module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.InnerProductSpace.TwoDim | {
"line": 425,
"column": 2
} | {
"line": 427,
"column": 41
} | {
"line": 429,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : Fact (finrank ℝ E = 2)\no : Orientation ℝ E (Fin 2)\nx y : E\n⊢ (o.kahler (o.rightAngleRotation x)) y = -Complex.I * (o.kahler x) y",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"instInner... | [] | simp only [o.areaForm_rightAngleRotation_left, o.inner_rightAngleRotation_left,
o.kahler_apply_apply, Complex.ofReal_neg, Complex.real_smul]
linear_combination ω x y * Complex.I_sq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MellinInversion | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 52
} | {
"line": 82,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nσ : ℝ\nf : ℂ → E\nx : ℝ\nhx : 0 < x\nhx0 : ↑x ≠ 0\n⊢ |(2 * π)⁻¹| • ↑x ^ (-↑σ) • ∫ (a : ℝ), ↑x ^ (-(↑a * I)) • f (↑σ + ↑a * I) =\n ↑x ^ (-↑σ) • ∫ (y : ℝ), cexp (2 * ↑π * (↑y * -↑(Real.log x)) * I) • f (↑σ + 2 * ↑π * ↑y * I)",
"... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nσ : ℝ\nf : ℂ → E\nx : ℝ\nhx : 0 < x\nhx0 : ↑x ≠ 0\n⊢ ↑x ^ (-↑σ) • ∫ (x_1 : ℝ), ↑x ^ (-(↑(2 * π * x_1) * I)) • f (↑σ + ↑(2 * π * x_1) * I) =\n ↑x ^ (-↑σ) • ∫ (y : ℝ), cexp (2 * ↑π * (↑y * -↑(Real.log x)) * I) • f (↑σ + 2 * ↑π * ↑y * I)"
] | rw [smul_comm, ← Measure.integral_comp_mul_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MellinTransform | {
"line": 120,
"column": 2
} | {
"line": 131,
"column": 98
} | {
"line": 133,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\ns : ℂ\na : ℝ\n⊢ mellin (fun t ↦ f (t ^ a)) s = |a|⁻¹ • mellin f (s / ↑a)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"add_sub_assoc",
"AddGroup.toSubtractionMonoid",
... | [] | rcases eq_or_ne a 0 with rfl | ha
· by_cases hE : CompleteSpace E
· simp [integral_smul_const, mellin, setIntegral_Ioi_zero_cpow]
· simp [integral, mellin, hE]
simp_rw [mellin]
conv_rhs => rw [← integral_comp_rpow_Ioi _ ha, ← integral_smul]
refine setIntegral_congr_fun measurableSet_Ioi fun t ht => ?_
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MellinTransform | {
"line": 120,
"column": 2
} | {
"line": 131,
"column": 98
} | {
"line": 133,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\ns : ℂ\na : ℝ\n⊢ mellin (fun t ↦ f (t ^ a)) s = |a|⁻¹ • mellin f (s / ↑a)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"add_sub_assoc",
"AddGroup.toSubtractionMonoid",
... | [] | rcases eq_or_ne a 0 with rfl | ha
· by_cases hE : CompleteSpace E
· simp [integral_smul_const, mellin, setIntegral_Ioi_zero_cpow]
· simp [integral, mellin, hE]
simp_rw [mellin]
conv_rhs => rw [← integral_comp_rpow_Ioi _ ha, ← integral_smul]
refine setIntegral_congr_fun measurableSet_Ioi fun t ht => ?_
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd | {
"line": 123,
"column": 4
} | {
"line": 124,
"column": 55
} | {
"line": 125,
"column": 4
} | [
{
"pp": "case e'_3.e_a\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 :\n ∀ x ∈ uIcc 0 (π / 2),\n HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x\nx : ℝ\nx✝ : x ∈ uIcc 0 (π / 2)\n⊢ ↑n * (Complex.cos (2 * z * ↑x) * Complex.cos ↑x ^ n) -\n (... | [
"case e'_3.e_a\nz : ℂ\nn : ℕ\nhn : 2 ≤ n\nhz : z ≠ 0\nder1 :\n ∀ x ∈ uIcc 0 (π / 2),\n HasDerivAt (fun y ↦ ↑(sin y) * ↑(cos y) ^ (n - 1)) (↑(cos x) ^ n - (↑n - 1) * ↑(sin x) ^ 2 * ↑(cos x) ^ (n - 2)) x\nx : ℝ\nx✝ : x ∈ uIcc 0 (π / 2)\nthis : Complex.cos ↑x ^ n = Complex.cos ↑x ^ (n - 2) * Complex.cos ↑x ^ 2\n⊢ ... | have : Complex.cos x ^ n = Complex.cos x ^ (n - 2) * Complex.cos x ^ 2 := by
conv_lhs => rw [← Nat.sub_add_cancel hn, pow_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 51
} | {
"line": 196,
"column": 2
} | [
{
"pp": "case hx\nx : ℝ\nn : ℕ\n⊢ ↑(n + 1) ≠ 0",
"ppTerm": "?hx",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"FloorRing.toFloorSemiring",
"Real.instZero",
"congrArg",
"AddMonoi... | [] | rw [Nat.cast_ne_zero]; exact Nat.succ_ne_zero n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 51
} | {
"line": 196,
"column": 2
} | [
{
"pp": "case hx\nx : ℝ\nn : ℕ\n⊢ ↑(n + 1) ≠ 0",
"ppTerm": "?hx",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"FloorRing.toFloorSemiring",
"Real.instZero",
"congrArg",
"AddMonoi... | [] | rw [Nat.cast_ne_zero]; exact Nat.succ_ne_zero n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 188,
"column": 24
} | {
"line": 188,
"column": 27
} | {
"line": 188,
"column": 28
} | [
{
"pp": "u v : ℂ\nhu : 0 < u.re\nF : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v\nhu' : 0 < (u + 1).re\nhv' : 0 < (v + 1).re\nhc : ContinuousOn F (Icc 0 1)\nhder : ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x\nh_int : IntervalIntegrable (fun x ↦ u * (↑x ^ (u -... | [
"u v : ℂ\nhu : 0 < u.re\nF : ℝ → ℂ := fun x ↦ ↑x ^ u * (1 - ↑x) ^ v\nhu' : 0 < (u + 1).re\nhv' : 0 < (v + 1).re\nhc : ContinuousOn F (Icc 0 1)\nhder : ∀ x ∈ Ioo 0 1, HasDerivAt F (u * (↑x ^ (u - 1) * (1 - ↑x) ^ v) - v * (↑x ^ u * (1 - ↑x) ^ (v - 1))) x\nh_int : IntervalIntegrable (fun x ↦ u * (↑x ^ (u - 1) * (1 - ↑... | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup | {
"line": 338,
"column": 62
} | {
"line": 338,
"column": 72
} | {
"line": 338,
"column": 73
} | [
{
"pp": "case refine_2\nA : 0 < 3 / 2\nthis : log (Γ (2 + 1 / 2)) ≤ log (Γ 2) + 1 / 2 * log 2\n⊢ Γ (3 / 2) < 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
... | [
"case refine_2\nA : 0 < 3 / 2\nthis : log (Γ (2 + 1 / 2)) ≤ log 1 + 1 / 2 * log 2\n⊢ Γ (3 / 2) < 1"
] | Gamma_two, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Affine.AsymptoticCone | {
"line": 51,
"column": 11
} | {
"line": 51,
"column": 20
} | {
"line": 51,
"column": 21
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : FiniteDimensional ℝ V\ns : Set P\nhs : s ∈ ⨆ v ∈ Metric.sphere 0 1, asymptoticNhds ℝ P v\np : P\n⊢ s ∈ cobounded P",
"ppTerm": "?m.76",
"assigned": ... | [
"V : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : MetricSpace P\ninst✝¹ : NormedAddTorsor V P\ninst✝ : FiniteDimensional ℝ V\ns : Set P\np : P\nhs : ∀ i ∈ Metric.sphere 0 1, s ∈ asymptoticNhds ℝ P i\n⊢ s ∈ cobounded P"
] | mem_iSup, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 149,
"column": 2
} | {
"line": 154,
"column": 76
} | {
"line": 156,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : LinearOrder k\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : AddTorsor V P\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V... | [] | have ⟨p⟩ : Nonempty P := inferInstance
simp_rw [asymptoticNhds_eq_smul_vadd _ p,
← show map (c • ·) (𝓝 v) = 𝓝 (c • v) from
(Homeomorph.smulOfNeZero c hc.ne').map_nhds_eq v,
← map₂_smul, map₂_map_right, smul_smul, ← map₂_map_left,
show map (· * c) atTop = atTop from (OrderIso.mulRight₀ _ hc).map_at... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 149,
"column": 2
} | {
"line": 154,
"column": 76
} | {
"line": 156,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹⁰ : Field k\ninst✝⁹ : LinearOrder k\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : AddTorsor V P\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V... | [] | have ⟨p⟩ : Nonempty P := inferInstance
simp_rw [asymptoticNhds_eq_smul_vadd _ p,
← show map (c • ·) (𝓝 v) = 𝓝 (c • v) from
(Homeomorph.smulOfNeZero c hc.ne').map_nhds_eq v,
← map₂_smul, map₂_map_right, smul_smul, ← map₂_map_left,
show map (· * c) atTop = atTop from (OrderIso.mulRight₀ _ hc).map_at... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 296,
"column": 2
} | {
"line": 297,
"column": 29
} | {
"line": 299,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝¹² : Field k\ninst✝¹¹ : LinearOrder k\ninst✝¹⁰ : AddCommGroup V\ninst✝⁹ : Module k V\ninst✝⁸ : AddTorsor V P\ninst✝⁷ : TopologicalSpace V\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : IsStrictOrderedRing k\ninst✝³ : IsTopologicalAddGroup... | [] | simp_rw [mem_asymptoticCone_iff, mem_closure_iff_frequently, ← frequently_bind,
asymptoticNhds_bind_nhds] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 335,
"column": 16
} | {
"line": 335,
"column": 28
} | {
"line": 335,
"column": 29
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝⁹ : Field k\ninst✝⁸ : LinearOrder k\ninst✝⁷ : IsStrictOrderedRing k\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul k V\ns : Set V... | [
"k : Type u_1\nV : Type u_2\ninst✝⁹ : Field k\ninst✝⁸ : LinearOrder k\ninst✝⁷ : IsStrictOrderedRing k\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul k V\ns : Set V\nc : k\nv p... | (ht : c ≤ t) | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.typeAscription |
Mathlib.LinearAlgebra.AffineSpace.Ceva | {
"line": 57,
"column": 8
} | {
"line": 57,
"column": 14
} | {
"line": 57,
"column": 14
} | [
{
"pp": "case refine_3\nk : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\nhp : AffineIndependent k p\ns : Set ι\nfs : ↑s → Finset ι\nhfs : ∀ (i : ↑s), ↑i ∈ fs i\nw : ↑s → ι → k\nhw : ∀ (i : ↑s), ∑ j ∈ fs i, ... | [
