module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 763,
"column": 4
} | {
"line": 764,
"column": 13
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\na : adicCompletion K v\nha : a ∈ adicCompletionIntegers K v\n⊢ ∃ b ∈ R⁰, a * ↑b ∈ adicCompletionIntegers K v",
"usedCo... | use 1
simp [ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 259,
"column": 14
} | {
"line": 259,
"column": 18
} | [
{
"pp": "case left\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf g : MvPowerSeries σ R\nn : ℕ\nH : ↑n < weightedOrder w g\nhn : ↑n = weightedOrder w f\nd : σ →₀ ℕ\nhd' : (coeff d) f ≠ 0\nhd : (weight w) d = n\n⊢ (coeff d) (f + g) ≠ 0",
"usedConstants": [
"Finsupp.instAddZeroClass",
... | ← hd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 383,
"column": 69
} | {
"line": 385,
"column": 64
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nf : MvPowerSeries σ R\n⊢ f ≠ 0 ↔ ∃ n d, (coeff d) f ≠ 0 ∧ degree d = n",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"MvPowerSeries.instZero"... | by
simp_rw [degree_eq_weight_one]
exact ne_zero_iff_exists_coeff_ne_zero_and_weight (fun _ => 1) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nφ : R⟦X⟧\nhf : φ ≠ 0\n⊢ ↑φ.order.toNat = φ.order",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MvPowerSeries.instZero",
"ENat.instNatCast",
"instTopENat",
"congrArg",
"id",
"Iff.not",
"Ne",
"Nat.cast",
... | ENat.coe_toNat_eq_self.mpr (order_eq_top.not.mpr hf) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 101,
"column": 31
} | {
"line": 101,
"column": 46
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\nφ : R⟦X⟧\nn : ℕ\nh : n < φ.order.toNat\nh' : ¬φ = 0\n⊢ ↑n < ↑φ.order.toNat",
"usedConstants": [
"Eq.mpr",
"ENat.instNatCast",
"congrArg",
"id",
"Nat.cast",
"ENat.coe_lt_coe",
"Nat",
"ENat",
"LT.lt",
... | ENat.coe_lt_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.PiTopology | {
"line": 223,
"column": 2
} | {
"line": 225,
"column": 24
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : TopologicalSpace R\ninst✝ : CommSemiring R\nf : MvPowerSeries σ R\nhf : constantCoeff f = 0\n⊢ Tendsto (fun n ↦ f ^ n) atTop (𝓝 0)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"MvPowerSeries",
"RingHom",
... | apply isTopologicallyNilpotent_of_constantCoeff_isNilpotent
rw [hf]
exact IsNilpotent.zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.PiTopology | {
"line": 223,
"column": 2
} | {
"line": 225,
"column": 24
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : TopologicalSpace R\ninst✝ : CommSemiring R\nf : MvPowerSeries σ R\nhf : constantCoeff f = 0\n⊢ Tendsto (fun n ↦ f ^ n) atTop (𝓝 0)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"MvPowerSeries",
"RingHom",
... | apply isTopologicallyNilpotent_of_constantCoeff_isNilpotent
rw [hf]
exact IsNilpotent.zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 349,
"column": 20
} | {
"line": 349,
"column": 35
} | [
{
"pp": "case neg\nR : Type u_2\ninst✝ : Semiring R\nφ : R⟦X⟧\nhφ : φ ≠ 0\nn : ℕ\nho : φ.order = ↑n\nhn : φ.order.toNat = n\nψ : R⟦X⟧\nH : φ = X ^ (φ.order.toNat + 1) * ψ\nthis : (coeff n) φ = (coeff n) (ψ * X ^ (φ.order.toNat + 1))\nh✝ : ¬1 = 0\n⊢ ↑φ.order.toNat < ↑(φ.order.toNat + 1)",
"usedConstants": [
... | ENat.coe_lt_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Evaluation | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 9
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : UniformSpace R\nS : Type u_3\ninst✝⁴ : CommRing S\ninst✝³ : UniformSpace S\nφ : R →+* S\na : σ → S\ninst✝² : IsUniformAddGroup R\ninst✝¹ : IsUniformAddGroup S\ninst✝ : IsLinearTopology S S\nhφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTop... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 318,
"column": 8
} | {
"line": 319,
"column": 34
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\na : σ → MvPowerSeries τ S\nha : HasSubst a\nha' : ∀ (i : σ), constantCoeff (a i) = 0\nf : MvPowerSeries σ R\nhf : constantCoeff f = 0\nd : σ →₀ ℕ\nhd : ¬d = 0\n⊢ ∃ i, d i ≠ 0",
"us... | by_contra! hc
exact hd <| Finsupp.ext hc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 318,
"column": 8
} | {
"line": 319,
"column": 34
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\na : σ → MvPowerSeries τ S\nha : HasSubst a\nha' : ∀ (i : σ), constantCoeff (a i) = 0\nf : MvPowerSeries σ R\nhf : constantCoeff f = 0\nd : σ →₀ ℕ\nhd : ¬d = 0\n⊢ ∃ i, d i ≠ 0",
"us... | by_contra! hc
exact hd <| Finsupp.ext hc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 428,
"column": 4
} | {
"line": 428,
"column": 89
} | [
{
