module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 518,
"column": 2
} | {
"line": 518,
"column": 66
} | {
"line": 520,
"column": 0
} | [
{
"pp": "case neg.hx\np k : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhtwo : ¬p ^ (k + 1) = 2\nthis : NumberField K\nH : (Algebra.norm ℤ) (hζ.toInteger - 1) = ↑p ^ 1\n⊢ Prime ((Algeb... | [] | · exact prime_norm_toInteger_sub_one_of_prime_pow_ne_two hζ htwo | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 536,
"column": 4
} | {
"line": 536,
"column": 30
} | {
"line": 537,
"column": 2
} | [
{
"pp": "case refine_3\np k : ℕ\nK : Type u\ninst✝² : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝¹ : CharZero K\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhodd : p ≠ 2\nthis : NumberField K\nh : ↑p ∣ 2\n⊢ p ∣ 2",
"ppTerm": "?refine_3",
"assigned": true,
... | [] | · exact Int.ofNat_dvd.mp h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 701,
"column": 8
} | {
"line": 701,
"column": 39
} | {
"line": 702,
"column": 8
} | [
{
"pp": "n₁ n₂ : ℕ\nhn₁ : 1 < n₁\nhn₂ : 1 < n₂\nh : n₁.Coprime n₂\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nhn : NeZero (n₁ * n₂)\nhK : IsCyclotomicExtension {n₁ * n₂} ℚ K\nthis✝⁵ : NumberField K\nthis✝⁴ : NeZero n₁\nthis✝³ : NeZero n₂\nζ : K := zeta (n₁ * n₂) ℚ K\nhζ : IsPrimitiveRoot (zeta (n₁ * n₂) ... | [
"n₁ n₂ : ℕ\nhn₁ : 1 < n₁\nhn₂ : 1 < n₂\nh : n₁.Coprime n₂\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nhn : NeZero (n₁ * n₂)\nhK : IsCyclotomicExtension {n₁ * n₂} ℚ K\nthis✝⁵ : NumberField K\nthis✝⁴ : NeZero n₁\nthis✝³ : NeZero n₂\nζ : K := zeta (n₁ * n₂) ℚ K\nhζ : IsPrimitiveRoot (zeta (n₁ * n₂) ℚ K) (n₁ * n... | rw [← Nat.Coprime.lcm_eq_mul h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 542,
"column": 2
} | {
"line": 542,
"column": 87
} | {
"line": 543,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra A B\ninst✝⁵ : IsDomain A\ninst✝⁴ : IsIntegrallyClosed A\ninst✝³ : IsDedekindDomain B\ninst✝² : IsTorsionFree A B\ninst✝¹ : Module.Finite A B\ninst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\nK : Type u_... | [
"A : Type u_1\nB : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra A B\ninst✝⁵ : IsDomain A\ninst✝⁴ : IsIntegrallyClosed A\ninst✝³ : IsDedekindDomain B\ninst✝² : IsTorsionFree A B\ninst✝¹ : Module.Finite A B\ninst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\nK : Type u_1 := Fractio... | rw [ne_eq, ← FractionalIdeal.coeIdeal_inj (K := L), coeIdeal_differentIdeal (K := K)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 793,
"column": 8
} | {
"line": 793,
"column": 39
} | {
"line": 794,
"column": 8
} | [
{
"pp": "n₁ n₂ : ℕ\nhn₁ : 1 < n₁\nhn₂ : 1 < n₂\nh : n₁.Coprime n₂\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nhn : NeZero (n₁ * n₂)\nhK : IsCyclotomicExtension {n₁ * n₂} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (n₁ * n₂)\nthis✝⁴ : NumberField K\nthis✝³ : NeZero n₁\nthis✝² : NeZero n₂\nthis✝¹ : IsCyclotomicExte... | [
"n₁ n₂ : ℕ\nhn₁ : 1 < n₁\nhn₂ : 1 < n₂\nh : n₁.Coprime n₂\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nhn : NeZero (n₁ * n₂)\nhK : IsCyclotomicExtension {n₁ * n₂} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (n₁ * n₂)\nthis✝⁴ : NumberField K\nthis✝³ : NeZero n₁\nthis✝² : NeZero n₂\nthis✝¹ : IsCyclotomicExtension {n₁} ℚ... | rw [← Nat.Coprime.lcm_eq_mul h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 40
} | {
"line": 102,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nh : ∑ i, ↑(↑i).mult * Real.log (↑i ((algebraMap (𝓞 K) K) ↑x)) = -↑w₀.mult * Real.log (w₀ ((algebraMap (𝓞 K) K) ↑x))\n⊢ ∑ w, (logEmbedding K) (Additive.ofMul x) w = -↑w₀.mult * Real.log (w₀ ((algebraMap (𝓞 K) K) ↑x))",
"ppTerm": ... | [] | simpa [logEmbedding_component] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.NumberField.Cyclotomic.Three | {
"line": 55,
"column": 50
} | {
"line": 55,
"column": 94
} | {
"line": 56,
"column": 6
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nu : (𝓞 K)ˣ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\n⊢ nrRealPlaces K + nrComplexPlaces K - 1 = 0",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.fintypeSingleton",
... | [
"K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nu : (𝓞 K)ˣ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\n⊢ 0 + nrComplexPlaces K - 1 = 0"
] | nrRealPlaces_eq_zero (n := 3) K (by decide), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 696,
"column": 2
} | {
"line": 696,
"column": 63
} | {
"line": 697,
"column": 2
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²¹ : CommRing A\ninst✝²⁰ : Field K\ninst✝¹⁹ : CommRing B\ninst✝¹⁸ : Field L\ninst✝¹⁷ : Algebra A K\ninst✝¹⁶ : Algebra B L\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Algebra A L\ninst✝¹² : IsScalarTower A K L\ninst✝¹¹ : IsScalarTow... | [
"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²¹ : CommRing A\ninst✝²⁰ : Field K\ninst✝¹⁹ : CommRing B\ninst✝¹⁸ : Field L\ninst✝¹⁷ : Algebra A K\ninst✝¹⁶ : Algebra B L\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Algebra A L\ninst✝¹² : IsScalarTower A K L\ninst✝¹¹ : IsScalarTower A B L\nin... | have habot : a ≠ ⊥ := fun ha' ↦ hp' (by simpa [ha'] using ha) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.Cyclotomic.Three | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 14
} | {
"line": 79,
"column": 15
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nu : (𝓞 K)ˣ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nhrank : rank K = 0\nx : ↥(torsion K)\ne : Fin (rank K) → ℤ\nhxu : u = ↑x\nn : ℕ\nhnpos : 0 < n\nhn : (algebraMap (𝓞 K) K) ↑u ^ ↑⟨n, hnpos⟩ = 1\nr : ℕ\nhr3 : r < ... | [
"case «0»\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nu : (𝓞 K)ˣ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nhrank : rank K = 0\nx : ↥(torsion K)\ne : Fin (rank K) → ℤ\nhxu : u = ↑x\nn : ℕ\nhnpos : 0 < n\nhn : (algebraMap (𝓞 K) K) ↑u ^ ↑⟨n, hnpos⟩ = 1\nhr3 : 0 < 3\nhru : ... | fin_cases hr | Lean.Elab.Tactic._aux_Mathlib_Tactic_FinCases___elabRules_Lean_Elab_Tactic_finCases_1 | Lean.Elab.Tactic.finCases |
Mathlib.NumberTheory.NumberField.Cyclotomic.Three | {
"line": 109,
"column": 2
} | {
"line": 114,
"column": 55
} | {
"line": 115,
"column": 2
} | [
{
"pp": "case «3»\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nhcong : ∃ n, 3 ∣ ↑(-⋯.unit) - ↑n\n⊢ False",
"ppTerm": "?«3»",
"assigned": true,
"usedConstants": [
"IsPrimitiveRoot.toInteger_isPrimitiveRoot",
... | [
"case «4»\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nhcong : ∃ n, 3 ∣ ↑(⋯.unit ^ 2) - ↑n\n⊢ False",
"case «5»\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtensio... | · apply hζ.not_exists_int_prime_dvd_sub_of_prime_ne_two' (by decide)
obtain ⟨n, x, hx⟩ := hcong
rw [sub_eq_iff_eq_add] at hx
refine ⟨-n, -x, sub_eq_iff_eq_add.2 ?_⟩
simp only [Nat.cast_ofNat, mul_neg, Int.cast_neg, ← neg_add, ← hx, Units.val_neg,
IsUnit.unit_spec, RingOfIntegers.neg_mk, neg_neg] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 231,
"column": 7
} | {
"line": 231,
"column": 97
} | {
"line": 231,
"column": 97
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw₁ : InfinitePlace K\nB : ℕ\nhB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B\nx : 𝓞 K\nhx : x ≠ 0\nhx' : (algebraMap (𝓞 K) K) x ≠ 0\nf : InfinitePlace K → ℝ≥0 := fun w ↦ ⟨w ↑x / 2, ⋯⟩\nthis : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0\ng : ... | [] | by rw [convexBodyLT_volume]; convert! hB; exact congr_arg ((↑) : NNReal → ENNReal) h_gprod | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.Cyclotomic.Three | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 60
} | {
"line": 138,
"column": 60
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nx : 𝓞 K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nthis✝ : Finite (𝓞 K ⧸ Ideal.span {λ})\nx✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})\nx✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := (Ide... | [
