module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.ModularForms.Derivative | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 9
} | {
"line": 160,
"column": 0
} | [
{
"pp": "k c : ℂ\nF : ℍ → ℂ\nhF : MDiff F\nz : ℍ\n⊢ c * D F z - k * 12⁻¹ * EisensteinSeries.E2 z * (c * F z) = c * (D F z - k * 12⁻¹ * EisensteinSeries.E2 z * F z)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Com... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.Derivative | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 9
} | {
"line": 167,
"column": 0
} | [
{
"pp": "k₁ k₂ : ℂ\nF G : ℍ → ℂ\nhF : MDiff F\nhG : MDiff G\nz : ℍ\n⊢ D F z * G z + F z * D G z - (k₁ + k₂) * 12⁻¹ * EisensteinSeries.E2 z * (F z * G z) =\n (D F z - k₁ * 12⁻¹ * EisensteinSeries.E2 z * F z) * G z + F z * (D G z - k₂ * 12⁻¹ * EisensteinSeries.E2 z * G z)",
"ppTerm": "?m.38",
"assigned... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 94,
"column": 6
} | {
"line": 94,
"column": 13
} | {
"line": 95,
"column": 4
} | [
{
"pp": "k l : ℕ\nf : ℕ → ℂ\np : ℝ\nhp : 0 < p\nhf : f =O[atTop] fun n ↦ ↑n ^ l\nK : Set ℂ\nhK : K ⊆ ℍₒ\nhKc : IsCompact K\nthis : CompactSpace ↑K\nc : C(↑K, ℂ) := { toFun := fun r ↦ cexp (2 * ↑π * I * ↑r / ↑p), continuous_toFun := ⋯ }\nr : ℝ := ‖mkOfCompact c‖\nx : ↑K\n⊢ cexp (2 * ↑π * I * (↑x / ↑p)) = cexp (2... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 9
} | {
"line": 62,
"column": 2
} | [
{
"pp": "z : ℍ\nN : ℕ\n⊢ ↑2 * ∑ k ∈ range N, e2Summand (↑(k + 1)) z + 2 • (2 * riemannZeta 2) - 2 * riemannZeta 2 =\n 2 * riemannZeta 2 + ∑ m ∈ range N, -8 * ↑π ^ 2 * ∑' (n : ℕ+), ↑↑n * cexp (2 * ↑π * I * ↑z) ^ ((m + 1) * ↑n)",
"ppTerm": "?m.130",
"assigned": true,
"usedConstants": [
"Mathl... | [
"z : ℍ\nN : ℕ\n⊢ (∑ x ∈ range N, e2Summand (↑(1 + x)) z) * 2 + riemannZeta 2 * 2 =\n riemannZeta 2 * 2 +\n ∑ x ∈ range N,\n -((↑π ^ 2 * ∑' (n : ℕ+), ↑↑n * cexp (↑π * I * ↑z * 2) ^ (x * ↑n) * cexp (↑π * I * ↑z * 2) ^ ↑n) * 8)"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 9
} | {
"line": 73,
"column": 2
} | [
{
"pp": "case e_a.e_f\nz : ℍ\nN a : ℕ\nH2 :\n ∑' (n : ℤ), ((↑n + ↑z * (↑a + 1)) ^ 2)⁻¹ =\n (I * (↑π * 2)) ^ 2 * ∑' (n : ℕ+), cexp (↑z * (↑a + 1) * (I * (↑π * 2))) ^ ↑n * ↑↑n\nb : ℕ+\n⊢ 2 * ((I * (↑π * 2)) ^ 2 * (cexp (↑b • (↑z * (↑a + 1) * (I * (↑π * 2)))) * ↑↑b)) =\n -(8 * (↑π ^ 2 * (cexp (↑b • (↑z * (I... | [
"case e_a.e_f\nz : ℍ\nN a : ℕ\nH2 :\n ∑' (n : ℤ), ((↑n + ↑z * (↑a + 1)) ^ 2)⁻¹ =\n (I * (↑π * 2)) ^ 2 * ∑' (n : ℕ+), cexp (↑z * (↑a + 1) * (I * (↑π * 2))) ^ ↑n * ↑↑n\nb : ℕ+\n⊢ I ^ 2 * ↑π ^ 2 * ↑↑b * cexp (I * ↑π * ↑z * ↑a * ↑↑b * 2 + I * ↑π * ↑z * ↑↑b * 2) * 8 =\n -(↑π ^ 2 * ↑↑b * cexp (I * ↑π * ↑z * ↑↑b * ... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 112,
"column": 82
} | {
"line": 117,
"column": 20
} | {
"line": 119,
"column": 0
} | [
{
"pp": "k : ℕ\n⊢ SummableLocallyUniformlyOn (fun n ↦ iteratedDerivWithin k (fun z ↦ cexp (2 * ↑π * I * z) ^ n) ℍₒ) ℍₒ",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Nat.cast_mul._simp_1",
"UniformSpace",
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
... | [] | by
have h0 : (fun n : ℕ ↦ (1 : ℂ)) =O[atTop] fun n ↦ ((n ^ 1) : ℝ) := by
simp only [Asymptotics.isBigO_iff, norm_one, norm_pow, Real.norm_natCast, eventually_atTop]
exact ⟨1, 1, fun b hb ↦ by norm_cast; simp [hb]⟩
simpa using summableLocallyUniformlyOn_iteratedDerivWithin_smul_cexp k 1 (p := 1)
(by norm... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 9
} | {
"line": 61,
"column": 2
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\n⊢ cexp (↑24 * (log z / 2)) = z ^ 12",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Common.div_congr",
"Complex.log"... | [
"z : ℂ\nhz : z ≠ 0\n⊢ cexp (log z * 12) = z ^ 12"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 61,
"column": 52
} | {
"line": 61,
"column": 64
} | {
"line": 61,
"column": 65
} | [
{
"pp": "z : ℂ\nhz : z ≠ 0\n⊢ cexp (↑12 * log z) = z ^ 12",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Complex.log",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instAtLeastTwoHAd... | [
"z : ℂ\nhz : z ≠ 0\n⊢ cexp (log z) ^ 12 = z ^ 12"
] | exp_nat_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Modular | {
"line": 683,
"column": 6
} | {
"line": 684,
"column": 55
} | {
"line": 685,
"column": 6
} | [
{
"pp": "case inr.inr.inl.inl\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(S * T) 1 0\nhc : ↑(S * T) 1 0 = 1\nhd : ↑(S * T) 1 1 = 1\nhz : ρ ∈ 𝒟\nhg : (S * T) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (S * T))) ↑ρ‖ ≤ 1\n⊢ normSq (denom (toGL ((SpecialLinearGroup.map... | [
"case inr.inr.inl.inr\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhc✝ : 0 ≤ ↑(T * S * T) 1 0\nhc : ↑(T * S * T) 1 0 = 1\nhd : ↑(T * S * T) 1 1 = 1\nhz : ρ ∈ 𝒟\nhg : (T * S * T) • ρ ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) (T * S * T))) ↑ρ‖ ≤ 1\n⊢ normSq (denom (toGL ((SpecialLinearG... | · rw [show S * T = ⟨!![0, -1; 1, 1], by simp⟩ by decide]
norm_num [ρ, denom, normSq, ← pow_two, div_pow] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 11
} | {
"line": 215,
"column": 4
} | [
{
"pp": "z : ℍ\nd : ℕ+\n⊢ 2 / ↑↑d +\n ((-↑↑d)⁻¹ - (↑↑d)⁻¹ + ∑' (n : ℕ+), ((↑↑n * ↑z - ↑↑d)⁻¹ - (↑↑n * ↑z + ↑↑d)⁻¹) +\n ∑' (n : ℕ+), ((-(↑↑n * ↑z) - ↑↑d)⁻¹ - (-(↑↑n * ↑z) + ↑↑d)⁻¹)) =\n ∑' (m : ℕ+), ((↑↑m * ↑z - ↑↑d)⁻¹ + (-(↑↑m * ↑z) + -↑↑d)⁻¹ - (↑↑m * ↑z + ↑↑d)⁻¹ - (-(↑↑m * ↑z) + ↑↑d)⁻¹)",
"p... | [
"z : ℍ\nd : ℕ+\n⊢ ∑' (n : ℕ+), ((-↑↑d + ↑↑n * ↑z)⁻¹ - (↑↑d + ↑↑n * ↑z)⁻¹) + ∑' (n : ℕ+), ((-↑↑d - ↑z * ↑↑n)⁻¹ - (↑↑d - ↑z * ↑↑n)⁻¹) =\n ∑' (m : ℕ+), ((-↑↑d + ↑z * ↑↑m)⁻¹ + (-↑↑d - ↑z * ↑↑m)⁻¹ - (↑↑d + ↑z * ↑↑m)⁻¹ - (↑↑d - ↑z * ↑↑m)⁻¹)"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 9
} | {
"line": 133,
"column": 0
} | [
{
"pp": "z : ℍ\n⊢ cexp (2 * ↑π * I * (1 + ↑z)) * ∏' (n : ℕ), (1 - cexp (2 * ↑π * I * (1 + ↑z)) ^ (n + 1)) ^ 24 =\n cexp (2 * ↑π * I * ↑z + 2 * ↑π * I) * ∏' (n : ℕ), (1 - cexp (2 * ↑π * I * ↑z + 2 * ↑π * I) ^ (n + 1)) ^ 24",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Mathlib.Ta... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 148,
"column": 2
} | {
"line": 149,
"column": 53
} | {
"line": 150,
"column": 2
} | [
{
"pp": "z : ℍ\n⊢ η (-(↑z)⁻¹) ^ 24 * (↑z ^ 12)⁻¹ = η ↑z ^ 24",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"DivInvMonoid.toInv",
"instHSMul",
"neg_div",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
"HMul.hMul",
... | [
"z : ℍ\nhe : η (-(↑z)⁻¹) = I.sqrt⁻¹ * ((↑z).sqrt * η ↑z)\n⊢ η (-(↑z)⁻¹) ^ 24 * (↑z ^ 12)⁻¹ = η ↑z ^ 24"
] | have he : η (-(↑z)⁻¹) = (sqrt I)⁻¹ * (sqrt z * η z) := by
simpa [neg_div] using eta_comp_eq_csqrt_I_inv z.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 9
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case e_f\nk : ℕ\nhk : 1 ≤ k\nz : ℍ\nthis :\n (-1) ^ k * ↑k ! * ∑' (n : ℤ), 1 / (↑z + ↑n) ^ (k + 1) =\n -(2 * ↑π * I) ^ (k + 1) * ∑' (n : ℕ), ↑n ^ k * cexp (2 * ↑π * I * ↑z) ^ n\nn : ℕ\n⊢ -(↑n ^ k / ↑k !) = (-1) ^ k * ↑n ^ k * (-1) ^ (k + 1) / ↑k !",
"ppTerm": "?e_f",
"assigned": true,
"... | [
"case e_f\nk : ℕ\nhk : 1 ≤ k\nz : ℍ\nthis :\n (-1) ^ k * ↑k ! * ∑' (n : ℤ), 1 / (↑z + ↑n) ^ (k + 1) =\n -(2 * ↑π * I) ^ (k + 1) * ∑' (n : ℕ), ↑n ^ k * cexp (2 * ↑π * I * ↑z) ^ n\nn : ℕ\n⊢ -(↑n ^ k * (↑k !)⁻¹) = -(↑n ^ k * (↑k !)⁻¹ * (-1) ^ (k * 2))"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 74
} | {
"line": 46,
"column": 2
} | [
{
"pp": "τ : ℍ\n⊢ jacobiTheta ↑(ModularGroup.S • τ) = (-I * ↑τ) ^ (1 / 2) * jacobiTheta ↑τ",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Real",
