module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 161,
"column": 10
} | {
"line": 161,
"column": 38
} | {
"line": 161,
"column": 39
} | [
{
"pp": "this :\n (Summable fun n ↦ (1 / 2) ^ (n + 1)) →\n (∀ (k : ℕ), Tendsto (fun x ↦ -x ^ (k + 1)) (𝓝 0) (𝓝 0)) →\n (∀ᶠ (n : ℂ) in 𝓝 0, ∀ (k : ℕ), ‖n‖ ^ (k + 1) ≤ (1 / 2) ^ (k + 1)) →\n Tendsto (fun n ↦ ∏' (k : ℕ), (1 + -n ^ (k + 1))) (𝓝 0) (𝓝 1)\n⊢ Summable fun n ↦ (1 / 2) ^ (n + 1)",
... | [
"this :\n (Summable fun n ↦ (1 / 2) ^ (n + 1)) →\n (∀ (k : ℕ), Tendsto (fun x ↦ -x ^ (k + 1)) (𝓝 0) (𝓝 0)) →\n (∀ᶠ (n : ℂ) in 𝓝 0, ∀ (k : ℕ), ‖n‖ ^ (k + 1) ≤ (1 / 2) ^ (k + 1)) →\n Tendsto (fun n ↦ ∏' (k : ℕ), (1 + -n ^ (k + 1))) (𝓝 0) (𝓝 1)\n⊢ Summable fun n ↦ 1 / 2 * (1 / 2) ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 707,
"column": 4
} | {
"line": 707,
"column": 44
} | {
"line": 707,
"column": 45
} | [
{
"pp": "case inr\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n ... | [
"case inr\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = T⁻... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 707,
"column": 65
} | {
"line": 707,
"column": 76
} | {
"line": 707,
"column": 77
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = ... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = T⁻¹ ∨ g = -T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 162,
"column": 18
} | {
"line": 162,
"column": 29
} | {
"line": 162,
"column": 30
} | [
{
"pp": "this :\n (Summable fun n ↦ (1 / 2) ^ (n + 1)) →\n (∀ (k : ℕ), Tendsto (fun x ↦ -x ^ (k + 1)) (𝓝 0) (𝓝 0)) →\n (∀ᶠ (n : ℂ) in 𝓝 0, ∀ (k : ℕ), ‖n‖ ^ (k + 1) ≤ (1 / 2) ^ (k + 1)) →\n Tendsto (fun n ↦ ∏' (k : ℕ), (1 + -n ^ (k + 1))) (𝓝 0) (𝓝 1)\nk : ℕ\n⊢ Tendsto (fun x ↦ -x ^ (k + 1)) ... | [
"this :\n (Summable fun n ↦ (1 / 2) ^ (n + 1)) →\n (∀ (k : ℕ), Tendsto (fun x ↦ -x ^ (k + 1)) (𝓝 0) (𝓝 0)) →\n (∀ᶠ (n : ℂ) in 𝓝 0, ∀ (k : ℕ), ‖n‖ ^ (k + 1) ≤ (1 / 2) ^ (k + 1)) →\n Tendsto (fun n ↦ ∏' (k : ℕ), (1 + -n ^ (k + 1))) (𝓝 0) (𝓝 1)\nk : ℕ\n⊢ Tendsto (fun x ↦ -x ^ (k + 1)) (𝓝 0) (𝓝 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 708,
"column": 10
} | {
"line": 708,
"column": 21
} | {
"line": 708,
"column": 22
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = ... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = T⁻¹ ∨ g = -T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 255,
"column": 10
} | {
"line": 255,
"column": 21
} | {
"line": 255,
"column": 22
} | [
{
"pp": "z : ℍ\nN : ℕ+\nh2 :\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N =\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N\n⊢ Summable (HPow.hPow (cexp (2 * ↑π * I * ↑{ coe := -↑... | [
"z : ℍ\nN : ℕ+\nh2 :\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N =\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N\n⊢ ‖cexp (2 * ↑π * I * (-↑↑N / ↑z))‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 708,
"column": 31
} | {
"line": 708,
"column": 42
} | {
"line": 708,
"column": 43
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = ... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : (g • z).im = z.im\nthis :\n ∀ {g : SL(2, ℤ)} {z : ℍ},\n z ∈ 𝒟 →\n g • z ∈ 𝒟 →\n (g • z).im = z.im →\n 0 ≤ ↑g 1 0 →\n (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = T⁻¹ ∨ g = -T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 255,
"column": 10
} | {
"line": 255,
"column": 21
} | {
"line": 255,
"column": 22
} | [
{
"pp": "z : ℍ\nN : ℕ+\nh2 :\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N =\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N\n⊢ Summable (HPow.hPow (cexp (2 * ↑π * I * ↑{ coe := -↑... | [
"z : ℍ\nN : ℕ+\nh2 :\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N =\n ↑π * I - 2 * ↑π * I * ∑' (n : ℕ), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ n - ↑z / -↑↑N\n⊢ ‖cexp (2 * ↑π * I * (-↑↑N / ↑z))‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 176,
"column": 50
} | {
"line": 189,
"column": 20
} | {
"line": 191,
"column": 0
} | [
{
"pp": "k : ℕ\nhk : 1 ≤ k\nz : ℍ\n⊢ ∑' (n : ℤ), 1 / (↑z + ↑n) ^ (k + 1) = (-2 * ↑π * I) ^ (k + 1) / ↑k ! * ∑' (n : ℕ), ↑n ^ k * cexp (2 * ↑π * I * ↑z) ^ n",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"one_pow",
... | [] | by
have : (-1) ^ k * k ! * ∑' n : ℤ, 1 / ((z : ℂ) + n) ^ (k + 1) =
-(2 * π * I) ^ (k + 1) * ∑' n : ℕ, n ^ k * cexp (2 * π * I * z) ^ n := by
rw [← iteratedDerivWithin_tsum_exp_aux_eq hk z,
← iteratedDerivWithin_cot_pi_mul_eq_mul_tsum_div_pow hk (by simpa using z.2)]
exact iteratedDerivWithin_congr (... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 194,
"column": 4
} | {
"line": 194,
"column": 15
} | {
"line": 194,
"column": 16
} | [
{
"pp": "k : ℕ\ne : ℕ+\nz : ℍ\n⊢ 0 < (↑↑e * ↑z).im",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"PNat.val",
"Complex.mul_im",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preo... | [
"k : ℕ\ne : ℕ+\nz : ℍ\n⊢ 0 < z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 204,
"column": 10
} | {
"line": 204,
"column": 39
} | {
"line": 204,
"column": 40
} | [
{
"pp": "q : ℂ\nhq : q ∈ Metric.ball 0 1\nhq0 : ¬q = 0\n⊢ ‖q‖ < 1",
"ppTerm": "?m.138",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"q : ℂ\nhq : q ∈ Metric.ball 0 1\nhq0 : ¬q = 0\n⊢ ‖q‖ < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 226,
"column": 10
} | {
"line": 226,
"column": 21
} | {
"line": 226,
"column": 22
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\n⊢ Summable fun b ↦ ∑' (c : ℤ), eisSummand ↑k ![(b, c).1, (b, c).2] z",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"AddCommGroup.toAddCommMonoid",
"PseudoMetricSpace.toUniformSpace"... | [
"k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\n⊢ Summable fun b ↦ ∑' (c : ℤ), eisSummand ↑k ![b, c] z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 230,
"column": 81
} | {
"line": 230,
"column": 92
} | {
"line": 230,
"column": 93
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\nb : ℕ+\n⊢ 0 < (↑↑b * ↑z).im",
"ppTerm": "?m.174",
"assigned": true,
"usedConstants": [
"PNat.val",
"Complex.mul_im",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder"... | [
"k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\nb : ℕ+\n⊢ 0 < z.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 261,
"column": 8
} | {
"line": 261,
"column": 19
} | {
"line": 261,
"column": 20
} | [
{
"pp": "k : ℤ\nf : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k\n⊢ (fun τ ↦ rexp (-2 * π * τ.im / 1)) =O[atImInfty] Δ",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"NonUnitalCommRi... | [
"k : ℤ\nf : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range k\n⊢ (fun τ ↦ rexp (-(2 * π * τ.im))) =O[atImInfty] Δ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 287,
"column": 2
} | {
"line": 288,
"column": 9
} | {
"line": 288,
"column": 10
} | [
{
