module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Calculus.VectorField | {
"line": 511,
"column": 2
} | {
"line": 512,
"column": 33
} | {
"line": 514,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nV : F → F\n⊢ pullbackWithin 𝕜 f V univ = pullback 𝕜 f V",
"ppTerm": "?m.23",
"assigned"... | [] | ext x
simp [pullbackWithin, pullback] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.VectorField | {
"line": 511,
"column": 2
} | {
"line": 512,
"column": 33
} | {
"line": 514,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nV : F → F\n⊢ pullbackWithin 𝕜 f V univ = pullback 𝕜 f V",
"ppTerm": "?m.23",
"assigned"... | [] | ext x
simp [pullbackWithin, pullback] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Norm | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 18
} | {
"line": 131,
"column": 19
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t < 0\nh : HasFDerivAt (fun x ↦ ‖x‖) f x\n⊢ HasFDerivAt (fun x ↦ ‖x‖) (-f) (t • x)",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t < 0\nh : HasFDerivAt (fun x ↦ ‖x‖) f x\n⊢ HasFDerivAt (fun x ↦ ‖x‖) (-f) (t • x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Norm | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 18
} | {
"line": 136,
"column": 19
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : 0 < t\nh : HasFDerivAt (fun x ↦ ‖x‖) f x\n⊢ HasFDerivAt (fun x ↦ ‖x‖) f (t • x)",
"ppTerm": "?m.44",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : 0 < t\nh : HasFDerivAt (fun x ↦ ‖x‖) f x\n⊢ HasFDerivAt (fun x ↦ ‖x‖) f (t • x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.VectorField | {
"line": 555,
"column": 4
} | {
"line": 555,
"column": 41
} | {
"line": 555,
"column": 42
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nx : E\nh'f : ContDiffWithinAt 𝕜 2 f s x\nhs : UniqueDiffOn ... | [
"𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nx : E\nh'f : ContDiffWithinAt 𝕜 2 f s x\nhs : UniqueDiffOn 𝕜 s\nhx : x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.VectorField | {
"line": 576,
"column": 4
} | {
"line": 576,
"column": 40
} | {
"line": 576,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nx : E\nh'f : ContDiffWithinAt 𝕜 2 f s x\nhs : UniqueDiffOn ... | [
"𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nx : E\nh'f : ContDiffWithinAt 𝕜 2 f s x\nhs : UniqueDiffOn 𝕜 s\nhx : x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Partial | {
"line": 79,
"column": 10
} | {
"line": 90,
"column": 31
} | {
"line": 91,
"column": 6
} | [] | [] | fun (v, w) => f v.1 w.2 - f w.1 w.2 - ↿f₁ u (v.1 - w.1)
_ =o[(𝓝 u.1 ×ˢ 𝓝 u.2) ×ˢ (𝓝 u.1 ×ˢ 𝓝 u.2)] (fun (v, w) => v.1 - w.1 : _ → E₁) := by
have h := tendsto_snd.prodMk <| tendsto_snd.comp <| tendsto_snd.comp <|
tendsto_fst (f := (𝓝 u.1 ×ˢ 𝓝 u.2) ×ˢ (𝓝 u.1 ×ˢ 𝓝 u.2)) (g := 𝓝... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 553,
"column": 4
} | {
"line": 553,
"column": 20
} | {
"line": 553,
"column": 21
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nx : E\nn : ℕ∞ω\nhf : ContDiffAt 𝕜 n f x\nh : ¬IsRCLikeNormedField 𝕜\nhn : n = ω\n⊢ Co... | [
"case neg\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nx : E\nn : ℕ∞ω\nhf : ContDiffAt 𝕜 n f x\nh : ¬IsRCLikeNormedField 𝕜\nhn : n = ω\n⊢ ContDiffAt 𝕜 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Gradient.Basic | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 45
} | {
"line": 146,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : F → 𝕜\nx : F\nh : DifferentiableAt 𝕜 f x\n⊢ HasGradientAt f (∇ f x) x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LinearIsomet... | [
"𝕜 : Type u_1\nF : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : F → 𝕜\nx : F\nh : DifferentiableAt 𝕜 f x\n⊢ HasFDerivAt f (fderiv 𝕜 f x) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Gradient.Basic | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 57
} | {
"line": 154,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : F → 𝕜\nx : F\ns : Set F\nh : DifferentiableWithinAt 𝕜 f s x\n⊢ HasGradientWithinAt f (gradientWithin f s x) s x",
"ppTerm": "?m.25",
"assigned": true,
... | [
"𝕜 : Type u_1\nF : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : F → 𝕜\nx : F\ns : Set F\nh : DifferentiableWithinAt 𝕜 f s x\n⊢ HasFDerivWithinAt f (fderivWithin 𝕜 f s x) s x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Gradient.Basic | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 13
} | {
"line": 195,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\ng : 𝕜 → 𝕜\ng' u : 𝕜\nh : HasDerivAt g (((toDual 𝕜 𝕜) g') 1) u\n⊢ HasDerivAt g ((starRingEnd 𝕜) g') u",
"ppTerm": "?m.48",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"𝕜 : Type u_1\ninst✝ : RCLike 𝕜\ng : 𝕜 → 𝕜\ng' u : 𝕜\nh : HasDerivAt g (((toDual 𝕜 𝕜) g') 1) u\n⊢ HasDerivAt g ((starRingEnd 𝕜) g') u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 1015,
"column": 39
} | {
"line": 1015,
"column": 57
} | {
"line": 1015,
"column": 57
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : MetricSpace α\ninst✝⁵ : SecondCountableTopology α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : OpensMeasurableSpace α\ninst✝² : HasBesicovitchCovering α\nμ : Measure α\ninst✝¹ : SFinite μ\ninst✝ : μ.OuterRegular\nε : ℝ≥0∞\nhε : ε ≠ 0\nf : α → Set ℝ\ns : Set α\nhf : ∀ x ∈ s, ∀ δ > 0, (f ... | [
"α : Type u_1\ninst✝⁶ : MetricSpace α\ninst✝⁵ : SecondCountableTopology α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : OpensMeasurableSpace α\ninst✝² : HasBesicovitchCovering α\nμ : Measure α\ninst✝¹ : SFinite μ\ninst✝ : μ.OuterRegular\nε : ℝ≥0∞\nhε : ε ≠ 0\nf : α → Set ℝ\ns : Set α\nhf : ∀ x ∈ s, ∀ δ > 0, (f x ∩ Ioo 0 δ)... | ENNReal.add_halves | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 1034,
"column": 72
} | {
"line": 1034,
"column": 83
} | {
"line": 1034,
"column": 84
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : MetricSpace α\nβ : Type u\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : HasBesicovitchCovering α\nμ : Measure α\ninst✝ : SFinite μ\ns : Set α\nf : α → Set (Set α)\nfsubset : ∀ x ∈ s, f x ⊆ (fun r ↦ closedBall x r) '' Io... | [