"case refine_3\nk : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np : ι → P\nhp : AffineIndependent k p\ns : Set ι\nfs : ↑s → Finset ι\nhfs : ∀ (i : ↑s), ↑i ∈ fs i\nw : ↑s → ι → k\nhw : ∀ (i : ↑s), ∑ j ∈ fs i, w i j = 1\np... | ← hri' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 354,
"column": 22
} | {
"line": 354,
"column": 24
} | {
"line": 354,
"column": 25
} | [
{
"pp": "case inr\nk : Type u_1\nV : Type u_2\ninst✝⁹ : Field k\ninst✝⁸ : LinearOrder k\ninst✝⁷ : IsStrictOrderedRing k\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul k V\... | [
"case inr\nk : Type u_1\nV : Type u_2\ninst✝⁹ : Field k\ninst✝⁸ : LinearOrder k\ninst✝⁷ : IsStrictOrderedRing k\ninst✝⁶ : TopologicalSpace k\ninst✝⁵ : OrderTopology k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul k V\ns✝ s : Set ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Normed.Affine.Simplex | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 28
} | {
"line": 53,
"column": 2
} | [
{
"pp": "case inl.inr\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : Module R V\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex R P n\nhs : s.Scalene\ni₁ i₂ i₃ i₄ : Fin (n + 1)\nh₁₂ : i₁ ≠ i₂\nh₃₄ : i₃ ≠ i₄\nh₁₂₃₄ : ¬i₁ = i₃... | [] | cases h₁₂₄₃ <;> simp [*] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Normed.Affine.Simplex | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 28
} | {
"line": 56,
"column": 2
} | [
{
"pp": "case inr.inl\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : Module R V\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex R P n\nhs : s.Scalene\ni₁ i₂ i₃ i₄ : Fin (n + 1)\nh₁₂ : i₁ ≠ i₂\nh₃₄ : i₃ ≠ i₄\nh₁₂₃₄ : ¬i₁ = i₃... | [] | cases h₁₂₄₃ <;> simp [*] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 12
} | {
"line": 136,
"column": 2
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : PreconnectedSpace X\ninst✝ : SeminormedAddCommGroup E\nf : X → E\nM : ℝ\nx : X\nhM : 0 < M\nhx : ‖f x‖ = M\nh : IsMinOn (fun x ↦ ‖f x‖) univ x\nhf : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nH : ∀ {y : X}... | [
"X : Type u_1\nE : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : PreconnectedSpace X\ninst✝ : SeminormedAddCommGroup E\nf : X → E\nM : ℝ\nx : X\nhM : 0 < M\nhx : ‖f x‖ = M\nh : IsMinOn (fun x ↦ ‖f x‖) univ x\nhf : Continuous[inst✝², PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nH : ∀ {y : X} (z : X), ‖f... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 334,
"column": 11
} | {
"line": 334,
"column": 13
} | {
"line": 334,
"column": 14
} | [
{
"pp": "F : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormedAlgebra ℝ F\ninst✝ : NormOneClass F\nx : F\nc : ℝ\nhc₀ : 0 < c\nhbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖\n⊢ Tendsto (fun x_1 ↦ φ x x_1) (cobounded (ℝ × ℝ)) (cobounded F)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
... | [
"F : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormedAlgebra ℝ F\ninst✝ : NormOneClass F\nx : F\nc : ℝ\nhc₀ : 0 < c\nhbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖\n⊢ Tendsto (fun x_1 ↦ x ^ 2 - x_1.1 • x + (algebraMap ℝ F) x_1.2) (cobounded (ℝ × ℝ)) (cobounded F)"
] | φ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 130,
"column": 10
} | {
"line": 130,
"column": 53
} | {
"line": 130,
"column": 53
} | [
{
"pp": "case h.inr\nR : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nm : ℕ\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\nhm✝ : max 1 (‖x‖ ^ m) = 1\nhx : ‖x‖ ^ m ≤ 1\ni : ℕ\nhi : i ∈ Finset.range (m + 1)\nhm : m ≠ 0\n⊢ ‖x‖ ^ i ≤ 1",
"ppTerm": "?h.inr",
"assigned": true,
"usedConstants": [
"Norm.nor... | [
"case h.inr\nR : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nm : ℕ\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\nhm✝ : max 1 (‖x‖ ^ m) = 1\nhx : ‖x‖ ≤ 1\ni : ℕ\nhi : i ∈ Finset.range (m + 1)\nhm : m ≠ 0\n⊢ ‖x‖ ^ i ≤ 1"
] | pow_le_one_iff_of_nonneg (norm_nonneg _) hm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 356,
"column": 6
} | {
"line": 360,
"column": 18
} | {
"line": 361,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormedAlgebra ℝ F\ninst✝ : NormOneClass F\nx : F\nc : ℝ\nhc₀ : 0 < c\nhbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖\nthis : Tendsto (fun y ↦ ‖y.1‖ * c) (cobounded ℝ ×ˢ ⊤) atTop\ny : ℝ × ℝ\nhy : y ∈ {0}ᶜ ×ˢ Set.univ\n⊢ ‖y.1‖ * c ≤ ‖y.1 • x - (algebraMap ℝ F)... | [] | calc ‖y.1‖ * c
_ ≤ ‖y.1‖ * ‖x - algebraMap ℝ F (y.1⁻¹ * y.2)‖ := by gcongr; exact hbd _
_ = ‖y.1 • x - algebraMap ℝ F y.2‖ := by
simp only [← norm_smul, smul_sub, smul_smul, Algebra.algebraMap_eq_smul_one]
simp_all | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Normed.Unbundled.AlgebraNorm | {
"line": 62,
"column": 4
} | {
"line": 67,
"column": 7
} | {
"line": 69,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : SeminormedCommRing R\nS : Type u_2\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nf✝ f f' : AlgebraNorm R S\nh : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) f'\n⊢ f = f'",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Seminormed... | [] | simp only [AddGroupSeminorm.toFun_eq_coe, RingSeminorm.toFun_eq_coe] at h
cases f; cases f'; congr
simp only at h
ext s
erw [h]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.AlgebraNorm | {
"line": 62,
"column": 4
} | {
"line": 67,
"column": 7
} | {
"line": 69,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : SeminormedCommRing R\nS : Type u_2\ninst✝¹ : Ring S\ninst✝ : Algebra R S\nf✝ f f' : AlgebraNorm R S\nh : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) f'\n⊢ f = f'",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Seminormed... | [] | simp only [AddGroupSeminorm.toFun_eq_coe, RingSeminorm.toFun_eq_coe] at h
cases f; cases f'; congr
simp only at h
ext s
erw [h]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.AlgebraNorm | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 48
} | {
"line": 116,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : SeminormedCommRing R\nS : Type u_2\ninst✝⁵ : Ring S\ninst✝⁴ : Algebra R S\nf✝ : AlgebraNorm R S\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra A S\ninst✝ : IsScalarTower R A S\nhinj : Function.Injective ⇑(algebraMap A S)\nf : AlgebraNorm R S\nx : A\nhx... | [
"R : Type u_1\ninst✝⁶ : SeminormedCommRing R\nS : Type u_2\ninst✝⁵ : Ring S\ninst✝⁴ : Algebra R S\nf✝ : AlgebraNorm R S\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra A S\ninst✝ : IsScalarTower R A S\nhinj : Function.Injective ⇑(algebraMap A S)\nf : AlgebraNorm R S\nx : A\nhx : f ((algeb... | rw [← map_eq_zero_iff (algebraMap A S) hinj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 30
} | {
"line": 143,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x ≠ 0\nL : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)\nhL0 : 0 ≤ L\nε : ℝ\nhε : ε > 0\nm1 : ℕ+\nhm1 : μ (x ^ ↑m1) ^ (1 / ↑↑m1) < (⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)) + ε / 2\nhε2 : 0 < ε / 2\nhL2 : ∀ ε_1 > 0, ∃ N, ∀ n ≥ N, dist ((L + ε /... | [
"R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : μ x ≠ 0\nL : ℝ := ⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)\nhL0 : 0 ≤ L\nε : ℝ\nhε : ε > 0\nm1 : ℕ+\nhm1 : μ (x ^ ↑m1) ^ (1 / ↑↑m1) < (⨅ n, μ (x ^ ↑n) ^ (1 / ↑↑n)) + ε / 2\nhε2 : 0 < ε / 2\nhL2 : ∀ ε_1 > 0, ∃ N, ∀ n ≥ N, dist ((L + ε / 2) ^ (-(↑(n... | obtain ⟨N, hN⟩ := hL2 δ hδ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Normed.Unbundled.SeminormFromBounded | {
"line": 361,
"column": 4
} | {
"line": 361,
"column": 84
} | {
"line": 362,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommRing R\nf : R → ℝ\nc : ℝ\nf_zero : f 0 = 0\nf_nonneg : 0 ≤ f\nf_mul : ∀ (x y : R), f (x * y) ≤ c * f x * f y\nf_add : ∀ (a b : R), f (a + b) ≤ f a + f b\nf_neg : ∀ (x : R), f (-x) = f x\nh0 : ∀ (x : R), (seminormFromBounded f_zero f_nonneg f_mul f_add f_neg).toF... | [] | exact ⟨fun h ↦ h0 x h, fun h ↦ by rw [h]; exact seminormFromBounded_zero f_zero⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 106,
"column": 12
} | {
"line": 106,
"column": 73
} | {
"line": 107,
"column": 12
} | [
{
"pp": "case a\nK : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : Is... | [
"case a\nK : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Normal K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint : IsIntegral K y... | apply IsConjRoot.of_isScalarTower (L := K⟮y⟯) xsep.isIntegral | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 118,
"column": 38
} | {
"line": 143,
"column": 98
} | {
"line": 145,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Algebra.IsAlgebraic K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint... | [] | by
-- Reduce to the case `L = algebraic closure of K` to apply the previous lemma.