"UpperHalfPlane.coe",
"Real.instZero",
"Complex.im",
"Complex.instZero",
"Eq.rec",
"ne_of_gt",
... | [
"τ : ℍ\nh0 : ↑τ ≠ 0\n⊢ jacobiTheta ↑(ModularGroup.S • τ) = (-I * ↑τ) ^ (1 / 2) * jacobiTheta ↑τ"
] | have h0 : (τ : ℂ) ≠ 0 := ne_of_apply_ne im (zero_im.symm ▸ ne_of_gt τ.2) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 241,
"column": 47
} | {
"line": 241,
"column": 54
} | {
"line": 241,
"column": 54
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\nH :\n ∀ (b : ℕ+),\n ∑' (n : ℤ), ((↑↑b * ↑z + ↑n) ^ k)⁻¹ =\n (-(2 * ↑π * I)) ^ k / ↑(k - 1)! * ∑' (n : ℕ+), ↑↑n ^ (k - 1) * cexp (2 * ↑π * I * (↑↑b * ↑z)) ^ ↑n\nm n : ℕ+\n⊢ ↑↑n ^ (k - 1) * cexp (↑↑n * (2 * ↑π * I * (↑↑m * ↑z))) = ↑↑n ^ (k - 1) * cexp (↑↑m... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 49
} | {
"line": 60,
"column": 2
} | [
{
"pp": "τ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\n⊢ ‖cexp (↑π * I * ↑n ^ 2 * τ)‖ ≤ rexp (-π * τ.im) ^ n.natAbs",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Int.cast",
"Real",
"Real.pi",
"HMul.hMul",
"Complex.im"... | [
"case refine_1\nτ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\n⊢ ‖cexp (↑π * I * ↑n ^ 2 * τ)‖ = y ^ n ^ 2",
"case refine_2\nτ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\n⊢ y ^ n ^ 2 ≤ rexp (-π * τ.im) ^ n.natAbs"
] | refine (le_of_eq ?_).trans (?_ : y ^ n ^ 2 ≤ _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 16
} | {
"line": 60,
"column": 16
} | [
{
"pp": "case refine_1\nτ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\n⊢ ‖cexp (↑π * I * ↑n ^ 2 * τ)‖ = y ^ n ^ 2",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Real",
"Real.pi",
"HMul.hMul",
... | [
"case refine_1\nτ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\n⊢ rexp (↑π * I * ↑n ^ 2 * τ).re = y ^ n ^ 2"
] | norm_exp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 245,
"column": 2
} | {
"line": 247,
"column": 50
} | {
"line": 249,
"column": 0
} | [
{
"pp": "k : ℤ\nz : ℍ\nn : ℕ\nv : ↑(gammaSet 1 n 0)\n⊢ eisSummand k (↑v) z = (↑n ^ k)⁻¹ * eisSummand k (divIntMap ↑n ↑v) z",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"Int.instDiv",
... | [] | simp_rw [eisSummand]
nth_rw 1 2 [gammaSet_eq_gcd_mul_divIntMap v.2]
simp [← mul_inv, ← mul_zpow, mul_add, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 245,
"column": 2
} | {
"line": 247,
"column": 50
} | {
"line": 249,
"column": 0
} | [
{
"pp": "k : ℤ\nz : ℍ\nn : ℕ\nv : ↑(gammaSet 1 n 0)\n⊢ eisSummand k (↑v) z = (↑n ^ k)⁻¹ * eisSummand k (divIntMap ↑n ↑v) z",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"Int.instDiv",
... | [] | simp_rw [eisSummand]
nth_rw 1 2 [gammaSet_eq_gcd_mul_divIntMap v.2]
simp [← mul_inv, ← mul_zpow, mul_add, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 66,
"column": 57
} | {
"line": 66,
"column": 69
} | {
"line": 66,
"column": 69
} | [
{
"pp": "case refine_1\nτ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\nthis : (↑π * I * ↑n ^ 2 * τ).re = -π * τ.im * ↑n ^ 2\nm : ℕ\nhm : n ^ 2 = ↑m\n⊢ rexp (-π * τ.im) ^ ↑m = y ^ ↑m",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr"... | [
"case refine_1\nτ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\nthis : (↑π * I * ↑n ^ 2 * τ).re = -π * τ.im * ↑n ^ 2\nm : ℕ\nhm : n ^ 2 = ↑m\n⊢ rexp (-π * τ.im) ^ m = rexp (-π * τ.im) ^ m"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 68,
"column": 14
} | {
"line": 68,
"column": 26
} | {
"line": 68,
"column": 26
} | [
{
"pp": "case refine_2\nτ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\nthis : n ^ 2 = ↑(n.natAbs ^ 2)\n⊢ y ^ ↑(n.natAbs ^ 2) ≤ rexp (-π * τ.im) ^ n.natAbs",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"Real.instLE",
... | [
"case refine_2\nτ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\nthis : n ^ 2 = ↑(n.natAbs ^ 2)\n⊢ y ^ n.natAbs ^ 2 ≤ rexp (-π * τ.im) ^ n.natAbs"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Modular | {
"line": 733,
"column": 12
} | {
"line": 733,
"column": 38
} | {
"line": 733,
"column": 39
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z = z\nhzρ : z ≠ ρ\nhzρ' : z ≠ 1 +ᵥ ρ\nthis✝¹ : T • z ≠ z\nthis✝ : T⁻¹ • z ≠ z\nthis : ↑z ≠ -↑I\nhzI : Complex.I ^ 2 = ↑z ^ 2\n⊢ z = I",
"ppTerm": "?m.159",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"NegZeroCla... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z = z\nhzρ : z ≠ ρ\nhzρ' : z ≠ 1 +ᵥ ρ\nthis✝¹ : T • z ≠ z\nthis✝ : T⁻¹ • z ≠ z\nthis : ↑z ≠ -↑I\nhzI : Complex.I = ↑z ∨ Complex.I = -↑z\n⊢ z = I"
] | sq_eq_sq_iff_eq_or_eq_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 117,
"column": 42
} | {
"line": 117,
"column": 49
} | {
"line": 117,
"column": 49
} | [
{
"pp": "y : ℝ\nhy : 1 ≤ y\nτ : ℂ\nhτ : τ.im = y\n⊢ 2 / (1 - rexp (-π * y)) * rexp (-π * y) ≤ 2 / (1 - rexp (-(π * 1))) * rexp (-π * y)",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [
"y : ℝ\nhy : 1 ≤ y\nτ : ℂ\nhτ : τ.im = y\n⊢ 2 / (1 - rexp (-(π * y))) * rexp (-(π * y)) ≤ 2 / (1 - rexp (-(π * 1))) * rexp (-(π * y))"
] | neg_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 338,
"column": 13
} | {
"line": 338,
"column": 25
} | {
"line": 338,
"column": 25
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nm : ℕ\nβ : ℂ := -(2 * ↑k / ↑(bernoulli k))\nc : ℕ → ℂ := fun m ↦ if m = 0 then 1 else β * ↑((σ (k - 1)) m)\nτ : ℍ\nhS : Summable fun n ↦ ↑((σ (k - 1)) (n + 1)) * cexp (2 * ↑π * I * ↑τ) ^ (n + 1)\nthis : (E hk) τ = 1 - 2 * ↑k / ↑(bernoulli k) * ∑' (n : ℕ+), ↑((σ (k - 1))... | [
"k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nm : ℕ\nβ : ℂ := -(2 * ↑k / ↑(bernoulli k))\nc : ℕ → ℂ := fun m ↦ if m = 0 then 1 else β * ↑((σ (k - 1)) m)\nτ : ℍ\nhS : Summable fun n ↦ ↑((σ (k - 1)) (n + 1)) * cexp (2 * ↑π * I * ↑τ) ^ (n + 1)\nthis : (E hk) τ = 1 - 2 * ↑k / ↑(bernoulli k) * ∑' (n : ℕ+), ↑((σ (k - 1)) ↑n) * cexp ... | zpow_natCast | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 99
} | {
"line": 120,
"column": 4
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝³ : FunLike F ℍ ℂ\nk : ℤ\ninst✝² : 𝒢.IsFiniteRelIndex ℋ\ninst✝¹ : ℋ.HasDetPlusMinusOne\ninst✝ : ModularFormClass F 𝒢 k\nγ : GL (Fin 2) ℝ\nh : IsCusp (γ • OnePoint.infty) ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.sub... | [
"𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝³ : FunLike F ℍ ℂ\nk : ℤ\ninst✝² : 𝒢.IsFiniteRelIndex ℋ\ninst✝¹ : ℋ.HasDetPlusMinusOne\ninst✝ : ModularFormClass F 𝒢 k\nγ : GL (Fin 2) ℝ\nh : IsCusp (γ • OnePoint.infty) ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)\n... | refine .finsetProd (Quotient.forall.mpr fun ⟨r, hr⟩ _ ↦ (translate f _).bdd_at_cusps' ?_ γ rfl) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Multiplicity | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 13
} | {
"line": 98,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\na b : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n (Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n (Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ∑ i ∈ range p, (Ideal.Quotient.mk (span {s})) ((a ^ (i - 1)... | [
"R : Type u_1\ninst✝ : CommRing R\na b : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n (Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n (Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\n⊢ ∑ x ∈ range p,\n (Ideal.Quotient.mk (span {s})) (a ^ (x - 1) * a ^... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Multiplicity | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 13
} | {