"α : Type u_1\ninst✝⁵ : MetricSpace α\nβ : Type u\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : HasBesicovitchCovering α\nμ : Measure α\ninst✝ : SFinite μ\ns : Set α\nf : α → Set (Set α)\nfsubset : ∀ x ∈ s, f x ⊆ (fun r ↦ closedBall x r) '' Ioi 0\nffine :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain | {
"line": 65,
"column": 6
} | {
"line": 65,
"column": 17
} | {
"line": 65,
"column": 18
} | [
{
"pp": "case disjoint\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAdd... | [
"case disjoint\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain | {
"line": 61,
"column": 17
} | {
"line": 70,
"column": 11
} | {
"line": 72,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\nin... | [] | by
constructor
· rw [LinearMap.disjoint_ker]
intro (_, y) h rfl
simpa using (injective_iff_map_eq_zero _).mp if₂u.injective y h
· rw [Submodule.codisjoint_iff_exists_add_eq]
intro v
have ⟨y, hy⟩ := if₂u.surjective (f'u v)
use v - (0, y), (0, y)
aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 58
} | {
"line": 128,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\nin... | [
"𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\ninst✝¹ : Norme... | have hψ := dfu.tendsto_implicitFunctionOfProdDomain if₂u | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain | {
"line": 129,
"column": 58
} | {
"line": 129,
"column": 69
} | {
"line": 129,
"column": 70
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\nin... | [
"𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\ninst✝¹ : Norme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ImplicitContDiff | {
"line": 44,
"column": 52
} | {
"line": 44,
"column": 88
} | {
"line": 44,
"column": 88
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpac... | [
"𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst... | ← HasStrictFDerivAt.localInverse_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ImplicitContDiff | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 19
} | {
"line": 96,
"column": 20
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpac... | [
"𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ImplicitContDiff | {
"line": 97,
"column": 10
} | {
"line": 97,
"column": 25
} | {
"line": 97,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpac... | [
"𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ImplicitContDiff | {
"line": 97,
"column": 35
} | {
"line": 97,
"column": 50
} | {
"line": 97,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpac... | [
"𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Implicit | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 71
} | {
"line": 394,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CompleteSpace E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivA... | [
"𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CompleteSpace E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivAt f f' a\nhf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Implicit | {
"line": 393,
"column": 2
} | {
"line": 395,
"column": 74
} | {
"line": 397,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CompleteSpace E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivA... | [] | simpa only [implicitToOpenPartialHomeomorphOfComplemented_self] using
(hf.implicitToOpenPartialHomeomorphOfComplemented f f' hf' hker).map_source <|
hf.mem_implicitToOpenPartialHomeomorphOfComplemented_source hf' hker | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.Implicit | {
"line": 393,
"column": 2
} | {
"line": 395,
"column": 74
} | {
"line": 397,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CompleteSpace E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivA... | [] | simpa only [implicitToOpenPartialHomeomorphOfComplemented_self] using
(hf.implicitToOpenPartialHomeomorphOfComplemented f f' hf' hker).map_source <|
hf.mem_implicitToOpenPartialHomeomorphOfComplemented_source hf' hker | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Implicit | {
"line": 393,
"column": 2
} | {
"line": 395,
"column": 74
} | {
"line": 397,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CompleteSpace E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\nf' : E →L[𝕜] F\na : E\nhf : HasStrictFDerivA... | [] | simpa only [implicitToOpenPartialHomeomorphOfComplemented_self] using
(hf.implicitToOpenPartialHomeomorphOfComplemented f f' hf' hker).map_source <|
hf.mem_implicitToOpenPartialHomeomorphOfComplemented_source hf' hker | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno | {
"line": 61,
"column": 2
} | {
"line": 63,
"column": 55
} | {
"line": 65,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ng : E → F\nf : 𝕜 → E\nx : 𝕜\nn : ℕ∞ω\ni : ℕ\nhg : ContDiffAt 𝕜 n g (f x)\nhf : ContDiffAt 𝕜 n f x\nhi : ... | [] | simp only [← iteratedDerivWithin_univ, ← iteratedFDerivWithin_univ]
exact iteratedDerivWithin_vcomp_eq_sum_orderedFinpartition hg hf uniqueDiffOn_univ
uniqueDiffOn_univ (mem_univ x) (mapsTo_univ f _) hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno | {
"line": 61,
"column": 2
} | {
"line": 63,
"column": 55
} | {
"line": 65,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ng : E → F\nf : 𝕜 → E\nx : 𝕜\nn : ℕ∞ω\ni : ℕ\nhg : ContDiffAt 𝕜 n g (f x)\nhf : ContDiffAt 𝕜 n f x\nhi : ... | [] | simp only [← iteratedDerivWithin_univ, ← iteratedFDerivWithin_univ]
exact iteratedDerivWithin_vcomp_eq_sum_orderedFinpartition hg hf uniqueDiffOn_univ
uniqueDiffOn_univ (mem_univ x) (mapsTo_univ f _) hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.LagrangeMultipliers | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 22
} | {
"line": 95,
"column": 23
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | f x = f x₀} x₀\nhf' : HasStrictFDerivAt f f' x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nΛ : Modul... | [
"case refine_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | f x = f x₀} x₀\nhf' : HasStrictFDerivAt f f' x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nΛ : Module.Dual ℝ ℝ\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LagrangeMultipliers | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 45
