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.SpecialFunctions.Trigonometric.InverseDeriv | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 24
} | [
{
"pp": "case refine_1\nx : ℝ\nn : ℕ∞ω\nh : ContDiffAt ℝ n arccos x\n⊢ ContDiffAt ℝ n arcsin x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.InverseDeriv | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 24
} | [
{
"pp": "case refine_2\nx : ℝ\nn : ℕ∞ω\nh : ContDiffAt ℝ n arcsin x\n⊢ ContDiffAt ℝ n arccos x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Stirling | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 48
} | [
{
"pp": "n : ℕ\n⊢ log (stirlingSeq 1) - log (stirlingSeq (n + 1)) ≤ 12⁻¹",
"usedConstants": [
"Real",
"instOfNatNat",
"Real.log",
"Stirling.stirlingSeq",
"instHAdd",
"HAdd.hAdd",
"Nat",
"instAddNat",
"OfNat.ofNat"
]
}
] | let f (k : ℕ) : ℝ := log (stirlingSeq (k + 1)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.SpecialFunctions.Stirling | {
"line": 180,
"column": 68
} | {
"line": 180,
"column": 93
} | [
{
"pp": "x : ℝ\nx_pos : 0 < x\nhx : ∀ (n : ℕ), x ≤ stirlingSeq (n + 1)\n⊢ x ∈ lowerBounds (Set.range (stirlingSeq ∘ succ))",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"lowerBounds",
"congrArg",
"setOf",
"Function.comp",
"Membership.mem",
"Exist... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.ContDiff | {
"line": 64,
"column": 42
} | {
"line": 70,
"column": 8
} | [
{
"pp": "E : 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 [[a, b]]\n⊢ ∫ (x : ℝ) in a..b, deriv f x = f b - f a",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NegZeroClass.toNeg",
... | by
rcases le_or_gt a b with hab | hab
· simp only [uIcc_of_le hab] at h
exact integral_deriv_of_contDiffOn_Icc h hab
· simp only [uIcc_of_ge hab.le] at h
rw [integral_symm, integral_deriv_of_contDiffOn_Icc h hab.le]
abel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Orthogonality | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : x ∈ Set.Ioo (min (-1) 1) (max (-1) 1)\n⊢ HasDerivAt (fun x ↦ -arccos x) (√(1 - x ^ 2)⁻¹) x",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 30
} | [
{
"pp": "L : PeriodPair\n⊢ L.ω₁ / 2 ∉ L.lattice",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 30
} | [
{
"pp": "L : PeriodPair\n⊢ L.ω₂ / 2 ∉ L.lattice",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 129,
"column": 2
} | {
"line": 132,
"column": 7
} | [
{
"pp": "L : PeriodPair\ns : Set ℂ\nhs : s ⊆ ↑L.lattice\n⊢ IsClosed s",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"SetLike.mem_coe._simp_1",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"HEq.refl",
"Comple... | convert!
L.isClosed_lattice.isClosedMap_subtype_val _ (isClosed_discrete (α := L.lattice) ((↑) ⁻¹' s))
convert! Set.image_preimage_eq_inter_range.symm using 1
simpa | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 129,
"column": 2
} | {
"line": 132,
"column": 7
} | [
{
"pp": "L : PeriodPair\ns : Set ℂ\nhs : s ⊆ ↑L.lattice\n⊢ IsClosed s",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"SetLike.mem_coe._simp_1",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"HEq.refl",
"Comple... | convert!
L.isClosed_lattice.isClosedMap_subtype_val _ (isClosed_discrete (α := L.lattice) ((↑) ⁻¹' s))
convert! Set.image_preimage_eq_inter_range.symm using 1
simpa | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 175,
"column": 61
} | {
"line": 175,
"column": 72
} | [
{
"pp": "L : PeriodPair\nf : ↥L.lattice → ℂ → ℂ\nu : ℝ → ↥L.lattice → ℝ\nhu : ∀ r > 0, Summable (u r)\nhf : ∀ r > 0, ∀ᶠ (R : ℝ) in atTop, ∀ (x : ℂ), ‖x‖ < r → ∀ (l : ↥L.lattice), ‖↑l‖ = R → ‖f l x‖ ≤ u r l\nx : ℂ\nr : ℝ\nhr : 0 < r\nhr' : 𝓝 x ≤ 𝓟 (Metric.ball 0 r)\nR : ℝ\nhR : ∀ b ≥ R, ∀ (x : ℂ), ‖x‖ < r → ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 224,
"column": 2
} | {
"line": 226,
"column": 43
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\n⊢ HasSumLocallyUniformly (fun l z ↦ if ↑l = l₀ then 0 else 1 / (z - ↑l) ^ 2 - 1 / ↑l ^ 2) ℘[L - l₀]",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"neg_lt_neg_iff._simp_1",
"Iff.mpr",
... | refine L.hasSumLocallyUniformly_aux (u := (10 * · * ‖·‖ ^ (-3 : ℝ))) _
(fun _ _ ↦ (ZLattice.summable_norm_rpow _ _ (by simp; norm_num)).mul_left _) fun r hr ↦
