module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.Cyclotomic.Discriminant | {
"line": 110,
"column": 15
} | {
"line": 110,
"column": 24
} | {
"line": 110,
"column": 25
} | [
{
"pp": "case e_a\np k : ℕ\nK : Type u\nL : Type v\nζ : L\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhk : p ^ (k + 1) ≠ 2\nhne : NeZero... | [
"case e_a\np k : ℕ\nK : Type u\nL : Type v\nζ : L\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhk : p ^ (k + 1) ≠ 2\nhne : NeZero ↑(p ^ (k + ... | cast_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Dioph | {
"line": 192,
"column": 18
} | {
"line": 192,
"column": 55
} | {
"line": 193,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx✝¹ x✝ : Poly α\n⊢ x✝¹ * x✝ = x✝ * x✝¹",
"ppTerm": "?m.191",
"assigned": true,
"usedConstants": [
"Poly.instMul",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
... | [] | by ext; simp_rw [mul_apply, mul_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 404,
"column": 4
} | {
"line": 404,
"column": 29
} | {
"line": 404,
"column": 29
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn k : ℕ\n⊢ yn a1 n ∣ yn a1 (n * (k + 1))",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.mul_succ",
"Dvd.dvd",
"HMul.hMul",
"Pell.yn_add",
"Pell.xn",
"congrArg",
"id",
"instMulNat",
"inst... | [
"a : ℕ\na1 : 1 < a\nn k : ℕ\n⊢ yn a1 n ∣ xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n"
] | rw [Nat.mul_succ, yn_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 498,
"column": 4
} | {
"line": 506,
"column": 33
} | {
"line": 507,
"column": 2
} | [
{
"pp": "case mp\nd : ℤ\nhd : d < 0\nz : ℤ√d\n⊢ z.norm = 0 → z = 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Zsqrtd.norm_def",
"mul_self_nonneg",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"add_eq_zero_iff_of_nonneg",
"In... | [] | intro h
rw [norm_def, sub_eq_add_neg, mul_assoc] at h
have left := mul_self_nonneg z.re
have right := neg_nonneg.mpr (mul_nonpos_of_nonpos_of_nonneg hd.le (mul_self_nonneg z.im))
obtain ⟨ha, hb⟩ := (add_eq_zero_iff_of_nonneg left right).mp h
ext <;> apply eq_zero_of_mul_self_eq_zero
· exact ha
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 498,
"column": 4
} | {
"line": 506,
"column": 33
} | {
"line": 507,
"column": 2
} | [
{
"pp": "case mp\nd : ℤ\nhd : d < 0\nz : ℤ√d\n⊢ z.norm = 0 → z = 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Zsqrtd.norm_def",
"mul_self_nonneg",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"add_eq_zero_iff_of_nonneg",
"In... | [] | intro h
rw [norm_def, sub_eq_add_neg, mul_assoc] at h
have left := mul_self_nonneg z.re
have right := neg_nonneg.mpr (mul_nonpos_of_nonpos_of_nonneg hd.le (mul_self_nonneg z.im))
obtain ⟨ha, hb⟩ := (add_eq_zero_iff_of_nonneg left right).mp h
ext <;> apply eq_zero_of_mul_self_eq_zero
· exact ha
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 445,
"column": 58
} | {
"line": 445,
"column": 65
} | {
"line": 446,
"column": 8
} | [
{
"pp": "case succ\na : ℕ\na1 : 1 < a\nn k : ℕ\nhx : xn a1 (n * (k + 1)) ≡ xn a1 n ^ (k + 1) [MOD yn a1 n ^ 2]\nhy : yn a1 (n * (k + 1)) ≡ (k + 1) * xn a1 n ^ (k + 1 - 1) * yn a1 n [MOD yn a1 n ^ 3]\nL :\n xn a1 (n * (k + 1)) * xn a1 n + d a1 * yn a1 (n * (k + 1)) * yn a1 n ≡ xn a1 n ^ (k + 1) * xn a1 n + 0 [M... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 471,