"pp": "z : ℍ\n⊢ HasSum (fun b ↦ ∑' (m : ℤ), (1 / (↑m * ↑z + ↑b) - 1 / (↑m * ↑z + ↑b + 1))) (-2 * ↑π * I / ↑z) (symmetricIco ℤ)",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"PNat.val",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"_private.... | [
"z : ℍ\n⊢ Tendsto (fun x ↦ ∑ x ∈ Ico (-↑↑x) ↑↑x, ∑' (m : ℤ), ((↑m * ↑z + ↑x)⁻¹ - (↑m * ↑z + ↑x + 1)⁻¹)) atTop\n (𝓝 (-(2 * ↑π * I) / ↑z))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 715,
"column": 4
} | {
"line": 718,
"column": 53
} | {
"line": 720,
"column": 0
} | [
{
"pp": "case inr\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ = 1\nhc✝ : 0 ≤ ↑g 1 0\nhc : ↑g 1 0 = 1\n⊢ (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨\... | [] | rcases Int.abs_le_one_iff.mp (cases_d_of_c_eq_one hz him.le hc) with hd | hd | hd
· grind [cases_c_one_d_zero hz hg him.le hc hd] -- ± S, T⁻¹S, TS
· grind [case_c_one_d_one hz hg him.le hc hd] -- ± ST, TST
· grind [case_c_one_d_neg_one hz hg him.le hc hd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 715,
"column": 4
} | {
"line": 718,
"column": 53
} | {
"line": 720,
"column": 0
} | [
{
"pp": "case inr\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ = 1\nhc✝ : 0 ≤ ↑g 1 0\nhc : ↑g 1 0 = 1\n⊢ (g = 1 ∨ g = -1) ∨\n (g = T ∨ g = -T) ∧ z.re = -1 / 2 ∨\n (g = T⁻¹ ∨ g = -T⁻¹) ∧ z.re = 1 / 2 ∨\... | [] | rcases Int.abs_le_one_iff.mp (cases_d_of_c_eq_one hz him.le hc) with hd | hd | hd
· grind [cases_c_one_d_zero hz hg him.le hc hd] -- ± S, T⁻¹S, TS
· grind [case_c_one_d_one hz hg him.le hc hd] -- ± ST, TST
· grind [case_c_one_d_neg_one hz hg him.le hc hd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 266,
"column": 4
} | {
"line": 266,
"column": 15
} | {
"line": 266,
"column": 16
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nz : ℍ\nhk1 : 1 < k\nhk2 : 3 ≤ ↑k\nb : ℕ\n⊢ Summable fun c ↦ (↑(b, c).1 ^ k)⁻¹ * eisSummand (↑k) (↑(b, c).2) z",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"DivInvMonoid.toInv",
"NormedRing.toRing",
"HMul.hMul",
"ZMod.commRing",
... | [
"k : ℕ\nhk : 3 ≤ k\nz : ℍ\nhk1 : 1 < k\nhk2 : 3 ≤ ↑k\nb : ℕ\n⊢ Summable fun c ↦ (↑b ^ k)⁻¹ * eisSummand (↑k) (↑c) z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 271,
"column": 4
} | {
"line": 272,
"column": 11
} | {
"line": 272,
"column": 12
} | [
{
"pp": "case inr\nk : ℕ\nhk : 3 ≤ k\nz : ℍ\nhk1 : 1 < k\nhk2 : 3 ≤ ↑k\nb : ℕ\nhb : b ≠ 0\nthis : NeZero b\n⊢ ∑' (c : ↑(gammaSet 1 b 0)), eisSummand (↑k) (↑c) z =\n ∑' (c : { x // x ∈ gammaSet 1 1 0 }), (↑(b, c).1 ^ k)⁻¹ * eisSummand (↑k) (↑(b, c).2) z",
"ppTerm": "?inr",
"assigned": true,
"usedC... | [
"case inr\nk : ℕ\nhk : 3 ≤ k\nz : ℍ\nhk1 : 1 < k\nhk2 : 3 ≤ ↑k\nb : ℕ\nhb : b ≠ 0\nthis : NeZero b\n⊢ ∑' (x : ↑(gammaSet 1 b 0)), eisSummand (↑k) (divIntMap ↑b ↑x) z =\n ∑' (x : { x // x ∈ gammaSet 1 1 0 }), eisSummand (↑k) (divIntMap 1 ↑x) z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 744,
"column": 4
} | {
"line": 744,
"column": 47
} | {
"line": 745,
"column": 4
} | [
{
"pp": "case mpr\ng : SL(2, ℤ)\n⊢ S • I = I",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"UpperHalfPlane.SLAction",
"UpperHalfPlane.coe",
"congrArg",
"UpperHalfPlane.ext_iff",
... | [
"case mpr\ng : SL(2, ℤ)\n⊢ ↑{ coe := (-↑I)⁻¹, coe_im_pos := ⋯ } = ↑I"
] | rw [modular_S_smul, UpperHalfPlane.ext_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Modular | {
"line": 784,
"column": 4
} | {
"line": 784,
"column": 15
} | {
"line": 785,
"column": 2
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟ᵒ\nhg : g • z ∈ 𝒟\nthis✝ : ρ ∉ 𝒟ᵒ\nh : 1 +ᵥ ρ ∈ 𝒟ᵒ\nthis : (1 +ᵥ ρ).re = 1 / 2\n⊢ False",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.Modular.0.ModularGroup.eq_one_or_neg_one_of_mem_fdo_mem_fd._proof_1_3"
... | [] | grind [h.2] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.NumberTheory.Modular | {
"line": 867,
"column": 8
} | {
"line": 867,
"column": 31
} | {
"line": 867,
"column": 32
} | [
{
"pp": "x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\n⊢ ↑1 * ↑x ∈ ofComplex.source",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.ofComplex",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"UpperHalfPlane.isOpenEmbedding_coe",
"Real",
... | [
"x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\n⊢ 0 < x.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 868,
"column": 6
} | {
"line": 868,
"column": 58
} | {
"line": 868,
"column": 59
} | [
{
"pp": "case h.refine_1\nx : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\nthis : ContinuousAt (fun a ↦ ↑(↑ofComplex (↑a * ↑x))) 1\n⊢ Filter.Tendsto (fun x_1 ↦ ‖↑(↑ofComplex (↑x_1 * ↑x))‖) (𝓝 1) (𝓝 ‖↑x‖)",
"ppTerm": "?h.refine_1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"case h.refine_1\nx : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\nthis : ContinuousAt (fun a ↦ ↑(↑ofComplex (↑a * ↑x))) 1\n⊢ Filter.Tendsto (fun x_1 ↦ ‖↑(↑ofComplex (↑x_1 * ↑x))‖) (𝓝 1) (𝓝 ‖↑x‖)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 871,
"column": 50
} | {
"line": 871,
"column": 61
} | {
"line": 871,
"column": 62
} | [
{
"pp": "x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\na : ℝ\nha : 0 < a\nha' : a ∈ Set.Iio 1\n⊢ 0 < (↑a * ↑x).im",
"ppTerm": "?m.207",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Eq.mpr",
"Real",
"HMul.hMul",
"UpperHalfPlane.coe",
"Real.instZero",
... | [
"x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\na : ℝ\nha : 0 < a\nha' : a ∈ Set.Iio 1\n⊢ 0 < a * x.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 13
} | {
"line": 121,
"column": 14
} | [
{
"pp": "hi : 0 < 1\n⊢ (PowerSeries.coeff 0) (qExpansion 1 Δ) = 0",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Semiring.toModule",
"UpperHalfPlane.qExpansion",
"congrArg",
"LinearMap.instFunLike",
"RingHom",
"id",
... | [
"hi : 0 < 1\n⊢ PowerSeries.constantCoeff (qExpansion 1 Δ) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 881,
"column": 38
} | {
"line": 881,
"column": 49
} | {
"line": 881,
"column": 50
} | [
{
"pp": "x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\nthis : Filter.Tendsto (fun t ↦ ↑t * ↑x) (𝓝 1) (𝓝 ↑x)\na : ℝ\nha : 0 < a\n⊢ 0 < (↑a * ↑x).im",
"ppTerm": "?m.380",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Eq.mpr",
"Real",
"HMul.hMul",
"UpperHalf... | [
"x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\nthis : Filter.Tendsto (fun t ↦ ↑t * ↑x) (𝓝 1) (𝓝 ↑x)\na : ℝ\nha : 0 < a\n⊢ 0 < a * x.im"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 927,
"column": 4
} | {
"line": 927,
"column": 66
} | {
"line": 927,
"column": 67
} | [
{
"pp": "ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)\nho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)\nx : ℍ\nhx : x ∈ interior 𝒟\nξ : ℍ\nhξ : ξ ∈ 𝒟\n⊢ ξ ∈ (fun τ ↦ ‖↑τ‖) ⁻¹' Set.Ici 1",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": ... | [
"ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)\nho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)\nx : ℍ\nhx : x ∈ interior 𝒟\nξ : ℍ\nhξ : ξ ∈ 𝒟\n⊢ 1 ≤ ‖↑ξ‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 932,
"column": 4
} | {
"line": 932,
"column": 55
} | {
"line": 932,
"column": 56
} | [
{
"pp": "ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)\nho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)\nx : ℍ\nhx : x ∈ interior 𝒟\nξ : ℍ\nhξ : ξ ∈ 𝒟\n⊢ ξ ∈ UpperHalfPlane.re ⁻¹' Set.Icc (-(1 / 2)) (1 / 2)",
"ppTerm": "?m.173",
"assigned": true,
... | [
"ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)\nho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)\nx : ℍ\nhx : x ∈ interior 𝒟\nξ : ℍ\nhξ : ξ ∈ 𝒟\n⊢ -2⁻¹ ≤ ξ.re ∧ ξ.re ≤ 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 949,
"column": 33
} | {
"line": 949,
"column": 72
} | {
"line": 949,
"column": 73
} | [
{
"pp": "y : ℝ\nz : ℂ\nhz : 0 < z.im\nh : { coe := z, coe_im_pos := hz } ∈ truncatedFundamentalDomain y\n⊢ 1 ≤ ‖↑{ coe := z, coe_im_pos := hz }‖",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"UpperHalfPlane.coe",
"Complex... | [
"y : ℝ\nz : ℂ\nhz : 0 < z.im\nh : { coe := z, coe_im_pos := hz } ∈ truncatedFundamentalDomain y\n⊢ 1 ≤ ‖z‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 954,
"column": 6
} | {
"line": 955,
"column": 25
} | {
"line": 955,
"column": 26
} | [
{
"pp": "y : ℝ\nz : ℂ\nhz : 0 ≤ z.im\nh1 : z.im ≤ y\nh2 : |z.re| ≤ 1 / 2\nh3 : 0 = z.im\n⊢ ‖z‖ < 1",
"ppTerm": "?m.175",
"assigned": true,
"usedConstants": [
"sq_lt_one_iff₀",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | [
"y : ℝ\nz : ℂ\nhz : 0 ≤ z.im\nh1 : z.im ≤ y\nh2 : |z.re| ≤ 1 / 2\nh3 : 0 = z.im\n⊢ |z.re| < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula | {
"line": 244,
"column": 2
} | {
"line": 245,
"column": 9
} | {
"line": 245,
"column": 10
} | [
{
"pp": "⊢ Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 2) = 0",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"ModularForm",
"Eq.mpr",
"Subgroup.instHasDetOneRangeSpecialLinearGroupGeneralLinearGroupMapGL",
"MonoidHom.range",
"Real",
... | [
"⊢ ∀ (x : ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 2), x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 13
} | {
"line": 349,
"column": 14
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\n⊢ (PowerSeries.coeff 0) (qExpansion 1 ⇑(E hk)) = 1",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"ModularForm",
"Eq.mpr",
"MonoidHom.range",
"Real",
"Matrix.SpecialLinearGroup",
"Semiring.toModule",
"Uppe... | [
"k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\n⊢ PowerSeries.constantCoeff (qExpansion 1 ⇑(E hk)) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 995,
"column": 4
} | {
"line": 996,
"column": 56
} | {
"line": 997,
"column": 2
} | [
{
"pp": "case inl\nτ : ℍ\nh : 1 / 2 ≤ τ.im\n⊢ ∃ γ, 1 / 2 ≤ (γ • τ).im ∧ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"Real.instLE",
... | [] | exact ⟨1, (one_smul SL(2, ℤ) τ).symm ▸ h, by
simp only [map_one, denom_one, norm_one, le_refl]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Modular | {
"line": 995,
"column": 4
} | {
"line": 996,
"column": 56
} | {
"line": 997,
"column": 2
} | [
{
"pp": "case inl\nτ : ℍ\nh : 1 / 2 ≤ τ.im\n⊢ ∃ γ, 1 / 2 ≤ (γ • τ).im ∧ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"Real.instLE",
... | [] | exact ⟨1, (one_smul SL(2, ℤ) τ).symm ▸ h, by
simp only [map_one, denom_one, norm_one, le_refl]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 995,
"column": 4
} | {
"line": 996,
"column": 56
} | {
"line": 997,
"column": 2
} | [
{
"pp": "case inl\nτ : ℍ\nh : 1 / 2 ≤ τ.im\n⊢ ∃ γ, 1 / 2 ≤ (γ • τ).im ∧ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"Real.instLE",
... | [] | exact ⟨1, (one_smul SL(2, ℤ) τ).symm ▸ h, by
simp only [map_one, denom_one, norm_one, le_refl]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 51
} | {
"line": 70,
"column": 52
} | [
{
"pp": "z : ℍ\ng : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 12\nhg : CuspForm.toModularFormₗ g = E₄CubeSubE₆SqForm\nc : ℂ\nhc : c • CuspForm.discriminant = g\nhgE : ⇑g = ⇑E₄CubeSubE₆SqForm\nhcΔ : c • ⇑CuspForm.discriminant = ⇑g\nhgΔ : qExpansion 1 ⇑E₄CubeSubE₆SqForm = c • qExpansion 1 ⇑CuspForm.discr... | [
"z : ℍ\ng : CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 12\nhg : CuspForm.toModularFormₗ g = E₄CubeSubE₆SqForm\nc : ℂ\nhc : c • CuspForm.discriminant = g\nhgE : ⇑g = ⇑E₄CubeSubE₆SqForm\nhcΔ : c • ⇑CuspForm.discriminant = ⇑g\nhgΔ : qExpansion 1 ⇑E₄CubeSubE₆SqForm = c • qExpansion 1 ⇑CuspForm.discriminant\n⊢ c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Modular | {
"line": 1001,
"column": 4
} | {
"line": 1001,
"column": 57
} | {
"line": 1002,
"column": 6
} | [
{
"pp": "case inr\nτ : ℍ\nh : τ.im ≤ 1 / 2\nγ : SL(2, ℤ)\nhγ : 1 / 2 ≤ (γ • τ).im\nh1 : τ.im * ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ^ 2 ≤ τ.im\n⊢ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1",
"ppTerm": "?inr",
"assigned": false,
"usedConstants... | [
"case inr\nτ : ℍ\nh : τ.im ≤ 1 / 2\nγ : SL(2, ℤ)\nhγ : 1 / 2 ≤ (γ • τ).im\nh1 : τ.im * ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ^ 2 ≤ τ.im\n⊢ ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑τ‖ ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula | {
"line": 258,
"column": 6
} | {
"line": 258,
"column": 29
} | {
"line": 258,
"column": 30
} | [
{
"pp": "case h.inl.«2»\nk : ℕ\nihn :\n ∀ m < 2,\n Even m →\n Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑m) =\n ↑(if m ≡ 2 [MOD 12] then m / 12 else m / 12 + 1)\nhk2 : Even 2\nhk : 2 < 3\n⊢ Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑2) =\n ... | [
"case h.inl.«2»\nk : ℕ\nihn :\n ∀ m < 2,\n Even m →\n Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑m) =\n ↑(if m ≡ 2 [MOD 12] then m / 12 else m / 12 + 1)\nhk2 : Even 2\nhk : 2 < 3\n⊢ Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 2) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LevelOne.GradedRing | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 72
} | {
"line": 85,
"column": 2
} | [
{
"pp": "⊢ (DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) (modularForm CuspForm.discriminant) =\n (1 / 1728) •\n ((DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) (E₄.pow 3) -\n (DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).rang... | [
"⊢ (DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) (modularForm CuspForm.discriminant) =\n (DirectSum.of (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range) 12) ((1 / 1728) • (E₄.pow 3 - E₆.pow 2))"
] | rw [← map_sub (DirectSum.of (ModularForm 𝒮ℒ) 12), ← DirectSum.of_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 59,
"column": 4
} | {
"line": 60,
"column": 11
} | {
"line": 60,
"column": 12
