module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 9
} | {
"line": 75,
"column": 10
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(f.eqLocus 1)\nhg_proj : ∀ (x : ↥(f.eqLocus 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑(f - 1).rang... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 17
} | {
"line": 105,
"column": 18
} | [
{
"pp": "case hg_ker.refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nhf : ‖f‖ ≤ 1\nx : E\nhx : x ∈ (↑f - 1).rangeᗮ\n⊢ ‖↑f x‖ ≤ ‖↑1 x‖",
"ppTerm": "?hg_ker.refine_1",
"assigned": true,
... | [
"case hg_ker.refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nhf : ‖f‖ ≤ 1\nx : E\nhx : x ∈ (↑f - 1).rangeᗮ\n⊢ ‖f x‖ ≤ ‖x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 75
} | {
"line": 107,
"column": 76
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nhf : ‖f‖ ≤ 1\nx : E\nhx : x ∈ (↑f - 1).rangeᗮ\n⊢ ∀ (y : E), ⟪f y, x⟫ = ⟪y, x⟫",
"ppTerm": "?m.219",
"assigned": false,
"usedConstants": [... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nhf : ‖f‖ ≤ 1\nx : E\nhx : x ∈ (↑f - 1).rangeᗮ\n⊢ ∀ (y : E), ⟪f y, x⟫ = ⟪y, x⟫"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.SingularValues | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 51
} | {
"line": 143,
"column": 52
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\ni j : ℕ\nhij : i ≤ j\... | [
"case pos\n𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\ni j : ℕ\nhij : i ≤ j\nhj : finran... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 89,
"column": 2
} | {
"line": 95,
"column": 78
} | {
"line": 97,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\nι : Type u_6\nι' : Type u_7\ninst✝¹ : Fintype ι\ninst✝ : Fintype ι'\nx : E ⊗[𝕜] F\ne : OrthonormalBasis ι 𝕜 E\... | [] | classical
have : x = ∑ i : ι, ∑ j : ι', (e.toBasis.tensorProduct f.toBasis).repr x (i, j) • e i ⊗ₜ f j := by
conv_lhs => rw [← (e.toBasis.tensorProduct f.toBasis).sum_repr x]
simp [← Finset.sum_product', Basis.tensorProduct_apply']
conv_lhs => rw [this]
simp only [inner_def, map_sum, LinearMap.sum_apply]
... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 89,
"column": 2
} | {
"line": 95,
"column": 78
} | {
"line": 97,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\nι : Type u_6\nι' : Type u_7\ninst✝¹ : Fintype ι\ninst✝ : Fintype ι'\nx : E ⊗[𝕜] F\ne : OrthonormalBasis ι 𝕜 E\... | [] | classical
have : x = ∑ i : ι, ∑ j : ι', (e.toBasis.tensorProduct f.toBasis).repr x (i, j) • e i ⊗ₜ f j := by
conv_lhs => rw [← (e.toBasis.tensorProduct f.toBasis).sum_repr x]
simp [← Finset.sum_product', Basis.tensorProduct_apply']
conv_lhs => rw [this]
simp only [inner_def, map_sum, LinearMap.sum_apply]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 89,
"column": 2
} | {
"line": 95,
"column": 78
} | {
"line": 97,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\nι : Type u_6\nι' : Type u_7\ninst✝¹ : Fintype ι\ninst✝ : Fintype ι'\nx : E ⊗[𝕜] F\ne : OrthonormalBasis ι 𝕜 E\... | [] | classical
have : x = ∑ i : ι, ∑ j : ι', (e.toBasis.tensorProduct f.toBasis).repr x (i, j) • e i ⊗ₜ f j := by
conv_lhs => rw [← (e.toBasis.tensorProduct f.toBasis).sum_repr x]
simp [← Finset.sum_product', Basis.tensorProduct_apply']
conv_lhs => rw [this]
simp only [inner_def, map_sum, LinearMap.sum_apply]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 15
} | {
"line": 112,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nE' : Submodule 𝕜 E\nF' : Submodule 𝕜 F\niE' : Module.Finite 𝕜 ↥E'\niF' : Module.Finite 𝕜 ↥F'\ny : ↥E' ⊗[𝕜] ↥... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nE' : Submodule 𝕜 E\nF' : Submodule 𝕜 F\niE' : Module.Finite 𝕜 ↥E'\niF' : Module.Finite 𝕜 ↥F'\ny : ↥E' ⊗[𝕜] ↥F'\nhz : {(m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 13
} | {
"line": 148,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx : E\ny : F\n⊢ ‖x ⊗ₜ[𝕜] y‖ = ‖x‖ * ‖y‖",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx : E\ny : F\n⊢ ‖x ⊗ₜ[𝕜] y‖ = ‖x‖ * ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 90
} | {
"line": 183,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx y : E ⊗[𝕜] F\n⊢ x = y ↔ ∀ (a : E) (b : F), inner 𝕜 (a ⊗ₜ[𝕜] b) x = inner 𝕜 (a ⊗ₜ[𝕜] b) y",
"ppTerm": "... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx y : E ⊗[𝕜] F\n⊢ x = y ↔ ∀ (a : E) (b : F), inner 𝕜 (a ⊗ₜ[𝕜] b) x = inner 𝕜 (a ⊗ₜ[𝕜] b) y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 548,
"column": 69
} | {
"line": 554,
"column": 14
} | {
"line": 555,
"column": 4
} | [
{
"pp": "E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ... | [] | by
rw [mul_assoc, ← lintegral_const_mul γ]
· gcongr
simp_rw [← mul_assoc]
exact enorm_fderiv_norm_rpow_le (hu.differentiable one_ne_zero) h1γ
dsimp [enorm]
fun_prop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 90
} | {
"line": 198,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] F ⊗[𝕜] G\n⊢ x = y ↔ ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] F ⊗[𝕜] G\n⊢ x = y ↔ ∀ (a : E) (b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 206,
"column": 8
} | {
"line": 206,
"column": 19
} | {
"line": 206,
"column": 20
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreS... | [
"X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreSeparated (S ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 206,
"column": 31
} | {
"line": 206,
"column": 42
} | {
"line": 206,
"column": 43
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreS... | [
"X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreSeparated (S ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 13
} | {
"line": 52,
"column": 14
} | [
{
"pp": "d : ℕ\n⊢ μH[↑d].IsAddHaarMeasure",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Real",
"fact_one_le_two_ennreal",
"MeasureTheory.Measure.hausdorffMeasure",