let C := AlgebraicClosure K
let : NontriviallyNormedField C := spectralNorm.nontriviallyNormedField K C
let : NormedAlgebra K C := spectralNorm.normedAlgebra K C
let iL : L →ₐ[K] C := IsAlgClosed.lift
algebraize [... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 388,
"column": 8
} | {
"line": 388,
"column": 10
} | {
"line": 388,
"column": 10
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\ns : ℕ → ℕ\nhs_le : ∀ (n : ℕ), s n ≤ n\nx : R\na : ℝ\na_in : a ∈ Set.Icc 0 1\nψ : ℕ → ℕ\nhψ_mono : StrictMono ψ\nhψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 a)\nha : a = 0\n⊢ limsup (fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ... | [
"case pos\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\ns : ℕ → ℕ\nhs_le : ∀ (n : ℕ), s n ≤ n\nx : R\na : ℝ\na_in : a ∈ Set.Icc 0 1\nψ : ℕ → ℕ\nhψ_mono : StrictMono ψ\nhψ_lim : Tendsto ((fun n ↦ ↑(s n) / ↑n) ∘ ψ) atTop (𝓝 0)\nha : a = 0\n⊢ limsup (fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n))) atT... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.ControlledClosure | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 48
} | {
"line": 124,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : CompleteSpace G\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : Type u_3\ninst✝ : SeminormedAddCommGroup K\nj : NormedAddGroupHom K H\nhj : ∀ (x : K), ‖j x‖ = ‖x‖\nC ε : ℝ\nhC : 0 < C\nhε : 0 < ε\nhyp : ∀ (k : K), ∃ g, f... | [
"G : Type u_1\ninst✝³ : NormedAddCommGroup G\ninst✝² : CompleteSpace G\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\nf : NormedAddGroupHom G H\nK : Type u_3\ninst✝ : SeminormedAddCommGroup K\nj : NormedAddGroupHom K H\nhj : ∀ (x : K), ‖j x‖ = ‖x‖\nC ε : ℝ\nhC : 0 < C\nhε : 0 < ε\nhyp : ∀ (k : K), ∃ g, f g = j k ∧ ‖... | rcases (j.mem_range _).mp h_in with ⟨k, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 477,
"column": 4
} | {
"line": 478,
"column": 84
} | {
"line": 479,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nhna : IsNonarchimedean ⇑μ\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nmu : ℕ → ℕ := fun n ↦ _root_.mu μ hn n\nnu : ℕ → ℕ := fun n ↦ n - mu n\nhnu : nu = fun n ↦ n - mu ... | [] | apply le_trans hxy' (mul_le_mul hx hy (le_limsup_of_frequently_le (Frequently.of_forall
(fun n ↦ by positivity)) h_bdd) (rpow_nonneg (smoothingFun_nonneg μ hμ1 x) _)) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 742,
"column": 2
} | {
"line": 753,
"column": 36
} | {
"line": 754,
"column": 2
} | [
{
"pp": "K : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nx : L\nE : Type v := id ↥K⟮x⟯\nthis✝ : Field E :=\n id\n (have this := i... | [
"K : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nx : L\nE : Type v := id ↥K⟮x⟯\nthis✝ : Field E :=\n id\n (have this := inferInstance... | let N2 : NormedSpace K K⟮x⟯ :=
{ one_smul e := by simp [one_smul]
mul_smul k1 k2 e := by simp [mul_smul]
smul_zero e := by simp
smul_add k e1 e2 := by simp [smul_add]
add_smul k1 k2 e := by simp [add_smul]
zero_smul e := by simp [zero_smul]
norm_smul_le k y := by
change (... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 882,
"column": 2
} | {
"line": 883,
"column": 16
} | {
"line": 885,
"column": 0
} | [
{
"pp": "R : Type u_1\nK : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\n⊢ SeminormedRing L",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NormedCom... | [] | letI : NormedField L := normedField K L
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 882,
"column": 2
} | {
"line": 883,
"column": 16
} | {
"line": 885,
"column": 0
} | [
{
"pp": "R : Type u_1\nK : Type u\ninst✝⁴ : NontriviallyNormedField K\nL : Type v\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsAlgebraic K L\nhu : IsUltrametricDist K\ninst✝ : CompleteSpace K\n⊢ SeminormedRing L",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NormedCom... | [] | letI : NormedField L := normedField K L
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 37
} | {
"line": 147,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : Semiring R\nE : Type u_2\nF : Type u_4\nG : Type u_6\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : Module R E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\nf : E →L[R] F\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommMonoid G\ninst... | [
"R : Type u_1\ninst✝⁹ : Semiring R\nE : Type u_2\nF : Type u_4\nG : Type u_6\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : Module R E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\nf : E →L[R] F\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommMonoid G\ninst✝ : Module R... | refine ⟨finv.comp ginv, fun x ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 226,
"column": 29
} | {
"line": 226,
"column": 43
} | {
"line": 227,
"column": 4
} | [
{
"pp": "R : Type u_7\nE : Type u_8\nF : Type u_10\ninst✝⁸ : Ring R\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module R E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nf : E →L[R] F\ninst✝¹ : T1Space F\nhf : f.HasLeftInverse\ninst✝ : IsTopologicalAddGroup F\n⊢ ... | [
"R : Type u_7\nE : Type u_8\nF : Type u_10\ninst✝⁸ : Ring R\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module R E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nf : E →L[R] F\ninst✝¹ : T1Space F\nhf : f.HasLeftInverse\ninst✝ : IsTopologicalAddGroup F\n⊢ IsClosed ↑(↑... | ← f.coe_range, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 37
} | {
"line": 314,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : Semiring R\nE : Type u_2\nF : Type u_4\nG : Type u_6\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : Module R E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\nf : E →L[R] F\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommMonoid G\ninst... | [
"R : Type u_1\ninst✝⁹ : Semiring R\nE : Type u_2\nF : Type u_4\nG : Type u_6\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommMonoid E\ninst✝⁶ : Module R E\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\nf : E →L[R] F\ninst✝² : TopologicalSpace G\ninst✝¹ : AddCommMonoid G\ninst✝ : Module R... | refine ⟨finv.comp ginv, fun x ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 194,
"column": 4
} | {
"line": 195,
"column": 99
} | {
"line": 197,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : Type u_2\ninst✝² : NormedCommRing R\ninst✝¹ : NormOneClass R\nf : ι → R\ninst✝ : CompleteSpace R\nhf : Summable fun i ↦ ‖f i‖\nε : ℝ\nhε : ε > 0\nr₁ : ℝ\nhr₁ : r₁ > 0\ns₁ : Finset ι\nhs₁ : ∀ (t : Finset ι), s₁ ⊆ t → ∏ i ∈ t, ‖1 + f i‖ ≤ r₁\ns₂ : Finset ι\nhs₂ : ∀ (t : F... | [] | intro x hx y hy
exact (dist_triangle_right _ _ (∏ i ∈ s, (1 + f i))).trans_lt (add_halves ε ▸ add_lt_add hx hy) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.Summable | {
"line": 194,
"column": 4
} | {
"line": 195,
"column": 99
} | {
"line": 197,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : Type u_2\ninst✝² : NormedCommRing R\ninst✝¹ : NormOneClass R\nf : ι → R\ninst✝ : CompleteSpace R\nhf : Summable fun i ↦ ‖f i‖\nε : ℝ\nhε : ε > 0\nr₁ : ℝ\nhr₁ : r₁ > 0\ns₁ : Finset ι\nhs₁ : ∀ (t : Finset ι), s₁ ⊆ t → ∏ i ∈ t, ‖1 + f i‖ ≤ r₁\ns₂ : Finset ι\nhs₂ : ∀ (t : F... | [] | intro x hx y hy