"pp": "σ : Type u_1\nτ : Type u_4\nS : Type u_5\ninst✝² : CommRing S\na : σ → MvPowerSeries τ S\nυ : Type u_7\nT : Type u_8\ninst✝¹ : CommRing T\ninst✝ : Algebra S T\nb : τ → MvPowerSeries υ T\nha : HasSubst a\nhb : HasSubst b\nthis✝ : UniformSpace S := ⋯\nthis : UniformSpace T := ⋯\n⊢ Filter.Tendsto (⇑(subst... | simpa [← map_zero (substAlgHom (R := S) hb)] using (continuous_subst hb).continuousAt | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Real.Cardinality | {
"line": 190,
"column": 36
} | {
"line": 190,
"column": 55
} | [
{
"pp": "case a\n⊢ ℵ₀ ^ ℵ₀ = 𝔠",
"usedConstants": [
"Eq.mpr",
"Cardinal.aleph0_power_aleph0",
"Cardinal.instPowCardinal",
"Cardinal",
"congrArg",
"id",
"Cardinal.aleph0",
"Cardinal.continuum",
"HPow.hPow",
"instHPow",
"Eq"
]
}
] | aleph0_power_aleph0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.Basic | {
"line": 129,
"column": 2
} | {
"line": 132,
"column": 81
} | [
{
"pp": "f : ℕ → ℂ\ns s' : ℂ\nh : s.re ≤ s'.re\nn : ℕ\n⊢ ‖term f s' n‖ ≤ ‖term f s n‖",
"usedConstants": [
"Iff.mpr",
"zero_le",
"Norm.norm",
"div_le_div₀",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Nat.instCanonicallyOrderedAdd",
"le_refl",
"Rea... | simp only [norm_term_eq]
split
· next => rfl
· next hn => gcongr; exact Nat.one_le_cast.mpr <| Nat.one_le_iff_ne_zero.mpr hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.Basic | {
"line": 129,
"column": 2
} | {
"line": 132,
"column": 81
} | [
{
"pp": "f : ℕ → ℂ\ns s' : ℂ\nh : s.re ≤ s'.re\nn : ℕ\n⊢ ‖term f s' n‖ ≤ ‖term f s n‖",
"usedConstants": [
"Iff.mpr",
"zero_le",
"Norm.norm",
"div_le_div₀",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Nat.instCanonicallyOrderedAdd",
"le_refl",
"Rea... | simp only [norm_term_eq]
split
· next => rfl
· next hn => gcongr; exact Nat.one_le_cast.mpr <| Nat.one_le_iff_ne_zero.mpr hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.EGauge | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\nI : Set ι\nhI : I.Finite\nU : (i : ι) → Set (E i)\nhU : ∀ i ∈ I, Balanced 𝕜 (U i)\nx : (i : ι) → E i\nhI₀ : I = univ ∨ (∃ i ∈ I, x i ≠ 0) ∨ (𝓝[≠] 0)... | rcases I.eq_empty_or_nonempty with rfl | hIne | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.Deriv.Basic | {
"line": 251,
"column": 2
} | {
"line": 253,
"column": 6
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\nh : ¬DifferentiableAt 𝕜 f x\n⊢ deriv f x = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRi... | unfold deriv
rw [fderiv_zero_of_not_differentiableAt h]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Basic | {
"line": 251,
"column": 2
} | {
"line": 253,
"column": 6
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\nh : ¬DifferentiableAt 𝕜 f x\n⊢ deriv f x = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRi... | unfold deriv
rw [fderiv_zero_of_not_differentiableAt h]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Basic | {
"line": 542,
"column": 4
} | {
"line": 542,
"column": 71
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx : ℝ\nH : DifferentiableWithinAt ℝ f (Ioi x) x\nA : HasDerivWithinAt f (derivWithin f (Ioi x) x) (Ici x) x\n⊢ derivWithin f (Ioi x) x = derivWithin f (Ici x) x",
"usedConstants": [
"NormedCommRing.toNo... | have B := (differentiableWithinAt_Ioi_iff_Ici.1 H).hasDerivWithinAt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 718,
"column": 2
} | {
"line": 718,
"column": 17
} | [
{
"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\nx : E\ns : Set E\ninst✝² : ContinuousAdd E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : T2Space E\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderiv 𝕜 id x = ContinuousLinear... | exact fderiv_id | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 738,
"column": 2
} | {
"line": 738,
"column": 22
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalSpace E\ninst✝² : Module 𝕜 E\nl : Filter α\nf : α → E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul 𝕜 E\nx : E\nh : Tendsto (fun x_1 ↦ f x_1 - x) l (𝓝 0)\nU : Set E\nhU₀ :... | use r, by positivity | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 819,
"column": 6
} | {
"line": 828,
"column": 39
} | [] | egauge 𝕜 (ball 0 r) (f x) ≤ ‖c‖ₑ * ‖f x‖ₑ / r :=
egauge_ball_le_of_one_lt_norm hc <| .inl hr₀.ne'
_ ≤ ‖c‖ₑ * (C * ‖g x‖ₑ) / r := by
gcongr
simp only [enorm_eq_nnnorm, ← coe_nnnorm] at hx ⊢
exact mod_cast hx
_ = ‖g x‖ₑ / (r / (C * ‖c‖₊) : ℝ≥0) := by
simp_all [pos_iff_... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Analytic.ChangeOrigin | {
"line": 281,
"column": 8
} | {
"line": 281,
"column": 74
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.... | simp only [changeOriginSeries, ContinuousMultilinearMap.sum_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.CPolynomialDef | {