"PseudoMetricSpace.toUniformSpace",
"AddCommGroup.toAddGroup",
"WithLp.instAddCommGroup... | [
"d : ℕ\n⊢ μH[↑d].IsAddHaarMeasure"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 686,
"column": 2
} | {
"line": 686,
"column": 90
} | {
"line": 687,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] (F ⊗[𝕜] G)\n⊢ x = y ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] (F ⊗[𝕜] G)\n⊢ x = y ↔ ∀ (a : E) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 572,
"column": 80
} | {
"line": 582,
"column": 16
} | {
"line": 584,
"column": 0
} | [
{
"pp": "E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ... | [] | by
suffices (C : ℝ) * γ = eLpNormLESNormFDerivOfEqInnerConst μ p by
rw [eLpNorm_nnreal_eq_lintegral h0p]
congr
norm_cast at this ⊢
simp_rw [eLpNormLESNormFDerivOfEqInnerConst, γ]
refold_let n n' C
rw [NNReal.coe_mul, NNReal.coe_mk, Real.coe_toNNReal', mul_eq_mul_left_iff,... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 41
} | {
"line": 99,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nbasis : OrthonormalBasis (Fin 0) ℝ (EuclideanSpace ℝ (Fin 0)) := EuclideanSpace.basisFun (Fin 0) ℝ\nheq : {0} = parallelepiped ⇑basis\nh✝ : volume = volume.addHaarScalarFactor μH[↑0] • μH[↑0]\nh : volume.addHaarSca... | [] | simp [euclideanHausdorffMeasure_def, h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 24
} | {
"line": 157,
"column": 25
} | [
{
"pp": "case a\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\ns : Set X\ni : ℝ≥0\nhi : μH[↑i] s = 0\nj : ℝ≥0\nhj : μH[↑j] s = ∞\nhij : i < j\n⊢ False",
"ppTerm": "?a✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case a\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\ns : Set X\ni : ℝ≥0\nhi : μH[↑i] s = 0\nj : ℝ≥0\nhj : μH[↑j] s = ∞\nhij : i < j\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 113,
"column": 8
} | {
"line": 113,
"column": 19
} | {
"line": 113,
"column": 20
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹⁰ : EMetricSpace X\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : BorelSpace X\ninst✝⁷ : EMetricSpace Y\ninst✝⁶ : MeasurableSpace Y\ninst✝⁵ : BorelSpace Y\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : Measurable... | [
"X : Type u_1\nY : Type u_2\ninst✝¹⁰ : EMetricSpace X\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : BorelSpace X\ninst✝⁷ : EMetricSpace Y\ninst✝⁶ : MeasurableSpace Y\ninst✝⁵ : BorelSpace Y\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 121,
"column": 70
} | {
"line": 121,
"column": 81
} | {
"line": 121,
"column": 82
} | [
{
"pp": "X : Type u_1\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nd₁ d₂ : ℕ\nh : d₁ < d₂\ns : Set X\n⊢ ↑d₁ < ↑d₂",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"FloorRing.toFloorSemiring",
... | [
"X : Type u_1\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nd₁ d₂ : ℕ\nh : d₁ < d₂\ns : Set X\n⊢ d₁ < d₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 219,
"column": 54
} | {
"line": 219,
"column": 87
} | {
"line": 219,
"column": 88
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreS... | [
"X : Type u_2\ninst✝ : EMetricSpace X\nμ : OuterMeasure X\nhm : μ.IsMetric\nt : Set X\nht : t ∈ {s | IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s}\ns : Set X\nS : ℕ → Set X := fun n ↦ {x | x ∈ s ∧ (↑n)⁻¹ ≤ infEDist x t}\nSsep : ∀ (n : ℕ), AreSeparated (S n) t\nSsep' : ∀ (n : ℕ), AreSeparated (S ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 294,
"column": 12
} | {
"line": 294,
"column": 23
} | {
"line": 294,
"column": 24
} | [
{
"pp": "V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nh... | [
"V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\ns : AffineSubspace ℝ P\nhs : Nonempty... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 330,
"column": 2
} | {
"line": 330,
"column": 49
} | {
"line": 330,
"column": 50
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nr : ℝ≥0\nf : X → Y\nhr : 0 < r\nhf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, HolderOnWith C r f s\nx : X\nx✝ : x ∈ univ\n⊢ ∃ C, ∃ t ∈ 𝓝[univ] x, HolderOnWith C r f t",
"ppTerm": "?m.49",
"assig... | [
"X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nr : ℝ≥0\nf : X → Y\nhr : 0 < r\nhf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, HolderOnWith C r f s\nx : X\nx✝ : x ∈ univ\n⊢ ∃ C, ∃ t ∈ 𝓝 x, HolderOnWith C r f t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 13
} | {
"line": 339,
"column": 14
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nK : ℝ≥0\nf : X → Y\ns : Set X\nh : LipschitzOnWith K f s\n⊢ dimH (f '' s) ≤ dimH s",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nK : ℝ≥0\nf : X → Y\ns : Set X\nh : LipschitzOnWith K f s\n⊢ dimH (f '' s) ≤ dimH s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 39
} | {
"line": 360,
"column": 40
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\ns : Set X\nhf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t\n⊢ ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, HolderOnWith C 1 f t",
"ppTerm": "?m.45",
"assigned": true,
"usedCon... | [
"X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\ns : Set X\nhf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t\n⊢ ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 45
} | {
"line": 361,
"column": 46
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\ns : Set X\nhf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t\nthis : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, HolderOnWith C 1 f t\n⊢ dimH (f '' s) ≤ dimH s",
"ppTerm": "?m.46",
... | [
"X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\ns : Set X\nhf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t\nthis : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, HolderOnWith C 1 f t\n⊢ dimH (f '' s) ≤ dimH s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 49