exact (dist_triangle_right _ _ (∏ i ∈ s, (1 + f i))).trans_lt (add_halves ε ▸ add_lt_add hx hy) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.MultipliableUniformlyOn | {
"line": 135,
"column": 33
} | {
"line": 135,
"column": 35
} | {
"line": 135,
"column": 36
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nK : Set α\nu : ι → ℝ\nR : Type u_3\ninst✝⁴ : NormedCommRing R\ninst✝³ : NormOneClass R\ninst✝² : CompleteSpace R\ninst✝¹ : TopologicalSpace α\nf : ι → α → R\ninst✝ : LocallyCompactSpace α\nhK : IsOpen[inst✝¹] K\nhu : Summable u\nh : ∀ᶠ (i : ι) in cofinite, ∀ x ∈ K, ‖f i x‖ ≤... | [
"α : Type u_1\nι : Type u_2\nK : Set α\nu : ι → ℝ\nR : Type u_3\ninst✝⁴ : NormedCommRing R\ninst✝³ : NormOneClass R\ninst✝² : CompleteSpace R\ninst✝¹ : TopologicalSpace α\nf : ι → α → R\ninst✝ : LocallyCompactSpace α\nhK : IsOpen[inst✝¹] K\nhu : Summable u\nh : ∀ᶠ (i : ι) in cofinite, ∀ x ∈ K, ‖f i x‖ ≤ u i\nhcts :... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Normed.Lp.lpHolder | {
"line": 238,
"column": 17
} | {
"line": 244,
"column": 30
} | {
"line": 246,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : ι → Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁶ : (i : ι) → NormedAddCommGroup (F i)\ninst✝⁵ : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝⁴ : (i : ι) → NormedAdd... | [] | by
rw [← Real.rpow_mul, ← Real.rpow_mul]
· simp only [← mul_div_assoc, ne_eq, hp.ne', not_false_eq_true, mul_div_cancel_left₀,
hq.ne', fieldLe]
rw [Real.mul_rpow, Real.mul_rpow]
all_goals positivity
all_goals positivity | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.Perturbation.StrictByFinite | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 56
} | {
"line": 263,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : CompleteSpace 𝕜\nE : Type u_2\nF : Type u_3\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module 𝕜 E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : Module 𝕜 F\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : IsTopologicalAddGroup E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³... | [
"𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : CompleteSpace 𝕜\nE : Type u_2\nF : Type u_3\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module 𝕜 E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : Module 𝕜 F\ninst✝⁶ : TopologicalSpace E\ninst✝⁵ : IsTopologicalAddGroup E\ninst✝⁴ : ContinuousSMul 𝕜 E\ninst✝³ : Topologic... | have range_eq : v.range = u.range := range_liftQ _ _ _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Order.UpperLower | {
"line": 133,
"column": 7
} | {
"line": 133,
"column": 11
} | {
"line": 133,
"column": 11
} | [
{
"pp": "ι : Type u_2\ninst✝ : Fintype ι\ny y₁ : ι → ℝ\nhy₁ : y₁ ∈ Ici y\ny₂ : ι → ℝ\nhy₂ : y₂ ∈ Ici y\nhy : y₁ ≤ y₂\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ (y₁ i - y i).toNNReal ≤ (y₂ i - y i).toNNReal",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"sub_le_sub_right",
"le_refl",
... | [
"ι : Type u_2\ninst✝ : Fintype ι\ny y₁ : ι → ℝ\nhy₁ : y₁ ∈ Ici y\ny₂ : ι → ℝ\nhy₂ : y₂ ∈ Ici y\nhy : y₁ ≤ y₂\ni : ι\nx✝ : i ∈ Finset.univ\n⊢ (y₂ i - y i).toNNReal ≤ (y₂ i - y i).toNNReal"
] | hy i | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 83,
"column": 2
} | {
"line": 84,
"column": 22
} | {
"line": 85,
"column": 2
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : NormedSpace 𝕜 W\ninst✝¹ : SeparatingDual 𝕜 V\ninst✝ : SeparatingDual 𝕜 W\nf : (V →L[𝕜] V) ≃A[𝕜] W →L[�... | [
"case neg\n𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup V\ninst✝⁴ : SeminormedAddCommGroup W\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : NormedSpace 𝕜 W\ninst✝¹ : SeparatingDual 𝕜 V\ninst✝ : SeparatingDual 𝕜 W\nf : (V →L[𝕜] V) ≃A[𝕜] W →L[𝕜] W\nhV : N... | set T' := apply' _ (.id 𝕜) u ∘L f.symm.toContinuousAlgHom.toContinuousLinearMap ∘L
smulRightL 𝕜 _ _ d | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Analysis.Normed.Operator.ContinuousAlgEquiv | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 91
} | {
"line": 209,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuous ⇑f\nh✝ :... | [
"𝕜 : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup V\ninst✝⁴ : InnerProductSpace 𝕜 V\ninst✝³ : CompleteSpace V\ninst✝² : NormedAddCommGroup W\ninst✝¹ : InnerProductSpace 𝕜 W\ninst✝ : CompleteSpace W\nf : (V →L[𝕜] V) ≃⋆ₐ[𝕜] W →L[𝕜] W\nhf : Continuous ⇑f\nh✝ : Nontrivial ... | rw [← LinearMap.isPositive_one.isPositive_smul_iff (E := V) (one_ne_zero' (V →ₗ[𝕜] V))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Polynomial.Basic | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 26
} | {
"line": 187,
"column": 4
} | [
{
"pp": "case neg.inl\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhQ : Q ≠ 0\nh✝ : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)\nhPQ : ¬P.leadingCoeff / Q.leadingCoeff = 0\nh : ↑P.natDegree - ↑Q.natDegree = 0 ∧ P... | [
"case neg.inr\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhQ : Q ≠ 0\nh✝ : Tendsto (fun x ↦ eval x P / eval x Q) atTop (𝓝 0)\nhPQ : ¬P.leadingCoeff / Q.leadingCoeff = 0\nh : ↑P.natDegree - ↑Q.natDegree < 0 ∧ 0 = 0\n⊢ P.de... | · exact absurd h.2 hPQ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Polynomial.Basic | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 85
} | {
"line": 248,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.degree < P.degree\nhQ : Q ≠ 0\nh : 0 ≤ P.leadingCoeff / Q.leadingCoeff\n⊢ Tendsto (fun x ↦ |eval x P / eval x Q|) atTop atTop",
"ppTerm": "?pos... | [] | exact tendsto_abs_atTop_atTop.comp (P.div_tendsto_atTop_of_degree_gt Q hdeg hQ h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Polynomial.Basic | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 85
} | {
"line": 248,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.degree < P.degree\nhQ : Q ≠ 0\nh : 0 ≤ P.leadingCoeff / Q.leadingCoeff\n⊢ Tendsto (fun x ↦ |eval x P / eval x Q|) atTop atTop",
"ppTerm": "?pos... | [] | exact tendsto_abs_atTop_atTop.comp (P.div_tendsto_atTop_of_degree_gt Q hdeg hQ h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Polynomial.Basic | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 85
} | {
"line": 248,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : Q.degree < P.degree\nhQ : Q ≠ 0\nh : 0 ≤ P.leadingCoeff / Q.leadingCoeff\n⊢ Tendsto (fun x ↦ |eval x P / eval x Q|) atTop atTop",
"ppTerm": "?pos... | [] | exact tendsto_abs_atTop_atTop.comp (P.div_tendsto_atTop_of_degree_gt Q hdeg hQ h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 54
} | {
"line": 93,
"column": 4
} | [
{
"pp": "R : Type u_1\nF : Type u_2\ninst✝² : Semiring R\ninst✝¹ : FunLike F R ℝ\nv : F\nc : ℝ\np : R[X]\ninst✝ : ZeroHomClass F R ℝ\nf : ↥p.support → ℝ := fun i ↦ v (p.coeff ↑i) * c ^ ↑i\n⊢ (f '' ⊤ ∪ {0}).Finite",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Set.Finite.union",
... | [
"R : Type u_1\nF : Type u_2\ninst✝² : Semiring R\ninst✝¹ : FunLike F R ℝ\nv : F\nc : ℝ\np : R[X]\ninst✝ : ZeroHomClass F R ℝ\nf : ↥p.support → ℝ := ⋯\n⊢ (f '' ⊤).Finite"
] | apply Set.Finite.union _ <| Set.finite_singleton 0 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 96
