompactOperat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 71,
"column": 20
} | {
"line": 71,
"column": 31
} | {
"line": 71,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni✝ j i : n\n⊢ A i i ≤ 0",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni✝ j i : n\n⊢ A i i ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 71,
"column": 52
} | {
"line": 71,
"column": 63
} | {
"line": 71,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni✝ j i : n\n⊢ 0 ≤ A i i",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni✝ j i : n\n⊢ 0 ≤ A i i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 13
} | {
"line": 76,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni j : n\nhdiag : True\nh1 : 0 ≤ A i j * star (A i j) + A i j * star (A i j)\nh2 : A i j * star (A i j) + A i j * star (A i j) ≤ 0\n⊢ A i j = 0",
"ppTerm": "?m.118",
"assigned": false,
... | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni j : n\nhdiag : True\nh1 : 0 ≤ A i j * star (A i j) + A i j * star (A i j)\nh2 : A i j * star (A i j) + A i j * star (A i j) ≤ 0\n⊢ A i j = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 38
} | {
"line": 81,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA B : Matrix n n 𝕜\nh₁ : A ≤ B\nh₂ : B ≤ A\n⊢ A = B",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA B : Matrix n n 𝕜\nh₁ : A ≤ B\nh₂ : B ≤ A\n⊢ A = B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 82,
"column": 9
} | {
"line": 82,
"column": 47
} | {
"line": 82,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA B : Matrix n n 𝕜\nh₁ : A ≤ B\nh₂ : B ≤ A\n⊢ (-(B - A)).PosSemidef",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA B : Matrix n n 𝕜\nh₁ : A ≤ B\nh₂ : B ≤ A\n⊢ (-(B - A)).PosSemidef"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 41
} | {
"line": 151,
"column": 42
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nx : n → 𝕜\n⊢ (((toLinearMap₂' 𝕜) A) (star x)) x = 0 ↔ A *ᵥ x = 0",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.m... | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nx : n → 𝕜\n⊢ star x ⬝ᵥ A *ᵥ x = 0 ↔ A *ᵥ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 44
} | {
"line": 177,
"column": 45
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nx✝ : A.IsHermitian ∧ spectrum 𝕜 A ⊆ {a | 0 ≤ a}\nh1 : A.IsHermitian\nh2 : spectrum 𝕜 A ⊆ {a | 0 ≤ a}\ni : n\n⊢ 0 i ≤ h1.eigenvalues i",
"ppTerm": "?refine_2",
"assigne... | [
"case refine_2\n𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nx✝ : A.IsHermitian ∧ spectrum 𝕜 A ⊆ {a | 0 ≤ a}\nh1 : A.IsHermitian\nh2 : spectrum 𝕜 A ⊆ {a | 0 ≤ a}\ni : n\n⊢ 0 ≤ h1.eigenvalues i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 670,
"column": 24
} | {
"line": 670,
"column": 35
} | {
"line": 670,
"column": 36
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn : ℕ∞\nK : Compacts E\ni : ℕ\nhin : n < ↑i\n⊢ ¬↑i ≤... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn : ℕ∞\nK : Compacts E\ni : ℕ\nhin : n < ↑i\n⊢ n < ↑i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 83
} | {
"line": 220,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Finite n\nm : Type u_3\ninst✝ : Finite m\nthis✝ : Fintype n\nthis : Fintype m\na : Matrix n n 𝕜\nhx : (star a * a).PosSemidef\nb : Matrix m m 𝕜\nhy : (star b * b).PosSemidef\n⊢ (kroneckerMap (fun x1 x2 ↦ x1 * x2) (star a * a) (star b * b)).Pos... | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Finite n\nm : Type u_3\ninst✝ : Finite m\nthis✝ : Fintype n\nthis : Fintype m\na : Matrix n n 𝕜\nhx : (star a * a).PosSemidef\nb : Matrix m m 𝕜\nhy : (star b * b).PosSemidef\n⊢ ((kroneckerMap (fun x1 x2 ↦ x1 * x2) a b)ᴴ * kroneckerMap (fun x1 x2 ↦ x1 * x2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 250,
"column": 2
} | {
"line": 251,
"column": 73
} | {
"line": 251,
"column": 74
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_3\nA B : Matrix ι ι 𝕜\nhA : A.PosSemidef\nhB : B.PosSemidef\nx : ι →₀ 𝕜\nhAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosSemidef\n⊢ 0 ≤ x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * (A ⊙ B) i j * xj",
"ppTerm": "?m.66",
"assigned": true,
"us... | [
"𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_3\nA B : Matrix ι ι 𝕜\nhA : A.PosSemidef\nhB : B.PosSemidef\nx : ι →₀ 𝕜\nhAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosSemidef\n⊢ 0 ≤\n ∑ x_1 ∈ (Finsupp.subtypeDomain (fun x_1 ↦ x_1 ∈ x.support) x).support,\n ∑ x_2 ∈ (Finsupp.subtypeDomain (fun x_2 ↦ x_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.GramMatrix | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 49
} | {
"line": 75,