"line": 241,
"column": 2
} | {
"line": 249,
"column": 47
} | [
{
"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∞\nhf : ∀ y ∈ Metric.eball x r, f y = 0\nr_pos ... | refine ⟨⟨?_, r_pos, ?_⟩, fun n _ ↦ hp n⟩
· rw [p.radius_eq_top_of_forall_image_add_eq_zero 0 (fun n ↦ by rw [add_zero]; exact hp n)]
exact le_top
· intro y hy
rw [hf (x + y)]
· convert! hasSum_zero
rw [hp, ContinuousMultilinearMap.zero_apply]
· rwa [Metric.mem_eball, edist_eq_enorm_sub, add_co... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.CPolynomialDef | {
"line": 241,
"column": 2
} | {
"line": 249,
"column": 47
} | [
{
"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∞\nhf : ∀ y ∈ Metric.eball x r, f y = 0\nr_pos ... | refine ⟨⟨?_, r_pos, ?_⟩, fun n _ ↦ hp n⟩
· rw [p.radius_eq_top_of_forall_image_add_eq_zero 0 (fun n ↦ by rw [add_zero]; exact hp n)]
exact le_top
· intro y hy
rw [hf (x + y)]
· convert! hasSum_zero
rw [hp, ContinuousMultilinearMap.zero_apply]
· rwa [Metric.mem_eball, edist_eq_enorm_sub, add_co... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.ConvergenceRadius | {
"line": 409,
"column": 4
} | {
"line": 409,
"column": 23
} | [
{
"pp": "case e_s.h.a.h\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\n⊢ ∃ i', ⨆ (_ : ∀ (n : ℕ), ‖p (n + 1)‖ * ... | use ‖p 0‖ ⊔ (C * r) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 123,
"column": 49
} | {
"line": 124,
"column": 18
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring E\ninst✝² : Algebra 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : IsTopologicalRing E\nc : ℕ → 𝕜\nx : E\nn : ℕ\n⊢ ((ofScalars E c n) fun x_1 ↦ x) = c n • x ^ n",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Li... | by
simp [ofScalars] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 128,
"column": 67
} | {
"line": 129,
"column": 18
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : Ring E\ninst✝² : Algebra 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : IsTopologicalRing E\nc : ℕ → 𝕜\nx : E\n⊢ (fun n ↦ (ofScalars E c n) fun x_1 ↦ x) = fun n ↦ c n • x ^ n",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
... | by
simp [ofScalars] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 209,
"column": 4
} | {
"line": 210,
"column": 14
} | [
{
"pp": "case neg.hs.refine_1.refine_1\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⊢ ∀ᶠ (n : ℕ) in atTop,... | · refine (hc.eventually_ne (NNReal.coe_ne_zero.mpr hr)).mp (Eventually.of_forall ?_)
simp_all | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 278,
"column": 4
} | {
"line": 278,
"column": 64
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing E\ninst✝¹ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\ninst✝ : NormOneClass E\nr : ℝ≥0\nhr : ↑r < (ofScalars E c).radius\nthis : 0 < r\nhc : ∀ᶠ (x : ℕ) in atTop, ‖‖ofScalars E c x‖ * ↑r ^ x‖ = 0\nn : ℕ\nhc' : ‖‖ofScalars... | rw [ofScalars_norm, norm_mul, norm_norm, mul_eq_zero] at hc' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Basic | {
"line": 424,
"column": 38
} | {
"line": 426,
"column": 23
} | [
{
"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\npf : FormalMultilinearSeries 𝕜 E F\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f pf x r\nv : Fi... | by
rw [← hasFPowerSeriesWithinOnBall_univ] at hf
exact hf.coeff_zero v | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Basic | {
"line": 524,
"column": 2
} | {
"line": 524,
"column": 36
} | [
{
"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\ns t : Set E\nx : E\nh : AnalyticWithinAt 𝕜 f s x\nhst : s =ᶠ[𝓝 x] t\n⊢ AnalyticWithinAt 𝕜 f t ... | refine h.mono_of_mem_nhdsWithin ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Analytic.Basic | {
"line": 708,
"column": 4
} | {
"line": 712,
"column": 62
} | [] | ‖(p n) fun _ : Fin n => y‖
_ ≤ ‖p n‖ * ∏ _i : Fin n, ‖y‖ := ContinuousMultilinearMap.le_opNorm _ _
_ = ‖p n‖ * (r' : ℝ) ^ n * (‖y‖ / r') ^ n := by simp [field, div_pow]
_ ≤ C * a ^ n * (‖y‖ / r') ^ n := by gcongr ?_ * _; apply hp
_ ≤ C * (a * (‖y‖ / r')) ^ n := by rw [mul_pow, mul_assoc] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Calculus.FDeriv.Comp | {
"line": 101,
"column": 11
} | {
"line": 101,
"column": 44
} | [
{
"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 → F\nf' : E →L[𝕜] F\nx : E\ng... | exact hg.comp_hasFDerivAt x hf ht | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.Inverse | {
"line": 392,
"column": 8
} | {
"line": 392,
"column": 80
} | [
{
"pp": "case hf\nn : ℕ\np : ℕ → ℝ\nhp : ∀ (k : ℕ), 0 ≤ p k\nr a : ℝ\nhr : 0 ≤ r\nha : 0 ≤ a\nx : (i : ℕ) × Composition i\na✝¹ : x ∈ compPartialSumTarget 2 (n + 1) n\na✝ : x ∉ (Ico 2 (n + 1)).sigma fun k ↦ {c | 1 < c.length}.toFinset\n⊢ 0 ≤ ∏ j, r * (a ^ x.snd.blocksFun j * p (x.snd.blocksFun j))",