"column": 32
} | {
"line": 471,
"column": 39
} | {
"line": 473,
"column": 0
} | [
{
"pp": "case re\na : ℕ\na1 : 1 < a\nn : ℕ\n⊢ (↑(xn a1 n) * ↑a + (↑a * ↑a - 1) * ↑(yn a1 n)) * ↑a + (↑a * ↑a - 1) * (↑(xn a1 n) + ↑(yn a1 n) * ↑a) + ↑(xn a1 n) =\n 2 * ↑a * (↑(xn a1 n) * ↑a + (↑a * ↑a - 1) * ↑(yn a1 n))",
"ppTerm": "?re",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 471,
"column": 32
} | {
"line": 471,
"column": 39
} | {
"line": 473,
"column": 0
} | [
{
"pp": "case im\na : ℕ\na1 : 1 < a\nn : ℕ\n⊢ ↑(xn a1 n) * ↑a + (↑a * ↑a - 1) * ↑(yn a1 n) + (↑(xn a1 n) + ↑(yn a1 n) * ↑a) * ↑a + ↑(yn a1 n) =\n 2 * ↑a * (↑(xn a1 n) + ↑(yn a1 n) * ↑a)",
"ppTerm": "?im",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.MulChar.Duality | {
"line": 45,
"column": 24
} | {
"line": 45,
"column": 37
} | {
"line": 45,
"column": 37
} | [
{
"pp": "case refine_1\nM : Type u_1\nR : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommRing R\na : Mˣ\nx✝ : ∃ χ, χ ↑a ≠ 1\nχ : MulChar M R\nhχ : ↑(χ.toUnitHom a) = ↑1\n⊢ χ ↑a = 1",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Units.val",
"MonoidHom.instFunLike",
"Mo... | [
"case refine_1\nM : Type u_1\nR : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommRing R\na : Mˣ\nx✝ : ∃ χ, χ ↑a ≠ 1\nχ : MulChar M R\nhχ : χ ↑a = ↑1\n⊢ χ ↑a = 1"
] | coe_toUnitHom | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MulChar.Duality | {
"line": 58,
"column": 2
} | {
"line": 61,
"column": 45
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case pos\nM : Type u_1\nR : Type u_2\ninst✝⁴ : CommMonoid M\ninst✝³ : CommRing R\ninst✝² : Finite M\ninst✝¹ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)\ninst✝ : Nontrivial R\na : M\nha : a ≠ 1\nhu : IsUnit a\n⊢ ∃ χ, χ a ≠ 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [
"case neg\nM : Type u_1\nR : Type u_2\ninst✝⁴ : CommMonoid M\ninst✝³ : CommRing R\ninst✝² : Finite M\ninst✝¹ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)\ninst✝ : Nontrivial R\na : M\nha : a ≠ 1\nhu : ¬IsUnit a\n⊢ ∃ χ, χ a ≠ 1"
] | · refine (exists_apply_ne_one_iff_exists_monoidHom hu.unit).mpr ?_
refine CommGroup.exists_apply_ne_one_of_hasEnoughRootsOfUnity Mˣ R ?_
contrapose ha
rw [← hu.unit_spec, ha, Units.val_eq_one] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 102,
"column": 6
} | {
"line": 110,
"column": 59
} | {
"line": 111,
"column": 4
} | [
{
"pp": "case pos.right\nR : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : ∀ {p : ℕ}, Nat.Prime p → Summable fun n ↦ ‖f (p ^ n)‖\np : ℕ\ns : Finset ℕ\nhp : p ∉ s\nih : (Summable fun m ↦ ‖f ↑m‖) ∧ HasSum (fu... | [] | · have hp' : p ∉ {p ∈ s | p.Prime} := mt (mem_of_mem_filter p) hp
rw [prod_insert hp', ← (equivProdNatFactoredNumbers hpp hp).hasSum_iff, Function.comp_def]
conv =>
enter [1, x]
rw [equivProdNatFactoredNumbers_apply', factoredNumbers.map_prime_pow_mul hmul hpp hp]
have : T3Sp... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 518,
"column": 10
} | {
"line": 518,
"column": 29
} | {
"line": 519,
"column": 10
} | [
{
"pp": "ξ : ℝ\nu : ℤ\nih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = ξ.convergent n\nleft✝ : IsCoprime u ↑1\nh₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹\nht : ξ < ↑u\nh₁ : ↑u + -(1 / 2) < ξ\nhξ₁ : ⌊ξ⌋ = u - 1\nHξ : ξ ≠ ↑⌊ξ⌋\n⊢ ↑u - (1 - 2⁻¹) < ξ",