Filter.eventually_atTop.mpr ⟨2 * r, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 15
} | [
{
"pp": "case pos\nL : PeriodPair\nl₀ : ℂ\nr : ℝ\nhr : r > 0\ns : ℂ\nhs : ‖s‖ < r\nl : ↥L.lattice\nh : ‖↑l‖ ≥ 2 * r\nh✝ : ↑l = l₀\n⊢ ‖0‖ ≤ (fun x1 x2 ↦ 10 * x1 * ‖x2‖ ^ (-3)) r l",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"Re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Niven | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 13
} | [
{
"pp": "q : ℚ\n⊢ IsIntegral ℤ (cexp (-(↑q * ↑π) * I))",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"congrArg",
"Complex.instNormedField",
"Complex.instMul",
"id",
"NonUnitalNonAssocRing.toNonUni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 33
} | [
{
"pp": "L : PeriodPair\nx : ℂ\nhx : x ∈ L.lattice\nH : ContinuousAt (fun z ↦ ℘[L - x] z + (1 / (z - x) ^ 2 - 1 / x ^ 2)) x\n⊢ ContinuousAt (fun x ↦ x ^ (-2)) 0",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMono... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 15
} | [
{
"pp": "case pos\nL : PeriodPair\nl₀ : ℂ\nr : ℝ\nhr : r > 0\ns : ℂ\nhs : ‖s‖ < r\nl : ↥L.lattice\nh : ‖↑l‖ ≥ 2 * r\nh✝ : ↑l = l₀\n⊢ ‖0‖ ≤ (fun x x_1 ↦ 16 * ‖x_1‖ ^ (-3)) r l",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Niven | {
"line": 168,
"column": 46
} | {
"line": 168,
"column": 57
} | [
{
"pp": "r : ℚ\nθ : ℝ\nh : ↑r * π = θ\nhcos : ∃ q, cos θ = ↑q\nh_bnd : θ ∈ Set.Icc (0 * π) (1 * π)\n⊢ θ ∈ Set.Icc 0 π",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"Real.instZero",
"Preorder.toLE",
"Membership.mem",
"id",
"LE.le",
"And",
"Set.I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 22
} | [
{
"pp": "case h.e'_4\nn : ℕ\nP : ℝ[X]\nhPdeg : P.degree ≤ ↑n\nhPbnd : ∀ x ∈ Set.Icc (-1) 1, |eval x P| ≤ 1\n⊢ 2 ^ (n - 1) = sumNodes n (fun i ↦ leadingCoeffC n i) (T ℝ ↑n)",
"usedConstants": [
"Eq.mpr",
"Real",
"Polynomial.Chebyshev.T",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat"... | · rw [sumNodes_T_eq] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Niven | {
"line": 178,
"column": 68
} | {
"line": 178,
"column": 87
} | [
{
"pp": "r q : ℚ\nhq : cos (↑r * π) = ↑q\n⊢ cos (↑r * π - ↑⌊r⌋ * π) = (-1) ^ ⌊r⌋ * ↑q",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"NormedRin... | cos_sub_int_mul_pi, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 393,
"column": 8
} | {
"line": 393,
"column": 19
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\nx : ↥L.lattice\nhl₁ : ¬↑x = l₀\nhl : ↑x ∈ (↑L.lattice \\ {l₀})ᶜ\ne : ↑x - ↑x = 0\n⊢ ↑x = l₀",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Extremal | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 22
} | [
{
"pp": "case h.e'_1.h.e'_3\nn : ℕ\nP : ℝ[X]\nhPdeg : P.degree ≤ ↑n\nhPbnd : ∀ x ∈ Set.Icc (-1) 1, |eval x P| ≤ 1\n⊢ 2 ^ (n - 1) = sumNodes n (fun i ↦ leadingCoeffC n i) (T ℝ ↑n)",
"usedConstants": [
"Eq.mpr",
"Real",
"Polynomial.Chebyshev.T",
"congrArg",
"Nat.instAtLeastTwoHAd... | · rw [sumNodes_T_eq] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 405,
"column": 8
} | {
"line": 405,
"column": 19
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\ns : Finset ↥L.lattice\nx : ↥L.lattice\nhxs : x ∈ s\nhl₁ : ¬↑x = l₀\nhl : ↑x ∈ (↑L.lattice \\ {l₀})ᶜ\ne : ↑x - ↑x = 0\n⊢ ↑x = l₀",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 422,
"column": 2
} | {
"line": 422,
"column": 40
} | [
{
"pp": "L : PeriodPair\n⊢ ℘'[L - 0] 0 = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 445,
"column": 17
} | {
"line": 445,
"column": 28
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∈ L.lattice\nl : ↥L.lattice\nhl : ↑l / 2 ∉ L.lattice\nx : ℂ\nh₁ : x ∈ (↑L.lattice \\ {l₀ - ↑l})ᶜ\nh₂ : x + ↑l ∈ ↑L.lattice\nh₃ : x + ↑l ≠ l₀\n⊢ x ∈ ↑L.lattice",
"usedConstants": [
"Eq.mpr",
"Submodule",
"SetLike.mem_coe._simp_1",
"NonUnitalCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Finset.Box | {
"line": 34,
"column": 51
} | {
"line": 34,
"column": 62
} | [
{
"pp": "case ha\nα : Type u_1\ninst✝³ : Ring α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedRing α\ninst✝ : LocallyFiniteOrder α\nm n : ℕ\nhmn : m ≤ n\n⊢ -↑n ≤ -↑m",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Ring.toNonAssocRing",
"AddGroupWithOne.toAddGroup",