"usedCon... | exact prod_nonneg fun j _ ↦ (by positivity [ha, hp (x.snd.blocksFun j)]) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.Inverse | {
"line": 448,
"column": 67
} | {
"line": 450,
"column": 60
} | [
{
"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\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 ... | by
congr! 2 with j hj
rw [rightInv_coeff _ _ _ _ (mem_Ico.1 hj).1, norm_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Inverse | {
"line": 494,
"column": 4
} | {
"line": 497,
"column": 39
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\nhp : 0 < p.radius\nC r : ℝ\nCpos : 0 < C\nrpos : ... | have :
Tendsto (fun a => 2 * I * C * r ^ 2 * (I + 1) ^ 2 * a) (𝓝 0)
(𝓝 (2 * I * C * r ^ 2 * (I + 1) ^ 2 * 0)) :=
tendsto_const_nhds.mul tendsto_id | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Analytic.Inverse | {
"line": 516,
"column": 14
} | {
"line": 516,
"column": 22
} | [
{
"pp": "case succ\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\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\nhp : 0 < p.radius\nC r : ℝ\nCpos : 0 <... | one_le_n | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Analytic.Inverse | {
"line": 628,
"column": 47
} | {
"line": 628,
"column": 51
} | [
{
"pp": "case h\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\nf : E → F\ng : F → G\nq : Fo... | h''y | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.MetricSpace.Completion | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 61
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\n⊢ 𝓤 (Completion α) = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε}",
"usedConstants": [
"Eq.mpr",
"Real",
"iInf",
"Real.instZero",
"Iff.of_eq",
"congrArg",
"Filter.instInfSet",
"Filter.instCompleteLatticeFilte... | simpa [iInf_subtype] using @Completion.uniformity_dist' α _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.MetricSpace.Completion | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 61
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\n⊢ 𝓤 (Completion α) = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε}",
"usedConstants": [
"Eq.mpr",
"Real",
"iInf",
"Real.instZero",
"Iff.of_eq",
"congrArg",
"Filter.instInfSet",
"Filter.instCompleteLatticeFilte... | simpa [iInf_subtype] using @Completion.uniformity_dist' α _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Completion | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 61
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\n⊢ 𝓤 (Completion α) = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | dist p.1 p.2 < ε}",
"usedConstants": [
"Eq.mpr",
"Real",
"iInf",
"Real.instZero",
"Iff.of_eq",
"congrArg",
"Filter.instInfSet",
"Filter.instCompleteLatticeFilte... | simpa [iInf_subtype] using @Completion.uniformity_dist' α _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 285,
"column": 6
} | {
"line": 285,
"column": 38
} | [
{
"pp": "case pos\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... | · apply h.1.cont m (mod_cast h') | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 720,
"column": 4
} | {
"line": 720,
"column": 30
} | [
{
"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\nf : E → F\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nH : HasFTaylorSeriesUpToOn n f p univ\n⊢ ∀ ... | simpa using H.fderivWithin | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 720,
"column": 4
} | {
"line": 720,
"column": 30
} | [
{
"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\nf : E → F\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nH : HasFTaylorSeriesUpToOn n f p univ\n⊢ ∀ ... | simpa using H.fderivWithin | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 720,
"column": 4
} | {
"line": 720,
"column": 30
} | [
{
"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\nf : E → F\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nH : HasFTaylorSeriesUpToOn n f p univ\n⊢ ∀ ... | simpa using H.fderivWithin | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Alternating.Basic | {
"line": 527,
"column": 39
} | {
"line": 527,
"column": 53
} | [
{
"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 : ... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Module.Alternating.Basic | {
"line": 527,
"column": 39
} | {
"line": 527,
"column": 53
} | [
{
"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 : ... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Alternating.Basic | {
"line": 527,
"column": 39
} | {
"line": 527,
"column": 53