} | {
"line": 370,
"column": 50
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\nhf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s\nx : X\nx✝ : x ∈ univ\n⊢ ∃ C, ∃ t ∈ 𝓝[univ] x, LipschitzOnWith C f t",
"ppTerm": "?m.40",
"assigned": true,
"us... | [
"X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\nhf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s\nx : X\nx✝ : x ∈ univ\n⊢ ∃ C, ∃ t ∈ 𝓝 x, LipschitzOnWith C f t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 420,
"column": 4
} | {
"line": 420,
"column": 41
} | {
"line": 420,
"column": 42
} | [
{
"pp": "𝕜 : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ne : E ≃L[𝕜] F\ns : Set E\n⊢ dimH s ≤ dimH (⇑e '' s)",
"ppTerm": "?m.52",
"assigned": false,
"us... | [
"𝕜 : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ne : E ≃L[𝕜] F\ns : Set E\n⊢ dimH s ≤ dimH (⇑e '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 13
} | {
"line": 344,
"column": 14
} | [
{
"pp": "X : Type u_2\ninst✝ : EMetricSpace X\nb : OuterMeasure X\nhb : ∀ (i : ℝ≥0∞), ⨆ (_ : i > 0), ⊤ ≤ b\n⊢ ⊤ ≤ b",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"CompleteLattice.toBoundedOrder",
"Measure... | [
"X : Type u_2\ninst✝ : EMetricSpace X\nb : OuterMeasure X\nhb : ∀ (i : ℝ≥0∞), ⨆ (_ : i > 0), ⊤ ≤ b\n⊢ b = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 470,
"column": 4
} | {
"line": 470,
"column": 49
} | {
"line": 470,
"column": 50
} | [
{
"pp": "case refine_2\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nx : E\ns : Set E\nh : s ∈ 𝓝 x\ne : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ\nthis : ⇑e '' s ∈ 𝓝 (e x)\nr : ℝ\nhr0 : 0 < r\nhr : Metric.ball (e x) r ⊆ ⇑e '' s\n⊢ ↑(finrank ℝ E) ≤ dimH (⇑e '' s)",
... | [
"case refine_2\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nx : E\ns : Set E\nh : s ∈ 𝓝 x\ne : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ\nthis : ⇑e '' s ∈ 𝓝 (e x)\nr : ℝ\nhr0 : 0 < r\nhr : Metric.ball (e x) r ⊆ ⇑e '' s\n⊢ ↑(finrank ℝ E) ≤ dimH (⇑e '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 689,
"column": 10
} | {
"line": 689,
"column": 21
} | {
"line": 689,
"column": 22
} | [
{
"pp": "case e'_4.h\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\nμ : Measure E\ninst✝¹ : μ.IsAddHaarMeasure\ninst✝ : FiniteDimensi... | [
"case e'_4.h\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\nμ : Measure E\ninst✝¹ : μ.IsAddHaarMeasure\ninst✝ : FiniteDimensional ℝ F\nu ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 487,
"column": 2
} | {
"line": 487,
"column": 33
} | {
"line": 487,
"column": 34
} | [
{
"pp": "E : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhcvx : Convex ℝ s\nhne : s.Nonempty\nthis : Nonempty ↑s\nφ : ↥(affineSpan ℝ s) ≃ᵃⁱ[ℝ] ↥(affineSpan ℝ s).direction := AffineIsometryEquiv.constVSub ℝ ⟨hne.some, ⋯⟩\nhs_eq : s = Subtype.val ''... | [
"E : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhcvx : Convex ℝ s\nhne : s.Nonempty\nthis : Nonempty ↑s\nφ : ↥(affineSpan ℝ s) ≃ᵃⁱ[ℝ] ↥(affineSpan ℝ s).direction := AffineIsometryEquiv.constVSub ℝ ⟨hne.some, ⋯⟩\nhs_eq : s = Subtype.val '' Subtype.val... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.Euclidean.Volume.Measure | {
"line": 349,
"column": 48
} | {
"line": 349,
"column": 64
} | {
"line": 349,
"column": 65
} | [
{
"pp": "V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\np : P\nv : V\nhv : v ≠ 0\... | [
"V : Type u_3\nP : Type u_4\ninst✝⁸ : NormedAddCommGroup V\ninst✝⁷ : InnerProductSpace ℝ V\ninst✝⁶ : MeasurableSpace V\ninst✝⁵ : BorelSpace V\ninst✝⁴ : FiniteDimensional ℝ V\ninst✝³ : MetricSpace P\ninst✝² : MeasurableSpace P\ninst✝¹ : BorelSpace P\ninst✝ : NormedAddTorsor V P\np : P\nv : V\nhv : v ≠ 0\nt : Set P\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 41
} | {
"line": 94,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\nι : Type u_5\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : U →ₗ[𝕜] V\nbu : O... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\nι : Type u_5\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : U →ₗ[𝕜] V\nbu : OrthonormalBa... | rw [← basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix (stdOrthonormalBasis 𝕜 U).toBasis
bu.toBasis h.some.toBasis bv.toBasis] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 117,
"column": 10
} | {
"line": 117,
"column": 21
} | {
"line": 117,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\n⊢ Function.Injective ⇑f.rangeRestrict",
"ppTe... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\n⊢ Function.Injective ⇑f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 528,
"column": 2
} | {
"line": 528,
"column": 33
} | {
"line": 528,
"column": 34
} | [
{
"pp": "X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nβ : Type u_4\nι : β → Type u_5\nhι : (n : β) → Fintype (ι n)\ns : Set X\nl : Filter β\nr : β → ℝ≥0∞\nhr : Tendsto r l (𝓝 0)\nt : (n : β) → ι n → Set X\nht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n\nhst :... | [
"X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nβ : Type u_4\nι : β → Type u_5\nhι : (n : β) → Fintype (ι n)\ns : Set X\nl : Filter β\nr : β → ℝ≥0∞\nhr : Tendsto r l (𝓝 0)\nt : (n : β) → ι n → Set X\nht : ∀ᶠ (n : β) in l, ∀ (i : ι n), ediam (t n i) ≤ r n\nhst : ∀ᶠ (n : β) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 62
} | {
"line": 123,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\ng : U ≃ₗ[𝕜] ↥f.range := LinearEquiv.ofBijective ... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\ng : U ≃ₗ[𝕜] ↥f.range := LinearEquiv.ofBijective f.rangeRestr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 124,
"column": 4
} | {
"line": 125,
"column": 44
} | {
