module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 64
} | {
"line": 90,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝² : Field k\ninst✝¹ : AddCommGroup E\ninst✝ : Module k E\nf : k → E\nx y : k\n⊢ slope (fun t ↦ -f t) x y = -slope f x y",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"_private.Mathlib.LinearAlgebra.AffineSpace.Slope.0.slope_neg._simp_1_1",
... | [] | simp only [slope_def_module, neg_sub_neg, ← smul_neg, neg_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 64
} | {
"line": 90,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝² : Field k\ninst✝¹ : AddCommGroup E\ninst✝ : Module k E\nf : k → E\nx y : k\n⊢ slope (fun t ↦ -f t) x y = -slope f x y",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"_private.Mathlib.LinearAlgebra.AffineSpace.Slope.0.slope_neg._simp_1_1",
... | [] | simp only [slope_def_module, neg_sub_neg, ← smul_neg, neg_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 64
} | {
"line": 90,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝² : Field k\ninst✝¹ : AddCommGroup E\ninst✝ : Module k E\nf : k → E\nx y : k\n⊢ slope (fun t ↦ -f t) x y = -slope f x y",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"_private.Mathlib.LinearAlgebra.AffineSpace.Slope.0.slope_neg._simp_1_1",
... | [] | simp only [slope_def_module, neg_sub_neg, ← smul_neg, neg_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 33
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case neg.refine_2\nk : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\nhxy : x ≤ y\nhxeqy : ¬x = y\nh : f ... | [] | rwa [vsub_eq_sub, sub_nonneg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 33
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case neg.refine_2\nk : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\nhxy : x ≤ y\nhxeqy : ¬x = y\nh : f ... | [] | rwa [vsub_eq_sub, sub_nonneg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 33
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case neg.refine_2\nk : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\nhxy : x ≤ y\nhxeqy : ¬x = y\nh : f ... | [] | rwa [vsub_eq_sub, sub_nonneg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.CompMul | {
"line": 43,
"column": 76
} | {
"line": 44,
"column": 64
} | {
"line": 45,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nf : 𝕜 → E\nx : 𝕜\n⊢ deriv (fun x ↦ f (c * x)) x = c • deriv f (c * x)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSemino... | [] | by
simp only [← smul_eq_mul, deriv, fderiv_comp_smul, smul_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 366,
"column": 44
} | {
"line": 367,
"column": 56
} | {
"line": 369,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → F\nf' : F\nx : 𝕜\ns : Set 𝕜\nl : F → E\nl' : F →L[𝕜] E\ny : F\nt : Set F\nhl : HasFDerivWithinAt l l' ... | [] | by
rw [hy] at hl; exact hl.comp_hasDerivWithinAt x hf hst | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 37
} | {
"line": 154,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nm n : ℤ\nhm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z\nhn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in ... | [
"case inr\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nm n : ℤ\nhm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z\nhn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝... | wlog! h_le : n ≤ m generalizing m n | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog!_1 | Mathlib.Tactic.wlog! |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 314,
"column": 2
} | {
"line": 319,
"column": 75
} | {
"line": 321,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nx : E\ng : F ≃ₗᵢ[𝕜] G\nf... | [] | have :
iteratedFDerivWithin 𝕜 i (g ∘ f) s x =
(g : F →L[𝕜] G).compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x) :=
g.toContinuousLinearEquiv.iteratedFDerivWithin_comp_left f hs hx i
rw [this]
apply LinearIsometry.norm_compContinuousMultilinearMap g.toLinearIsometry | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 314,
"column": 2
} | {
"line": 319,
"column": 75
} | {
"line": 321,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nx : E\ng : F ≃ₗᵢ[𝕜] G\nf... | [] | have :
iteratedFDerivWithin 𝕜 i (g ∘ f) s x =
(g : F →L[𝕜] G).compContinuousMultilinearMap (iteratedFDerivWithin 𝕜 i f s x) :=
g.toContinuousLinearEquiv.iteratedFDerivWithin_comp_left f hs hx i
rw [this]
apply LinearIsometry.norm_compContinuousMultilinearMap g.toLinearIsometry | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 38
} | {
"line": 113,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn✝ : ℕ∞ω\ns : Set E\nt : Set F\ng : ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nn✝ : ℕ∞ω\ns : Set E\nt : Set F\ng : F → G\nf : E... | rcases hg m hm with ⟨v, hv, q, hq⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 275,
"column": 2
} | {
"line": 277,
"column": 73
} | {
"line": 279,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → F\ng : F → G\nx : E\nn : ℕ∞ω... | [] | simp only [← iteratedFDerivWithin_univ, ← ftaylorSeriesWithin_univ]
exact iteratedFDerivWithin_comp hg.contDiffWithinAt hf.contDiffWithinAt
uniqueDiffOn_univ uniqueDiffOn_univ (mem_univ _) (mapsTo_univ _ _) hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 275,
"column": 2
} | {
"line": 277,
"column": 73
} | {
"line": 279,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → F\ng : F → G\nx : E\nn : ℕ∞ω... | [] | simp only [← iteratedFDerivWithin_univ, ← ftaylorSeriesWithin_univ]
exact iteratedFDerivWithin_comp hg.contDiffWithinAt hf.contDiffWithinAt
uniqueDiffOn_univ uniqueDiffOn_univ (mem_univ _) (mapsTo_univ _ _) hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 554,
"column": 4
} | {
"line": 554,
"column": 38
} | {
"line": 555,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nx : E\nn✝ : ℕ∞ω\ns : Set E\nf : E → ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nx : E\nn✝ : ℕ∞ω\ns : Set E\nf : E → F\ng : E → G... | rcases hg m hm with ⟨v, hv, q, hq⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.Deriv.Inverse | {
"line": 133,
"column": 4
} | {
"line": 134,
"column": 45
} | {
