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.Analysis.Calculus.Rademacher | {
"line": 296,
"column": 50
} | {
"line": 296,
"column": 78
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nC : ℝ≥0\ninst✝ : FiniteDimensional ℝ E\nf : E → F\nhf : LipschitzWith C f\ns : Set E\nhs : sphere 0 1 ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 312,
"column": 8
} | {
"line": 312,
"column": 34
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nF : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nC : ℝ≥0\ninst✝ : FiniteDimensional ℝ E\nf : E → F\nhf : LipschitzWith C f\ns : Set E\nhs : sphere 0 1 ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 190,
"column": 51
} | {
"line": 190,
"column": 62
} | [
{
"pp": "F : Type u_2\ninst✝ : SeminormedAddCommGroup F\na b : ℝ\nf g : ℝ → F\nhf : AbsolutelyContinuousOnInterval f a b\nhg : AbsolutelyContinuousOnInterval g a b\n⊢ Tendsto ?m.14 (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 199,
"column": 51
} | {
"line": 199,
"column": 62
} | [
{
"pp": "F : Type u_2\ninst✝ : SeminormedAddCommGroup F\na b : ℝ\nf : ℝ → F\nhf : AbsolutelyContinuousOnInterval f a b\n⊢ Tendsto ?m.10 (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 30
} | [
{
"pp": "F : Type u_2\ninst✝ : SeminormedAddCommGroup F\na b : ℝ\nf g : ℝ → F\nhf : AbsolutelyContinuousOnInterval f a b\nhg : AbsolutelyContinuousOnInterval g a b\n⊢ AbsolutelyContinuousOnInterval (f - g) a b",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"AddMonoid.toAddZeroCla... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 212,
"column": 51
} | {
"line": 212,
"column": 62
} | [
{
"pp": "F : Type u_2\ninst✝³ : SeminormedAddCommGroup F\na b : ℝ\nf : ℝ → F\nM : Type u_3\ninst✝² : SeminormedRing M\ninst✝¹ : Module M F\ninst✝ : NormSMulClass M F\nα : M\nhf : AbsolutelyContinuousOnInterval f a b\n⊢ Tendsto ?m.18 (totalLengthFilter ⊓ 𝓟 (disjWithin a b)) (𝓝 0)",
"usedConstants": []
}
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 90
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhf : MonotoneOn f (Icc a b)\ng : ℝ → ℝ := fun x ↦ f (max a (min x b))\nhg : Monotone g\nhfg : EqOn (deriv f) (deriv g) (Ioo a b)\nh₁ : ∀ᵐ (x : ℝ), x ≠ a\nh₂ : ∀ᵐ (x : ℝ), x ≠ b\nG : ℝ → ℝ → ℝ := fun c x ↦ slope g x (x + c)\nG_integrable : ∀ (n : ℕ), Integrable (G (↑n)⁻¹... | refine ⟨fun n x ↦ G (n : ℝ)⁻¹ x, ?_, fun n ↦ G_integrable n |>.aestronglyMeasurable, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 265,
"column": 8
} | {
"line": 265,
"column": 19
} | [
{
"pp": "F : Type u_2\ninst✝³ : SeminormedAddCommGroup F\na b : ℝ\nM : Type u_3\ninst✝² : SeminormedRing M\ninst✝¹ : Module M F\ninst✝ : NormSMulClass M F\nf : ℝ → M\ng : ℝ → F\nhf :\n Tendsto (fun E ↦ ∑ i ∈ Finset.range E.1, dist (f (E.2 i).1) (f (E.2 i).2)) (totalLengthFilter ⊓ 𝓟 (disjWithin a b))\n (𝓝 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.AbsolutelyContinuous | {
"line": 301,
"column": 6
} | {
"line": 301,
"column": 29
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoMetricSpace X\na b : ℝ\nf : ℝ → X\nK : ℝ≥0\nhfK : LipschitzOnWith K f (uIcc a b)\nε : ℝ\nhε : ε > 0\nx✝ : ℕ × (ℕ → ℝ × ℝ)\nn : ℕ\nI : ℕ → ℝ × ℝ\nhnI₁ : (n, I) ∈ disjWithin a b\nhnI₂ : ∑ i ∈ Finset.range (n, I).1, dist ((n, I).2 i).1 ((n, I).2 i).2 < ε / (↑K + 1)\n⊢ ∑ i ∈ Fin... | apply Finset.sum_le_sum | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 366,
"column": 69
} | {
"line": 371,
"column": 71
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\nF : Type u_2\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nC : ℝ≥0\ns : Set E\nμ : Measure E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : FiniteDimensional ℝ F\ninst✝ : μ.I... | by
filter_upwards [H] with x hx xs
have : f = (A.symm ∘ A) ∘ f := by
simp only [ContinuousLinearEquiv.symm_comp_self, Function.id_comp]
rw [this]
exact A.symm.differentiableAt.comp_differentiableWithinAt x (hx xs) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 43