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : SlashInvariantFormClass F 𝒢 k\ninst✝ : 𝒢.IsFiniteRelIndex ℋ\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)\n⊢ (let this := Fintype.ofFinite (↥ℋ... | [
"𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : SlashInvariantFormClass F 𝒢 k\ninst✝ : 𝒢.IsFiniteRelIndex ℋ\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)\n⊢ ∑ x, quotientFunc f (⟨h, hh⟩⁻¹ • x) = ∑ q, qu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 68,
"column": 4
} | {
"line": 70,
"column": 53
} | {
"line": 70,
"column": 54
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝³ : FunLike F ℍ ℂ\nk : ℤ\ninst✝² : SlashInvariantFormClass F 𝒢 k\ninst✝¹ : 𝒢.IsFiniteRelIndex ℋ\ninst✝ : ℋ.HasDetPlusMinusOne\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)\n⊢ (l... | [
"𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝³ : FunLike F ℍ ℂ\nk : ℤ\ninst✝² : SlashInvariantFormClass F 𝒢 k\ninst✝¹ : 𝒢.IsFiniteRelIndex ℋ\ninst✝ : ℋ.HasDetPlusMinusOne\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)\n⊢ ∏ x, quotientF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 15
} | {
"line": 90,
"column": 16
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : 𝒢.IsFiniteRelIndex ℋ\ninst✝ : ModularFormClass F 𝒢 k\nγ : GL (Fin 2) ℝ\nh : IsCusp (γ • OnePoint.infty) ℋ\nx✝¹ : ↥ℋ\nr : GL (Fin 2) ℝ\nhr : r ∈ ℋ\nx✝ : ⟦⟨r, hr⟩⟧ ∈ Finset.univ\n⊢ IsCusp (γ • OnePoint.infty) (... | [
"𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : 𝒢.IsFiniteRelIndex ℋ\ninst✝ : ModularFormClass F 𝒢 k\nγ : GL (Fin 2) ℝ\nh : IsCusp (γ • OnePoint.infty) ℋ\nx✝¹ : ↥ℋ\nr : GL (Fin 2) ℝ\nhr : r ∈ ℋ\nx✝ : ⟦⟨r, hr⟩⟧ ∈ Finset.univ\n⊢ IsCusp (γ • OnePoint.infty) (ConjAct.toCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 15
} | {
"line": 104,
"column": 16
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : 𝒢.IsFiniteRelIndex ℋ\ninst✝ : CuspFormClass 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)\nx✝¹ : ↥ℋ\nr : GL (Fin ... | [
"𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : 𝒢.IsFiniteRelIndex ℋ\ninst✝ : CuspFormClass 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)\nx✝¹ : ↥ℋ\nr : GL (Fin 2) ℝ\nhr : r... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 15
} | {
"line": 120,
"column": 16
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 41
} | {
"line": 127,
"column": 42
} | [
{
"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\nhf : ∃ i ∈ Finset.univ, quotientFunc f i = 0\n⊢ ⇑f = 0",
"ppTerm": "?m.79",
"assigned": false,
"usedConstants":... | [
"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\nhf : ∃ i ∈ Finset.univ, quotientFunc f i = 0\n⊢ ⇑f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 136,
"column": 4
} | {
"line": 137,
"column": 11
} | {
"line": 137,
"column": 12
} | [
{
"pp": "case refine_2\n𝒢 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\nhf : ⇑f = 0\nτ : ℍ\n⊢ (ModularForm.norm ℋ f) τ = 0 τ",
"ppTerm": "?refine_2",
"assigned": true,
... | [
"case refine_2\n𝒢 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\nhf : ⇑f = 0\nτ : ℍ\n⊢ ∃ a ∈ ℋ, (⇑f ∣[k] a⁻¹) τ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 137,
"column": 19
} | {
"line": 137,
"column": 30
} | {
"line": 137,
"column": 31
} | [
{
"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\nhf : ⇑f = 0\nτ : ℍ\n⊢ 1 ∈ ℋ ∧ (⇑f ∣[k] 1⁻¹) τ = 0",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
... | [
"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\nhf : ⇑f = 0\nτ : ℍ\n⊢ f τ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 66
} | {
"line": 158,
"column": 67
} | [
{
"pp": "𝒢 : Subgroup (GL (Fin 2) ℝ)\ninst✝¹ : 𝒢.IsArithmetic\ninst✝ : 𝒢.HasDetOne\nf : ModularForm 𝒢 0\nthis : ModularFormClass\n (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range\n (0 *\n ↑(Nat.card\n (↥(Matrix.SpecialLinearGroup.mapGL ℝ).range ⧸ 𝒢.subgroupOf (Matrix.SpecialLinearG... | [
"𝒢 : Subgroup (GL (Fin 2) ℝ)\ninst✝¹ : 𝒢.IsArithmetic\ninst✝ : 𝒢.HasDetOne\nf : ModularForm 𝒢 0\nthis : ModularFormClass\n (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range\n (0 *\n ↑(Nat.card\n (↥(Matrix.SpecialLinearGroup.mapGL ℝ).range ⧸ 𝒢.subgroupOf (Matrix.SpecialLinearGroup.mapGL ℝ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 47,
"column": 53
} | {
"line": 47,
"column": 64
} | {
"line": 47,
"column": 65
} | [
{
"pp": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\nx y p : R\nh : p ∣ x - y\n⊢ p ∣ y - x",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nn : ℕ\ninst✝ : CommRing R\nx y p : R\nh : p ∣ x - y\n⊢ p ∣ y - x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfiniteAdeleRing | {
"line": 124,
"column": 60
} | {
"line": 124,
"column": 92
} | {
"line": 124,
"column": 93
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : InfiniteAdeleRing K\nv : InfinitePlace K\nhv : ¬IsUnit (x v)\n⊢ ‖x v‖ ^ v.mult = 0",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Norm.norm... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : InfiniteAdeleRing K\nv : InfinitePlace K\nhv : ¬IsUnit (x v)\n⊢ x v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 18
} | {
"line": 261,
"column": 19
} | [
{
"pp": "m : ℤ\neq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))\n⊢ 8 ∣ (2 * m + 1) ^ 2 - 1",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"HSub.hSub",
"id",
... | [
"m : ℤ\neq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))\n⊢ 8 ∣ 4 * (m * (m + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 18
} | {
"line": 266,
"column": 19
} | [
{
"pp": "m : ℕ\neq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))\n⊢ 8 ∣ (2 * m + 1) ^ 2 - 1",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"HSub... | [
"m : ℕ\neq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))\n⊢ 8 ∣ 4 * (m * (m + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 272,
"column": 28
} | {
"line": 272,
"column": 39
} | {
"line": 272,
"column": 40
} | [
{
"pp": "x y : ℤ\nhx : ¬2 ∣ x\nhxy : 4 ∣ x - y\ni : ℕ\nhx_odd : Odd x\nhxy_even : Even (x - y)\n⊢ Odd y",
"ppTerm": "?m.95",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℤ\nhx : ¬2 ∣ x\nhxy : 4 ∣ x - y\ni : ℕ\nhx_odd : Odd x\nhxy_even : Even (x - y)\n⊢ Odd y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 295,
"column": 28
} | {
"line": 295,
"column": 39
} | {
"line": 295,
"column": 40
} | [
{
"pp": "x y : ℤ\nn : ℕ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\nhxy_even : Even (x - y)\n⊢ Odd y",
"ppTerm": "?m.79",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℤ\nn : ℕ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\nhxy_even : Even (x - y)\n⊢ Odd y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 277,
"column": 6
} | {
"line": 277,
"column": 67
} | {
"line": 277,
"column": 68