"K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nx : 𝓞 K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nthis✝ : Finite (𝓞 K ⧸ Ideal.span {λ})\nx✝² : Fintype (𝓞 K ⧸ Ideal.span {λ}) := Fintype.ofFinite (𝓞 K ⧸ Ideal.span {λ})\nx✝¹ : Ring (𝓞 K ⧸ Ideal.span {λ}) := (Ideal.Quotient.... | hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 755,
"column": 4
} | {
"line": 755,
"column": 12
} | {
"line": 756,
"column": 4
} | [
{
"pp": "case pos\nA : Type u_1\nB : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra A B\ninst✝⁵ : IsDomain A\ninst✝⁴ : IsDedekindDomain A\ninst✝³ : IsDedekindDomain B\ninst✝² : IsTorsionFree A B\ninst✝¹ : Module.Finite A B\ninst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np :... | [
"case pos\nA : Type u_1\nB : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra A B\ninst✝⁵ : IsDomain A\ninst✝⁴ : IsDedekindDomain A\ninst✝³ : IsDedekindDomain B\ninst✝² : IsTorsionFree A B\ninst✝¹ : Module.Finite A B\ninst✝ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\nP Q : Ideal B\n... | subst hp | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 446,
"column": 48
} | {
"line": 446,
"column": 80
} | {
"line": 446,
"column": 80
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ (Subgroup.toAddSubgroup (torsion K)).FG",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.toAddSubgroup",
"NumberField.instCommRingRingOfIntegers",
"congrArg",
... | [
"case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ AddGroup.FG ↥(Subgroup.toAddSubgroup (torsion K))"
] | ← AddGroup.fg_iff_addSubgroup_fg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 35
} | {
"line": 263,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\nhF : ringChar F ≠ 2\nh : (quadraticChar F) (-1) = if (-1) ^ (Fintype.card F / 2) = 1 then 1 else -1\n⊢ (if (-1) ^ (Fintype.card F / 2) = 1 then 1 else -1) = (-1) ^ (Fintype.card F / 2)",
"ppTerm": "?m.39",
"assigned": tr... | [
"F : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\nhF : ringChar F ≠ 2\nh : (quadraticChar F) (-1) = if (-1) ^ (Fintype.card F / 2) = 1 then 1 else -1\nn : ℕ\n⊢ (if (-1) ^ n = 1 then 1 else -1) = (-1) ^ n"
] | generalize Fintype.card F / 2 = n | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum | {
"line": 63,
"column": 75
} | {
"line": 72,
"column": 7
} | {
"line": 74,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\n⊢ IsSquare (-2) ↔ Fintype.card F % 8 ≠ 5 ∧ Fintype.card F % 8 ≠ 7",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | by
classical
by_cases hF : ringChar F = 2
· have h := FiniteField.even_card_of_char_two hF
simp only [FiniteField.isSquare_of_char_two hF, true_iff]
lia
· have h := FiniteField.odd_card_of_char_ne_two hF
rw [← quadraticChar_one_iff_isSquare (neg_ne_zero.mpr (Ring.two_ne_zero hF)),
quadraticCha... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.FLT.Three | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 18
} | {
"line": 118,
"column": 2
} | [
{
"pp": "case inr.inl\na b c : ℤ\nha : a ≠ 0\nh3a : 3 ∣ a\nHgcd : {a, b, c}.gcd id = 1\nH : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3\nHF : a ^ 3 + b ^ 3 + c ^ 3 = 0\nx : ℤ\nh3b : 3 ∣ b\nhx : x = b\n⊢ 3 ∣ id x",
"ppTerm": "?inr.inl",
"assigned": true,
"us... | [
"case inr.inr\na b c : ℤ\nha : a ≠ 0\nh3a : 3 ∣ a\nHgcd : {a, b, c}.gcd id = 1\nH : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3\nHF : a ^ 3 + b ^ 3 + c ^ 3 = 0\nx : ℤ\nh3b : 3 ∣ b\nhx : x = c\n⊢ 3 ∣ id x"
] | · exact hx ▸ h3b | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.FLT.Three | {
"line": 173,
"column": 6
} | {
"line": 173,
"column": 60
} | {
"line": 173,
"column": 60
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nH : FermatLastTheoremForThreeGen hζ\na b c : ℤ\nhc : c ≠ 0\nha : ¬3 ∣ a\nhb : ¬3 ∣ b\nx✝ : 3 ∣ c\nhcoprime : IsCoprime a b\nh : a ^ 3 + b ^ 3 = c ^ 3\nx : ℤ\nhx... | [
"case refine_1\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nH : FermatLastTheoremForThreeGen hζ\na b c : ℤ\nhc : c ≠ 0\nha : ¬3 ∣ a\nhb : ¬3 ∣ b\nx✝ : 3 ∣ c\nhcoprime : IsCoprime a b\nh : a ^ 3 + b ^ 3 = c ^ 3\nx : ℤ\nhx : c = 3 * x... | hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 172,
"column": 2
} | {
"line": 173,
"column": 69
} | {
"line": 174,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nH : FermatLastTheoremForThreeGen hζ\na b c : ℤ\nhc : c ≠ 0\nha : ¬3 ∣ a\nhb : ¬3 ∣ b\nx✝ : 3 ∣ c\nhcoprime : IsCoprime a b\nh : a ^ 3 + b ^ 3 = c ^ 3\nx : ℤ\nhx... | [
"case refine_2\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nH : FermatLastTheoremForThreeGen hζ\na b c : ℤ\nhc : c ≠ 0\nha : ¬3 ∣ a\nhb : ¬3 ∣ b\nx✝ : 3 ∣ c\nhcoprime : IsCoprime a b\nh : a ^ 3 + b ^ 3 = c ^ 3\nx : ℤ\nhx : c = 3 * x... | · rwa [← Ideal.norm_dvd_iff (hζ.prime_norm_toInteger_sub_one_of_prime_ne_two' (by decide)),
hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)] at hdvd | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.FLT.Three | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 60
} | {
"line": 175,
"column": 60
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nH : FermatLastTheoremForThreeGen hζ\na b c : ℤ\nhc : c ≠ 0\nha : ¬3 ∣ a\nhb : ¬3 ∣ b\nx✝ : 3 ∣ c\nhcoprime : IsCoprime a b\nh : a ^ 3 + b ^ 3 = c ^ 3\nx : ℤ\nhx... | [
"case refine_2\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nH : FermatLastTheoremForThreeGen hζ\na b c : ℤ\nhc : c ≠ 0\nha : ¬3 ∣ a\nhb : ¬3 ∣ b\nx✝ : 3 ∣ c\nhcoprime : IsCoprime a b\nh : a ^ 3 + b ^ 3 = c ^ 3\nx : ℤ\nhx : c = 3 * x... | hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 174,
"column": 2
} | {
"line": 175,
"column": 69
} | {
"line": 176,
"column": 2
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nH : FermatLastTheoremForThreeGen hζ\na b c : ℤ\nhc : c ≠ 0\nha : ¬3 ∣ a\nhb : ¬3 ∣ b\nx✝ : 3 ∣ c\nhcoprime : IsCoprime a b\nh : a ^ 3 + b ^ 3 = c ^ 3\nx : ℤ\nhx... | [
"case refine_3\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nH : FermatLastTheoremForThreeGen hζ\na b c : ℤ\nhc : c ≠ 0\nha : ¬3 ∣ a\nhb : ¬3 ∣ b\nx✝ : 3 ∣ c\nhcoprime : IsCoprime a b\nh : a ^ 3 + b ^ 3 = c ^ 3\nx : ℤ\nhx : c = 3 * x... | · rwa [← Ideal.norm_dvd_iff (hζ.prime_norm_toInteger_sub_one_of_prime_ne_two' (by decide)),
hζ.norm_toInteger_sub_one_of_prime_ne_two' (by decide)] at hdvd | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 862,
"column": 2
} | {
"line": 862,
"column": 63
} | {
"line": 863,
"column": 2
} | [
{
"pp": "case neg\nA : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\n... | [
"case neg\nA : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal A\... | have habot : a ≠ ⊥ := fun ha' ↦ hp' (by simpa [ha'] using ha) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.FermatPsp | {
"line": 77,
"column": 2
} | {
"line": 99,
"column": 8
} | {
"line": 101,
"column": 0
} | [
{
"pp": "n b : ℕ\nh : n.ProbablePrime b\nh₁ : 1 ≤ n\nh₂ : 1 ≤ b\n⊢ n.Coprime b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"Nat.Coprime",
"False",
"Nat.Prime",
"Dvd.dvd",
"HMul.hMul",
"dvd_of_mul_right_dvd",
"... | [] | by_cases h₃ : 2 ≤ n
· -- To prove that `n` is coprime with `b`, we need to show that for all prime factors of `n`,
-- we can derive a contradiction if `n` divides `b`.
apply Nat.coprime_of_dvd
-- If `k` is a prime number that divides both `n` and `b`, then we know that `n = m * k` and
-- `b = j * k` f... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FermatPsp | {
"line": 77,
"column": 2
} | {
"line": 99,
"column": 8
} | {
"line": 101,
"column": 0
} | [
{
"pp": "n b : ℕ\nh : n.ProbablePrime b\nh₁ : 1 ≤ n\nh₂ : 1 ≤ b\n⊢ n.Coprime b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"Nat.Coprime",
"False",
"Nat.Prime",
"Dvd.dvd",
"HMul.hMul",
"dvd_of_mul_right_dvd",
"... | [] | by_cases h₃ : 2 ≤ n
· -- To prove that `n` is coprime with `b`, we need to show that for all prime factors of `n`,
-- we can derive a contradiction if `n` divides `b`.
