module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.LSeries.Basic | {
"line": 290,
"column": 2
} | {
"line": 291,
"column": 18
} | {
"line": 293,
"column": 0
} | [
{
"pp": "f : ℕ → ℂ\nn : ℕ\n⊢ (f * δ) n = (f 1 • δ) n",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"eq_false",
"LSeries.delta",
"congrArg",
"Complex.instZero",
"Complex.instMul",
"Function.ha... | [] | by_cases hn : n = 1 <;>
simp [hn, delta] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 582,
"column": 2
} | {
"line": 582,
"column": 87
} | {
"line": 583,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝³ : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\na : σ → MvPowerSeries τ S\nf : MvPowerSeries σ R\ninst✝ : Finite τ\nk : ℕ\nha : HasSubst a\nha₁ : ∀ (i : σ), constantCoeff (a i) = 0\n⊢ (truncTotal k) (subst a f) = (truncTotal k) (∑ ... | [
"σ : Type u_1\nR : Type u_3\ninst✝³ : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\na : σ → MvPowerSeries τ S\nf : MvPowerSeries σ R\ninst✝ : Finite τ\nk : ℕ\nha : HasSubst a\nha₁ : ∀ (i : σ), constantCoeff (a i) = 0\n⊢ (truncTotal k) (∑ x ∈ range k, (substAlgHom ha) ((homogeneo... | rw [truncTotal_subst_eq_truncTotal_subst_sum ha ha₁, ← substAlgHom_apply ha, map_sum] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 667,
"column": 6
} | {
"line": 667,
"column": 20
} | {
"line": 668,
"column": 4
} | [
{
"pp": "case e_a.e_a\nσ : Type u_1\nA : Type u_2\ninst✝²¹ : CommSemiring A\nR✝ : Type u_3\ninst✝²⁰ : CommRing R✝\ninst✝¹⁹ : Algebra A R✝\nτ : Type u_4\nS : Type u_5\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra A S\ninst✝¹⁶ : Algebra R✝ S\ninst✝¹⁵ : IsScalarTower A R✝ S\na✝¹ a✝ : σ → MvPowerSeries τ S\nT✝ : Type u_... | [] | simp [pow_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 520,
"column": 6
} | {
"line": 520,
"column": 16
} | {
"line": 521,
"column": 6
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommRing R\nP : R⟦X⟧\nhP : constantCoeff P = 0\ninst✝ : Invertible ((coeff 1) P)\nn : ℕ\nthis : (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P) = (coeff n) X\nm : ℕ\nB : R⟦X⟧\nhB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B\n⊢ ∀ m < n + 1, (coeff m) (mk P.substInvFun - B) = ... | [
"R : Type u_2\ninst✝¹ : CommRing R\nP : R⟦X⟧\nhP : constantCoeff P = 0\ninst✝ : Invertible ((coeff 1) P)\nn : ℕ\nthis : (coeff n) (subst (∑ i, C (P.substInvFun ↑i) * X ^ ↑i) P) = (coeff n) X\nm✝ : ℕ\nB : R⟦X⟧\nhB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B\nm : ℕ\nhm : m < n + 1\n⊢ (coeff m) (mk P.substInvFun - B) = 0... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 578,
"column": 71
} | {
"line": 579,
"column": 41
} | {
"line": 581,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\nP : R⟦X⟧\nhP' : IsUnit ((coeff 1) P)\n⊢ HasSubst (P.substInvOfIsUnit hP')",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"IsNilpotent.zero._simp_1",
"MvPowerSeries",
"RingHom",
... | [] | by
simp [HasSubst, ← constantCoeff.eq_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.TangentCone.Basic | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 31
} | {
"line": 263,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\nx : E\ns : Set E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousConstSMul 𝕜 E\n⊢ UniqueDiffWithinAt 𝕜 (closure[inst✝²] s) x ↔ UniqueDiffWithinAt 𝕜 s x",
"ppTerm": "?m.24",... | [] | simp [uniqueDiffWithinAt_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.TangentCone.Basic | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 31
} | {
"line": 263,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\nx : E\ns : Set E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousConstSMul 𝕜 E\n⊢ UniqueDiffWithinAt 𝕜 (closure[inst✝²] s) x ↔ UniqueDiffWithinAt 𝕜 s x",
"ppTerm": "?m.24",... | [] | simp [uniqueDiffWithinAt_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.TangentCone.Basic | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 31
} | {
"line": 263,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\nx : E\ns : Set E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousConstSMul 𝕜 E\n⊢ UniqueDiffWithinAt 𝕜 (closure[inst✝²] s) x ↔ UniqueDiffWithinAt 𝕜 s x",
"ppTerm": "?m.24",... | [] | simp [uniqueDiffWithinAt_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Congr | {
"line": 107,
"column": 2
} | {
"line": 108,
"column": 76
} | {
"line": 110,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\nF : Type u_3\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf₀ f₁ : E → F\nf₀' f₁' : E →L[𝕜] F\nL : Filter (E × E)\nh₀ : Prod.map f... | [] | simp only [hasFDerivAtFilter_iff_isLittleOTVS]
exact isLittleOTVS_congr (h₀.mono fun y hy => by simp_all [Prod.map]) .rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Congr | {
"line": 107,
"column": 2
} | {
"line": 108,
"column": 76
} | {
"line": 110,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\nF : Type u_3\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf₀ f₁ : E → F\nf₀' f₁' : E →L[𝕜] F\nL : Filter (E × E)\nh₀ : Prod.map f... | [] | simp only [hasFDerivAtFilter_iff_isLittleOTVS]
exact isLittleOTVS_congr (h₀.mono fun y hy => by simp_all [Prod.map]) .rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Const | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 25
} | {
"line": 294,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\nF : Type u_3\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf : E → F\nx : E\n⊢ HasFDerivWithinAt f 0 {x} x",
"ppTerm": "?m.31",... | [
"𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\nF : Type u_3\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf : E → F\nx : E\n⊢ ¬AccPt x (𝓟 {x})"
] | refine .of_not_accPt ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Analytic.ConvergenceRadius | {
"line": 293,
"column": 6
} | {
"line": 293,
"column": 16
} | {
"line": 293,
"column": 16
} | [
{
"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\np q : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nhr : ↑r < min p.radius q.radius\n⊢ ↑r ≤ (p + q).radius",