"ppTerm": "?m.653",
"a... | [
"case e'_3\nξ : ℝ\nu : ℤ\nih : ∀ m < 1, ∀ {ξ : ℝ} {u : ℤ}, ContfracLegendre.Ass ξ u ↑m → ∃ n, ↑u / ↑m = ξ.convergent n\nleft✝ : IsCoprime u ↑1\nh₂ : |ξ - ↑u / ↑↑1| < (↑↑1 * (2 * ↑↑1 - 1))⁻¹\nht : ξ < ↑u\nh₁ : ↑u + -(1 / 2) < ξ\nhξ₁ : ⌊ξ⌋ = u - 1\nHξ : ξ ≠ ↑⌊ξ⌋\n⊢ ↑u - (1 - 2⁻¹) = ↑u + -(1 / 2)"
] | convert! h₁ using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.NumberTheory.LSeries.Deriv | {
"line": 69,
"column": 2
} | {
"line": 70,
"column": 68
} | {
"line": 71,
"column": 2
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\nh : abscissaOfAbsConv f < ↑s.re\nx : ℝ\nhxs : x < s.re\nhf : LSeriesSummable f ↑x\ny : ℝ\nhxy : x < y\nhys : y < s.re\nS : Set ℂ := {z | y < z.re}\nh₀ : Summable fun n ↦ ‖term f (↑x) n‖\nh₁ : ∀ (n : ℕ), DifferentiableOn ℂ (fun x ↦ term f x n) S\nh₂ : IsOpen S\n⊢ LSeriesSummable (logMu... | [
"f : ℕ → ℂ\ns : ℂ\nh : abscissaOfAbsConv f < ↑s.re\nx : ℝ\nhxs : x < s.re\nhf : LSeriesSummable f ↑x\ny : ℝ\nhxy : x < y\nhys : y < s.re\nS : Set ℂ := {z | y < z.re}\nh₀ : Summable fun n ↦ ‖term f (↑x) n‖\nh₁ : ∀ (n : ℕ), DifferentiableOn ℂ (fun x ↦ term f x n) S\nh₂ : IsOpen S\nh₃ : ∀ (n : ℕ), ∀ z ∈ S, ‖term f z n... | have h₃ (n z) (hz : z ∈ S) : ‖term f z n‖ ≤ ‖term f x n‖ :=
norm_term_le_of_re_le_re f (by simpa using! (hxy.trans hz).le) n | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 210,
"column": 88
} | {
"line": 216,
"column": 19
} | {
"line": 218,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : Summable fun x ↦ ‖f x‖\nhf₀ : f 0 = 0\n⊢ Tendsto (fun n ↦ ∏ p ∈ n.primesBelow, ∑' (e : ℕ), f (p ^ e)) atTop (𝓝 (∑' (n : ℕ), f n))",
"ppTerm": ... | [] | by
have := (eulerProduct_hasProd_mulIndicator hf₁ hmul hsum hf₀).tendsto_prod_nat
let F : ℕ → R := fun p ↦ ∑' (e : ℕ), f (p ^ e)
have H (n : ℕ) : ∏ i ∈ range n, Set.mulIndicator {p | Nat.Prime p} F i =
∏ p ∈ primesBelow n, ∑' (e : ℕ), f (p ^ e) :=
prod_mulIndicator_eq_prod_filter (range n... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.Deriv | {
"line": 72,
"column": 11
} | {
"line": 72,
"column": 40
} | {
"line": 72,
"column": 40
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\nh : abscissaOfAbsConv f < ↑s.re\nx : ℝ\nhxs : x < s.re\nhf : LSeriesSummable f ↑x\ny : ℝ\nhxy : x < y\nhys : y < s.re\nS : Set ℂ := {z | y < z.re}\nh₀ : Summable fun n ↦ ‖term f (↑x) n‖\nh₁ : ∀ (n : ℕ), DifferentiableOn ℂ (fun x ↦ term f x n) S\nh₂ : IsOpen S\nh₃ : ∀ (n : ℕ), ∀ z ∈ S,... | [
"f : ℕ → ℂ\ns : ℂ\nh : abscissaOfAbsConv f < ↑s.re\nx : ℝ\nhxs : x < s.re\nhf : LSeriesSummable f ↑x\ny : ℝ\nhxy : x < y\nhys : y < s.re\nS : Set ℂ := {z | y < z.re}\nh₀ : Summable fun n ↦ ‖term f (↑x) n‖\nh₁ : ∀ (n : ℕ), DifferentiableOn ℂ (fun x ↦ term f x n) S\nh₂ : IsOpen S\nh₃ : ∀ (n : ℕ), ∀ z ∈ S, ‖term f z n... | (hasDerivAt_term f _ _).deriv | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.Positivity | {
"line": 76,
"column": 2
} | {
"line": 77,
"column": 76
} | {
"line": 78,
"column": 2
} | [
{
"pp": "a : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nf : ℂ → ℂ\nhf : Differentiable ℂ f\nx : ℝ\nhx : abscissaOfAbsConv a ≤ ↑x\nhf' : Set.EqOn f (LSeries a) {s | x < s.re}\ny : ℝ\nhxy : x < max x y + 1\nhxy' : abscissaOfAbsConv a < ↑(max x y) + 1\nhys : ↑(max x y) + 1 ∈ {s | x < s.re}\n⊢ 0 < f ↑y",