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\nF : Type u_2\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nC : ℝ≥0\nμ : Measure E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : FiniteDimensional ℝ F\ninst✝ : μ.IsAddHaarMea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable | {
"line": 152,
"column": 59
} | {
"line": 152,
"column": 73
} | [
{
"pp": "a b : ℝ\np q : ℝ → ℝ\nhp : MonotoneOn p (uIcc a b)\nhq : MonotoneOn q (uIcc a b)\nhf : BoundedVariationOn (p - q) (uIcc a b)\nh₂ : ∀ᵐ (x : ℝ), x ≠ max a b\n⊢ MeasurableSet (uIoc a b)",
"usedConstants": [
"Set.Ioc",
"instClosedIicTopology",
"Real",
"Lattice.toSemilatticeSup",... | by simp [uIoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 32
} | [
{
"pp": "case h\na b : ℝ\np q : ℝ → ℝ\nhp : MonotoneOn p (uIcc a b)\nhq : MonotoneOn q (uIcc a b)\nhf : BoundedVariationOn (p - q) (uIcc a b)\nh₂ : ∀ᵐ (x : ℝ), x ≠ max a b\nx : ℝ\nhx₃ : x ≠ max a b\nhx₄ : min a b < x ∧ x ≤ max a b\nhx₅ : x ∈ uIcc a b\nhx₆ : uIcc a b ∈ 𝓝 x\nhx₁ : HasDerivAt p (deriv p x) x\nhx₂... | exact (hx₁.sub hx₂).deriv.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 77,
"column": 61
} | {
"line": 97,
"column": 47
} | [
{
"pp": "s : ℝ\n⊢ ∃ M ε, 0 < M ∧ 0 < ε ∧ ∀ (x y : ℝ), 0 < x → x < ε → |y| < s * x → √(x ^ 2 + y ^ 2) < M * (1 - √((1 - x) ^ 2 + y ^ 2))",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Mathlib.T... | by
refine ⟨2 * √(1 + s ^ 2) + 1, 1 / (1 + s ^ 2), by positivity, by positivity,
fun x y hx₀ hx₁ hy ↦ ?_⟩
have H : √((1 - x) ^ 2 + y ^ 2) ≤ 1 - x / 2 := by
calc √((1 - x) ^ 2 + y ^ 2)
_ ≤ √((1 - x) ^ 2 + (s * x) ^ 2) := sqrt_le_sqrt <| by rw [← sq_abs y]; gcongr
_ = √(1 - 2 * x + (1 + s ^ 2) * x ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Taylor | {
"line": 177,
"column": 80
} | {
"line": 198,
"column": 35
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx y : ℝ\nn : ℕ\ns s' : Set ℝ\nhs_unique : UniqueDiffOn ℝ s\nhs' : s' ∈ 𝓝[s] y\nhy : y ∈ s'\nh : s' ⊆ s\nhf : ContDiffOn ℝ (↑n) f s\nhf' : DifferentiableWithinAt ℝ (iteratedDerivWithin n f s) s y\n⊢ HasDerivWithinAt (fun t... | by
have hs'_unique : UniqueDiffWithinAt ℝ s' y :=
UniqueDiffWithinAt.mono_nhds (hs_unique _ (h hy)) (nhdsWithin_le_iff.mpr hs')
induction n with
| zero =>
simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDeri... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Taylor | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 13
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx₀ : ℝ\nn : ℕ\nhf : ContDiff ℝ (↑n) f\n⊢ (fun x ↦ f x - taylorWithinEval f n univ x₀ x) =o[𝓝 x₀] fun x ↦ (x - x₀) ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Taylor | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 43
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx₀ : ℝ\nn : ℕ\ns : Set ℝ\nhs : Convex ℝ s\nhx₀s : x₀ ∈ s\nhf : ContDiffOn ℝ (↑n) f s\nh_isLittleO : Filter.Tendsto (fun x ↦ ‖f x - taylorWithinEval f n s x₀ x‖ / ‖(x - x₀) ^ n‖) (𝓝[s] x₀) (𝓝 0)\n⊢ Filter.Tendsto (fun x ↦... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 40
} | [
{
"pp": "case hf\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nz : ℂ\nhz : ‖z‖ < 1\ns : ℕ → ℂ := ⋯\nk :\n Tendsto (fun n ↦ (1 - z) * ∑ j ∈ range n, (∑ k ∈ range n, f k - ∑ k ∈ range (j + 1), f k) * z ^ j) atTop\n (𝓝 (l - ∑' (i : ℕ), f i * z ^ i))\n⊢ Tendsto (fun x ↦ l - s x) atT... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ray | {
"line": 50,
"column": 2
} | {
"line": 52,
"column": 46
} | [
{
"pp": "E : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx y : E\nh : SameRay ℝ x y\n⊢ ‖x‖ • y = ‖y‖ • x",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real",
"instHSMul",
"HMul.hMul",
"congrArg",
"DistribMulAction.... | rcases h.exists_eq_smul with ⟨u, a, b, ha, hb, -, rfl, rfl⟩
simp only [norm_smul_of_nonneg, *, mul_smul]
rw [smul_comm, smul_comm b, smul_comm a b u] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Ray | {
"line": 50,
"column": 2
} | {
"line": 52,
"column": 46
} | [
{