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\ninst✝¹ : (i : ι) → TopologicalSpace (R i)\nS : Set ι\ninst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)\nhS : Sᶜ.Finite\nhAcompact : ∀ i ∈ S, IsCompact (A i)\nx : Πʳ (i : ι), [R i, A i]_[𝓟 S]\nK : (i : ι) → Set (R i)\nK_compact :... | [
"case refine_2\nι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\ninst✝¹ : (i : ι) → TopologicalSpace (R i)\nS : Set ι\ninst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)\nhS : Sᶜ.Finite\nhAcompact : ∀ i ∈ S, IsCompact (A i)\nx : Πʳ (i : ι), [R i, A i]_[𝓟 S]\nK : (i : ι) → Set (R i)\nK_compact : ∀ (i : ι), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 39
} | {
"line": 306,
"column": 40
} | [
{
"pp": "case succ.hxy\nx y : ℤ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : ℕ\nh : FiniteMultiplicity 2 (n + 1)\nhpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1\nk : ℕ\nhk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k\n⊢ 2 ∣ x ^ 2 ^ multiplicity 2 (n + 1) - y ^ 2 ^ mul... | [
"case succ.hxy\nx y : ℤ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : ℕ\nh : FiniteMultiplicity 2 (n + 1)\nhpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1\nk : ℕ\nhk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k\n⊢ 2 ∣ x ^ 2 ^ multiplicity 2 (n + 1) - y ^ 2 ^ multiplicity 2 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 63
} | {
"line": 307,
"column": 64
} | [
{
"pp": "case succ.hx\nx y : ℤ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : ℕ\nh : FiniteMultiplicity 2 (n + 1)\nhpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1\nk : ℕ\nhk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k\n⊢ ¬2 ∣ x ^ 2 ^ multiplicity 2 (n + 1)",
"ppTerm... | [
"case succ.hx\nx y : ℤ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : ℕ\nh : FiniteMultiplicity 2 (n + 1)\nhpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1\nk : ℕ\nhk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k\n⊢ ¬2 ∣ x ^ 2 ^ multiplicity 2 (n + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.InfinitePlace | {
"line": 404,
"column": 4
} | {
"line": 404,
"column": 45
} | {
"line": 405,
"column": 6
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\nL : Type u_2\ninst✝² : Field L\ninst✝¹ : Algebra K L\nw : InfinitePlace L\nv : InfinitePlace K\ninst✝ : (↑w).LiesOver ↑v\nx : WithAbs ↑v\n⊢ ‖(algebraMap (WithAbs ↑v) (WithAbs ↑w)) x‖ = ‖x‖",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Norm.... | [
"K : Type u_1\ninst✝³ : Field K\nL : Type u_2\ninst✝² : Field L\ninst✝¹ : Algebra K L\nw : InfinitePlace L\nv : InfinitePlace K\ninst✝ : (↑w).LiesOver ↑v\nx : WithAbs ↑v\n⊢ ↑w ((algebraMap (WithAbs ↑v) (WithAbs ↑w)) x).ofAbs = ↑v x.ofAbs"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 376,
"column": 2
} | {
"line": 376,
"column": 13
} | {
"line": 376,
"column": 14
} | [
{
"pp": "x n : ℕ\nh1x : 1 < x\nhx : ¬2 ∣ x\nhn : n ≠ 0\nhneven : Even n\n⊢ padicValNat 2 (x ^ n - 1) + 1 = padicValNat 2 (x + 1) + padicValNat 2 (x - 1) + padicValNat 2 n",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x n : ℕ\nh1x : 1 < x\nhx : ¬2 ∣ x\nhn : n ≠ 0\nhneven : Even n\n⊢ padicValNat 2 (x ^ n - 1) + 1 = padicValNat 2 (x + 1) + padicValNat 2 (x - 1) + padicValNat 2 n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 637,
"column": 6
} | {
"line": 637,
"column": 17
} | {
"line": 637,
"column": 18
} | [
{
"pp": "ι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\n𝓕✝ 𝓖 : Filter ι\nS : ι → Type u_3\ninst✝⁴ : (i : ι) → SetLike (S i) (R i)\nB : (i : ι) → S i\nT : Set ι\n𝓕 : Filter ι\ninst✝³ : (i : ι) → TopologicalSpace (R i)\nhBopen : Fact (∀ (i : ι), IsOpen[inst✝³ i] ↑(B i))\ninst✝² : (i : ι) → Group (R i... | [
"ι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\n𝓕✝ 𝓖 : Filter ι\nS : ι → Type u_3\ninst✝⁴ : (i : ι) → SetLike (S i) (R i)\nB : (i : ι) → S i\nT : Set ι\n𝓕 : Filter ι\ninst✝³ : (i : ι) → TopologicalSpace (R i)\nhBopen : Fact (∀ (i : ι), IsOpen[inst✝³ i] ↑(B i))\ninst✝² : (i : ι) → Group (R i)\ninst✝¹ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 377,
"column": 19
} | {
"line": 377,
"column": 30
} | {
"line": 377,
"column": 31
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nh : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w\na : mixedSpace K\nha : a ∈ A\n⊢ normAtComplexPlaces a ∈ Set.univ.pi fun w ↦ if w.IsReal then Set.u... | [
"K : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nh : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w\na : mixedSpace K\nha : a ∈ A\n⊢ ∀ (i : InfinitePlace K), i.IsComplex → 0 ≤ normAtComplexPlaces a i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 17
} | {
"line": 167,
"column": 18
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\n⊢ regOfFamily u ≠ 0 → IsMaxRank u",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"NumberField.Units.IsMaxRa... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\n⊢ ¬regOfFamily u = 0 → IsMaxRank u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 58
} | {
"line": 382,
"column": 59
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nh : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w\nx : realSpace K\nx✝ : x ∈ ⇑mixedSpaceOfRealSpace ⁻¹' A ∩ Set.univ.pi fun w ↦ if w.I... | [
"case refine_2\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nh : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w\nx : realSpace K\nx✝ : x ∈ ⇑mixedSpaceOfRealSpace ⁻¹' A ∩ Set.univ.pi fun w ↦ if w.IsReal then S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 210,
"column": 17
} | {
"line": 210,
"column": 59
} | {
"line": 210,
"column": 60
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx y : mixedSpace K\nhx : x ∈ fundamentalCone K\nhy : ∀ (w : InfinitePlace K), (normAtPlace w) y = (normAtPlace w) x\n⊢ y ∉ {x | mixedEmbedding.norm x = 0}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx y : mixedSpace K\nhx : x ∈ fundamentalCone K\nhy : ∀ (w : InfinitePlace K), (normAtPlace w) y = (normAtPlace w) x\n⊢ ¬mixedEmbedding.norm x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 393,
"column": 92
} | {
"line": 411,
"column": 100
} | {
"line": 413,
"column": 0
} | [
{
"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\n⊢ volume A =\n ENNReal.ofReal (2 * π) ^ nrComplexPlaces K *\n ∫⁻ (x : realSpace K) in normAtComplexPlaces '' A, ∏ w, ENNReal.ofReal (x ... | [] | by
have hA' {x} : (A.indicator 1 x : ℝ≥0∞) =
(normAtComplexPlaces '' A).indicator 1 (normAtComplexPlaces x) := by
simp_rw [← Set.indicator_comp_right, Function.comp_def, Pi.one_def, hA]
rw [← lintegral_indicator_one hm, ← lintegral_comp_polarSpaceCoord_symm, polarSpaceCoord_target',
Measure.volume_eq_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 429,
"column": 8
} | {
"line": 430,
"column": 15
} | {
"line": 430,
"column": 16
} | [
{