"line": 136,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\na b : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n (Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n (Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x ∈ range p, ↑x = ↑(∑ x ∈ range p, x)\n⊢ (Ideal.Quot... | [
"R : Type u_1\ninst✝ : CommRing R\na b : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n (Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n (Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nthis : ∑ x ∈ range p, ↑x = ↑(∑ x ∈ range p, x)\n⊢ (Ideal.Quotient.mk (spa... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Multiplicity | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 32
} | {
"line": 205,
"column": 4
} | [
{
"pp": "case succ.hn\np : ℕ\nhp : Prime ↑p\nhp1 : Odd p\nx y : ℤ\nhxy : ↑p ∣ x - y\nhx : ¬↑p ∣ x\nn : ℕ\nh : FiniteMultiplicity p (n + 1)\nhpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1\nk : ℕ\nhk : n + 1 = p ^ multiplicity p (n + 1) * k\n⊢ ¬↑p ∣ ↑k",
"ppTerm": "?succ.hn",
"assigned": true,
"usedC... | [
"case succ.hn\np : ℕ\nhp : Prime ↑p\nhp1 : Odd p\nx y : ℤ\nhxy : ↑p ∣ x - y\nhx : ¬↑p ∣ x\nn : ℕ\nh : FiniteMultiplicity p (n + 1)\nhpn : ¬p ^ (multiplicity p (n + 1) + 1) ∣ n + 1\nk : ℕ\nhk : n + 1 = p ^ multiplicity p (n + 1) * k\n⊢ ¬p ∣ k"
] | rw [Int.natCast_dvd_natCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Multiplicity | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 9
} | {
"line": 255,
"column": 2
} | [
{
"pp": "w✝ : ℤ\n⊢ (2 * w✝ + 1) ^ 2 % 4 = 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Mathlib.Tactic.RingNF.add_assoc_rev",
... | [
"w✝ : ℤ\n⊢ (1 + w✝ * 4 + w✝ ^ 2 * 4) % 4 = 1"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 59,
"column": 65
} | {
"line": 59,
"column": 82
} | {
"line": 59,
"column": 83
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ Fintype.card (Fin (rank K)) = Fintype.card (InfinitePlace K) - 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.card_fin",
"congrArg",
"NumberField.Units.rank",
"HSub.hSub",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ rank K = Fintype.card (InfinitePlace K) - 1"
] | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 128,
"column": 2
} | {
"line": 129,
"column": 86
} | {
"line": 130,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realMixedSpace K\n⊢ (FDerivPolarCoordRealSymm K x).det = ∏ w, (x.2 w).1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Pi.Function.module",
"Real",
"Semiring.toModule",
"Pi.a... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realMixedSpace K\nthis : ↑(FDerivPolarCoordRealSymm K x) = LinearMap.id.prodMap ↑(fderivPiPolarCoordSymm x.2)\n⊢ (FDerivPolarCoordRealSymm K x).det = ∏ w, (x.2 w).1"
] | have : (FDerivPolarCoordRealSymm K x).toLinearMap =
LinearMap.prodMap (LinearMap.id) (fderivPiPolarCoordSymm x.2).toLinearMap := rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 404,
"column": 36
} | {
"line": 404,
"column": 89
} | {
"line": 404,
"column": 89
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nA : Set (mixedSpace K)\ninst✝ : NumberField K\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nhm : MeasurableSet A\nhA' : ∀ {x : mixedSpace K}, A.indicator 1 x = (normAtComplexPlaces '' A).indicator 1 (normAtComplexPlaces x)\n⊢ MeasurableSet (⇑mixedSpaceOfRea... | [] | convert! hm.preimage mixedSpaceOfRealSpace.measurable | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 404,
"column": 36
} | {
"line": 404,
"column": 89
} | {
"line": 404,
"column": 89
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nA : Set (mixedSpace K)\ninst✝ : NumberField K\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nhm : MeasurableSet A\nhA' : ∀ {x : mixedSpace K}, A.indicator 1 x = (normAtComplexPlaces '' A).indicator 1 (normAtComplexPlaces x)\n⊢ MeasurableSet (⇑mixedSpaceOfRea... | [] | convert! hm.preimage mixedSpaceOfRealSpace.measurable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 404,
"column": 36
} | {
"line": 404,
"column": 89
} | {
"line": 404,
"column": 89
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nA : Set (mixedSpace K)\ninst✝ : NumberField K\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nhm : MeasurableSet A\nhA' : ∀ {x : mixedSpace K}, A.indicator 1 x = (normAtComplexPlaces '' A).indicator 1 (normAtComplexPlaces x)\n⊢ MeasurableSet (⇑mixedSpaceOfRea... | [] | convert! hm.preimage mixedSpaceOfRealSpace.measurable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 335,
"column": 65
} | {
"line": 335,
"column": 82
} | {
"line": 335,
"column": 83
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ Fintype.card (Fin (rank K)) = Fintype.card (InfinitePlace K) - 1",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.card_fin",
"congrArg",
"NumberField.Units.rank",
"HSub.hSub",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ rank K = Fintype.card (InfinitePlace K) - 1"
] | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Completion.Ramification | {
"line": 62,
"column": 8
} | {
"line": 63,
"column": 95
} | {
"line": 64,
"column": 6
} | [
{
"pp": "case inl\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nv : InfinitePlace K\nw : InfinitePlace L\ninst✝ : (↑w).LiesOver ↑v\nh : IsUnramified K w\nhv : v.IsReal\nthis : ComplexEmbedding.LiesOver (extensionEmbedding w) (extensionEmbedding v)\n⊢ Module.finrank v.Com... | [
"case inl\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nv : InfinitePlace K\nw : InfinitePlace L\ninst✝ : (↑w).LiesOver ↑v\nh : IsUnramified K w\nhv : v.IsReal\nthis : ComplexEmbedding.LiesOver (extensionEmbedding w) (extensionEmbedding v)\n⊢ Module.finrank ℝ ℝ = 1"
] | Algebra.finrank_eq_of_equiv_equiv (ringEquivRealOfIsReal hv) (ringEquivRealOfIsReal
(h.liesOver_isReal_over _ _ hv)) (RingHom.ext fun _ ↦ Complex.ofReal_inj.1 <| by simp), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 277,
"column": 2
} | {
"line": 278,
"column": 83
} | {
"line": 280,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ basisOfIsMaxRank ⋯ = Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"NumberField.Units.basisUnitLattice",
"Eq.mpr",
"Pi.Function.module",
... | [] | ext
rw [Basis.ofZLatticeBasis_apply, basisOfIsMaxRank_apply, logEmbedding_fundSystem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 277,
"column": 2
} | {
"line": 278,
"column": 83
} | {
"line": 280,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ basisOfIsMaxRank ⋯ = Basis.ofZLatticeBasis ℝ (unitLattice K) (basisUnitLattice K)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"NumberField.Units.basisUnitLattice",
"Eq.mpr",
"Pi.Function.module",
... | [] | ext
rw [Basis.ofZLatticeBasis_apply, basisOfIsMaxRank_apply, logEmbedding_fundSystem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 38
} | {
"line": 451,
"column": 2
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw i : InfinitePlace K\nhw : w = w₀\n⊢ (completeBasis K) w i = ↑i.mult",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Real",
"NumberField.mixedEmbedding.realSp... | [
"case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw i : InfinitePlace K\nhw : ¬w = w₀\n⊢ (completeBasis K) w i = ↑i.mult * Real.log (i ((algebraMap (𝓞 K) K) ↑(fundSystem K (equivFinRank.symm ⟨w, hw⟩))))"
] | · rw [hw, completeBasis_apply_of_eq] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 74,
"column": 4
} | {
"line": 76,
"column": 39
} | {