"covaria... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Interval.Finset.Box | {
"line": 34,
"column": 51
} | {
"line": 34,
"column": 62
} | [
{
"pp": "case hb\nα : Type u_1\ninst✝³ : Ring α\ninst✝² : PartialOrder α\ninst✝¹ : IsOrderedRing α\ninst✝ : LocallyFiniteOrder α\nm n : ℕ\nhmn : m ≤ n\n⊢ ↑m ≤ ↑n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 451,
"column": 6
} | {
"line": 451,
"column": 17
} | [
{
"pp": "case a\nL : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∈ L.lattice\nl : ↥L.lattice\nhl : ↑l / 2 ∉ L.lattice\nx : ℂ\nhx : x ∈ (↑L.lattice \\ {l₀ - ↑l})ᶜ\n⊢ x ∈ (↑L.lattice \\ {l₀ - ↑l})ᶜ",
"usedConstants": [
"Eq.mpr",
"Submodule",
"SetLike.mem_coe._simp_1",
"NonUnitalCommRing.toNonUnitalNo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 454,
"column": 27
} | {
"line": 454,
"column": 38
} | [
{
"pp": "L : PeriodPair\nl₀ : ℂ\nhl₀ : l₀ ∈ L.lattice\nl : ↥L.lattice\nhl : ↑l / 2 ∉ L.lattice\nx : ℂ\nhx : x ∈ L.lattice → x + ↑l = l₀\nH : x + ↑l ∈ L.lattice\n⊢ x ∈ L.lattice",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 108,
"column": 2
} | {
"line": 109,
"column": 60
} | [
{
"pp": "c d : ℝ\nhc : 1 ≤ c ^ 2\nz : ℂ\nhz : 0 < z.im\nH1 : z.im ≤ √((c * z.re + d) ^ 2 + (↑c * z).im ^ 2)\n⊢ r { coe := z, coe_im_pos := hz } ≤ ‖↑c * ↑{ coe := z, coe_im_pos := hz } + ↑d‖",
"usedConstants": [
"Complex.mul_im",
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 114,
"column": 2
} | {
"line": 115,
"column": 73
} | [
{
"pp": "z : ℍ\nc d : ℝ\nhd : 1 ≤ d ^ 2\nH1 : √(r1 z) ≤ √((c * z.re + d) ^ 2 + (c * z.im) ^ 2)\n⊢ r z ≤ ‖↑c * ↑z + ↑d‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"Lattice.toSemilatticeSup",
"HMul.hMul",
"_private.Mathlib.NumberTheory.ModularF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 118,
"column": 49
} | {
"line": 129,
"column": 14
} | [
{
"pp": "x : Fin 2 → ℤ\nhx : x ≠ 0\n⊢ 1 ≤ (↑(x 0) / ‖x‖) ^ 2 ∨ 1 ≤ (↑(x 1) / ‖x‖) ^ 2",
"usedConstants": [
"max_choice",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Int.cast",
"GroupWithZero.toMonoidWithZero",
"False",
"Int.cas... | by
refine (max_choice (x 0).natAbs (x 1).natAbs).imp (fun H0 ↦ ?_) (fun H1 ↦ ?_)
· have : x 0 ≠ 0 := by
rwa [← norm_ne_zero_iff, norm_eq_max_natAbs, H0, Nat.cast_ne_zero, Int.natAbs_ne_zero] at hx
simp only [norm_eq_max_natAbs, H0, Nat.cast_natAbs, Int.cast_abs, div_pow, sq_abs, ne_eq,
OfNat.ofNat_n... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 71
} | [
{
"pp": "case hbc.inl\nz : ℍ\nx : Fin 2 → ℤ\nhx : x ≠ 0\nhn0 : ‖x‖ ≠ 0\nh11 : ↑(x 0) * ↑z + ↑(x 1) = (↑(x 0) / ↑‖x‖ * ↑z + ↑(x 1) / ↑‖x‖) * ↑‖x‖\nH1 : 1 ≤ (↑(x 0) / ‖x‖) ^ 2\n⊢ r z ≤ ‖↑(x 0) / ↑↑(max (x 0).natAbs (x 1).natAbs) * ↑z + ↑(x 1) / ↑↑(max (x 0).natAbs (x 1).natAbs)‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 139,
"column": 6
} | {
"line": 139,
"column": 71
} | [
{
"pp": "case hbc.inr\nz : ℍ\nx : Fin 2 → ℤ\nhx : x ≠ 0\nhn0 : ‖x‖ ≠ 0\nh11 : ↑(x 0) * ↑z + ↑(x 1) = (↑(x 0) / ↑‖x‖ * ↑z + ↑(x 1) / ↑‖x‖) * ↑‖x‖\nH2 : 1 ≤ (↑(x 1) / ‖x‖) ^ 2\n⊢ r z ≤ ‖↑(x 0) / ↑↑(max (x 0).natAbs (x 1).natAbs) * ↑z + ↑(x 1) / ↑↑(max (x 0).natAbs (x 1).natAbs)‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 32
} | [
{
"pp": "case hΘ\nc e : ℤ\nz : ℂ\n⊢ (fun d ↦ ↑c * z) =o[cofinite] fun n ↦ ↑n",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"HMul.hMul",
"Complex.instNormedAddCommGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 32
} | [
{
"pp": "case ho\nc e : ℤ\nz : ℂ\n⊢ (fun d ↦ ↑e) =o[cofinite] fun n ↦ ↑n",
"usedConstants": [
"Norm.norm",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Complex.instNormedAddCommGroup",
"congrArg",
"Complex.instNormedField",
"Func... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 13
} | [
{
"pp": "c : ℤ\nz : ℂ\n⊢ (fun d ↦ ↑c * z + ↑d) =Θ[cofinite] fun n ↦ ↑n",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"HMul.hMul",
"instConditionallyCompleteLinearOrder",
"congrArg",
"PartialOrder.toPreorder",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 23
} | [
{
"pp": "z : ℂ\nc₁ c₂ : ℤ\n⊢ (fun n ↦ ((↑c₁ * z - ↑n) * (↑c₂ * z + ↑n))⁻¹) =O[cofinite] fun n ↦ (|↑n| ^ 2)⁻¹",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"HMul.hMul",
"pow_two",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 23
} | [
{