"line": 135,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\nc : F\nh : ∃ᶠ (y : 𝕜) in 𝓝[s \\ {x}] x, f y = c\nhf : DifferentiableWithinAt 𝕜 f s x\n⊢ derivWithin f s x = 0",
"ppTerm": "?pos✝",
... | [] | contrapose! h
exact hf.hasDerivWithinAt.eventually_ne h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Inverse | {
"line": 133,
"column": 4
} | {
"line": 134,
"column": 45
} | {
"line": 135,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\nc : F\nh : ∃ᶠ (y : 𝕜) in 𝓝[s \\ {x}] x, f y = c\nhf : DifferentiableWithinAt 𝕜 f s x\n⊢ derivWithin f s x = 0",
"ppTerm": "?pos✝",
... | [] | contrapose! h
exact hf.hasDerivWithinAt.eventually_ne h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 176,
"column": 9
} | {
"line": 176,
"column": 32
} | {
"line": 177,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhn : ω ≠ ∞\n⊢ ContDiffWithinAt 𝕜 ω f s x ↔\n ∃ u ∈ 𝓝[insert x s] x, ∃... | [] | simp [ContDiffWithinAt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 176,
"column": 9
} | {
"line": 176,
"column": 32
} | {
"line": 177,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhn : ω ≠ ∞\n⊢ ContDiffWithinAt 𝕜 ω f s x ↔\n ∃ u ∈ 𝓝[insert x s] x, ∃... | [] | simp [ContDiffWithinAt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 176,
"column": 9
} | {
"line": 176,
"column": 32
} | {
"line": 177,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhn : ω ≠ ∞\n⊢ ContDiffWithinAt 𝕜 ω f s x ↔\n ∃ u ∈ 𝓝[insert x s] x, ∃... | [] | simp [ContDiffWithinAt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 324,
"column": 9
} | {
"line": 324,
"column": 32
} | {
"line": 325,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\n⊢ ContDiffWithinAt 𝕜 ω f (insert x s) x ↔ ContDiffWithinAt 𝕜 ω f s x",
... | [] | simp [ContDiffWithinAt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 324,
"column": 9
} | {
"line": 324,
"column": 32
} | {
"line": 325,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\n⊢ ContDiffWithinAt 𝕜 ω f (insert x s) x ↔ ContDiffWithinAt 𝕜 ω f s x",
... | [] | simp [ContDiffWithinAt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 324,
"column": 9
} | {
"line": 324,
"column": 32
} | {
"line": 325,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\n⊢ ContDiffWithinAt 𝕜 ω f (insert x s) x ↔ ContDiffWithinAt 𝕜 ω f s x",
... | [] | simp [ContDiffWithinAt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 37
} | {
"line": 330,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\ny : E\n⊢ ContDiffWithinAt 𝕜 n f (insert y s) x ↔ ContDiffWithinAt 𝕜 n f ... | [
"case inl\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\n⊢ ContDiffWithinAt 𝕜 n f (insert x s) x ↔ ContDiffWithinAt 𝕜 n f s x",
"... | rcases eq_or_ne x y with (rfl | hx) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 396,
"column": 6
} | {
"line": 396,
"column": 52
} | {
"line": 397,
"column": 4
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhn : n ≠ ∞\nh'n : n + 1 ≠ ∞\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\nhf : n ... | [] | exact nhdsWithin_mono _ (subset_insert x u) hv | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 261,
"column": 4
} | {
"line": 265,
"column": 61
} | {
"line": 266,
"column": 2
} | [
{
"pp": "case inl\nn : ℕ\nc : OrderedFinpartition (n + 1)\nhc : range (c.emb 0) ≠ {0}\nthis : c.partSize (c.index 0) = Nat.card ↑(range (c.emb (c.index 0)))\nh : c.index 0 = 0\n⊢ 1 < Nat.card ↑(range (c.emb (c.index 0)))",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← h] at hc
have : {0} ⊂ range (c.emb (c.index 0)) := by
apply ssubset_of_subset_of_ne ?_ hc.symm
simpa only [singleton_subset_iff, mem_range] using ⟨0, emb_zero c⟩
simpa using Set.Finite.card_lt_card (finite_range _) this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 261,
"column": 4
} | {
"line": 265,
"column": 61
} | {
"line": 266,
"column": 2
} | [
{
"pp": "case inl\nn : ℕ\nc : OrderedFinpartition (n + 1)\nhc : range (c.emb 0) ≠ {0}\nthis : c.partSize (c.index 0) = Nat.card ↑(range (c.emb (c.index 0)))\nh : c.index 0 = 0\n⊢ 1 < Nat.card ↑(range (c.emb (c.index 0)))",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [← h] at hc
have : {0} ⊂ range (c.emb (c.index 0)) := by
apply ssubset_of_subset_of_ne ?_ hc.symm
simpa only [singleton_subset_iff, mem_range] using ⟨0, emb_zero c⟩
simpa using Set.Finite.card_lt_card (finite_range _) this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 304,
"column": 6
} | {
"line": 304,
"column": 53
} | {
"line": 305,
"column": 2
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\... | [] | exact strictMono_succ.comp (c.emb_strictMono i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 304,
"column": 6
} | {
"line": 304,
"column": 53
} | {
"line": 305,
"column": 2
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\... | [] | exact strictMono_succ.comp (c.emb_strictMono i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 304,
"column": 6
} | {
"line": 304,
"column": 53
} | {
"line": 305,
"column": 2
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\... | [] | exact strictMono_succ.comp (c.emb_strictMono i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 533,
"column": 2
} | {
"line": 537,
"column": 50
} | {
"line": 539,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nh : ContDiffWithinAt 𝕜 n f s x\nhn : n ≠ ∞\n⊢ ∀ᶠ (y : E) in 𝓝[insert x s... | [] | rcases h.contDiffOn le_rfl (by simp [hn]) with ⟨u, hu, _, hd⟩
have : ∀ᶠ y : E in 𝓝[insert x s] x, u ∈ 𝓝[insert x s] y ∧ y ∈ u :=
(eventually_eventually_nhdsWithin.2 hu).and hu
refine this.mono fun y hy => (hd y hy.2).mono_of_mem_nhdsWithin ?_
exact nhdsWithin_mono y (subset_insert _ _) hy.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 533,
"column": 2
} | {
"line": 537,
"column": 50
} | {
"line": 539,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nh : ContDiffWithinAt 𝕜 n f s x\nhn : n ≠ ∞\n⊢ ∀ᶠ (y : E) in 𝓝[insert x s... | [] | rcases h.contDiffOn le_rfl (by simp [hn]) with ⟨u, hu, _, hd⟩
have : ∀ᶠ y : E in 𝓝[insert x s] x, u ∈ 𝓝[insert x s] y ∧ y ∈ u :=
(eventually_eventually_nhdsWithin.2 hu).and hu
refine this.mono fun y hy => (hd y hy.2).mono_of_mem_nhdsWithin ?_
exact nhdsWithin_mono y (subset_insert _ _) hy.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 860,
