module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Convex.EGauge | {
"line": 225,
"column": 15
} | {
"line": 225,
"column": 56
} | {
"line": 226,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝⁴ : NormedDivisionRing 𝕜\nE : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nF : Type u_3\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nU : Set E\nV : Set F\nhU : Balanced 𝕜 U\nhV : Balanced 𝕜 V\na : E\nb : F\nr : ℝ≥0∞\nx : 𝕜\nhx : a ∈ x • U\nhxr : ‖x‖ₑ < r\... | [] | exact ⟨y, ⟨hU.smul_mono hle hx, hy⟩, hyr⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 491,
"column": 14
} | {
"line": 492,
"column": 50
} | {
"line": 493,
"column": 12
} | [
{
"pp": "case e_a.succ.succ.h₀.succ.succ\nR : Type u_2\ninst✝¹ : CommRing R\nP : R⟦X⟧\nhP : constantCoeff P = 0\ninst✝ : Invertible ((coeff 1) P)\nn : ℕ\nB : R⟦X⟧\nhB : ∑ i, C (P.substInvFun ↑i) * X ^ ↑i = B\nhB' : constantCoeff B = 0\nk : R\nhk : ⅟((coeff 1) P) * (coeff (n + 1 + 1)) (subst B P) = k\ni j : ℕ\nh... | [] | · rw [← neg_mul, mul_pow, ← pow_mul, mul_comm (_ ^ _)]
simp [mul_assoc, coeff_X_pow_mul'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 631,
"column": 29
} | {
"line": 631,
"column": 41
} | {
"line": 631,
"column": 42
} | [
{
"pp": "R : Type u_2\ninst✝ : CommRing R\nf : R⟦X⟧\ne : Fin 2 →₀ ℕ\n⊢ ∑ᶠ (d : Unit →₀ ℕ), (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (∏ a, (X₀ + X₁) ^ d a) =\n ↑((e 0 + e 1).choose (e 0)) * (coeff (e 0 + e 1)) f",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"Finsupp.i... | [
"R : Type u_2\ninst✝ : CommRing R\nf : R⟦X⟧\ne : Fin 2 →₀ ℕ\n⊢ ∑ᶠ (d : Unit →₀ ℕ), (MvPowerSeries.coeff d) f • (MvPowerSeries.coeff e) (∏ x ∈ {default}, (X₀ + X₁) ^ d x) =\n ↑((e 0 + e 1).choose (e 0)) * (coeff (e 0 + e 1)) f"
] | univ_unique, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Calculus.FDeriv.Const | {
"line": 379,
"column": 21
} | {
"line": 379,
"column": 35
} | {
"line": 379,
"column": 35
} | [
{
"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⊢ x ∉ tsupport f → x ∉ support (fderiv 𝕜 f)",
"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⊢ x ∉ tsupport f → fderiv 𝕜 f x = 0"
] | notMem_support | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Deriv.Support | {
"line": 54,
"column": 21
} | {
"line": 54,
"column": 35
} | {
"line": 54,
"column": 35
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\n⊢ x ∉ tsupport f → x ∉ support (deriv f)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCom... | [
"𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\n⊢ x ∉ tsupport f → deriv f x = 0"
] | notMem_support | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 40
} | {
"line": 178,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module 𝕜 E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : ContinuousAdd E\ninst✝⁶ : ContinuousSMul 𝕜 E\nF : Type u_3\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜 F\ninst✝³ : TopologicalSpace F\ninst✝² : Cont... | [] | exact uniqueDiffWithinAt_univ.eq h₀ h₁ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.Deriv.Basic | {
"line": 639,
"column": 6
} | {
"line": 639,
"column": 30
} | {
"line": 639,
"column": 30
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf f₁ : 𝕜 → F\nx : 𝕜\ns : Set 𝕜\nhs : EqOn f₁ f s\nhx : f₁ x = f x\n⊢ (fderivWithin 𝕜 f₁ s x) 1 = (fderivWithin 𝕜 f s x) 1",
"ppTerm": "?m.20",
"assigned": true,
"usedC... | [
"𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf f₁ : 𝕜 → F\nx : 𝕜\ns : Set 𝕜\nhs : EqOn f₁ f s\nhx : f₁ x = f x\n⊢ (fderivWithin 𝕜 f s x) 1 = (fderivWithin 𝕜 f s x) 1"
] | fderivWithin_congr hs hx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 272,
"column": 6
} | {
"line": 272,
"column": 38
} | {
"line": 272,
"column": 39
} | [
{
"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\nf' : E →L[𝕜] F\nx : E\ns : Set E\n⊢ HasFDerivWithinAt f f' (... | [
"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\nf' : E →L[𝕜] F\nx : E\ns : Set E\n⊢ HasFDerivWithinAt f f' (insert x (s ... | ← hasFDerivWithinAt_insert_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 53
} | {
"line": 349,
"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\nf : E → F\nf' : E →L[𝕜] F\nx : E\ns t : Set E\nh : t ∈ 𝓝[s] x\n⊢ HasFD... | [] | simp [HasFDerivWithinAt, nhdsWithin_restrict'' s h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 53
} | {
"line": 349,
"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\nf : E → F\nf' : E →L[𝕜] F\nx : E\ns t : Set E\nh : t ∈ 𝓝[s] x\n⊢ HasFD... | [] | simp [HasFDerivWithinAt, nhdsWithin_restrict'' s h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 53
} | {
"line": 349,
"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\nf : E → F\nf' : E →L[𝕜] F\nx : E\ns t : Set E\nh : t ∈ 𝓝[s] x\n⊢ HasFD... | [] | simp [HasFDerivWithinAt, nhdsWithin_restrict'' s h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 470,
"column": 2
} | {
"line": 470,
"column": 36
} | {
"line": 471,
"column": 2
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : Module 𝕜 E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : Module 𝕜 F\ninst✝⁴ : AddCommGroup G\ninst✝³ : Topolog... | [
"α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : Module 𝕜 E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : Module 𝕜 F\ninst✝⁴ : AddCommGroup G\ninst✝³ : TopologicalSpace G\... | rintro ⟨U, V⟩ ⟨⟨hU, hUb⟩, hV, hVb⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 493,