apply Nat.coprime_of_dvd
-- If `k` is a prime number that divides both `n` and `b`, then we know that `n = m * k` and
-- `b = j * k` f... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FermatPsp | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 9
} | {
"line": 155,
"column": 0
} | [
{
"pp": "case e'_5\nb p : ℕ\nx✝ : 2 ≤ b\nhp : Odd p\nq₁ : b - 1 ∣ b ^ p - 1\nq₂ : b + 1 ∣ b ^ p + 1\n⊢ b ^ (2 * p) - 1 = (b ^ p) ^ 2 - 1 ^ 2",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"instPowNat",
"Eq.mpr",
"NonAssocSe... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Fermat | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 9
} | {
"line": 61,
"column": 2
} | [
{
"pp": "case succ\nn : ℕ\nhn : ∏ k ∈ range n, k.fermatNumber = n.fermatNumber - 2\n⊢ (2 ^ 2 ^ n) ^ 2 - 1 ^ 2 = 2 ^ 2 ^ (n + 1) + 1 - 2",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"instPowNat",
"Eq.mpr",
"NonAssocSemirin... | [
"case succ\nn : ℕ\nhn : ∏ k ∈ range n, k.fermatNumber = n.fermatNumber - 2\n⊢ 2 ^ (2 ^ n * 2) - 1 = 1 + 2 ^ (2 ^ n * 2) - 2"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.FermatPsp | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 51
} | {
"line": 220,
"column": 2
} | [
{
"pp": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : ¬p ∣ b * (b ^ 2 - 1)\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nhA : p < A\nhi_A : 1 < A\nhi_B : 1 < B\nhi_b : 0 < b\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 ≤ b ^ (2 * p)\nhi_bpowpsubone : 1 ≤ b ^... | [
"b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : ¬p ∣ b * (b ^ 2 - 1)\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nhA : p < A\nhi_A : 1 < A\nhi_B : 1 < B\nhi_b : 0 < b\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 ≤ b ^ (2 * p)\nhi_bpowpsubone : 1 ≤ b ^ (p - 1)\np_... | refine ⟨?_, AB_not_prime, one_lt_mul'' hi_A hi_B⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Fermat | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 9
} | {
"line": 76,
"column": 2
} | [
{
"pp": "n : ℕ\nthis : 0 ≤ 1 + 2 ^ (2 ^ n * 4)\n⊢ 2 * (2 ^ 2 ^ n) ^ 2 ≤ (2 ^ 2 ^ (n + 1) + 1) ^ 2",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isN... | [
"n : ℕ\nthis : 0 ≤ 1 + 2 ^ (2 ^ n * 4)\n⊢ 2 ^ (2 ^ n * 2) * 2 ≤ 1 + 2 ^ (2 ^ n * 2) * 2 + 2 ^ (2 ^ n * 4)"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Fermat | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 9
} | {
"line": 84,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ 2 ^ 2 ^ (n + 2) + 1 + 2 * 2 ^ 2 ^ (n + 1) - 2 * 2 ^ 2 ^ (n + 1) = (2 ^ 2 ^ (n + 1) + 1) ^ 2 - 2 * (2 ^ 2 ^ n) ^ 2",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOn... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Fermat | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 58
} | {
"line": 113,
"column": 0
} | [
{
"pp": "m n : ℕ\nhmn : m ≠ n\nhmn' : m < n\nd : ℕ := m.fermatNumber.gcd n.fermatNumber\nh_n : d ∣ n.fermatNumber\nh_m : d ∣ 2\nh_two : d = 2\n⊢ False",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"Nat.odd_fermatNumber",
"Nat.fermatNumber",
"Dvd.dvd",
"Eq.rec",
... | [] | exact (odd_fermatNumber _).not_two_dvd_nat (h_two ▸ h_n) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.FermatPsp | {
"line": 325,
"column": 4
} | {
"line": 325,
"column": 43
} | {
"line": 326,
"column": 4
} | [
{
"pp": "case pos\nb : ℕ\nh : 1 ≤ b\nm : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\nhp₁ : b * (b ^ 2 - 1) + 1 + m ≤ p\nhp₂ : Prime p\nh₁ : 0 < b\nh₂ : 4 ≤ b ^ 2\nh₃ : 0 < b ^ 2 - 1\nh₄ : 0 < b * (b ^ 2 - 1)\nh₅ : b * (b ^ 2 - 1) < p\nh₆ : ¬p ∣ b * (b ^ 2 - 1)\nh₇ : b ≤ b * (b ^ 2 - 1)\nh₈ : 2 ≤ b * (b ^ 2 - 1)\n⊢ ∃ n, n.Ferma... | [
"case pos\nb : ℕ\nh : 1 ≤ b\nm : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\nhp₁ : b * (b ^ 2 - 1) + 1 + m ≤ p\nhp₂ : Prime p\nh₁ : 0 < b\nh₂ : 4 ≤ b ^ 2\nh₃ : 0 < b ^ 2 - 1\nh₄ : 0 < b * (b ^ 2 - 1)\nh₅ : b * (b ^ 2 - 1) < p\nh₆ : ¬p ∣ b * (b ^ 2 - 1)\nh₇ : b ≤ b * (b ^ 2 - 1)\nh₈ : 2 ≤ b * (b ^ 2 - 1)\nh₉ : 2 < p\n⊢ ∃ n, n.Ferma... | have h₉ : 2 < p := lt_of_le_of_lt h₈ h₅ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Harmonic.Int | {
"line": 55,
"column": 41
} | {
"line": 55,
"column": 53
} | {
"line": 55,
"column": 53
} | [
{
"pp": "n : ℕ\nh : 2 ≤ n\n⊢ 1 < ↑2 ^ ↑(Nat.log 2 n)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Rat.instOfNat",
"Eq.mpr",
"congrArg",
"Rat",
"DivInvMonoid.toZPow",
"DivisionRing.toDivInvMonoid",
"id",
"DivInvMonoid... | [
"n : ℕ\nh : 2 ≤ n\n⊢ 1 < ↑2 ^ Nat.log 2 n"
] | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 452,
"column": 40
} | {
"line": 452,
"column": 84
} | {
"line": 453,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nS : Solution hζ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\np : 𝓞 K\nhp : Prime p\nhpaηb : p ∣ 1 * S.a + ↑η * S.b\nhpaηsqb : p ∣ 1 * S.a + ↑η ^ 2 * S.b\nthis : p ∣ ↑η ^ 2 - ↑η\n⊢ (↑η ^ 2 - ↑η) * ↑η * ↑η = (-↑η - 1 - ↑... | [] | by rw [eta_sq, mul_assoc, ← pow_two, eta_sq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Height.Basic | {
"line": 372,
"column": 2
} | {
"line": 387,
"column": 65
} | {
"line": 389,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Finite ι\ninst✝ : Finite ι'\nf : ι → ι'\nx : ι' → K\n⊢ mulHeight (x ∘ f) ≤ mulHeight x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mp... | [] | rcases eq_or_ne (x ∘ f) 0 with h₀ | h₀
· simpa [h₀] using one_le_mulHeight _
rcases eq_or_ne x 0 with rfl | hx
· simp
have : Nonempty ι := .intro (ne_iff.mp h₀).choose
rw [mulHeight_eq h₀, mulHeight_eq hx]
have H (v : AbsoluteValue K ℝ) : ⨆ i, v ((x ∘ f) i) ≤ ⨆ i, v (x i) :=
ciSup_le fun i ↦ Finite.le_c... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Height.Basic | {
"line": 372,
"column": 2
} | {
"line": 387,
"column": 65
} | {
"line": 389,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Finite ι\ninst✝ : Finite ι'\nf : ι → ι'\nx : ι' → K\n⊢ mulHeight (x ∘ f) ≤ mulHeight x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mp... | [] | rcases eq_or_ne (x ∘ f) 0 with h₀ | h₀
· simpa [h₀] using one_le_mulHeight _
rcases eq_or_ne x 0 with rfl | hx
· simp
have : Nonempty ι := .intro (ne_iff.mp h₀).choose
rw [mulHeight_eq h₀, mulHeight_eq hx]
have H (v : AbsoluteValue K ℝ) : ⨆ i, v ((x ∘ f) i) ≤ ⨆ i, v (x i) :=
ciSup_le fun i ↦ Finite.le_c... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 87,
"column": 67
} | {
"line": 87,
"column": 91
} | {
"line": 87,
"column": 91
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝² : Field K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : Finite ι'\nv : AbsoluteValue K ℝ\nA : ι' × ι → K\nx : ι → K\nh✝ : Nonempty ι'\nj : ι'\n⊢ ↑(Fintype.card ι) * ((⨆ ji, v (A ji)) * ⨆ i, v (x i)) ≤ ↑(Nat.card ι) * ((⨆ ji, v (A ji)) * ⨆ i, v (x i))",
"pp... | [
"case inr\nK : Type u_1\ninst✝² : Field K\nι : Type u_2\nι' : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : Finite ι'\nv : AbsoluteValue K ℝ\nA : ι' × ι → K\nx : ι → K\nh✝ : Nonempty ι'\nj : ι'\n⊢ ↑(Fintype.card ι) * ((⨆ ji, v (A ji)) * ⨆ i, v (x i)) ≤ ↑(Fintype.card ι) * ((⨆ ji, v (A ji)) * ⨆ i, v (x i))"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 11
} | {
"line": 142,
"column": 0
} | [
{