"line": 77,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\np f n : ℕ\nP : K[X]\nhK : Fintype.card K = p ^ f\nhn : p.Coprime n\nhp : Fact (Nat.Prime p)\nhP : P ∣ cyclotomic n K\nhPirr : Irreducible P\nhPmo : P.Monic\nthis✝¹ : Fact (Irreducible P)\nthis✝ : Module.Finite K (AdjoinRoot P)\nthis : Fi... | [] | simpa only [Nat.cast_pow, Nat.cast_one, coe_unitOfCoprime, Units.val_one,
Units.val_pow_eq_pow_val] using! Units.val_inj.mpr <| pow_orderOf_eq_one
(unitOfCoprime _ (hn.pow_left f)) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 53
} | {
"line": 88,
"column": 0
} | [
{
"pp": "A : Type u_1\nζ : A\ninst✝⁴ : CommRing A\ninst✝³ : IsDomain A\nR : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Algebra R A\nn : ℕ\ninst✝ : NeZero n\nhζ : IsPrimitiveRoot ζ n\nσ : A ≃ₐ[R] A\n⊢ (↑((autToPow R hζ) σ)).val.Coprime n",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"M... | [] | exact ZMod.val_coe_unit_coprime ((autToPow R hζ) σ) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 567,
"column": 81
} | {
"line": 568,
"column": 68
} | {
"line": 568,
"column": 69
} | [
{
"pp": "case e_a\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\n⊢ (∏ x_1, Real.exp ((↑1)⁻¹ * (completeBasis K).equivFun.symm x ↑x_1)) *\n ∏ x_1, Real.exp ((↑2)⁻¹ * (completeBasis K).equivFun.symm x ↑x_1) =\n (∏ x_1, Real.exp ((↑1)⁻¹ * (completeBasis K).equivFun.symm x ↑x_1)) *... | [
"case e_a\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\n⊢ (∏ x_1, Real.exp ((↑1)⁻¹ * (completeBasis K).equivFun.symm x ↑x_1)) *\n ∏ x_1, Real.exp ((↑2)⁻¹ * (completeBasis K).equivFun.symm x ↑x_1) =\n (∏ x_1, Real.exp ((↑1)⁻¹ * (completeBasis K).equivFun.symm x ↑x_1)) *\n ((∏ ... | mul_inv_cancel₀
(Finset.prod_ne_zero_iff.mpr <| fun _ _ ↦ Real.exp_ne_zero _), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.Ideal.Basic | {
"line": 77,
"column": 52
} | {
"line": 88,
"column": 74
} | {
"line": 90,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nI : Ideal (𝓞 K)\ninst✝¹ : NumberField K\nn : ℕ\ninst✝ : NeZero n\nhI₁ : absNorm I ≠ 1\nhI₂ : (absNorm I).Coprime n\n⊢ Function.Injective ⇑(I.rootsOfUnityMapQuot n)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Iff.mpr",
... | [] | by
refine (injective_iff_map_eq_one _).mpr fun ⟨ζ, hζ⟩ h ↦ ?_
obtain ⟨t, ht₀, ht, hζ⟩ := isPrimitiveRoot_of_mem_rootsOfUnity hζ
suffices ¬ (2 ≤ t) by
simpa [show t = 1 by grind] using hζ
intro ht'
let μ : K := ζ.val
have hμ : IsPrimitiveRoot μ t :=
(IsPrimitiveRoot.coe_units_iff.mpr hζ).map_of_injec... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 672,
"column": 4
} | {
"line": 673,
"column": 49
} | {
"line": 675,
"column": 0
} | [
{
"pp": "case neg.refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : ¬∀ (w : InfinitePlace K), 0 < x w\n⊢ x ∈ ↑expMapBasis '' paramSet K → x ∈ normAtAllPlaces '' normLeOne K",
"ppTerm": "?neg.refine_2✝",
"assigned": true,
"usedConstants": [
"Real",
"Nu... | [] | · rintro ⟨a, _, rfl⟩
exact (hx fun w ↦ expMapBasis_pos a w).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 731,
"column": 2
} | {
"line": 731,
"column": 29
} | {
"line": 732,
"column": 2
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : InfinitePlace K\nh✝ : w = w₀\n⊢ IsCompact {0}",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"PseudoMetricSpace.toUniformSpace",
"isCompact_singleton",
"Zer... | [
"case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : InfinitePlace K\nh✝ : ¬w = w₀\n⊢ IsCompact (Set.Icc 0 1)"
] | · exact isCompact_singleton | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 871,
"column": 31
} | {
"line": 871,
"column": 65
} | {
"line": 871,
"column": 66
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ volume (closure (normLeOne K)) - volume (interior (normLeOne K)) = 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ volume (closure (normLeOne K)) - volume (closure (normLeOne K)) = 0",
"case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ interior (normLeOne K) ⊆ closure (normLeOne K)",
"case h₂\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ Nu... | volume_interior_eq_volume_closure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.ExistsRamified | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 81
} | {
"line": 85,
"column": 2
} | [
{
"pp": "K : Type u_1\n𝒪 : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : NumberField K\ninst✝³ : CommRing 𝒪\ninst✝² : Algebra 𝒪 K\ninst✝¹ : IsIntegralClosure 𝒪 ℤ K\ninst✝ : IsGalois ℚ K\nH : 1 < Module.finrank ℚ K\nthis✝⁵ : IsDomain 𝒪\nthis✝⁴ : IsDedekindDomain 𝒪\nthis✝³ : IsFractionRing 𝒪 K\nthis✝² : Module.Fini... | [
"K : Type u_1\n𝒪 : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : NumberField K\ninst✝³ : CommRing 𝒪\ninst✝² : Algebra 𝒪 K\ninst✝¹ : IsIntegralClosure 𝒪 ℤ K\ninst✝ : IsGalois ℚ K\nH : 1 < Module.finrank ℚ K\nthis✝⁵ : IsDomain 𝒪\nthis✝⁴ : IsDedekindDomain 𝒪\nthis✝³ : IsFractionRing 𝒪 K\nthis✝² : Module.Finite ℤ 𝒪\nthi... | obtain ⟨p, hp : _ = Ideal.span _⟩ := IsPrincipalIdealRing.principal (P.under ℤ) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.NumberField.House | {
"line": 84,
"column": 2
} | {
"line": 85,
"column": 58
} | {
"line": 87,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα : K\nσ : K →+* ℂ\n⊢ ‖σ α‖ ≤ house α",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
... | [] | rw [house_eq_sup']
exact Finset.le_sup' (f := (‖· α‖₊)) (Finset.mem_univ σ) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.House | {
"line": 84,
"column": 2
} | {
"line": 85,
"column": 58
} | {
"line": 87,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα : K\nσ : K →+* ℂ\n⊢ ‖σ α‖ ≤ house α",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Finset.mem_univ",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
... | [] | rw [house_eq_sup']
exact Finset.le_sup' (f := (‖· α‖₊)) (Finset.mem_univ σ) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 11
} | {
"line": 114,
"column": 4
} | [
{
"pp": "case e'_5\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nh₁ :\n ∀ (s : ℝ),\n {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =\n ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}\nh₂ : {x | x ∈ fundam... | [
"case e'_5\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nh₁ :\n ∀ (s : ℝ),\n {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =\n ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}\nh₂ : {x | x ∈ fundamentalCone K ... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Topology.Algebra.InfiniteSum.GroupCompletion | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 29
} | {
"line": 45,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝² : AddCommGroup α\ninst✝¹ : UniformSpace α\ninst✝ : IsUniformAddGroup α\nf : β → α\n⊢ (CauchySeq fun s ↦ ∑ b ∈ s, f b) ∧ ∑' (i : β), toCompl (f i) ∈ Set.range ⇑toCompl → Summable f",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Ca... | [
"case mpr\nα : Type u_1\nβ : Type u_2\ninst✝² : AddCommGroup α\ninst✝¹ : UniformSpace α\ninst✝ : IsUniformAddGroup α\nf : β → α\nh_cauchy : CauchySeq fun s ↦ ∑ b ∈ s, f b\nh_tsum : ∑' (i : β), toCompl (f i) ∈ Set.range ⇑toCompl\n⊢ Summable f"
] | rintro ⟨h_cauchy, h_tsum⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 88,
"column": 92
} | {
"line": 90,
"column": 10
} | {
"line": 92,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : ℕ\n⊢ Continuous fun x ↦ Ring.multichoose x k",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Continuous.comp'",
"Norm.norm",
"Polynomial.eval",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
... | [] | by
simp only [Ring.multichoose, BinomialRing.multichoose]
fun_prop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Ostrowski | {