} | {
"line": 98,
"column": 46
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | f x = f x₀} x₀\nhf' : HasStrictFDerivAt f f' x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nΛ : Module.Dual ℝ ℝ\nΛ₀ ... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | f x = f x₀} x₀\nhf' : HasStrictFDerivAt f f' x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nΛ : Module.Dual ℝ ℝ\nΛ₀ : ℝ\nhΛ : (Λ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LagrangeMultipliers | {
"line": 99,
"column": 47
} | {
"line": 99,
"column": 81
} | {
"line": 99,
"column": 82
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | f x = f x₀} x₀\nhf' : HasStrictFDerivAt f f' x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nΛ : Module.Dual ℝ ℝ\nΛ₀ ... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | f x = f x₀} x₀\nhf' : HasStrictFDerivAt f f' x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nΛ : Module.Dual ℝ ℝ\nΛ₀ : ℝ\nhΛ : (Λ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LagrangeMultipliers | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 26
} | {
"line": 100,
"column": 27
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | f x = f x₀} x₀\nhf' : HasStrictFDerivAt f f' x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nΛ : Modul... | [
"case refine_2\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | f x = f x₀} x₀\nhf' : HasStrictFDerivAt f f' x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nΛ : Module.Dual ℝ ℝ\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LagrangeMultipliers | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 33
} | {
"line": 116,
"column": 34
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → E → ℝ\nf' : ι → StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivA... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → E → ℝ\nf' : ι → StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LagrangeMultipliers | {
"line": 123,
"column": 11
} | {
"line": 123,
"column": 33
} | {
"line": 123,
"column": 34
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → E → ℝ\nf' : ι → StrongDual ℝ E\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ ... | [
"case refine_2\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nι : Type u_3\ninst✝ : Fintype ι\nf : ι → E → ℝ\nf' : ι → StrongDual ℝ E\nhf' : ∀ (i : ι), HasStrictFDerivAt (f i) (f' i) x₀\nhφ' : HasStrictFDerivAt φ φ' x₀\nthis ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LagrangeMultipliers | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 26
} | {
"line": 141,
"column": 27
} | [
{
"pp": "case intro.refine_1\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nι : Type u_3\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ... | [
"case intro.refine_1\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nφ : E → ℝ\nx₀ : E\nφ' : StrongDual ℝ E\nι : Type u_3\ninst✝ : Finite ι\nf : ι → E → ℝ\nf' : ι → StrongDual ℝ E\nhextr : IsLocalExtrOn φ {x | ∀ (i : ι), f i x = f i x₀} x₀\nhf' : ∀ (i : ι), HasStric... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 40
} | {
"line": 49,
"column": 4
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : ℝ → ℝ≥0∞\n⊢ ∫⁻ (x : ℝ) in f '' s, g x = ∫⁻ (x : ℝ) in s, ENNReal.ofReal |f' x| * g (f x)",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"us... | [
"s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : ℝ → ℝ≥0∞\n⊢ ∫⁻ (x : ℝ) in f '' s, g x = ∫⁻ (x : ℝ) in s, ENNReal.ofReal |f' x| * g (f x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 40
} | {
"line": 60,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : ℝ → F\n⊢ IntegrableOn g (f '' s) volume ↔ IntegrableOn (fun x ↦ |f' x| • g (f x)) s volume",
"ppTerm": "?m.44",
... | [
"F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : ℝ → F\n⊢ IntegrableOn g (f '' s) volume ↔ IntegrableOn (fun x ↦ |f' x| • g (f x)) s volume"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 40
} | {
"line": 70,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : ℝ → F\n⊢ ∫ (x : ℝ) in f '' s, g x = ∫ (x : ℝ) in s, |f' x| • g (f x)",
"ppTerm": "?m.48",
"assigned": false,
... | [
"F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : InjOn f s\ng : ℝ → F\n⊢ ∫ (x : ℝ) in f '' s, g x = ∫ (x : ℝ) in s, |f' x| • g (f x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 26
} | {
"line": 107,
"column": 27
} | [
{
"pp": "case refine_2\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x... | [
"case refine_2\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LHopital | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 34
} | {
"line": 260,
"column": 2
} | [
{
"pp": "a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, HasDerivAt f (f' y) y\nhgg' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, HasDerivAt g (g' y) y\nhg' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, g' y ≠ 0\nhfa : Tendsto f (𝓝[>] a) (𝓝 0)\nhga : Tendsto g (𝓝[>] a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[>] a) ... | [
"a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhgg' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, HasDerivAt g (g' y) y\nhg' : ∃ v ∈ 𝓝[>] a, ∀ y ∈ v, g' y ≠ 0\nhfa : Tendsto f (𝓝[>] a) (𝓝 0)\nhga : Tendsto g (𝓝[>] a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[>] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[>] a\nhff' : ∀ y ∈ s₁, HasDerivAt f (... | rcases hff' with ⟨s₁, hs₁, hff'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 19
} | {
"line": 108,
"column": 20
} | [
{
"pp": "case refine_3\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x... | [
"case refine_3\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LHopital | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 34
} | {
"line": 276,
"column": 2
} | [
{
"pp": "a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, HasDerivAt f (f' y) y\nhgg' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, HasDerivAt g (g' y) y\nhg' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, g' y ≠ 0\nhfa : Tendsto f (𝓝[<] a) (𝓝 0)\nhga : Tendsto g (𝓝[<] a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] a) ... | [