} | [
{
"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 : ... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 358,
"column": 2
} | {
"line": 360,
"column": 73
} | [
{
"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 : AnalyticOn 𝕜 f s\nhu : UniqueDiffOn 𝕜 s\n⊢ AnalyticOn 𝕜 (fderivWithin 𝕜 f s) s",
... | intro x hx
rcases h x hx with ⟨p, r, hr⟩
refine ⟨p.derivSeries, r, hr.fderivWithin_of_mem_of_analyticOn h hu hx⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 358,
"column": 2
} | {
"line": 360,
"column": 73
} | [
{
"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 : AnalyticOn 𝕜 f s\nhu : UniqueDiffOn 𝕜 s\n⊢ AnalyticOn 𝕜 (fderivWithin 𝕜 f s) s",
... | intro x hx
rcases h x hx with ⟨p, r, hr⟩
refine ⟨p.derivSeries, r, hr.fderivWithin_of_mem_of_analyticOn h hu hx⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 49
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\ns : Set 𝕜\nc : F\n⊢ derivWithin (fun y ↦ f y - c) s x = derivWithin f s x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.toSem... | simp only [derivWithin, fderivWithin_sub_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 49
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\ns : Set 𝕜\nc : F\n⊢ derivWithin (fun y ↦ f y - c) s x = derivWithin f s x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.toSem... | simp only [derivWithin, fderivWithin_sub_const] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 49
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\ns : Set 𝕜\nc : F\n⊢ derivWithin (fun y ↦ f y - c) s x = derivWithin f s x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.toSem... | simp only [derivWithin, fderivWithin_sub_const] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 443,
"column": 2
} | {
"line": 443,
"column": 55
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\nc : F\n⊢ deriv (fun x ↦ c - f x) x = -deriv f x",
"usedConstants": [
"NegZeroClass.toNeg",
"congrArg",
"deriv",
"Set.univ",
"Norme... | simp only [← derivWithin_univ, derivWithin_const_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 443,
"column": 2
} | {
"line": 443,
"column": 55
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\nc : F\n⊢ deriv (fun x ↦ c - f x) x = -deriv f x",
"usedConstants": [
"NegZeroClass.toNeg",
"congrArg",
"deriv",
"Set.univ",
"Norme... | simp only [← derivWithin_univ, derivWithin_const_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 443,
"column": 2
} | {
"line": 443,
"column": 55
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\nc : F\n⊢ deriv (fun x ↦ c - f x) x = -deriv f x",
"usedConstants": [
"NegZeroClass.toNeg",
"congrArg",
"deriv",
"Set.univ",
"Norme... | simp only [← derivWithin_univ, derivWithin_const_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 457,
"column": 79
} | {
"line": 458,
"column": 56
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\na b : 𝕜\n⊢ DifferentiableAt 𝕜 (fun x ↦ f (x - b)) a ↔ DifferentiableAt 𝕜 f (a - b)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.to... | by
simp [sub_eq_add_neg, differentiableAt_comp_add_const] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 469,
"column": 79
} | {
"line": 470,
"column": 56
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\na b : 𝕜\n⊢ DifferentiableAt 𝕜 f a ↔ DifferentiableAt 𝕜 (fun x ↦ f (x - b)) (a + b)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.to... | by
simp [sub_eq_add_neg, differentiableAt_comp_add_const] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Deriv.Polynomial | {
"line": 121,
"column": 2
} | {
"line": 122,
"column": 44
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R 𝕜\nq : R[X]\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (fun x ↦ (aeval x) q) s x = (aeval x) (derivative q)",
"usedConstants": [
"NormedCommRing.toNormedRing",
... | simpa only [aeval_def, eval₂_eq_eval_map, derivative_map] using
(q.map (algebraMap R 𝕜)).derivWithin hxs | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.Deriv.Polynomial | {
"line": 121,
"column": 2
} | {
"line": 122,
"column": 44
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R 𝕜\nq : R[X]\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (fun x ↦ (aeval x) q) s x = (aeval x) (derivative q)",
"usedConstants": [
"NormedCommRing.toNormedRing",
... | simpa only [aeval_def, eval₂_eq_eval_map, derivative_map] using
(q.map (algebraMap R 𝕜)).derivWithin hxs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Polynomial | {
"line": 121,
"column": 2
} | {
"line": 122,
"column": 44
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R 𝕜\nq : R[X]\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (fun x ↦ (aeval x) q) s x = (aeval x) (derivative q)",