"pp": "case succ\nN : ℕ\nhN : ∑ n ∈ Finset.range N, term (n + 1) 1 = log (↑N + 1) - ↑(harmonic (N + 1)) + 1\n⊢ log (↑N + 1) - ↑(harmonic (N + 1)) + 1 + (log (↑N + 1 + 1) - log (↑N + 1) - 1 / (↑N + 1 + 1)) =\n log (↑N + 1 + 1) - (↑(harmonic (N + 1)) + (↑N + 1 + 1)⁻¹) + 1",
"ppTerm": "?succ",
"assign... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 207,
"column": 6
} | {
"line": 207,
"column": 13
} | {
"line": 208,
"column": 2
} | [
{
"pp": "case e_a.succ\ns : ℝ\nhs : 1 < s\nN : ℕ\nhN :\n ∑ x ∈ Finset.range N, 1 / (s - 1) * (1 / ↑(x + 1) ^ (s - 1) - 1 / (↑(x + 1) + 1) ^ (s - 1)) =\n 1 / (s - 1) * (1 - 1 / (↑N + 1) ^ (s - 1))\n⊢ 1 / (s - 1) * (1 - 1 / (↑N + 1) ^ (s - 1)) + 1 / (s - 1) * (1 / (↑N + 1) ^ (s - 1) - 1 / (↑N + 1 + 1) ^ (s - ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 215,
"column": 6
} | {
"line": 215,
"column": 13
} | {
"line": 217,
"column": 0
} | [
{
"pp": "case e_a.e_a.succ.e_a\ns : ℝ\nhs : 1 < s\nN : ℕ\nhN :\n ∑ i ∈ Finset.range N, (1 / ↑(i + 1) ^ s - 1 / (↑(i + 1) + 1) ^ s) * ↑(i + 1) =\n ∑ n ∈ Finset.range N, 1 / (↑n + 1) ^ s - ↑N / (↑N + 1) ^ s\n⊢ -(↑N / (↑N + 1) ^ s) + (1 / (↑N + 1) ^ s + -(1 / (↑N + (1 + 1)) ^ s)) * (↑N + 1) =\n 1 / (↑N + 1)... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 366,
"column": 12
} | {
"line": 366,
"column": 19
} | {
"line": 366,
"column": 20
} | [
{
"pp": "f : ℂ → ℂ := fun s ↦ riemannZeta s - 1 / (s - 1)\n⊢ 𝓝 0 = 𝓝 (1 - 1 - f 1 * (1 - 1))",
"ppTerm": "?m.352",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"NonAs... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 366,
"column": 12
} | {
"line": 366,
"column": 19
} | {
"line": 366,
"column": 20
} | [
{
"pp": "f : ℂ → ℂ := fun s ↦ riemannZeta s - 1 / (s - 1)\n⊢ 𝓝 0 = 𝓝 (1 - 1 - f 1 * (1 - 1))",
"ppTerm": "?m.352",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"NonAs... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 366,
"column": 12
} | {
"line": 366,
"column": 19
} | {
"line": 366,
"column": 20
} | [
{
"pp": "f : ℂ → ℂ := fun s ↦ riemannZeta s - 1 / (s - 1)\n⊢ 𝓝 0 = 𝓝 (1 - 1 - f 1 * (1 - 1))",
"ppTerm": "?m.352",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"NonAs... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 9
} | {
"line": 409,
"column": 0
} | [
{
"pp": "this : Tendsto (fun x ↦ riemannZeta x - 1 / x.Gammaℝ / (x - 1)) (𝓝[≠] 1) (𝓝 (↑γ + -(↑γ + Complex.log (4 * ↑π)) / 2))\n⊢ (↑γ - Complex.log (4 * ↑π)) / 2 = ↑γ + -(↑γ + Complex.log (4 * ↑π)) / 2",
"ppTerm": "?m.184",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 268,
"column": 16
} | {
"line": 268,
"column": 20
} | {
"line": 268,
"column": 21
} | [
{
"pp": "case inr.inr\nF : Type u_1\nR : Type u_2\ninst✝⁴ : Field F\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Finite F\nn : ℕ\ninst✝ : NeZero n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhμ : IsPrimitiveRoot μ n\nx : F\nhx₀ : x ≠ 0\nhx₁ : x ≠ 1\nz₁ : R\nhz₁ : z₁ ∈ ℤ[μ]\nHz₁ : χ x - 1 =... | [
"case inr.inr\nF : Type u_1\nR : Type u_2\ninst✝⁴ : Field F\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : Finite F\nn : ℕ\ninst✝ : NeZero n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhμ : IsPrimitiveRoot μ n\nx : F\nhx₀ : x ≠ 0\nhx₁ : x ≠ 1\nz₁ : R\nhz₁ : z₁ ∈ ℤ[μ]\nHz₁ : χ x - 1 = z₁ * (μ - 1... | Hz₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 325,
"column": 6
} | {
"line": 325,
"column": 88
} | {
"line": 326,
"column": 6
} | [
{
"pp": "case b0\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nn : ℕ\nhn : n ≠ 0\nB : ℝ\nx : 𝓞 K\nh : ∏ v, (⨆ i, v (![↑x, ↑n] i)) ^ v.mult ≤ B\nv : InfinitePlace K\n⊢ 0 ≤ ∏ v' ∈ univ.erase v, (⨆ i, v' (![↑x, ↑n] i)) ^ v'.mult",
"ppTerm": "?b0",
"assigned": true,
"usedConstants": [
"... | [
"case hab\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nn : ℕ\nhn : n ≠ 0\nB : ℝ\nx : 𝓞 K\nh : ∏ v, (⨆ i, v (![↑x, ↑n] i)) ^ v.mult ≤ B\nv w : InfinitePlace K\na✝ : w ∈ univ.erase v\n⊢ ↑n ≤ ⨆ i, w (![↑x, ↑n] i)",
"case hab\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nn : ℕ\nhn : n ≠ 0\nB : ℝ... | · exact prod_nonneg fun _ _ ↦ pow_nonneg (Real.iSup_nonneg_of_nonnegHomClass ..) _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Height.NumberField | {
"line": 452,
"column": 2
} | {
"line": 461,
"column": 71
} | {
"line": 463,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nx : ι → ℤ\nv : FinitePlace ℚ\nhx : Finset.univ.gcd x = 1\n⊢ ⨆ i, v ↑(x i) = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Real.instIsOrderedRing",
"Int.cast",
"Real.partialOrder... | [] | have hv : IsNonarchimedean (v ·) := FinitePlace.add_le v
have H (n : ℤ) : v n ≤ 1 := IsNonarchimedean.apply_intCast_le_one hv
obtain ⟨f, hf⟩ := Finset.gcd_eq_sum_mul .univ x
apply_fun v at hf
simp_rw [hx, Int.cast_one, map_one, Int.cast_sum, Int.cast_mul] at hf
replace hf := hf.trans_le hv.apply_sum_univ_le
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Height.NumberField | {
"line": 452,
"column": 2
} | {
"line": 461,
"column": 71
} | {
"line": 463,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nx : ι → ℤ\nv : FinitePlace ℚ\nhx : Finset.univ.gcd x = 1\n⊢ ⨆ i, v ↑(x i) = 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Real.instIsOrderedRing",
"Int.cast",
"Real.partialOrder... | [] | have hv : IsNonarchimedean (v ·) := FinitePlace.add_le v
have H (n : ℤ) : v n ≤ 1 := IsNonarchimedean.apply_intCast_le_one hv
obtain ⟨f, hf⟩ := Finset.gcd_eq_sum_mul .univ x
apply_fun v at hf
simp_rw [hx, Int.cast_one, map_one, Int.cast_sum, Int.cast_mul] at hf
replace hf := hf.trans_le hv.apply_sum_univ_le
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Height.NumberField | {
"line": 471,
"column": 12
} | {
"line": 471,
"column": 48
} | {
"line": 471,
"column": 48
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nx : ι → ℤ\nhx : x = 0\n⊢ Finset.univ.gcd 0 ≠ 1",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Finset.univ",
"congrArg",
"Int.instStrongNormalizedGCDMonoid",
"Finset",
... | [
"ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nx : ι → ℤ\nhx : x = 0\n⊢ 0 ≠ 1"
] | Finset.gcd_eq_zero_iff.mpr (by simp) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 219,
"column": 39
} | {
"line": 219,
"column": 46
} | {
"line": 219,
"column": 46
} | [
{
"pp": "case e_f.e_f\nN : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\ns : ℂ\nhs : Φ 0 = 0 ∨ s ≠ 1\nthis : ∀ (j : ZMod N), Φ j * LFunction (fun k ↦ 𝕖 (-j * k)) s = Φ j * expZeta (toAddCircle (-j)) s\nj k : ZMod N\n⊢ ↑N ^ (-s) * ↑(toCircle (-(k * j))) * Φ k * hurwitzZeta (toAddCircle j) s =\n ↑N ^ (-s) * ↑(toCircle... | [
"case e_f.e_f\nN : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\ns : ℂ\nhs : Φ 0 = 0 ∨ s ≠ 1\nthis : ∀ (j : ZMod N), Φ j * LFunction (fun k ↦ 𝕖 (-j * k)) s = Φ j * expZeta (toAddCircle (-j)) s\nj k : ZMod N\n⊢ ↑N ^ (-s) * ↑(toCircle (-(k * j))) * Φ k * hurwitzZeta (toAddCircle j) s =\n ↑N ^ (-s) * ↑(toCircle (-(k * j)))... | neg_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 37
} | {
"line": 232,
"column": 4
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : Φ 0 = 0 ∨ s ≠ 1\nj : ZMod N\n⊢ Φ j * hurwitzZeta (toAddCircle j) (1 - s) =\n Φ j *\n ((2 * ↑π) ^ (-s) * Gamma s *\n (cexp (-↑π * I * s / 2) * expZeta (toAddCircle j) s + cexp (↑π * I * s / 2) * expZeta (-toAdd... | [