"a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhgg' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, HasDerivAt g (g' y) y\nhg' : ∃ v ∈ 𝓝[<] a, ∀ y ∈ v, g' y ≠ 0\nhfa : Tendsto f (𝓝[<] a) (𝓝 0)\nhga : Tendsto g (𝓝[<] a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[<] a\nhff' : ∀ y ∈ s₁, HasDerivAt f (... | rcases hff' with ⟨s₁, hs₁, hff'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 169,
"column": 18
} | {
"line": 169,
"column": 29
} | {
"line": 169,
"column": 30
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ... | [
"s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LHopital | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 34
} | {
"line": 334,
"column": 2
} | [
{
"pp": "l : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v ∈ atTop, ∀ y ∈ v, HasDerivAt f (f' y) y\nhgg' : ∃ v ∈ atTop, ∀ y ∈ v, HasDerivAt g (g' y) y\nhg' : ∃ v ∈ atTop, ∀ y ∈ v, g' y ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atTop l\n⊢ Tendsto (fun x ↦... | [
"l : Filter ℝ\nf f' g g' : ℝ → ℝ\nhgg' : ∃ v ∈ atTop, ∀ y ∈ v, HasDerivAt g (g' y) y\nhg' : ∃ v ∈ atTop, ∀ y ∈ v, g' y ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y\n⊢ Tendsto (f... | rcases hff' with ⟨s₁, hs₁, hff'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 84,
"column": 61
} | {
"line": 176,
"column": 44
} | {
"line": 178,
"column": 0
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\n⊢ ∃ a b c,\n a ∪ (b ∪ c) = s ∧\n MeasurableSet a ∧\n MeasurableSet b ∧\n MeasurableSet c ∧\n Disjoint a (b ∪ c) ∧\n Disjoint b c ∧ a.Countabl... | [] | by
let a := {x ∈ s | 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x ∈ s | 𝓝[s ∩ Iio x] x = ⊥}
have a_count : a.Countable :=
countable_setOf_isolated_right_within.union countable_setOf_isolated_left_within
let s₁ := s \ a
have hs₁ : MeasurableSet s₁ := hs.diff a_count.measurableSet
let u : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.LHopital | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 34
} | {
"line": 350,
"column": 2
} | [
{
"pp": "l : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v ∈ atBot, ∀ y ∈ v, HasDerivAt f (f' y) y\nhgg' : ∃ v ∈ atBot, ∀ y ∈ v, HasDerivAt g (g' y) y\nhg' : ∃ v ∈ atBot, ∀ y ∈ v, g' y ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atBot l\n⊢ Tendsto (fun x ↦... | [
"l : Filter ℝ\nf f' g g' : ℝ → ℝ\nhgg' : ∃ v ∈ atBot, ∀ y ∈ v, HasDerivAt g (g' y) y\nhg' : ∃ v ∈ atBot, ∀ y ∈ v, g' y ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atBot l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atBot\nhff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y\n⊢ Tendsto (f... | rcases hff' with ⟨s₁, hs₁, hff'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 347,
"column": 6
} | {
"line": 347,
"column": 37
} | {
"line": 347,
"column": 38
} | [
{
"pp": "case hf'_nonneg\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\nz : ℝ\nhz : z ∈ Ioo (min a b) (max a b)\n⊢ 0... | [
"case hf'_nonneg\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\nz : ℝ\nhz : z ∈ Ioo (min a b) (max a b)\n⊢ 0 ≤ f' z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 13
} | {
"line": 225,
"column": 14
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : MonotoneOn f s\n⊢ ∫⁻ (x : ℝ) in s, ENNReal.ofReal (f' x) = volume (f '' s)",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : MonotoneOn f s\n⊢ ∫⁻ (x : ℝ) in s, ENNReal.ofReal (f' x) = volume (f '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 376,
"column": 6
} | {
"line": 376,
"column": 37
} | {
"line": 376,
"column": 38
} | [
{
"pp": "case hf'_nonneg\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\nz : ℝ\nhz : z ∈ Ioo (min a b) (max a b)\n⊢ 0... | [
"case hf'_nonneg\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\nz : ℝ\nhz : z ∈ Ioo (min a b) (max a b)\n⊢ 0 ≤ f' z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 136,
"column": 70
} | {
"line": 136,
"column": 93
} | {
"line": 136,
"column": 94
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝¹ : PseudoMetricSpace α\ninst✝ : OpensMeasurableSpace α\nx : α\nr : ι → ℝ\nhr : Tendsto r l atTop\ny : α\na : ι\nha : a ∈ r ⁻¹' Ioi (dist x y)\n⊢ y ∈ Metric.ball x (r a)",
"ppTerm": "?m.60",
"assigned": tr... | [
"α : Type u_1\nι : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝¹ : PseudoMetricSpace α\ninst✝ : OpensMeasurableSpace α\nx : α\nr : ι → ℝ\nhr : Tendsto r l atTop\ny : α\na : ι\nha : a ∈ r ⁻¹' Ioi (dist x y)\n⊢ dist x y < r a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 143,
"column": 70
} | {
"line": 143,
"column": 93
} | {
"line": 143,
"column": 94
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝¹ : PseudoMetricSpace α\ninst✝ : OpensMeasurableSpace α\nx : α\nr : ι → ℝ\nhr : Tendsto r l atTop\ny : α\na : ι\nha : a ∈ r ⁻¹' Ici (dist x y)\n⊢ y ∈ Metric.closedBall x (r a)",
"ppTerm": "?m.59",
"assigne... | [
"α : Type u_1\nι : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝¹ : PseudoMetricSpace α\ninst✝ : OpensMeasurableSpace α\nx : α\nr : ι → ℝ\nhr : Tendsto r l atTop\ny : α\na : ι\nha : a ∈ r ⁻¹' Ici (dist x y)\n⊢ dist x y ≤ r a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 432,
"column": 8
} | {
"line": 432,
"column": 52
} | {
"line": 432,
"column": 53
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : l.NeBot\ninst✝ : l.IsCountablyGenerated\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ≥0\nhfm : AEStronglyMeasurable f μ\nhbounded : ∀ᶠ (i : ι) in l, ∫⁻ (x : ... | [
"α : Type u_1\nι : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : l.NeBot\ninst✝ : l.IsCountablyGenerated\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ≥0\nhfm : AEStronglyMeasurable f μ\nhbounded : ∀ᶠ (i : ι) in l, ∫⁻ (x : α) in φ i, ‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 438,
"column": 8
} | {
"line": 438,
"column": 52
} | {
"line": 438,
"column": 53
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : l.NeBot\ninst✝ : l.IsCountablyGenerated\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ≥0\nhfm : AEStronglyMeasurable f μ\nhtendsto : Tendsto (fun i ↦ ∫⁻ (x : ... | [