"pp": "z : ℂ\nc₁ c₂ : ℤ\n⊢ (fun n ↦ ((↑c₁ * z + ↑n + 1) * (↑c₂ * z + ↑n))⁻¹) =O[cofinite] fun n ↦ (|↑n| ^ 2)⁻¹",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"HMul.hMul",
"pow_t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 13
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\nc₁ c₂ : ℤ\n⊢ (fun n ↦ ((↑n * z + ↑c₁) * (↑n * z + ↑c₂))⁻¹) =O[cofinite] fun n ↦ (↑n * ↑n)⁻¹",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"DivInvMonoid.toInv",
"HMul.hMul",
"instConditionallyCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 275,
"column": 21
} | {
"line": 275,
"column": 63
} | [
{
"pp": "z : ℍ\na b : ℤ\nh0 : z ∈ verticalStrip |z.re| z.im\nm : Fin 2 → ℤ\n⊢ ‖((↑(m 0) + ↑a) * ↑z + ↑(m 1) + ↑b)⁻¹‖ ≤ ?m.98",
"usedConstants": [
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"UpperHalfPlane.coe",
"congrAr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 55
} | [
{
"pp": "α : Type u_1\na : α\ninst✝² : NormedAddCommGroup α\ninst✝¹ : DiscreteTopology α\ninst✝ : ProperSpace α\n⊢ (fun x ↦ a) =o[cofinite] fun x ↦ ‖x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"Function.comp",
"id",
"Real.normedAddCommGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Summable | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 57
} | [
{
"pp": "a b : ℤ\nthis : ∀ (x : Fin 2 → ℤ), ![x 0 + a, x 1 + b] = x + ![a, b]\n⊢ (fun m ↦ ‖![m 0 + a, m 1 + b]‖⁻¹) =Θ[cofinite] fun m ↦ ‖m‖⁻¹",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrAr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 55
} | [
{
"pp": "case inr\nn : ℤ\nx : ℝ\nhx : 1 ≤ |x|\nthis : ∀ (n : ℤ) {x : ℝ}, 1 ≤ |x| → 0 ≤ x → 1 ≤ |eval x (T ℝ n)|\nh : x < 0\n⊢ 1 ≤ |eval x (T ℝ n)|",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 55
} | [
{
"pp": "case inr\nn : ℤ\nhn : n ≠ 0\nx : ℝ\nhx : 1 < |x|\nthis : ∀ {n : ℤ}, n ≠ 0 → ∀ {x : ℝ}, 1 < |x| → 0 ≤ x → 1 < |eval x (T ℝ n)|\nh : x < 0\n⊢ 1 < |eval x (T ℝ n)|",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 87,
"column": 32
} | {
"line": 87,
"column": 43
} | [
{
"pp": "n : ℤ\nhn : n ≠ 0\nx : ℝ\n⊢ |eval x (T ℝ n)| ≤ 1 → |x| ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 666,
"column": 10
} | {
"line": 666,
"column": 71
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\n⊢ 1 ∈ (fun x_1 ↦ x_1 * ‖z - x‖) ⁻¹' (↑(upperClosure (dist x '' (↑L.lattice \\ {l₀}))))ᶜ",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"not_exists._simp_1",
"No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 669,
"column": 6
} | {
"line": 669,
"column": 17
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : κ > 0\nhκ' : Metric.ball 1 κ ⊆ (fun x_1 ↦ x_1 * ‖z - x‖) ⁻¹' (↑(upperClosure (dist x '' (↑L.lattice \\ {l₀}))))ᶜ\nl : ↥L.lattice\nhl : ↑l ≠ l₀\n⊢ ∀ l ∈ L.lattice, l ≠ l₀ → (κ / 2 + 1) * ‖z - x‖ < dist x l",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 104,
"column": 18
} | {
"line": 104,
"column": 29
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nx : ℝ\nhx : |x| ≤ 1\nk : ℕ\nhk : ↑k * π = ↑n * arccos x\nhk' : ↑k = ↑n * (arccos x / π)\nhkn : ↑k ≤ ↑n\n⊢ k ≤ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 670,
"column": 4
} | {
"line": 670,
"column": 64
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : κ > 0\nhκ' : Metric.ball 1 κ ⊆ (fun x_1 ↦ x_1 * ‖z - x‖) ⁻¹' (↑(upperClosure (dist x '' (↑L.lattice \\ {l₀}))))ᶜ\nl : ↥L.lattice\nhl : ↑l ≠ l₀\nthis : ∀ l ∈ L.lattice, l ≠ l₀ → (κ / 2 + 1) * ‖z - x‖ < dist x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 677,
"column": 4
} | {
"line": 678,
"column": 28
} | [
{
"pp": "case h₁\nL : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : 1 < κ\nhκ' : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice := ⋯\n⊢ Summable (((Function.uncurry fun b c ↦ L.weierstrassPExceptSummand l₀ x b c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 683,
"column": 6
} | {
"line": 683,
"column": 28
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : 1 < κ\nhκ' : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.ArithmeticGeometric | {
"line": 90,
"column": 2
} | {
"line": 91,
"column": 57
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝ : Field R\nha : a ≠ 1\nn : ℕ\n⊢ arithGeom a b b n = b * (a ^ (n + 1) - 1) / (a - 1)",
"usedConstants": [
"Eq.mpr",
"neg_sub",