"line": 125,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\ng : U ≃ₗ[𝕜] ↥f.range := LinearEquiv.ofBijective ... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nh : f.ker = ⊥\ng : U ≃ₗ[𝕜] ↥f.range := LinearEquiv.ofBijective f.rangeRestr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 17
} | {
"line": 132,
"column": 18
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nb : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range\n⊢ finrank... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nb : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range\n⊢ finrank 𝕜 ↥f.range... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 23
} | {
"line": 133,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nb : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range\nhrank : f... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nb : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 ↥f.range\nhrank : finrank 𝕜 ↥f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 593,
"column": 32
} | {
"line": 593,
"column": 74
} | {
"line": 593,
"column": 75
} | [
{
"pp": "X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nd₁ d₂ : ℝ\nh : d₁ < d₂\ns : Set X\nH : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ∞\nc : ℝ≥0\nhc : c ≠ 0\nthis : 0 < ↑c ^ (d₂ - d₁)⁻¹\nr : ℝ≥0\nhr₀✝ : 0 ≤ ↑r\nhrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹\nhr₀ : r ≠ 0\n⊢ ↑r ≠ 0",
"ppTerm": "?m.2... | [
"X : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nd₁ d₂ : ℝ\nh : d₁ < d₂\ns : Set X\nH : μH[d₂] s ≠ 0 ∧ μH[d₁] s ≠ ∞\nc : ℝ≥0\nhc : c ≠ 0\nthis : 0 < ↑c ^ (d₂ - d₁)⁻¹\nr : ℝ≥0\nhr₀✝ : 0 ≤ ↑r\nhrc : ↑r < ↑c ^ (d₂ - d₁)⁻¹\nhr₀ : r ≠ 0\n⊢ ¬r = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 15
} | {
"line": 153,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\ntfae_1_iff_2 : f.normDet ≠ 0 ↔ f.ker = ⊥\ntfae_1_iff_3 : f.normD... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\ntfae_1_iff_2 : f.normDet ≠ 0 ↔ f.ker = ⊥\ntfae_1_iff_3 : f.normDet ≠ 0 ↔ fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 172,
"column": 24
} | {
"line": 172,
"column": 35
} | {
"line": 172,
"column": 36
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\ntfae_1_iff_2 : f.normDet = 0 ↔ f.ker ≠ ⊥\ntfae_1_iff_3 : f.normD... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\ntfae_1_iff_2 : f.normDet = 0 ↔ f.ker ≠ ⊥\ntfae_1_iff_3 : f.normDet = 0 ↔ fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 15
} | {
"line": 175,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\ntfae_1_iff_2 : f.normDet = 0 ↔ f.ker ≠ ⊥\ntfae_1_iff_3 : f.normD... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\ntfae_1_iff_2 : f.normDet = 0 ↔ f.ker ≠ ⊥\ntfae_1_iff_3 : f.normDet = 0 ↔ fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 624,
"column": 4
} | {
"line": 624,
"column": 33
} | {
"line": 624,
"column": 34
} | [
{
"pp": "case a\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nx : X\nr : ℕ → ℝ≥0∞ := fun x ↦ 0\nt : ℕ → Unit → Set X := fun x_1 x_2 ↦ {x}\nht : ∀ᶠ (n : ℕ) in atTop, ∀ (i : Unit), ediam (t n i) ≤ r n\n⊢ μH[0] {x} ≤ 1",
"ppTerm": "?a✝",
"assigned": true,
"us... | [
"case a\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nx : X\nr : ℕ → ℝ≥0∞ := fun x ↦ 0\nt : ℕ → Unit → Set X := fun x_1 x_2 ↦ {x}\nht : ∀ᶠ (n : ℕ) in atTop, ∀ (i : Unit), ediam (t n i) ≤ r n\n⊢ μH[0] {x} ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 198,
"column": 6
} | {
"line": 198,
"column": 17
} | {
"line": 198,
"column": 18
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\nι : Type u_5\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : U →ₗ[𝕜] V\nbu : O... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\nι : Type u_5\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : U →ₗ[𝕜] V\nbu : OrthonormalBa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 15
} | {
"line": 202,
"column": 16
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\nι : Type u_5\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : U →ₗ[𝕜]... | [
"case neg\n𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\nι : Type u_5\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : U →ₗ[𝕜] V\nbu : Ort... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 685,
"column": 41
} | {
"line": 685,
"column": 72
} | {
"line": 685,
"column": 73
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nr : ℝ≥0\nf : X → Y\ns : Set X\nhr : 0 < r\nd : ℝ\nhd : 0 ≤ d\nh : HolderOnWith 0 r f s\nx : X\nhx : x ∈ s\n⊢ (f '' s).Subsin... | [
"X : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nr : ℝ≥0\nf : X → Y\ns : Set X\nhr : 0 < r\nd : ℝ\nhd : 0 ≤ d\nh : HolderOnWith 0 r f s\nx : X\nhx : x ∈ s\n⊢ (f '' s).Subsingleton"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 692,
"column": 6
} | {
"line": 693,
"column": 44
} | {
"line": 695,
"column": 2
} | [
{
"pp": "case inl.inr.inr\nX : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nr : ℝ≥0\nf : X → Y\ns : Set X\nhr : 0 < r\nd : ℝ\nhd : 0 ≤ d\nh : HolderOnWith 0 r f s\nx : X\nhx : x ∈ s\... | [] | haveI := nullSingletonClass_hausdorff Y h'd
simp only [zero_le, measure_singleton] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 692,
"column": 6
} | {
"line": 693,
"column": 44
} | {
"line": 695,
"column": 2
} | [
{
"pp": "case inl.inr.inr\nX : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nr : ℝ≥0\nf : X → Y\ns : Set X\nhr : 0 < r\nd : ℝ\nhd : 0 ≤ d\nh : HolderOnWith 0 r f s\nx : X\nhx : x ∈ s\... | [] | haveI := nullSingletonClass_hausdorff Y h'd
simp only [zero_le, measure_singleton] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.OfNorm | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 28
} | {