"case inl\nN : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : Φ 0 = 0 ∨ s ≠ 1\n⊢ Φ 0 * hurwitzZeta (toAddCircle 0) (1 - s) =\n Φ 0 *\n ((2 * ↑π) ^ (-s) * Gamma s *\n (cexp (-↑π * I * s / 2) * expZeta (toAddCircle 0) s + cexp (↑π * I * s / 2) * expZeta (-toAddCircle 0) s))"... | rcases eq_or_ne j 0 with rfl | hj | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 13
} | {
"line": 62,
"column": 4
} | [
{
"pp": "case refine_1\nk : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\nn : ℕ\n⊢ Real.cos (2 * π * x * ↑n) / ↑(n ^ (2 * k)) = 1 / ↑(n ^ (2 * k)) * Real.cos (2 * π * ↑n * x)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 87,
"column": 6
} | {
"line": 87,
"column": 13
} | {
"line": 88,
"column": 4
} | [
{
"pp": "case refine_1\nk : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\nn : ℕ\n⊢ Real.sin (2 * π * x * ↑n) / ↑(n ^ (2 * k + 1)) = 1 / ↑(n ^ (2 * k + 1)) * Real.sin (2 * π * ↑n * x)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 325,
"column": 2
} | {
"line": 327,
"column": 57
} | {
"line": 329,
"column": 0
} | [
{
"pp": "Φ : ZMod 1 → ℂ\ns : ℂ\n⊢ completedLFunction Φ s = Φ 1 * completedRiemannZeta s",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ZMod.completedLFunction",
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Real",
"AddMonoidHom.instAddMonoidHomClass... | [] | rw [completedLFunction_def_even (show Φ.Even from fun _ ↦ congr_arg Φ (Subsingleton.elim ..)),
Nat.cast_one, one_cpow, one_mul, ← singleton_eq_univ 0, sum_singleton, map_zero,
completedHurwitzZetaEven_zero, Subsingleton.elim 0 1] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 325,
"column": 2
} | {
"line": 327,
"column": 57
} | {
"line": 329,
"column": 0
} | [
{
"pp": "Φ : ZMod 1 → ℂ\ns : ℂ\n⊢ completedLFunction Φ s = Φ 1 * completedRiemannZeta s",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ZMod.completedLFunction",
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Real",
"AddMonoidHom.instAddMonoidHomClass... | [] | rw [completedLFunction_def_even (show Φ.Even from fun _ ↦ congr_arg Φ (Subsingleton.elim ..)),
Nat.cast_one, one_cpow, one_mul, ← singleton_eq_univ 0, sum_singleton, map_zero,
completedHurwitzZetaEven_zero, Subsingleton.elim 0 1] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 325,
"column": 2
} | {
"line": 327,
"column": 57
} | {
"line": 329,
"column": 0
} | [
{
"pp": "Φ : ZMod 1 → ℂ\ns : ℂ\n⊢ completedLFunction Φ s = Φ 1 * completedRiemannZeta s",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ZMod.completedLFunction",
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Real",
"AddMonoidHom.instAddMonoidHomClass... | [] | rw [completedLFunction_def_even (show Φ.Even from fun _ ↦ congr_arg Φ (Subsingleton.elim ..)),
Nat.cast_one, one_cpow, one_mul, ← singleton_eq_univ 0, sum_singleton, map_zero,
completedHurwitzZetaEven_zero, Subsingleton.elim 0 1] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ZetaValues | {
"line": 375,
"column": 8
} | {
"line": 375,
"column": 18
} | {
"line": 375,
"column": 18
} | [
{
"pp": "case e'_3\nk : ℕ\nhk : k ≠ 0\nx : ℝ\nhx : x ∈ Icc 0 1\nthis :\n HasSum (fun n ↦ 1 / ↑n ^ (2 * k) * ((fourier ↑n) ↑x + (fourier (-↑n)) ↑x))\n ((-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / ↑(2 * k)! * ↑(bernoulliFun (2 * k) x))\nofReal_two : ↑2 = 2\nn : ℕ\n⊢ 1 / ↑n ^ (2 * k) * ((fourier ↑n) ↑x + (fourier (-... | [
"case e'_3\nk : ℕ\nhk : k ≠ 0\nx : ℝ\nhx : x ∈ Icc 0 1\nthis :\n HasSum (fun n ↦ 1 / ↑n ^ (2 * k) * ((fourier ↑n) ↑x + (fourier (-↑n)) ↑x))\n ((-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / ↑(2 * k)! * ↑(bernoulliFun (2 * k) x))\nofReal_two : ↑2 = 2\nn : ℕ\n⊢ 1 / ↑n ^ (2 * k) * ((fourier ↑n) ↑x + (fourier (-↑n)) ↑x) / 2... | ofReal_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 413,
"column": 8
} | {
"line": 413,
"column": 18
} | {
"line": 413,
"column": 18
} | [
{
"pp": "case e'_3\nk : ℕ\nhk : k ≠ 0\nx : ℝ\nhx : x ∈ Icc 0 1\nthis :\n HasSum (fun n ↦ 1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x - (fourier (-↑n)) ↑x))\n ((-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x))\nofReal_two : ↑2 = 2\nx✝ : ℕ\n⊢ 1 / ↑x✝ ^ (2 * k + 1) * ((fo... | [
"case e'_3\nk : ℕ\nhk : k ≠ 0\nx : ℝ\nhx : x ∈ Icc 0 1\nthis :\n HasSum (fun n ↦ 1 / ↑n ^ (2 * k + 1) * ((fourier ↑n) ↑x - (fourier (-↑n)) ↑x))\n ((-1) ^ (k + 1) * I * (2 * ↑π) ^ (2 * k + 1) / ↑(2 * k + 1)! * ↑(bernoulliFun (2 * k + 1) x))\nofReal_two : ↑2 = 2\nx✝ : ℕ\n⊢ 1 / ↑x✝ ^ (2 * k + 1) * ((fourier ↑x✝) ↑... | ofReal_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 254,
"column": 18
} | {
"line": 254,
"column": 32
} | {
"line": 254,
"column": 33
} | [
{
"pp": "k : ℕ\n⊢ -↑(bernoulli' (k + 1)) / (↑k + 1) = (-1) ^ k * ((-1) ^ (k + 1) * ↑(bernoulli' (k + 1))) / (↑k + 1)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAs... | [
"k : ℕ\n⊢ -1 * ↑(bernoulli' (k + 1)) / (↑k + 1) = (-1) ^ k * ((-1) ^ (k + 1) * ↑(bernoulli' (k + 1))) / (↑k + 1)"
] | ← neg_one_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 471,
"column": 4
} | {
"line": 471,
"column": 11
} | {
"line": 472,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ 1 / ↑n ^ (2 * 1 + 1) * Real.sin (2 * π * ↑n * (1 / 4)) = 1 / ↑n ^ 3 * Real.sin (π * ↑n / 2)",
"ppTerm": "?m.114",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.RingNF.nnrat_rawCast",
"Eq.mpr",
"NonAssocSemirin... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 360,
"column": 2
} | {
"line": 369,
"column": 95
} | {
"line": 371,
"column": 0
} | [
{
"pp": "q : ℕ\na : ZMod q\ninst✝ : NeZero q\nha : IsUnit a\ns : ℂ\nhs : 1 < s.re\nha' : IsUnit a⁻¹\n⊢ deriv (LFunctionTrivChar₁ q) s / LFunctionTrivChar₁ q s + ∑ χ ∈ {1}ᶜ, χ a⁻¹ * deriv (LFunction χ) s / LFunction χ s =\n ∑ χ, χ a⁻¹ * (deriv (LFunction χ) s / LFunction χ s) + 1 / (s - 1)",
"ppTerm": "?m... | [] | classical -- for `Fintype.sum_eq_add_sum_compl`
rw [Fintype.sum_eq_add_sum_compl 1, MulChar.one_apply ha', one_mul, add_right_comm]
simp only [mul_div_assoc]
congrm (?_ + _)
have hs₁ : s ≠ 1 := fun h ↦ ((h ▸ hs).trans_eq one_re).false
rw [deriv_LFunctionTrivChar₁_apply_of_ne_one _ hs₁, LFunctionTrivChar₁,
... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 13
} | {
"line": 120,
"column": 4
} | [
{
"pp": "case e'_5\nf : ℕ → ℂ\nhf : f 0 = 0\nr : ℝ\nhr : 0 ≤ r\ns : ℂ\nhs : r < s.re\nhS : LSeriesSummable f s\nh₁ : (-s - 1).re + r < -1\nh₂ : s ≠ 0\nh₃ : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ↑x ^ (-s)) t\nh₄ : ∀ (n : ℕ), ∑ k ∈ Icc 0 n, f k = ∑ k ∈ Icc 1 n, f k\nhO : (fun n ↦ ∑ k ∈ Icc 0 n, f k) =O[atT... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 13
} | {
"line": 120,
"column": 4
} | [
{
"pp": "case e'_5\nf : ℕ → ℂ\nhf : f 0 = 0\nr : ℝ\nhr : 0 ≤ r\ns : ℂ\nhs : r < s.re\nhS : LSeriesSummable f s\nh₁ : (-s - 1).re + r < -1\nh₂ : s ≠ 0\nh₃ : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ↑x ^ (-s)) t\nh₄ : ∀ (n : ℕ), ∑ k ∈ Icc 0 n, f k = ∑ k ∈ Icc 1 n, f k\nhO : (fun n ↦ ∑ k ∈ Icc 0 n, f k) =O[atT... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 13
} | {
"line": 120,
"column": 4
} | [
{
"pp": "case e'_5\nf : ℕ → ℂ\nhf : f 0 = 0\nr : ℝ\nhr : 0 ≤ r\ns : ℂ\nhs : r < s.re\nhS : LSeriesSummable f s\nh₁ : (-s - 1).re + r < -1\nh₂ : s ≠ 0\nh₃ : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ↑x ^ (-s)) t\nh₄ : ∀ (n : ℕ), ∑ k ∈ Icc 0 n, f k = ∑ k ∈ Icc 1 n, f k\nhO : (fun n ↦ ∑ k ∈ Icc 0 n, f k) =O[atT... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 236,