"column": 2
} | [
{
"pp": "E : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : Fintype n\nv : n → E\nx y : n → 𝕜\n⊢ star x ⬝ᵥ gram 𝕜 v *ᵥ y = ⟪∑ i, x i • v i, ∑ i, y i • v i⟫_𝕜",
"ppTerm": "?m.41",
"assigned": true,
"usedConstan... | [
"E : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : Fintype n\nv : n → E\nx y : n → 𝕜\n⊢ star x ⬝ᵥ gram 𝕜 v *ᵥ y = ∑ i, ∑ j, (starRingEnd 𝕜) (x i) * y j * ⟪v i, v j⟫_𝕜",
"E : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝³ :... | trans ∑ i, ∑ j, conj (x i) * y j * ⟪v i, v j⟫_𝕜 | Batteries.Tactic._aux_Batteries_Tactic_Trans___elabRules_Batteries_Tactic_tacticTrans____1 | Batteries.Tactic.tacticTrans___ |
Mathlib.Analysis.Matrix.Order | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 46
} | {
"line": 272,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_3\nA B : Matrix ι ι 𝕜\nhA : A.PosDef\nhB : B.PosDef\nx : ι →₀ 𝕜\nhx : x ≠ 0\nhAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosDef\n⊢ Finsupp.subtypeDomain (fun x_1 ↦ x_1 ∈ x.support) x ≠ 0",
"ppTerm": "?m.210",
"assigned": true,
"usedConsta... | [
"𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_3\nA B : Matrix ι ι 𝕜\nhA : A.PosDef\nhB : B.PosDef\nx : ι →₀ 𝕜\nhx : x ≠ 0\nhAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosDef\n⊢ x.support.Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 330,
"column": 6
} | {
"line": 330,
"column": 31
} | {
"line": 330,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : Fintype n\nx : Matrix n n 𝕜\ny : (Matrix n n 𝕜)ˣ\nhM : (star ↑y * ↑y).PosDef\n__spread✝⁻⁰ : PreInnerProductSpace.Core 𝕜 (Matrix n n 𝕜) := ⋯.matrixPreInnerProductSpace\nhx : ↑y * xᴴ = 0\n⊢ x = 0",
"ppTerm": "?m.176",
"assigned": false,... | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : Fintype n\nx : Matrix n n 𝕜\ny : (Matrix n n 𝕜)ˣ\nhM : (star ↑y * ↑y).PosDef\n__spread✝⁻⁰ : PreInnerProductSpace.Core 𝕜 (Matrix n n 𝕜) := ⋯.matrixPreInnerProductSpace\nhx : ↑y * xᴴ = 0\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 13
} | {
"line": 124,
"column": 14
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | [
"A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : NonnegSpectrum... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 24
} | {
"line": 139,
"column": 4
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | [
"A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : NonnegSpectrum... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 24
} | {
"line": 143,
"column": 4
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | [
"A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : NonnegSpectrum... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 154,
"column": 49
} | {
"line": 154,
"column": 60
} | {
"line": 154,
"column": 61
} | [
{
"pp": "A : Type u_2\ninst✝¹⁶ : NonUnitalRing A\ninst✝¹⁵ : StarRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module ℝ A\ninst✝¹² : SMulCommClass ℝ A A\ninst✝¹¹ : IsScalarTower ℝ A A\ninst✝¹⁰ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁹ : PartialOrder A\ninst✝⁸ : StarOrderedRing A\ninst✝⁷... | [
"A : Type u_2\ninst✝¹⁶ : NonUnitalRing A\ninst✝¹⁵ : StarRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module ℝ A\ninst✝¹² : SMulCommClass ℝ A A\ninst✝¹¹ : IsScalarTower ℝ A A\ninst✝¹⁰ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁹ : PartialOrder A\ninst✝⁸ : StarOrderedRing A\ninst✝⁷ : NonnegSpe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 42
} | {
"line": 43,
"column": 43
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nh : MemLp f 2 μ\n⊢ Integrable (fun x ↦ f x ^ 2) μ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"PseudoMetricSpace.toUniformSpace",
... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nh : MemLp f 2 μ\n⊢ MemLp (fun x ↦ f x ^ 2) 1 μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.SmoothApprox | {
"line": 87,
"column": 52
} | {
"line": 87,
"column": 80
} | {
"line": 87,
"column": 81
} | [
{
"pp": "E : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : Chart... | [
"E : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 64
} | {
"line": 117,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf g : ↥(Lp E 2 μ)\nx : α\n⊢ ‖⟪↑↑f x, ↑↑g x⟫‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖",
"ppTerm": "?m.65",
"assigned": true,
"usedConstant... | [
"α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf g : ↥(Lp E 2 μ)\nx : α\n⊢ ‖⟪↑↑f x, ↑↑g x⟫‖ ≤ ‖‖↑↑f x‖ ^ 2 + ‖↑↑g x‖ ^ 2‖"
] | rw [← @Nat.cast_two ℝ, Real.rpow_natCast, Real.rpow_natCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 204,
"column": 55
} | {