"line": 166,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nr : ℚ\nx y : E\na b : 𝕜\n⊢ inner_ 𝕜 ((a + b) • x) y = inner_ 𝕜 (a • x) y + inner_ 𝕜 (b • x) y",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nr : ℚ\nx y : E\na b : 𝕜\n⊢ inner_ 𝕜 (a • x + b • x) y = inner_ 𝕜 (a • x) y + inner_ 𝕜 (b • x) y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.OfNorm | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 33
} | {
"line": 167,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nr : ℚ\nx y : E\nhom : 𝕜 →ₗ[ℚ] 𝕜 := (AddMonoidHom.mk' (fun r ↦ inner_ 𝕜 (r • x) y) ⋯).toRatLinearMap\n⊢ inner_ 𝕜 (↑r • x) y = (starRingEnd 𝕜) ↑r * inner_ 𝕜 x ... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nr : ℚ\nx y : E\nhom : 𝕜 →ₗ[ℚ] 𝕜 := (AddMonoidHom.mk' (fun r ↦ inner_ 𝕜 (r • x) y) ⋯).toRatLinearMap\n⊢ inner_ 𝕜 (↑r • x) y = ↑r * inner_ 𝕜 x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.OfNorm | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 15
} | {
"line": 181,
"column": 16
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nhI : I = 0\n⊢ innerProp' E 0",
"ppTerm": "?pos✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case pos\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nhI : I = 0\n⊢ innerProp' E 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 258,
"column": 36
} | {
"line": 258,
"column": 65
} | {
"line": 258,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nc : 𝕜\nhc : ¬c = 0\nh : ¬f.ker = ⊥\n⊢ (c • f).ker ≠ ⊥",
"pp... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nc : 𝕜\nhc : ¬c = 0\nh : ¬f.ker = ⊥\n⊢ ¬f.ker = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 243,
"column": 55
} | {
"line": 259,
"column": 86
} | {
"line": 261,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\nc : 𝕜\n⊢ (c • f).normDet = ‖c‖ ^ finrank 𝕜 U * f.normDet",
... | [] | by
by_cases hc : c = 0
· nontriviality U
simp [hc, zero_pow finrank_pos.ne.symm]
by_cases h : f.ker = ⊥
· obtain ⟨bv⟩ := (f.normDet_ne_zero_tfae.out 1 3).mp h
let bu : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 U := stdOrthonormalBasis 𝕜 U
let bv' : OrthonormalBasis (Fin (finrank 𝕜 U)) 𝕜 (c • f).ra... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 13
} | {
"line": 263,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\n⊢ (-f).normDet = f.normDet",
"ppTerm": "?m.40",
"assigne... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup U\ninst✝³ : InnerProductSpace 𝕜 U\ninst✝² : FiniteDimensional 𝕜 U\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace 𝕜 V\nf : U →ₗ[𝕜] V\n⊢ (-f).normDet = f.normDet"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 727,
"column": 2
} | {
"line": 727,
"column": 44
} | {
"line": 727,
"column": 45
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nK : ℝ≥0\nf : X → Y\ns : Set X\nh : LipschitzOnWith K f s\nd : ℝ\nhd : 0 ≤ d\n⊢ μH[d] (f '' s) ≤ ↑K ^ d * μH[d] s",
"ppTe... | [
"X : Type u_2\nY : Type u_3\ninst✝⁵ : EMetricSpace X\ninst✝⁴ : EMetricSpace Y\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : MeasurableSpace Y\ninst✝ : BorelSpace Y\nK : ℝ≥0\nf : X → Y\ns : Set X\nh : LipschitzOnWith K f s\nd : ℝ\nhd : 0 ≤ d\n⊢ μH[d] (f '' s) ≤ ↑K ^ d * μH[d] s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 750,
"column": 4
} | {
"line": 751,
"column": 11
} | {
"line": 751,
"column": 12
} | [
{
"pp": "𝕜 : Type u_4\nE : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : NormSMulClass 𝕜 E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nd : ℝ\nhd : 0 ≤ d\nr✝ : 𝕜\nhr : r✝ ≠ 0\ns✝ : Set E\nr : 𝕜\ns : Set E\n⊢ μH[d] (r • s) ≤ ‖r‖₊ ^ d • μH[d] s",... | [
"𝕜 : Type u_4\nE : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : NormSMulClass 𝕜 E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nd : ℝ\nhd : 0 ≤ d\nr✝ : 𝕜\nhr : r✝ ≠ 0\ns✝ : Set E\nr : 𝕜\ns : Set E\n⊢ μH[d] (r • s) ≤ ↑‖r‖₊ ^ d * μH[d] s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 46
} | {
"line": 137,
"column": 47
} | [
{
"pp": "case mp\nH : Type u_1\ninst✝ : NormedAddCommGroup H\nipc : InnerProductSpace ℂ H\nS : ClosedSubmodule ℝ H\nx : H\nh : ∀ (a : ℝ), a • (scalarSMulCLE H UnitI).symm x ∈ S.mulI\n⊢ x ∈ S",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nH : Type u_1\ninst✝ : NormedAddCommGroup H\nipc : InnerProductSpace ℂ H\nS : ClosedSubmodule ℝ H\nx : H\nh : ∀ (a : ℝ), a • (scalarSMulCLE H UnitI).symm x ∈ S.mulI\n⊢ x ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 46
} | {
"line": 140,
"column": 47
} | [
{
"pp": "case mpr\nH : Type u_1\ninst✝ : NormedAddCommGroup H\nipc : InnerProductSpace ℂ H\nS : ClosedSubmodule ℝ H\nx : H\nh : ∀ (a : ℝ), a • x ∈ S\n⊢ x ∈ S.mulI.mulI",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Units.val",
"Eq.mpr",
... | [
"case mpr\nH : Type u_1\ninst✝ : NormedAddCommGroup H\nipc : InnerProductSpace ℂ H\nS : ClosedSubmodule ℝ H\nx : H\nh : ∀ (a : ℝ), a • x ∈ S\n⊢ -x ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 46
} | {
"line": 163,
"column": 0
} | [
{
"pp": "H : Type u_1\ninst✝ : NormedAddCommGroup H\nipc : InnerProductSpace ℂ H\nS T : ClosedSubmodule ℝ H\n⊢ (S.mulI ⊔ T.mulI)ᗮ = S.mulIᗮ ⊓ T.mulIᗮ",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"NormedSpace.toIsBoundedSMul",
... | [] | exact Eq.symm (inf_orthogonal S.mulI T.mulI) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 206,
"column": 21
} | {
"line": 206,
"column": 57
} | {
"line": 206,
"column": 58
} | [
{
"pp": "H : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℂ H\nS : StandardSubspace H\n⊢ S.toClosedSubmodule.mulI ⊓ S.toClosedSubmodule.mulI.mulI = ⊥",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",... | [
"H : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℂ H\nS : StandardSubspace H\n⊢ S.toClosedSubmodule ⊓ S.toClosedSubmodule.mulI = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 207,
"column": 17
} | {
"line": 207,