"pp": "E : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx y : E\nh : SameRay ℝ x y\n⊢ ‖x‖ • y = ‖y‖ • x",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real",
"instHSMul",
"HMul.hMul",
"congrArg",
"DistribMulAction.... | rcases h.exists_eq_smul with ⟨u, a, b, ha, hb, -, rfl, rfl⟩
simp only [norm_smul_of_nonneg, *, mul_smul]
rw [smul_comm, smul_comm b, smul_comm a b u] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Ray | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 41
} | [
{
"pp": "F : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ny : F\nhy : y ≠ 0\n⊢ Set.InjOn Norm.norm {x | SameRay ℝ x y}",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"NormedSp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Taylor | {
"line": 435,
"column": 23
} | {
"line": 435,
"column": 54
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na : ℝ\nn : ℕ\nhab : a ≤ a\nhf : ContDiffOn ℝ (↑n + 1) f (Icc a a)\nx : ℝ\nhx : x ∈ Icc a a\n⊢ x = a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.StrictConvexSpace | {
"line": 87,
"column": 18
} | {
"line": 87,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : LinearMap.CompatibleSMul E E 𝕜 ℝ\nh : StrictConvex 𝕜 (closedBall 0 1)\nr : ℝ\nhr : 0 < r\n⊢ StrictConvex 𝕜 (closedBall 0 r)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.AbelLimit | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 65
} | [
{
"pp": "case right\nf : ℕ → ℂ\nl : ℂ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, f i) atTop (𝓝 l)\nM : ℝ\nhM : 1 < M\ns : ℕ → ℂ := fun n ↦ ∑ i ∈ range n, f i\ng : ℂ → ℂ := fun z ↦ ∑' (n : ℕ), f n * z ^ n\nhm : ∀ ε > 0, ∃ N, ∀ n ≥ N, ‖∑ i ∈ range n, f i - l‖ < ε\nε : ℝ\nεpos : ε > 0\nB₁ : ℕ\nhB₁ : ∀ n ≥ B₁, ‖∑ i ∈ ra... | simp_rw [Metric.tendsto_atTop, dist_eq_norm, norm_sub_rev] at p | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Convex.StrictConvexSpace | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 60
} | [
{
"pp": "E : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y : E\nh : ¬SameRay ℝ x y\n⊢ |‖x‖ - ‖y‖| < ‖x - y‖",
"usedConstants": [
"lt_norm_sub_of_not_sameRay",
"Iff.mpr",
"Norm.norm",
"Real",
"Preorder.toLT",
"abs",
... | refine abs_sub_lt_iff.2 ⟨lt_norm_sub_of_not_sameRay h, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.StrictConvexSpace | {
"line": 212,
"column": 69
} | {
"line": 212,
"column": 86
} | [
{
"pp": "E : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y : E\nh : ‖x‖ = ‖y‖\n⊢ 1 / 2 * ‖x + y‖ < ‖x‖ ↔ x ≠ y",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
... | ← inv_eq_one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.AbsMax | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 53
} | [
{
"pp": "F : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : ℂ → F\nz w : ℂ\nhd : DiffContOnCl ℂ f (ball z (dist w z))\nhz : IsMaxOn (norm ∘ f) (closedBall z (dist w z)) z\ne : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL\nhe : ∀ (x : F), ‖e x‖ = ‖x‖\n⊢ IsMaxOn (norm ∘ ⇑e ∘ f) (closedBall ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.AbsMax | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 9
} | [
{
"pp": "F : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : ℂ → F\nz w : ℂ\nhd : DiffContOnCl ℂ f (ball z (dist w z))\ne : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL\nhe : ∀ (x : F), ‖e x‖ = ‖x‖\nhz : IsMaxOn (norm ∘ ⇑e ∘ f) (closedBall z (dist w z)) z\n⊢ ‖f w‖ = ‖f z‖",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.AbsMax | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 62
} | [
{
"pp": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nz : E\nr : ℝ\nhd : DiffContOnCl ℂ f (ball z r)\nhz : IsMaxOn (norm ∘ f) (ball z r) z\nw : E\nhw : dist z w ≤ r\nhne : z ≠ w\ne : ℂ → E := ⇑(lineMap z w)\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.AbsMax | {
"line": 237,
"column": 4
} | {
"line": 238,
"column": 11
} | [
{
"pp": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nU : Set E\nc : E\nhc : IsPreconnected U\nho : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U\nhd : DifferentiableOn ℂ f U\nhcU : c ∈ U\nhm :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 34