"column": 2
} | {
"line": 493,
"column": 36
} | {
"line": 494,
"column": 2
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : Module 𝕜 E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : Module 𝕜 F\ninst✝⁴ : AddCommGroup G\ninst✝³ : Topolog... | [
"α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : Module 𝕜 E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : Module 𝕜 F\ninst✝⁴ : AddCommGroup G\ninst✝³ : TopologicalSpace G\... | rintro ⟨U, V⟩ ⟨⟨hU, hUb⟩, hV, hVb⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 802,
"column": 4
} | {
"line": 802,
"column": 25
} | {
"line": 803,
"column": 2
} | [
{
"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\nL : Filter (E × E)\n⊢ Tendsto (fun x ↦ ‖f x.1 - f x.2 - f' (x.1 - x.2)‖ / ‖x.1 -... | [] | simp [div_eq_inv_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 802,
"column": 4
} | {
"line": 802,
"column": 25
} | {
"line": 803,
"column": 2
} | [
{
"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\nL : Filter (E × E)\n⊢ Tendsto (fun x ↦ ‖f x.1 - f x.2 - f' (x.1 - x.2)‖ / ‖x.1 -... | [] | simp [div_eq_inv_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 802,
"column": 4
} | {
"line": 802,
"column": 25
} | {
"line": 803,
"column": 2
} | [
{
"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\nL : Filter (E × E)\n⊢ Tendsto (fun x ↦ ‖f x.1 - f x.2 - f' (x.1 - x.2)‖ / ‖x.1 -... | [] | simp [div_eq_inv_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.ConvergenceRadius | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 33
} | {
"line": 122,
"column": 4
} | [
{
"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\nr : ℝ≥0\nh : Summable fun n ↦ ‖p n‖ * ↑r ^ n\n⊢ Summable fun n ↦ ‖p 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\nr : ℝ≥0\nh : Summable fun n ↦ ↑‖p n‖₊ * ↑r ^ n\n⊢ Summable fun n ↦ ‖p n‖₊ * r ^ n"
] | simp only [← coe_nnnorm] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.ConvergenceRadius | {
"line": 187,
"column": 25
} | {
"line": 187,
"column": 41
} | {
"line": 187,
"column": 41
} | [
{
"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\nr : ℝ≥0\nh₀ : 0 < r\na✝ : ℝ\nha✝ : a✝ ∈ Ioo (-1) 1\nhp✝ : (fun n ↦ ‖p 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\nr : ℝ≥0\nh₀ : 0 < ↑r\na✝ : ℝ\nha✝ : a✝ ∈ Ioo (-1) 1\nhp✝ : (fun n ↦ ‖p n‖ * ↑r ^ n) ... | ← NNReal.coe_pos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.ConvergenceRadius | {
"line": 353,
"column": 10
} | {
"line": 353,
"column": 26
} | {
"line": 353,
"column": 26
} | [
{
"pp": "case hb\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\np : FormalMultilinearSeries... | [
"case hb\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\np : FormalMultilinearSeries 𝕜 F G\nu :... | ← NNReal.coe_pos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.ConvergenceRadius | {
"line": 432,
"column": 10
} | {
"line": 432,
"column": 26
} | {
"line": 432,
"column": 26
} | [
{
"pp": "case h.inr\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\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nC : ℝ\nh : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C\nn : ℕ\nhr... | [
"case h.inr\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\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0\nC : ℝ\nh : ∀ (n : ℕ), ‖p n‖ * ↑r ^ n ≤ C\nn : ℕ\nhr : 0 < ↑r\n⊢... | ← NNReal.coe_pos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.ChangeOrigin | {
"line": 67,
"column": 31
} | {
"line": 67,
"column": 48
} | {
"line": 67,
"column": 49
} | [
{
"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\nx y : E\nr : ℝ≥0\nk l : ℕ\ns : Finset (Fin (k + l))\nhs : s.card = l\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\nx y : E\nr : ℝ≥0\nk l : ℕ\ns : Finset (Fin (k + l))\nhs : s.card = l\n⊢ k + l - s.ca... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 794,
"column": 2
} | {
"line": 828,
"column": 39
} | {
"line": 830,
"column": 0
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\n⊢ f =O[𝕜; l] g ↔ f =O[l] g",
"ppTerm": "?m.20... | [] | rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc : 1 < ‖c‖₊⟩
constructor
· rw [nhds_basis_ball.isBigOTVS_iff nhds_basis_ball, isBigO_iff]
intro h
rcases h 1 one_pos with ⟨r, hr₀, hr⟩
lift r to ℝ≥0 using hr₀.le
norm_cast at hr₀
refine ⟨(‖c‖₊ / r : ℝ≥0), hr.mono fun x hx ↦ ?_⟩
suffices ‖f ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 794,
"column": 2
} | {
"line": 828,
"column": 39
} | {
"line": 830,
"column": 0
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\n⊢ f =O[𝕜; l] g ↔ f =O[l] g",
"ppTerm": "?m.20... | [] | rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc : 1 < ‖c‖₊⟩
constructor
· rw [nhds_basis_ball.isBigOTVS_iff nhds_basis_ball, isBigO_iff]
intro h
rcases h 1 one_pos with ⟨r, hr₀, hr⟩
lift r to ℝ≥0 using hr₀.le
norm_cast at hr₀
refine ⟨(‖c‖₊ / r : ℝ≥0), hr.mono fun x hx ↦ ?_⟩
suffices ‖f ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.CPolynomialDef | {
"line": 217,
"column": 20
} | {
"line": 218,
"column": 65
} | {
"line": 218,
"column": 65
} | [
{
"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\nx : E\nr : ℝ≥0∞\nn : ℕ\nhf : HasFiniteFPowerSeriesOnBall f p ... | [] | simp only [Finset.mem_range, not_lt] at hN
rw [hf.finite _ (le_trans hm hN), zero_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.CPolynomialDef | {