"line": 204,
"column": 66
} | {
"line": 204,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : NonUnitalRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module 𝕜 A\ninst✝¹² : StarRing A\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : StarOrderedRing A\ninst✝⁹ : IsScalarTower 𝕜 A A\ninst✝⁸ : SMulCommClass 𝕜 A A\ninst✝⁷ : NonUnitalCo... | [
"𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : NonUnitalRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module 𝕜 A\ninst✝¹² : StarRing A\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : StarOrderedRing A\ninst✝⁹ : IsScalarTower 𝕜 A A\ninst✝⁸ : SMulCommClass 𝕜 A A\ninst✝⁷ : NonUnitalContinuousFunc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 811,
"column": 4
} | {
"line": 811,
"column": 15
} | {
"line": 811,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : SMulCommClass ℝ 𝕜 F\ninst✝⁴ : NormedAddCommGroup F'\nin... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedSpace 𝕜 F\ninst✝⁵ : SMulCommClass ℝ 𝕜 F\ninst✝⁴ : NormedAddCommGroup F'\ninst✝³ : Norme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 46
} | {
"line": 242,
"column": 47
} | [
{
"pp": "A : Type u_2\ninst✝¹⁰ : Ring A\ninst✝⁹ : StarRing A\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : Algebra ℝ A\ninst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : IsTopologicalRing A\ninst✝¹ : T2Space A\ninst✝... | [
"A : Type u_2\ninst✝¹⁰ : Ring A\ninst✝⁹ : StarRing A\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : Algebra ℝ A\ninst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : IsTopologicalRing A\ninst✝¹ : T2Space A\ninst✝ : StarModul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 850,
"column": 4
} | {
"line": 850,
"column": 15
} | {
"line": 850,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : SMulCommClass ℝ 𝕜 F\ninst✝³ : NormedAddCommGroup F'\nins... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : SMulCommClass ℝ 𝕜 F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : Normed... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.SmoothApprox | {
"line": 103,
"column": 8
} | {
"line": 103,
"column": 25
} | {
"line": 103,
"column": 26
} | [
{
"pp": "case pos\nE : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst... | [
"case pos\nE : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : Charted... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 873,
"column": 4
} | {
"line": 873,
"column": 48
} | {
"line": 874,
"column": 6
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn k : ℕ∞\nK : Compacts E\ni : ℕ\nf : 𝓓^{n... | [
"case pos\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn k : ℕ∞\nK : Compacts E\ni : ℕ\nf : 𝓓^{n}_{K}(E, F)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Manifold.SmoothApprox | {
"line": 104,
"column": 8
} | {
"line": 104,
"column": 25
} | {
"line": 104,
"column": 26
} | [
{
"pp": "case neg\nE : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst... | [
"case neg\nE : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : Charted... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 892,
"column": 4
} | {
"line": 892,
"column": 15
} | {
"line": 892,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : SMulCommClass ℝ 𝕜 F\ninst✝³ : NormedAddCommGroup F'\nins... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : SMulCommClass ℝ 𝕜 F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : Normed... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.SmoothApprox | {
"line": 48,
"column": 4
} | {
"line": 48,
"column": 37
} | {
"line": 48,
"column": 38
} | [
{
"pp": "case inl\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\nε : ℝ\nhε : 0 < ... | [
"case inl\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\nε : ℝ\nhε : 0 < ε\nf : E → F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 112,
"column": 8
} | {
"line": 113,
"column": 37
} | {
"line": 113,
"column": 38
} | [
{
"pp": "case pos.inl\nα : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.OuterRegular\nhp : p ≠ ∞\ns u : Set α\ns_closed : IsClosed[inst✝⁶... | [
"case pos.inl\nα : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.OuterRegular\nhp : p ≠ ∞\ns u : Set α\ns_closed : IsClosed[inst✝⁶] s\nu_open ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.SmoothApprox | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 82
} | {
"line": 102,
"column": 2
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ ... | [
"E : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ : Fact (1 ≤ ... | refine (mem_closure_iff_nhds_basis Metric.nhds_basis_closedBall).2 fun ε hε ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 1081,