"pp": "case refine_2.refine_1.inl\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtAllPlaces ⁻¹' normAtAllPlaces '' A = A\na : mixedSpace K\nha₁ : a ∈ A\nha₂ : a ∈ {x | ∀ (w : { w // w.IsReal }), 0 < x.1 w}\nw : InfinitePlace K\nhw : w.IsReal\n⊢ normAtAllPlaces a w = normAtComplexPlaces a w"... | [
"case refine_2.refine_1.inl\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtAllPlaces ⁻¹' normAtAllPlaces '' A = A\na : mixedSpace K\nha₁ : a ∈ A\nha₂ : a ∈ {x | ∀ (w : { w // w.IsReal }), 0 < x.1 w}\nw : InfinitePlace K\nhw : w.IsReal\n⊢ 0 ≤ a.1 ⟨w, hw⟩"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 70
} | {
"line": 433,
"column": 71
} | [
{
"pp": "case refine_2.refine_2\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtAllPlaces ⁻¹' normAtAllPlaces '' A = A\na : mixedSpace K\nha₁ : a ∈ A\nha₂ : a ∈ {x | ∀ (w : { w // w.IsReal }), 0 < x.1 w}\nw : { w // w.IsReal }\n⊢ normAtComplexPlaces a ∈ {x | x ↑w ≠ 0}",
"ppTerm": "?refine... | [
"case refine_2.refine_2\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtAllPlaces ⁻¹' normAtAllPlaces '' A = A\na : mixedSpace K\nha₁ : a ∈ A\nha₂ : a ∈ {x | ∀ (w : { w // w.IsReal }), 0 < x.1 w}\nw : { w // w.IsReal }\n⊢ ¬a.1 w = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 386,
"column": 63
} | {
"line": 386,
"column": 74
} | {
"line": 386,
"column": 75
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na b : ↑(integerSet K)\nx✝ : ∃ u, (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b\nu : (↥(𝓞 K)⁰)ˣ\nh : (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b\n⊢ ↑⟨unitsNonZeroDivisorsEquiv u, ?m.50⟩ • ↑a = ↑b",
"ppTerm": "?m.51",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na b : ↑(integerSet K)\nx✝ : ∃ u, (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b\nu : (↥(𝓞 K)⁰)ˣ\nh : (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b\n⊢ (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 387,
"column": 60
} | {
"line": 387,
"column": 71
} | {
"line": 387,
"column": 72
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na b : ↑(integerSet K)\nx✝ : ∃ ζ, ↑ζ • ↑a = ↑b\nu : (𝓞 K)ˣ\nproperty✝ : u ∈ torsion K\nh : ↑⟨u, property✝⟩ • ↑a = ↑b\n⊢ (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑(unitsNonZeroDivisorsEquiv.symm u)) * ↑a = ↑b",
"ppTerm": "?m.85",
"assigned... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na b : ↑(integerSet K)\nx✝ : ∃ ζ, ↑ζ • ↑a = ↑b\nu : (𝓞 K)ˣ\nproperty✝ : u ∈ torsion K\nh : ↑⟨u, property✝⟩ • ↑a = ↑b\n⊢ (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑u) * ↑a = ↑b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 388,
"column": 58
} | {
"line": 388,
"column": 73
} | {
"line": 388,
"column": 74
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na b : ↑(integerSet K)\nx✝ : ∃ u, (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b\nu : (↥(𝓞 K)⁰)ˣ\nh : (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b\n⊢ unitsNonZeroDivisorsEquiv u • ↑a ∈ fundamentalCone K",
"ppTerm": "?m.130... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na b : ↑(integerSet K)\nx✝ : ∃ u, (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b\nu : (↥(𝓞 K)⁰)ˣ\nh : (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b\n⊢ ↑b ∈ fundamentalCone K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 466,
"column": 6
} | {
"line": 466,
"column": 21
} | {
"line": 466,
"column": 22
} | [
{
"pp": "case refine_2\nK : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : CharZero K\ninst✝³ : Algebra.IsIntegral ℚ K\ninst✝² : IsTotallyComplex K\nE : Subfield K\ninst✝¹ : IsTotallyReal ↥E\ninst✝ : IsQuadraticExtension (↥E) K\nh : ¬maximalRealSubfield K ≤ E\nL : IntermediateField (↥E) K := (E ⊔ maximalRealSubfield K).t... | [
"case refine_2\nK : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : CharZero K\ninst✝³ : Algebra.IsIntegral ℚ K\ninst✝² : IsTotallyComplex K\nE : Subfield K\ninst✝¹ : IsTotallyReal ↥E\ninst✝ : IsQuadraticExtension (↥E) K\nh : ¬maximalRealSubfield K ≤ E\nL : IntermediateField (↥E) K := (E ⊔ maximalRealSubfield K).toIntermediat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.Ramification | {
"line": 117,
"column": 51
} | {
"line": 117,
"column": 62
} | {
"line": 117,
"column": 63
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nv : InfinitePlace K\ninst✝¹ : NumberField K\ninst✝ : NumberField L\nx✝ : InfinitePlace L\nh : x✝ ∈ (ramifiedPlacesOver L v).toFinset\n⊢ x✝ ∈ ramifiedPlacesOver L v",
"ppTerm": "?m.92",
"assigned": false,
"... | [
"K : Type u_1\nL : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nv : InfinitePlace K\ninst✝¹ : NumberField K\ninst✝ : NumberField L\nx✝ : InfinitePlace L\nh : x✝ ∈ (ramifiedPlacesOver L v).toFinset\n⊢ x✝ ∈ ramifiedPlacesOver L v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.Ramification | {
"line": 118,
"column": 51
} | {
"line": 118,
"column": 62
} | {
"line": 118,
"column": 63
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nv : InfinitePlace K\ninst✝¹ : NumberField K\ninst✝ : NumberField L\nx✝ : InfinitePlace L\nh : x✝ ∈ (unramifiedPlacesOver L v).toFinset\n⊢ x✝ ∈ unramifiedPlacesOver L v",
"ppTerm": "?m.136",
"assigned": false,
... | [
"K : Type u_1\nL : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nv : InfinitePlace K\ninst✝¹ : NumberField K\ninst✝ : NumberField L\nx✝ : InfinitePlace L\nh : x✝ ∈ (unramifiedPlacesOver L v).toFinset\n⊢ x✝ ∈ unramifiedPlacesOver L v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 574,
"column": 4
} | {
"line": 574,
"column": 44
} | {
"line": 575,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : CharZero K\nS : Set ℕ\ninst✝ : IsCyclotomicExtension S ℚ K\nthis✝² : Algebra.IsIntegral ℚ K\nn : ℕ\nhn₁ : n ∈ S\nhn₂ : 2 < n\nthis✝¹ : NeZero n\nζ : K\nhζ : IsPrimitiveRoot ζ n\nthis✝ : IsCyclotomicExtension {n} ℚ ↥ℚ⟮ζ⟯\nthis : IsTotallyComplex ↥ℚ⟮ζ⟯\n⊢ IsTotall... | [] | exact isTotallyComplex_of_algebra ℚ⟮ζ⟯ K | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 39,
"column": 50
} | {
"line": 39,
"column": 66
} | {
"line": 39,
"column": 67
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\np f : ℕ\nhK : Fintype.card K = p ^ f\nh0 : f = 0\n⊢ Fintype.card K ≤ 1",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\np f : ℕ\nhK : Fintype.card K = p ^ f\nh0 : f = 0\n⊢ Fintype.card K ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 524,
"column": 2
} | {
"line": 525,
"column": 69
} | {
"line": 525,
"column": 70
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\n⊢ mixedEmbedding.norm (mixedSpaceOfRealSpace (↑expMapBasis x)) = Real.exp (x w₀) ^ finrank ℚ K",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"P... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\n⊢ ∏ x_1, ↑expMapBasis x x_1 ^ x_1.mult = Real.exp (x w₀) ^ finrank ℚ K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 53,
"column": 35
} | {
"line": 53,
"column": 46
} | {
"line": 53,
"column": 47
} | [
{