"column": 4
} | {
"line": 860,
"column": 46
} | {
"line": 861,
"column": 4
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nn : ℕ∞ω\nhs : UniqueDiffOn 𝕜 s\nH : ContDiffOn 𝕜 (n + 1) f s\nx : E\nhx : x ∈ s\nm : ℕ\n... | [
"𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nn : ℕ∞ω\nhs : UniqueDiffOn 𝕜 s\nH : ContDiffOn 𝕜 (n + 1) f s\nx : E\nhx : x ∈ s\nm : ℕ\nhm : ↑m ≤ n\... | rw [inter_comm, insert_eq_of_mem hx] at ho | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 1100,
"column": 6
} | {
"line": 1102,
"column": 61
} | {
"line": 1104,
"column": 0
} | [
{
"pp": "case mpr\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nn✝ : ℕ∞ω\nn : ℕ∞\n⊢ ContDiff 𝕜 (↑n) f → ContDiffOn 𝕜 (↑n) f univ",
"ppTerm": "?mpr",
... | [] | rintro ⟨p, hp⟩ x _ m hm
exact ⟨univ, Filter.univ_sets _, p,
(hp.hasFTaylorSeriesUpToOn univ).of_le (mod_cast hm)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 1100,
"column": 6
} | {
"line": 1102,
"column": 61
} | {
"line": 1104,
"column": 0
} | [
{
"pp": "case mpr\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nn✝ : ℕ∞ω\nn : ℕ∞\n⊢ ContDiff 𝕜 (↑n) f → ContDiffOn 𝕜 (↑n) f univ",
"ppTerm": "?mpr",
... | [] | rintro ⟨p, hp⟩ x _ m hm
exact ⟨univ, Filter.univ_sets _, p,
(hp.hasFTaylorSeriesUpToOn univ).of_le (mod_cast hm)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 990,
"column": 2
} | {
"line": 991,
"column": 86
} | {
"line": 992,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq : F → Formal... | [
"𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq : F → FormalMultilinearS... | change AnalyticOn 𝕜
((fun p ↦ B p.1 p.2) ∘ (fun x ↦ (q (f x) c.length, fun m ↦ p x (c.partSize m)))) s | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 345,
"column": 2
} | {
"line": 348,
"column": 40
} | {
"line": 349,
"column": 2
} | [
{
"pp": "𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\nhy : y ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\n⊢ ∑' ... | [
"𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx y : 𝔸\nhxy : Commute x y\nhx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\nhy : y ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\n⊢ ∑' (x_1 : ℕ), ∑... | conv_lhs =>
congr
ext
rw [hxy.add_pow' _, Finset.smul_sum] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.Topology.MetricSpace.CauSeqFilter | {
"line": 53,
"column": 79
} | {
"line": 62,
"column": 51
} | {
"line": 64,
"column": 0
} | [
{
"pp": "β : Type v\ninst✝ : NormedField β\nf : ℕ → β\nhf : CauchySeq f\n⊢ IsCauSeq norm f",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Set.instSProd",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"le_refl",
... | [] | by
obtain ⟨hf1, hf2⟩ := cauchy_iff.1 hf
intro ε hε
rcases hf2 { x | dist x.1 x.2 < ε } (dist_mem_uniformity hε) with ⟨t, ⟨ht, htsub⟩⟩
simp only [mem_map, mem_atTop_sets, mem_preimage] at ht; obtain ⟨N, hN⟩ := ht
exists N
intro j hj
rw [← dist_eq_norm]
apply @htsub (f j, f N)
apply Set.mk_mem_prod <;> ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 647,
"column": 8
} | {
"line": 647,
"column": 17
} | {
"line": 647,
"column": 18
} | [
{
"pp": "case inr\n𝔸 : Type u_1\ninst✝² : NormedDivisionRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nn : ℕ\n⊢ exp (-↑n • x) = exp x ^ (-↑n)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"GroupWithZero.toDivisionMonoid",
... | [
"case inr\n𝔸 : Type u_1\ninst✝² : NormedDivisionRing 𝔸\ninst✝¹ : NormedAlgebra ℚ 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nn : ℕ\n⊢ exp (-↑n • x) = (exp x ^ ↑n)⁻¹"
] | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.ExtendFrom | {
"line": 49,
"column": 2
} | {
"line": 51,
"column": 50
} | {
"line": 53,
"column": 0
} | [
{
"pp": "case neg.inr\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : DenselyOrdered α\ninst✝² : OrderTopology α\ninst✝¹ : TopologicalSpace β\nf : α → β\na b : α\nla lb : β\ninst✝ : RegularSpace β\nhf : ContinuousOn f (uIoo a b)\nha : Tendsto f (𝓝[uIoo a b] a) (𝓝 la)... | [] | · simp only [hba', uIoo_of_gt, nhdsWithin_Ioo_eq_nhdsGT, nhdsWithin_Ioo_eq_nhdsLT,
uIcc_of_gt] at ha hb hf ⊢
exact continuousOn_Icc_extendFrom_Ioo hf hb ha | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 120,
"column": 4
} | {
"line": 121,
"column": 35
} | {
"line": 123,
"column": 0
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ns : Set 𝕜\n𝕝 : Type u_4\ninst✝³ : DivisionSemiring 𝕝\ninst✝² : Module 𝕝 F\ninst✝¹ : SMulCommClass 𝕜 𝕝 F\ninst✝ : ContinuousConstSMul 𝕝 F\nc : 𝕝\nn : ℕ\nIH : ∀ {... | [] | simp_rw [iteratedDerivWithin_succ, funext (@IH · f), ← Pi.smul_def,
derivWithin_const_smul_field] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 120,
"column": 4
} | {
"line": 121,
"column": 35
} | {
"line": 123,
"column": 0
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ns : Set 𝕜\n𝕝 : Type u_4\ninst✝³ : DivisionSemiring 𝕝\ninst✝² : Module 𝕝 F\ninst✝¹ : SMulCommClass 𝕜 𝕝 F\ninst✝ : ContinuousConstSMul 𝕝 F\nc : 𝕝\nn : ℕ\nIH : ∀ {... | [] | simp_rw [iteratedDerivWithin_succ, funext (@IH · f), ← Pi.smul_def,
derivWithin_const_smul_field] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 120,
"column": 4
} | {
"line": 121,
"column": 35
} | {
"line": 123,
"column": 0
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ns : Set 𝕜\n𝕝 : Type u_4\ninst✝³ : DivisionSemiring 𝕝\ninst✝² : Module 𝕝 F\ninst✝¹ : SMulCommClass 𝕜 𝕝 F\ninst✝ : ContinuousConstSMul 𝕝 F\nc : 𝕝\nn : ℕ\nIH : ∀ {... | [] | simp_rw [iteratedDerivWithin_succ, funext (@IH · f), ← Pi.smul_def,
derivWithin_const_smul_field] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 395,
"column": 10
} | {
"line": 395,
"column": 15
} | {
"line": 395,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nι : Type u_3\nf : ι → E → F\nu : Finset ι\ni : ℕ\nx : E\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\nh : ∀ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 395,
"column": 10
} | {
"line": 395,
"column": 15
} | {
"line": 395,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nι : Type u_3\nf : ι → E → F\nu : Finset ι\ni : ℕ\nx : E\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\nh : ∀ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 395,