"α : Type u_1\nι : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : l.NeBot\ninst✝ : l.IsCountablyGenerated\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nI : ℝ≥0\nhfm : AEStronglyMeasurable f μ\nhtendsto : Tendsto (fun i ↦ ∫⁻ (x : α) in φ i, ‖... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 409,
"column": 6
} | {
"line": 409,
"column": 37
} | {
"line": 409,
"column": 38
} | [
{
"pp": "case hf'_nonpos\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0\nz : ℝ\nhz : z ∈ Ioo (min a b) (max a b)\n⊢ d... | [
"case hf'_nonpos\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0\nz : ℝ\nhz : z ∈ Ioo (min a b) (max a b)\n⊢ f' z ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 320,
"column": 6
} | {
"line": 320,
"column": 37
} | {
"line": 320,
"column": 38
} | [
{
"pp": "case hf'_nonneg\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, 0 ≤ f' x\nhab : a ≤ b\nz : ℝ\nhz : z ∈ Ioo a b\n⊢ 0 ≤ deriv f z",
"ppTerm": "?hf'... | [
"case hf'_nonneg\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, 0 ≤ f' x\nhab : a ≤ b\nz : ℝ\nhz : z ∈ Ioo a b\n⊢ 0 ≤ f' z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 344,
"column": 6
} | {
"line": 344,
"column": 37
} | {
"line": 344,
"column": 38
} | [
{
"pp": "case hf'_nonneg\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, 0 ≤ f' x\nhab : a ≤ b\nz : ℝ\nhz : z ∈ Ioo a b\n⊢ 0 ≤ deriv f z",
"ppTerm": "?hf'... | [
"case hf'_nonneg\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, 0 ≤ f' x\nhab : a ≤ b\nz : ℝ\nhz : z ∈ Ioo a b\n⊢ 0 ≤ f' z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 727,
"column": 6
} | {
"line": 727,
"column": 64
} | {
"line": 727,
"column": 65
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nε : ℝ\nεpos : ε > 0\nL : Tendsto (fun n ↦ ∫ (x : ℝ) in Ici ↑n, ‖f' x‖) atTop (𝓝 (∫ (x : ℝ) in ⋂ n, I... | [
"E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nε : ℝ\nεpos : ε > 0\nL : Tendsto (fun n ↦ ∫ (x : ℝ) in Ici ↑n, ‖f' x‖) atTop (𝓝 (∫ (x : ℝ) in ⋂ n, Ici ↑n, ‖f' x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 332,
"column": 8
} | {
"line": 332,
"column": 90
} | {
"line": 332,
"column": 91
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ENNReal.ofReal |A.det| < ↑m\nd : ℝ≥0∞ := ⋯\nε : ℝ\nhε : μ (closedBall 0 ε + ⇑... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ENNReal.ofReal |A.det| < ↑m\nd : ℝ≥0∞ := ENNReal.ofReal |A.det|\nε : ℝ\nhε : μ (closedBal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 439,
"column": 6
} | {
"line": 439,
"column": 37
} | {
"line": 439,
"column": 38
} | [
{
"pp": "case hf'_nonpos\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0\nz : ℝ\nhz : z ∈ Ioo (min a b) (max a b)\n⊢ d... | [
"case hf'_nonpos\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0\nz : ℝ\nhz : z ∈ Ioo (min a b) (max a b)\n⊢ f' z ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 383,
"column": 2
} | {
"line": 383,
"column": 13
} | {
"line": 383,
"column": 14
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : AntitoneOn f s\n⊢ ∫⁻ (x : ℝ) in s, ENNReal.ofReal (-f' x) = volume (f '' s)",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : AntitoneOn f s\n⊢ ∫⁻ (x : ℝ) in s, ENNReal.ofReal (-f' x) = volume (f '' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 436,
"column": 6
} | {
"line": 436,
"column": 37
} | {
"line": 436,
"column": 38
} | [
{
"pp": "case hf'_nonpos\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, f' x ≤ 0\nhab : a ≤ b\nz : ℝ\nhz : z ∈ Ioo a b\n⊢ deriv f z ≤ 0",
"ppTerm": "?hf'... | [
"case hf'_nonpos\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, f' x ≤ 0\nhab : a ≤ b\nz : ℝ\nhz : z ∈ Ioo a b\n⊢ f' z ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 502,
"column": 69
} | {
"line": 502,
"column": 91
} | {
"line": 502,
"column": 92
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntegrableOn g (f '' [[a, b]]) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) [[a, b]] volume\n⊢ Integr... | [
"a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntegrableOn g (f '' [[a, b]]) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) [[a, b]] volume\n⊢ IntegrableOn (fun ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 503,
"column": 2
} | {
"line": 503,
"column": 24
} | {
"line": 503,
"column": 25
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntegrableOn g (f '' [[a, b]]) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) [[a, b]] volume\nhg2' : I... | [
"a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioo (min a b) (max a b))\nhg1 : IntegrableOn g (f '' [[a, b]]) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) [[a, b]] volume\nhg2' : IntegrableOn ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 513,
"column": 2
} | {
"line": 513,
"column": 24
} | {
"line": 513,
"column": 25
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g (f '' [[a, b]])\n⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (u : ℝ) in f a..f b, g u",
"ppTerm": "?m.60",
"assigne... | [
"a b : ℝ\nf f' g : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g (f '' [[a, b]])\n⊢ ∫ (x : ℝ) in a..b, f' x * g (f x) = ∫ (u : ℝ) in f a..f b, g u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 524,
"column": 2
} | {
"line": 524,
"column": 24
} | {
"line": 524,
"column": 25
} | [
{
"pp": "a b : ℝ\nf f' g : ℝ → ℝ\nh : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x\nh' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g (f '' [[a, b]])\n⊢ ∫ (x : ℝ) in a..b, (g ∘ f) x * f' x = ∫ (x : ℝ) in f a..f b, g x",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRi... | [
"a b : ℝ\nf f' g : ℝ → ℝ\nh : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x\nh' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g (f '' [[a, b]])\n⊢ ∫ (x : ℝ) in a..b, f' x * g (f x) = ∫ (x : ℝ) in f a..f b, g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 461,
"column": 6
} | {
"line": 461,
"column": 37
} | {
"line": 461,
"column": 38
} | [
{
"pp": "case hf'_nonpos\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, f' x ≤ 0\nhab : a ≤ b\nz : ℝ\nhz : z ∈ Ioo a b\n⊢ deriv f z ≤ 0",