"pp": "n : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : ζ ^ n = 1\nhζ1 : ζ ≠ 1\nkey : ↑n = ∑ i ∈ range n, (1 - ζ ^ i)\ni : ℕ\nx✝ : i ∈ range n\n⊢ ζ - 1 ∣ ζ ^ i - 1",
"ppTerm": "?m.148",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : ζ ^ n = 1\nhζ1 : ζ ≠ 1\nkey : ↑n = ∑ i ∈ range n, (1 - ζ ^ i)\ni : ℕ\nx✝ : i ∈ range n\n⊢ ζ - 1 ∣ ζ ^ i - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 55
} | {
"line": 98,
"column": 4
} | [
{
"pp": "K : 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)\nhPirr : Irreducible P\nA : K[X]\nhA : cyclotomic n K = P * A\nhQ : P * C P.leadingCoeff⁻¹ ∣ cyclotomic n K\n⊢ P.natDegree = orderOf (unitOfCoprime (p ^ f) ⋯)",... | [
"K : 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)\nhPirr : Irreducible P\nA : K[X]\nhA : cyclotomic n K = P * A\nhQ : P * C P.leadingCoeff⁻¹ ∣ cyclotomic n K\n⊢ (P * C P.leadingCoeff⁻¹).natDegree = orderOf (unitOfCoprime (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 127,
"column": 8
} | {
"line": 127,
"column": 19
} | {
"line": 127,
"column": 20
} | [
{
"pp": "p n : ℕ\nhp : Fact (Nat.Prime p)\nP : (ZMod p)[X]\nhpn : ¬p ∣ n\nhP : P ∣ cyclotomic n (ZMod p)\nhPdeg : P.natDegree = orderOf (unitOfCoprime p ⋯)\n⊢ p.Coprime n",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p n : ℕ\nhp : Fact (Nat.Prime p)\nP : (ZMod p)[X]\nhpn : ¬p ∣ n\nhP : P ∣ cyclotomic n (ZMod p)\nhPdeg : P.natDegree = orderOf (unitOfCoprime p ⋯)\n⊢ p.Coprime n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 15
} | {
"line": 143,
"column": 16
} | [
{
"pp": "case inr\np : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : IsPrimitiveRoot ζ p\nhp : Nat.Prime p\nthis✝ : NeZero p\ni : ℕ\nhi : i < p\nhη₁ : ζ ^ i ∈ ↑(nthRootsFinset p 1)\nj : ℕ\nhj : j < p\nhη₂ : ζ ^ j ∈ ↑(nthRootsFinset p 1)\ne : ζ ^ i ≠ ζ ^ j\nthis :\n ∀ {p : ℕ} {A : Type u... | [
"case inr\np : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : IsPrimitiveRoot ζ p\nhp : Nat.Prime p\nthis✝ : NeZero p\ni : ℕ\nhi : i < p\nhη₁ : ζ ^ i ∈ ↑(nthRootsFinset p 1)\nj : ℕ\nhj : j < p\nhη₂ : ζ ^ j ∈ ↑(nthRootsFinset p 1)\ne : ζ ^ i ≠ ζ ^ j\nthis :\n ∀ {p : ℕ} {A : Type u_1} {ζ : A} ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Ideal.Basic | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 37
} | {
"line": 81,
"column": 38
} | [
{
"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\nx✝ : ↥(rootsOfUnity n (𝓞 K))\nζ : (𝓞 K)ˣ\nhζ✝ : ζ ∈ rootsOfUnity n (𝓞 K)\nh : (I.rootsOfUnityMapQuot n) ⟨ζ, hζ✝⟩ = 1\nt : ℕ\nht₀ : t ≠ 0\nht : t ∣ n\nh... | [
"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\nx✝ : ↥(rootsOfUnity n (𝓞 K))\nζ : (𝓞 K)ˣ\nhζ✝ : ζ ∈ rootsOfUnity n (𝓞 K)\nh : (I.rootsOfUnityMapQuot n) ⟨ζ, hζ✝⟩ = 1\nt : ℕ\nht₀ : t ≠ 0\nht : t ∣ n\nhζ : IsPrimit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 665,
"column": 56
} | {
"line": 665,
"column": 72
} | {
"line": 665,
"column": 73
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : ∀ (w : InfinitePlace K), 0 < x w\nh : ∀ (i : InfinitePlace K), ↑expMapBasis.symm x i ∈ if i = w₀ then Set.Iic 0 else Set.Ico 0 1\nw : InfinitePlace K\nhw : ¬w = w₀\n⊢ ↑expMapBasis.symm x w ∈ Set.Ico 0 1",
"ppTerm": "?m.87"... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : ∀ (w : InfinitePlace K), 0 < x w\nh : ∀ (i : InfinitePlace K), ↑expMapBasis.symm x i ∈ if i = w₀ then Set.Iic 0 else Set.Ico 0 1\nw : InfinitePlace K\nhw : ¬w = w₀\n⊢ 0 ≤ ↑expMapBasis.symm x w ∧ ↑expMapBasis.symm x w < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 665,
"column": 81
} | {
"line": 665,
"column": 92
} | {
"line": 665,
"column": 93
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : ∀ (w : InfinitePlace K), 0 < x w\nh : ∀ (i : InfinitePlace K), ↑expMapBasis.symm x i ∈ if i = w₀ then Set.Iic 0 else Set.Ico 0 1\n⊢ ↑expMapBasis.symm x w₀ ≤ 0",
"ppTerm": "?m.89",
"assigned": false,
"usedConstants"... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : ∀ (w : InfinitePlace K), 0 < x w\nh : ∀ (i : InfinitePlace K), ↑expMapBasis.symm x i ∈ if i = w₀ then Set.Iic 0 else Set.Ico 0 1\n⊢ ↑expMapBasis.symm x w₀ ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 660,
"column": 4
} | {
"line": 668,
"column": 21
} | {
"line": 669,
"column": 2
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : ∀ (w : InfinitePlace K), 0 < x w\n⊢ x ∈ normAtAllPlaces '' normLeOne K ↔ x ∈ ↑expMapBasis '' paramSet K",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
... | [] | rw [← expMapBasis.right_inv (Set.mem_univ_pi.mpr hx), (injective_expMapBasis K).mem_set_image]
simp only [normAtAllPlaces_normLeOne, Set.mem_inter_iff, Set.mem_setOf_eq, expMapBasis_nonneg,
Set.mem_preimage, logMap_expMapBasis, implies_true, and_true, norm_expMapBasis,
pow_le_one_iff_of_nonneg (Real.exp... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 660,
"column": 4
} | {
"line": 668,
"column": 21
} | {
"line": 669,
"column": 2
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : ∀ (w : InfinitePlace K), 0 < x w\n⊢ x ∈ normAtAllPlaces '' normLeOne K ↔ x ∈ ↑expMapBasis '' paramSet K",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
... | [] | rw [← expMapBasis.right_inv (Set.mem_univ_pi.mpr hx), (injective_expMapBasis K).mem_set_image]
simp only [normAtAllPlaces_normLeOne, Set.mem_inter_iff, Set.mem_setOf_eq, expMapBasis_nonneg,
Set.mem_preimage, logMap_expMapBasis, implies_true, and_true, norm_expMapBasis,
pow_le_one_iff_of_nonneg (Real.exp... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 755,
"column": 6
} | {
"line": 755,
"column": 21
} | {
"line": 755,
"column": 22
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nc : ℝ\nhc : c ∈ Set.Icc 0 1\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then {0} else Set.Icc 0 1\nhx₀ : (fun x1 x2 ↦ x1 • x2) c (↑expMapBasis y) ≠ 0\nw : InfinitePlace K\nh : ¬w = w₀\n⊢ y w ∈ Set.Icc 0 1",
"ppTerm": "?ne... | [
"case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nc : ℝ\nhc : c ∈ Set.Icc 0 1\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then {0} else Set.Icc 0 1\nhx₀ : (fun x1 x2 ↦ x1 • x2) c (↑expMapBasis y) ≠ 0\nw : InfinitePlace K\nh : ¬w = w₀\n⊢ 0 ≤ y w ∧ y w ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 762,
"column": 6
} | {
"line": 762,
"column": 30
} | {
"line": 762,
"column": 31
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nc : ℝ\nhc : c ∈ Set.Icc 0 1\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then {0} else Set.Icc 0 1\nhx₀ : (fun x1 x2 ↦ x1 • x2) c (↑expMapBasis y) ≠ 0\nhc' : 0 < c\nw : InfinitePlace K\nh : w = w₀\n⊢ 0 = y w",
"ppTerm": "?... | [
"case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nc : ℝ\nhc : c ∈ Set.Icc 0 1\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then {0} else Set.Icc 0 1\nhx₀ : (fun x1 x2 ↦ x1 • x2) c (↑expMapBasis y) ≠ 0\nhc' : 0 < c\nw : InfinitePlace K\nh : w = w₀\n⊢ 0 = y w₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 775,