... | [
"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\np q : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nhr : ↑r < p.radius ∧ ↑r < q.radius\n⊢ ↑r ≤ (p + q).radius"
] | lt_min_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Multilinear.Curry | {
"line": 453,
"column": 2
} | {
"line": 455,
"column": 58
} | {
"line": 456,
"column": 2
} | [
{
"pp": "𝕜 : Type u\nG : Type wG\nG' : Type wG'\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : G [×0]→L[𝕜] G'\n⊢ ‖f‖ ≤ ‖f 0‖",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
... | [
"𝕜 : Type u\nG : Type wG\nG' : Type wG'\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : G [×0]→L[𝕜] G'\nthis : ‖uncurry0 𝕜 G f.curry0‖ ≤ ‖f.curry0‖\n⊢ ‖f‖ ≤ ‖f 0‖"
] | have : ‖ContinuousMultilinearMap.uncurry0 𝕜 G f.curry0‖ ≤ ‖f.curry0‖ :=
ContinuousMultilinearMap.opNorm_le_bound (norm_nonneg _) fun m => by
simp [-ContinuousMultilinearMap.apply_zero_uncurry0] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Analytic.CPolynomialDef | {
"line": 352,
"column": 2
} | {
"line": 352,
"column": 12
} | {
"line": 353,
"column": 2
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → p m = 0\nk : ℕ\n⊢ ∀ {m : ℕ}, n ≤ k + m → ... | [
"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\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → p m = 0\nk m : ℕ\nhm : n ≤ k + m\n⊢ p.changeOriginSer... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Normed.Module.Multilinear.Curry | {
"line": 681,
"column": 10
} | {
"line": 681,
"column": 22
} | {
"line": 682,
"column": 8
} | [
{
"pp": "case inr\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nEi : Fin n.succ → Type wEi\nG✝ : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst... | [] | · simp [hij] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Analytic.CPolynomialDef | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 12
} | {
"line": 363,
"column": 2
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → p m = 0\nk : ℕ\nx : E\nx✝ : Fin k → E\n⊢ ... | [
"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\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → p m = 0\nk : ℕ\nx : E\nx✝ : Fin k → E\nm : ℕ\nhm : m ... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Analytic.CPolynomialDef | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 12
} | {
"line": 375,
"column": 2
} | [
{
"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\nx : E\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → p m = 0\nk : ℕ\nhk : n ≤ k\n⊢ ∀ x_... | [
"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\nx : E\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhn : ∀ (m : ℕ), n ≤ m → p m = 0\nk : ℕ\nhk : n ≤ k\nm : ℕ\nhm : m ∈ Fi... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Normed.Module.Multilinear.Curry | {
"line": 694,
"column": 10
} | {
"line": 694,
"column": 22
} | {
"line": 695,
"column": 8
} | [
{
"pp": "case inr\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nEi : Fin n.succ → Type wEi\nG✝ : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst... | [] | · simp [hij] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Analytic.Linear | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 14
} | {
"line": 46,
"column": 4
} | [
{
"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 →L[𝕜] F\nx : E\n⊢ ∀ (m : ℕ), 2 ≤ m → f.fpowerSeries x m = 0",
"ppTerm": "?m.70",
"assigned": ... | [
"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 →L[𝕜] F\nx : E\nm : ℕ\nhm : 2 ≤ m\n⊢ f.fpowerSeries x m = 0"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 14
} | {
"line": 206,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing E\ninst✝ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\nr : ℝ≥0\nhr : r ≠ 0\nhc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)\nr' : ℝ≥0\nhr' : r' * r < 1\nhrz : r' = 0\n⊢ ↑r' ≤ (ofScalars E c).radius",
"ppTerm... | [
"case neg\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing E\ninst✝ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\nr : ℝ≥0\nhr : r ≠ 0\nhc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)\nr' : ℝ≥0\nhr' : r' * r < 1\nhrz : ¬r' = 0\n⊢ ↑r' ≤ (ofScalars E c).radius"
] | · simp [hrz] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Analytic.Composition | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝¹⁵ : CommRing 𝕜\ninst✝¹⁴ : AddCommGroup E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : AddCommGroup G\ninst✝¹¹ : Module 𝕜 E\ninst✝¹⁰ : Module 𝕜 F\ninst✝⁹ : Module 𝕜 G\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : Topologica... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝¹⁵ : CommRing 𝕜\ninst✝¹⁴ : AddCommGroup E\ninst✝¹³ : AddCommGroup F\ninst✝¹² : AddCommGroup G\ninst✝¹¹ : Module 𝕜 E\ninst✝¹⁰ : Module 𝕜 F\ninst✝⁹ : Module 𝕜 G\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : TopologicalSpace G\nin... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.Analytic.Composition | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 90
} | {
"line": 458,
"column": 2
} | [
{
"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\nq : FormalMultilinearSeries 𝕜 F G\n... | [
"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\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMu... | simp only [lt_min_iff, ENNReal.coe_lt_one_iff, ENNReal.coe_pos] at hrp hrq rp_pos rq_pos | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.Basic | {
"line": 785,
"column": 2
} | {
"line": 785,
"column": 44
} | {
"line": 786,
"column": 2
} | [
{
"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\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesWithinOnBall ... | [
"case inl\n𝕜 : 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\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesWithinOnBall f ... | rcases eq_zero_or_pos r' with (rfl | hr'0) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Analytic.Composition | {
"line": 754,
"column": 4