"line": 217,
"column": 20
} | {
"line": 218,
"column": 65
} | {
"line": 218,
"column": 65
} | [
{
"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\nx : E\nr : ℝ≥0∞\nn : ℕ\nhf : HasFiniteFPowerSeriesOnBall f p ... | [] | simp only [Finset.mem_range, not_lt] at hN
rw [hf.finite _ (le_trans hm hN), zero_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Multilinear.Curry | {
"line": 684,
"column": 6
} | {
"line": 684,
"column": 24
} | {
"line": 685,
"column": 6
} | [
{
"pp": "𝕜 : 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✝⁹ : (i : ... | [
"𝕜 : 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✝⁹ : (i : Fin n.succ) ... | intro inst v j c x | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Analytic.Composition | {
"line": 721,
"column": 4
} | {
"line": 721,
"column": 95
} | {
"line": 722,
"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... | have := Hf.analyticWithinAt.continuousWithinAt_insert.tendsto_nhdsWithin (hs.insert x) this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Analytic.Inverse | {
"line": 409,
"column": 10
} | {
"line": 409,
"column": 41
} | {
"line": 409,
"column": 41
} | [
{
"pp": "case e_5\nn : ℕ\np : ℕ → ℝ\nhp : ∀ (k : ℕ), 0 ≤ p k\nr a : ℝ\nhr : 0 ≤ r\nha : 0 ≤ a\nk : ℕ\nblocksFun : Fin k → ℕ\nH : ⟨k, blocksFun⟩ ∈ compPartialSumSource 2 (n + 1) n\nK : (compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.length = k\nj : Fin k\n⊢ r * (a ^ ⟨k, blocksFun⟩.snd j * p (⟨k, blocksF... | [
"case e_5\nn : ℕ\np : ℕ → ℝ\nhp : ∀ (k : ℕ), 0 ≤ p k\nr a : ℝ\nhr : 0 ≤ r\nha : 0 ≤ a\nk : ℕ\nblocksFun : Fin k → ℕ\nH : ⟨k, blocksFun⟩ ∈ compPartialSumSource 2 (n + 1) n\nK : (compChangeOfVariables 2 (n + 1) n ⟨k, blocksFun⟩ H).snd.length = k\nj : Fin k\n⊢ r * (a ^ ⟨k, blocksFun⟩.snd j * p (⟨k, blocksFun⟩.snd j)) ... | compChangeOfVariables_blocksFun | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Inverse | {
"line": 460,
"column": 6
} | {
"line": 460,
"column": 35
} | {
"line": 461,
"column": 6
} | [
{
"pp": "case hbc\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\nx : E\nn : ℕ\nhn : 2 ≤ n + 1\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nr a C : ℝ\nhr : ... | [
"case hbc\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\nx : E\nn : ℕ\nhn : 2 ≤ n + 1\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nr a C : ℝ\nhr : 0 ≤ r\nha : ... | apply (norm_sum_le _ _).trans | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Analytic.Composition | {
"line": 1223,
"column": 4
} | {
"line": 1223,
"column": 36
} | {
"line": 1224,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCom... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup H\nin... | rw [sigma_pi_composition_eq_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Within | {
"line": 83,
"column": 8
} | {
"line": 83,
"column": 35
} | {
"line": 83,
"column": 36
} | [
{
"pp": "case a\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\ns : Set E\nh : ∀ x ∈ s, ∃ u, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ... | [] | · simp only [mem_eball, yr] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Analytic.Inverse | {
"line": 576,
"column": 2
} | {
"line": 577,
"column": 72
} | {
"line": 578,
"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\nf : E → G\nq : FormalMultilinearSeri... | [
"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\nf : E → G\nq : FormalMultilinearSeries 𝕜 F G\np... | have : Metric.eball (0 : E) r ∈ 𝓝 0 :=
Metric.eball_mem_nhds 0 (lt_min h0.r_pos (by exact_mod_cast r1_pos)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Analytic.Constructions | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 18
} | {
"line": 312,
"column": 0
} | [
{
"pp": "𝕜 : 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\ne : E\nf : E → F\ng : E → G\nr s : ℝ... | [] | exact hf.prod hg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.FDeriv.Bilinear | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 63
} | {
"line": 97,
"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\nb : E × F → G\nu : Set (E × F)\nh : ... | [
"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\nb : E × F → G\nu : Set (E × F)\nh : IsBoundedBil... | rw [DifferentiableAt.fderivWithin (h.differentiableAt p) hxs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 74
} | {
"line": 131,
"column": 2
} | [
{
"pp": "case pos\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\niso : E ≃L[𝕜] F\nf : G → ... | [] | rw [fderiv_comp_fderivWithin x iso.differentiableAt h hxs, iso.fderiv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 74
} | {
"line": 131,
"column": 2
} | [
{
"pp": "case pos\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\niso : E ≃L[𝕜] F\nf : G → ... | [] | rw [fderiv_comp_fderivWithin x iso.differentiableAt h hxs, iso.fderiv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 74
} | {
"line": 131,
"column": 2
} | [
{
"pp": "case pos\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\niso : E ≃L[𝕜] F\nf : G → ... | [] | rw [fderiv_comp_fderivWithin x iso.differentiableAt h hxs, iso.fderiv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 887,
"column": 2
} | {
"line": 887,
"column": 67
} | {
"line": 889,
"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 a : E\n⊢ HasFDerivAt (fun x ↦ f (x + a)) f' x ↔ HasFDerivAt f f' (x + a)",