"column": 53
} | {
"line": 207,
"column": 54
} | [
{
"pp": "H : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℂ H\nS : StandardSubspace H\n⊢ S.toClosedSubmodule.mulI ⊔ S.toClosedSubmodule.mulI.mulI = ⊤",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",... | [
"H : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℂ H\nS : StandardSubspace H\n⊢ S.toClosedSubmodule ⊔ S.toClosedSubmodule.mulI = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StarOrder | {
"line": 48,
"column": 6
} | {
"line": 49,
"column": 13
} | {
"line": 49,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nH : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup H\ninst✝³ : InnerProductSpace 𝕜 H\ninst✝² : CompleteSpace H\ninst✝¹ : Algebra ℝ (H →L[𝕜] H)\ninst✝ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)\nf : H →L[𝕜] H\nhf : f.IsPositive\nc✝ : ℝ\nc : ℝ := -c✝\nhc : 0 < c\nx : H\n⊢ re ⟪((algebr... | [
"𝕜 : Type u_1\nH : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup H\ninst✝³ : InnerProductSpace 𝕜 H\ninst✝² : CompleteSpace H\ninst✝¹ : Algebra ℝ (H →L[𝕜] H)\ninst✝ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)\nf : H →L[𝕜] H\nhf : f.IsPositive\nc✝ : ℝ\nc : ℝ := -c✝\nhc : 0 < c\nx : H\n⊢ 0 ≤ re ⟪f x, x⟫_𝕜"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 48
} | {
"line": 216,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : CompleteSpace V\nK : Matrix X X (V →L[𝕜] V)\nthis : ∀ {h p1 p2 p3 : Prop}, (h → [p1, p2, p3].TFAE) → [h ∧ p1, h ∧ p2, h ∧ p3].TFAE\nhHerm : K.IsHermitian\nx✝ : Nontriv... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : CompleteSpace V\nK : Matrix X X (V →L[𝕜] V)\nthis : ∀ {h p1 p2 p3 : Prop}, (h → [p1, p2, p3].TFAE) → [h ∧ p1, h ∧ p2, h ∧ p3].TFAE\nhHerm : K.IsHermitian\nx✝ : Nontrivial V\nv : V... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 72
} | {
"line": 220,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : CompleteSpace V\nK : Matrix X X (V →L[𝕜] V)\nthis : ∀ {h p1 p2 p3 : Prop}, (h → [p1, p2, p3].TFAE) → [h ∧ p1, h ∧ p2, h ∧ p3].TFAE\nhHerm : K.IsHermitian\nx✝ : Nontriv... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace 𝕜 V\ninst✝ : CompleteSpace V\nK : Matrix X X (V →L[𝕜] V)\nthis : ∀ {h p1 p2 p3 : Prop}, (h → [p1, p2, p3].TFAE) → [h ∧ p1, h ∧ p2, h ∧ p3].TFAE\nhHerm : K.IsHermitian\nx✝ : Nontrivial V\nv✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 346,
"column": 6
} | {
"line": 346,
"column": 24
} | {
"line": 346,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : InnerProductSpace 𝕜 W\nf ... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup U\ninst✝⁵ : InnerProductSpace 𝕜 U\ninst✝⁴ : FiniteDimensional 𝕜 U\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace 𝕜 V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : InnerProductSpace 𝕜 W\nf : U →ₗ[𝕜] V... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 359,
"column": 6
} | {
"line": 359,
"column": 71
} | {
"line": 359,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup U\ninst✝⁶ : InnerProductSpace 𝕜 U\ninst✝⁵ : FiniteDimensional 𝕜 U\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace 𝕜 V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ni... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup U\ninst✝⁶ : InnerProductSpace 𝕜 U\ninst✝⁵ : FiniteDimensional 𝕜 U\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace 𝕜 V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : Finit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 362,
"column": 6
} | {
"line": 362,
"column": 24
} | {
"line": 362,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup U\ninst✝⁶ : InnerProductSpace 𝕜 U\ninst✝⁵ : FiniteDimensional 𝕜 U\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace 𝕜 V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ni... | [
"𝕜 : Type u_1\nU : Type u_2\nV : Type u_3\nW : Type u_4\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup U\ninst✝⁶ : InnerProductSpace 𝕜 U\ninst✝⁵ : FiniteDimensional 𝕜 U\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace 𝕜 V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : Finit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.NormDet | {
"line": 391,
"column": 2
} | {
"line": 391,
"column": 13
} | {
"line": 391,
"column": 14
} | [
{
"pp": "U : Type u_5\ninst✝² : NormedAddCommGroup U\ninst✝¹ : InnerProductSpace ℝ U\ninst✝ : FiniteDimensional ℝ U\nf : U →ₗ[ℝ] U\n⊢ f.normDet = |LinearMap.det f|",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"U : Type u_5\ninst✝² : NormedAddCommGroup U\ninst✝¹ : InnerProductSpace ℝ U\ninst✝ : FiniteDimensional ℝ U\nf : U →ₗ[ℝ] U\n⊢ f.normDet = |LinearMap.det f|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 904,
"column": 4
} | {
"line": 904,
"column": 46
} | {
"line": 904,
"column": 47
} | [
{
"pp": "ι : Type u_4\ninst✝ : Fintype ι\na b : ι → ℚ\nH : ∀ (i : ι), a i < b i\ni : ι\n⊢ 0 ≤ ↑(b i) - ↑(a i)",
"ppTerm": "?m.140",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"Real.instSub... | [
"ι : Type u_4\ninst✝ : Fintype ι\na b : ι → ℚ\nH : ∀ (i : ι), a i < b i\ni : ι\n⊢ a i ≤ b i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Reproducing | {
"line": 303,
"column": 15
} | {
"line": 303,
"column": 65
} | {
"line": 303,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace 𝕜 V\nH : Type u_4\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : RKHS 𝕜 H X V\ninst✝² : CompleteSpace H\ninst✝¹ : CompleteSpace V\nK : Matrix X X (V →L[�... | [
"𝕜 : Type u_1\ninst✝⁸ : RCLike 𝕜\nX : Type u_2\nV : Type u_3\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace 𝕜 V\nH : Type u_4\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : RKHS 𝕜 H X V\ninst✝² : CompleteSpace H\ninst✝¹ : CompleteSpace V\nK : Matrix X X (V →L[𝕜] V)\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 928,