"line": 415,
"column": 2
} | {
"line": 416,
"column": 88
} | {
"line": 418,
"column": 0
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\nm n : ℕ\nhm : 1 < m\nhn : 1 < n\nnotbdd : ¬∀ (n : ℕ), f ↑n ≤ 1\ns t : ℝ\nhfm : f ↑m = ↑m ^ s\nhfn : f ↑n = ↑n ^ t\n⊢ f ↑m ^ logb ↑m ↑n ≤ ↑n ^ s",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",... | [] | rw [hfm, ← rpow_mul (Nat.cast_nonneg m), mul_comm, rpow_mul (Nat.cast_nonneg m),
rpow_logb (mod_cast zero_lt_of_lt hm) (mod_cast hm.ne') (mod_cast zero_lt_of_lt hn)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 196,
"column": 82
} | {
"line": 196,
"column": 94
} | {
"line": 196,
"column": 94
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : Module ℤ_[p] E\ninst✝¹ : IsBoundedSMul ℤ_[p] E\ninst✝ : IsUltrametricDist E\nf : C(ℤ_[p], E)\ns t : ℕ\nhst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s\nn i : ℕ\nx✝ : i ∈ range (n + 1)\n⊢ (↑p ^... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : Module ℤ_[p] E\ninst✝¹ : IsBoundedSMul ℤ_[p] E\ninst✝ : IsUltrametricDist E\nf : C(ℤ_[p], E)\ns t : ℕ\nhst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s\nn i : ℕ\nx✝ : i ∈ range (n + 1)\n⊢ (↑p ^ t)⁻¹ ≤ (↑p ... | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 35
} | {
"line": 170,
"column": 35
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℚ\n⊢ ‖↑x‖ ≤ 1 ↔ (Rat.padicValuation p) x ≤ 1",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Int.instAddCommMonoid",
"Multiplicative.linearOrder",
"Int.instIsStrictOrderedRing",
"Rea... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℚ\n⊢ ‖↑x‖ ≤ 1 ↔ ¬p ∣ x.den"
] | Rat.padicValuation_le_one_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Pell | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 9
} | {
"line": 240,
"column": 2
} | [
{
"pp": "d : ℤ\na b : Solution₁ d\nha : 0 < a.x\nhb : 0 < b.x\n⊢ 0 < (1 + d * a.y ^ 2) * (1 + d * b.y ^ 2) - (d * (a.y * b.y)) ^ 2",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.instCommMonoid",
"AddGroup.toSubtractionMonoid... | [
"d : ℤ\na b : Solution₁ d\nha : 0 < a.x\nhb : 0 < b.x\n⊢ 0 < 1 + d * a.y ^ 2 + d * b.y ^ 2"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Pell | {
"line": 279,
"column": 30
} | {
"line": 279,
"column": 42
} | {
"line": 279,
"column": 42
} | [
{
"pp": "case ofNat\nd : ℤ\na : Solution₁ d\nhax : 0 < a.x\nn : ℕ\n⊢ 0 < (a ^ ↑n).x",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg",
"Pell.Solution₁.x",
"DivInvMonoid.toZPow",
"id",
"DivInvMonoid.toMonoid",
... | [
"case ofNat\nd : ℤ\na : Solution₁ d\nhax : 0 < a.x\nn : ℕ\n⊢ 0 < (a ^ n).x"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Pell | {
"line": 291,
"column": 30
} | {
"line": 291,
"column": 42
} | {
"line": 291,
"column": 42
} | [
{
"pp": "case ofNat.succ\nd : ℤ\na : Solution₁ d\nhax : 0 < a.x\nhay : 0 < a.y\nn : ℕ\n⊢ (a ^ ↑(n + 1)).y.sign = (↑(n + 1)).sign",
"ppTerm": "?ofNat.succ",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg",
"DivInvMonoid.toZPow",
"Int.sign",
... | [
"case ofNat.succ\nd : ℤ\na : Solution₁ d\nhax : 0 < a.x\nhay : 0 < a.y\nn : ℕ\n⊢ (a ^ (n + 1)).y.sign = (↑(n + 1)).sign"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 110,
"column": 2
} | {
"line": 112,
"column": 58
} | {
"line": 113,
"column": 2
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nha : (Polynomial.aeval a) F = 0\nz' : ℤ_[p]\nhz' : (Polynomial.aeval z') F = 0\nhnormz' : ‖z' - a‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq ... | [
"p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nha : (Polynomial.aeval a) F = 0\nz' : ℤ_[p]\nhz' : (Polynomial.aeval z') F = 0\nhnormz' : ‖z' - a‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖\nh : ℤ_[p] := z' - a\nq : ℤ_[p]\nhq ... | have : (F.derivative.aeval a + q * h) * h = 0 := by calc
_ = F.aeval (a + h) := by rw [hq, ha, zero_add, sq, right_distrib, mul_assoc]
_ = _ := show F.aeval (a + (z' - a)) = 0 by simp [hz'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.RamificationInertia.HilbertTheory | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 63
} | {
"line": 195,
"column": 2
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝²¹ : Field K\ninst✝²⁰ : Field L\ninst✝¹⁹ : Algebra K L\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : CommRing B\ninst✝¹⁶ : Algebra A B\np : Ideal A\nP : Ideal B\ninst✝¹⁵ : P.LiesOver p\ninst✝¹⁴ : FiniteDimensional K L\ninst✝¹³ : MulSemiringAction Gal(L/K)... | [
"A : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝²¹ : Field K\ninst✝²⁰ : Field L\ninst✝¹⁹ : Algebra K L\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : CommRing B\ninst✝¹⁶ : Algebra A B\np : Ideal A\nP : Ideal B\ninst✝¹⁵ : P.LiesOver p\ninst✝¹⁴ : FiniteDimensional K L\ninst✝¹³ : MulSemiringAction Gal(L/K) B\ninst✝¹² ... | have : FiniteDimensional D L := FiniteDimensional.right K D L | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Pell | {
"line": 545,
"column": 2
} | {
"line": 545,
"column": 9
} | {
"line": 546,
"column": 2
} | [
{
"pp": "d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ 0 ≤ a.y ^ 2 * (1 + d * a₁.y ^ 2) - (1 + d * a.y ^ 2) * a₁.y ^ 2",
"ppTerm": "?m.111",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.instCommMonoi... | [
"d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ 0 ≤ a.y ^ 2 - a₁.y ^ 2"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.RamificationInertia.Valuation | {
"line": 106,
"column": 34
} | {
"line": 106,
"column": 60
} | {
"line": 107,
"column": 4
} | [
{
"pp": "case h.inr\nA : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁶ : CommRing A\ninst✝¹⁵ : IsDedekindDomain A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : IsDedekindDomain B\ninst✝¹² : Algebra A B\ninst✝¹¹ : Module.IsTorsionFree A B\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : Alge... | [
"case h.inr\nA : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁶ : CommRing A\ninst✝¹⁵ : IsDedekindDomain A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : IsDedekindDomain B\ninst✝¹² : Algebra A B\ninst✝¹¹ : Module.IsTorsionFree A B\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : Algebra A K\nins... | ← σL.strictMono.lt_iff_lt, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Valuation | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 34
} | {
"line": 110,
"column": 35
} | [
{
"pp": "case h.inr\nA : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁶ : CommRing A\ninst✝¹⁵ : IsDedekindDomain A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : IsDedekindDomain B\ninst✝¹² : Algebra A B\ninst✝¹¹ : Module.IsTorsionFree A B\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : Alge... | [
"case h.inr\nA : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁶ : CommRing A\ninst✝¹⁵ : IsDedekindDomain A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : IsDedekindDomain B\ninst✝¹² : Algebra A B\ninst✝¹¹ : Module.IsTorsionFree A B\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : Algebra A K\nins... | WithVal.algebraMap_left_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Rayleigh | {
"line": 94,
"column": 74
} | {
"line": 102,
"column": 38
} | {
"line": 104,
"column": 0
} | [
{
"pp": "r : ℝ\nj : ℤ\nh : r > 0\n⊢ j ∈ {x | ∃ k, beattySeq r k = x} ∨ ∃ k, ↑k < ↑j / r ∧ (↑j + 1) / r ≤ ↑k + 1",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Real.instIsOrderedRing",
"Int.cast",
"Eq.mpr",
... | [] | by
-- for both cases, the candidate is `k = ⌈(j + 1) / r⌉ - 1`