"ppTerm": "?hf'... | [
"case hf'_nonpos\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, f' x ≤ 0\nhab : a ≤ b\nz : ℝ\nhz : z ∈ Ioo a b\n⊢ f' z ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 941,
"column": 4
} | {
"line": 941,
"column": 15
} | {
"line": 941,
"column": 16
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\ng : ℝ → E := f ∘ fun x ↦ -x\nx : ℝ\nhx : x ∈ Ioi (-a)\nthis : -x ∈ Iic a\n⊢ HasDerivAt g (-f' (-x)) x... | [
"E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\ng : ℝ → E := f ∘ fun x ↦ -x\nx : ℝ\nhx : x ∈ Ioi (-a)\nthis : -x ∈ Iic a\n⊢ HasDerivAt g (-f' (-x)) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 574,
"column": 2
} | {
"line": 574,
"column": 24
} | {
"line": 574,
"column": 25
} | [
{
"pp": "a b : ℝ\nf f' g g' : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g [[f a, f b]]\nhgg' : ∀ x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)), HasDerivWithinAt g (g' x) (Ioi x) x\nhg' : Continu... | [
"a b : ℝ\nf f' g g' : ℝ → ℝ\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : ContinuousOn f' [[a, b]]\nhg : ContinuousOn g [[f a, f b]]\nhgg' : ∀ x ∈ Ioo (min (f a) (f b)) (max (f a) (f b)), HasDerivWithinAt g (g' x) (Ioi x) x\nhg' : ContinuousOn g' (f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 580,
"column": 2
} | {
"line": 580,
"column": 24
} | {
"line": 580,
"column": 25
} | [
{
"pp": "a b : ℝ\nf f' g g' : ℝ → ℝ\nhf : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x\nhg : ∀ x ∈ [[a, b]], HasDerivAt g (g' (f x)) (f x)\nhf' : ContinuousOn f' [[a, b]]\nhg' : Continuous g'\n⊢ ∫ (x : ℝ) in a..b, (g' ∘ f) x * f' x = (g ∘ f) b - (g ∘ f) a",
"ppTerm": "?m.61",
"assigned": true,
"usedConstan... | [
"a b : ℝ\nf f' g g' : ℝ → ℝ\nhf : ∀ x ∈ [[a, b]], HasDerivAt f (f' x) x\nhg : ∀ x ∈ [[a, b]], HasDerivAt g (g' (f x)) (f x)\nhf' : ContinuousOn f' [[a, b]]\nhg' : Continuous g'\n⊢ ∫ (x : ℝ) in a..b, f' x * g' (f x) = g (f b) - g (f a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 948,
"column": 2
} | {
"line": 948,
"column": 17
} | {
"line": 948,
"column": 18
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\ng : ℝ → E := f ∘ fun x ↦ -x\nhdg : ∀ x ∈ Ioi (-a), HasDerivAt g (-f' (-x)) x\nL : Tendsto g atTop (𝓝... | [
"E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\ng : ℝ → E := f ∘ fun x ↦ -x\nhdg : ∀ x ∈ Ioi (-a), HasDerivAt g (-f' (-x)) x\nL : Tendsto g atTop (𝓝 (atTop.limU... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 430,
"column": 4
} | {
"line": 430,
"column": 43
} | {
"line": 430,
"column": 44
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃L... | [
"case neg\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃L[ℝ] E := A.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 437,
"column": 8
} | {
"line": 437,
"column": 38
} | {
"line": 437,
"column": 39
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃L... | [
"case inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃L[ℝ] E := A.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LocalExtr.LineDeriv | {
"line": 29,
"column": 64
} | {
"line": 29,
"column": 75
} | {
"line": 29,
"column": 76
} | [
{
"pp": "E : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nf : E → ℝ\na b : E\nf' : ℝ\nl : Filter E\nh : IsExtrFilter f l a\nhd : HasLineDerivAt ℝ f f' a b\nh' : Tendsto (fun t ↦ a + t • b) (𝓝 0) l\n⊢ IsExtrFilter f l (a + 0 • b)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
... | [
"E : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nf : E → ℝ\na b : E\nf' : ℝ\nl : Filter E\nh : IsExtrFilter f l a\nhd : HasLineDerivAt ℝ f f' a b\nh' : Tendsto (fun t ↦ a + t • b) (𝓝 0) l\n⊢ IsExtrFilter f l a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 440,
"column": 6
} | {
"line": 440,
"column": 51
} | {
"line": 440,
"column": 52
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃L... | [
"case inr\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃L[ℝ] E := A.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1079,
"column": 2
} | {
"line": 1079,
"column": 13
} | {
"line": 1079,
"column": 14
} | [
{
"pp": "case pos\nE : Type u_1\nf f' : ℝ → E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x\nhf' : Integrable f' volume\nhf : Integrable f volume\nhE : CompleteSpace E\nA : Tendsto f atBot (𝓝 0)\nB : Tendsto f atTop (𝓝 0)\n⊢ ∫ (x : ℝ), f' x = 0",
"ppTer... | [
"case pos\nE : Type u_1\nf f' : ℝ → E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x\nhf' : Integrable f' volume\nhf : Integrable f volume\nhE : CompleteSpace E\nA : Tendsto f atBot (𝓝 0)\nB : Tendsto f atTop (𝓝 0)\n⊢ ∫ (x : ℝ), f' x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1127,
"column": 69
} | {
"line": 1127,
"column": 91
} | {
"line": 1127,
"column": 92
} | [
{
"pp": "f f' g : ℝ → ℝ\na : ℝ\nhf : ContinuousOn f (Ici a)\nhft : Tendsto f atTop atTop\nhff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) (Ici a) volume\n⊢ IntegrableOn (fun x ... | [
"f f' g : ℝ → ℝ\na : ℝ\nhf : ContinuousOn f (Ici a)\nhft : Tendsto f atTop atTop\nhff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) (Ici a) volume\n⊢ IntegrableOn (fun x ↦ f' x * g (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1128,
"column": 2
} | {
"line": 1128,
"column": 24
} | {
"line": 1128,
"column": 25
} | [
{
"pp": "f f' g : ℝ → ℝ\na : ℝ\nhf : ContinuousOn f (Ici a)\nhft : Tendsto f atTop atTop\nhff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) (Ici a) volume\nhg2' : IntegrableOn (f... | [
"f f' g : ℝ → ℝ\na : ℝ\nhf : ContinuousOn f (Ici a)\nhft : Tendsto f atTop atTop\nhff' : ∀ x ∈ Ioi a, HasDerivWithinAt f (f' x) (Ioi x) x\nhg_cont : ContinuousOn g (f '' Ioi a)\nhg1 : IntegrableOn g (f '' Ici a) volume\nhg2 : IntegrableOn (fun x ↦ (g ∘ f) x * f' x) (Ici a) volume\nhg2' : IntegrableOn (fun x ↦ f' x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 454,
"column": 6
} | {
"line": 454,
"column": 48
} | {
"line": 454,
"column": 49
} | [
{
"pp": "case h0\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃L[... | [
"case h0\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃L[ℝ] E := A.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LocalExtr.Polynomial | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 70