"column": 10
} | {
"line": 928,
"column": 49
} | {
"line": 928,
"column": 50
} | [
{
"pp": "case h'\nι : Type u_4\ninst✝ : Fintype ι\na b : ι → ℚ\nH : ∀ (i : ι), a i < b i\nI : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)\nγ : ℕ → Type u_4 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊\nt : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n... | [
"case h'\nι : Type u_4\ninst✝ : Fintype ι\na b : ι → ℚ\nH : ∀ (i : ι), a i < b i\nI : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)\nγ : ℕ → Type u_4 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊\nt : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)\nA : Tends... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 947,
"column": 8
} | {
"line": 951,
"column": 39
} | {
"line": 952,
"column": 6
} | [
{
"pp": "case refine_1\nι : Type u_4\ninst✝ : Fintype ι\na b : ι → ℚ\nH : ∀ (i : ι), a i < b i\nI : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)\nγ : ℕ → Type u_4 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊\nt : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1... | [] | filter_upwards [B] with _ hn
apply Finset.sum_le_sum fun i _ => _
simp only [ENNReal.rpow_natCast]
intro i _
exact pow_le_pow_left' (hn i) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 947,
"column": 8
} | {
"line": 951,
"column": 39
} | {
"line": 952,
"column": 6
} | [
{
"pp": "case refine_1\nι : Type u_4\ninst✝ : Fintype ι\na b : ι → ℚ\nH : ∀ (i : ι), a i < b i\nI : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)\nγ : ℕ → Type u_4 := fun n ↦ (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊\nt : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f ↦ univ.pi fun i ↦ Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1... | [] | filter_upwards [B] with _ hn
apply Finset.sum_le_sum fun i _ => _
simp only [ENNReal.rpow_natCast]
intro i _
exact pow_le_pow_left' (hn i) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.ContinuousOfBounded | {
"line": 52,
"column": 27
} | {
"line": 52,
"column": 38
} | {
"line": 52,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : No... | [
"𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : NormedField 𝕜... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 1038,
"column": 4
} | {
"line": 1039,
"column": 44
} | {
"line": 1039,
"column": 45
} | [
{
"pp": "E : Type u_5\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nv : E\ns : Set ℝ\nhv : v ≠ 0\nhn : ‖v‖ ≠ 0\nthis : μH[1] ((fun x ↦ ‖v‖ • x) '' ⇑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s) = ‖v‖₊ • μH[1] s\n⊢ μH[1] ((fun r ↦ r • v) '' s) = ‖... | [
"E : Type u_5\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nv : E\ns : Set ℝ\nhv : v ≠ 0\nhn : ‖v‖ ≠ 0\nthis : μH[1] ((fun x ↦ ‖v‖ • x) '' ⇑(LinearMap.toSpanSingleton ℝ E (‖v‖⁻¹ • v)) '' s) = ‖v‖₊ • μH[1] s\n⊢ μH[1] ((fun r ↦ r • v) '' s) = ‖v‖₊ • μH[1] ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.ContinuousOfBounded | {
"line": 93,
"column": 6
} | {
"line": 93,
"column": 17
} | {
"line": 93,
"column": 18
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜... | [
"case refine_2\n𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Hausdorff | {
"line": 1116,
"column": 2
} | {
"line": 1116,
"column": 13
} | {
"line": 1116,
"column": 14
} | [
{
"pp": "𝕜 : Type u_4\nE : Type u_5\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nd : ℝ\ns : Set E\nhs : 0 ≤ d\n⊢ μH[d] (⇑K.orthogonalProjectionOnto '' s) ≤ μH[d] s",... | [
"𝕜 : Type u_4\nE : Type u_5\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nd : ℝ\ns : Set E\nhs : 0 ≤ d\n⊢ μH[d] (⇑K.orthogonalProjectionOnto '' s) ≤ μH[d] s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.ContinuousOfBounded | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 78
} | {
"line": 105,
"column": 79
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : No... | [
"𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : NormedField 𝕜... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.ContinuousOfBounded | {
"line": 115,
"column": 53
} | {
"line": 115,
"column": 64
} | {
"line": 115,
"column": 65
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : No... | [
"𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : TopologicalSpace E\ninst✝⁹ : IsTopologicalAddGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : NormedField 𝕜... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 58
} | {
"line": 132,
"column": 59
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\n⊢ |(o.areaForm x) y| ≤ ‖x‖ * ‖y‖",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"AlternatingMap",
"Norm.norm",
"Eq.... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\n⊢ |o.volumeForm ![x, y]| ≤ ‖x‖ * ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 58
} | {
"line": 135,
"column": 59
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\n⊢ (o.areaForm x) y ≤ ‖x‖ * ‖y‖",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"AlternatingMap",
"Norm.norm",
"Eq.mp... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\n⊢ o.volumeForm ![x, y] ≤ ‖x‖ * ‖y‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 15
} | {
"line": 143,
"column": 16
} | [
{
"pp": "case «0».«1»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nh : ⟪x, y⟫ = 0\nhij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩\n⊢ ⟪![x, y] ((fun i ↦ i) ⟨0, ⋯⟩), ![x, y] ((fun i ↦ i) ⟨1, ⋯⟩)⟫ = 0",
"ppTer... | [
"case «0».«1»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nh : ⟪x, y⟫ = 0\nhij : (fun i ↦ i) ⟨0, ⋯⟩ ≠ (fun i ↦ i) ⟨1, ⋯⟩\n⊢ ⟪x, y⟫ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 33
} | {
"line": 144,
"column": 34
} | [
{