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : DifferentiableOn ℝ f (interior D)\nh_anti : AntitoneOn (deriv f) (interior D)\n⊢ MonotoneOn (deriv (-f)) (interior D)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 40,
"column": 2
} | {
"line": 41,
"column": 49
} | [
{
"pp": "x y : ℝ\nh :\n ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * π / 2) ∧ ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * π / 2) ∨\n (∃ k, x = (2 * ↑k + 1) * π / 2) ∧ ∃ l, y = (2 * ↑l + 1) * π / 2\n⊢ tan (x + y) = (tan x + tan y) / (1 - tan x * tan y)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.AbsMax | {
"line": 426,
"column": 51
} | {
"line": 426,
"column": 76
} | [
{
"pp": "E : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : Nontrivial E\nf g : E → F\nU : Set E\nhU : Bornology.IsBounded U\nhf : DiffContOnCl ℂ f U\nhg : DiffContOnCl ℂ g U\nhfg : EqOn f g (frontier U)\nH : ∀ z ∈ cl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 53,
"column": 2
} | {
"line": 54,
"column": 49
} | [
{
"pp": "x y : ℝ\nh :\n ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * π / 2) ∧ ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * π / 2) ∨\n (∃ k, x = (2 * ↑k + 1) * π / 2) ∧ ∃ l, y = (2 * ↑l + 1) * π / 2\n⊢ tan (x - y) = (tan x - tan y) / (1 + tan x * tan y)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.ComplexDeriv | {
"line": 41,
"column": 36
} | {
"line": 41,
"column": 55
} | [
{
"pp": "x : ℂ\nhx : cos x = 0\nh : sin x = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 71,
"column": 4
} | {
"line": 72,
"column": 22
} | [
{
"pp": "this : ContinuousOn (fun x ↦ sin x / cos x) {x | cos x ≠ 0}\n⊢ ContinuousOn tan {x | cos x ≠ 0}",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.instZero",
"Real.cos",
"congrArg",
"Real.instDivInvMonoid",
"setOf",
"PseudoMetricSpace.toUn... | have h_eq : (fun x => sin x / cos x) = tan := by ext1 x; rw [tan_eq_sin_div_cos]
rwa [h_eq] at this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 71,
"column": 4
} | {
"line": 72,
"column": 22
} | [
{
"pp": "this : ContinuousOn (fun x ↦ sin x / cos x) {x | cos x ≠ 0}\n⊢ ContinuousOn tan {x | cos x ≠ 0}",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.instZero",
"Real.cos",
"congrArg",
"Real.instDivInvMonoid",
"setOf",
"PseudoMetricSpace.toUn... | have h_eq : (fun x => sin x / cos x) = tan := by ext1 x; rw [tan_eq_sin_div_cos]
rwa [h_eq] at this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 29
} | [
{
"pp": "x : ℝ\nA : cos (arctan x) ≠ 0\n⊢ HasStrictDerivAt arctan (1 / (1 + x ^ 2)) x",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 92,
"column": 4
} | {
"line": 99,
"column": 61
} | [
{
"pp": "case neg\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : ℝ\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : ℝ\nha : (f w - f x) / (w - x) < deriv f a\nhxa : x < a\nhaw : a < w\n⊢ ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a",
"... | obtain ⟨b, ⟨hwb, hby⟩, hb⟩ : ∃ b ∈ Ioo w y, (f y - f w) / (y - w) < deriv f b := by
apply StrictMonoOn.exists_slope_lt_deriv_aux _ hwy _ _
· refine hf.mono (Icc_subset_Icc hxw.le le_rfl)
· exact hf'_mono.mono (Ioo_subset_Ioo hxw.le le_rfl)
· intro z hz
rw [← hw]
apply ne_of_gt
... | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 32
} | [
{
"pp": "x : ℝ\n⊢ 0 < arctan x ↔ 0 < x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 32
} | [
{
"pp": "x : ℝ\n⊢ arctan x < 0 ↔ x < 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 32
} | [
{
"pp": "x : ℝ\n⊢ 0 ≤ arctan x ↔ 0 ≤ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 32
} | [
{
"pp": "x : ℝ\n⊢ arctan x ≤ 0 ↔ x ≤ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 15
} | [
{
"pp": "case hx₁\nx : ℝ\nh : 0 < x\n⊢ -(π / 2) < π / 2 - arctan x",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"instHDiv",
"Real.pi",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 35
} | [
{