"column": 10
} | {
"line": 395,
"column": 15
} | {
"line": 395,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nι : Type u_3\nf : ι → E → F\nu : Finset ι\ni : ℕ\nx : E\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\nh : ∀ ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 401,
"column": 8
} | {
"line": 401,
"column": 80
} | {
"line": 401,
"column": 81
} | [
{
"pp": "case cons\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nι : Type u_3\nf : ι → E → F\ni : ℕ\nx : E\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\na : ι\nu ... | [
"case cons\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nι : Type u_3\nf : ι → E → F\ni : ℕ\nx : E\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\na : ι\nu : Finset ι\n... | fun_iteratedFDerivWithin_add_apply h.1 (ContDiffWithinAt.sum h.2) hs hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 695,
"column": 8
} | {
"line": 695,
"column": 42
} | {
"line": 695,
"column": 42
} | [
{
"pp": "case succ.h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ni✝ : ℕ\na : 𝕜\ni : ℕ\nhi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z... | [
"case succ.h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ni✝ : ℕ\na : 𝕜\ni : ℕ\nhi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) = fun x ↦... | ← Function.comp_def (g := (a • ·)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 698,
"column": 6
} | {
"line": 698,
"column": 44
} | {
"line": 700,
"column": 0
} | [
{
"pp": "case hf\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ni✝ : ℕ\na : 𝕜\ni : ℕ\nhi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) =... | [] | exact differentiableAt_id.const_smul _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 698,
"column": 6
} | {
"line": 698,
"column": 44
} | {
"line": 700,
"column": 0
} | [
{
"pp": "case hf\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ni✝ : ℕ\na : 𝕜\ni : ℕ\nhi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) =... | [] | exact differentiableAt_id.const_smul _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 698,
"column": 6
} | {
"line": 698,
"column": 44
} | {
"line": 700,
"column": 0
} | [
{
"pp": "case hf\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ni✝ : ℕ\na : 𝕜\ni : ℕ\nhi : ContDiff 𝕜 (↑i) f → (iteratedFDeriv 𝕜 i fun z ↦ f (a • z)) =... | [] | exact differentiableAt_id.const_smul _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 919,
"column": 8
} | {
"line": 919,
"column": 47
} | {
"line": 920,
"column": 8
} | [
{
"pp": "case h.refine_1\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type uF\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nn✝ : ℕ∞ω\ninst✝ : CompleteSpace E\nf : OpenPartialHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nh... | [
"case h.refine_2\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type uF\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nn✝ : ℕ∞ω\ninst✝ : CompleteSpace E\nf : OpenPartialHomeomorph E F\nf₀' : E ≃L[𝕜] F\na : F\nha : a ∈ f.ta... | · exact f.isOpen_inter_preimage_symm ht | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 119,
"column": 2
} | {
"line": 120,
"column": 43
} | {
"line": 122,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\nx f' : ℝ\nhf : HasStrictDerivAt f f' x\nhx : f x ≠ 0\n⊢ HasStrictDerivAt (fun y ↦ Real.log (f y)) (f' / f x) x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"NonUnitalCo... | [] | rw [div_eq_inv_mul]
exact (hasStrictDerivAt_log hx).comp x hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 119,
"column": 2
} | {
"line": 120,
"column": 43
} | {
"line": 122,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\nx f' : ℝ\nhf : HasStrictDerivAt f f' x\nhx : f x ≠ 0\n⊢ HasStrictDerivAt (fun y ↦ Real.log (f y)) (f' / f x) x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"NonUnitalCo... | [] | rw [div_eq_inv_mul]
exact (hasStrictDerivAt_log hx).comp x hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Convex.Cone.Pointed | {
"line": 351,
"column": 26
} | {
"line": 351,
"column": 31
} | {
"line": 351,
"column": 31
} | [
{
"pp": "R : Type u_1\nE : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : LinearOrder R\ninst✝² : IsOrderedRing R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nx✝³ : Submodule R E\nx✝² : PointedCone R E\nx✝¹ : ↑x✝³ ≤ x✝²\nx✝ : E\n⊢ x✝ ∈ x✝³ → x✝ ∈ x✝².lineal",
"ppTerm": "?m.31",
"assigned": true,
"usedConstant... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Geometry.Convex.Cone.Pointed | {
"line": 351,
"column": 26
} | {
"line": 351,
"column": 31
} | {
"line": 351,
"column": 31
} | [
{
"pp": "R : Type u_1\nE : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : LinearOrder R\ninst✝² : IsOrderedRing R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nx✝³ : Submodule R E\nx✝² : PointedCone R E\nx✝¹ : ↑x✝³ ≤ x✝²\nx✝ : E\n⊢ x✝ ∈ x✝³ → x✝ ∈ x✝².lineal",
"ppTerm": "?m.31",
"assigned": true,
"usedConstant... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Convex.Cone.Pointed | {
"line": 351,
"column": 26
} | {
"line": 351,
"column": 31
} | {
"line": 351,
"column": 31
} | [
{
"pp": "R : Type u_1\nE : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : LinearOrder R\ninst✝² : IsOrderedRing R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nx✝³ : Submodule R E\nx✝² : PointedCone R E\nx✝¹ : ↑x✝³ ≤ x✝²\nx✝ : E\n⊢ x✝ ∈ x✝³ → x✝ ∈ x✝².lineal",
"ppTerm": "?m.31",
"assigned": true,
"usedConstant... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Gauge | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 56
} | {
"line": 98,
"column": 0
} | [
{
"pp": "case neg\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nh : 0 ∉ s\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • 0 ∈ s} = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"False",
"Real",
"Set.Ioi",
"instHSMul",
"eq_false",
"Real.instZe... | [] | · simp only [smul_zero, sep_false, h, Real.sInf_empty] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Gauge | {
"line": 346,
"column": 15
} | {
"line": 346,
"column": 25
} | {
"line": 346,
"column": 25
} | [
{