cases lt_or_ge ((⌈(j + 1) / r⌉ - 1) * r) j
· refine Or.inr ⟨⌈(j + 1) / r⌉ - 1, ?_⟩
rw [Int.cast_sub, Int.cast_one, lt_div_iff₀ h, sub_add_cancel]
exact ⟨‹_›, Int.le_ceil _⟩
· refine Or.inl ⟨⌈(j + 1) / r⌉ - 1, ?_⟩
rw [beattySeq, Int.floor... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.SelbergSieve | {
"line": 131,
"column": 4
} | {
"line": 137,
"column": 41
} | {
"line": 139,
"column": 0
} | [] | [] | 0 < ∏ p ∈ d.primeFactors, s.nu p := by
apply prod_pos
intro p hpd
have hp_prime : p.Prime := prime_of_mem_primeFactors hpd
have hp_dvd : p ∣ s.prodPrimes := (dvd_of_mem_primeFactors hpd).trans hd
exact s.nu_pos_of_prime p hp_prime hp_dvd
_ = s.nu d := prod_primeFactors_nu hd | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.NumberTheory.SelbergSieve | {
"line": 197,
"column": 7
} | {
"line": 201,
"column": 67
} | {
"line": 202,
"column": 2
} | [] | [
"case caseA\ns : BoundingSieve\nmuPlus : ℕ → ℝ\nh : IsUpperMoebius muPlus\nhμ : ∀ (n : ℕ), (if n = 1 then 1 else 0) ≤ ∑ d ∈ n.divisors, muPlus d\n⊢ siftedSum ≤ ∑ n ∈ s.support, s.weights n * ∑ d ∈ (s.prodPrimes.gcd n).divisors, muPlus d",
"case caseB\ns : BoundingSieve\nmuPlus : ℕ → ℝ\nh : IsUpperMoebius muPlus\n... | siftedSum ≤
∑ n ∈ s.support, s.weights n * ∑ d ∈ (Nat.gcd s.prodPrimes n).divisors, muPlus d := ?caseA
_ = ∑ n ∈ s.support, ∑ d ∈ divisors s.prodPrimes,
if d ∣ n then s.weights n * muPlus d else 0 := ?caseB
_ = ∑ d ∈ divisors s.prodPrimes, muPlus d * multSum d := ?caseC | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 198,
"column": 4
} | {
"line": 203,
"column": 83
} | {
"line": 205,
"column": 0
} | [
{
"pp": "case hH.refine_2\nK : Type u_1\nΓ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝¹ : v.IsNontrivial\ninst✝ : IsTrivialOn K v\nhle : v X ≤ 1\nhv : v.IsRankOneDiscrete\nh0 : v ↑πᵥ ≠ 0\nγ : Γˣ\nx✝ : ∃ a, ¬a = 0 ∧ ∃ x, (MonoidWithZeroHom.ofClass v) a * ↑... | [] | · obtain ⟨ka, hka⟩ := exists_zpow_uniformizingPolynomial hle ha
obtain ⟨kb, hkb⟩ := exists_zpow_uniformizingPolynomial hle (f := b) (by aesop)
rw [MonoidWithZeroHom.coe_ofClass, hka, hkb] at hab
use kb - ka
have : v ↑πᵥ ^ ka ≠ 0 := zpow_ne_zero _ h0
simp [zpow_sub, ← Units.val_inj, ← coePo... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 161,
"column": 8
} | {
"line": 161,
"column": 28
} | {
"line": 162,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝¹⁶ : CommRing R\nS : Type v\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹³ : Field K\ninst✝¹² : Algebra R K\nL : Type u_2\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra S L\ninst✝⁹ : IsFractionRing S L\ninst✝⁸ : IsDomain R\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra... | [] | rw [RingHom.map_det] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 161,
"column": 8
} | {
"line": 161,
"column": 28
} | {
"line": 162,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝¹⁶ : CommRing R\nS : Type v\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹³ : Field K\ninst✝¹² : Algebra R K\nL : Type u_2\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra S L\ninst✝⁹ : IsFractionRing S L\ninst✝⁸ : IsDomain R\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra... | [] | rw [RingHom.map_det] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 161,
"column": 8
} | {
"line": 161,
"column": 28
} | {
"line": 162,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝¹⁶ : CommRing R\nS : Type v\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹³ : Field K\ninst✝¹² : Algebra R K\nL : Type u_2\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra S L\ninst✝⁹ : IsFractionRing S L\ninst✝⁸ : IsDomain R\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra... | [] | rw [RingHom.map_det] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 166,
"column": 53
} | {
"line": 166,
"column": 70
} | {
"line": 166,
"column": 71
} | [
{
"pp": "R : Type u\ninst✝¹⁶ : CommRing R\nS : Type v\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹³ : Field K\ninst✝¹² : Algebra R K\nL : Type u_2\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra S L\ninst✝⁹ : IsFractionRing S L\ninst✝⁸ : IsDomain R\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra... | [
"R : Type u\ninst✝¹⁶ : CommRing R\nS : Type v\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹³ : Field K\ninst✝¹² : Algebra R K\nL : Type u_2\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra S L\ninst✝⁹ : IsFractionRing S L\ninst✝⁸ : IsDomain R\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra K L\ninst✝⁵... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 195,
"column": 9
} | {
"line": 195,
"column": 96
} | {
"line": 195,
"column": 96
} | [
{
"pp": "x y : ℤ[i]\nthis : |2⁻¹| = 2⁻¹\n⊢ |(↑(toComplex x / toComplex y).re - ↑(toComplex (x / y)).re +\n (↑(toComplex x / toComplex y).im - ↑(toComplex (x / y)).im) * I).im| ≤\n |(1 / 2 + 1 / 2 * I).im|",
"ppTerm": "?m.145",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
... | [] | by rw [toComplex_im_div]; simp [normSq, this]; simpa using abs_sub_round (x / y : ℂ).im | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 73
} | {
"line": 209,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝¹⁵ : CommRing R\nS : Type v\ninst✝¹⁴ : CommRing S\ninst✝¹³ : Algebra R S\nK : Type u_1\ninst✝¹² : Field K\ninst✝¹¹ : Algebra R K\nV : Type u_3\nV' : Type u_4\nV'' : Type u_5\ninst✝¹⁰ : AddCommGroup V\ninst✝⁹ : Module R V\ninst✝⁸ : Module K V\ninst✝⁷ : IsScalarTower R K V\ninst✝⁶ : AddC... | [
"R : Type u\ninst✝¹⁵ : CommRing R\nS : Type v\ninst✝¹⁴ : CommRing S\ninst✝¹³ : Algebra R S\nK : Type u_1\ninst✝¹² : Field K\ninst✝¹¹ : Algebra R K\nV : Type u_3\nV' : Type u_4\nV'' : Type u_5\ninst✝¹⁰ : AddCommGroup V\ninst✝⁹ : Module R V\ninst✝⁸ : Module K V\ninst✝⁷ : IsScalarTower R K V\ninst✝⁶ : AddCommGroup V'\... | simp only [FractionalIdeal.mem_coeIdeal, not_exists, not_and'] at hgI | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 72,
"column": 74
} | {
"line": 72,
"column": 82
} | {
"line": 72,
"column": 82
} | [
{
"pp": "f : ℕ → ℂ[X]\ns : ℂ\nc : ℝ\nhc : ∀ (p : ℕ), ∀ x ∈ Set.Ioc 0 1, ‖eval (x • s) (f p)‖ ≤ c ^ p\np : ℕ\n⊢ ‖cexp s‖ * (‖∫ (x : ℝ) in 0..1, cexp (-(x • s)) * eval (x • s) (f p)‖ * ‖s‖) ≤\n Real.exp s.re * (Real.exp ‖s‖ * |c| ^ p * ‖s‖)",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
... | [
"f : ℕ → ℂ[X]\ns : ℂ\nc : ℝ\nhc : ∀ (p : ℕ), ∀ x ∈ Set.Ioc 0 1, ‖eval (x • s) (f p)‖ ≤ c ^ p\np : ℕ\n⊢ Real.exp s.re * (‖∫ (x : ℝ) in 0..1, cexp (-(x • s)) * eval (x • s) (f p)‖ * ‖s‖) ≤\n Real.exp s.re * (Real.exp ‖s‖ * |c| ^ p * ‖s‖)"
] | norm_exp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 79,
"column": 16
} | {
"line": 79,
"column": 24
} | {
"line": 79,
"column": 24
} | [
{
"pp": "case hbc.convert_11\nf : ℕ → ℂ[X]\ns : ℂ\nc : ℝ\nhc : ∀ (p : ℕ), ∀ x ∈ Set.Ioc 0 1, ‖eval (x • s) (f p)‖ ≤ c ^ p\np : ℕ\nx : ℝ\nhx : x ∈ Set.Ioc 0 1\n⊢ ‖cexp (-(x • s))‖ * ‖eval (x • s) (f p)‖ ≤ Real.exp ‖s‖ * |c| ^ p",
"ppTerm": "?hbc.convert_11",
"assigned": true,
"usedConstants": [
... | [
"case hbc.convert_11\nf : ℕ → ℂ[X]\ns : ℂ\nc : ℝ\nhc : ∀ (p : ℕ), ∀ x ∈ Set.Ioc 0 1, ‖eval (x • s) (f p)‖ ≤ c ^ p\np : ℕ\nx : ℝ\nhx : x ∈ Set.Ioc 0 1\n⊢ Real.exp (-(x • s)).re * ‖eval (x • s) (f p)‖ ≤ Real.exp ‖s‖ * |c| ^ p"
] | norm_exp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.Irrational | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 23
} | {