} | {
"line": 754,
"column": 35
} | {
"line": 755,
"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\ng : F → G\nf : E → F\nq : FormalMult... | [
"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\ng : F → G\nf : E → F\nq : FormalMultilinearSerie... | apply Tendsto.prodMk tendsto_id | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 14
} | {
"line": 165,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\nEm : ι → Type u_6\ninst✝² : (i : ι) → NormedAddCommGroup (Em i)\ninst✝¹ : (i : ι) → NormedSpa... | [
"𝕜 : Type u_1\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\nEm : ι → Type u_6\ninst✝² : (i : ι) → NormedAddCommGroup (Em i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (Em i)... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Analytic.Composition | {
"line": 758,
"column": 6
} | {
"line": 758,
"column": 32
} | {
"line": 759,
"column": 6
} | [
{
"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\ng : F → G\nf : E → F\nq : FormalMult... | [
"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\ng : F → G\nf : E → F\nq : FormalMultilinearSerie... | refine ⟨1, fun n hn => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Analytic.Composition | {
"line": 759,
"column": 6
} | {
"line": 759,
"column": 10
} | {
"line": 760,
"column": 6
} | [
{
"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\ng : F → G\nf : E → F\nq : FormalMult... | [
"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\ng : F → G\nf : E → F\nq : FormalMultilinearSerie... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.Analytic.Inverse | {
"line": 402,
"column": 6
} | {
"line": 402,
"column": 10
} | {
"line": 403,
"column": 6
} | [
{
"pp": "n : ℕ\np : ℕ → ℝ\nhp : ∀ (k : ℕ), 0 ≤ p k\nr a : ℝ\nhr : 0 ≤ r\nha : 0 ≤ a\n⊢ ∑ d ∈ compPartialSumTarget 2 (n + 1) n, ∏ j, r * (a ^ d.snd.blocksFun j * p (d.snd.blocksFun j)) =\n ∑ e ∈ compPartialSumSource 2 (n + 1) n, ∏ j, r * (a ^ e.snd j * p (e.snd j))",
"ppTerm": "?m.321",
"assigned": tr... | [
"n : ℕ\np : ℕ → ℝ\nhp : ∀ (k : ℕ), 0 ≤ p k\nr a : ℝ\nhr : 0 ≤ r\nha : 0 ≤ a\n⊢ ∑ e ∈ compPartialSumSource 2 (n + 1) n, ∏ j, r * (a ^ e.snd j * p (e.snd j)) =\n ∑ d ∈ compPartialSumTarget 2 (n + 1) n, ∏ j, r * (a ^ d.snd.blocksFun j * p (d.snd.blocksFun j))"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.Calculus.FDeriv.Linear | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 16
} | {
"line": 132,
"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\nx : E\ns : Set E\nh : IsBoundedLinearMap 𝕜 f\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderiv 𝕜 f x =... | [] | exact h.fderiv | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.Constructions | {
"line": 280,
"column": 8
} | {
"line": 280,
"column": 18
} | {
"line": 280,
"column": 18
} | [
{
"pp": "case a\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\nG : Type u_5\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\np : FormalMultilinearSeries ... | [
"case a\n𝕜 : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\nG : Type u_5\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\np : FormalMultilinearSeries 𝕜 E F\nq : ... | lt_min_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Constructions | {
"line": 1260,
"column": 4
} | {
"line": 1264,
"column": 69
} | {
"line": 1266,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → E →L[𝕜] F\npf : FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)\ns : Set E\nx : E\nr : ℝ≥0∞\nz : F\nhf : ... | [] | intro y hy h'y
apply HasSum.zero_add
simp only [FormalMultilinearSeries.unshift, Nat.succ_eq_add_one,
continuousMultilinearCurryRightEquiv_symm_apply', add_sub_cancel_left]
exact (ContinuousLinearMap.apply 𝕜 F y).hasSum (hf.hasSum hy h'y) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Constructions | {
"line": 1260,
"column": 4
} | {
"line": 1264,
"column": 69
} | {
"line": 1266,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → E →L[𝕜] F\npf : FormalMultilinearSeries 𝕜 E (E →L[𝕜] F)\ns : Set E\nx : E\nr : ℝ≥0∞\nz : F\nhf : ... | [] | intro y hy h'y
apply HasSum.zero_add
simp only [FormalMultilinearSeries.unshift, Nat.succ_eq_add_one,
continuousMultilinearCurryRightEquiv_symm_apply', add_sub_cancel_left]
exact (ContinuousLinearMap.apply 𝕜 F y).hasSum (hf.hasSum hy h'y) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 864,
"column": 53
} | {
"line": 865,
"column": 64
} | {
"line": 867,
"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\nx : E\ns : Set E\na : E\n⊢ DifferentiableWithinAt 𝕜 (fun x ↦ f (a + x)) s x ↔ DifferentiableWith... | [] | by
simp [DifferentiableWithinAt, hasFDerivWithinAt_comp_add_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 87
} | {
"line": 359,
"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\nf' : E →L[𝕜] F\nx : E\nc : F\nh : HasFDerivAt f f' x\nhf' : ∃ C, AntilipschitzWith C ⇑f'\n⊢ ∀ᶠ (... | [] | simpa only [compl_eq_univ_sdiff] using (hasFDerivWithinAt_univ.2 h).eventually_ne hf' | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 87
} | {
"line": 359,
"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\nf' : E →L[𝕜] F\nx : E\nc : F\nh : HasFDerivAt f f' x\nhf' : ∃ C, AntilipschitzWith C ⇑f'\n⊢ ∀ᶠ (... | [] | simpa only [compl_eq_univ_sdiff] using (hasFDerivWithinAt_univ.2 h).eventually_ne hf' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 87
} | {
"line": 359,
"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\nf' : E →L[𝕜] F\nx : E\nc : F\nh : HasFDerivAt f f' x\nhf' : ∃ C, AntilipschitzWith C ⇑f'\n⊢ ∀ᶠ (... | [] | simpa only [compl_eq_univ_sdiff] using (hasFDerivWithinAt_univ.2 h).eventually_ne hf' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 379,
"column": 2
} | {
"line": 379,
"column": 6
} | {
"line": 380,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nf' : E →L[ℝ] F\nx : E\nL : Filter E\n⊢ Tendsto (fun x' ↦ ‖x' - x‖⁻¹ * ‖f x' - f x - f' (x' - x)‖) L (𝓝 0) ↔\n Tendsto (fun x' ↦ ‖x' - x‖⁻¹ • (f x' ... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nf' : E →L[ℝ] F\nx : E\nL : Filter E\n⊢ Tendsto (fun x' ↦ ‖x' - x‖⁻¹ • (f x' - f x - f' (x' - x))) L (𝓝 0) ↔\n Tendsto (fun x' ↦ ‖x' - x‖⁻¹ * ‖f x' - f x - f' (... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 273,