"pp": "E : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns : Set E\nx : E\ninst✝¹ : TopologicalSpace E\ninst✝ : T1Space E\nhs : Absorbent ℝ s\nhb : Bornology.IsVonNBounded ℝ s\nh₀ : gauge s x = 0\nhne : x ≠ 0\nthis : {x}ᶜ ∈ comap (gauge s) (𝓝 0)\nr : ℝ\nhr₀ : 0 < r\nhr : gauge s ⁻¹' {b | |b| < r} ⊆... | [] | simpa [h₀] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.Gauge | {
"line": 346,
"column": 15
} | {
"line": 346,
"column": 25
} | {
"line": 346,
"column": 25
} | [
{
"pp": "E : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns : Set E\nx : E\ninst✝¹ : TopologicalSpace E\ninst✝ : T1Space E\nhs : Absorbent ℝ s\nhb : Bornology.IsVonNBounded ℝ s\nh₀ : gauge s x = 0\nhne : x ≠ 0\nthis : {x}ᶜ ∈ comap (gauge s) (𝓝 0)\nr : ℝ\nhr₀ : 0 < r\nhr : gauge s ⁻¹' {b | |b| < r} ⊆... | [] | simpa [h₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Gauge | {
"line": 346,
"column": 15
} | {
"line": 346,
"column": 25
} | {
"line": 346,
"column": 25
} | [
{
"pp": "E : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns : Set E\nx : E\ninst✝¹ : TopologicalSpace E\ninst✝ : T1Space E\nhs : Absorbent ℝ s\nhb : Bornology.IsVonNBounded ℝ s\nh₀ : gauge s x = 0\nhne : x ≠ 0\nthis : {x}ᶜ ∈ comap (gauge s) (𝓝 0)\nr : ℝ\nhr₀ : 0 < r\nhr : gauge s ⁻¹' {b | |b| < r} ⊆... | [] | simpa [h₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 82,
"column": 4
} | {
"line": 84,
"column": 61
} | {
"line": 86,
"column": 0
} | [
{
"pp": "case refine_1.inr\nE : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : Module ℝ E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhs₀ : 0 ∈ s\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\nx₀ : E\nhx₀ : x₀ ∉ s\nf : E →ₗ.[ℝ] ℝ := LinearPMap.mkSpanSingleton x₀ 1 ⋯\n... | [] | exact
one_le_gauge_of_notMem (hs₁.starConvex hs₀)
(absorbent_nhds_zero <| hs₂.mem_nhds hs₀).absorbs hx₀ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Gauge | {
"line": 528,
"column": 4
} | {
"line": 528,
"column": 44
} | {
"line": 529,
"column": 2
} | [
{
"pp": "case inr.refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\np : Seminorm ℝ E\nr : ℝ\nhr : 0 < r\ny : E\nhy : p y < 1\nhp : {r_1 | 0 < r_1 ∧ (fun x ↦ r • x) y ∈ r_1 • p.ball 0 1}.Nonempty\n⊢ r * p y ≤ r",
"ppTerm": "?inr.refine_1",
"assigned": true,
"usedConstants": [
... | [] | exact mul_le_of_le_one_right hr.le hy.le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 172,
"column": 13
} | {
"line": 172,
"column": 18
} | {
"line": 172,
"column": 18
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝¹⁵ : Field R\ninst✝¹⁴ : AddCommGroup V\ninst✝¹³ : TopologicalSpace R\ninst✝¹² : TopologicalSpace V\ninst✝¹¹ : IsTopologicalRing R\ninst✝¹⁰ : Module R V\ninst✝⁹ : SeparatingDual R V\ninst✝⁸ : IsTopologicalAddGroup V\ninst✝⁷ : ContinuousSMul R V\nS : Type u_3\ninst✝⁶ : Co... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 172,
"column": 13
} | {
"line": 172,
"column": 18
} | {
"line": 172,
"column": 18
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝¹⁵ : Field R\ninst✝¹⁴ : AddCommGroup V\ninst✝¹³ : TopologicalSpace R\ninst✝¹² : TopologicalSpace V\ninst✝¹¹ : IsTopologicalRing R\ninst✝¹⁰ : Module R V\ninst✝⁹ : SeparatingDual R V\ninst✝⁸ : IsTopologicalAddGroup V\ninst✝⁷ : ContinuousSMul R V\nS : Type u_3\ninst✝⁶ : Co... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 172,
"column": 13
} | {
"line": 172,
"column": 18
} | {
"line": 172,
"column": 18
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝¹⁵ : Field R\ninst✝¹⁴ : AddCommGroup V\ninst✝¹³ : TopologicalSpace R\ninst✝¹² : TopologicalSpace V\ninst✝¹¹ : IsTopologicalRing R\ninst✝¹⁰ : Module R V\ninst✝⁹ : SeparatingDual R V\ninst✝⁸ : IsTopologicalAddGroup V\ninst✝⁷ : ContinuousSMul R V\nS : Type u_3\ninst✝⁶ : Co... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 305,
"column": 65
} | {
"line": 305,
"column": 85
} | {
"line": 305,
"column": 85
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ns t : Set E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nhs : Convex ℝ s\nht : Convex ℝ t\nhst : Disjoint (... | [] | by simpa using hzero | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 333,
"column": 65
} | {
"line": 333,
"column": 85
} | {
"line": 333,
"column": 85
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\nx : E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nA : Set E\nhA : Convex ℝ A\nhxA : x ∉ interior A\nhAint ... | [] | by simpa using hzero | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 516,
"column": 2
} | {
"line": 517,
"column": 8
} | {
"line": 518,
"column": 2
} | [
{
"pp": "case refine_3\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ\nhf : Integrable f μ\nε : ℝ\nεpos : 0 < ε\ng : α → EReal\ng_lt_f : ∀ (x : α), ↑(-f x) < g x\ngcont : LowerSemicontinuous... | [
"case refine_4\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ\nhf : Integrable f μ\nε : ℝ\nεpos : 0 < ε\ng : α → EReal\ng_lt_f : ∀ (x : α), ↑(-f x) < g x\ngcont : LowerSemicontinuous g\ng_integr... | · convert! g_integrable.neg
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 405,
"column": 4
} | {
"line": 405,
"column": 88
} | {
"line": 406,
"column": 4
} | [
{
"pp": "case inr\nε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nhf : IntegrableOn f [[a, b]] volume\nc : ℝ\nh : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞\nhc : c ≠ 0\nA : MeasurableEmbedding fun x ↦ x * c⁻¹\n⊢ Inte... | [
"case inr\nε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nhf : IntegrableOn f [[a, b]] volume\nc : ℝ\nh : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞\nhc : c ≠ 0\nA : MeasurableEmbedding fun x ↦ x * c⁻¹\n⊢ Integrable (fun ... | integrable_smul_measure (by simpa : ENNReal.ofReal |c⁻¹| ≠ 0) ENNReal.ofReal_ne_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 543,
"column": 95
} | {
"line": 545,
"column": 92
} | {
"line": 547,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf : ℝ → E\nc : E\nla la' : Filter ℝ\nlt : Filter ι\na b : ℝ\nu v : ι → ℝ\ninst✝ : FTCFilter a la la'\nhab : IntervalIntegrable f volume a b\nhmeas : StronglyMeasurableAtFilter f la' volume\nhf... | [] | by
simpa only [integral_const, smul_eq_mul, mul_one] using!