"usedConstants": [
"NormedCommRing.toNormedRing",
... | simpa only [aeval_def, eval₂_eq_eval_map, derivative_map] using
(q.map (algebraMap R 𝕜)).derivWithin hxs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 630,
"column": 4
} | {
"line": 630,
"column": 76
} | [
{
"pp": "case pos\n𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nx : 𝕜\ns : Set 𝕜\nG : Type u_2\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →... | exact (hc.hasDerivWithinAt.clm_comp hd.hasDerivWithinAt).derivWithin hsx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 630,
"column": 4
} | {
"line": 630,
"column": 76
} | [
{
"pp": "case pos\n𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nx : 𝕜\ns : Set 𝕜\nG : Type u_2\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →... | exact (hc.hasDerivWithinAt.clm_comp hd.hasDerivWithinAt).derivWithin hsx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 630,
"column": 4
} | {
"line": 630,
"column": 76
} | [
{
"pp": "case pos\n𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nx : 𝕜\ns : Set 𝕜\nG : Type u_2\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →... | exact (hc.hasDerivWithinAt.clm_comp hd.hasDerivWithinAt).derivWithin hsx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 628,
"column": 73
} | {
"line": 631,
"column": 57
} | [
{
"pp": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nx : 𝕜\ns : Set 𝕜\nG : Type u_2\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nc : 𝕜 → F →L[𝕜] G\nd... | by
by_cases hsx : UniqueDiffWithinAt 𝕜 s x
· exact (hc.hasDerivWithinAt.clm_comp hd.hasDerivWithinAt).derivWithin hsx
· simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 87,
"column": 4
} | {
"line": 88,
"column": 57
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhpd : deriv f z₀ = p.coeff 1\nhp0 : p.coeff 0 = f z₀\nhp : ∀ᶠ (z : 𝕜) in 𝓝 0, HasSum (fun n ↦ z ^ n • p.coeff ... | suffices HasSum (fun n => x⁻¹ • x ^ (n + 1) • p.coeff (n + 1)) (x⁻¹ • (f (z₀ + x) - f z₀)) by
simpa [dslope, slope, h, smul_smul, hxx] using this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 428,
"column": 17
} | {
"line": 428,
"column": 48
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → F\nx : 𝕜\nl : F → E\ny : F\nhl : DifferentiableAt 𝕜 l (f x)\nhf : DifferentiableAt 𝕜 f x\nhy : y = f x... | exact fderiv_comp_deriv x hl hf | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 536,
"column": 2
} | {
"line": 540,
"column": 92
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → G\nC : ℝ\ns : Set E\nx y : E\nf' : E → E ... | calc
‖f y - f x - φ (y - x)‖ = ‖f y - f x - (φ y - φ x)‖ := by simp
_ = ‖f y - φ y - (f x - φ x)‖ := by congr 1; abel
_ = ‖g y - g x‖ := by simp [g]
_ ≤ C * ‖y - x‖ := Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le hg bound hs xs ys | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 101,
"column": 8
} | {
"line": 102,
"column": 71
} | [
{
"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... | have : AnalyticOn 𝕜 (fun y ↦ (continuousMultilinearCurryFin0 𝕜 E F).symm (f y)) w :=
((h'p 0).mono wu).congr fun y hy ↦ (hp.zero_eq' (wu hy)).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 195,
"column": 52
} | {
"line": 196,
"column": 36
} | [
{
"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
subst hy; exact hg.comp_inter x hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 566,
"column": 81
} | {
"line": 568,
"column": 98
} | [
{
"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\nc : E → F →L[𝕜] G\nhc : Co... | by
simp only [← iteratedFDerivWithin_univ]
exact iteratedFDerivWithin_clm_apply_const_apply uniqueDiffOn_univ hc.contDiffOn hi (mem_univ _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 503,
"column": 2
} | {
"line": 503,
"column": 72
} | [
{
"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\nn : ℕ∞ω\ne : G ≃L[... | rw [← contDiffWithinAt_univ, ← contDiffWithinAt_univ, ← preimage_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.OpenPartialHomeomorph.IsImage | {
"line": 354,
"column": 4
} | {
"line": 354,
"column": 41
} | [
{
"pp": "case left\nX : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne e' : OpenPartialHomeomorph X Y\nh : EqOn (↑e) (↑e') (e.source ∩ e'.source)\n⊢ (e.restr e'.source).source = (e'.restr e.source).source",
"usedConstants": [
"Eq.mpr",
"OpenPartialHomeomorph.o... | rw [e'.restr_source' _ e.open_source] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 388,
"column": 6
} | {
"line": 388,
"column": 86
} | [
{