"ppTerm": "?m.... | [
"a : ℕ → ℂ\nha₀ : 0 ≤ a\nha₁ : 0 < a 1\nf : ℂ → ℂ\nhf : Differentiable ℂ f\nx : ℝ\nhx : abscissaOfAbsConv a ≤ ↑x\nhf' : Set.EqOn f (LSeries a) {s | x < s.re}\ny : ℝ\nhxy : x < max x y + 1\nhxy' : abscissaOfAbsConv a < ↑(max x y) + 1\nhys : ↑(max x y) + 1 ∈ {s | x < s.re}\nhfx : 0 < f (↑(max x y) + 1)\n⊢ 0 < f ↑y"
] | have hfx : 0 < f (max x y + 1) := by
simpa only [hf' hys, ofReal_add, ofReal_one] using positive ha₀ ha₁ hxy' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 549,
"column": 81
} | {
"line": 549,
"column": 88
} | {
"line": 549,
"column": 88
} | [
{
"pp": "ξ : ℝ\nq : ℚ\nh : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)\nhq₀ : 0 < ↑q.den\nhq₁ : 0 < ↑q.den * (2 * ↑q.den - 1)\nhq₂ : 0 < 2 * (↑q.den * ↑q.den)\n⊢ 0 < 2 * (↑q.den * ↑q.den) - ↑q.den * (2 * ↑q.den - 1)",
"ppTerm": "?m.217",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul... | [
"ξ : ℝ\nq : ℚ\nh : |ξ - ↑q| < 1 / (2 * ↑q.den ^ 2)\nhq₀ : 0 < ↑q.den\nhq₁ : 0 < ↑q.den * (2 * ↑q.den - 1)\nhq₂ : 0 < 2 * (↑q.den * ↑q.den)\n⊢ 0 < ↑q.den"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 9
} | {
"line": 124,
"column": 0
} | [
{
"pp": "case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nn : ℕ\n⊢ (↑n + a) ^ 0 * rexp (-π * (↑n + a ^ 2) * t) = rexp (-π * a ^ 2 * t + ↑n * (-π * t))",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero"... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 77,
"column": 25
} | {
"line": 77,
"column": 42
} | {
"line": 77,
"column": 43
} | [
{
"pp": "n : ℤ\nz τ : ℂ\n⊢ ‖cexp (2 * ↑π * I * ↑n * z + ↑π * I * ↑n ^ 2 * τ)‖ = rexp (-π * ↑n ^ 2 * τ.im - 2 * π * ↑n * z.im)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Real",
"Real.pi",
"HMul.hMul",
"congrA... | [
"n : ℤ\nz τ : ℂ\n⊢ rexp (2 * ↑π * I * ↑n * z + ↑π * I * ↑n ^ 2 * τ).re = rexp (-π * ↑n ^ 2 * τ.im - 2 * π * ↑n * z.im)"
] | Complex.norm_exp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 9
} | {
"line": 82,
"column": 0
} | [
{
"pp": "n : ℤ\nz τ : ℂ\n⊢ rexp (-(π * (2 * ↑n) * z.im) + -(π * ↑n ^ 2 * τ.im)) = rexp (-π * ↑n ^ 2 * τ.im - 2 * π * ↑n * z.im)",
"ppTerm": "?m.156",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 172,
"column": 6
} | {
"line": 172,
"column": 13
} | {
"line": 173,
"column": 4
} | [
{
"pp": "case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nh0' : ‖rexp (-π * t)‖ < 1\nn : ℕ\n⊢ ↑n * rexp (-π * (↑n + a ^ 2) * t) = ↑n * rexp (↑n * (-π * t) + -π * a ^ 2 * t)",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 11
} | {
"line": 179,
"column": 0
} | [
{
"pp": "case e'_5\na : ℝ\nha : 0 ≤ a\nt : ℝ\nht : 0 < t\nn : ℕ\n⊢ a * rexp (-π * (↑n + a ^ 2) * t) = a * rexp (-π * a ^ 2 * t + ↑n * (-π * t))",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 83
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case refine_1\nz τ : ℂ\nhτ : 0 < τ.im\n⊢ ∀ (i : ℤ), ‖jacobiTheta₂_term i z τ‖ ≤ ↑|i| ^ 0 * rexp (-π * (τ.im * ↑i ^ 2 - 2 * |z.im| * ↑|i|))",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCom... | [] | simpa only [pow_zero, one_mul] using norm_jacobiTheta₂_term_le hτ le_rfl le_rfl | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 49,
"column": 61
} | {
"line": 49,
"column": 71
} | {
"line": 49,
"column": 71
} | [
{