measure_integral_sub_integral_sub_linear_isLittleO_of_tendsto_ae_left hab hmeas hf hu hv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 66
} | {
"line": 479,
"column": 66
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → E\nh : ‖f (min a b)‖ₑ ≠ ∞\n⊢ IntervalIntegrable f volume a b ↔ IntervalIntegrable (fun x ↦ f (-x)) volume (-a) (-b)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NegZeroClass.toN... | [
"E : Type u_5\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → E\nh : ‖f (min a b)‖ₑ ≠ ∞\n⊢ IntervalIntegrable (fun x ↦ f (-1 * x)) volume (a / -1) (b / -1) ↔\n IntervalIntegrable (fun x ↦ f (-x)) volume (-a) (-b)"
] | rw [← comp_mul_left_iff (neg_ne_zero.2 one_ne_zero) h (by simp)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 33
} | {
"line": 565,
"column": 2
} | [
{
"pp": "case inl\nE : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nh₁f : ∀ (x : ℝ), f x = f (-x)\nh₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x\nt : ℝ\nht : ‖f (min 0 t)‖ₑ ≠ ∞\nh : t < 0\n⊢ IntervalIntegrable f volume (-0) (-t)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants":... | [] | simp [h₂f (-t) (by simp [h])] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 592,
"column": 4
} | {
"line": 592,
"column": 33
} | {
"line": 593,
"column": 2
} | [
{
"pp": "case inl\nE : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nh₁f : ∀ (x : ℝ), -f x = f (-x)\nh₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x\nt : ℝ\nht : ‖f (min 0 t)‖ₑ ≠ ∞\nh : t < 0\n⊢ IntervalIntegrable f volume (-0) (-t)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants"... | [] | simp [h₂f (-t) (by simp [h])] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1163,
"column": 2
} | {
"line": 1169,
"column": 90
} | {
"line": 1170,
"column": 2
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : CompleteSpace E\nf f' : ℝ → E\nhab : a < b\nfa fb : E\nhderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhint : IntervalIntegrable f' volume a b\nha : Tendsto f (𝓝[>] a) (𝓝 fa)\nhb : Tendsto f (𝓝[<] b) (𝓝 fb)\nF : ... | [
"E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : CompleteSpace E\nf f' : ℝ → E\nhab : a < b\nfa fb : E\nhderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhint : IntervalIntegrable f' volume a b\nha : Tendsto f (𝓝[>] a) (𝓝 fa)\nhb : Tendsto f (𝓝[<] b) (𝓝 fb)\nF : ℝ → E := upd... | have hcont : ContinuousOn F (Icc a b) := by
rw [continuousOn_update_iff, continuousOn_update_iff, Icc_sdiff_right, Ico_sdiff_left]
refine ⟨⟨fun z hz => (hderiv z hz).continuousAt.continuousWithinAt, ?_⟩, ?_⟩
· exact fun _ => ha.mono_left (nhdsWithin_mono _ Ioo_subset_Ioi_self)
· rintro -
refine (h... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1185,
"column": 2
} | {
"line": 1185,
"column": 32
} | {
"line": 1187,
"column": 0
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : CompleteSpace E\nf' f : ℝ → E\nhderiv : deriv f = f'\nhdiff : ∀ x ∈ [[a, b]], DifferentiableAt ℝ f x\nhcont : ContinuousOn f' [[a, b]]\n⊢ IntervalIntegrable f' volume a b",
"ppTerm": "?m.57",
"assigned": tru... | [] | exact hcont.intervalIntegrable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1206,
"column": 2
} | {
"line": 1206,
"column": 34
} | {
"line": 1207,
"column": 2
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : IsScalarTower ℝ 𝕜 E\nf f' : 𝕜 → E\nz₀ z₁ : 𝕜\nhcont : ContinuousOn (fun t ↦ f' (z₀ + t • z₁)) (Icc 0 1)\nhderiv : ∀ t ∈ Icc 0 1, HasD... | [
"𝕜 : Type u_2\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : CompleteSpace E\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : IsScalarTower ℝ 𝕜 E\nf f' : 𝕜 → E\nz₀ z₁ : 𝕜\nhcont : ContinuousOn (fun t ↦ f' (z₀ + t • z₁)) (Icc 0 1)\nhderiv : ∀ t ∈ Icc 0 1, HasDerivAt f (f'... | let γ (t : ℝ) : 𝕜 := z₀ + t • z₁ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1198,
"column": 76
} | {
"line": 1202,
"column": 29
} | {
"line": 1204,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\nhf : IntegrableOn f (Ici a) μ\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in Ici a, f x ∂μ - ∫ (x : ℝ) in Ici b, f x ∂μ = ∫ (x : ℝ) in Ico a b, f x ∂μ",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants... | [] | by
have ha : IntegrableOn f (Ici b) μ := hf.mono_set (Ici_subset_Ici.2 hab)
have h : IntegrableOn f (Ico a b) μ := hf.mono_set Ico_subset_Ici_self
rw [sub_eq_iff_eq_add', ← setIntegral_union (by grind) measurableSet_Ico ha h, union_comm,
Ico_union_Ici_eq_Ici hab] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1213,
"column": 75
} | {
"line": 1215,
"column": 43
} | {
"line": 1217,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ℝ\nf : ℝ → E\nμ : Measure ℝ\nh_left : IntegrableOn f (Iic b) μ\nh_right : IntegrableOn f (Ioi b) μ\n⊢ ∫ (x : ℝ) in Iic b, f x ∂μ + ∫ (x : ℝ) in Ioi b, f x ∂μ = ∫ (x : ℝ), f x ∂μ",
"ppTerm": "?m.50",
"assigned": true,
... | [] | by
convert! (setIntegral_union (Iic_disjoint_Ioi <| Eq.le rfl) measurableSet_Ioi h_left h_right).symm
rw [Iic_union_Ioi, Measure.restrict_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 390,