"column": 10
} | {
"line": 1081,
"column": 21
} | {
"line": 1081,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝²⁰ : NontriviallyNormedField 𝕜\ninst✝¹⁹ : NormedAddCommGroup E\ninst✝¹⁸ : NormedSpace ℝ E\ninst✝¹⁷ : NormedAddCommGroup F\ninst✝¹⁶ : NormedSpace ℝ F\ninst✝¹⁵ : NormedSpace 𝕜 F\ninst✝¹⁴ : SMulCommClass ℝ 𝕜 F\ninst✝¹³ : NormedAddCommGroup ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝²⁰ : NontriviallyNormedField 𝕜\ninst✝¹⁹ : NormedAddCommGroup E\ninst✝¹⁸ : NormedSpace ℝ E\ninst✝¹⁷ : NormedAddCommGroup F\ninst✝¹⁶ : NormedSpace ℝ F\ninst✝¹⁵ : NormedSpace 𝕜 F\ninst✝¹⁴ : SMulCommClass ℝ 𝕜 F\ninst✝¹³ : NormedAddCommGroup F'\ninst✝¹² ... | 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.Bases | {
"line": 247,
"column": 27
} | {
"line": 247,
"column": 72
} | {
"line": 247,
"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.Bases | {
"line": 336,
"column": 27
} | {
"line": 336,
"column": 50
} | {
"line": 336,
"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.MeasureTheory.Function.ContinuousMapDense | {
"line": 199,
"column": 38
} | {
"line": 199,
"column": 91
} | {
"line": 199,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : BorelSpace α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\nμ : Measure α\ninst✝³ : NormedSpace ℝ E\ninst✝² : R1Space α\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf :... | [
"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : BorelSpace α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\nμ : Measure α\ninst✝³ : NormedSpace ℝ E\ninst✝² : R1Space α\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\np : ℝ\nhp : 0 < p\nf : α → E\nhf :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Deriv | {
"line": 65,
"column": 6
} | {
"line": 66,
"column": 79
} | {
"line": 67,
"column": 8
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF✝ : Type u_6\nV : Type u_7\nF : Type u_8\nF₁ : Type u_9\nF₂ : Type u_10\nF₃ : Type u_11\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedSpace ℝ E\ninst✝¹ : RCLike 𝕜\ninst✝ ... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF✝ : Type u_6\nV : Type u_7\nF : Type u_8\nF₁ : Type u_9\nF₂ : Type u_10\nF₃ : Type u_11\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedSpace ℝ E\ninst✝¹ : RCLike 𝕜\ninst✝ : NormedSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Deriv | {
"line": 89,
"column": 6
} | {
"line": 90,
"column": 60
} | {
"line": 90,
"column": 61
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF✝ : Type u_6\nV : Type u_7\nF : Type u_8\nF₁ : Type u_9\nF₂ : Type u_10\nF₃ : Type u_11\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedSpace ℝ E\ninst✝² : RCLike 𝕜\ninst✝¹... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF✝ : Type u_6\nV : Type u_7\nF : Type u_8\nF₁ : Type u_9\nF₂ : Type u_10\nF₃ : Type u_11\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedSpace ℝ E\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 13
} | {
"line": 229,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : BorelSpace α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\nμ : Measure α\ninst✝³ : NormedSpace ℝ E\ninst✝² : R1Space α\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\nf : α → E\nε : ℝ\nhε :... | [
"α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : BorelSpace α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\nμ : Measure α\ninst✝³ : NormedSpace ℝ E\ninst✝² : R1Space α\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\nf : α → E\nε : ℝ\nhε : 0 < ε\nhf :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 290,
"column": 38
} | {
"line": 290,
"column": 91
} | {
"line": 290,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.WeaklyRegular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : MemLp f (ENNReal.ofReal p) μ\nε : ℝ\nhε... | [
"α : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.WeaklyRegular\np : ℝ\nhp : 0 < p\nf : α → E\nhf : MemLp f (ENNReal.ofReal p) μ\nε : ℝ\nhε : 0 < ε\nI ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 13
} | {
"line": 314,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.WeaklyRegular\nf : α → E\nε : ℝ\nhε : 0 < ε\nhf : MemLp f (ENNReal.ofReal 1) μ\n⊢ ∃ g, ∫ ... | [
"α : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.WeaklyRegular\nf : α → E\nε : ℝ\nhε : 0 < ε\nhf : MemLp f (ENNReal.ofReal 1) μ\n⊢ ∃ g, ∫ (x : α), ‖f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 31
} | {
"line": 172,
"column": 32