"neg_div",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUn... | rw [arithGeom_same_eq_mul_div' ha n, ← neg_sub _ a, div_neg,
← neg_sub _ (a ^ (n + 1)), mul_neg, neg_div, neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecificLimits.ArithmeticGeometric | {
"line": 90,
"column": 2
} | {
"line": 91,
"column": 57
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝ : Field R\nha : a ≠ 1\nn : ℕ\n⊢ arithGeom a b b n = b * (a ^ (n + 1) - 1) / (a - 1)",
"usedConstants": [
"Eq.mpr",
"neg_sub",
"neg_div",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUn... | rw [arithGeom_same_eq_mul_div' ha n, ← neg_sub _ a, div_neg,
← neg_sub _ (a ^ (n + 1)), mul_neg, neg_div, neg_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.ArithmeticGeometric | {
"line": 90,
"column": 2
} | {
"line": 91,
"column": 57
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝ : Field R\nha : a ≠ 1\nn : ℕ\n⊢ arithGeom a b b n = b * (a ^ (n + 1) - 1) / (a - 1)",
"usedConstants": [
"Eq.mpr",
"neg_sub",
"neg_div",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUn... | rw [arithGeom_same_eq_mul_div' ha n, ← neg_sub _ a, div_neg,
← neg_sub _ (a ^ (n + 1)), mul_neg, neg_div, neg_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 692,
"column": 33
} | {
"line": 692,
"column": 58
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nκ : ℝ\nhκ : 1 < κ\nhκ' : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ * κ < ‖↑l - x‖\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 693,
"column": 6
} | {
"line": 693,
"column": 42
} | [
{
"pp": "L : PeriodPair\nl₀ z : ℂ\nκ : ℝ\nhκ : 1 < κ\ne : ℕ × ↥L.lattice ≃ ↥L.lattice ⊕ ℕ × ↥L.lattice :=\n (Equiv.prodCongrLeft fun x ↦ (Denumerable.eqv (Option ℕ)).symm).trans optionProdEquiv\nH₁ : Summable fun i ↦ (↑i + 2) * κ ^ (-↑i)\np : ℕ × ↥L.lattice\nhp : ¬↑p.2 = l₀\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.Fib.Basic | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 25
} | [
{
"pp": "case inr\nn : ℕ\nhn : Odd n\n⊢ fib (-↑n) = (-1) ^ (n + 1) * ↑(Nat.fib n)",
"usedConstants": [
"one_pow",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing... | · simp [fib_of_odd, hn] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 716,
"column": 4
} | {
"line": 716,
"column": 26
} | [
{
"pp": "case neg\nL : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nl : ↥L.lattice\nh : ¬↑l = l₀\n⊢ 1 / (z - ↑l) ^ 2 - 1 / ↑l ^ 2 =\n ∑' (i : ℕ), ((↑i + 1) * (↑l - x) ^ (-↑(i + 2)) - Nat.casesOn i (↑l ^ (-2)) 0) * (z - x) ^ i",
"usedConstants": [
"neg_add_rev",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 717,
"column": 10
} | {
"line": 717,
"column": 35
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nl : ↥L.lattice\nh : ¬↑l = l₀\n⊢ ↑l ≠ x",
"usedConstants": [
"Submodule",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Complex.instNormedField",
"Membership.mem",
"id",
"NonUni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 718,
"column": 10
} | {
"line": 718,
"column": 39
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\nl : ↥L.lattice\nh : ¬↑l = l₀\n⊢ z - x ∈ Metric.eball 0 ‖↑l - x‖ₑ",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Int.Fib.Basic | {
"line": 153,
"column": 71
} | {
"line": 156,
"column": 62
} | [
{
"pp": "m n : ℤ\n⊢ (fib m).gcd (fib n) = Nat.fib (m.gcd n)",
"usedConstants": [
"Nat.gcd",
"Int.gcd",
"congrArg",
"ite_self",
"Int.fib",
"Int.gcd_neg",
"Exists",
"Int.instDecidablePredEven",
"apply_ite",
"Nat.fib_gcd",
"Int.instNegInt",
... | by
obtain ⟨m, (rfl | rfl)⟩ := m.eq_nat_or_neg
<;> obtain ⟨n, (rfl | rfl)⟩ := n.eq_nat_or_neg
<;> simp [fib_neg, Nat.fib_gcd, apply_ite, apply_ite_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Int.Fib.Basic | {
"line": 159,
"column": 70
} | {
"line": 159,
"column": 81
} | [
{
"pp": "m n : ℤ\n⊢ fib ↑(m.gcd n) = ↑((fib m).gcd (fib n))",
"usedConstants": [
"Int.gcd",
"Eq.mpr",
"Int.fib",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"Int",
"Nat.cast",
"Nat.fib",
"Int.instRing",
"Nat",
"Nat.cast_inj._simp_1",
"i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 709,
"column": 87
} | {
"line": 723,
"column": 56
} | [
{
"pp": "L : PeriodPair\nl₀ z x : ℂ\nhx : ∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z - x‖ < ‖↑l - x‖\n⊢ ℘[L - l₀] z = ∑' (i : ℕ), (L.weierstrassPExceptSeries l₀ x).coeff i * (z - x) ^ i",