"column": 38
} | {
"line": 775,
"column": 49
} | {
"line": 775,
"column": 50
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then Set.Iic 0 else Set.Icc 0 1\nhx₀ : ↑expMapBasis y ≠ 0\n⊢ y w₀ ≤ 0",
"ppTerm": "?m.96",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then Set.Iic 0 else Set.Icc 0 1\nhx₀ : ↑expMapBasis y ≠ 0\n⊢ y w₀ ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 778,
"column": 6
} | {
"line": 778,
"column": 21
} | {
"line": 778,
"column": 22
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then Set.Iic 0 else Set.Icc 0 1\nhx₀ : ↑expMapBasis y ≠ 0\nw : InfinitePlace K\nh : ¬w = w₀\n⊢ y w ∈ Set.Icc 0 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [... | [
"case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then Set.Iic 0 else Set.Icc 0 1\nhx₀ : ↑expMapBasis y ≠ 0\nw : InfinitePlace K\nh : ¬w = w₀\n⊢ 0 ≤ y w ∧ y w ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 776,
"column": 4
} | {
"line": 778,
"column": 43
} | {
"line": 780,
"column": 0
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then Set.Iic 0 else Set.Icc 0 1\nhx₀ : ↑expMapBasis y ≠ 0\nw : InfinitePlace K\n⊢ (if w = w₀ then 0 else y w) ∈ if w = w₀ then {0} else Set.Icc 0 1",
"ppTerm": "?refine_2",
... | [] | split_ifs with h
· rfl
· simpa [h] using hy w (Set.mem_univ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 776,
"column": 4
} | {
"line": 778,
"column": 43
} | {
"line": 780,
"column": 0
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then Set.Iic 0 else Set.Icc 0 1\nhx₀ : ↑expMapBasis y ≠ 0\nw : InfinitePlace K\n⊢ (if w = w₀ then 0 else y w) ∈ if w = w₀ then {0} else Set.Icc 0 1",
"ppTerm": "?refine_2",
... | [] | split_ifs with h
· rfl
· simpa [h] using hy w (Set.mem_univ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 785,
"column": 4
} | {
"line": 785,
"column": 21
} | {
"line": 785,
"column": 22
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx₀ : x = 0\n⊢ x ∈ compactSet K ↔ x ∈ ↑expMapBasis '' closure (paramSet K) ∪ {0}",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NumberField.mixedEmbedding.realSp... | [
"case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx₀ : x = 0\n⊢ 0 ∈ compactSet K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 788,
"column": 4
} | {
"line": 788,
"column": 41
} | {
"line": 790,
"column": 0
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx₀ : ¬x = 0\nhx : x ∈ ↑expMapBasis '' closure (paramSet K)\n⊢ x ∈ compactSet K",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"NumberField.mixedEmbedding.fundamentalCone.compactSet_eq_union_a... | [] | exact compactSet_eq_union_aux₂ hx₀ hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 824,
"column": 8
} | {
"line": 824,
"column": 32
} | {
"line": 824,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : IsBounded (↑expMapBasis '' paramSet K)\nC : ℝ\nhC : ∀ x ∈ ↑expMapBasis '' paramSet K, ‖x‖ ≤ C\nx : mixedSpace K\nhx : x ∈ normAtAllPlaces ⁻¹' ↑expMapBasis '' paramSet K\n⊢ ‖x‖ ≤ C",
"ppTerm": "?m.49",
"assigned": true,
"usedConst... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : IsBounded (↑expMapBasis '' paramSet K)\nC : ℝ\nhC : ∀ x ∈ ↑expMapBasis '' paramSet K, ‖x‖ ≤ C\nx : mixedSpace K\nhx : x ∈ normAtAllPlaces ⁻¹' ↑expMapBasis '' paramSet K\n⊢ (univ.sup' ⋯ fun w ↦ (normAtPlace w) x) ≤ C"
] | norm_eq_sup'_normAtPlace | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 826,
"column": 4
} | {
"line": 827,
"column": 11
} | {
"line": 827,
"column": 12
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : IsBounded (↑expMapBasis '' paramSet K)\nC : ℝ\nhC : ∀ x ∈ ↑expMapBasis '' paramSet K, ‖x‖ ≤ C\nx : mixedSpace K\nhx : x ∈ normAtAllPlaces ⁻¹' ↑expMapBasis '' paramSet K\nw : InfinitePlace K\nx✝ : w ∈ univ\n⊢ (normAtPlace w) x ≤ C",
"ppTe... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : IsBounded (↑expMapBasis '' paramSet K)\nC : ℝ\nhC : ∀ x ∈ ↑expMapBasis '' paramSet K, ‖x‖ ≤ C\nx : mixedSpace K\nhx : x ∈ normAtAllPlaces ⁻¹' ↑expMapBasis '' paramSet K\nw : InfinitePlace K\nx✝ : w ∈ univ\n⊢ (normAtPlace w) x ≤ C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 124,
"column": 10
} | {
"line": 124,
"column": 21
} | {
"line": 124,
"column": 22
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nK : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Ideal (𝓞 K)\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver (span {↑p})\nhn : p.Coprime n\nσ : Gal(K/ℚ)\nhσ : σ ∈ stabilizer Gal(K/ℚ) P\nhζ : IsPrimitiveR... | [
"n : ℕ\ninst✝⁴ : NeZero n\nK : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Ideal (𝓞 K)\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver (span {↑p})\nhn : p.Coprime n\nσ : Gal(K/ℚ)\nhσ : σ ∈ stabilizer Gal(K/ℚ) P\nhζ : IsPrimitiveRoot (zeta n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 123,
"column": 4
} | {
"line": 124,
"column": 56
} | {
"line": 125,
"column": 4
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nK : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Ideal (𝓞 K)\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver (span {↑p})\nhn : p.Coprime n\nσ : Gal(K/ℚ)\nhσ : σ ∈ stabilizer Gal(K/ℚ) P\nhζ : IsPrimitiveR... | [
"n : ℕ\ninst✝⁴ : NeZero n\nK : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Ideal (𝓞 K)\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver (span {↑p})\nhn : p.Coprime n\nσ : Gal(K/ℚ)\nhσ : σ ∈ stabilizer Gal(K/ℚ) P\nhζ : IsPrimitiveRoot (zeta n ... | refine hζ.toInteger_isPrimitiveRoot.idealQuotient_mk
(by simpa using IsMaximal.ne_top inferInstance) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.NumberField.ExistsRamified | {
"line": 90,
"column": 64
} | {
"line": 90,
"column": 75
} | {
"line": 90,
"column": 76
} | [
{
"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✝⁶ : IsDomain 𝒪\nthis✝⁵ : IsDedekindDomain 𝒪\nthis✝⁴ : IsFractionRing 𝒪 K\nthis✝³ : 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✝⁶ : IsDomain 𝒪\nthis✝⁵ : IsDedekindDomain 𝒪\nthis✝⁴ : IsFractionRing 𝒪 K\nthis✝³ : Module.Finite ℤ 𝒪\nthi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 13
} | {
"line": 84,
"column": 14
} | [
{
"pp": "p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\n⊢ absNorm (span {hζ.toInteger - 1}) = p",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\n⊢ ((Algebra.norm ℤ) (hζ.toInteger - 1)).natAbs = p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 43
} | {
"line": 91,
"column": 0
} | [
{
"pp": "case e'_5\np k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\n⊢ p = absNorm (span {hζ.toInteger - 1})",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
... | [] | exact (absNorm_span_zeta_sub_one ..).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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