"pp": "case «1».«0»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nh : ⟪x, y⟫ = 0\nhij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩\n⊢ ⟪![x, y] ((fun i ↦ i) ⟨1, ⋯⟩), ![x, y] ((fun i ↦ i) ⟨0, ⋯⟩)⟫ = 0",
"ppTer... | [
"case «1».«0»\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nh : ⟪x, y⟫ = 0\nhij : (fun i ↦ i) ⟨1, ⋯⟩ ≠ (fun i ↦ i) ⟨0, ⋯⟩\n⊢ ⟪x, y⟫ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 48,
"column": 36
} | {
"line": 48,
"column": 47
} | {
"line": 48,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : Convex ℝ s\nx ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : (t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 20
} | {
"line": 57,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : Convex ℝ s\nx ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : (t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 46
} | {
"line": 362,
"column": 47
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\na b : E\n⊢ ⟪a, b⟫ ^ 2 + (o.areaForm a) b ^ 2 = ‖a‖ ^ 2 * ‖b‖ ^ 2",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.m... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\na b : E\n⊢ ⟪a, b⟫ * ⟪a, b⟫ + (o.areaForm a) b * (o.areaForm a) b = ‖a‖ * ‖a‖ * (‖b‖ * ‖b‖)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 67
} | {
"line": 68,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\ns : Set E\n⊢ ⇑(toWeakSpace 𝕜 ... | [] | refine LinearMap.image_convexHull (toWeakSpace 𝕜 E).toLinearMap s | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 77,
"column": 4
} | {
"line": 78,
"column": 11
} | {
"line": 78,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : TopologicalSpace E\ninst✝⁶... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : IsTopolog... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 94,
"column": 2
} | {
"line": 95,
"column": 9
} | {
"line": 95,
"column": 10
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : TopologicalSpace E\ninst✝⁶... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : IsTopolog... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 22
} | {
"line": 115,
"column": 23
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : Topological... | [
"case refine_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : TopologicalSpace E\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.WeakSpace | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 23
} | {
"line": 116,
"column": 24
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : Topological... | [
"case refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : AddCommGroup E\ninst✝¹⁴ : Module 𝕜 E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : Module 𝕜 F\ninst✝¹¹ : Module ℝ E\ninst✝¹⁰ : IsScalarTower ℝ 𝕜 E\ninst✝⁹ : Module ℝ F\ninst✝⁸ : IsScalarTower ℝ 𝕜 F\ninst✝⁷ : TopologicalSpace E\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 463,
"column": 2
} | {
"line": 463,
"column": 54
} | {
"line": 463,
"column": 55
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\n⊢ Complex.normSq ((o.kahler x) y) = ‖x‖ ^ 2 * ‖y‖ ^ 2",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceReal... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\n⊢ ⟪x, y⟫ * ⟪x, y⟫ + (o.areaForm x) y * (o.areaForm x) y = ‖x‖ * ‖x‖ * (‖y‖ * ‖y‖)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.WeakOperatorTopology | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 26
} | {
"line": 347,
"column": 27
} | [
{
"pp": "𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝¹⁰ : NormedField 𝕜₁\ninst✝⁹ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : Module 𝕜₂ F\ninst✝² : IsTopologi... | [
"𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝¹⁰ : NormedField 𝕜₁\ninst✝⁹ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : Module 𝕜₂ F\ninst✝² : IsTopologicalAddGroup ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.LocallyConvex.WeakOperatorTopology | {
"line": 344,
"column": 87
} | {
"line": 347,
"column": 30
} | {
"line": 349,
"column": 0
} | [
{
"pp": "𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝¹⁰ : NormedField 𝕜₁\ninst✝⁹ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : Module 𝕜₁ E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : TopologicalSpace F\ninst✝³ : Module 𝕜₂ F\ninst✝² : IsTopologi... | [] | by
refine Function.Injective.isEmbedding_induced fun A B hAB => ?_
rw [ContinuousLinearMapWOT.ext_dual_iff]
simpa [funext_iff] using hAB | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 472,
"column": 29
} | {
"line": 472,
"column": 45
} | {
"line": 472,
"column": 46
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nhx : (o.kahler x) y = 0\n⊢ ‖x‖ * ‖y‖ = 0",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"A... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nhx : (o.kahler x) y = 0\n⊢ x = 0 ∨ y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 475,
"column": 4
} | {
"line": 475,
"column": 15
} | {
"line": 475,
"column": 16
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nhx : (o.kahler x) y = 0\nthis : ‖x‖ * ‖y‖ = 0\nh : ‖x‖ = 0\n⊢ x = 0",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"u... | [
"case inl\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nhx : (o.kahler x) y = 0\nthis : ‖x‖ * ‖y‖ = 0\nh : ‖x‖ = 0\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 477,
"column": 4
} | {
"line": 477,
"column": 15
} | {
"line": 477,
"column": 16
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nhx : (o.kahler x) y = 0\nthis : ‖x‖ * ‖y‖ = 0\nh : ‖y‖ = 0\n⊢ y = 0",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"u... | [
"case inr\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\nhx : (o.kahler x) y = 0\nthis : ‖x‖ * ‖y‖ = 0\nh : ‖y‖ = 0\n⊢ y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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