"usedConstants": [
"neg_add_rev",
"Int.instAddCommGroup",
"AddGroup.toSubtractionMonoid",
"NonUnitalNonAss... | by
trans ∑' (l : L.lattice) (i : ℕ), if l.1 = l₀ then 0 else
((i + 1) * (l.1 - x) ^ (- ↑(i + 2) : ℤ) - i.casesOn (l.1 ^ (-2 : ℤ)) 0) * (z - x) ^ i
· delta weierstrassPExcept
congr 1 with l
split_ifs with h
· simp
simpa [mul_comm] using ((Complex.one_div_sub_sq_sub_one_div_sq_hasFPowerSeriesOnB... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 744,
"column": 6
} | {
"line": 745,
"column": 13
} | [
{
"pp": "case r_le.h\nL : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\nl : ↥L.lattice\nhl : ↑l ≠ l₀\n⊢ ‖x + ↑↑r - x‖ < ‖↑l - x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Submodule",
"Real",
"NonUnitalCommRing.toNonUnital... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 748,
"column": 31
} | {
"line": 748,
"column": 42
} | [
{
"pp": "L : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\nz : ℂ\nhz : z ∈ Metric.eball 0 ↑r\n⊢ ‖z‖ < ↑r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 752,
"column": 6
} | {
"line": 752,
"column": 67
} | [
{
"pp": "L : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\nz : ℂ\nhz : ‖z‖ < ↑r\nthis :\n (∀ (l : ↥L.lattice), ↑l ≠ l₀ → ‖z‖ < ‖↑l - x‖) →\n HasSum (fun i ↦ (L.weierstrassPExceptSeries l₀ x).coeff i * z ^ i) (℘[L - l₀] (x + z))\nl : ↥L.lattice\nhl : ↑l ≠... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 763,
"column": 2
} | {
"line": 763,
"column": 30
} | [
{
"pp": "L : PeriodPair\nl : ℂ\nr : ℝ\nh₁ : 0 < r\nh₂ : Metric.closedBall l r ⊆ (↑L.lattice \\ {l})ᶜ\n⊢ HasFPowerSeriesAt ℘[L - l]\n (FormalMultilinearSeries.ofScalars ℂ fun i ↦ if i = 0 then ℘[L - l] l else (↑i + 1) * L.sumInvPow l (i + 2)) l",
"usedConstants": [
"Real.instLE",
"Real",
... | lift r to NNReal using h₁.le | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 764,
"column": 2
} | {
"line": 764,
"column": 40
} | [
{
"pp": "L : PeriodPair\nl : ℂ\nr : NNReal\nh₁ : 0 < ↑r\nh₂ : Metric.closedBall l ↑r ⊆ (↑L.lattice \\ {l})ᶜ\n⊢ HasFPowerSeriesAt ℘[L - l]\n (FormalMultilinearSeries.ofScalars ℂ fun i ↦ if i = 0 then ℘[L - l] l else (↑i + 1) * L.sumInvPow l (i + 2)) l",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 781,
"column": 4
} | {
"line": 781,
"column": 15
} | [
{
"pp": "L : PeriodPair\nl : ℂ\nn : ℕ\n⊢ iteratedDeriv n ℘[L - l] l / ↑n ! = if n = 0 then ℘[L - l] l else (↑n + 1) * L.sumInvPow l (n + 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 50
} | [
{
"pp": "case mp.e_x.e_a\nn : ℕ\nhn : 2 ≤ n\nx : ℝ\nhx✝ : x ∈ Finset.image (fun k ↦ cos ((↑k + 1) * π / (↑(n - 1) + 1))) (Finset.range (n - 1))\nk : ℕ\nhk₁ : k ∈ Finset.range (n - 1)\nhx : cos ((↑k + 1) * π / (↑(n - 1) + 1)) = x\n⊢ n - 1 + 1 = n",
"usedConstants": [
"instOfNatNat",
"Nat.one_le_o... | exact Nat.sub_add_cancel (Nat.one_le_of_lt hn) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 806,
"column": 4
} | {
"line": 806,
"column": 15
} | [
{
"pp": "case refine_2\nL : PeriodPair\nl₀ x : ℂ\nr : NNReal\nhr0 : 0 < r\nhr : Metric.closedBall x ↑r ⊆ (↑L.lattice \\ {l₀})ᶜ\n⊢ Metric.eball x ↑r ⊆ Metric.closedBall x ↑r",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"ENNReal.ofNNReal",
"Complex.instNormedAd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 813,
"column": 2
} | {
"line": 813,
"column": 45
} | [
{
"pp": "L : PeriodPair\nl : ℂ\nr : ℝ\nh₁ : 0 < r\nh₂ : Metric.closedBall l r ⊆ (↑L.lattice \\ {l})ᶜ\n⊢ HasFPowerSeriesAt ℘'[L - l] (FormalMultilinearSeries.ofScalars ℂ fun i ↦ (↑i + 1) * (↑i + 2) * L.sumInvPow l (i + 3))\n l",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 828,
"column": 4
} | {
"line": 828,
"column": 15
} | [
{
"pp": "L : PeriodPair\nl : ℂ\nn : ℕ\n⊢ iteratedDeriv n ℘'[L - l] l / ↑n ! = (↑n + 1) * (↑n + 2) * L.sumInvPow l (n + 3)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 836,
"column": 2
} | {
"line": 836,
"column": 13
} | [
{
"pp": "L : PeriodPair\nl : ℂ\n⊢ deriv ℘'[L - l] l = 6 * L.sumInvPow l 4",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 923,
"column": 8
} | {
"line": 923,
"column": 25
} | [
{