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nG : ι → Type u_4\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝ : DecidableEq ι\ni : ι\na : G i\nf : ↥(lp G 2)\n⊢ ⟪f, lp.single 2 i a⟫ = ⟪↑f i, a⟫",
"ppTerm": "?m.24",
"assigned": false,
"us... | [
"ι : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nG : ι → Type u_4\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝ : DecidableEq ι\ni : ι\na : G i\nf : ↥(lp G 2)\n⊢ ⟪f, lp.single 2 i a⟫ = ⟪↑f i, a⟫"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 221,
"column": 2
} | {
"line": 227,
"column": 12
} | {
"line": 229,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁶ : RCLike 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝¹ : CompleteSpace E\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : Orthog... | [] | rw [hV.linearIsometry_apply, ← tsum_ite_eq i (fun _ ↦ V i x)]
congr
ext j
rw [lp.single_apply]
split_ifs with h
· subst h; simp
· simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 221,
"column": 2
} | {
"line": 227,
"column": 12
} | {
"line": 229,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁶ : RCLike 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝¹ : CompleteSpace E\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : Orthog... | [] | rw [hV.linearIsometry_apply, ← tsum_ite_eq i (fun _ ↦ V i x)]
congr
ext j
rw [lp.single_apply]
split_ifs with h
· subst h; simp
· simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 41
} | {
"line": 181,
"column": 0
} | [
{
"pp": "T : ℝ\nm n : ℤ\nx : AddCircle T\n⊢ ↑((m + n) • x).toCircle = (fourier m) x * (fourier n) x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"HMul.hMul",
"congrArg",
"Cont... | [] | rw [← fourier_apply]; exact fourier_add | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 41
} | {
"line": 181,
"column": 0
} | [
{
"pp": "T : ℝ\nm n : ℤ\nx : AddCircle T\n⊢ ↑((m + n) • x).toCircle = (fourier m) x * (fourier n) x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"HMul.hMul",
"congrArg",
"Cont... | [] | rw [← fourier_apply]; exact fourier_add | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 248,
"column": 4
} | {
"line": 253,
"column": 80
} | {
"line": 255,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\n𝕜 : Type u_2\ninst✝⁶ : RCLike 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝¹ : CompleteSpace E\nV : (i : ι) → G i →ₗᵢ[𝕜]... | [] | apply topologicalClosure_minimal
· refine iSup_le ?_
rintro i x ⟨x, rfl⟩
use lp.single 2 i x
exact hV.linearIsometry_apply_single x
exact hV.linearIsometry.isometry.isUniformInducing.isComplete_range.isClosed | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 248,
"column": 4
} | {
"line": 253,
"column": 80
} | {
"line": 255,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\n𝕜 : Type u_2\ninst✝⁶ : RCLike 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝¹ : CompleteSpace E\nV : (i : ι) → G i →ₗᵢ[𝕜]... | [] | apply topologicalClosure_minimal
· refine iSup_le ?_
rintro i x ⟨x, rfl⟩
use lp.single 2 i x
exact hV.linearIsometry_apply_single x
exact hV.linearIsometry.isometry.isUniformInducing.isComplete_range.isClosed | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 293,
"column": 30
} | {
"line": 293,
"column": 80
} | {
"line": 293,
"column": 81
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nF : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace ↥(F i)\nhFortho : OrthogonalFamily 𝕜 (fun i ↦ ↥(F i)) fun i ↦ (F i).subtypeₗᵢ\nhFtotal : ⊤ ≤... | [
"ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nF : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace ↥(F i)\nhFortho : OrthogonalFamily 𝕜 (fun i ↦ ↥(F i)) fun i ↦ (F i).subtypeₗᵢ\nhFtotal : ⊤ ≤ (⨆ i, F i).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 41
} | {
"line": 271,
"column": 0
} | [
{
"pp": "case e'_9\nT : ℝ\nhT : Fact (0 < T)\np : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\nhp : p ≠ ∞\ne_3✝ : Complex.instSemiring = NormedField.toNormedCommRing.toSemiring\ne_6✝ : Lp.instModule ≍ Lp.instModule\n⊢ span ℂ (⇑(toLp p haarAddCircle ℂ) '' range fourier) = span ℂ (⇑↑(toLp p haarAddCircle ℂ) '' range fourier)",
... | [] | simp only [ContinuousLinearMap.coe_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 275,
"column": 6
} | {
"line": 275,
"column": 76
} | {
"line": 275,
"column": 76
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\ni j : ℤ\n⊢ inner ℂ (fourierLp 2 i) (fourierLp 2 j) = if i = j then 1 else 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCom... | [
"T : ℝ\nhT : Fact (0 < T)\ni j : ℤ\n⊢ ∫ (x : AddCircle T), (fourier j) x * (starRingEnd ℂ) ((fourier i) x) ∂haarAddCircle = if i = j then 1 else 0"