... | [] | simp [← hasFDerivWithinAt_univ, hasFDerivWithinAt_comp_add_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 887,
"column": 2
} | {
"line": 887,
"column": 67
} | {
"line": 889,
"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 a : E\n⊢ HasFDerivAt (fun x ↦ f (x + a)) f' x ↔ HasFDerivAt f f' (x + a)",
... | [] | simp [← hasFDerivWithinAt_univ, hasFDerivWithinAt_comp_add_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 887,
"column": 2
} | {
"line": 887,
"column": 67
} | {
"line": 889,
"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 a : E\n⊢ HasFDerivAt (fun x ↦ f (x + a)) f' x ↔ HasFDerivAt f f' (x + a)",
... | [] | simp [← hasFDerivWithinAt_univ, hasFDerivWithinAt_comp_add_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 43
} | {
"line": 321,
"column": 0
} | [
{
"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\niso : E ≃ₗᵢ[𝕜] F\nf : G → E\nx : G\... | [] | exact LinearIsometryEquiv.comp_fderiv iso | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 492,
"column": 2
} | {
"line": 493,
"column": 93
} | {
"line": 495,
"column": 0
} | [
{
"pp": "case inr\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\ns : Set E\nx : E\nc : 𝕜\nhc : c ≠ 0\n⊢ fderivWithin 𝕜 (fun x ↦ f (c • x)) s x = fderi... | [] | · classical
simp only [fderivWithin, DifferentiableWithinAt, hasFDerivWithinAt_comp_smul_iff_smul hc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Module.Alternating.Basic | {
"line": 469,
"column": 4
} | {
"line": 482,
"column": 25
} | {
"line": 484,
"column": 0
} | [
{
"pp": "𝕜 : Type u\nn : ℕ\nE : Type wE\nF : Type wF\nG : Type wG\nι : Type v\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : SeminormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : ... | [] | intro dg v a b heq hne
trans ∑ i, f fun j ↦ Function.update (fun _ ↦ g) i dg j (v j)
· simp
· rw [← Finset.sum_add_sum_compl {a, b}, Finset.sum_pair hne, Finset.sum_eq_zero, add_zero]
· convert! f.map_add_swap _ hne with i
rcases eq_or_ne i a with rfl | hia
· simp [heq, hne, hne.symm]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Alternating.Basic | {
"line": 469,
"column": 4
} | {
"line": 482,
"column": 25
} | {
"line": 484,
"column": 0
} | [
{
"pp": "𝕜 : Type u\nn : ℕ\nE : Type wE\nF : Type wF\nG : Type wG\nι : Type v\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : SeminormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : ... | [] | intro dg v a b heq hne
trans ∑ i, f fun j ↦ Function.update (fun _ ↦ g) i dg j (v j)
· simp
· rw [← Finset.sum_add_sum_compl {a, b}, Finset.sum_pair hne, Finset.sum_eq_zero, add_zero]
· convert! f.map_add_swap _ hne with i
rcases eq_or_ne i a with rfl | hia
· simp [heq, hne, hne.symm]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 14
} | {
"line": 160,
"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 f₁ : E → F\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpToOn n f p s\nh₁... | [
"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\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpToOn n f p s\nh₁ : ∀ x ∈ s, ... | rw [h₁ x hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 348,
"column": 14
} | {
"line": 348,
"column": 39
} | {
"line": 349,
"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\nHzero_eq : ∀ x ∈ s, (p x 0).... | [
"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\nHzero_eq : ∀ x ∈ s, (p x 0).curry0 = f x... | (hm : (m : ℕ∞ω) ≤ n.succ) | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.typeAscription |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 39
} | {
"line": 345,
"column": 4
} | [
{
"pp": "case neg.inl\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝕜' : Type u_2\ninst✝¹ : NormedDivisionRing 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nu : 𝕜 → 𝕜'\nhu : ¬DifferentiableAt 𝕜 u x\n⊢ deriv (fun y ↦ u y * 0) x = 0",
"ppTerm": "?neg.inl✝",
"assigned": true,
"usedConstants": [
... | [
"case neg.inr\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝕜' : Type u_2\ninst✝¹ : NormedDivisionRing 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nu : 𝕜 → 𝕜'\nv : 𝕜'\nhu : ¬DifferentiableAt 𝕜 u x\nhd : v ≠ 0\n⊢ deriv (fun y ↦ u y * v) x = 0"
] | · simp only [mul_zero, deriv_const] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 362,
"column": 4
} | {
"line": 362,
"column": 36
} | {
"line": 363,
"column": 4
} | [
{
"pp": "case hf'\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∞\nf : E → F\nx : E\ns : Set E\nhr : HasFPowerSeriesWithinO... | [
"case hf'\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∞\nf : E → F\nx : E\ns : Set E\nhr : HasFPowerSeriesWithinOnBall f p s ... | rw [insert_eq_of_mem hx] at hy ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 544,
"column": 38
} | {
"line": 547,
"column": 32
} | {
"line": 549,
"column": 0
} | [
{
"pp": "𝕜 : 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\nf : E → F\ns : Set E\nh : CPolynomialOn 𝕜 f s\n⊢ CPolynomialOn 𝕜 (_root_.fderiv 𝕜 f) s",
"ppTerm": "?m.39... | [] | by
intro y hy
rcases h y hy with ⟨p, r, n, hp⟩
exact hp.fderiv'.cpolynomialAt | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 722,
"column": 2
} | {
"line": 722,
"column": 24
} | {
"line": 724,
"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\np : E → FormalMultilinearSeries 𝕜 E F\nH : HasFTaylorSeriesUpTo n f p\n⊢ ∀ (m : ℕ... | [] | · simpa using H.fderiv | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 621,