"line": 88,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : Irrational x\nn : ℕ\nA : IsClosed (range fun m ↦ (↑n)⁻¹ * ↑m)\nB : x ∉ range fun m ↦ (↑n)⁻¹ * ↑m\nε : ℝ\nε0 : ε > 0\nhε : ball x ε ⊆ (range fun m ↦ (↑n)⁻¹ * ↑m)ᶜ\nδ : ℝ\nhδ : δ ≤ ε\nm : ℤ\nhlt : dist (↑m / ↑n) x < δ\n⊢ (fun m ↦ (↑n)⁻¹ * ↑m) m = ↑m / ↑n",
"ppTerm": "?m.159",
"assigne... | [] | simp [div_eq_inv_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 429,
"column": 2
} | {
"line": 431,
"column": 94
} | {
"line": 433,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁵ : CommRing R\nS : Type v\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal R\nP : Ideal S\nhfp : NeZero e\ninst✝² : IsDedekindDomain S\ninst✝¹ : p.IsMaximal\ninst✝ : P.IsPrime\nhP0 : P ≠ ⊥\ni : ℕ\nhi : i < e\n⊢ Module.rank (R ⧸ p) ↥(map (Quotient.mk (P ^ e)) (P ^ i)) =\n Modul... | [] | rw [← rank_range_of_injective _ (powQuotSuccInclusion_injective p P i),
(quotientRangePowQuotSuccInclusionEquiv p P hP0 hi).symm.rank_eq]
exact (Submodule.rank_quotient_add_rank (LinearMap.range (powQuotSuccInclusion p P i))).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 429,
"column": 2
} | {
"line": 431,
"column": 94
} | {
"line": 433,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁵ : CommRing R\nS : Type v\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal R\nP : Ideal S\nhfp : NeZero e\ninst✝² : IsDedekindDomain S\ninst✝¹ : p.IsMaximal\ninst✝ : P.IsPrime\nhP0 : P ≠ ⊥\ni : ℕ\nhi : i < e\n⊢ Module.rank (R ⧸ p) ↥(map (Quotient.mk (P ^ e)) (P ^ i)) =\n Modul... | [] | rw [← rank_range_of_injective _ (powQuotSuccInclusion_injective p P i),
(quotientRangePowQuotSuccInclusionEquiv p P hP0 hi).symm.rank_eq]
exact (Submodule.rank_quotient_add_rank (LinearMap.range (powQuotSuccInclusion p P i))).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Bounded | {
"line": 321,
"column": 2
} | {
"line": 322,
"column": 19
} | {
"line": 324,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Unbounded (fun x1 x2 ↦ x1 < x2) (s ∩ {b | a ≤ b}) ↔ Unbounded (fun x1 x2 ↦ x1 < x2) s",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"HEq.refl",... | [] | convert! @unbounded_lt_inter_not_lt _ s _ a
exact not_lt.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Bounded | {
"line": 321,
"column": 2
} | {
"line": 322,
"column": 19
} | {
"line": 324,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Unbounded (fun x1 x2 ↦ x1 < x2) (s ∩ {b | a ≤ b}) ↔ Unbounded (fun x1 x2 ↦ x1 < x2) s",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"HEq.refl",... | [] | convert! @unbounded_lt_inter_not_lt _ s _ a
exact not_lt.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.CompleteLattice.PiLex | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 38
} | {
"line": 56,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearOrder ι\ninst✝¹ : (i : ι) → CompleteLinearOrder (α i)\ninst✝ : WellFoundedLT ι\ns : Set (Lex ((i : ι) → (fun i ↦ α i) i))\n⊢ IsGLB s (sInf s)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"lowerBounds",
"Lex",
"Part... | [
"case refine_1\nι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearOrder ι\ninst✝¹ : (i : ι) → CompleteLinearOrder (α i)\ninst✝ : WellFoundedLT ι\ns : Set (Lex ((i : ι) → (fun i ↦ α i) i))\ne : Lex ((i : ι) → (fun i ↦ α i) i)\nhe : e ∈ s\n⊢ sInf s ≤ e",
"case refine_2\nι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearO... | refine ⟨fun e he ↦ ?_, fun e h ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.CompleteLattice.PiLex | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 38
} | {
"line": 84,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearOrder ι\ninst✝¹ : (i : ι) → CompleteLinearOrder (α i)\ninst✝ : WellFoundedLT ι\ns : Set (Lex ((i : ι) → (fun i ↦ α i) i))\n⊢ IsLUB s (sSup s)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"lowerBounds",
"Lex",
"Pi.L... | [
"case refine_1\nι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearOrder ι\ninst✝¹ : (i : ι) → CompleteLinearOrder (α i)\ninst✝ : WellFoundedLT ι\ns : Set (Lex ((i : ι) → (fun i ↦ α i) i))\ne : Lex ((i : ι) → (fun i ↦ α i) i)\nhe : e ∈ s\n⊢ e ≤ sSup s",
"case refine_2\nι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearO... | refine ⟨fun e he ↦ ?_, fun e h ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Completion | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 37
} | {
"line": 89,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\nhf : Monotone f\nA : DedekindCut α\nx : α\nhx : x ∈ A.right\ny : α\nhy : y ∈ A.left\n⊢ f y ≤ f x",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Preorder.toLE",
"Concept.rel_extent_intent"... | [] | exact hf <| rel_extent_intent hy hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Completion | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 65
} | {
"line": 233,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b : DedekindCut α\n⊢ a < b ↔ ∃ c, a ≤ principal c ∧ principal c < b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Order.Completion.0.DedekindCut.lt_iff_exists'.match_1_1",
"lt_of_le_of_lt",
"Preorder.toLT... | [
"α : Type u_1\ninst✝ : LinearOrder α\na b : DedekindCut α\nh : a < b\n⊢ ∃ c, a ≤ principal c ∧ principal c < b"
] | refine ⟨fun h ↦ ?_, fun ⟨c, hca, hcb⟩ ↦ lt_of_le_of_lt hca hcb⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Concept | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 13
} | {
"line": 214,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → β → Prop\ns s' : Set α\nh : IsExtent r s\nhs' : s' ⊆ s\n⊢ lowerPolar r (upperPolar r s') ⊆ s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Order.IsExtent.eq",
"Eq.mpr",
"congrArg",
"id",
"LE.le",
"upperPola... | [
"α : Type u_2\nβ : Type u_3\nr : α → β → Prop\ns s' : Set α\nh : IsExtent r s\nhs' : s' ⊆ s\n⊢ lowerPolar r (upperPolar r s') ⊆ lowerPolar r (upperPolar r s)"
] | rw [← h.eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Concept | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 13
} | {
"line": 249,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → β → Prop\nt t' : Set β\nh : IsIntent r t\nht' : t' ⊆ t\n⊢ upperPolar r (lowerPolar r t') ⊆ t",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"LE.le",
"upperPolar",
"Set.instLE",
... | [
"α : Type u_2\nβ : Type u_3\nr : α → β → Prop\nt t' : Set β\nh : IsIntent r t\nht' : t' ⊆ t\n⊢ upperPolar r (lowerPolar r t') ⊆ upperPolar r (lowerPolar r t)"
] | rw [← h.eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.CompleteSublattice | {
"line": 167,
"column": 2
} | {
"line": 168,
"column": 33
} | {
"line": 170,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\nL : CompleteSublattice α\na b : ↥L\n⊢ Codisjoint a b ↔ Codisjoint ↑a ↑b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Sublattice",
"Eq.mpr",
"Codisjoint",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLatti... | [] | rw [codisjoint_iff, codisjoint_iff, ← Sublattice.coe_sup, ← coe_top (L := L),
Subtype.coe_injective.eq_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.CompleteSublattice | {
"line": 167,
"column": 2
} | {
"line": 168,
"column": 33
} | {
"line": 170,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\nL : CompleteSublattice α\na b : ↥L\n⊢ Codisjoint a b ↔ Codisjoint ↑a ↑b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Sublattice",
"Eq.mpr",
"Codisjoint",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLatti... | [] | rw [codisjoint_iff, codisjoint_iff, ← Sublattice.coe_sup, ← coe_top (L := L),
Subtype.coe_injective.eq_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.CompleteSublattice | {
"line": 167,
"column": 2
} | {
"line": 168,
"column": 33