} | {
"line": 89,
"column": 71
} | [
{
"pp": "p : ℝ[X]\nx : ℝ\na✝ : x ∈ p.roots.toFinset ∪ (derivative p).roots.toFinset\nhx₂ : x ∉ (derivative p).roots.toFinset\n⊢ Multiset.count x (derivative p).roots = 0",
"ppTerm": "?m.411",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Real",
"Pol... | [
"p : ℝ[X]\nx : ℝ\na✝ : x ∈ p.roots.toFinset ∪ (derivative p).roots.toFinset\nhx₂ : x ∉ (derivative p).roots.toFinset\n⊢ x ∉ (derivative p).roots"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LocalExtr.Polynomial | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 87
} | {
"line": 96,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : CommRing F\ninst✝ : Algebra F ℝ\np : F[X]\n⊢ Fintype.card ↑(p.rootSet ℝ) ≤ Fintype.card ↑((derivative p).rootSet ℝ) + 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Polynomial.derivative",
"Eq.mpr",
"Real",
... | [
"F : Type u_1\ninst✝¹ : CommRing F\ninst✝ : Algebra F ℝ\np : F[X]\n⊢ (p.aroots ℝ).toFinset.card ≤ ((derivative p).aroots ℝ).toFinset.card + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 137,
"column": 40
} | {
"line": 137,
"column": 62
} | {
"line": 137,
"column": 63
} | [
{
"pp": "E : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\ninst✝³ : Measurab... | [
"E : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\ninst✝³ : MeasurableSpace E\nμ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 141,
"column": 8
} | {
"line": 141,
"column": 19
} | {
"line": 141,
"column": 20
} | [
{
"pp": "case hx\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\ninst✝³ :... | [
"case hx\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\ninst✝³ : MeasurableS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1172,
"column": 2
} | {
"line": 1172,
"column": 29
} | {
"line": 1172,
"column": 30
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : ℝ → E\na b : ℝ\nhb : 0 < b\n⊢ ∫ (x : ℝ) in Ioi a, g (x * b) = b⁻¹ • ∫ (x : ℝ) in Ioi (a * b), g x",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemir... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : ℝ → E\na b : ℝ\nhb : 0 < b\n⊢ ∫ (x : ℝ) in Ioi a, g (b * x) = b⁻¹ • ∫ (x : ℝ) in Ioi (a * b), g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RemovableSingularity | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 96
} | {
"line": 58,
"column": 6
} | [
{
"pp": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\ns : Set ℂ\nc : ℂ\nhs : s ∈ 𝓝 c\nhd : DifferentiableOn ℂ f (s \\ {c})\nhc : ContinuousAt f c\nx : ℂ\nhx : x ∈ s\nhne : x ≠ c\n⊢ DifferentiableWithinAt ℂ f s x",
"ppTerm": "?inr",
"... | [
"case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\ns : Set ℂ\nc : ℂ\nhs : s ∈ 𝓝 c\nhd : DifferentiableOn ℂ f (s \\ {c})\nhc : ContinuousAt f c\nx : ℂ\nhx : x ∈ s\nhne : x ≠ c\n⊢ ∃ f', HasFDerivAtFilter f f' (𝓝[s] x ×ˢ pure x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RemovableSingularity | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 46
} | {
"line": 86,
"column": 47
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\ns : Set ℂ\nc : ℂ\nhc : s ∈ 𝓝 c\nhd : DifferentiableOn ℂ f (s \\ {c})\nho : (fun z ↦ f z - f c) =o[𝓝[≠] c] fun z ↦ (z - c)⁻¹\nF : ℂ → E := fun z ↦ (z - c) • f z\nH : Tendsto (fun x ↦ (x - c)⁻¹⁻¹ • ... | [
"E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\ns : Set ℂ\nc : ℂ\nhc : s ∈ 𝓝 c\nhd : DifferentiableOn ℂ f (s \\ {c})\nho : (fun z ↦ f z - f c) =o[𝓝[≠] c] fun z ↦ (z - c)⁻¹\nF : ℂ → E := fun z ↦ (z - c) • f z\nH : Tendsto (fun x ↦ (x - c)⁻¹⁻¹ • (f x - f c))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 39
} | {
"line": 150,
"column": 40
} | [
{
"pp": "E : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\ninst✝³ : Measurab... | [
"E : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\ninst✝³ : MeasurableSpace E\nμ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.RemovableSingularity | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 71
} | {
"line": 121,
"column": 0
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nc : ℂ\nhd : ∀ᶠ (x : ℂ) in 𝓝 c, x ∈ {c}ᶜ → DifferentiableAt ℂ f x\nho : (fun z ↦ f z - f c) =o[𝓝[≠] c] fun z ↦ (z - c)⁻¹\nthis : DifferentiableOn ℂ f ({z | z ≠ c → DifferentiableAt ℂ f z} \\ {c})\n... | [] | exact continuousAt_update_same.1 (H.differentiableAt hd).continuousAt | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 64
} | {
"line": 153,
"column": 65
} | [
{
"pp": "case pos.inr.hf'g\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W... | [
"case pos.inr.hf'g\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\ninst✝³ : M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1227,
"column": 2
} | {
"line": 1227,
"column": 39
} | {
"line": 1227,
"column": 40
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nc a : ℝ\nha : 0 < a\n⊢ IntegrableOn (fun x ↦ f (x * a)) (Ioi c) volume ↔ IntegrableOn f (Ioi (c * a)) volume",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
... | [
"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nc a : ℝ\nha : 0 < a\n⊢ IntegrableOn (fun x ↦ f (a * x)) (Ioi c) volume ↔ IntegrableOn f (Ioi (c * a)) volume"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 64
} | {
"line": 154,
"column": 65
} | [
{
"pp": "case pos.inr.hfg'\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W... | [
"case pos.inr.hfg'\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\ninst✝³ : M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 64
} | {
"line": 155,
"column": 65
} | [
{
"pp": "case pos.inr.hfg\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\... | [
"case pos.inr.hfg\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\ninst✝³ : Me... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.LocallyUniformLimit | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 43
} | {
"line": 74,
"column": 4
} | [
{
"pp": "case e_a\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nr : ℝ\nf g : ℂ → E\nhr : 0 < r\nhf : ContinuousOn f (sphere z r)\nhg : ContinuousOn g (sphere z r)\nh1 : ContinuousOn (fun w ↦ ((w - z) ^ 2)⁻¹) (sphere z r)\n⊢ ∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • (f - g) w =\n ... | [