"pp": "case pos\nL : PeriodPair\nl₀ : ℂ\nh : l₀ ∈ L.lattice\nthis : AnalyticAt ℂ ℘[L - l₀] l₀\nhl₀ : l₀ = 0\n⊢ AnalyticAt ℂ (fun z ↦ (z - l₀) ^ 2 / l₀ ^ 2) l₀",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNormedSpace",
"False",
"instHDi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 37
} | [
{
"pp": "x : ℂ\nhx : x ∈ ℂ_ℤ\nn : ℕ\n⊢ (↑n + 1) ^ 2 ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"GroupWithZero.toMonoidWithZero",
"False",
"eq_false",
"AddMonoid.toAddSemigroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"NormedDivisionRing.toNormMul... | · simp [Nat.cast_add_one_ne_zero n] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 57
} | [
{
"pp": "case hf\nx : ℂ\n⊢ Summable fun i ↦ ‖sineTerm x i‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Real",
"instHDiv",
"NormedRing.toRing",
"Nat.one_lt_two",
"AddGroupWithOne.toAddMonoidWithOne",
"PseudoMetricSpace.toUniformSpace",
... | have := summable_pow_div_add (x ^ 2) 2 1 Nat.one_lt_two | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 24
} | [
{
"pp": "case hf\nx : ℂ\nthis : Summable fun n ↦ ‖x ^ 2 / (↑n + ↑1) ^ 2‖\n⊢ Summable fun i ↦ ‖sineTerm x i‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 15
} | [
{
"pp": "case refine_1\nZ : Set ℂ\nhZ : IsCompact Z\nhf : ContinuousOn (fun x ↦ ‖-x ^ 2‖) Z\ns : ℝ\nhs : ∀ y ∈ (fun x ↦ ‖-x ^ 2‖) '' Z, y ≤ s\n⊢ Summable fun n ↦ ‖↑s / (↑n + 1) ^ 2‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"SeminormedRing.toN... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 51
} | [
{
"pp": "Z : Set ℂ\nhZ2 : Z ⊆ ℂ_ℤ\nhZC : IsCompact Z\n⊢ Set.EqOn (fun x ↦ ∏' (i : ℕ), (1 + sineTerm x i)) (fun x ↦ Complex.sin (↑π * x) / (↑π * x)) Z",
"usedConstants": [
"Membership.mem",
"euler_sineTerm_tprod",
"Complex",
"Set.instMembership",
"Set"
]
}
] | exact fun x hx => euler_sineTerm_tprod (by aesop) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1013,
"column": 4
} | {
"line": 1013,
"column": 48
} | [
{
"pp": "L : PeriodPair\nthis : 7 ≤ meromorphicOrderAt L.relation 0 + ↑6 * meromorphicOrderAt id 0\n⊢ 1 + 6 ≤ meromorphicOrderAt L.relation 0 + 6",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1018,
"column": 4
} | {
"line": 1018,
"column": 82
} | [
{
"pp": "case pos\nL : PeriodPair\ni : ℕ\nhi₁ : i < 7\nhi₂ : Odd i\n⊢ iteratedDeriv i (L.relation * id ^ 6) 0 = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NegZeroClass.toNeg",
"_private.Mathlib.Analysis.SpecialFunctions.E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.NormNum.Prime | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 56
} | [
{
"pp": "case refine_2.inr.inl\nn k k' : ℕ\ne : k + 2 = k'\nh : MinFacHelper n k\nnp : n.minFac ≠ k\nh2✝ : k < n.minFac\nh2 : k.succ = n.minFac\nh3 : 2 ∣ n.minFac\n⊢ 2 = n.minFac",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1038,
"column": 2
} | {
"line": 1038,
"column": 13
} | [
{
"pp": "case e_c\nL : PeriodPair\nx : ℂ\nl : ↥L.lattice\n⊢ (x + ↑l ∈ L.lattice) = (x ∈ L.lattice)",
"usedConstants": [
"Eq.mpr",
"Submodule",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Complex.instNormedField",
"eq_iff_iff._simp_1",
"Membership.mem",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1054,
"column": 6
} | {
"line": 1054,
"column": 52
} | [
{
"pp": "L : PeriodPair\nx : ℂ\nhx : x ∉ L.lattice\n⊢ AnalyticAt ℂ (fun z ↦ ℘'[L] z ^ 2 - 4 * ℘[L] z ^ 3 + L.g₂ * ℘[L] z + L.g₃) x",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Complex.instNormedAddCommGroup",
"Complex.instDenselyNormedField",
"Complex.instRCLike",
"... | have := L.analyticOnNhd_derivWeierstrassP _ hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 179,
"column": 4
} | {
"line": 179,
"column": 32
} | [
{
"pp": "x : ℂ\nhx : x ∈ ℂ_ℤ\ni : ℕ\nh1 : x + ↑(↑i + 1) ≠ 0\n⊢ x - (↑i + 1) ≠ 0",
"usedConstants": [
"neg_add_rev",
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
"Complex.instZero",
"AddZer... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Elliptic.Weierstrass | {
"line": 1071,
"column": 2
} | {
"line": 1071,
"column": 50
} | [
{
"pp": "L : PeriodPair\nz : ℂ\nhz : z ∉ L.lattice\n⊢ ℘'[L] z ^ 2 = 4 * ℘[L] z ^ 3 - L.g₂ * ℘[L] z - L.g₃",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 13
} | [
{