} | {
"line": 207,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\nF : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : FunLike F R ℝ\nv : F\nc : ℝ\ninst✝² : ZeroHomClass F R ℝ\ninst✝¹ : NonnegHomClass F R ℝ\ninst✝ : MulHomClass F R ℝ\nhna : IsNonarchimedean ⇑v\np q : R[X]\nhc : 0 ≤ c\nhpq : p * q ≠ 0\nh_supp_p : p.support.Nonempty\nh_supp_q : q.support... | [
"case inr\nR : Type u_1\nF : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : FunLike F R ℝ\nv : F\nc : ℝ\ninst✝² : ZeroHomClass F R ℝ\ninst✝¹ : NonnegHomClass F R ℝ\ninst✝ : MulHomClass F R ℝ\nhna : IsNonarchimedean ⇑v\np q : R[X]\nhc : 0 ≤ c\nhpq : p * q ≠ 0\nh_supp_p : p.support.Nonempty\nh_supp_q : q.support.Nonempty\ni... | obtain ⟨j, _, _⟩ := IsNonarchimedean.finset_image_add_of_nonempty hna _ nonempty_range_add_one | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 216,
"column": 6
} | {
"line": 216,
"column": 47
} | {
"line": 217,
"column": 6
} | [
{
"pp": "R : Type u_1\nF : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : FunLike F R ℝ\nv : F\nc : ℝ\ninst✝² : ZeroHomClass F R ℝ\ninst✝¹ : NonnegHomClass F R ℝ\ninst✝ : MulHomClass F R ℝ\nhna : IsNonarchimedean ⇑v\np q : R[X]\nhc : 0 ≤ c\nhpq : p * q ≠ 0\nh_supp_p : p.support.Nonempty\nh_supp_q : q.support.Nonempty\... | [
"R : Type u_1\nF : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : FunLike F R ℝ\nv : F\nc : ℝ\ninst✝² : ZeroHomClass F R ℝ\ninst✝¹ : NonnegHomClass F R ℝ\ninst✝ : MulHomClass F R ℝ\nhna : IsNonarchimedean ⇑v\np q : R[X]\nhc : 0 ≤ c\nhpq : p * q ≠ 0\nh_supp_p : p.support.Nonempty\nh_supp_q : q.support.Nonempty\ni : ℕ\na✝ :... | have hq_le := q.le_gaussNorm v hc (i - j) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Polynomial.Order | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 28
} | {
"line": 101,
"column": 2
} | [
{
"pp": "case hn\nP : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\nhlc' : 0 ≤ ↑↑(↑(P.comp (-X)).natDegree).negOnePow * (P.comp (-X)).leadingCoeff\nh : Even P.natDegree\n⊢ Even ↑(P.comp (-X)).natDegree",
"ppTerm": "?hn✝",... | [] | · simpa [natDegree_comp] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Polynomial.Order | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 28
} | {
"line": 109,
"column": 0
} | [
{
"pp": "case hn\nP : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\nhlc' : 0 ≤ ↑↑(↑(P.comp (-X)).natDegree).negOnePow * (P.comp (-X)).leadingCoeff\nh : Odd P.natDegree\n⊢ Odd ↑(P.comp (-X)).natDegree",
"ppTerm": "?hn✝",
... | [] | · simpa [natDegree_comp] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 31
} | {
"line": 140,
"column": 2
} | [
{
"pp": "p q : ℂ[X]\nhpq : ¬p = 0 ∧ ¬q = 0\n⊢ {a |\n ¬(a ∈ Set.uIoc 0 (2 * π) →\n log (‖eval (circleMap 0 1 a) p‖ * ‖eval (circleMap 0 1 a) q‖) =\n log ‖eval (circleMap 0 1 a) p‖ + log ‖eval (circleMap 0 1 a) q‖)}.Finite",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants... | [
"p q : ℂ[X]\nhpq : ¬p = 0 ∧ ¬q = 0\n⊢ {a |\n a ∈ Set.uIoc 0 (2 * π) ∧\n ¬log (‖eval (circleMap 0 1 a) p‖ * ‖eval (circleMap 0 1 a) q‖) =\n log ‖eval (circleMap 0 1 a) p‖ + log ‖eval (circleMap 0 1 a) q‖}.Finite"
] | simp only [Classical.not_imp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.RCLike.BoundedContinuous | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 43
} | {
"line": 53,
"column": 4
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\nE : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : PseudoEMetricSpace E\nA : StarSubalgebra 𝕜 (E →ᵇ 𝕜)\ng : C(E, ℝ)\nx : E →ᵇ 𝕜\nhxA : x ∈ A\nhxg : (toContinuousMapStarₐ 𝕜) x = (AlgHom.compLeftContinuous ℝ ofRealAm ⋯) g\nhg_apply : ∀ (a : E), x a = ↑(g a)\nh_comp_eq : (AlgHom.compLe... | [
"case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : PseudoEMetricSpace E\nA : StarSubalgebra 𝕜 (E →ᵇ 𝕜)\ng : C(E, ℝ)\nx : E →ᵇ 𝕜\nhxA : x ∈ A\nhxg : (toContinuousMapStarₐ 𝕜) x = (AlgHom.compLeftContinuous ℝ ofRealAm ⋯) g\nhg_apply : ∀ (a : E), x a = ↑(g a)\nh_comp_eq : (AlgHom.compLeftContinuousBo... | use x.comp reCLM (@reCLM 𝕜 _).lipschitz | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 61
} | {
"line": 256,
"column": 0
} | [
{
"pp": "case ha\np : ℂ[X]\nhlc : 1 ≤ ‖p.leadingCoeff‖\n⊢ 0 ≤ (Multiset.map (fun a ↦ max 1 ‖a‖) p.roots).prod",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"Polynomial.roots",
"Polynomial.one_le_prod_max_one_norm_roots",
"Complex.commRing"... | [] | exact zero_le_one.trans <| one_le_prod_max_one_norm_roots p | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Rat.NatSqrt.Real | {
"line": 33,
"column": 2
} | {
"line": 34,
"column": 14
} | {
"line": 35,
"column": 2
} | [
{
"pp": "x prec : ℕ\nh : 0 < prec\nthis✝¹ : ↑x < (x.ratSqrt prec + 1 / ↑prec) ^ 2\nthis✝ : ↑x < ↑((x.ratSqrt prec + 1 / ↑prec) ^ 2)\nthis : √↑x < ↑(x.ratSqrt prec + 1 / ↑prec)\n⊢ √↑x < ↑(x.ratSqrt prec) + 1 / ↑prec",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
... | [
"x prec : ℕ\nh : 0 < prec\nthis✝¹ : ↑x < (x.ratSqrt prec + 1 / ↑prec) ^ 2\nthis✝ : ↑x < ↑((x.ratSqrt prec + 1 / ↑prec) ^ 2)\nthis : √↑x < √(↑(x.ratSqrt prec + 1 / ↑prec) ^ 2)\n⊢ 0 ≤ ↑(x.ratSqrt prec + 1 / ↑prec)"
] | · push_cast at this
exact this | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Real.OfDigits | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 47
} | {
"line": 73,
"column": 0
} | [
{
"pp": "b : ℕ\ndigits : ℕ → Fin b\n⊢ 0 ≤ ∑' (n : ℕ), ofDigitsTerm digits n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Real",
"Real.lattice",
"instHasSolidNormReal",
"PseudoMetricSpace.toUniformSpace",
"Real.normedAddCommGroup",
"Real.ofDigitsTerm_n... | [] | exact tsum_nonneg fun _ ↦ ofDigitsTerm_nonneg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Real.OfDigits | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 96
} | {
"line": 80,
"column": 4
} | [
{
"pp": "case e'_4\nb : ℕ\ndigits : ℕ → Fin b\nhb : 1 < ↑b\n⊢ 1 = ∑' (n : ℕ), (↑b - 1) * (↑b)⁻¹ * (↑b)⁻¹ ^ n",
"ppTerm": "?e'_4",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Semigroup.t... | [
"case e'_4\nb : ℕ\ndigits : ℕ → Fin b\nhb : 1 < ↑b\n⊢ 1 = (↑b - 1) * (↑b)⁻¹ * (1 - (↑b)⁻¹)⁻¹"
] | rw [tsum_mul_left, tsum_geometric_of_lt_one (by positivity) (by simp [inv_lt_one_iff₀, hb])] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Real.Hyperreal | {
"line": 561,
"column": 61
} | {
"line": 561,
"column": 75
} | {
"line": 563,
"column": 0
} | [
{
"pp": "r s t : ℝ\n⊢ r = s → s = t → r = t",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Real",
"Eq.trans"
],
"usedFVars": [
"r",
"s",
"t"
],
"usedGoals": []
}
] | [] | exact Eq.trans | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Real.Pi.Leibniz | {
"line": 36,
"column": 2
} | {
"line": 43,
"column": 88
} | {
"line": 44,
"column": 2
} | [
{
"pp": "l : ℝ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i / (2 * ↑i + 1)) atTop (𝓝 l)\nabel : Tendsto (fun x ↦ ∑' (n : ℕ), (-1) ^ n / (2 * ↑n + 1) * x ^ n) (𝓝[<] 1) (𝓝 l)\nm : 𝓝[<] 1 ≤ 𝓝 1\n⊢ Tendsto (fun k ↦ ∑ i ∈ range k, (-1) ^ i / (2 * ↑i + 1)) atTop (𝓝 (π / 4))",
"ppTerm": "?m.234",
"assig... | [