"column": 2
} | {
"line": 391,
"column": 24
} | {
"line": 394,
"column": 0
} | [
{
"pp": "case hf'\nG : Type u_1\ninst✝¹⁰ : Group G\ninst✝⁹ : MeasurableSpace G\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : IsTopologicalGroup G\ninst✝⁶ : BorelSpace G\nμ : Measure G\nΓ : Subgroup G\n𝓕 : Set G\nh𝓕 : IsFundamentalDomain (↥Γ.op) 𝓕 μ\ninst✝⁵ : Countable ↥Γ\ninst✝⁴ : MeasurableSpace (G ⧸ Γ)\ninst✝³ : ... | [] | · rw [← h𝓕.lintegral_eq_tsum'' (‖f ·‖ₑ)]
exact ne_of_lt hf₁.2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 209,
"column": 28
} | {
"line": 209,
"column": 33
} | {
"line": 209,
"column": 34
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nc : ℂ\nR : ℝ\nι : Type u_3\ns : Finset ι\nf : ι → ℂ → E\nh : ∀ i ∈ s, CircleIntegrable (f i) c R\n⊢ (fun θ ↦ (∑ i ∈ s, f i) (circleMap c R θ)) = ∑ i ∈ s, fun θ ↦ f i (circleMap c R θ)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 209,
"column": 28
} | {
"line": 209,
"column": 33
} | {
"line": 209,
"column": 34
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nc : ℂ\nR : ℝ\nι : Type u_3\ns : Finset ι\nf : ι → ℂ → E\nh : ∀ i ∈ s, CircleIntegrable (f i) c R\n⊢ (fun θ ↦ (∑ i ∈ s, f i) (circleMap c R θ)) = ∑ i ∈ s, fun θ ↦ f i (circleMap c R θ)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 209,
"column": 28
} | {
"line": 209,
"column": 33
} | {
"line": 209,
"column": 34
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nc : ℂ\nR : ℝ\nι : Type u_3\ns : Finset ι\nf : ι → ℂ → E\nh : ∀ i ∈ s, CircleIntegrable (f i) c R\n⊢ (fun θ ↦ (∑ i ∈ s, f i) (circleMap c R θ)) = ∑ i ∈ s, fun θ ↦ f i (circleMap c R θ)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 68
} | {
"line": 226,
"column": 2
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nt : ℝ\nf : ℝ → ℂ\np : ℝ≥0∞\nhLp : MemLp f p (volume.restrict (Ioc t (t + T)))\n⊢ MemLp (AddCircle.liftIoc T t f) p volume",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.instArchimedea... | [
"T : ℝ\nhT : Fact (0 < T)\nt : ℝ\nf : ℝ → ℂ\np : ℝ≥0∞\nhLp : MemLp f p (volume.restrict (Ioc t (t + T)))\n⊢ MemLp (fun x ↦ f ↑((AddCircle.equivIoc T t) x)) p volume"
] | simp only [AddCircle.liftIoc, Set.restrict_def, Function.comp_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 286,
"column": 30
} | {
"line": 286,
"column": 35
} | {
"line": 287,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t : ℝ\nh₁f : Periodic f T\nhT✝ : T ≠ 0\nh₂f : IntervalIntegrable f volume t (t + T)\na₁ a₂ : ℝ\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T t : ℝ},\n Periodic f T →\n T ≠ 0 → IntervalIntegrable f volume t (t + ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 308,
"column": 69
} | {
"line": 308,
"column": 74
} | {
"line": 308,
"column": 74
} | [
{
"pp": "E✝ : Type u_1\ninst✝¹ : NormedAddCommGroup E✝\nf✝ : ℝ → E✝\nT✝ : ℝ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t : ℝ\nh₁f : Periodic f T\nhT✝ : T ≠ 0\nh₂f : IntervalIntegrable f volume t (t + T)\na₁ a₂ : ℝ\nhT : 0 < T\nn₁ : ℕ\nhn₁ : (t - min a₁ a₂) / T ≤ ↑n₁\nn₂ : ℕ\nhn₂ : (max a₁ a₂ - t)... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 308,
"column": 69
} | {
"line": 308,
"column": 74
} | {
"line": 308,
"column": 74
} | [
{
"pp": "E✝ : Type u_1\ninst✝¹ : NormedAddCommGroup E✝\nf✝ : ℝ → E✝\nT✝ : ℝ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t : ℝ\nh₁f : Periodic f T\nhT✝ : T ≠ 0\nh₂f : IntervalIntegrable f volume t (t + T)\na₁ a₂ : ℝ\nhT : 0 < T\nn₁ : ℕ\nhn₁ : (t - min a₁ a₂) / T ≤ ↑n₁\nn₂ : ℕ\nhn₂ : (max a₁ a₂ - t)... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 308,
"column": 69
} | {
"line": 308,
"column": 74
} | {
"line": 308,
"column": 74
} | [
{
"pp": "E✝ : Type u_1\ninst✝¹ : NormedAddCommGroup E✝\nf✝ : ℝ → E✝\nT✝ : ℝ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t : ℝ\nh₁f : Periodic f T\nhT✝ : T ≠ 0\nh₂f : IntervalIntegrable f volume t (t + T)\na₁ a₂ : ℝ\nhT : 0 < T\nn₁ : ℕ\nhn₁ : (t - min a₁ a₂) / T ≤ ↑n₁\nn₂ : ℕ\nhn₂ : (max a₁ a₂ - t)... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 322,
"column": 2
} | {
"line": 324,
"column": 92
} | {
"line": 326,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t₁ t₂ : ℝ\nhf : Periodic f T\n⊢ IntervalIntegrable f volume t₁ (t₁ + T) ↔ IntervalIntegrable f volume t₂ (t₂ + T)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Function.Periodic.intervalIntegrable",
"Real",
... | [] | wlog hT : T ≠ 0
· simp_all
exact ⟨(hf.intervalIntegrable hT · t₂ (t₂ + T)), (hf.intervalIntegrable hT · t₁ (t₁ + T))⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 322,
"column": 2
} | {
"line": 324,
"column": 92
} | {
"line": 326,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t₁ t₂ : ℝ\nhf : Periodic f T\n⊢ IntervalIntegrable f volume t₁ (t₁ + T) ↔ IntervalIntegrable f volume t₂ (t₂ + T)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Function.Periodic.intervalIntegrable",
"Real",
... | [] | wlog hT : T ≠ 0
· simp_all
exact ⟨(hf.intervalIntegrable hT · t₂ (t₂ + T)), (hf.intervalIntegrable hT · t₁ (t₁ + T))⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 541,