"pp": "z w : ℂ\n⊢ cos z - w = 0 ↔ cexp (z * I) ^ 2 - 2 * w * cexp (z * I) + 1 = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Complex.exp_ne_zero._simp_1",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"instDecidableNot",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 59,
"column": 44
} | {
"line": 59,
"column": 55
} | [
{
"pp": "x : ℝ\nhx : x < 0\n⊢ x < sin x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 60,
"column": 44
} | {
"line": 60,
"column": 55
} | [
{
"pp": "x : ℝ\nhx : x ≤ 0\n⊢ x ≤ sin x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 26
} | [
{
"pp": "x : ℝ\nhx : 0 < x\nhx' : x < 1\n⊢ x < sin (π / 2 * x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 26
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\nhx' : x ≤ 1\n⊢ x ≤ sin (π / 2 * x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 65
} | [
{
"pp": "x : ℝ\nhx : 0 < x\nhx' : x < π / 2\n⊢ (π / 2)⁻¹ * x < sin x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 65
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\nhx' : x ≤ π / 2\n⊢ (π / 2)⁻¹ * x ≤ sin x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : -(π / 2) ≤ x\nhx₀ : x ≤ 0\n⊢ sin x ≤ 2 / π * x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 95,
"column": 14
} | {
"line": 95,
"column": 25
} | [
{
"pp": "x : ℝ\nhx : |x| ≤ π / 2\nthis : ∀ {x : ℝ}, |x| ≤ π / 2 → 0 ≤ x → 2 / π * |x| ≤ |sin x|\nhx₀ : ¬0 ≤ x\n⊢ 2 / π * |x| ≤ |sin x|",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 15
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\nthis : ∀ {x : ℝ}, x ≠ 0 → 0 < x → sin x ^ 2 < x ^ 2\nhx₀ : x ≤ 0\n⊢ sin x ^ 2 < x ^ 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 336,
"column": 2
} | {
"line": 337,
"column": 34
} | [
{
"pp": "⊢ 4 * arctan 5⁻¹ - arctan 239⁻¹ = π / 4",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHDiv",
"Real.pi",
"Mathlib.Tactic.Ring.Common.mul_congr",
"HMul.hMul",
"Real.arctan",
"Nat.rawCast",
"congrArg",... | rw [show 4 * arctan _ = 2 * (2 * _) by ring, two_mul_arctan, two_mul_arctan, ← arctan_one,
sub_eq_iff_eq_add, arctan_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 13
} | [
{
"pp": "x : ℝ\n⊢ 0 < sin (arctan x) ↔ 0 < x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 13
} | [
{
"pp": "x : ℝ\n⊢ sin (arctan x) < 0 ↔ x < 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 183,
"column": 54
} | {
"line": 183,
"column": 84
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nh_anti : StrictAntiOn (deriv f) (interior D)\n⊢ StrictMonoOn (deriv (-f)) (interior D)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 13
} | [
{
"pp": "x : ℝ\n⊢ 0 ≤ sin (arctan x) ↔ 0 ≤ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 13
} | [
{
"pp": "x : ℝ\n⊢ sin (arctan x) ≤ 0 ↔ x ≤ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 130,
"column": 2
} | {
"line": 131,
"column": 9
} | [
{
"pp": "x : ℝ\nhx₀ : 0 ≤ x\nhx : x ≤ π / 2\n⊢ 1 - 2 / π * x ≤ cos x",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.cos",
"Real.instDivInvMonoid",
"Real.instSub",
"covariant_swap_add_of_covariant_add... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 13
} | [
{
"pp": "x : ℝ\nhx₀ : -(π / 2) ≤ x\nhx : x ≤ 0\n⊢ 1 + 2 / π * x ≤ cos x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 139,
"column": 14
} | {
"line": 139,
"column": 25
} | [
{
"pp": "x : ℝ\nhx : |x| ≤ π\nthis : ∀ {x : ℝ}, |x| ≤ π → 0 ≤ x → cos x ≤ 1 - 2 / π ^ 2 * x ^ 2\nhx₀ : ¬0 ≤ x\n⊢ cos x ≤ 1 - 2 / π ^ 2 * x ^ 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 141,
"column": 35
} | {
"line": 141,
"column": 46
} | [
{
"pp": "x✝ x : ℝ\nhx : x ≤ π\nhx₀ : 0 ≤ x\n⊢ x / π ≤ sin (x / 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 165,
"column": 25
} | {
"line": 165,
"column": 36
} | [
{
"pp": "x : ℝ\nh : cos x ≠ 0\n⊢ HasDerivAt (fun y ↦ tan y - y) (1 / cos x ^ 2 - 1) x",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"NormedSpace.toIsBoundedSMul",
"Real.denselyNormedField",
"Rea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 193,
"column": 6