"pp": "case convert_3.inr\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\np : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhp : ∀ (i : ι), a i = 0 ∨ 0 < p i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-p i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / p i ^ s.re\ni : ι\nhpi : 0 < p i\nthis : IntegrableOn (fun... | [
"case convert_3.inr\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\np : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhp : ∀ (i : ι), a i = 0 ∨ 0 < p i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-p i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / p i ^ s.re\ni : ι\nhpi : 0 < p i\nthis : IntegrableOn (fun x ↦ ↑(rexp ... | ofReal_mul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 193,
"column": 18
} | {
"line": 193,
"column": 52
} | {
"line": 194,
"column": 6
} | [
{
"pp": "a : ℝ\nha : 0 ≤ a\naux' : (fun t ↦ ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x ↦ 1\nha' : 0 < a\n| fun t ↦ rexp (-(π * a ^ 2) * t)",
"ppTerm": "?m.464",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Real",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
... | [
"a : ℝ\nha : 0 ≤ a\naux' : (fun t ↦ ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x ↦ 1\nha' : 0 < a\nt : ℝ\n| rexp (-(π * a ^ 2) * t) * 1"
] | enter [t]; rw [← mul_one (rexp _)] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 193,
"column": 18
} | {
"line": 193,
"column": 52
} | {
"line": 194,
"column": 6
} | [
{
"pp": "a : ℝ\nha : 0 ≤ a\naux' : (fun t ↦ ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x ↦ 1\nha' : 0 < a\n| fun t ↦ rexp (-(π * a ^ 2) * t)",
"ppTerm": "?m.464",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Real",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
... | [
"a : ℝ\nha : 0 ≤ a\naux' : (fun t ↦ ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x ↦ 1\nha' : 0 < a\nt : ℝ\n| rexp (-(π * a ^ 2) * t) * 1"
] | enter [t]; rw [← mul_one (rexp _)] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 15
} | {
"line": 72,
"column": 8
} | [
{
"pp": "a : UnitAddCircle\nx ξ : ℝ\n⊢ cexp (-↑π * I * (I * ↑x + 2 * (↑ξ * I * ↑x))) = ↑(rexp (π * (x + 2 * ξ * x)))",
"ppTerm": "?m.125",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroC... | [
"a : UnitAddCircle\nx ξ : ℝ\n⊢ cexp (-(↑π * I ^ 2 * ↑x) - ↑π * I ^ 2 * ↑x * ↑ξ * 2) = ↑(rexp (π * x + π * x * ξ * 2))"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 135,
"column": 8
} | {
"line": 135,
"column": 30
} | {
"line": 135,
"column": 31
} | [
{
"pp": "case inl\na : UnitAddCircle\nx : ℝ\nhx : x ≤ 0\n⊢ evenKernel a x = 1 / x ^ (1 / 2) * cosKernel a (1 / x)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"HMul.hMul",
"Real.instZero",
"congrArg",... | [
"case inl\na : UnitAddCircle\nx : ℝ\nhx : x ≤ 0\n⊢ 0 = 1 / x ^ (1 / 2) * cosKernel a (1 / x)"
] | evenKernel_undef _ hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 11
} | {
"line": 84,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\ni : ι\nthis : a i / ↑(π... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 11
} | {
"line": 179,
"column": 4
} | [
{
"pp": "a t : ℝ\nht : 0 < t\nn : ℤ\n⊢ cexp (2 * ↑π * I * ↑a * ↑n + -(↑π * ↑n ^ 2 * ↑t)) = cexp (2 * ↑π * I * ↑n * ↑a + ↑π * I * ↑n ^ 2 * (I * ↑t))",
"ppTerm": "?m.126",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast",