"column": 8
} | {
"line": 236,
"column": 74
} | {
"line": 237,
"column": 6
} | [
{
"pp": "case refine_1\nl : ℂ\ns T ε : ℝ\nS : ℝ → ℂ\nhS : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume\nhε : 0 < ε\nhs : 1 < s\nhT₁ : 1 ≤ T\nhT : ∀ t ≥ T, ‖S t - l * ↑t‖ ≤ ε * t\nhT₀ : 0 < T\nh : ∀ {t : ℝ}, 0 < t → t ^ (-s) = t * t ^ (-s - 1)\n⊢ IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - ... | [] | exact (lemma₂ hs hS hT).mono_set <| Set.Ioi_subset_Ici_iff.mpr hT₁ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 236,
"column": 8
} | {
"line": 236,
"column": 74
} | {
"line": 237,
"column": 6
} | [
{
"pp": "case refine_1\nl : ℂ\ns T ε : ℝ\nS : ℝ → ℂ\nhS : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume\nhε : 0 < ε\nhs : 1 < s\nhT₁ : 1 ≤ T\nhT : ∀ t ≥ T, ‖S t - l * ↑t‖ ≤ ε * t\nhT₀ : 0 < T\nh : ∀ {t : ℝ}, 0 < t → t ^ (-s) = t * t ^ (-s - 1)\n⊢ IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - ... | [] | exact (lemma₂ hs hS hT).mono_set <| Set.Ioi_subset_Ici_iff.mpr hT₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 236,
"column": 8
} | {
"line": 236,
"column": 74
} | {
"line": 237,
"column": 6
} | [
{
"pp": "case refine_1\nl : ℂ\ns T ε : ℝ\nS : ℝ → ℂ\nhS : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume\nhε : 0 < ε\nhs : 1 < s\nhT₁ : 1 ≤ T\nhT : ∀ t ≥ T, ‖S t - l * ↑t‖ ≤ ε * t\nhT₀ : 0 < T\nh : ∀ {t : ℝ}, 0 < t → t ^ (-s) = t * t ^ (-s - 1)\n⊢ IntegrableOn (fun t ↦ ‖S t - l * ↑t‖ * t ^ (-s - ... | [] | exact (lemma₂ hs hS hT).mono_set <| Set.Ioi_subset_Ici_iff.mpr hT₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 242,
"column": 10
} | {
"line": 242,
"column": 29
} | {
"line": 242,
"column": 30
} | [
{
"pp": "l : ℂ\ns T ε : ℝ\nS : ℝ → ℂ\nhS : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume\nhε : 0 < ε\nhs : 1 < s\nhT₁ : 1 ≤ T\nhT : ∀ t ≥ T, ‖S t - l * ↑t‖ ≤ ε * t\nhT₀ : 0 < T\nh : ∀ {t : ℝ}, 0 < t → t ^ (-s) = t * t ^ (-s - 1)\n⊢ (s - 1) * ∫ (t : ℝ) in Set.Ioi T, ε * t ^ (-s) ≤ ε * ((s - 1) * ... | [
"l : ℂ\ns T ε : ℝ\nS : ℝ → ℂ\nhS : LocallyIntegrableOn (fun t ↦ S t - l * ↑t) (Set.Ici 1) volume\nhε : 0 < ε\nhs : 1 < s\nhT₁ : 1 ≤ T\nhT : ∀ t ≥ T, ‖S t - l * ↑t‖ ≤ ε * t\nhT₀ : 0 < T\nh : ∀ {t : ℝ}, 0 < t → t ^ (-s) = t * t ^ (-s - 1)\n⊢ (s - 1) * (ε * ∫ (a : ℝ) in Set.Ioi T, a ^ (-s)) ≤ ε * ((s - 1) * ∫ (t : ℝ) ... | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 294,
"column": 6
} | {
"line": 296,
"column": 88
} | {
"line": 297,
"column": 4
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Se... | [] | rw [h₃, mul_one, mul_comm l, LSeries_eq_mul_integral _ zero_le_one (by rwa [ofReal_re])
(hfS _ hs), neg_add', mul_assoc]
exact isBigO_atTop_natCast_rpow_of_tendsto_div_rpow (a := l) (by simpa using hlim) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 294,
"column": 6
} | {
"line": 296,
"column": 88
} | {
"line": 297,
"column": 4
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nhfS : ∀ (s : ℝ), 1 < s → LSeriesSummable f ↑s\nε : ℝ\nhε : ε > 0\nT : ℝ\nhT₁ : 1 ≤ T\nhT : ∀ (y : ℝ), T ≤ y → ‖∑ k ∈ Icc 1 ⌊y⌋₊, f k - l * ↑y‖ < ε * y\nS : ℝ → ℂ := fun t ↦ ∑ k ∈ Icc 1 ⌊t⌋₊, f k\nC : ℝ := ∫ (t : ℝ) in Se... | [] | rw [h₃, mul_one, mul_comm l, LSeries_eq_mul_integral _ zero_le_one (by rwa [ofReal_re])
(hfS _ hs), neg_add', mul_assoc]
exact isBigO_atTop_natCast_rpow_of_tendsto_div_rpow (a := l) (by simpa using hlim) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 185,
"column": 2
} | {
"line": 186,
"column": 56
} | {
"line": 189,
"column": 2
} | [
{
"pp": "K : Type u_1\nΓ₀ : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Valued K Γ₀\nhc : IsCompact ↑𝒪[K]\n⊢ Nonempty (LocallyFiniteOrder (↥(MonoidHom.mrange v))ˣ)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
... | [
"K : Type u_1\nΓ₀ : Type u_2\ninst✝² : Field K\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : Valued K Γ₀\nhc : IsCompact ↑𝒪[K]\n⊢ Nonempty (LocallyFiniteOrder (↥(MonoidHom.mrange (MonoidWithZeroHom.ofClass v)))ˣ)"
] | change Nonempty (LocallyFiniteOrder (MonoidHom.mrange
(MonoidWithZeroHom.ofClass (Valued.v (R := K))))ˣ) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.NumberTheory.LocalField.Basic | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 84
} | {
"line": 78,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : ValuativeRel K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsNonarchimedeanLocalField K\nγ : K\nhγ : ¬γ = 0\nthis✝ : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K\nthis : IsUniformAddGroup K := isUniformAddGroup_of_addCommGroup\ns : Set K\nhs : s ∈ nhds ... | [
"K : Type u_1\ninst✝³ : Field K\ninst✝² : ValuativeRel K\ninst✝¹ : TopologicalSpace K\ninst✝ : IsNonarchimedeanLocalField K\nγ : K\nhγ : ¬γ = 0\nthis✝ : UniformSpace K := IsTopologicalAddGroup.rightUniformSpace K\nthis : IsUniformAddGroup K := isUniformAddGroup_of_addCommGroup\ns : Set K\nhs : s ∈ nhds 0\nhs' : IsC... | obtain ⟨r, hr, hrs⟩ := (IsValuativeTopology.hasBasis_nhds_zero' K).mem_iff.mp hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 94
} | {
"line": 114,
"column": 2
} | [
{
"pp": "n✝ : Type u_1\ninst✝¹ : Fintype n✝\ninst✝ : DecidableEq n✝\nΓ : Subgroup (GL (Fin 2) ℝ)\nh : Γ.IsArithmetic\ng : GL (Fin 2) ℝ\nhg : g ∈ Γ\nn : ℕ\nhn : 0 < n\nleft✝ : n ≤ (mapGL ℝ).range.relIndex Γ\nhgn : g ^ n ∈ (mapGL ℝ).range ⊓ Γ\n⊢ |↑(GeneralLinearGroup.det g)| = 1",
"ppTerm": "?m.57",
"assi... | [
"n✝ : Type u_1\ninst✝¹ : Fintype n✝\ninst✝ : DecidableEq n✝\nΓ : Subgroup (GL (Fin 2) ℝ)\nh : Γ.IsArithmetic\ng : GL (Fin 2) ℝ\nhg : g ∈ Γ\nn : ℕ\nhn : 0 < n\nleft✝ : n ≤ (mapGL ℝ).range.relIndex Γ\nhgn : g ^ n ∈ (mapGL ℝ).range ⊓ Γ\n⊢ |↑(GeneralLinearGroup.det g ^ n)| = 1"
] | suffices |(g.det ^ n).val| = 1 by simpa [← abs_pow, abs_pow_eq_one _ (Nat.ne_zero_of_lt hn)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Topology.Algebra.Order.ArchimedeanDiscrete | {
"line": 46,
"column": 4
} | {
"line": 46,
"column": 44
} | {
"line": 47,
"column": 4
} | [
{
"pp": "case inr.mp\nG✝ : Type u_1\ninst✝⁶ : CommGroup G✝\ninst✝⁵ : TopologicalSpace G✝\nG : Type u_1\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : TopologicalSpace G\ninst✝ : OrderTopology G\ng : G\nha✝ : 1 ≤ g\nha : 1 < g\nn : ℤ\nhn : g ^ (-1) < g ^ n\nhn' : g ^ n < g ^ ... | [
"case inr.mp\nG✝ : Type u_1\ninst✝⁶ : CommGroup G✝\ninst✝⁵ : TopologicalSpace G✝\nG : Type u_1\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : TopologicalSpace G\ninst✝ : OrderTopology G\ng : G\nha✝ : 1 ≤ g\nha : 1 < g\nn : ℤ\nhn : -1 < n\nhn' : n < 1\n⊢ ⟨g ^ n, ⋯⟩ = 1"
] | rw [zpow_lt_zpow_iff_right ha] at hn hn' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.CongruenceSubgroups | {
"line": 218,
"column": 67
} | {
"line": 222,
"column": 10
} | {
"line": 224,
"column": 0
} | [
{
"pp": "g : ConjAct SL(2, ℤ)\nΓ : Subgroup SL(2, ℤ)\nh : IsCongruenceSubgroup Γ\n⊢ IsCongruenceSubgroup (g • Γ)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Subgroup.pointwise_smul_le_pointwise_smul_iff",
"Eq.mpr",
"instHSMul",
"Matrix.SpecialLinearGroup",
... | [] | by
obtain ⟨N, HN⟩ := h
refine ⟨N, ?_⟩
rw [← Gamma_cong_eq_self N g, Subgroup.pointwise_smul_le_pointwise_smul_iff]
exact HN | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.SlashActions | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 11