"column": 47
} | {
"line": 621,
"column": 64
} | {
"line": 621,
"column": 65
} | [
{
"pp": "𝕜 : 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... | [
"𝕜 : 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 : ι) → E i\... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 1007,
"column": 2
} | {
"line": 1007,
"column": 9
} | {
"line": 1008,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\ns : Set 𝕜\nn : ℕ\nc a : 𝕜\nthis : (fun z ↦ f (c - z)) = fun z ↦ (fun w ↦ f (c + w)) (-z)\n⊢ (((-1) ^ n • iteratedFDerivWithin 𝕜 n f (c +ᵥ -s) (c + -a)) fun x ↦ 1) =\n ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 83
} | {
"line": 87,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na b : k\n⊢ slope f a b = slope f b a",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",... | [] | rw [slope, slope, ← neg_vsub_eq_vsub_rev, smul_neg, ← neg_smul, neg_inv, neg_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 83
} | {
"line": 87,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na b : k\n⊢ slope f a b = slope f b a",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",... | [] | rw [slope, slope, ← neg_vsub_eq_vsub_rev, smul_neg, ← neg_smul, neg_inv, neg_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 83
} | {
"line": 87,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\nPE : Type u_3\ninst✝³ : Field k\ninst✝² : AddCommGroup E\ninst✝¹ : Module k E\ninst✝ : AddTorsor E PE\nf : k → PE\na b : k\n⊢ slope f a b = slope f b a",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",... | [] | rw [slope, slope, ← neg_vsub_eq_vsub_rev, smul_neg, ← neg_smul, neg_inv, neg_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 144,
"column": 4
} | {
"line": 145,
"column": 38
} | {
"line": 146,
"column": 2
} | [
{
"pp": "case neg.refine_1\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 : 0 ... | [] | rwa [slope, ← mul_smul, mul_inv_cancel₀ (mt sub_eq_zero.1 (Ne.symm hxeqy)), one_smul,
vsub_eq_sub, sub_nonneg] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 162,
"column": 29
} | {
"line": 162,
"column": 56
} | {
"line": 162,
"column": 57
} | [
{
"pp": "k : 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\n⊢ 0 ≤ slope f x y ∧ 0 ≠ slope f x y ↔ f x... | [
"k : 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\n⊢ f x ≤ f y ∧ 0 ≠ slope f x y ↔ f x ≤ f y ∧ f x ≠ f y... | slope_nonneg_iff_of_le hxy, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 76
} | {
"line": 170,
"column": 0
} | [
{
"pp": "k : 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\ns : Set k\nhf : StrictMonoOn f s\nhx : x ∈ s\nhy : y ∈... | [] | rcases lt_or_gt_of_ne hxy with hxy | hxy
· exact (slope_pos_iff_of_le hxy.le).mpr (hf hx hy hxy)
· exact slope_comm f x y ▸ (slope_pos_iff_of_le hxy.le).mpr (hf hy hx hxy) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 76
} | {
"line": 170,
"column": 0
} | [
{
"pp": "k : 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\ns : Set k\nhf : StrictMonoOn f s\nhx : x ∈ s\nhy : y ∈... | [] | rcases lt_or_gt_of_ne hxy with hxy | hxy
· exact (slope_pos_iff_of_le hxy.le).mpr (hf hx hy hxy)
· exact slope_comm f x y ▸ (slope_pos_iff_of_le hxy.le).mpr (hf hy hx hxy) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Slope | {
"line": 109,
"column": 8
} | {
"line": 109,
"column": 41
} | {
"line": 109,
"column": 42
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\ns t : Set 𝕜\nh : s ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s ∩ t)\nx : 𝕜\nH : UniqueDiffWithinAt 𝕜 s x\nH' : DifferentiableWithinAt 𝕜 f s x\n⊢ (... | [
"𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\ns t : Set 𝕜\nh : s ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s ∩ t)\nx : 𝕜\nH : UniqueDiffWithinAt 𝕜 s x\nH' : DifferentiableWithinAt 𝕜 f s x\n⊢ AccPt x (𝓟 (... | ← accPt_principal_iff_nhdsWithin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.CircleMap | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 38
} | {
"line": 140,
"column": 2
} | [
{
"pp": "a b R : ℝ\nc : ℂ\nh_R : R ≠ 0\nn : ℤ\nhn : a = b + ↑(n * 2) * π\nh_dist : |↑n| * (2 * π) < 2 * π\n⊢ a = b",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Int.cast",
"MulOne.toOne",
"Real.partialOrder",
"Real",
"Preorde... | [
"a b R : ℝ\nc : ℂ\nh_R : R ≠ 0\nn : ℤ\nhn : a = b + ↑(n * 2) * π\nh_dist : |↑n| < 1\n⊢ a = b"
] | simp (disch := positivity) at h_dist | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.Deriv.Shift | {
"line": 87,
"column": 27
} | {
"line": 87,
"column": 47
} | {
"line": 87,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\na x : 𝕜\n⊢ deriv (fun x ↦ f (x + -a)) x = deriv f (x + -a)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"AddGroupWithOne.toAddGroup",
... | [] | deriv_comp_add_const | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Calculus.Deriv.ZPow | {
"line": 46,
"column": 42
} | {
"line": 46,
"column": 54
} | {
"line": 46,
"column": 54
} | [
{
"pp": "𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nm✝ : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m✝\nm : ℕ\nhm : 0 < ↑m\n⊢ x ^ ↑(m - 1) = x ^ (m - 1)",
"ppTerm": "?m.228",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg",
"HSub.hSub",
"DivInvMonoid.toZ... | [
"𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nm✝ : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m✝\nm : ℕ\nhm : 0 < ↑m\n⊢ x ^ (m - 1) = x ^ (m - 1)",
"𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nm✝ : ℤ\nx : 𝕜\nh : x ≠ 0 ∨ 0 ≤ m✝\nm : ℕ\nhm : 0 < ↑m\n⊢ 1 ≤ m"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Deriv.ZPow | {
"line": 162,
"column": 6
} | {
"line": 162,
"column": 13
} | {
"line": 164,
"column": 0
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nc d : 𝕜\nk : ℕ\nihk : (deriv^[k] fun x ↦ (c * x + d)⁻¹) = fun x ↦ (-1) ^ k * ↑k ! * c ^ k * (c * x + d) ^ (-1 - ↑k)\nz : 𝕜\nhd : ¬c = 0\nthis : deriv (fun x ↦ (c * x + d) ^ (-1 - ↑k)) z = c • deriv (fun x ↦ x ^ (-1 - ↑k)) (c * (z + d / c))\nh... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 167,
"column": 59
} | {
"line": 167,
"column": 71
} | {
"line": 167,
"column": 71
} | [
{
"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 ... | [
"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 𝓝[≠] z₀, f ... | zpow_natCast | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 178,
"column": 2
} | {
"line": 180,
"column": 79
} | {
"line": 182,
"column": 0
} | [
{
"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 : ℕ\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... | [] | exact Int.ofNat_inj.mp <| unique_eventuallyEq_zpow_smul_nonzero
(let ⟨g, h₁, h₂, h₃⟩ := hm; ⟨g, h₁, h₂, h₃.filter_mono nhdsWithin_le_nhds⟩)
(let ⟨g, h₁, h₂, h₃⟩ := hn; ⟨g, h₁, h₂, h₃.filter_mono nhdsWithin_le_nhds⟩) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 296,
"column": 13
} | {
"line": 296,
"column": 87
} | {
"line": 298,
"column": 0
} | [
{
"pp": "case neg.inr\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nU : Set 𝕜\nA : Type u_3\ninst✝⁶ : NormedRing A\ninst✝⁵ : IsDomain A\ninst✝⁴ : NormedAlgebra 𝕜 A\nB : Type u_4\ninst✝³ : NormedAddCommGroup B\ninst✝² : NormedSpace 𝕜 B\ninst✝¹ : Module A B\ninst✝ : IsTorsionFree A B\nf : 𝕜 → A\ng : 𝕜... | [] | exact Or.inr <| hg.eqOn_zero_of_preconnected_of_frequently_eq_zero hU hz h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 296,
"column": 13
} | {
"line": 296,
"column": 87
} | {
"line": 298,
"column": 0
} | [
{
"pp": "case neg.inr\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nU : Set 𝕜\nA : Type u_3\ninst✝⁶ : NormedRing A\ninst✝⁵ : IsDomain A\ninst✝⁴ : NormedAlgebra 𝕜 A\nB : Type u_4\ninst✝³ : NormedAddCommGroup B\ninst✝² : NormedSpace 𝕜 B\ninst✝¹ : Module A B\ninst✝ : IsTorsionFree A B\nf : 𝕜 → A\ng : 𝕜... | [] | exact Or.inr <| hg.eqOn_zero_of_preconnected_of_frequently_eq_zero hU hz h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 296,
"column": 13
} | {
"line": 296,
"column": 87
} | {
"line": 298,
"column": 0
} | [
{
"pp": "case neg.inr\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nU : Set 𝕜\nA : Type u_3\ninst✝⁶ : NormedRing A\ninst✝⁵ : IsDomain A\ninst✝⁴ : NormedAlgebra 𝕜 A\nB : Type u_4\ninst✝³ : NormedAddCommGroup B\ninst✝² : NormedSpace 𝕜 B\ninst✝¹ : Module A B\ninst✝ : IsTorsionFree A B\nf : 𝕜 → A\ng : 𝕜... | [] | exact Or.inr <| hg.eqOn_zero_of_preconnected_of_frequently_eq_zero hU hz h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.LogDeriv | {
"line": 92,
"column": 8
} | {
"line": 92,
"column": 23
} | {
"line": 92,
"column": 24
} | [
{
"pp": "case inr.inr\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nf : 𝕜 → 𝕜'\nx : 𝕜\nhdf : DifferentiableAt 𝕜 f x\nn : ℤ\nhn : n ≠ 0\nhf : f x ≠ 0\n⊢ logDeriv (fun x ↦ f x ^ n) x = ↑n * logDeriv f x",
"ppTerm": ... | [
"case inr.inr\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nf : 𝕜 → 𝕜'\nx : 𝕜\nhdf : DifferentiableAt 𝕜 f x\nn : ℤ\nhn : n ≠ 0\nhf : f x ≠ 0\n⊢ deriv (fun x ↦ f x ^ n) x / f x ^ n = ↑n * logDeriv f x"
] | logDeriv_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 248,
"column": 6
} | {
"line": 248,
"column": 47
} | {
"line": 248,
"column": 48
} | [
{
"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\nn : ℕ∞ω\nf : E → F... | [
"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\nn : ℕ∞ω\nf : E → F\ng : F →L[�... | ← iteratedFDerivWithin_inter_open hU hxU, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 84,
"column": 40
} | {
"line": 122,
"column": 33
} | {
"line": 124,
"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\nn : ℕ∞ω\ns : Set E\nt : Set F\ng : F... | [] | by
match n with
| ω =>
have h'f : ContDiffWithinAt 𝕜 ω f s x := hf
obtain ⟨u, hu, p, hp, h'p⟩ := h'f
obtain ⟨v, hv, q, hq, h'q⟩ := hg
let w := insert x s ∩ (u ∩ f ⁻¹' v)
have wv : w ⊆ f ⁻¹' v := fun y hy => hy.2.2
have wu : w ⊆ u := fun y hy => hy.2.1
refine ⟨w, ?_, fun y ↦ (q (f y)).ta... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 249,
"column": 4
} | {
"line": 249,
"column": 52
} | {
"line": 250,
"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\ng : F → G\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\ng : F → G\nx : E\nn : ℕ∞ω\... | have hxt : f x ∈ t := hst.self_of_nhdsWithin hxs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 250,
"column": 4
} | {
"line": 251,
"column": 61
} | {
"line": 252,
"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\ng : F → G\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\ng : F → G\nx : E\nn : ℕ∞ω\... | have hf_tendsto : Tendsto f (𝓝[s] x) (𝓝[t] (f x)) :=
tendsto_nhdsWithin_iff.mpr ⟨hf.continuousWithinAt, hst⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 559,
"column": 4