"Mathlib.Tactic.Ring.... | [
"a t : ℝ\nht : 0 < t\nn : ℤ\n⊢ cexp (↑π * I * ↑a * ↑n * 2 - ↑π * ↑n ^ 2 * ↑t) = cexp (↑π * I * ↑a * ↑n * 2 + ↑π * I ^ 2 * ↑n ^ 2 * ↑t)"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 49
} | {
"line": 207,
"column": 2
} | [
{
"pp": "a t : ℝ\nht : 0 < t\n⊢ HasSum (fun x ↦ ↑(2 * Real.cos (2 * π * a * (↑x + 1)) * rexp (-π * (↑x + 1) ^ 2 * t))) (↑(cosKernel (↑a) t) - 1)",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalComm... | [
"a t : ℝ\nht : 0 < t\nthis :\n HasSum\n (fun n ↦\n cexp (2 * ↑π * I * ↑a * ↑↑n) * ↑(rexp (-π * ↑↑n ^ 2 * t)) +\n cexp (2 * ↑π * I * ↑a * ↑(-↑n)) * ↑(rexp (-π * ↑(-↑n) ^ 2 * t)))\n (↑(cosKernel (↑a) t) + cexp (2 * ↑π * I * ↑a * ↑0) * ↑(rexp (-π * ↑0 ^ 2 * t)))\n⊢ HasSum (fun x ↦ ↑(2 * Real.cos (2 ... | have := (hasSum_int_cosKernel a ht).nat_add_neg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 13
} | {
"line": 128,
"column": 4
} | [
{
"pp": "case e'_6\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nr : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhs : 0 < s.re\nh_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re\nhs' : 0 < (s / 2).re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)\ni : ι\n⊢ ↑|r i| ^ s = ↑|r i| ^ ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 13
} | {
"line": 128,
"column": 4
} | [
{
"pp": "case e'_6\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nr : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhs : 0 < s.re\nh_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re\nhs' : 0 < (s / 2).re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)\ni : ι\n⊢ ↑|r i| ^ s = ↑|r i| ^ ... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 13
} | {
"line": 128,
"column": 4
} | [
{
"pp": "case e'_6\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nr : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhs : 0 < s.re\nh_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re\nhs' : 0 < (s / 2).re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i ^ 2 = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)\ni : ι\n⊢ ↑|r i| ^ s = ↑|r i| ^ ... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 9
} | {
"line": 401,
"column": 0
} | [
{
"pp": "z τ : ℂ\nn : ℤ\n⊢ cexp (2 * ↑π * I * ↑n * (z + τ) + ↑π * I * ↑n ^ 2 * τ) =\n cexp (-↑π * I * (τ + 2 * z) + (2 * ↑π * I * (↑n + ↑1) * z + ↑π * I * (↑n + ↑1) ^ 2 * τ))",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast"... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 452,
"column": 2
} | {
"line": 452,
"column": 9
} | {
"line": 454,
"column": 0
} | [
{
"pp": "case e_f\nz τ : ℂ\nn : ℤ\n⊢ 2 * ↑π * I * -↑n * cexp (2 * ↑π * I * -↑n * -z + ↑π * I * (-↑n) ^ 2 * τ) =\n -(2 * ↑π * I * ↑n) * cexp (2 * ↑π * I * ↑n * z + ↑π * I * ↑n ^ 2 * τ)",
"ppTerm": "?e_f",
"assigned": true,
"usedConstants": [
"Mathlib.Meta.NormNum.isInt_pow",
"Mathlib.T... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 460,
"column": 2
} | {
"line": 460,
"column": 9
} | {
"line": 462,
"column": 0
} | [
{
"pp": "case e_f\nz τ : ℂ\nn : ℤ\n⊢ -(↑2 * ↑π * -I * ↑↑n * cexp (↑2 * ↑π * -I * ↑↑n * (starRingEnd ℂ) z + ↑π * -I * ↑↑n ^ 2 * (starRingEnd ℂ) τ)) =\n ↑2 * ↑π * I * ↑↑n * cexp (↑2 * ↑π * I * ↑↑n * -(starRingEnd ℂ) z + ↑π * I * ↑↑n ^ 2 * -(starRingEnd ℂ) τ)",