} | {
"line": 208,
"column": 2
} | [
{
"pp": "k1 k2 : ℤ\nA : GL (Fin 2) ℝ\nf g : ℍ → ℂ\nx : ℍ\nd : ℂ := ↑|↑(Matrix.GeneralLinearGroup.det A)|\nthis : d ≠ 0\n⊢ d ^ (k1 + k2 - 1) = d ^ (1 + (k1 - 1) + (k2 - 1))",
"ppTerm": "?m.155",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"N... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 59
} | {
"line": 286,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\n𝒢 : Subgroup (GL (Fin 2) R)\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalRing R\nhG : DiscreteTopology ↥𝒢\n⊢ DiscreteTopology ↥𝒢.strictPeriods",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Matrix",
"instDecidableEqFin",
... | [
"R : Type u_1\ninst✝² : Ring R\n𝒢 : Subgroup (GL (Fin 2) R)\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalRing R\nhG : DiscreteTopology ↥𝒢\nH : Set (GL (Fin 2) R) := ↑𝒢 ∩ Set.range ⇑upperRightHom\n⊢ DiscreteTopology ↥𝒢.strictPeriods"
] | let H : Set (GL (Fin 2) R) := 𝒢 ∩ Set.range upperRightHom | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.ModularForms.BoundedAtCusp | {
"line": 37,
"column": 43
} | {
"line": 37,
"column": 73
} | {
"line": 37,
"column": 73
} | [
{
"pp": "g : GL (Fin 2) ℝ\nf : ℍ → ℂ\nk : ℤ\nhg : ↑g 1 0 = 0\nhf : f =O[atImInfty] 1\n⊢ (f ∣[k] g) =O[atImInfty] 1",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"Matrix",... | [
"g : GL (Fin 2) ℝ\nf : ℍ → ℂ\nk : ℤ\nhg : ↑g 1 0 = 0\nhf : (fun x ↦ ‖f x‖) =O[atImInfty] 1\n⊢ (fun x ↦ ‖(f ∣[k] g) x‖) =O[atImInfty] 1"
] | ← Asymptotics.isBigO_norm_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 381,
"column": 2
} | {
"line": 384,
"column": 35
} | {
"line": 385,
"column": 2
} | [
{
"pp": "case mp\n𝒢 : Subgroup (GL (Fin 2) ℝ)\ninst✝¹ : DiscreteTopology ↥𝒢.strictPeriods\ninst✝ : 𝒢.HasDetPlusMinusOne\n⊢ 0 < 𝒢.strictWidthInfty → IsCusp ∞ 𝒢",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"OnePoint.instGLAction",
"Iff.mpr",
"Units.val",
"Eq.mpr... | [
"case mpr\n𝒢 : Subgroup (GL (Fin 2) ℝ)\ninst✝¹ : DiscreteTopology ↥𝒢.strictPeriods\ninst✝ : 𝒢.HasDetPlusMinusOne\n⊢ IsCusp ∞ 𝒢 → 0 < 𝒢.strictWidthInfty"
] | · refine fun h ↦ ⟨_, mem_strictPeriods_iff.mpr 𝒢.strictWidthInfty_mem_strictPeriods, ?_, ?_⟩
· rw [GeneralLinearGroup.isParabolic_iff_of_upperTriangular (by simp)]
simpa using h.ne'
· simp [smul_infty_eq_self_iff] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.Petersson | {
"line": 108,
"column": 51
} | {
"line": 108,
"column": 73
} | {
"line": 108,
"column": 74
} | [
{
"pp": "case right\nF : Type u_1\nF' : Type u_2\ninst✝⁶ : FunLike F ℍ ℂ\ninst✝⁵ : FunLike F' ℍ ℂ\nk : ℤ\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝⁴ : Fact (IsCusp OnePoint.infty Γ)\ninst✝³ : Γ.HasDetPlusMinusOne\ninst✝² : DiscreteTopology ↥Γ\ninst✝¹ : ModularFormClass F Γ k\ninst✝ : ModularFormClass F' Γ k\nf : F\nh_... | [
"case right\nF : Type u_1\nF' : Type u_2\ninst✝⁶ : FunLike F ℍ ℂ\ninst✝⁵ : FunLike F' ℍ ℂ\nk : ℤ\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝⁴ : Fact (IsCusp OnePoint.infty Γ)\ninst✝³ : Γ.HasDetPlusMinusOne\ninst✝² : DiscreteTopology ↥Γ\ninst✝¹ : ModularFormClass F Γ k\ninst✝ : ModularFormClass F' Γ k\nf : F\nh_bd : IsZeroA... | mul_comm ‖f _‖ ‖f' _‖, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 49
} | {
"line": 189,
"column": 0
} | [
{
"pp": "case e'_5\nh : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nq : ℂ\nhq : ‖q‖ < 1\nm : ℕ\n⊢ (PowerSeries.coeff m) (qExpansion h f) • q ^ m = (↑m.factorial)⁻¹ • (q - 0) ^ m • iteratedDeriv m (cuspFunction h f) 0",
"ppTerm": "?e'_5",
... | [] | grind [qExpansion_coeff, sub_zero, smul_eq_mul] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.NumberTheory.Modular | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 11
} | {
"line": 245,
"column": 2
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhp : IsCoprime (↑g 1 0) (↑g 1 1)\nnonZ1 : ↑(↑g 1 0) ^ 2 + ↑(↑g 1 1) ^ 2 ≠ 0\nthis : Int.cast ∘ ↑g 1 ≠ 0\nnonZ2 : ↑z * ↑(↑g 1 0) + ↑(↑g 1 1) ≠ 0\nH : ↑(↑g).det = ↑(↑g 0 0 * ↑g 1 1 - ↑g 0 1 * ↑g 1 0)\n⊢ (↑((algebraMap ℤ ℝ) (↑g 0 0)) * ↑z + ↑((algebraMap ℤ ℝ) (↑g 0 1))) /\n (↑((a... | [
"g : SL(2, ℤ)\nz : ℍ\nhp : IsCoprime (↑g 1 0) (↑g 1 1)\nnonZ1 : ↑(↑g 1 0) ^ 2 + ↑(↑g 1 1) ^ 2 ≠ 0\nthis : Int.cast ∘ ↑g 1 ≠ 0\nnonZ2 : ↑z * ↑(↑g 1 0) + ↑(↑g 1 1) ≠ 0\nH : 1 = ↑(↑g 0 0) * ↑(↑g 1 1) - ↑(↑g 0 1) * ↑(↑g 1 0)\n⊢ (↑((algebraMap ℤ ℝ) (↑g 0 0)) * ↑z + ↑((algebraMap ℤ ℝ) (↑g 0 1))) /\n (↑((algebraMap ℤ... | simp at H | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ModularForms.LevelOne.Basic | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 62
} | {
"line": 94,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nk : ℤ\ninst✝ : ModularFormClass F (Matrix.SpecialLinearGroup.mapGL ℝ).range k\nhk : k ≤ 0\nf : F\nz : ℍ\n⊢ 𝕢 1 ↑z ∈ Metric.ball 0 1",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"SeminormedAddGroup.... | [] | simpa using (norm_qParam_lt_iff zero_lt_one 0 z.1).mpr z.2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs | {
"line": 74,
"column": 41
} | {
"line": 78,
"column": 7
} | {
"line": 80,
"column": 0
} | [
{
"pp": "r : ℕ\ninst✝ : NeZero r\nv : Fin 2 → ℤ\nhv : finGcdMap v = r\n⊢ IsCoprime ((v / ↑r) 0) ((v / ↑r) 1)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instDiv",
"False",
"Nat.instMulZeroClass",
"instHDiv",
"eq_false",
"congrArg... | [] | by
rw [← hv]
apply isCoprime_div_gcd_div_gcd_of_gcd_ne_zero
have := NeZero.ne r
aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 11
} | {
"line": 198,
"column": 2
} | [
{
"pp": "k : ℤ\ni : Fin 2 → ℤ\nA : SL(2, ℤ)\nz : ℍ\na b c d u v : ℂ\nhc : ↑z * c + d ≠ 0\n⊢ (↑z * c + d) ^ k / (u * (a * ↑z + b) + (↑z * c + d) * v) ^ k =\n (↑z * c + d) ^ k * ((↑z * (u * a + c * v) + (u * b + d * v)) ^ k)⁻¹",
"ppTerm": "?m.207",
"assigned": true,
"usedConstants": [
"Mathli... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 41
} | {
"line": 96,
"column": 4
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t\nhf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ\nF : ℝ\nτ : ℍ\ng : SL(2, ℤ)\nhF𝒟 : ... | [
"case pos\nE : Type u_1\ninst✝ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t\nhf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ\nF : ℝ\nτ : ℍ\ng : SL(2, ℤ)\nhF𝒟 : ‖f τ‖ ≤ F * ... | refine le_trans ?_ <| le_max_left _ _ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Modular | {
"line": 444,
"column": 4
} | {
"line": 444,
"column": 31
} | {
"line": 446,
"column": 4
} | [
{
"pp": "case right\nz : ℍ\ng₀ : SL(2, ℤ)\nhg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im\ng : SL(2, ℤ)\nhg : ↑g 1 = ↑g₀ 1\nhg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|\nhg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im\n⊢ |(g • z).re| ≤ 1 / 2",
"ppTerm": "?right",
"assigned... | [
"case right\nz : ℍ\ng₀ : SL(2, ℤ)\nhg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im\ng : SL(2, ℤ)\nhg : ↑g 1 = ↑g₀ 1\nhg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|\nhg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im\n⊢ |(g • z).re| ≤ 1 / 2"
] | change |(g • z).re| ≤ 1 / 2 | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 9
} | {
"line": 400,
"column": 0
} | [
{