"column": 4
} | {
"line": 274,
"column": 70
} | {
"line": 275,
"column": 2
} | [
{
"pp": "case mp\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\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\n⊢ HasFTaylorSeriesUpToOn (↑n + 1) ... | [] | exact fun h ↦ ⟨h.of_le (mod_cast Nat.le_succ n),
h.fderivWithin _ (mod_cast lt_add_one n), h.cont (n + 1) le_rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 273,
"column": 4
} | {
"line": 274,
"column": 70
} | {
"line": 275,
"column": 2
} | [
{
"pp": "case mp\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\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\n⊢ HasFTaylorSeriesUpToOn (↑n + 1) ... | [] | exact fun h ↦ ⟨h.of_le (mod_cast Nat.le_succ n),
h.fderivWithin _ (mod_cast lt_add_one n), h.cont (n + 1) le_rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 273,
"column": 4
} | {
"line": 274,
"column": 70
} | {
"line": 275,
"column": 2
} | [
{
"pp": "case mp\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\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\n⊢ HasFTaylorSeriesUpToOn (↑n + 1) ... | [] | exact fun h ↦ ⟨h.of_le (mod_cast Nat.le_succ n),
h.fderivWithin _ (mod_cast lt_add_one n), h.cont (n + 1) le_rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 278,
"column": 6
} | {
"line": 278,
"column": 16
} | {
"line": 279,
"column": 6
} | [
{
"pp": "case fderivWithin\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\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nh :\n HasFTaylorSeriesU... | [
"case fderivWithin\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\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nh :\n HasFTaylorSeriesUpToOn (↑n) f... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 284,
"column": 6
} | {
"line": 284,
"column": 16
} | {
"line": 285,
"column": 6
} | [
{
"pp": "case mpr.cont\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\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nh :\n HasFTaylorSeriesUpToO... | [
"case mpr.cont\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\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nh :\n HasFTaylorSeriesUpToOn (↑n) f p s... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 540,
"column": 4
} | {
"line": 540,
"column": 18
} | {
"line": 542,
"column": 0
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nn : ℕ\nf : E → F\nx : E\nh : HasFiniteFPowerSeriesOnBall... | [] | exact h.fderiv | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 610,
"column": 4
} | {
"line": 610,
"column": 14
} | {
"line": 611,
"column": 4
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E... | [
"case inr\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\nx : (i :... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 233,
"column": 71
} | {
"line": 236,
"column": 57
} | {
"line": 238,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nx : 𝕜\ns : Set 𝕜\nι : Type u_1\nu : Finset ι\nA : ι → 𝕜 → F\nh : ∀ i ∈ u, DifferentiableWithinAt 𝕜 (A i) s x\n⊢ derivWithin (∑ i ∈ u, A i) s x = ∑ i ∈ u, derivWithin (A i) s x",
... | [] | by
by_cases hsx : UniqueDiffWithinAt 𝕜 s x
· exact (HasDerivWithinAt.sum fun i hi ↦ (h i hi).hasDerivWithinAt).derivWithin hsx
· simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 166,
"column": 2
} | {
"line": 198,
"column": 25
} | {
"line": 200,
"column": 0
} | [
{
"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 F\nf : E → F\nU : Set E\nhf : AnalyticOnNhd 𝕜 f U\nhU : IsPreconnected U\nz₀ : E\nh₀... | [] | let u := {x | f =ᶠ[𝓝 x] 0}
suffices main : closure u ∩ U ⊆ u by
have Uu : U ⊆ u :=
hU.subset_of_closure_inter_subset isOpen_setOf_eventually_nhds ⟨z₀, h₀, hfz₀⟩ main
intro z hz
simpa using mem_of_mem_nhds (Uu hz)
/- Take a limit point `x`, then a ball `B (x, r)` on which it has a power series exp... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 166,
"column": 2
} | {
"line": 198,
"column": 25
} | {
"line": 200,
"column": 0
} | [
{
"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 F\nf : E → F\nU : Set E\nhf : AnalyticOnNhd 𝕜 f U\nhU : IsPreconnected U\nz₀ : E\nh₀... | [] | let u := {x | f =ᶠ[𝓝 x] 0}
suffices main : closure u ∩ U ⊆ u by
have Uu : U ⊆ u :=
hU.subset_of_closure_inter_subset isOpen_setOf_eventually_nhds ⟨z₀, h₀, hfz₀⟩ main
intro z hz
simpa using mem_of_mem_nhds (Uu hz)
/- Take a limit point `x`, then a ball `B (x, r)` on which it has a power series exp... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 138,
"column": 78
} | {
"line": 141,
"column": 57
} | {
"line": 143,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nx : 𝕜\ns : Set 𝕜\n𝕜' : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nt' : Set 𝕜'\nh : ... | [] | by
by_cases hsx : UniqueDiffWithinAt 𝕜 s x
· exact (HasDerivWithinAt.scomp x hg.hasDerivWithinAt hh.hasDerivWithinAt hs).derivWithin hsx
· simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.LogDeriv | {
"line": 133,
"column": 8
} | {
"line": 135,
"column": 44
} | {
"line": 136,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NontriviallyNormedField 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : IsRCLikeNormedField 𝕜\nf g : 𝕜 → 𝕜'\ns : Set 𝕜\nhf : DifferentiableOn 𝕜 f s\nhg : DifferentiableOn 𝕜 g s\nhs2 : IsOpen[PseudoMetricSpace.toUniformSpace.... | [] | simp only [Pi.sub_apply, Pi.mul_apply, Pi.inv_apply, Pi.div_apply, Pi.pow_apply,
Pi.zero_apply]
grind [logDeriv_apply, Pi.div_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.LogDeriv | {
"line": 133,
"column": 8
} | {
"line": 135,
"column": 44
} | {
"line": 136,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NontriviallyNormedField 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : IsRCLikeNormedField 𝕜\nf g : 𝕜 → 𝕜'\ns : Set 𝕜\nhf : DifferentiableOn 𝕜 f s\nhg : DifferentiableOn 𝕜 g s\nhs2 : IsOpen[PseudoMetricSpace.toUniformSpace.... | [] | simp only [Pi.sub_apply, Pi.mul_apply, Pi.inv_apply, Pi.div_apply, Pi.pow_apply,