"case e_a\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nr : ℝ\nf g : ℂ → E\nhr : 0 < r\nhf : ContinuousOn f (sphere z r)\nhg : ContinuousOn g (sphere z r)\nh1 : ContinuousOn (fun w ↦ ((w - z) ^ 2)⁻¹) (sphere z r)\n⊢ ∮ (w : ℂ) in C(z, r), ((w - z) ^ 2)⁻¹ • f w - ((w - z) ^ 2)⁻¹ • g w ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1345,
"column": 2
} | {
"line": 1345,
"column": 13
} | {
"line": 1345,
"column": 14
} | [
{
"pp": "A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x\nhuv : IntegrableOn (u' * v + u * v') (Ioi a) volume\nh_zero : Tendsto (u * v) (𝓝[Ici a \\ {a}... | [
"A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x\nhuv : IntegrableOn (u' * v + u * v') (Ioi a) volume\nh_zero : Tendsto (u * v) (𝓝[Ici a \\ {a}] a) (𝓝 a')... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1357,
"column": 2
} | {
"line": 1357,
"column": 29
} | {
"line": 1357,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x\nhuv' : IntegrableOn (fun i ↦ u i * v' i) (Ioi a) volume\nhu'v : IntegrableOn (fun i ↦ u' i * v... | [
"A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Ioi a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Ioi a, HasDerivAt v (v' x) x\nhuv' : IntegrableOn (fun i ↦ u i * v' i) (Ioi a) volume\nhu'v : IntegrableOn (fun i ↦ u' i * v i) (Ioi a) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1376,
"column": 2
} | {
"line": 1376,
"column": 13
} | {
"line": 1376,
"column": 14
} | [
{
"pp": "A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x\nhuv : IntegrableOn (u' * v + u * v') (Iic a) volume\nh_zero : Tendsto (u * v) (𝓝[Iic a \\ {a}... | [
"A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x\nhuv : IntegrableOn (u' * v + u * v') (Iic a) volume\nh_zero : Tendsto (u * v) (𝓝[Iic a \\ {a}] a) (𝓝 a')... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1388,
"column": 2
} | {
"line": 1388,
"column": 29
} | {
"line": 1388,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x\nhuv' : IntegrableOn (fun i ↦ u i * v' i) (Iic a) volume\nhu'v : IntegrableOn (fun i ↦ u' i * v... | [
"A : Type u_1\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : ℝ\na' b' : A\nu v u' v' : ℝ → A\ninst✝ : CompleteSpace A\nhu : ∀ x ∈ Iio a, HasDerivAt u (u' x) x\nhv : ∀ x ∈ Iio a, HasDerivAt v (v' x) x\nhuv' : IntegrableOn (fun i ↦ u i * v' i) (Iic a) volume\nhu'v : IntegrableOn (fun i ↦ u' i * v i) (Iic a) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 59
} | {
"line": 175,
"column": 60
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝² : CommMonoid α\nf : ι → β → α\ng : β → α\ns : Set β\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nh : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, HasProdUniformlyOn f g t\n⊢ ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop t",
"p... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝² : CommMonoid α\nf : ι → β → α\ng : β → α\ns : Set β\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nh : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, HasProdUniformlyOn f g t\n⊢ ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 57
} | {
"line": 199,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝³ : CommMonoid α\nf : ι → β → α\ng : β → α\ns : Set β\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalSpace β\nhs : IsOpen[inst✝¹] s\ninst✝ : LocallyCompactSpace β\nh : ∀ K ⊆ s, IsCompact K → HasProdUniformlyOn f g K\n⊢ ∀ K ⊆ s, IsCompact K → TendstoUniforml... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝³ : CommMonoid α\nf : ι → β → α\ng : β → α\ns : Set β\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalSpace β\nhs : IsOpen[inst✝¹] s\ninst✝ : LocallyCompactSpace β\nh : ∀ K ⊆ s, IsCompact K → HasProdUniformlyOn f g K\n⊢ ∀ K ⊆ s, IsCompact K → TendstoUniformlyOn (fun x1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 444,
"column": 4
} | {
"line": 444,
"column": 59
} | {
"line": 444,
"column": 60
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝² : CommMonoid α\nf : ι → β → α\ng : β → α\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nh : ∀ (x : β), ∃ t ∈ 𝓝 x, HasProdUniformlyOn f g t\n⊢ ∀ (x : β), ∃ t ∈ 𝓝 x, TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop t",
"ppTerm": "?m.3... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝² : CommMonoid α\nf : ι → β → α\ng : β → α\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nh : ∀ (x : β), ∃ t ∈ 𝓝 x, HasProdUniformlyOn f g t\n⊢ ∀ (x : β), ∃ t ∈ 𝓝 x, TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, f i x2) g atTop t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 57
} | {
"line": 456,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝³ : CommMonoid α\nf : ι → β → α\ng : β → α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : LocallyCompactSpace β\nh : ∀ (K : Set β), IsCompact K → HasProdUniformlyOn f g K\n⊢ ∀ (K : Set β), IsCompact K → TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝³ : CommMonoid α\nf : ι → β → α\ng : β → α\ninst✝² : UniformSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : LocallyCompactSpace β\nh : ∀ (K : Set β), IsCompact K → HasProdUniformlyOn f g K\n⊢ ∀ (K : Set β), IsCompact K → TendstoUniformlyOn (fun x1 x2 ↦ ∏ i ∈ x1, f i x2)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.UniformOn | {
"line": 492,
"column": 2
} | {
"line": 492,
"column": 51
} | {
"line": 493,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ng : β → α\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nf : ℕ → β → α\nh : HasProdLocallyUniformly f g\n⊢ TendstoLocallyUniformly (fun x1 x2 ↦ ∏ i ∈ Finset.range x1, f i x2) g atTop",
"ppTerm": "?m.22",
"assigned": false,
"usedConst... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ng : β → α\ninst✝¹ : UniformSpace α\ninst✝ : TopologicalSpace β\nf : ℕ → β → α\nh : HasProdLocallyUniformly f g\n⊢ TendstoLocallyUniformly (fun x1 x2 ↦ ∏ i ∈ Finset.range x1, f i x2) g atTop"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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