"pp": "case h.e'_5.h.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... | simp only [↓reduceDIte, update_self, add_tsub_cancel_right, comp_apply, cast_mk] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 509,
"column": 56
} | {
"line": 509,
"column": 76
} | [
{
"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 mem_range_self | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 529,
"column": 4
} | {
"line": 529,
"column": 49
} | [
{
"pp": "case 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\nq : ... | · simpa using c.one_lt_partSize_index_zero hc | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 385,
"column": 29
} | {
"line": 389,
"column": 47
} | [
{
"pp": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕂\ninst✝⁷ : NormedRing 𝔸\ninst✝⁶ : NormedRing 𝔹\ninst✝⁵ : NormedAlgebra 𝕂 𝔸\ninst✝⁴ : CompleteSpace 𝔸\ninst✝³ : Algebra 𝕂 𝔹\ninst✝² : CharZero 𝕂\nF : Type u_4\ninst✝¹ : FunLike F 𝔸 𝔹\ninst✝ : RingHomClass F 𝔸 𝔹\n... | by
rw [exp_eq_tsum 𝕂, exp_eq_tsum 𝕂]
refine ((expSeries_summable_of_mem_ball' _ hx).hasSum.map f hf).tsum_eq.symm.trans ?_
dsimp only [Function.comp_def]
simp_rw [map_inv_natCast_smul f 𝕂 𝕂, map_pow] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 682,
"column": 18
} | {
"line": 682,
"column": 95
} | [
{
"pp": "case pos.partSize.zero\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... | change 1 = c.partSize 0; simp [c.partSize_eq_one_of_range_emb_eq_singleton h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 682,
"column": 18
} | {
"line": 682,
"column": 95
} | [
{
"pp": "case pos.partSize.zero\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... | change 1 = c.partSize 0; simp [c.partSize_eq_one_of_range_emb_eq_singleton h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 717,
"column": 53
} | {
"line": 717,
"column": 67
} | [
{
"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... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 717,
"column": 53
} | {
"line": 717,
"column": 67
} | [
{
"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... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 717,
"column": 53
} | {
"line": 717,
"column": 67
} | [
{
"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... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.LocalExtr.Basic | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 21
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\ns : Set E\na y : E\nh : IsLocalMaxOn f s a\nhf : HasFDerivWithinAt f f' s a\nhy : y ∈ posTangentConeAt s a\nι : Type u\nl : Filter ι\nhl : l.NeBot\nc : ι → ℝ≥0\nd : ι → E\nhd₀ : Tendsto d l (𝓝 0)\nhcd :... | rw [add_zero] at hd | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Order.Rolle | {
"line": 55,
"column": 43
} | {
"line": 55,
"column": 57
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝⁶ : ConditionallyCompleteLinearOrder X\ninst✝⁵ : DenselyOrdered X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : OrderTopology X\ninst✝² : LinearOrder Y\ninst✝¹ : TopologicalSpace Y\ninst✝ : OrderTopology Y\nf : X → Y\na b : X\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI :... | by rw [h, hfI] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.Rolle | {
"line": 57,
"column": 41
} | {
"line": 57,
"column": 55
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝⁶ : ConditionallyCompleteLinearOrder X\ninst✝⁵ : DenselyOrdered X\ninst✝⁴ : TopologicalSpace X\ninst✝³ : OrderTopology X\ninst✝² : LinearOrder Y\ninst✝¹ : TopologicalSpace Y\ninst✝ : OrderTopology Y\nf : X → Y\na b : X\nhab : a < b\nhfc : ContinuousOn f (Icc a b)\nhfI :... | by rw [h, hfI] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 432,
"column": 4
} | {
"line": 432,
"column": 95
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nn m : ℕ\nh : n < m\n⊢ iteratedDeriv n (fun x ↦ x ^ m) 0 = ↑(if n = m then m.factorial else 0)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Nat.instCanonicallyOrderedAdd",
"tsub_pos_iff_lt._simp_1",
"False",... | simp_all [Nat.descFactorial_self, Nat.descFactorial_eq_zero_iff_lt.mpr, ne_of_lt, ne_of_gt] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 432,
"column": 4
} | {
"line": 432,
"column": 95
} | [
{
"pp": "case inr.inl\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nn m : ℕ\nh : n = m\n⊢ iteratedDeriv n (fun x ↦ x ^ m) 0 = ↑(if n = m then m.factorial else 0)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Nat.instCanonicallyOrderedAdd",
"MulOne.toOne",
"Nat.instOrdere... | simp_all [Nat.descFactorial_self, Nat.descFactorial_eq_zero_iff_lt.mpr, ne_of_lt, ne_of_gt] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 432,
"column": 4
} | {
"line": 432,
"column": 95
} | [
{