} | {
"line": 559,
"column": 87
} | {
"line": 560,
"column": 4
} | [
{
"pp": "case succ\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\nn : ℕ∞ω\ns : Set E\nhs : ... | [
"case succ\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\nn : ℕ∞ω\ns : Set E\nhs : UniqueDiffOn... | simp only [ContinuousLinearMap.flip_apply, ContinuousLinearMap.comp_zero, zero_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.OpenPartialHomeomorph.IsImage | {
"line": 115,
"column": 53
} | {
"line": 115,
"column": 77
} | {
"line": 115,
"column": 77
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\ns : Set X\nt : Set Y\n⊢ e.target ∩ ↑e.symm ⁻¹' s = e.target ∩ t ↔ ↑e '' (e.source ∩ s) = e.target ∩ t",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\ns : Set X\nt : Set Y\n⊢ ↑e '' (e.source ∩ s) = e.target ∩ t ↔ ↑e '' (e.source ∩ s) = e.target ∩ t"
] | ← image_source_inter_eq' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 210,
"column": 19
} | {
"line": 210,
"column": 46
} | {
"line": 212,
"column": 0
} | [
{
"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... | [] | apply c.emb_injective; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 210,
"column": 19
} | {
"line": 210,
"column": 46
} | {
"line": 212,
"column": 0
} | [
{
"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... | [] | apply c.emb_injective; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 306,
"column": 4
} | {
"line": 312,
"column": 53
} | {
"line": 313,
"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... | [] | induction j using Fin.induction with
| zero => simp at hij
| succ j => induction i using Fin.induction with
| zero => simp
| succ i =>
simp only [cons_succ, cases_succ, comp_apply, succ_lt_succ_iff]
exact c.parts_strictMono (by simpa using hij) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 306,
"column": 4
} | {
"line": 312,
"column": 53
} | {
"line": 313,
"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... | [] | induction j using Fin.induction with
| zero => simp at hij
| succ j => induction i using Fin.induction with
| zero => simp
| succ i =>
simp only [cons_succ, cases_succ, comp_apply, succ_lt_succ_iff]
exact c.parts_strictMono (by simpa using hij) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 306,
"column": 4
} | {
"line": 312,
"column": 53
} | {
"line": 313,
"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... | [] | induction j using Fin.induction with
| zero => simp at hij
| succ j => induction i using Fin.induction with
| zero => simp
| succ i =>
simp only [cons_succ, cases_succ, comp_apply, succ_lt_succ_iff]
exact c.parts_strictMono (by simpa using hij) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 480,
"column": 2
} | {
"line": 480,
"column": 36
} | {
"line": 482,
"column": 0
} | [
{
"pp": "case h\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∞\nf' : E → FormalMultilinearSeries 𝕜 E F\nhf : HasFTaylorSeriesUpToOn (↑n) ... | [] | exact ⟨f', hf.of_le (mod_cast hm)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 572,
"column": 6
} | {
"line": 580,
"column": 23
} | {
"line": 581,
"column": 6
} | [
{
"pp": "case e'_4.inl\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\... | [
"case e'_4.inr\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 → For... | · simp only [↓reduceDIte, update_self, succ_mk, cast_mk, val_pred]
have A := c.one_lt_partSize_index_zero hc
rw [Nat.sub_add_cancel]
· congr; lia
· rw [Order.one_le_iff_pos]
conv_lhs => rw [show (0 : ℕ) = c.emb (c.index 0) 0 by simp [emb_zero]]
rw [← lt_def]
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 715,
"column": 12
} | {
"line": 715,
"column": 45
} | {
"line": 716,
"column": 12
} | [
{
"pp": "case neg.emb.refine_2.inl\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... | [
"case neg.emb.refine_2.inl.inl\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 : Se... | rcases eq_or_ne j 0 with rfl | hj | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Complex.RealDeriv | {
"line": 104,
"column": 2
} | {
"line": 105,
"column": 35
} | {
"line": 107,
"column": 0
} | [
{
"pp": "z : ℝ\nf : ℝ → ℝ\nu : ℝ\nhf : HasDerivAt f u z\n⊢ HasDerivAt (fun y ↦ ↑(f y)) (↑u) z",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"Semiring.toModule",
"NormedSpace.toIsBoundedSMul",
... | [] | simpa only [ofRealCLM_apply, ofReal_one, real_smul, mul_one] using!
ofRealCLM.hasDerivAt.scomp z hf | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Analysis.Complex.RealDeriv | {
"line": 104,
"column": 2
} | {
"line": 105,
"column": 35
} | {
"line": 107,
"column": 0
} | [
{
"pp": "z : ℝ\nf : ℝ → ℝ\nu : ℝ\nhf : HasDerivAt f u z\n⊢ HasDerivAt (fun y ↦ ↑(f y)) (↑u) z",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"Semiring.toModule",
"NormedSpace.toIsBoundedSMul",
... | [] | simpa only [ofRealCLM_apply, ofReal_one, real_smul, mul_one] using!
ofRealCLM.hasDerivAt.scomp z hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.RealDeriv | {
"line": 104,
"column": 2
} | {
"line": 105,
"column": 35
} | {
"line": 107,
"column": 0
} | [
{
"pp": "z : ℝ\nf : ℝ → ℝ\nu : ℝ\nhf : HasDerivAt f u z\n⊢ HasDerivAt (fun y ↦ ↑(f y)) (↑u) z",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHSMul",
"Semiring.toModule",
"NormedSpace.toIsBoundedSMul",
... | [] | simpa only [ofRealCLM_apply, ofReal_one, real_smul, mul_one] using!