"l : ℝ\nh : Tendsto (fun n ↦ ∑ i ∈ range n, (-1) ^ i / (2 * ↑i + 1)) atTop (𝓝 l)\nabel : Tendsto (fun x ↦ ∑' (n : ℕ), (-1) ^ n / (2 * ↑n + 1) * x ^ n) (𝓝[<] 1) (𝓝 l)\nm : 𝓝[<] 1 ≤ 𝓝 1\nq : Tendsto (fun x ↦ x ^ 2) (𝓝[<] 1) (𝓝[<] 1)\n⊢ Tendsto (fun k ↦ ∑ i ∈ range k, (-1) ^ i / (2 * ↑i + 1)) atTop (𝓝 (π / 4))... | have q : Tendsto (fun x : ℝ ↦ x ^ 2) (𝓝[<] 1) (𝓝[<] 1) := by
apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within
· nth_rw 3 [← one_pow 2]
exact Tendsto.pow ‹_› _
· rw [eventually_iff_exists_mem]
use Set.Ioo (-1) 1
exact ⟨Ioo_mem_nhdsLT <| by simp,
fun _ _ ↦ by rwa [Set.... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 41,
"column": 71
} | {
"line": 41,
"column": 86
} | {
"line": 41,
"column": 86
} | [
{
"pp": "x y : ℝ≥0\nh : x ≠ y\nhl : y < x\n⊢ 0 < x - y",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"tsub_pos_iff_lt",
"congrArg",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"HSub.hSub",
"NNReal.inst... | [
"x y : ℝ≥0\nh : x ≠ y\nhl : y < x\n⊢ y < x"
] | tsub_pos_iff_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 34
} | {
"line": 133,
"column": 2
} | [
{
"pp": "p : ℝ\nh : p ≠ 0 ∧ p ≠ 1\n⊢ binEntropy p ≠ 0",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Real.instZero",
"PartialOrder.toPreorder",
"Real.binEntropy",
"Ne",
"Or.casesOn",
"Real.instOne",
"And.left"... | [
"case inl\np : ℝ\nh : p ≠ 0 ∧ p ≠ 1\nhp₀ : p < 0\n⊢ binEntropy p ≠ 0",
"case inr\np : ℝ\nh : p ≠ 0 ∧ p ≠ 1\nhp₀ : 0 < p\n⊢ binEntropy p ≠ 0"
] | obtain hp₀ | hp₀ := h.1.lt_or_gt | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.ConjSqrt | {
"line": 51,
"column": 55
} | {
"line": 51,
"column": 62
} | {
"line": 51,
"column": 62
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : PartialOrder A\ninst✝⁷ : Ring A\ninst✝⁶ : StarRing A\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : Algebra ℝ A\ninst✝² : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : SeparatelyContinuousMul A\nc a b : A\nhab : a ≤ ... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.ConjSqrt | {
"line": 51,
"column": 55
} | {
"line": 51,
"column": 62
} | {
"line": 51,
"column": 62
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : PartialOrder A\ninst✝⁷ : Ring A\ninst✝⁶ : StarRing A\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : Algebra ℝ A\ninst✝² : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : SeparatelyContinuousMul A\nc a b : A\nhab : a ≤ ... | [] | cfc_tac | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.ConjSqrt | {
"line": 51,
"column": 55
} | {
"line": 51,
"column": 62
} | {
"line": 51,
"column": 62
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : PartialOrder A\ninst✝⁷ : Ring A\ninst✝⁶ : StarRing A\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : Algebra ℝ A\ninst✝² : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : SeparatelyContinuousMul A\nc a b : A\nhab : a ≤ ... | [] | cfc_tac | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.ConjSqrt | {
"line": 66,
"column": 42
} | {
"line": 66,
"column": 49
} | {
"line": 67,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹⁰ : PartialOrder A\ninst✝⁹ : Ring A\ninst✝⁸ : StarRing A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : StarOrderedRing A\ninst✝⁵ : Algebra ℝ A\ninst✝⁴ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : SeparatelyContinuousMul A\ninst✝¹ : IsSemitopo... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 21
} | {
"line": 67,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np : ℝ≥0\nhp : p ∈ Icc 0 1\na b : A\nhab : a ≤ b\n⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Real",
"NonUnitalCommRing.toNon... | [
"case pos\nA : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np : ℝ≥0\nhp : p ∈ Icc 0 1\na b : A\nhab : a ≤ b\nha : 0 ≤ a\n⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b",
"case neg\nA : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrde... | by_cases ha : 0 ≤ a | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 23
} | {
"line": 138,
"column": 4
} | [
{
"pp": "case inl\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np : ℝ\nhp : p ∈ Icc 0 1\nq : ℝ≥0 := ⟨p, ⋯⟩\nhq : q = 0\na b : A\nhab : a ≤ b\n⊢ (fun a ↦ a ^ ↑0) a ≤ (fun a ↦ a ^ ↑0) b",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"CStarA... | [
"case pos\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np : ℝ\nhp : p ∈ Icc 0 1\nq : ℝ≥0 := ⟨p, ⋯⟩\nhq : q = 0\na b : A\nhab : a ≤ b\nha : 0 ≤ a\n⊢ (fun a ↦ a ^ ↑0) a ≤ (fun a ↦ a ^ ↑0) b",
"case neg\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\nins... | by_cases ha : 0 ≤ a | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 301,
"column": 2
} | {
"line": 305,
"column": 24
} | {
"line": 306,
"column": 2
} | [
{
"pp": "h' : ¬Irrational (π / 2)\na : ℤ\nb : ℕ\nhb : 0 < b\nh : π / 2 = ↑a / ↑b\nha : 0 < ↑a\nk : ∀ (n : ℕ), 0 < ↑a ^ (2 * n + 1) / ↑n !\nj : ∀ᶠ (n : ℕ) in atTop, ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1\nn : ℕ\nhn : ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1\nhn' : 0 < ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2)\nz... | [
"h' : ¬Irrational (π / 2)\na : ℤ\nb : ℕ\nhb : 0 < b\nh : π / 2 = ↑a / ↑b\nha : 0 < ↑a\nk : ∀ (n : ℕ), 0 < ↑a ^ (2 * n + 1) / ↑n !\nj : ∀ᶠ (n : ℕ) in atTop, ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1\nn : ℕ\nhn : ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1\nhn' : 0 < ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2)\nz : ℤ\nhz : e... | have : a ^ (2 * n + 1) / n ! * I n (π / 2) =
eval₂ (Int.castRingHom ℝ) (π / 2) (sinPoly n) * b ^ (2 * n + 1) := by
nth_rw 2 [h] at e
simp [field, div_pow] at e ⊢
linear_combination e | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order | {
"line": 68,
"column": 10
} | {
"line": 68,
"column": 12
} | {
"line": 69,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np : ℝ\na : A\n⊢ a ∈ {a | IsStrictlyPositive a} →\n (fun a ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ p⁻¹ * (x ^ p - 1)) a else 0) a =\n (fun a ↦ p⁻¹ • (a ^ p - 1)) a",
"ppTerm": "?m.77",
... | [
"A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np : ℝ\na : A\nha : a ∈ {a | IsStrictlyPositive a}\n⊢ (fun a ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ p⁻¹ * (x ^ p - 1)) a else 0) a =\n (fun a ↦ p⁻¹ • (a ^ p - 1)) a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order | {
"line": 67,
"column": 94
} | {
"line": 72,
"column": 42
} | {
"line": 74,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np : ℝ\n⊢ Set.EqOn (fun a ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ p⁻¹ * (x ^ p - 1)) a else 0)\n (fun a ↦ p⁻¹ • (a ^ p - 1)) {a | IsStrictlyPositive a}",
"ppTerm": "?m.76",
"assigned": tru... | [] | by
intro a ha
simp only [ha, ↓reduceIte, ← smul_eq_mul]
rw [cfc_smul _ (hf := by fun_prop (disch := grind)),
cfc_sub _ _ (hf := by fun_prop (disch := grind)),
cfc_const_one .., rpow_eq_cfc_real ..] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder | {
"line": 37,
"column": 76
} | {
"line": 37,
"column": 78
} | {
"line": 37,
"column": 79
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nxpos : IsStrictlyPositive x\ny : A\nypos : IsStrictlyPositive y\na b : ℝ\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 → (a • x + b • y)⁻¹ʳ ≤ a • x⁻¹ʳ + b • y⁻¹ʳ",