"ppTerm": "?e_f",
"assigned": true,
"u... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 481,
"column": 27
} | {
"line": 481,
"column": 34
} | {
"line": 481,
"column": 34
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nh2 : 0 < (-I * τ).re\nn : ℤ\n⊢ cexp (2 * ↑π * I * ↑n * z + ↑π * I * ↑n ^ 2 * τ) = cexp (-↑π * (-I * τ) * ↑n ^ 2 + 2 * ↑π * (I * z) * ↑n)",
"ppTerm": "?m.242",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast",
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 481,
"column": 27
} | {
"line": 481,
"column": 34
} | {
"line": 481,
"column": 34
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nh2 : 0 < (-I * τ).re\nn : ℤ\n⊢ cexp (2 * ↑π * I * ↑n * z + ↑π * I * ↑n ^ 2 * τ) = cexp (-↑π * (-I * τ) * ↑n ^ 2 + 2 * ↑π * (I * z) * ↑n)",
"ppTerm": "?m.242",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast",
... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 481,
"column": 27
} | {
"line": 481,
"column": 34
} | {
"line": 481,
"column": 34
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nh2 : 0 < (-I * τ).re\nn : ℤ\n⊢ cexp (2 * ↑π * I * ↑n * z + ↑π * I * ↑n ^ 2 * τ) = cexp (-↑π * (-I * τ) * ↑n ^ 2 + 2 * ↑π * (I * z) * ↑n)",
"ppTerm": "?m.242",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast",
... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 489,
"column": 4
} | {
"line": 489,
"column": 11
} | {
"line": 490,
"column": 4
} | [
{
"pp": "case e_a.e_f\nz τ : ℂ\nhτ : 0 < τ.im\nh2 : 0 < (-I * τ).re\nn : ℤ\n⊢ (↑n + I ^ 2 * z) ^ 2 / τ = I ^ 2 * (-z ^ 2 + ↑n * (z * 2 + -↑n)) / τ",
"ppTerm": "?e_a.e_f✝",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast",
"Mathlib.Tactic.Rin... | [
"case e_a.e_f\nz τ : ℂ\nhτ : 0 < τ.im\nh2 : 0 < (-I * τ).re\nn : ℤ\n⊢ ↑n * I ^ 2 * z * τ⁻¹ * 2 + ↑n ^ 2 * τ⁻¹ + I ^ 4 * z ^ 2 * τ⁻¹ =\n ↑n * I ^ 2 * z * τ⁻¹ * 2 - ↑n ^ 2 * I ^ 2 * τ⁻¹ - I ^ 2 * z ^ 2 * τ⁻¹"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 491,
"column": 4
} | {
"line": 491,
"column": 11
} | {
"line": 492,
"column": 2
} | [
{
"pp": "case e_a.e_f\nz τ : ℂ\nhτ : 0 < τ.im\nh2 : 0 < (-I * τ).re\nn : ℤ\n⊢ ↑n * -1 * z * τ⁻¹ * 2 + ↑n ^ 2 * τ⁻¹ + 1 * z ^ 2 * τ⁻¹ = ↑n * -1 * z * τ⁻¹ * 2 - ↑n ^ 2 * -1 * τ⁻¹ - -1 * z ^ 2 * τ⁻¹",
"ppTerm": "?e_a.e_f✝",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_l... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.HurwitzZeta | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 9
} | {
"line": 60,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\n⊢ hurwitzZetaEven a s = (hurwitzZetaEven a s + hurwitzZetaOdd a s + (hurwitzZetaEven a s + -hurwitzZetaOdd a s)) / 2",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.HurwitzZeta | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 9
} | {
"line": 65,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\n⊢ hurwitzZetaOdd a s = (hurwitzZetaEven a s + hurwitzZetaOdd a s - (hurwitzZetaEven a s + -hurwitzZetaOdd a s)) / 2",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero"... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.HurwitzZeta | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 9
} | {