"pp": "k : ℤ\nF : Type u_1\ninst✝⁴ : FunLike F ℍ ℂ\nΓ : Subgroup (GL (Fin 2) ℝ)\nf : F\ninst✝³ : ModularFormClass F Γ k\ninst✝² : Γ.HasDetPlusMinusOne\ninst✝¹ : DiscreteTopology ↥Γ\ninst✝ : Fact (IsCusp OnePoint.infty Γ)\nhh : 0 < Γ.strictWidthInfty\nhΓ : Γ.strictWidthInfty ∈ Γ.strictPeriods\nτ : ℍ\n⊢ -(2 * π... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Modular | {
"line": 550,
"column": 2
} | {
"line": 550,
"column": 41
} | {
"line": 551,
"column": 2
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : T ^ ↑g 0 0 • S • z ∈ 𝒟\nhg'✝ : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 0\nhb : ↑g 0 1 = -1\nhz' : ‖↑z‖ = 1\nhg' : g = T ^ ↑g 0 0 * S\nhSre : (S • z).re = -z.re\nthis✝ : |↑(↑g 0 0) + -z.re| ≤ 1 / 2\nth... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : T ^ ↑g 0 0 • S • z ∈ 𝒟\nhg'✝ : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 0\nhb : ↑g 0 1 = -1\nhz' : ‖↑z‖ = 1\nhg' : g = T ^ ↑g 0 0 * S\nhSre : (S • z).re = -z.re\nthis✝ : |↑(↑g 0 0) + -z.re| ≤ 1 / 2\nthis : |↑(↑g 0... | rw [abs_neg, sub_le_iff_le_add] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 9
} | {
"line": 50,
"column": 0
} | [
{
"pp": "n : ℕ\nz : ℂ\n⊢ cexp ((↑n + 1) * (2 * ↑π * I * z)) = cexp (2 * ↑π * I * (↑n + 1) * z)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.RingNF.add_a... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Modular | {
"line": 574,
"column": 4
} | {
"line": 574,
"column": 11
} | {
"line": 575,
"column": 4
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 1\n⊢ ↑g =\n !![(1 * 0 + ↑g 0 0 * 1) * 1 + (1 * -1 + ↑g 0 0 * 0) * 0, (1 * 0 + ↑g 0 0 * 1) * 1 + (1 * -1 + ↑g 0 0 * 0) * 1;\n (0 * 0 + 1... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = 1\n⊢ ↑g = !![↑g 0 0, -1 + ↑g 0 0; 1, 1]"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 83
} | {
"line": 166,
"column": 2
} | [
{
"pp": "z : ℂ\nhz : z ∈ ℍₒ\nthis :\n Summable fun n ↦ ↑n ^ 1 * 𝕢 ↑1 ↑{ coe := z, coe_im_pos := hz } ^ n / (1 - 𝕢 ↑1 ↑{ coe := z, coe_im_pos := hz } ^ n)\n⊢ Summable fun i ↦ logDeriv (fun x ↦ 1 - eta_q i x) z",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"logDeriv",
"Iff.m... | [
"case e'_5\nz : ℂ\nhz : z ∈ ℍₒ\nthis :\n Summable fun n ↦ ↑n ^ 1 * 𝕢 ↑1 ↑{ coe := z, coe_im_pos := hz } ^ n / (1 - 𝕢 ↑1 ↑{ coe := z, coe_im_pos := hz } ^ n)\n⊢ (fun i ↦ logDeriv (fun x ↦ 1 - eta_q i x) z) = fun i ↦\n -2 * ↑π * I *\n (↑(i + 1) ^ 1 * 𝕢 ↑1 ↑{ coe := z, coe_im_pos := hz } ^ (i + 1) /\n ... | convert! ((summable_nat_add_iff 1).mpr this).mul_left (-2 * π * I) using 1 with n | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 28
} | {
"line": 112,
"column": 4
} | [
{
"pp": "case neg\nz : ℍ\nb n : ℤ\nh : ¬(b = 0 ∧ n = 0)\n⊢ (↑b * ↑z + ↑n + 1)⁻¹ * ((↑b * ↑z + ↑n) ^ 2)⁻¹ + δ ![b, n] + ((↑b * ↑z + ↑n)⁻¹ - (↑b * ↑z + ↑n + 1)⁻¹) =\n ((↑b * ↑z + ↑n) ^ 2)⁻¹",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mp",
"Int",
"And",
"ins... | [
"case neg\nz : ℍ\nb n : ℤ\nh : b = 0 → ¬n = 0\n⊢ (↑b * ↑z + ↑n + 1)⁻¹ * ((↑b * ↑z + ↑n) ^ 2)⁻¹ + δ ![b, n] + ((↑b * ↑z + ↑n)⁻¹ - (↑b * ↑z + ↑n + 1)⁻¹) =\n ((↑b * ↑z + ↑n) ^ 2)⁻¹"
] | simp only [not_and] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Modular | {
"line": 613,
"column": 4
} | {
"line": 613,
"column": 11
} | {
"line": 614,
"column": 4
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = -1\nthis : ↑g 0 1 = -↑g 0 0 - 1\n⊢ ↑g =\n !![(1 * 0 + ↑g 0 0 * 1) * 1 + (1 * -1 + ↑g 0 0 * 0) * 0, (1 * 0 + ↑g 0 0 * 1) * -1 + (1 * -1 + ↑g ... | [
"g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = -1\nthis : ↑g 0 1 = -↑g 0 0 - 1\n⊢ ↑g = !![↑g 0 0, -1 - ↑g 0 0; 1, -1]"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Modular | {
"line": 617,
"column": 4
} | {
"line": 617,
"column": 11
} | {
"line": 618,
"column": 2
} | [
{
"pp": "case «0».«1»\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhg' : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : ↑g 1 0 = 1\nhd : ↑g 1 1 = -1\nthis : ↑g 0 1 = -↑g 0 0 - 1\n⊢ -↑g 0 0 - 1 = -1 - ↑g 0 0",
"ppTerm": "?«0».«1»",
"assigned": true,
"usedConstants... | [] | 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.Transform | {
"line": 145,
"column": 2
} | {
"line": 146,
"column": 83
} | {
"line": 147,
"column": 2
} | [
{
"pp": "z : ℍ\n⊢ (↑z ^ 2)⁻¹ * G2 (S • z) - -2 * ↑π * I / ↑z = ∑' (n : ℤ) (m : ℤ), G2Term z ![m, n]",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"_private.Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform.0.EisensteinSeries.G2Term",
... | [
"z : ℍ\n⊢ ∑'[symmetricIco ℤ] (b : ℤ),\n (∑' (m : ℤ), 1 / (↑m * ↑z + ↑b) ^ 2 - ∑' (m : ℤ), (1 / (↑m * ↑z + ↑b) - 1 / (↑m * ↑z + ↑b + 1))) =\n ∑'[symmetricIco ℤ] (b : ℤ), ∑' (m : ℤ), G2Term z ![m, b]",
"case hf\nz : ℍ\n⊢ Summable (fun n ↦ ∑' (m : ℤ), 1 / (↑m * ↑z + ↑n) ^ 2) (symmetricIco ℤ)",
"case hg\nz ... | rw [← tsum_symmetricIco_tsum_sub_eq z, ← tsum_symmetricIco_tsum_eq_S_act z,
← tsum_eq_of_summable_unconditional (L := symmetricIco ℤ), ← Summable.tsum_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 208,
"column": 8
} | {
"line": 208,
"column": 15
} | {
"line": 209,
"column": 8
} | [
{
"pp": "case mem.inl\nγ g : SL(2, ℤ)\nh1 : g = S\nz : ℍ\n⊢ ↑z ^ (-2) * G2 { coe := (-↑z)⁻¹, coe_im_pos := ⋯ } =\n (↑z ^ 2)⁻¹ * G2 { coe := (-↑z)⁻¹, coe_im_pos := ⋯ } - I * (↑π * -2) / ↑z - I * (↑π * 2) / ↑z",
"ppTerm": "?mem.inl",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.C... | [
"case mem.inl\nγ g : SL(2, ℤ)\nh1 : g = S\nz : ℍ\n⊢ ↑z ^ (-2) * G2 { coe := -(↑z)⁻¹, coe_im_pos := ⋯ } = (↑z)⁻¹ ^ 2 * G2 { coe := -(↑z)⁻¹, coe_im_pos := ⋯ }"
] | 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": 59,
"column": 4
} | {
"line": 59,
"column": 11
} | {
"line": 60,
"column": 2
} | [
{
"pp": "k m : ℕ\np : ℝ\nS : Set ℂ\nhs : IsOpen S\nx : ℂ\nhx : x ∈ S\nz : ℂ\n⊢ cexp (2 * ↑π * I * ↑m * z / ↑p) = cexp (2 * ↑π * I * ↑m / ↑p * z)",
"ppTerm": "?m.160",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCom... | [] | 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": 61,
"column": 2
} | {
"line": 61,
"column": 9
} | {
"line": 63,
"column": 0
} | [
{
"pp": "k m : ℕ\np : ℝ\nS : Set ℂ\nhs : IsOpen S\nx : ℂ\nhx : x ∈ S\nthis : (fun s ↦ cexp (2 * ↑π * I * ↑m * s / ↑p)) = fun s ↦ cexp (2 * ↑π * I * ↑m / ↑p * s)\n⊢ (2 * ↑π * I * ↑m / ↑p) ^ k * cexp (2 * ↑π * I * ↑m / ↑p * x) =\n (2 * ↑π * I * ↑m / ↑p) ^ k * cexp (2 * ↑π * I * ↑m * x / ↑p)",
"ppTerm": "?m... | [] | 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": 146,
"column": 2
} | {
"line": 146,
"column": 9
} | {
"line": 148,
"column": 0
} | [
{
"pp": "k : ℂ\nF G : ℍ → ℂ\nhF : MDiff F\nhG : MDiff G\nz : ℍ\n⊢ D F z + D G z - k * 12⁻¹ * EisensteinSeries.E2 z * (F z + G z) =\n D F z - k * 12⁻¹ * EisensteinSeries.E2 z * F z + (D G z - k * 12⁻¹ * EisensteinSeries.E2 z * G z)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"M... | [] | 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": 152,
"column": 2
} | {
"line": 152,
"column": 9
} | {
"line": 154,
"column": 0
} | [
{
"pp": "k : ℂ\nF G : ℍ → ℂ\nhF : MDiff F\nhG : MDiff G\nz : ℍ\n⊢ D F z - D G z - k * 12⁻¹ * EisensteinSeries.E2 z * (F z - G z) =\n D F z - k * 12⁻¹ * EisensteinSeries.E2 z * F z - (D G z - k * 12⁻¹ * EisensteinSeries.E2 z * G z)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"M... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
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