Pi.zero_apply]
grind [logDeriv_apply, Pi.div_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 14
} | {
"line": 223,
"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\ns : Set E\nf : E → F\nx : E\nn✝ : ℕ∞... | [
"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\nf : E → F\nx : E\nn✝ : ℕ∞ω\ng : F →L[... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 68
} | {
"line": 289,
"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\nx : E\ng : F ≃L[𝕜] G\ni ... | [] | apply g.iteratedFDerivWithin_comp_left f uniqueDiffOn_univ trivial | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 14
} | {
"line": 380,
"column": 4
} | [
{
"pp": "case cont\n𝕜 : 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\nf : E → F\nn :... | [
"case cont\n𝕜 : 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\nf : E → F\nn : ℕ∞ω\np : E ... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 14
} | {
"line": 111,
"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... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 408,
"column": 4
} | {
"line": 408,
"column": 14
} | {
"line": 409,
"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\ns : Set E\nf : E → F\nn✝ : ℕ∞ω\nx : ... | [
"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\nf : E → F\nn✝ : ℕ∞ω\nx : G\ng : G →L[... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 534,
"column": 4
} | {
"line": 534,
"column": 14
} | {
"line": 535,
"column": 4
} | [
{
"pp": "case cont\n𝕜 : 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\nf : E → F\np :... | [
"case cont\n𝕜 : 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\nf : E → F\np : E → FormalM... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 552,
"column": 4
} | {
"line": 552,
"column": 14
} | {
"line": 553,
"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... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.Deriv.Inverse | {
"line": 159,
"column": 2
} | {
"line": 160,
"column": 29
} | {
"line": 162,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\nt : Set F\nht : ¬AccPt (f x) (𝓟 t)\nh : ∃ᶠ (y : 𝕜) in 𝓝[≠] x, f y ∈ t\n⊢ deriv f x = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
... | [] | rw [← derivWithin_univ, derivWithin_zero_of_frequently_mem t ht]
rwa [← compl_eq_univ_sdiff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Inverse | {
"line": 159,
"column": 2
} | {
"line": 160,
"column": 29
} | {
"line": 162,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\nt : Set F\nht : ¬AccPt (f x) (𝓟 t)\nh : ∃ᶠ (y : 𝕜) in 𝓝[≠] x, f y ∈ t\n⊢ deriv f x = 0",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
... | [] | rw [← derivWithin_univ, derivWithin_zero_of_frequently_mem t ht]
rwa [← compl_eq_univ_sdiff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 14
} | {
"line": 228,
"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 f₁ : E → F\nx : E\nn✝ : ℕ∞ω\nh₁ : f₁ =ᶠ[𝓝[s] x] f\nhx : f₁ x = f x\nn : ℕ∞\nh : ContDiffWithinAt �... | [
"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 f₁ : E → F\nx : E\nn✝ : ℕ∞ω\nh₁ : f₁ =ᶠ[𝓝[s] x] f\nhx : f₁ x = f x\nn : ℕ∞\nh : ContDiffWithinAt 𝕜 (↑n) f s x... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 14
} | {
"line": 292,
"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\nt : Set E\nhst : s ∈ 𝓝[t] x\nn : ℕ∞\nh : ContDiffWithinAt 𝕜 (↑n) f s x\... | [
"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\nt : Set E\nhst : s ∈ 𝓝[t] x\nn : ℕ∞\nh : ContDiffWithinAt 𝕜 (↑n) f s x\nm : ℕ\nhm :... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 404,
"column": 8
} | {
"line": 404,
"column": 33
} | {
"line": 405,
"column": 8
} | [
{
"pp": "case e'_12\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\nhn : n ≠ ∞\nh'n : n + 1 ≠ ∞\nu : Set E\nhu : u ∈ 𝓝[insert x s... | [
"case e'_12\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\nhn : n ≠ ∞\nh'n : n + 1 ≠ ∞\nu : Set E\nhu : u ∈ 𝓝[insert x s] x\nhf : n ... | rw [← Hp'.zero_eq y hy.1] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 336,
"column": 6
} | {
"line": 337,
"column": 38
} | {
"line": 338,
"column": 4
} | [
{
"pp": "case refine_1\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\... | [] | simp only [mem_range]
exact ⟨0, ⟨0, by simp⟩, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 336,
"column": 6
} | {
"line": 337,
"column": 38
} | {
"line": 338,
"column": 4
} | [
{
"pp": "case refine_1\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\... | [] | simp only [mem_range]
exact ⟨0, ⟨0, by simp⟩, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 457,
"column": 4
} | {
"line": 461,
"column": 33
} | {
"line": 463,
"column": 0
} | [
{
"pp": "case inr\nn : ℕ\nc : OrderedFinpartition n\ni j : Fin c.length\nh : range ((c.extendMiddle i).emb j) = {0}\nhij : j ≠ i\n⊢ False",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Set.mem_range_self",
"of_eq_false",
"instNeZeroNatHAdd_1",
"Function.update",
... | [] | have : (c.extendMiddle i).emb j 0 ∈ range ((c.extendMiddle i).emb j) :=
mem_range_self 0
rw [h] at this
simp only [extendMiddle, hij, ↓reduceDIte, comp_apply, mem_singleton_iff] at this
exact Fin.succ_ne_zero _ this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 457,
"column": 4
} | {
"line": 461,
"column": 33
} | {
"line": 463,
"column": 0
} | [
{
"pp": "case inr\nn : ℕ\nc : OrderedFinpartition n\ni j : Fin c.length\nh : range ((c.extendMiddle i).emb j) = {0}\nhij : j ≠ i\n⊢ False",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Set.mem_range_self",
"of_eq_false",
"instNeZeroNatHAdd_1",
"Function.update",
... | [] | have : (c.extendMiddle i).emb j 0 ∈ range ((c.extendMiddle i).emb j) :=
mem_range_self 0
rw [h] at this
simp only [extendMiddle, hij, ↓reduceDIte, comp_apply, mem_singleton_iff] at this
exact Fin.succ_ne_zero _ this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 671,