"pp": "case inr.inr\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nn m : ℕ\nh : m < n\n⊢ iteratedDeriv n (fun x ↦ x ^ m) 0 = ↑(if n = m then m.factorial else 0)",
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
"iteratedDeriv_pow",
"NormedRing.toRing",
"HMul... | simp_all [Nat.descFactorial_self, Nat.descFactorial_eq_zero_iff_lt.mpr, ne_of_lt, ne_of_gt] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 35
} | [
{
"pp": "case refine_2\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\ng : ↥⊤ →ₗ[ℝ] ℝ\nhfg : ∀ ⦃x : ↥f.domain⦄ ⦃y : ↥{ domain := ⊤, toFun := g }.domain⦄, ↑x = ↑y → ↑f x = ↑{ domain ... | exact fun x hx => hgs ⟨x, _⟩ hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 35
} | [
{
"pp": "case refine_2\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\ng : ↥⊤ →ₗ[ℝ] ℝ\nhfg : ∀ ⦃x : ↥f.domain⦄ ⦃y : ↥{ domain := ⊤, toFun := g }.domain⦄, ↑x = ↑y → ↑f x = ↑{ domain ... | exact fun x hx => hgs ⟨x, _⟩ hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 35
} | [
{
"pp": "case refine_2\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\ng : ↥⊤ →ₗ[ℝ] ℝ\nhfg : ∀ ⦃x : ↥f.domain⦄ ⦃y : ↥{ domain := ⊤, toFun := g }.domain⦄, ↑x = ↑y → ↑f x = ↑{ domain ... | exact fun x hx => hgs ⟨x, _⟩ hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 104,
"column": 64
} | {
"line": 104,
"column": 84
} | [
{
"pp": "E : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : AddCommGroup E\ninst✝² : Module ℝ E\ns t : Set E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsOpen s\nht : Convex ℝ t\ndisj : Disjoint s t\na₀ : E\nha₀ : a₀ ∈ s\nb₀ : E\nhb₀ : b₀ ∈ t\nx₀ : E := b₀ - a₀\nC : ... | sub_add_sub_cancel', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Module.Dual | {
"line": 85,
"column": 2
} | {
"line": 92,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nr : ℝ\nhr : 0 < r\n⊢ polar 𝕜 (ball 0 r) ⊆ closedBall 0 (‖c‖ / r)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
... | intro x' hx'
rw [StrongDual.mem_polar_iff] at hx'
simp only [mem_closedBall_zero_iff, mem_ball_zero_iff] at *
have hcr : 0 < ‖c‖ / r := div_pos (zero_lt_one.trans hc) hr
refine ContinuousLinearMap.opNorm_le_of_shell hr hcr.le hc fun x h₁ h₂ => ?_
calc
‖x' x‖ ≤ 1 := hx' _ h₂
_ ≤ ‖c‖ / r * ‖x‖ := (inv_l... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Dual | {
"line": 85,
"column": 2
} | {
"line": 92,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nr : ℝ\nhr : 0 < r\n⊢ polar 𝕜 (ball 0 r) ⊆ closedBall 0 (‖c‖ / r)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
... | intro x' hx'
rw [StrongDual.mem_polar_iff] at hx'
simp only [mem_closedBall_zero_iff, mem_ball_zero_iff] at *
have hcr : 0 < ‖c‖ / r := div_pos (zero_lt_one.trans hc) hr
refine ContinuousLinearMap.opNorm_le_of_shell hr hcr.le hc fun x h₁ h₂ => ?_
calc
‖x' x‖ ≤ 1 := hx' _ h₂
_ ≤ ‖c‖ / r * ‖x‖ := (inv_l... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 659,
"column": 10
} | {
"line": 660,
"column": 76
} | [
{
"pp": "case hz\nF : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nx : ℝ\nhx : x ∈ D f K\nn : ℕ → ℕ\nL : ℕ → ℕ → ℕ → F\nhn :\n ∀ (e p q : ℕ),\n n e ≤ p →\n n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ ... | · simp only [pow_add, tsub_le_iff_left] at h'k
simpa only [hy.1, mem_Icc, true_and, one_div, pow_one] using h'k | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 297,
"column": 14
} | {
"line": 297,
"column": 65
} | [
{
"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\nfint : Integrable (fun x ↦ ↑(f x)) μ\nfmeas : AEMeasurable f μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nδ : ℝ≥0\nδpos : 0 < δ\nhδε : δ < ε\nint_f_ne... | ENNReal.toReal_add int_f_ne_top ENNReal.coe_ne_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 693,
"column": 2
} | {
"line": 693,
"column": 45
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\n⊢ MeasurableSet {x | DifferentiableWithinAt ℝ f (Ici x) x ∧ derivWithin f (Ici x) x ∈ K}",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Semiring.toModule",
... | simp only [differentiable_set_eq_D K hK, D] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 65,
"column": 28
} | {
"line": 65,
"column": 36
} | [
{
"pp": "case h.e'_2\nX : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : RCLike 𝕜'\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\ninst✝⁴ : Norme... | clear hf | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 65,
"column": 28
} | {
"line": 65,
"column": 36
} | [
{
"pp": "case h.e'_3\nX : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : RCLike 𝕜'\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\ninst✝⁴ : Norme... | clear hf | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.