"ppTerm": "?m.46",
"assigned": true,
"usedCons... | [
"A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nxpos : IsStrictlyPositive x\ny : A\nypos : IsStrictlyPositive y\na b : ℝ\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → (a • x + b • y)⁻¹ʳ ≤ a • x⁻¹ʳ + b • y⁻¹ʳ"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 105,
"column": 8
} | {
"line": 105,
"column": 31
} | {
"line": 105,
"column": 32
} | [
{
"pp": "⊢ Tendsto (fun z ↦ z / 2) (𝓝[≠] 0) (𝓝[≠] 0)",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"instHDiv",
"tendsto_nhdsWithin_iff",
"congrArg",
"Compl.compl",
"nhdsWithin",
"Filter.Eve... | [
"⊢ Tendsto (fun z ↦ z / 2) (𝓝[≠] 0) (𝓝 0) ∧ ∀ᶠ (n : ℂ) in 𝓝[≠] 0, n / 2 ∈ {0}ᶜ"
] | tendsto_nhdsWithin_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 42
} | {
"line": 377,
"column": 0
} | [
{
"pp": "p t x : ℝ\nhx : 0 ≤ x\nht : 0 < t\n⊢ p.rpowIntegrand₁₂ t x = x * (p - 1).rpowIntegrand₀₁ t x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation.0.Real.rpowIntegrand₁₂_eq_mul_rpo... | [] | grind [rpowIntegrand₁₂, rpowIntegrand₀₁] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 42
} | {
"line": 377,
"column": 0
} | [
{
"pp": "p t x : ℝ\nhx : 0 ≤ x\nht : 0 < t\n⊢ p.rpowIntegrand₁₂ t x = x * (p - 1).rpowIntegrand₀₁ t x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation.0.Real.rpowIntegrand₁₂_eq_mul_rpo... | [] | grind [rpowIntegrand₁₂, rpowIntegrand₀₁] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 42
} | {
"line": 377,
"column": 0
} | [
{
"pp": "p t x : ℝ\nhx : 0 ≤ x\nht : 0 < t\n⊢ p.rpowIntegrand₁₂ t x = x * (p - 1).rpowIntegrand₀₁ t x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation.0.Real.rpowIntegrand₁₂_eq_mul_rpo... | [] | grind [rpowIntegrand₁₂, rpowIntegrand₀₁] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 404,
"column": 62
} | {
"line": 404,
"column": 64
} | {
"line": 405,
"column": 4
} | [
{
"pp": "p x : ℝ\nhp : p ∈ Ioo 1 2\nhx : 0 ≤ x\na : ℝ\n⊢ a ∈ Ioi 0 → p.rpowIntegrand₁₂ a x = x * (p - 1).rpowIntegrand₀₁ a x",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"Real",
"Set.Ioi",
"Real.instZero",
"Membership.mem",
"Zero.toOfNat0",
"OfNat.ofN... | [
"p x : ℝ\nhp : p ∈ Ioo 1 2\nhx : 0 ≤ x\na : ℝ\nha : a ∈ Ioi 0\n⊢ p.rpowIntegrand₁₂ a x = x * (p - 1).rpowIntegrand₀₁ a x"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 416,
"column": 62
} | {
"line": 416,
"column": 64
} | {
"line": 417,
"column": 4
} | [
{
"pp": "p x : ℝ\nhp : p ∈ Ioo 1 2\nhx : 0 ≤ x\na : ℝ\n⊢ a ∈ Ioi 0 → p.rpowIntegrand₁₂ a x = x * (p - 1).rpowIntegrand₀₁ a x",
"ppTerm": "?m.115",
"assigned": true,
"usedConstants": [
"Real",
"Set.Ioi",
"Real.instZero",
"Membership.mem",
"Zero.toOfNat0",
"OfNat.of... | [
"p x : ℝ\nhp : p ∈ Ioo 1 2\nhx : 0 ≤ x\na : ℝ\nha : a ∈ Ioi 0\n⊢ p.rpowIntegrand₁₂ a x = x * (p - 1).rpowIntegrand₀₁ a x"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.Log.Monotone | {
"line": 46,
"column": 33
} | {
"line": 46,
"column": 65
} | {
"line": 47,
"column": 2
} | [
{
"pp": "x : ℝ\nhx : x ∈ interior (Icc 0 (rexp (-1)))\n⊢ x < rexp (-1)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"MulZeroClass.toMul",
"instNoMaxOrderOfNontrivial",
"Real... | [] | simp_all [interior_Icc, mem_Ioo] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Log.Monotone | {
"line": 47,
"column": 26
} | {
"line": 47,
"column": 58
} | {
"line": 48,
"column": 2
} | [
{
"pp": "x : ℝ\nhx : x ∈ interior (Icc 0 (rexp (-1)))\nhgt : x < rexp (-1)\n⊢ 0 < x",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"and_true",
"MulZeroClass.toMul",
"instNoMax... | [] | simp_all [interior_Icc, mem_Ioo] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 538,
"column": 56
} | {
"line": 538,
"column": 58
} | {
"line": 538,
"column": 59
} | [
{
"pp": "c : ℝ\nhc : c ≠ 0\na : ℝ\n⊢ 0 < a → log (c * a) = log c + log a",
"ppTerm": "?m.117",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Real.instZero",
"LT.lt",
"Zero.toOfNat0",
"OfNat.ofNat",
"Real.instPreorder"
],
"usedFVars":... | [
"c : ℝ\nhc : c ≠ 0\na : ℝ\nha : 0 < a\n⊢ log (c * a) = log c + log a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 622,
"column": 4
} | {
"line": 623,
"column": 19
} | {
"line": 624,
"column": 4
} | [
{
"pp": "P : ℝ → Prop\nx₀ r : ℝ\nhr : 1 < r\nhx₀ : 0 < x₀\nbase : ∀ x ∈ Ico x₀ (r * x₀), P x\nstep : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z\nthis : ∀ (n : ℕ), ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x\nx : ℝ\nhx : x ≥ x₀\nhx' : 0 < x / x₀\n⊢ x ∈ Ico x₀ (r ^ (⌊logb ... | [
"P : ℝ → Prop\nx₀ r : ℝ\nhr : 1 < r\nhx₀ : 0 < x₀\nbase : ∀ x ∈ Ico x₀ (r * x₀), P x\nstep : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z\nthis : ∀ (n : ℕ), ∀ x ∈ Ico x₀ (r ^ (n + 1) * x₀), P x\nx : ℝ\nhx : x ≥ x₀\nhx' : 0 < x / x₀\n⊢ x₀ ≤ x ∧ logb r (x / x₀) < ↑⌊logb r... | rw [mem_Ico, ← div_lt_iff₀ hx₀, ← rpow_natCast, ← logb_lt_iff_lt_rpow hr hx', Nat.cast_add,
Nat.cast_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.RegularityCompacts | {
"line": 177,
"column": 22
} | {
"line": 181,
"column": 84
} | {
"line": 183,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : SecondCountableTopology α\ninst✝² : IsCompletelyPseudoMetrizableSpace α\ninst✝¹ : BorelSpace α\nP : Measure α\ninst✝ : IsFiniteMeasure P\n⊢ P.InnerRegular",
"ppTerm": "?m.12",
"assigned": true,
"u... | [] | by
suffices P.InnerRegularCompactLTTop from inferInstance
refine ⟨Measure.InnerRegularWRT.measurableSet_of_isOpen ?_ ?_⟩
· exact innerRegularWRT_isCompact_isOpen P
· exact fun s t hs_compact ht_open ↦ hs_compact.inter_right ht_open.isClosed_compl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.RegularityCompacts | {
"line": 191,
"column": 2
} | {
"line": 200,
"column": 45
} | {
"line": 202,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : MeasurableSpace α\nμ✝ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : IsCompletelyPseudoMetrizableSpace α\ninst✝ : BorelSpace α\nμ : Measure α\n⊢ μ.InnerRegularCompactLTTop",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": ... | [] | constructor
intro A ⟨hA1, hA2⟩ r hr
have := Fact.mk hA2.lt_top
have hA2' : (μ.restrict A) A ≠ ⊤ := by
rwa [Measure.restrict_apply_self]
have hr' : r < μ.restrict A A := by
rwa [Measure.restrict_apply_self]
obtain ⟨K, ⟨hK1, hK2, hK3⟩⟩ := MeasurableSet.exists_lt_isCompact_of_ne_top hA1 hA2' hr'
use K,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.RegularityCompacts | {
"line": 191,
"column": 2
} | {
"line": 200,
"column": 45
} | {
"line": 202,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : MeasurableSpace α\nμ✝ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : IsCompletelyPseudoMetrizableSpace α\ninst✝ : BorelSpace α\nμ : Measure α\n⊢ μ.InnerRegularCompactLTTop",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": ... | [] | constructor
intro A ⟨hA1, hA2⟩ r hr
have := Fact.mk hA2.lt_top
have hA2' : (μ.restrict A) A ≠ ⊤ := by
rwa [Measure.restrict_apply_self]
have hr' : r < μ.restrict A A := by
rwa [Measure.restrict_apply_self]
obtain ⟨K, ⟨hK1, hK2, hK3⟩⟩ := MeasurableSet.exists_lt_isCompact_of_ne_top hA1 hA2' hr'