"column": 4
} | {
"line": 671,
"column": 14
} | {
"line": 672,
"column": 4
} | [
{
"pp": "case cont\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\nn : ℕ∞ω\nh : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\n⊢ ∀ (m : ℕ), ↑m ≤ n →... | [
"case cont\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\nn : ℕ∞ω\nh : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\nm : ℕ\nhm : ↑m ≤ n\n⊢ ContinuousO... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 552,
"column": 8
} | {
"line": 552,
"column": 43
} | {
"line": 552,
"column": 43
} | [
{
"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... | ← Nat.add_lt_add_iff_right (k := 1) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 783,
"column": 59
} | {
"line": 787,
"column": 29
} | {
"line": 789,
"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\nn : ℕ∞ω\nm : ℕ\nh : ContDiffOn 𝕜 n f s\nhmn : ↑m < n\nhs : UniqueDiffOn 𝕜 s\n⊢ Different... | [] | by
intro x hx
have : (m + 1 : ℕ) ≤ n := ENat.add_one_natCast_le_withTop_of_lt hmn
apply (((h.of_le this).ftaylorSeriesWithin hs).fderivWithin m ?_ x hx).differentiableWithinAt
exact_mod_cast lt_add_one m | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 790,
"column": 8
} | {
"line": 791,
"column": 50
} | {
"line": 791,
"column": 50
} | [
{
"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... | [] | simp only [applyOrderedFinpartition_update_right,
ContinuousMultilinearMap.map_update_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 790,
"column": 8
} | {
"line": 791,
"column": 50
} | {
"line": 791,
"column": 50
} | [
{
"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... | [] | simp only [applyOrderedFinpartition_update_right,
ContinuousMultilinearMap.map_update_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 790,
"column": 8
} | {
"line": 791,
"column": 50
} | {
"line": 791,
"column": 50
} | [
{
"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... | [] | simp only [applyOrderedFinpartition_update_right,
ContinuousMultilinearMap.map_update_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 1105,
"column": 4
} | {
"line": 1105,
"column": 14
} | {
"line": 1106,
"column": 4
} | [
{
"pp": "case cont\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\nq :... | [
"case cont\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\nq : F → FormalM... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.SpecialFunctions.Exponential | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 33
} | {
"line": 276,
"column": 2
} | [
{
"pp": "𝕂 : Type u_1\n𝕊 : Type u_2\n𝔸 : Type u_3\ninst✝⁹ : NontriviallyNormedField 𝕂\ninst✝⁸ : CharZero 𝕂\ninst✝⁷ : NormedCommRing 𝕊\ninst✝⁶ : NormedRing 𝔸\ninst✝⁵ : NormedSpace 𝕂 𝕊\ninst✝⁴ : NormedAlgebra 𝕂 𝔸\ninst✝³ : Algebra 𝕊 𝔸\ninst✝² : ContinuousSMul 𝕊 𝔸\ninst✝¹ : IsScalarTower 𝕂 𝕊 𝔸\ni... | [] | exact tendsto_id.smul_const x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 14
} | {
"line": 75,
"column": 4
} | [
{
"pp": "case refine_4\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ns : Set E\nι : Type u_3\ninst✝² : Fintype ι\nF' : ι → Type u_5\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → ... | [
"case refine_4\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ns : Set E\nι : Type u_3\ninst✝² : Fintype ι\nF' : ι → Type u_5\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np'... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.ExtendFrom | {
"line": 69,
"column": 4
} | {
"line": 71,
"column": 49
} | {
"line": 72,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : RegularSpace Y\nf : X → Y\nA B : Set X\nhB : B ⊆ closure[inst✝²] A\nhf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)\nφ : X → Y := extendFrom A f\nx : X\nx_in : x ∈ B\nV' : Set Y\nV'_in : V' ∈ 𝓝 (φ x)\nV'_closed... | [] | have := tendsto_extendFrom (hf x x_in)
rcases (nhdsWithin_basis_open x A).tendsto_left_iff.mp this V' V'_in with ⟨V, ⟨hxV, V_op⟩, hV⟩
exact ⟨V, IsOpen.mem_nhds V_op hxV, V_op, hV⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ExtendFrom | {
"line": 69,
"column": 4
} | {
"line": 71,
"column": 49
} | {
"line": 72,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : RegularSpace Y\nf : X → Y\nA B : Set X\nhB : B ⊆ closure[inst✝²] A\nhf : ∀ x ∈ B, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)\nφ : X → Y := extendFrom A f\nx : X\nx_in : x ∈ B\nV' : Set Y\nV'_in : V' ∈ 𝓝 (φ x)\nV'_closed... | [] | have := tendsto_extendFrom (hf x x_in)
rcases (nhdsWithin_basis_open x A).tendsto_left_iff.mp this V' V'_in with ⟨V, ⟨hxV, V_op⟩, hV⟩
exact ⟨V, IsOpen.mem_nhds V_op hxV, V_op, hV⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 14
} | {
"line": 101,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ns : Set E\nx : E\nn✝ : ℕ∞ω\nι : Type u_3\ninst✝² : Fintype ι\nF' : ι → Type u_5\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nΦ : E → (i :... | [
"𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ns : Set E\nx : E\nn✝ : ℕ∞ω\nι : Type u_3\ninst✝² : Fintype ι\nF' : ι → Type u_5\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nΦ : E → (i : ι) → F' i\n... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 37
} | {
"line": 421,
"column": 2
} | [
{
"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\nι : Type u_3\nf : ι → E → F\nu : Finset ι\nn : ℕ\nx : E\nh : ∀ j ∈ u, ContDiffAt 𝕜 (↑n) (f j) x\n⊢ iteratedFD... | [
"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\nι : Type u_3\nf : ι → E → F\nu : Finset ι\nn : ℕ\nx : E\nh : ∀ j ∈ u, ContDiffAt 𝕜 (↑n) (f j) x\nx✝ : E\n⊢ ∑ j ∈ u, f j x... | convert! iteratedFDeriv_sum_apply h | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 1045,
"column": 4
} | {
"line": 1045,
"column": 14
} | {
"line": 1046,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn✝ : ℕ∞ω\n𝕜' : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜'\ninst✝⁴ : Normed... | [
"𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn✝ : ℕ∞ω\n𝕜' : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜'\ninst✝⁴ : NormedAlgebra 𝕜 �... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 532,
"column": 6
} | {
"line": 532,
"column": 31
} | {
"line": 532,
"column": 32
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\ns : Set E\nx y : E\nf' : E → StrongDual ℝ E\nhf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x\nhs : Convex ℝ s\nxs : x ∈ s\nys : y ∈ s\ng : ℝ → E := fun t ↦ (AffineMap.lineMap x y) t\nI : Set ℝ := Icc 0 1\nhsub : Ioo 0 1 ⊆ I\n... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\ns : Set E\nx y : E\nf' : E → StrongDual ℝ E\nhf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x\nhs : Convex ℝ s\nxs : x ∈ s\nys : y ∈ s\ng : ℝ → E := fun t ↦ (AffineMap.lineMap x y) t\nI : Set ℝ := Icc 0 1\nhsub : Ioo 0 1 ⊆ I\nhmaps : Maps... | segment_eq_image_lineMap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 208,
"column": 29
} | {
"line": 209,
"column": 89
} | {
"line": 210,
"column": 6
} | [
{
"pp": "α : 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≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : ∫... | [] | by
simpa only [← ENNReal.coe_lt_coe, add_zero] using add_lt_add_right (wpos x) (f x) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 187,
"column": 6
} | {
"line": 193,
"column": 79
} | {
"line": 194,
"column": 4
} | [
{
"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\nc : 𝕜\nhc : 1 < ‖c‖\nr ε : ℝ\nhε : 0 < ε\nhr : 0 < r\nx : E\nL₁ L₂ : E →L[𝕜] F\nh₁ : x ∈ A f L₁... | [] | apply add_le_add
· apply le_of_mem_A h₂
· simp only [le_of_lt (half_pos hr), mem_closedBall, dist_self]
· simp only [dist_eq_norm, add_sub_cancel_left, mem_closedBall, ylt.le]
· apply le_of_mem_A h₁
· simp only [le_of_lt (half_pos hr), mem_closedBall, dist_self]
· simp only [... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 187,
"column": 6
} | {
"line": 193,
"column": 79
} | {
"line": 194,
"column": 4
} | [
{
"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\nc : 𝕜\nhc : 1 < ‖c‖\nr ε : ℝ\nhε : 0 < ε\nhr : 0 < r\nx : E\nL₁ L₂ : E →L[𝕜] F\nh₁ : x ∈ A f L₁... | [] | apply add_le_add
· apply le_of_mem_A h₂
· simp only [le_of_lt (half_pos hr), mem_closedBall, dist_self]
· simp only [dist_eq_norm, add_sub_cancel_left, mem_closedBall, ylt.le]
· apply le_of_mem_A h₁
· simp only [le_of_lt (half_pos hr), mem_closedBall, dist_self]
· simp only [... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 390,
"column": 4
} | {
"line": 390,
"column": 14
} | {
"line": 391,
"column": 4
} | [
{
"pp": "x : ℝ\nh : |x| < 1\nterm : ℕ → ℝ := fun n ↦ -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)\nh_term_eq_goal : (term ∘ fun x ↦ 2 * x) = fun k ↦ 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)\n⊢ ∀ x ∉ Set.range fun x ↦ 2 * x, term x = 0",
"ppTerm": "?m.249",
"assigned": true,
"usedConstan... | [
"x : ℝ\nh : |x| < 1\nterm : ℕ → ℝ := fun n ↦ -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)\nh_term_eq_goal : (term ∘ fun x ↦ 2 * x) = fun k ↦ 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)\nm : ℕ\nhm : m ∉ Set.range fun x ↦ 2 * x\n⊢ term m = 0"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 391,
"column": 23
} | {
"line": 391,
"column": 40
} | {
"line": 391,
"column": 41
} | [
{
"pp": "x : ℝ\nh : |x| < 1\nterm : ℕ → ℝ := fun n ↦ -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)\nh_term_eq_goal : (term ∘ fun x ↦ 2 * x) = fun k ↦ 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)\nm : ℕ\nhm : m ∉ {a | Even a}\n⊢ term m = 0",
"ppTerm": "?m.391",
"assigned": true,
"usedConstant... | [
"x : ℝ\nh : |x| < 1\nterm : ℕ → ℝ := fun n ↦ -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)\nh_term_eq_goal : (term ∘ fun x ↦ 2 * x) = fun k ↦ 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)\nm : ℕ\nhm : ¬Even m\n⊢ term m = 0"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Gauge | {
"line": 82,
"column": 10
} | {
"line": 82,
"column": 60
} | {
"line": 84,
"column": 0
} | [
{
"pp": "case ht\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns t : Set E\nhs : Absorbent ℝ s\nh : s ⊆ t\nx✝ : E\n⊢ BddBelow {r | 0 < r ∧ x✝ ∈ r • t}",
"ppTerm": "?ht",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.Convex.Gauge.0.bddBelow_gauge_set"
],
... | [] | exacts [bddBelow_gauge_set, hs.gauge_set_nonempty] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 339,
"column": 4
} | {
"line": 346,
"column": 29
} | {
"line": 347,
"column": 2
} | [
{
"pp": "case neg\n𝕜 : 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\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1... | [] | calc
‖f (x + y) - f x - f' y‖ = ‖f (x + y) - f x - L e (n e) m y + (L e (n e) m - f') y‖ :=
congr_arg _ (by simp)
_ ≤ 4 * (1 / 2) ^ e * ‖y‖ + 12 * ‖c‖ * (1 / 2) ^ e * ‖y‖ :=
norm_add_le_of_le J2 <| (le_opNorm _ _).trans <| by gcongr; exact Lf' _ _ m_ge
_ = (4 + 12 * ‖c‖) * ‖y‖ * (1 / 2... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 445,
"column": 8
} | {
"line": 445,
"column": 24
} | {
"line": 445,
"column": 24
} | [
{
"pp": "α : 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\nδ : ℝ≥0 := NNReal.mk (ε / 2) ⋯\nδpos : 0 < δ\nfp : α → ℝ≥0 := fun x ↦ (f x).toNNReal\nint_fp... | [
"α : 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\nδ : ℝ≥0 := NNReal.mk (ε / 2) ⋯\nδpos : 0 < δ\nfp : α → ℝ≥0 := fun x ↦ (f x).toNNReal\nint_fp : Integrabl... | EReal.toReal_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Gauge | {
"line": 442,
"column": 2
} | {
"line": 444,
"column": 72
} | {
"line": 445,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ns : Set E\nx : E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\nhc : Convex ℝ s\nhs₀ : s ∈ 𝓝 0\nha : Absorbent ℝ s\nε : ℝ\nhε₀ : 0 < ε\n⊢ ∀ᶠ (a : E) in 𝓝 0, gauge s (x + a) ∈ Icc (gauge s x - ε) (... | [
"E : Type u_2\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ns : Set E\nx : E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\nhc : Convex ℝ s\nhs₀ : s ∈ 𝓝 0\nha : Absorbent ℝ s\nε : ℝ\nhε₀ : 0 < ε\nthis : ε • s ∩ -(ε • s) ∈ 𝓝 0\n⊢ ∀ᶠ (a : E) in 𝓝 0, gauge s (x + a) ∈ I... | have : ε • s ∩ -(ε • s) ∈ 𝓝 0 :=