} | {
"line": 193,
"column": 56
} | [
{
"pp": "x : ℝ\nh1 : 0 < x\nh2 : x < π / 2\nU : Set ℝ := Ico 0 (π / 2)\nintU : interior U = Ioo 0 (π / 2)\nhalf_pi_pos : 0 < π / 2\ncos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y\nsin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y\ntan_cts_U : ContinuousOn tan U\ntan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y) U\ny : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 39
} | [
{
"pp": "case hb\nx : ℝ\nh1 : 0 < x\nh2 : x < π / 2\nU : Set ℝ := Ico 0 (π / 2)\nintU : interior U = Ioo 0 (π / 2)\nhalf_pi_pos : 0 < π / 2\ncos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y\nsin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y\ntan_cts_U : ContinuousOn tan U\ntan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 48
} | [
{
"pp": "x : ℝ\nh1 : 0 < x\nh2 : x < π / 2\nU : Set ℝ := Ico 0 (π / 2)\nintU : interior U = Ioo 0 (π / 2)\nhalf_pi_pos : 0 < π / 2\ncos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y\nsin_pos : ∀ {y : ℝ}, y ∈ interior U → 0 < sin y\ntan_cts_U : ContinuousOn tan U\ntan_minus_id_cts : ContinuousOn (fun y ↦ tan y - y) U\nderi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 43
} | [
{
"pp": "x : ℝ\nh1 : 0 ≤ x\nh2 : x < π / 2\n⊢ x ≤ tan x",
"usedConstants": [
"Real.partialOrder",
"Real",
"Real.instZero",
"eq_or_lt_of_le",
"Zero.toOfNat0",
"OfNat.ofNat"
]
}
] | rcases eq_or_lt_of_le h1 with (rfl | h1') | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Convex.Deriv | {
"line": 438,
"column": 10
} | {
"line": 438,
"column": 21
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx : ℝ\nhfc : ConvexOn ℝ S f\nhxs : x ∈ interior S\na b : ℝ\nhxab : x ∈ Ioo a b\nhabs : Ioo a b ⊆ S\nh : Ioo x b ⊆ {y | y ∈ S ∧ x < y}\n⊢ (Ioo x b).Nonempty",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"MulZeroClass.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 216,
"column": 34
} | {
"line": 216,
"column": 56
} | [
{
"pp": "x : ℝ\nhx1 : -(3 * π / 2) ≤ x\nhx2 : x ≤ 3 * π / 2\nhx3 : x ≠ 0\ny : ℝ\nhy1 : 0 < y\nhy2 : y ≤ 3 * π / 2\n⊢ 0 < y ^ 2 + 1",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroC... | linarith [sq_nonneg y] | Mathlib.Tactic._aux_Mathlib_Tactic_Linarith_Frontend___elabRules_Mathlib_Tactic_linarith_1 | Mathlib.Tactic.linarith |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 237,
"column": 60
} | {
"line": 237,
"column": 71
} | [
{
"pp": "⊢ ∀ (x : ℝ), ‖deriv sin x‖₊ ≤ 1",
"usedConstants": [
"Eq.mpr",
"Real",
"Semiring.toModule",
"Real.denselyNormedField",
"Real.cos",
"Real.instRCLike",
"congrArg",
"deriv",
"SeminormedAddGroup.toNNNorm",
"NNNorm.nnnorm",
"NormedSpace.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 240,
"column": 60
} | {
"line": 240,
"column": 71
} | [
{
"pp": "⊢ ∀ (x : ℝ), ‖deriv cos x‖₊ ≤ 1",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Semiring.toModule",
"Real.denselyNormedField",
"Real.cos",
"Real.instRCLike",
"congrArg",
"deriv",
"SeminormedAddGroup.toNNNorm"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 26
} | [
{
"pp": "x y : ℝ\n⊢ |sin x - sin y| ≤ |x - y|",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 26
} | [
{
"pp": "x y : ℝ\n⊢ |cos x - cos y| ≤ |x - y|",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 456,
"column": 10
} | {
"line": 456,
"column": 21
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx : ℝ\nhfc : ConvexOn ℝ S f\nhxs : x ∈ interior S\na b : ℝ\nhxab : x ∈ Ioo a b\nhabs : Ioo a b ⊆ S\nh : Ioo a x ⊆ {y | y ∈ S ∧ y < x}\n⊢ (Ioo a x).Nonempty",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"MulZeroClass.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 505,
"column": 4
} | {
"line": 506,
"column": 24
} | [
{
"pp": "case inr.refine_2\nS : Set ℝ\nf : ℝ → ℝ\nhfc : ConvexOn ℝ S f\nx : ℝ\nhxs : x ∈ interior S\ny : ℝ\nhys : y ∈ interior S\nhxy✝ : x ≤ y\nhxy : x < y\nz : ℝ\nhzs : z ∈ S\nhyz : y < z\n⊢ slope f y x ≤ slope f y z",
"usedConstants": [
"Real",
"interior_subset",