ofRealCLM.hasDerivAt.scomp z hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 65,
"column": 2
} | {
"line": 66,
"column": 37
} | {
"line": 67,
"column": 2
} | [
{
"pp": "𝕜 : 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' : ... | [
"𝕜 : 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' : (i : ι) → E ... | set L : ∀ m : ℕ, (∀ i, E [×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E [×m]→L[𝕜] ∀ i, F' i := fun m =>
ContinuousMultilinearMap.piₗᵢ _ _ | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 380,
"column": 2
} | {
"line": 381,
"column": 61
} | {
"line": 383,
"column": 0
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : ∀ x ∈ interior D, 0 < deriv f x\nx : ℝ\nhx : x ∈ D\ny : ℝ\nhy : y ∈ D\nthis : DifferentiableOn ℝ f (interior D)\n⊢ x < y → f x < f y",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRight... | [] | simpa only [zero_mul, sub_pos] using
hD.mul_sub_lt_image_sub_of_lt_deriv hf this hf' x hx y hy | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Convex.Cone.Basic | {
"line": 312,
"column": 15
} | {
"line": 312,
"column": 33
} | {
"line": 312,
"column": 33
} | [
{
"pp": "R : Type u_2\nG : Type u_3\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : AddCommGroup G\ninst✝ : SMul R G\nC : ConvexCone R G\nx : G\nhx : x ∈ C\nleft✝ : x ≠ 0\nhxneg : -x ∈ C\n⊢ 0 ∈ C",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"add_neg_cancel"... | [
"R : Type u_2\nG : Type u_3\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : AddCommGroup G\ninst✝ : SMul R G\nC : ConvexCone R G\nx : G\nhx : x ∈ C\nleft✝ : x ≠ 0\nhxneg : -x ∈ C\n⊢ x + -x ∈ C"
] | ← add_neg_cancel x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Convex.Cone.Basic | {
"line": 707,
"column": 2
} | {
"line": 708,
"column": 83
} | {
"line": 709,
"column": 2
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nM : Type u_4\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup M\ninst✝ : Module 𝕜 M\ns : Set M\nhs : Convex 𝕜 s\nx : M\n⊢ (∃ c, 0 < c ∧ ∃ y ∈ s, c • y = x) → ∃ c, 0 < c ∧ c • x ∈ s",
"ppTerm": "?refine_1",
"assig... | [
"case refine_2\n𝕜 : Type u_1\nM : Type u_4\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup M\ninst✝ : Module 𝕜 M\ns : Set M\nhs : Convex 𝕜 s\nx : M\n⊢ (∃ c, 0 < c ∧ c • x ∈ s) → ∃ c, 0 < c ∧ ∃ y ∈ s, c • y = x"
] | · rintro ⟨c, hc, y, hy, rfl⟩
exact ⟨c⁻¹, inv_pos.2 hc, by rwa [smul_smul, inv_mul_cancel₀ hc.ne', one_smul]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 385,
"column": 4
} | {
"line": 385,
"column": 11
} | {
"line": 386,
"column": 2
} | [
{
"pp": "x : ℝ\nh : |x| < 1\nterm : ℕ → ℝ := fun n ↦ -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)\nn : ℕ\n⊢ -1 * (-x ^ (2 * n + 1) / (2 * ↑n + 1)) + x ^ (2 * n + 1) / (2 * ↑n + 1) = 2 * (1 / (2 * ↑n + 1)) * x ^ (2 * n + 1)",
"ppTerm": "?m.231",
"assigned": true,
"usedConstants": [
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 11
} | {
"line": 390,
"column": 2
} | [
{
"pp": "case e'_6\nx : 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)\nh₁ : HasSum (fun i ↦ -1 * ((-x) ^ (i + 1) / (↑i + 1))) (-1 * -log (1 - -x))\n⊢ log (1 + x) - log ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Geometry.Convex.Cone.Pointed | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 67
} | {
"line": 358,
"column": 0
} | [
{
"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\nC : PointedCone R E\n⊢ C.lineal = sSup {S | ↑S ≤ C}",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
... | [] | simp_rw [gc_ofSubmodule_lineal.le_iff_le, Set.Iic_def, csSup_Iic] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Convex.Cone.Pointed | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 67
} | {
"line": 358,
"column": 0
} | [
{
"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\nC : PointedCone R E\n⊢ C.lineal = sSup {S | ↑S ≤ C}",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
... | [] | simp_rw [gc_ofSubmodule_lineal.le_iff_le, Set.Iic_def, csSup_Iic] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Convex.Cone.Pointed | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 67
} | {
"line": 358,
"column": 0
} | [
{
"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\nC : PointedCone R E\n⊢ C.lineal = sSup {S | ↑S ≤ C}",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
... | [] | simp_rw [gc_ofSubmodule_lineal.le_iff_le, Set.Iic_def, csSup_Iic] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 86,
"column": 2
} | {
"line": 89,
"column": 16
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : PointedCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\nhdom : f.domain ≠ ⊤\ny : E\nhy : y ∉ f.domain\nc : ℝ\nle_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c\nc_le ... | [
"case refine_2\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : PointedCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\nhdom : f.domain ≠ ⊤\ny : E\nhy : y ∉ f.domain\nc : ℝ\nle_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c\nc_le : ∀ (x : ↥f.... | · refine lt_iff_le_not_ge.2 ⟨f.left_le_sup _ _, fun H => ?_⟩
replace H := LinearPMap.domain_mono.monotone H
rw [LinearPMap.domain_supSpanSingleton, sup_le_iff, span_le, singleton_subset_iff] at H
exact hy H.2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 383,
"column": 4
} | {
"line": 385,
"column": 67
} | {
"line": 387,
"column": 0
} | [] | [] | (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 := int_fs
_ ≤ (∫⁻ x, g x ∂μ) + ε / 2 + ε / 2 := add_le_add gint le_rfl
_ = (∫⁻ x, g x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 52
} | {
"line": 172,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nf : E →ₗ.[ℝ] ℝ\nN : E → ℝ\nN_hom : ∀ (c : ℝ), 0 < c → ∀ (x : E), N (c • x) = c * N x\nN_add : ∀ (x y : E), N (x + y) ≤ N x + N y\nhf : ∀ (x : ↥f.domain), ↑f x ≤ N ↑x\nN_0 : N 0 = 0\ns : PointedCone ℝ (E × ℝ) := { carrier := {p | N p.1 ≤ p.2}, a... | [] | simpa [f'] using le_trans (hf ⟨x.1.1, x.2.1⟩) hx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 52
} | {
"line": 172,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nf : E →ₗ.[ℝ] ℝ\nN : E → ℝ\nN_hom : ∀ (c : ℝ), 0 < c → ∀ (x : E), N (c • x) = c * N x\nN_add : ∀ (x y : E), N (x + y) ≤ N x + N y\nhf : ∀ (x : ↥f.domain), ↑f x ≤ N ↑x\nN_0 : N 0 = 0\ns : PointedCone ℝ (E × ℝ) := { carrier := {p | N p.1 ≤ p.2}, a... | [] | simpa [f'] using le_trans (hf ⟨x.1.1, x.2.1⟩) hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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