inter_mem ((set_smul_mem_nhds_zero_iff hε₀.ne').2 hs₀)
(neg_mem_nhds_zero _ ((set_smul_mem_nhds_zero_iff hε₀.ne').2 hs₀)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 81
} | {
"line": 294,
"column": 2
} | [
{
"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\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\nht₁ : Convex ℝ t... | [
"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\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\nht₁ : Convex ℝ t\nht₃ : IsOp... | obtain ⟨f, u, h⟩ := _root_.geometric_hahn_banach_open_open hs₁ hs₂ ht₁ ht₃ disj | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 252,
"column": 2
} | {
"line": 293,
"column": 52
} | {
"line": 295,
"column": 0
} | [
{
"pp": "X : Type u_1\nE : Type u_3\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → ℝ≥0\nf_meas : Measurable f\ng : X → E\n⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ",
"ppTerm": "?m.36",
"assigned": true,
"us... | [] | by_cases hE : CompleteSpace E; swap; · simp [integral, hE]
by_cases hg : Integrable g (μ.withDensity fun x => f x); swap
· rw [integral_undef hg, integral_undef]
rwa [← integrable_withDensity_iff_integrable_smul f_meas]
refine Integrable.induction
(P := fun g => ∫ x, g x ∂μ.withDensity (fun x => f x) = ∫ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 252,
"column": 2
} | {
"line": 293,
"column": 52
} | {
"line": 295,
"column": 0
} | [
{
"pp": "X : Type u_1\nE : Type u_3\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → ℝ≥0\nf_meas : Measurable f\ng : X → E\n⊢ (∫ (x : X), g x ∂μ.withDensity fun x ↦ ↑(f x)) = ∫ (x : X), f x • g x ∂μ",
"ppTerm": "?m.36",
"assigned": true,
"us... | [] | by_cases hE : CompleteSpace E; swap; · simp [integral, hE]
by_cases hg : Integrable g (μ.withDensity fun x => f x); swap
· rw [integral_undef hg, integral_undef]
rwa [← integrable_withDensity_iff_integrable_smul f_meas]
refine Integrable.induction
(P := fun g => ∫ x, g x ∂μ.withDensity (fun x => f x) = ∫ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 319,
"column": 8
} | {
"line": 320,
"column": 72
} | {
"line": 321,
"column": 4
} | [
{
"pp": "case h₂\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b₀ b₁ b₂ : ℝ\nμ : Measure ℝ\nf : ℝ → E\nhb₀ : μ {b₀} = 0\nh_int : IntervalIntegrable f μ (min a b₁) (max a b₂)\nh₀ : b₀ ∈ Icc b₁ b₂\nh₁₂ : b₁ ≤ b₂\nmin₁₂ : min b₁ b₂ = b₁\nh_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegr... | [] | exact ⟨min_le_of_left_le (min_le_right _ _),
le_max_of_le_right (h₁.trans <| h₂.trans (le_max_right a b₂))⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 319,
"column": 8
} | {
"line": 320,
"column": 72
} | {
"line": 321,
"column": 4
} | [
{
"pp": "case h₂\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b₀ b₁ b₂ : ℝ\nμ : Measure ℝ\nf : ℝ → E\nhb₀ : μ {b₀} = 0\nh_int : IntervalIntegrable f μ (min a b₁) (max a b₂)\nh₀ : b₀ ∈ Icc b₁ b₂\nh₁₂ : b₁ ≤ b₂\nmin₁₂ : min b₁ b₂ = b₁\nh_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegr... | [] | exact ⟨min_le_of_left_le (min_le_right _ _),
le_max_of_le_right (h₁.trans <| h₂.trans (le_max_right a b₂))⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 319,
"column": 8
} | {
"line": 320,
"column": 72
} | {
"line": 321,
"column": 4
} | [
{
"pp": "case h₂\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b₀ b₁ b₂ : ℝ\nμ : Measure ℝ\nf : ℝ → E\nhb₀ : μ {b₀} = 0\nh_int : IntervalIntegrable f μ (min a b₁) (max a b₂)\nh₀ : b₀ ∈ Icc b₁ b₂\nh₁₂ : b₁ ≤ b₂\nmin₁₂ : min b₁ b₂ = b₁\nh_int' : ∀ {x : ℝ}, x ∈ Icc b₁ b₂ → IntervalIntegr... | [] | exact ⟨min_le_of_left_le (min_le_right _ _),
le_max_of_le_right (h₁.trans <| h₂.trans (le_max_right a b₂))⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 94,
"column": 63
} | {
"line": 95,
"column": 82
} | {
"line": 97,
"column": 0
} | [
{
"pp": "ε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\ninst✝ : PseudoMetrizableSpace ε\nf : ℝ → ε\na b : ℝ\nμ : Measure ℝ\ng : ℝ → ε\nh : f =ᵐ[μ.restrict (Ι a b)] g\n⊢ IntervalIntegrable f μ a b ↔ IntervalIntegrable g μ a b",
"ppTerm": "?m.21",
"assigned": true,
"usedConsta... | [] | by
rw [intervalIntegrable_iff, integrableOn_congr_fun_ae h, intervalIntegrable_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 417,
"column": 2
} | {
"line": 418,
"column": 27
} | {
"line": 420,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → E\nc : ℝ\nhc : c ≠ 0\nh : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞\n⊢ IntervalIntegrable (fun x ↦ f (c * x)) volume (a / c) (b / c) ↔ IntervalIntegrable f volume a b",
"ppTerm": "?m.57",
"assigned": true,
"usedC... | [] | exact ⟨fun h ↦ by simpa [hc] using h.comp_mul_left (c := c⁻¹) h' (by simp),
(comp_mul_left · h h')⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 417,
"column": 2
} | {
"line": 418,
"column": 27
} | {
"line": 420,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → E\nc : ℝ\nhc : c ≠ 0\nh : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞\n⊢ IntervalIntegrable (fun x ↦ f (c * x)) volume (a / c) (b / c) ↔ IntervalIntegrable f volume a b",
"ppTerm": "?m.57",
"assigned": true,
"usedC... | [] | exact ⟨fun h ↦ by simpa [hc] using h.comp_mul_left (c := c⁻¹) h' (by simp),
(comp_mul_left · h h')⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 417,
"column": 2
} | {
"line": 418,
"column": 27
} | {
"line": 420,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedAddCommGroup E\na b : ℝ\nf : ℝ → E\nc : ℝ\nhc : c ≠ 0\nh : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞\n⊢ IntervalIntegrable (fun x ↦ f (c * x)) volume (a / c) (b / c) ↔ IntervalIntegrable f volume a b",
"ppTerm": "?m.57",
"assigned": true,
"usedC... | [] | exact ⟨fun h ↦ by simpa [hc] using h.comp_mul_left (c := c⁻¹) h' (by simp),
(comp_mul_left · h h')⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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