] | ContinuousMap.inner_toLp (@haarAddCircle T hT) (fourier i) (fourier j) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 44
} | {
"line": 324,
"column": 45
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : AddCircle T → E\nhf : Integrable f haarAddCircle\nhg : Integrable g haarAddCircle\nx : ℤ\n⊢ fourierCoeff (f + g) x = (fourierCoeff f + fourierCoeff g) x",
"ppTerm": "?m.55",
"assigned": true,
... | [
"T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : AddCircle T → E\nhf : Integrable f haarAddCircle\nhg : Integrable g haarAddCircle\nx : ℤ\n⊢ ∫ (t : AddCircle T), (fourier (-x)) t • f t + (fourier (-x)) t • g t ∂haarAddCircle =\n ∫ (t : AddCircle T), (fourier ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 333,
"column": 30
} | {
"line": 333,
"column": 41
} | {
"line": 333,
"column": 42
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nι : Type u_2\nf : ι → AddCircle T → E\na : ι\ns : Finset ι\nha : a ∉ s\niha : (∀ i ∈ s, Integrable (f i) haarAddCircle) → fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)\nhf : ∀ i ∈ insert a s, Int... | [
"T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nι : Type u_2\nf : ι → AddCircle T → E\na : ι\ns : Finset ι\nha : a ∉ s\niha : (∀ i ∈ s, Integrable (f i) haarAddCircle) → fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)\nhf : ∀ i ∈ insert a s, Integrable (f i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 447,
"column": 47
} | {
"line": 447,
"column": 58
} | {
"line": 447,
"column": 59
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nb : HilbertBasis ι 𝕜 E\nx : E\n⊢ HasSum (fun i ↦ ↑(b.repr x) i • b i) x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"LinearIsometryEquiv.instEqu... | [
"ι : Type u_1\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nb : HilbertBasis ι 𝕜 E\nx : E\n⊢ HasSum (fun i ↦ ↑(b.repr x) i • b i) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 328,
"column": 65
} | {
"line": 335,
"column": 68
} | {
"line": 338,
"column": 0
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nι : Type u_2\ns : Finset ι\nf : ι → AddCircle T → E\nhf : ∀ i ∈ s, Integrable (f i) haarAddCircle\n⊢ fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)",
"ppTerm": "?m.42",
"assigned": true,
... | [] | by
classical
induction s using Finset.induction_on with
| empty => ext; simp [fourierCoeff]
| insert a s ha iha =>
obtain ⟨hf₁, hf₂⟩ := by simpa using hf
rw [s.sum_insert ha, s.sum_insert ha,
fourierCoeff.add hf₁ (integrable_finsetSum' s hf₂), iha hf₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 485,
"column": 6
} | {
"line": 485,
"column": 99
} | {
"line": 486,
"column": 8
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝ : Fintype ι\nb : HilbertBasis ι 𝕜 E\nthis : IsClosed[Pseudo... | [
"ι : Type u_1\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝ : Fintype ι\nb : HilbertBasis ι 𝕜 E\nthis : IsClosed[PseudoMetricSpace.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 503,
"column": 4
} | {
"line": 503,
"column": 54
} | {
"line": 503,
"column": 55
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\ni j : ι\n⊢ ((innerSL 𝕜) (b i)) (b j) = Pi... | [
"ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\ni j : ι\n⊢ ⟪b i, b j⟫ = if i = j then 1 else 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 505,
"column": 4
} | {
"line": 505,
"column": 64
} | {
"line": 505,
"column": 65
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\nx : E\n⊢ HasSum (fun i ↦ ((innerSL 𝕜) (b ... | [
"ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\nx : E\n⊢ HasSum (fun i ↦ ↑(b.repr x) i • b i) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 519,
"column": 2
} | {
"line": 519,
"column": 86
} | {
"line": 520,
"column": 4
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\nb : HilbertBasis ι 𝕜 ↥U\nx : E\n⊢ HasSum (fun i ↦ ⟪↑(b i), x⟫ • b i) (U.orthogonalProjectionOnto x)",
"ppTerm": "?m.34",
... | [
"ι : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU : Submodule 𝕜 E\ninst✝ : CompleteSpace ↥U\nb : HilbertBasis ι 𝕜 ↥U\nx : E\n⊢ HasSum (fun i ↦ ⟪↑(b i), x⟫ • b i) (U.orthogonalProjectionOnto x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 587,
"column": 8
} | {
"line": 588,
"column": 73
} | {
"line": 588,
"column": 74
} | [
{
"pp": "𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nw : Set E\nhws : w ⊇ s\nhw_ortho : Orthonormal 𝕜 Subtype.val\nhw_max : ∀ u ⊇ w, Orthonormal 𝕜 Subtype.val → u = w\n⊢ (s... | [
"𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\nw : Set E\nhws : w ⊇ s\nhw_ortho : Orthonormal 𝕜 Subtype.val\nhw_max : ∀ u ⊇ w, Orthonormal 𝕜 Subtype.val → u = w\n⊢ (span 𝕜 w)ᗮ =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 546,