"PseudoMetricSpace.toUni... | exact slope_mono hfc (interior_subset hys) ⟨interior_subset hxs, hxy.ne⟩ ⟨hzs, hyz.ne'⟩
(hxy.trans hyz).le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Deriv | {
"line": 866,
"column": 2
} | {
"line": 866,
"column": 51
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : ConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhf' : HasDerivWithinAt f f' (Ioi x) x\n⊢ slope f x y ≤ f'",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 905,
"column": 2
} | {
"line": 905,
"column": 51
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : ConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhf' : HasDerivWithinAt f f' (Iio y) y\n⊢ f' ≤ slope f x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Angle | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 13
} | [
{
"pp": "x : ℝ\n⊢ angle (cexp (↑x * I)) 1 = |toIocMod Real.two_pi_pos (-π) x|",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 951,
"column": 2
} | {
"line": 951,
"column": 13
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nhfc : ConcaveOn ℝ S f\nhfd : ∀ x ∈ S, DifferentiableAt ℝ f x\n⊢ AntitoneOn (deriv f) S",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 965,
"column": 2
} | {
"line": 965,
"column": 51
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : StrictConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhf' : HasDerivWithinAt f f' (Ioi x) x\n⊢ slope f x y < f'",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 976,
"column": 2
} | {
"line": 976,
"column": 51
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : StrictConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhfd : HasDerivWithinAt f f' S x\n⊢ slope f x y < f'",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 987,
"column": 2
} | {
"line": 987,
"column": 51
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : StrictConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhfd : HasDerivAt f f' x\n⊢ slope f x y < f'",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 1005,
"column": 2
} | {
"line": 1005,
"column": 51
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : StrictConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhf' : HasDerivWithinAt f f' (Iio y) y\n⊢ f' < slope f x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 1016,
"column": 2
} | {
"line": 1016,
"column": 51
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : StrictConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhf' : HasDerivWithinAt f f' S y\n⊢ f' < slope f x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Deriv | {
"line": 1047,
"column": 2
} | {
"line": 1047,
"column": 13
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nhfc : StrictConcaveOn ℝ S f\nhfd : ∀ x ∈ S, DifferentiableAt ℝ f x\n⊢ StrictAntiOn (deriv f) S",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.BorelCaratheodory | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 21
} | [
{
"pp": "f : ℂ → ℂ\nM R : ℝ\nz : ℂ\nhM : 0 < M\nhf : DifferentiableOn ℂ f (ball 0 R)\nhf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}\nhz : z ∈ ball 0 R\nhf₂ : f 0 = 0\nx : ℂ\nhx : x ∈ ball 0 R\n⊢ f x / (2 * ↑M - f x) ∈ closedBall (f 0 / (2 * ↑M - f 0)) 1",
"usedConstants": [
"AddGroup.toSubtractionMonoi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.BorelCaratheodory | {
"line": 91,
"column": 17
} | {
"line": 92,
"column": 9
} | [
{
"pp": "f : ℂ → ℂ\nM R : ℝ\nz : ℂ\nhM : 0 < M\nhf : DifferentiableOn ℂ f (ball 0 R)\nhf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}\nhR : 0 < R\nhz : z ∈ ball 0 R\nhf₂ : f 0 = 0\nw : ℂ := f z / (2 * ↑M - f z)\nhzR : ‖z‖ < R\n⊢ ?m.101",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.BorelCaratheodory | {
"line": 93,
"column": 67
} | {
"line": 93,
"column": 88
} | [
{
"pp": "f : ℂ → ℂ\nM R : ℝ\nz : ℂ\nhM : 0 < M\nhf : DifferentiableOn ℂ f (ball 0 R)\nhf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}\nhR : 0 < R\nhz : z ∈ ball 0 R\nhf₂ : f 0 = 0\nw : ℂ := f z / (2 * ↑M - f z)\nhzR : ‖z‖ < R\nhwR : ‖f z / (2 * ↑M - f z)‖ ≤ ‖z‖ / R\nh : 2 * ↑M = f z\n⊢ False",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.BorelCaratheodory | {