} | {
"line": 170,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\nL : CompleteSublattice α\na b : ↥L\n⊢ Codisjoint a b ↔ Codisjoint ↑a ↑b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Sublattice",
"Eq.mpr",
"Codisjoint",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLatti... | [] | rw [codisjoint_iff, codisjoint_iff, ← Sublattice.coe_sup, ← coe_top (L := L),
Subtype.coe_injective.eq_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.CardinalInter | {
"line": 172,
"column": 46
} | {
"line": 178,
"column": 32
} | {
"line": 180,
"column": 0
} | [
{
"pp": "ι α β : Type u\nc : Cardinal.{u}\nl✝ : Filter α\ninst✝ : CardinalInterFilter l✝ c\nl : Set (Set α)\nhc : 2 < c\nhl : ∀ (S : Set (Set α)), #↑S < c → S ⊆ l → ⋂₀ S ∈ l\nh_mono : ∀ (s t : Set α), s ∈ l → s ⊆ t → t ∈ l\ns t : Set α\nhs : s ∈ l\nht : t ∈ l\n⊢ ⋂₀ {s, t} ∈ l",
"ppTerm": "?m.53",
"assig... | [] | by
apply hl _ (?_) (insert_subset_iff.2 ⟨hs, singleton_subset_iff.2 ht⟩)
have : #({s, t} : Set (Set α)) ≤ 2 := by
calc
_ ≤ #({t} : Set (Set α)) + 1 := Cardinal.mk_insert_le
_ = 2 := by norm_num
exact lt_of_le_of_lt this hc | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Height | {
"line": 164,
"column": 2
} | {
"line": 166,
"column": 26
} | {
"line": 168,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\n⊢ (univ.chainHeight fun x1 x2 ↦ r ↑x1 ↑x2) = s.chainHeight r",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.chainHeight_eq_of_relEmbedding",
"Set.chainHeight",
"Set.image_univ",
"Subtype.relEmbedding",
"... | [] | have hc := Set.chainHeight_eq_of_relEmbedding univ <| Subtype.relEmbedding (r · ·) (· ∈ s)
have hs : Subtype.val ⁻¹'o (r · ·) = (fun x y : s ↦ r x y) := by funext; simp
simpa [hs] using hc.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Height | {
"line": 164,
"column": 2
} | {
"line": 166,
"column": 26
} | {
"line": 168,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\nr : α → α → Prop\n⊢ (univ.chainHeight fun x1 x2 ↦ r ↑x1 ↑x2) = s.chainHeight r",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.chainHeight_eq_of_relEmbedding",
"Set.chainHeight",
"Set.image_univ",
"Subtype.relEmbedding",
"... | [] | have hc := Set.chainHeight_eq_of_relEmbedding univ <| Subtype.relEmbedding (r · ·) (· ∈ s)
have hs : Subtype.val ⁻¹'o (r · ·) = (fun x y : s ↦ r x y) := by funext; simp
simpa [hs] using hc.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Cocardinal | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 56
} | {
"line": 53,
"column": 4
} | [
{
"pp": "α : Type u\nc : Cardinal.{u}\nhreg : c.IsRegular\nS : Set (Set α)\nhS : #↑S < c\nhSs : ∀ s ∈ S, s ∈ cocardinal α hreg\n⊢ ⋂₀ S ∈ cocardinal α hreg",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Cardinal.mk_sUnion_le",
"Eq.mpr",
"le_... | [
"α : Type u\nc : Cardinal.{u}\nhreg : c.IsRegular\nS : Set (Set α)\nhS : #↑S < c\nhSs : ∀ s ∈ S, s ∈ cocardinal α hreg\n⊢ #↑(compl '' S) * ⨆ s, #↑↑s < c"
] | grw [mem_cocardinal, Set.compl_sInter, mk_sUnion_le] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Order.Interval.Set.SurjOn | {
"line": 31,
"column": 2
} | {
"line": 31,
"column": 31
} | {
"line": 32,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\np : β\nhp : p ∈ Ioo (f a) (f b)\n⊢ p ∈ f '' Ioo a b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"M... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b x : α\nhp : f x ∈ Ioo (f a) (f b)\n⊢ f x ∈ f '' Ioo a b"
] | rcases h_surj p with ⟨x, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Order.Nucleus | {
"line": 216,
"column": 66
} | {
"line": 216,
"column": 86
} | {
"line": 217,
"column": 12
} | [
{
"pp": "X : Type u_1\ninst✝ : Frame X\nn✝ m✝ : Nucleus X\nx✝ y✝ : X\nm n : Nucleus X\nx y : X\nthis : Nonempty X\nk : X\nhxyk : k ≥ x ⊓ y\nl : X\nhlx : ∀ x_1 ≥ x, l ⊓ m x_1 ≤ n x_1\nhly : ∀ i ≥ y, l ⊓ m i ≤ n i\nhlk : l ≤ m k\n⊢ l = l ⊓ m (x ⊓ y ⊔ k)",
"ppTerm": "?m.218",
"assigned": true,
"usedCon... | [
"X : Type u_1\ninst✝ : Frame X\nn✝ m✝ : Nucleus X\nx✝ y✝ : X\nm n : Nucleus X\nx y : X\nthis : Nonempty X\nk : X\nhxyk : k ≥ x ⊓ y\nl : X\nhlx : ∀ x_1 ≥ x, l ⊓ m x_1 ≤ n x_1\nhly : ∀ i ≥ y, l ⊓ m i ≤ n i\nhlk : l ≤ m k\n⊢ l = l ⊓ m k"
] | sup_eq_right.2 hxyk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.PrimeIdeal | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 17
} | {
"line": 161,
"column": 0
} | [
{
"pp": "P : Type u_1\ninst✝ : BooleanAlgebra P\nx : P\nI : Ideal P\nhI : I.IsPrime\n⊢ ⊥ ∈ I",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"BiheytingAlgebra.toHeytingAlgebra",
"Bihe... | [] | exact I.bot_mem | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Partition.Basic | {
"line": 205,
"column": 11
} | {
"line": 205,
"column": 28
} | {
"line": 205,
"column": 28
} | [
{
"pp": "case refine_1\nα : Type u_1\ns t x✝ y✝ z : α\nS : Set α\ninst✝ : CompleteLattice α\nP✝ Q✝ P Q : Partition s\nhp : ∀ ⦃x : α⦄, x ∈ P → ∃ y ∈ Q, x ≤ y\nhq : ∀ ⦃x : α⦄, x ∈ Q → ∃ y ∈ P, x ≤ y\nx : α\nh : x ∈ P\ny : α\nhy : y ∈ Q\nhxy : x ≤ y\nhx' : x ∈ P\nhyx' : y ≤ x\n⊢ x ∈ Q",
"ppTerm": "?refine_1",
... | [
"case refine_1\nα : Type u_1\ns t x✝ y✝ z : α\nS : Set α\ninst✝ : CompleteLattice α\nP✝ Q✝ P Q : Partition s\nhp : ∀ ⦃x : α⦄, x ∈ P → ∃ y ∈ Q, x ≤ y\nhq : ∀ ⦃x : α⦄, x ∈ Q → ∃ y ∈ P, x ≤ y\nx : α\nh : x ∈ P\ny : α\nhy : y ∈ Q\nhxy : x ≤ y\nhx' : x ∈ P\nhyx' : y ≤ x\n⊢ y ∈ Q"
] | hxy.antisymm hyx' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Partition.Basic | {
"line": 209,
"column": 9
} | {
"line": 209,
"column": 26
} | {
"line": 209,
"column": 26
} | [
{
"pp": "case refine_2\nα : Type u_1\ns t x✝ y✝ z : α\nS : Set α\ninst✝ : CompleteLattice α\nP✝ Q✝ P Q : Partition s\nhp : ∀ ⦃x : α⦄, x ∈ P → ∃ y ∈ Q, x ≤ y\nhq : ∀ ⦃x : α⦄, x ∈ Q → ∃ y ∈ P, x ≤ y\nx : α\nh : x ∈ Q\ny : α\nhy : y ∈ P\nhxy : x ≤ y\nhx' : x ∈ Q\nhyx' : y ≤ x\n⊢ x ∈ P",
"ppTerm": "?refine_2",
... | [
"case refine_2\nα : Type u_1\ns t x✝ y✝ z : α\nS : Set α\ninst✝ : CompleteLattice α\nP✝ Q✝ P Q : Partition s\nhp : ∀ ⦃x : α⦄, x ∈ P → ∃ y ∈ Q, x ≤ y\nhq : ∀ ⦃x : α⦄, x ∈ Q → ∃ y ∈ P, x ≤ y\nx : α\nh : x ∈ Q\ny : α\nhy : y ∈ P\nhxy : x ≤ y\nhx' : x ∈ Q\nhyx' : y ≤ x\n⊢ y ∈ P"
] | hxy.antisymm hyx' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Std | {
"line": 286,
"column": 6
} | {
"line": 286,
"column": 39
} | {
"line": 287,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\nargs : OfStdArgs α\nthis✝⁷ : LE α := args.le\nthis✝⁶ : LT α := args.lt\nthis✝⁵ :\n let this := args.le;\n let this_1 := args.lt;\n Std.LawfulOrderLT α\nthis✝⁴ :\n let this := args.le;\n Std.IsLinearOrder α\nthis✝³ : DecidableLE α := args.decidableLE\nthis✝² : Std.LawfulOrde... | [
"case pos\nα : Type u_1\nargs : OfStdArgs α\nthis✝⁷ : LE α := args.le\nthis✝⁶ : LT α := args.lt\nthis✝⁵ :\n let this := args.le;\n let this_1 := args.lt;\n Std.LawfulOrderLT α\nthis✝⁴ :\n let this := args.le;\n Std.IsLinearOrder α\nthis✝³ : DecidableLE α := args.decidableLE\nthis✝² : Std.LawfulOrderLeftLeaning... | case _ => rwa [Std.compare_eq_lt] | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Order.Sublocale | {
"line": 207,
"column": 48
} | {
"line": 207,
"column": 79
} | {
"line": 209,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nm n : Nucleus X\n⊢ m.toSublocale ≤ n.toSublocale ↔ n ≤ m",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Sublocale.instPartialOrder",
"CompleteLattice.toLattice",
"congrArg",
"Nucleus",
"PartialOrder.toPreorder",
... | [] | simp [← SetLike.coe_subset_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.