"column": 2
} | {
"line": 546,
"column": 13
} | {
"line": 546,
"column": 14
} | [
{
"pp": "T : ℝ\nn : ℤ\nx : ℝ\n⊢ HasDerivAt (fun y ↦ (fourier (-n)) ↑y) (-2 * ↑π * I * ↑n / ↑T * (fourier (-n)) ↑x) x",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Int.cast",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"N... | [
"T : ℝ\nn : ℤ\nx : ℝ\n⊢ HasDerivAt (fun y ↦ (starRingEnd ℂ) (cexp (2 * ↑π * I * ↑n * ↑y / ↑T)))\n (-(2 * ↑π * I * ↑n) / ↑T * (starRingEnd ℂ) (cexp (2 * ↑π * I * ↑n * ↑x / ↑T))) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 554,
"column": 16
} | {
"line": 554,
"column": 34
} | {
"line": 554,
"column": 34
} | [
{
"pp": "case e'_8\nT : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn : n ≠ 0\nx y : ℝ\n⊢ ↑T / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑y = (fourier (-n)) ↑y / (-2 * ↑π * I * ↑n / ↑T)",
"ppTerm": "?e'_8",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing... | [
"case e'_8\nT : ℝ\nhT : Fact (0 < T)\nn : ℤ\nhn : n ≠ 0\nx y : ℝ\n⊢ ↑T / (-2 * ↑π * I * ↑n) * (fourier (-n)) ↑y = (fourier (-n)) ↑y * ↑T / (-2 * ↑π * I * ↑n)"
] | div_div_eq_mul_div | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 577,
"column": 4
} | {
"line": 577,
"column": 15
} | {
"line": 577,
"column": 16
} | [
{
"pp": "a b : ℝ\nhab : a < b\nf f' : ℝ → ℂ\nn : ℤ\nhn : n ≠ 0\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : IntervalIntegrable f' volume a b\nhT : Fact (0 < b - a)\nthis : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)\n⊢ ↑b = ↑a... | [
"a b : ℝ\nhab : a < b\nf f' : ℝ → ℂ\nn : ℤ\nhn : n ≠ 0\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : IntervalIntegrable f' volume a b\nhT : Fact (0 < b - a)\nthis : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)\n⊢ ↑b = ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 52
} | {
"line": 78,
"column": 53
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\na : E\nhs : MeasurableSet s\nh'st : t ∈ 𝓝[s] x₀\nhlφ... | [
"α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\na : E\nhs : MeasurableSet s\nh'st : t ∈ 𝓝[s] x₀\nhlφ : ∀ (u : Se... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 164,
"column": 8
} | {
"line": 164,
"column": 56
} | {
"line": 164,
"column": 57
} | [
{
"pp": "case hbc\nα : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\nhs : MeasurableSet s\nht : MeasurableSet t\... | [
"case hbc\nα : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\nhs : MeasurableSet s\nht : MeasurableSet t\nhts : t ⊆ s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 67
} | {
"line": 104,
"column": 68
} | [
{
"pp": "x : ℝ\nh1 : ∀ (y : ℝ), ↑(𝐞 y) = (fourier 1) ↑y\n⊢ HasDerivAt (fun x ↦ ↑(𝐞 x)) (2 * ↑π * I * ↑(𝐞 x)) x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSemin... | [
"x : ℝ\nh1 : ∀ (y : ℝ), ↑(𝐞 y) = (fourier 1) ↑y\n⊢ HasDerivAt (fun x ↦ (fourier 1) ↑x) (2 * ↑π * I * (fourier 1) ↑x) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 38
} | {
"line": 166,
"column": 39
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nf : 𝓢(E, F)\ninst✝ : ProperSpace E\nk : ℤ\n⊢ ⇑f =O[cocompact E] fun x ↦ ‖x‖ ^ k",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"us... | [
"E : Type u_5\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nf : 𝓢(E, F)\ninst✝ : ProperSpace E\nk : ℤ\n⊢ ⇑f =O[cocompact E] fun x ↦ ‖x‖ ^ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 234,
"column": 33
} | {
"line": 234,
"column": 63
} | {
"line": 234,
"column": 64
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : NormedSpace ℝ V\ninst✝⁴ : NormedAddCommGroup W\ninst✝³ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\ninst✝... | [
"E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : NormedSpace ℝ V\ninst✝⁴ : NormedAddCommGroup W\ninst✝³ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\ninst✝ : SecondCou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 233,
"column": 8
} | {
"line": 233,
"column": 43
} | {
"line": 233,
"column": 44
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : BorelSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\nφ : ι → α → ℝ\na : E\ninst✝ : CompleteSpace E\nt : Set α\nht : MeasurableSet t\n... | [
"α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : BorelSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\nφ : ι → α → ℝ\na : E\ninst✝ : CompleteSpace E\nt : Set α\nht : MeasurableSet t\nh'ts : t ∈ �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 549,
"column": 18
} | {
"line": 549,
"column": 29
} | {
"line": 549,