"line": 100,
"column": 8
} | {
"line": 100,
"column": 19
} | [
{
"pp": "case hdb\nf : ℂ → ℂ\nM R : ℝ\nz : ℂ\nhM : 0 < M\nhf : DifferentiableOn ℂ f (ball 0 R)\nhf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}\nhR : 0 < R\nhz : z ∈ ball 0 R\nhf₂ : f 0 = 0\nw : ℂ := ⋯\nhzR : ‖z‖ < R\nhwR : ‖f z / (2 * ↑M - f z)‖ ≤ ‖z‖ / R\nh_denom : 2 * ↑M - f z ≠ 0\n⊢ 1 - ‖w‖ ≤ ‖1 + w‖",
"us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.Action | {
"line": 35,
"column": 8
} | {
"line": 35,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\ninst✝⁴ : NormedField 𝕜\ninst✝³ : NormedField 𝕜'\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜' E\nr : ℝ\nc : ↑(closedBall 0 1)\nx : ↑(ball 0 r)\n⊢ ‖↑c • ↑x‖ < r",
"usedConstants": [
"Norm.norm",
"Semi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Ball.Action | {
"line": 50,
"column": 8
} | {
"line": 50,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\nE : Type u_3\ninst✝⁴ : NormedField 𝕜\ninst✝³ : NormedField 𝕜'\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜' E\nr : ℝ\nc : ↑(closedBall 0 1)\nx : ↑(closedBall 0 r)\n⊢ ‖↑c • ↑x‖ ≤ r",
"usedConstants": [
"Norm.norm",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 49
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\np : 𝕜[X]\nx : 𝕜\nhx : x ∈ ((fun x ↦ eval x p) '' {k | eval k (derivative p) = 0})ᶜ\nne : p ≠ C x\n⊢ ((fun x ↦ eval x p) ⁻¹' {x}).Finite",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 59
} | [
{
"pp": "case convert_5\n𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nn : ℕ\nhn : ↑n ≠ 0\nx' : 𝕜\nh : x' ∈ {0}ᶜ\n⊢ x' ∈ ((fun x ↦ eval x (X ^ n)) '' {k | eval k (derivative (X ^ n)) = 0})ᶜ",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Polynomial.derivative"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 17
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nn : ℕ\nhn : ↑(-↑n) ≠ 0\n⊢ ↑n ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedRing.toRing",
"AddGroupWithOne.toAddMonoidWithOne",
"NormedField.toField",
"id",
"Add... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 94,
"column": 9
} | {
"line": 94,
"column": 20
} | [
{
"pp": "case refine_2.convert_2.h.a\n𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nn : ℤ\nhn : ↑n ≠ 0\nthis : ∀ (x : 𝕜), x ^ n = 0 ↔ x = 0\nx✝ : 𝕜\n⊢ x✝ ∈ (fun x ↦ x ^ n) ⁻¹' {0}ᶜ ↔ x✝ ≠ 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Div... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Schwarz | {
"line": 114,
"column": 33
} | {
"line": 114,
"column": 58
} | [
{
"pp": "f : ℂ → ℂ\nc z : ℂ\nR₂ : ℝ\nn : ℕ\nhn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n\nR₁ : ℝ\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nhd : DifferentiableOn ℂ f (closedBall c R₁)\nh_maps : MapsTo f (closedBall c R₁) (closedBall (f c) R₂)\nhne : z ≠ c\ng : ℂ → ℂ := fun w ↦ ((w - c) ^ (n + 1))⁻¹ * (f w - f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Schwarz | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 54
} | [
{
"pp": "f : ℂ → ℂ\nc z : ℂ\nR₂ : ℝ\nn : ℕ\nhn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n\nR₁ : ℝ\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nhd : DifferentiableOn ℂ f (closedBall c R₁)\nh_maps : MapsTo f (closedBall c R₁) (closedBall (f c) R₂)\nhne : z ≠ c\ng : ℂ → ℂ := fun w ↦ ((w - c) ^ (n + 1))⁻¹ * (f w - f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Schwarz | {
"line": 122,
"column": 6
} | {
"line": 122,
"column": 32
} | [
{
"pp": "f : ℂ → ℂ\nc z : ℂ\nR₂ : ℝ\nn : ℕ\nhn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n\nR₁ : ℝ\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nhd : DifferentiableOn ℂ f (closedBall c R₁)\nh_maps : MapsTo f (closedBall c R₁) (closedBall (f c) R₂)\nhne : z ≠ c\ng : ℂ → ℂ := fun w ↦ ((w - c) ^ (n + 1))⁻¹ * (f w - f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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