use K,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pochhammer | {
"line": 59,
"column": 12
} | {
"line": 59,
"column": 14
} | {
"line": 59,
"column": 15
} | [
{
"pp": "case succ\nn : ℕ\nih : MonotoneOn (deriv fun x ↦ Polynomial.eval x (descPochhammer ℝ n)) (Set.Ioi (↑n - 1))\na : ℝ\n⊢ a ∈ Set.Ioi (↑(n + 1) - 1) →\n ∀ ⦃b : ℝ⦄,\n b ∈ Set.Ioi (↑(n + 1) - 1) →\n a ≤ b →\n deriv (fun x ↦ Polynomial.eval x (descPochhammer ℝ (n + 1))) a ≤\n ... | [
"case succ\nn : ℕ\nih : MonotoneOn (deriv fun x ↦ Polynomial.eval x (descPochhammer ℝ n)) (Set.Ioi (↑n - 1))\na : ℝ\nha : a ∈ Set.Ioi (↑(n + 1) - 1)\n⊢ ∀ ⦃b : ℝ⦄,\n b ∈ Set.Ioi (↑(n + 1) - 1) →\n a ≤ b →\n deriv (fun x ↦ Polynomial.eval x (descPochhammer ℝ (n + 1))) a ≤\n deriv (fun x ↦ Poly... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 42
} | {
"line": 189,
"column": 2
} | [
{
"pp": "r : ℝ\nhr : 0 < r\ns : ℂ\nhs : ‖s‖ < r\nl : ℂ\nh : 2 * r ≤ ‖l‖\n⊢ ‖1 / (s - l) ^ 2 - 1 / l ^ 2‖ ≤ 10 * r * ‖l‖ ^ (-3)",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | [
"r : ℝ\nhr : 0 < r\ns : ℂ\nhs : ‖s‖ < r\nl : ℂ\nh : 2 * r ≤ ‖l‖\nthis : s ≠ l\n⊢ ‖1 / (s - l) ^ 2 - 1 / l ^ 2‖ ≤ 10 * r * ‖l‖ ^ (-3)"
] | have : s ≠ ↑l := by rintro rfl; linarith | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 31
} | {
"line": 190,
"column": 2
} | [
{
"pp": "r : ℝ\nhr : 0 < r\ns : ℂ\nhs : ‖s‖ < r\nl : ℂ\nh : 2 * r ≤ ‖l‖\nthis : s ≠ l\n⊢ ‖1 / (s - l) ^ 2 - 1 / l ^ 2‖ ≤ 10 * r * ‖l‖ ^ (-3)",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | [
"r : ℝ\nhr : 0 < r\ns : ℂ\nhs : ‖s‖ < r\nl : ℂ\nh : 2 * r ≤ ‖l‖\nthis✝ : s ≠ l\nthis : 0 < ‖l‖\n⊢ ‖1 / (s - l) ^ 2 - 1 / l ^ 2‖ ≤ 10 * r * ‖l‖ ^ (-3)"
] | have : 0 < ‖l‖ := by linarith | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.RegularizedHypergeometric | {
"line": 143,
"column": 23
} | {
"line": 143,
"column": 38
} | {
"line": 143,
"column": 39
} | [
{
"pp": "case e_a\nn : ℕ\na b : Multiset ℂ\nhn : n ≠ 0\n⊢ ↑n ^ a.card = ↑n ^ (b.card + 1) * ↑n ^ (↑a.card - ↑b.card - 1)",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"HSub.hSub",
"DivInvMonoid.toZPow"... | [
"case e_a\nn : ℕ\na b : Multiset ℂ\nhn : n ≠ 0\n⊢ ↑n ^ ↑a.card = ↑n ^ (b.card + 1) * ↑n ^ (↑a.card - ↑b.card - 1)"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.RegularizedHypergeometric | {
"line": 143,
"column": 39
} | {
"line": 143,
"column": 54
} | {
"line": 143,
"column": 55
} | [
{
"pp": "case e_a\nn : ℕ\na b : Multiset ℂ\nhn : n ≠ 0\n⊢ ↑n ^ ↑a.card = ↑n ^ (b.card + 1) * ↑n ^ (↑a.card - ↑b.card - 1)",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"HSub.hSub",
"DivInvMonoid.toZPow... | [
"case e_a\nn : ℕ\na b : Multiset ℂ\nhn : n ≠ 0\n⊢ ↑n ^ ↑a.card = ↑n ^ ↑(b.card + 1) * ↑n ^ (↑a.card - ↑b.card - 1)"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.RegularizedHypergeometric | {
"line": 230,
"column": 28
} | {
"line": 230,
"column": 43
} | {
"line": 230,
"column": 44
} | [
{
"pp": "a b : Multiset ℂ\nh : a.card ≤ b.card\nha : ∀ j ∈ a, ∀ (k : ℕ), j ≠ -↑k\nthis : Tendsto (fun x ↦ (1 / ↑x) ^ (b.card + 1 - a.card)) atTop (𝓝 0)\nn : ℕ\n⊢ (↑n ^ (b.card + 1 - a.card))⁻¹ = ↑n ^ (↑a.card - ↑b.card - 1)",
"ppTerm": "?m.155",
"assigned": true,
"usedConstants": [
"zpow_natC... | [
"a b : Multiset ℂ\nh : a.card ≤ b.card\nha : ∀ j ∈ a, ∀ (k : ℕ), j ≠ -↑k\nthis : Tendsto (fun x ↦ (1 / ↑x) ^ (b.card + 1 - a.card)) atTop (𝓝 0)\nn : ℕ\n⊢ (↑n ^ ↑(b.card + 1 - a.card))⁻¹ = ↑n ^ (↑a.card - ↑b.card - 1)"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 401,
"column": 15
} | {
"line": 401,
"column": 30
} | {
"line": 401,
"column": 31
} | [
{
"pp": "case neg\nL : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\nl : ℂ\nhl : l ∈ (↑L.lattice \\ {l₀})ᶜ\nx : ↥L.lattice\nhl₁✝ : ¬↑x = l₀\nhl₁ : l - ↑x ≠ 0\n⊢ -2 / (l - ↑x) ^ 3 = deriv (fun x2 ↦ 1 / (x2 - ↑x) ^ 2) l - 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case neg\nL : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\nl : ℂ\nhl : l ∈ (↑L.lattice \\ {l₀})ᶜ\nx : ↥L.lattice\nhl₁✝ : ¬↑x = l₀\nhl₁ : l - ↑x ≠ 0\n⊢ -2 / (l - ↑x) ^ ↑3 = deriv (fun x2 ↦ 1 / (x2 - ↑x) ^ ↑2) l - 0"
] | ← zpow_natCast, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 447,
"column": 4
} | {
"line": 447,
"column": 77
} | {
"line": 448,
"column": 2
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∈ L.lattice\nl : ↥L.lattice\nhl : ↑l / 2 ∉ L.lattice\n⊢ (↑L.lattice \\ {l₀ - ↑l}).Countable",
"ppTerm": "?m.147",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"CompleteBooleanAlgebra.toCompleteDis... | [] | exact .mono sdiff_le (countable_of_Lindelof_of_discrete (X := L.lattice)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Sigmoid | {
"line": 72,
"column": 2
} | {
"line": 73,
"column": 12
} | {
"line": 75,
"column": 0
} | [
{
"pp": "x : ℝ\n⊢ 0 < x.sigmoid",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real",
"MulZeroClass.toMul",
"Nat.ble",
"Real.instZero",
"Real.instAddMonoid",
"AddMonoid.toAddZeroClass",
"P... | [] | change 0 < (1 + exp (-x))⁻¹
positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Sigmoid | {
"line": 72,
"column": 2
} | {
"line": 73,
"column": 12
} | {
"line": 75,
"column": 0
} | [
{
"pp": "x : ℝ\n⊢ 0 < x.sigmoid",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real",
"MulZeroClass.toMul",
"Nat.ble",
"Real.instZero",
"Real.instAddMonoid",
"AddMonoid.toAddZeroClass",
"P... | [] | change 0 < (1 + exp (-x))⁻¹
positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 462,
"column": 6
} | {
"line": 462,
"column": 62
} | {
"line": 462,
"column": 63
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∈ L.lattice\nl : ↥L.lattice\nhl : ↑l / 2 ∉ L.lattice\n⊢ ℘[L] (-(↑l / 2) + ↑l) + (1 / ↑⟨l₀, hl₀⟩ ^ 2 - 1 / (-(↑l / 2) + ↑l - ↑⟨l₀, hl₀⟩) ^ 2) =\n ℘[L - -(l₀ - ↑l)] (↑l / 2) + (1 / l₀ ^ 2 - 1 / (l₀ - ↑l) ^ 2)",
"ppTerm": "?m.337",
"assigned": true,
"usedCon... | [
"L : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∈ L.lattice\nl : ↥L.lattice\nhl : ↑l / 2 ∉ L.lattice\n⊢ ℘[L] (-(↑l / 2) + ↑l) + (1 / ↑⟨l₀, hl₀⟩ ^ 2 - 1 / (-(↑l / 2) + ↑l - ↑⟨l₀, hl₀⟩) ^ 2) =\n ℘[L] (↑l / 2) + (1 / ↑⟨-(l₀ - ↑l), ⋯⟩ ^ 2 - 1 / (↑l / 2 - ↑⟨-(l₀ - ↑l), ⋯⟩) ^ 2) + (1 / l₀ ^ 2 - 1 / (l₀ - ↑l) ^ 2)"
] | L.weierstrassPExcept_def ⟨_, neg_mem (sub_mem hl₀ l.2)⟩, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.ContDiff | {
"line": 46,
"column": 4
} | {
"line": 46,
"column": 39
} | {
"line": 47,
"column": 4
} | [
{
"pp": "case inr.hint\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\ninst✝ : CompleteSpace E\nh : ContDiffOn ℝ 1 f (Icc a b)\nhab : a ≤ b\nh'ab : a < b\nthis : ContinuousOn (derivWithin f (Icc a b)) (Icc a b)\n⊢ derivWithin f (Icc a b) =ᵐ[volume.restrict (Ioc a b)] ... | [
"case inr.hint\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\ninst✝ : CompleteSpace E\nh : ContDiffOn ℝ 1 f (Icc a b)\nhab : a ≤ b\nh'ab : a < b\nthis : ContinuousOn (derivWithin f (Icc a b)) (Icc a b)\n⊢ derivWithin f (Icc a b) =ᵐ[volume.restrict (Ioo a b)] deriv f"
] | rw [← restrict_Ioo_eq_restrict_Ioc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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