"column": 30
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : 𝓢(E, F)\nn : ℕ\nC : ℝ\nCpos : 0 < C\nhC : ∀ (x : E), ‖x‖ ^ 0 * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C\n⊢ ∀ (x : E), ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C * (1 + ‖x‖) ^ 0"... | [
"E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : 𝓢(E, F)\nn : ℕ\nC : ℝ\nCpos : 0 < C\nhC : ∀ (x : E), ‖x‖ ^ 0 * ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C\n⊢ ∀ (x : E), ‖iteratedFDeriv ℝ n (⇑f) x‖ ≤ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 655,
"column": 93
} | {
"line": 657,
"column": 78
} | {
"line": 658,
"column": 4
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedField 𝕜\ninst✝³ : NormedAddCommGroup G\ninst... | [] | by
gcongr
exact norm_iteratedFDeriv_clm_apply_const (f.smooth _).contDiffAt le_rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 821,
"column": 4
} | {
"line": 821,
"column": 38
} | {
"line": 822,
"column": 6
} | [
{
"pp": "case pos\n𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAlgebra ℝ 𝕜\ninst✝ : NormedSpace 𝕜 F\ng : E → 𝕜\nf : 𝓢(E, F)\nhg : Function.HasT... | [
"case pos\n𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAlgebra ℝ 𝕜\ninst✝ : NormedSpace 𝕜 F\ng : E → 𝕜\nf : 𝓢(E, F)\nhg : Function.HasTemperateGrow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 862,
"column": 28
} | {
"line": 862,
"column": 39
} | {
"line": 862,
"column": 40
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAlgebra... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAlgebra ℝ 𝕜\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 299,
"column": 4
} | {
"line": 318,
"column": 21
} | {
"line": 319,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nhs : IsCo... | [
"α : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nhs : IsCompact s\nhμ ... | have M : ∀ n, ∀ x ∈ s \ u, φ n x ≤ (μ.real (v ∩ s))⁻¹ * (t / t') ^ n := by
intro n x hx
have B : t' ^ n * μ.real (v ∩ s) ≤ ∫ y in s, c y ^ n ∂μ :=
calc
t' ^ n * μ.real (v ∩ s) = ∫ _ in v ∩ s, t' ^ n ∂μ := by simp [mul_comm]
_ ≤ ∫ y in v ∩ s, c y ^ n ∂μ := by
apply set... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 898,
"column": 17
} | {
"line": 901,
"column": 14
} | {
"line": 903,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAlgebra... | [] | by
simp only [Finset.sup_insert, schwartzSeminormFamily_apply, Finset.sup_singleton,
Seminorm.coe_sup, Pi.sup_apply]
ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 365,
"column": 32
} | {
"line": 365,
"column": 53
} | {
"line": 367,
"column": 0
} | [
{
"pp": "case h₂\nα : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : BorelSpace α\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝³ : CompleteSpace E\ninst✝² : MetrizableSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\... | [] | apply interior_subset | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 414,
"column": 6
} | {
"line": 414,
"column": 17
} | {
"line": 414,
"column": 18
} | [
{
"pp": "E : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : FiniteDimensional ℝ F\ninst✝² : MeasurableSpace F\ninst✝¹ : BorelSpace F\nμ : Measure F\ninst✝ : μ.IsAddHaarMeasure\nφ : F → ℝ... | [
"E : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : FiniteDimensional ℝ F\ninst✝² : MeasurableSpace F\ninst✝¹ : BorelSpace F\nμ : Measure F\ninst✝ : μ.IsAddHaarMeasure\nφ : F → ℝ\nhφ : ∀ (x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1148,
"column": 4
} | {
"line": 1148,
"column": 15
} | {
"line": 1148,
"column": 16
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 1149,
"column": 27
} | {
"line": 1149,
"column": 38
} | {
"line": 1149,
"column": 39
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶... | [
"ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.FourierTransform | {
"line": 192,
"column": 52
} | {
"line": 210,
"column": 8
} | {
"line": 212,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝²² : CommRing 𝕜\nV : Type u_2\ninst✝²¹ : AddCommGroup V\ninst✝²⁰ : Module 𝕜 V\ninst✝¹⁹ : MeasurableSpace V\nW : Type u_3\ninst✝¹⁸ : AddCommGroup W\ninst✝¹⁷ : Module 𝕜 W\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace ℂ E\ninst✝¹⁴ :... | [] | by
rw [integral_integral_swap]
have : Integrable (fun (p : W × V) ↦ ‖M‖ * (‖g p.1‖ * ‖f p.2‖)) (ν.prod μ) :=
(hg.norm.mul_prod hf.norm).const_mul _
apply this.mono
· change AEStronglyMeasurable (fun p : W × V ↦ (M (g p.1) (e (-(L p.2) p.1) • f p.2))) _
have A : AEStronglyMeasurable (fun (p : W × V) ↦ e ... | [anonymous] | Lean.Parser.Term.byTactic |
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