"column": 4
} | {
"line": 544,
"column": 77
} | {
"line": 545,
"column": 2
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf f' : ℂ → E\nc : ℂ\nR : ℝ\nh : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z\nhi : CircleIntegrable f' c R\n⊢ ∮ (z : ℂ) in C(c, R), f' z = 0",
"ppTerm": "?pos✝",
"assigned... | [] | rw [← sub_eq_zero.2 ((periodic_circleMap c R).comp f).eq]
refine intervalIntegral.integral_eq_sub_of_hasDerivAt (fun θ _ => ?_) hi.out
exact (h _ (circleMap_mem_sphere' _ _ _)).scomp_hasDerivAt θ
(differentiable_circleMap _ _ _).hasDerivAt (circleMap_mem_sphere' _ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 541,
"column": 4
} | {
"line": 544,
"column": 77
} | {
"line": 545,
"column": 2
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf f' : ℂ → E\nc : ℂ\nR : ℝ\nh : ∀ z ∈ sphere c |R|, HasDerivWithinAt f (f' z) (sphere c |R|) z\nhi : CircleIntegrable f' c R\n⊢ ∮ (z : ℂ) in C(c, R), f' z = 0",
"ppTerm": "?pos✝",
"assigned... | [] | rw [← sub_eq_zero.2 ((periodic_circleMap c R).comp f).eq]
refine intervalIntegral.integral_eq_sub_of_hasDerivAt (fun θ _ => ?_) hi.out
exact (h _ (circleMap_mem_sphere' _ _ _)).scomp_hasDerivAt θ
(differentiable_circleMap _ _ _).hasDerivAt (circleMap_mem_sphere' _ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 348,
"column": 6
} | {
"line": 348,
"column": 11
} | {
"line": 349,
"column": 4
} | [
{
"pp": "case inr.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℝ → E\nT : ℝ\ninst✝ : NormedSpace ℝ E\nhf : Periodic f T\nt s : ℝ\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T : ℝ} [inst_1 : NormedSpace ℝ E],\n Periodic f T → ∀ (t s : ℝ), 0 < T → ∫ (x : ℝ) in t..t + T, f x ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 348,
"column": 6
} | {
"line": 348,
"column": 11
} | {
"line": 349,
"column": 4
} | [
{
"pp": "case inr.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℝ → E\nT : ℝ\ninst✝ : NormedSpace ℝ E\nhf : Periodic f T\nt s : ℝ\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T : ℝ} [inst_1 : NormedSpace ℝ E],\n Periodic f T → ∀ (t s : ℝ), 0 < T → ∫ (x : ℝ) in t..t + T, f x ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 348,
"column": 6
} | {
"line": 348,
"column": 11
} | {
"line": 349,
"column": 4
} | [
{
"pp": "case inr.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℝ → E\nT : ℝ\ninst✝ : NormedSpace ℝ E\nhf : Periodic f T\nt s : ℝ\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T : ℝ} [inst_1 : NormedSpace ℝ E],\n Periodic f T → ∀ (t s : ℝ), 0 < T → ∫ (x : ℝ) in t..t + T, f x ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 351,
"column": 40
} | {
"line": 351,
"column": 45
} | {
"line": 351,
"column": 46
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℝ → E\nT : ℝ\ninst✝ : NormedSpace ℝ E\nhf : Periodic f T\nt s : ℝ\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T : ℝ} [inst_1 : NormedSpace ℝ E],\n Periodic f T → ∀ (t s : ℝ), 0 < T → ∫ (x : ℝ) in t..t + T, f x = ∫ (x : ℝ) in... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 351,
"column": 40
} | {
"line": 351,
"column": 45
} | {
"line": 351,
"column": 46
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℝ → E\nT : ℝ\ninst✝ : NormedSpace ℝ E\nhf : Periodic f T\nt s : ℝ\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T : ℝ} [inst_1 : NormedSpace ℝ E],\n Periodic f T → ∀ (t s : ℝ), 0 < T → ∫ (x : ℝ) in t..t + T, f x = ∫ (x : ℝ) in... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 351,
"column": 40
} | {
"line": 351,
"column": 45
} | {
"line": 351,
"column": 46
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℝ → E\nT : ℝ\ninst✝ : NormedSpace ℝ E\nhf : Periodic f T\nt s : ℝ\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T : ℝ} [inst_1 : NormedSpace ℝ E],\n Periodic f T → ∀ (t s : ℝ), 0 < T → ∫ (x : ℝ) in t..t + T, f x = ∫ (x : ℝ) in... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 401,
"column": 6
} | {
"line": 404,
"column": 22
} | {
"line": 405,
"column": 4
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc : ℂ\nR : ℝ\nh0 : 0 < R\nf : ℂ → E\ny : E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : ∀ z ∈ (ball c R \\ {c}) \\ s, DifferentiableAt ℂ f z\nhy : Tendsto f (𝓝[≠] c) (𝓝 y)\... | [] | congr 2
· exact circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable hr0
hrR hs (hc.mono hsub) fun z hz => hd z ⟨hsub' hz.1, hz.2⟩
· simp [hr0.ne'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 401,
"column": 6
} | {
"line": 404,
"column": 22
} | {
"line": 405,
"column": 4
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nc : ℂ\nR : ℝ\nh0 : 0 < R\nf : ℂ → E\ny : E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c R \\ {c})\nhd : ∀ z ∈ (ball c R \\ {c}) \\ s, DifferentiableAt ℂ f z\nhy : Tendsto f (𝓝[≠] c) (𝓝 y)\... | [] | congr 2
· exact circleIntegral_sub_center_inv_smul_eq_of_differentiable_on_annulus_off_countable hr0
hrR hs (hc.mono hsub) fun z hz => hd z ⟨hsub' hz.1, hz.2⟩
· simp [hr0.ne'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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