"pp": "x : ℂ\nhx : x ∈ ℂ_ℤ\n⊢ Tendsto (fun n ↦ logDeriv (fun z ↦ ∏ j ∈ Finset.range n, (1 + sineTerm z j)) x) atTop\n (𝓝 (logDeriv (fun t ↦ Complex.sin (↑π * t) / (↑π * t)) x))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 32
} | [
{
"pp": "case ha\nx : ℂ\nhz : x ∈ ℂ_ℤ\nn : ℕ\n⊢ x - (↑n + 1) ≠ 0",
"usedConstants": [
"neg_add_rev",
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
"AddZeroClass.toAddZero",
"Complex.addGrou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 15
} | [
{
"pp": "case hb\nx : ℂ\nhz : x ∈ ℂ_ℤ\nn : ℕ\n⊢ x + (↑n + 1) ≠ 0",
"usedConstants": [
"id",
"Ne",
"Complex.instNatCast",
"Nat.cast",
"Field.toSemifield",
"instHAdd",
"Semifield.toDivisionSemiring",
"HAdd.hAdd",
"DivisionSemiring.toSemiring",
"One.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 15
} | [
{
"pp": "x : ℂ\nhz : x ∈ ℂ_ℤ\nthis : Summable fun n ↦ (x - ↑(n + 1))⁻¹ * (x + ↑(n + 1))⁻¹\n⊢ Summable fun i ↦ 1 / ((x + (↑i + 1)) * (x - (↑i + 1)))",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"Non... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 13
} | [
{
"pp": "k : ℕ\nd : ℤ\n⊢ ContDiffOn ℂ (↑k) (fun z ↦ 1 / (z + ↑d)) ℂ_ℤ",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"Complex.instNormedAddCommGroup",
"ENat.instNatCast",
"M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 32
} | [
{
"pp": "case hf\nk d : ℕ\nz : ℂ\nhz : z ∈ ℂ_ℤ\n⊢ ContDiffAt ℂ (↑k) (fun z ↦ 1 / (z - (↑d + 1))) z",
"usedConstants": [
"neg_add_rev",
"ContDiffAt",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"DivInvMonoid.toInv",
"instHDiv",
"Complex.instNormedAddCommGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 15
} | [
{
"pp": "case hg\nk d : ℕ\nz : ℂ\nhz : z ∈ ℂ_ℤ\n⊢ ContDiffAt ℂ (↑k) (fun z ↦ 1 / (z + (↑d + 1))) z",
"usedConstants": [
"ContDiffAt",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"Complex.instNormedAddCommGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 13
} | [
{
"pp": "k d : ℕ\nz : ℂ\nhz : z ∈ ℍₒ\n⊢ iteratedDerivWithin k (fun z ↦ cotTerm z d) (range Int.cast)ᶜ z =\n (fun z ↦ (-1) ^ k * ↑k ! * ((z + (↑d + 1)) ^ (-1 - ↑k) + (z - (↑d + 1)) ^ (-1 - ↑k))) z",
"usedConstants": [
"Int.cast",
"InnerProductSpace.toNormedSpace",
"HMul.hMul",
"Com... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 315,
"column": 4
} | {
"line": 316,
"column": 26
} | [
{
"pp": "case hbc.h₁\nk : ℕ\nK : Set ℂ\nA B : ℝ\nhB : 0 < B\nhKAB : K ⊆ UpperHalfPlane.coe '' verticalStrip A B\nn : ℕ\na : ℍ\nhaAB : a ∈ verticalStrip A B\nha : ↑a ∈ K\nh1 :\n ‖↑(![1, ↑n + 1] 0) * ↑a + ↑(![1, ↑n + 1] 1)‖ ^ (-(↑k + 1)) ≤\n r { coe := { re := A, im := B }, coe_im_pos := hB } ^ (-(↑k + 1)) * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 317,
"column": 4
} | {
"line": 318,
"column": 26
} | [
{
"pp": "case hbc.h₂\nk : ℕ\nK : Set ℂ\nA B : ℝ\nhB : 0 < B\nhKAB : K ⊆ UpperHalfPlane.coe '' verticalStrip A B\nn : ℕ\na : ℍ\nhaAB : a ∈ verticalStrip A B\nha : ↑a ∈ K\nh1 :\n ‖↑(![1, ↑n + 1] 0) * ↑a + ↑(![1, ↑n + 1] 1)‖ ^ (-(↑k + 1)) ≤\n r { coe := { re := A, im := B }, coe_im_pos := hB } ^ (-(↑k + 1)) * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 337,
"column": 4
} | {
"line": 337,
"column": 41
} | [
{
"pp": "case ha.hf\nn l : ℕ\nx : ℂ\nhx : x ∈ ℍₒ\n⊢ x + (↑n + 1) ≠ 0",
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"_private.Mathlib.Ana... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 29
} | [
{
"pp": "case ha.hg\nn l : ℕ\nx : ℂ\nhx : x ∈ ℍₒ\n⊢ x - (↑n + 1) ≠ 0",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent.0.differentiableOn_iteratedDerivWithin_cotTerm._simp_1_2",
"AddGroupWithOne.toAddGroup",
"congrArg",
"HSub.hSu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.GoldenRatio | {
"line": 225,
"column": 2
} | {
"line": 233,
"column": 51
} | [
{
"pp": "n : ℕ\n⊢ φ * ↑(Nat.fib (n + 1)) + ↑(Nat.fib n) = φ ^ (n + 1)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Common.div_congr",
"Mathlib.Meta.NormNum.is... | induction n with
| zero => simp
| succ n ih =>
calc
_ = φ * (Nat.fib n) + φ ^ 2 * (Nat.fib (n + 1)) := by
simp only [Nat.fib_add_one (Nat.succ_ne_zero n), Nat.succ_sub_succ_eq_sub,
Nat.cast_add, goldenRatio_sq, Nat.sub_zero]; ring
_ = φ * ((Nat.fib n) + φ * (Nat.fib (n + 1))) := by... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
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