"line": 123,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\n⊢ cosZeta a s = (cosZeta a s + I * sinZeta a s + (cosZeta a s + I * -sinZeta a s)) / 2",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.HurwitzZeta | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 9
} | {
"line": 171,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : a ≠ 0 ∨ s ≠ 1\nx y z : ℂ\n⊢ 2 * x * Complex.Gamma s * ((y + z) / 2) * cosZeta a s + 2 * x * Complex.Gamma s * ((z - y) * I / 2) * sinZeta a s =\n x * Complex.Gamma s * (z * (cosZeta a s + I * sinZeta a s) + y * (cosZeta a s + I * -sinZeta a s)... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 532,
"column": 4
} | {
"line": 532,
"column": 11
} | {
"line": 533,
"column": 0
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nhτ' : 0 < (-1 / τ).im\nhj : HasDerivAt (fun w ↦ jacobiTheta₂ (w / τ) (-1 / τ)) (1 / τ * jacobiTheta₂' (z / τ) (-1 / τ)) z\n⊢ (cexp (-↑π * I * z ^ 2 * τ⁻¹) * (-↑π * I * (↑2 * z ^ (2 - 1)) * τ⁻¹) * jacobiTheta₂ (z * τ⁻¹) (-1 * τ⁻¹) +\n cexp (-↑π * I * z ^ 2 * τ⁻¹) * (1 * τ⁻... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.HurwitzZeta | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 9
} | {
"line": 189,
"column": 2
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nhs' : ∀ (n : ℕ), s ≠ -↑n\nx y z : ℂ\n⊢ 2 * x * Complex.Gamma s * ((y + z) / 2) * hurwitzZetaEven a s +\n I * (2 * x * Complex.Gamma s * ((z - y) * I / 2) * hurwitzZetaOdd a s) =\n x * Complex.Gamma s *\n (y * (hurwitzZetaEven a s + hurw... | [
"a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nhs' : ∀ (n : ℕ), s ≠ -↑n\nx y z : ℂ\n⊢ x * Complex.Gamma s * y * hurwitzZetaEven a s - x * Complex.Gamma s * y * I ^ 2 * hurwitzZetaOdd a s +\n x * Complex.Gamma s * z * hurwitzZetaEven a s +\n x * Complex.Gamma s * z * I ^ 2 * hurwitzZetaOdd a s =\... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.HurwitzZeta | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 9
} | {
"line": 192,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nhs' : ∀ (n : ℕ), s ≠ -↑n\nx y z : ℂ\n⊢ x * Complex.Gamma s * y * hurwitzZetaEven a s - x * Complex.Gamma s * y * -1 * hurwitzZetaOdd a s +\n x * Complex.Gamma s * z * hurwitzZetaEven a s +\n x * Complex.Gamma s * z * -1 * hurwitzZetaOdd ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.SumPrimeReciprocals | {
"line": 57,
"column": 26
} | {
"line": 57,
"column": 35
} | {
"line": 57,
"column": 36
} | [
{
"pp": "k : ℕ\nm : ℕ := 2 ^ k.primesBelow.card\nN₀ : ℕ := 2 * m ^ 2\nhN₀ : N₀ = 2 * m ^ 2\nS : ℝ := ∑ p ∈ (2 * N₀).succ.primesBelow \\ k.primesBelow, 1 / ↑p\nthis : 2 * (↑2 * ↑(m ^ 2)) - ↑m * (↑2 * ↑m) ≤ 2 * (↑2 * ↑(m ^ 2)) * S\n⊢ 1 / 2 ≤ S",
"ppTerm": "?m.466",
"assigned": true,
"usedConstants": [... | [
"k : ℕ\nm : ℕ := 2 ^ k.primesBelow.card\nN₀ : ℕ := 2 * m ^ 2\nhN₀ : N₀ = 2 * m ^ 2\nS : ℝ := ∑ p ∈ (2 * N₀).succ.primesBelow \\ k.primesBelow, 1 / ↑p\nthis : 2 * (↑2 * ↑m ^ 2) - ↑m * (↑2 * ↑m) ≤ 2 * (↑2 * ↑m ^ 2) * S\n⊢ 1 / 2 ≤ S"
] | cast_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 57
} | {
"line": 196,
"column": 2
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\n⊢ completedRiemannZeta s = ↑π ^ (-s / 2) * Gamma (s / 2) * ∑' (n : ℕ), 1 / ↑n ^ s",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.inst... | [
"s : ℂ\nhs : 1 < s.re\nthis : completedCosZeta (↑0) s = ∑' (b : ℕ), if b = 0 then 0 else s.Gammaℝ * ↑(Real.cos (2 * π * 0 * ↑b)) / ↑b ^ s\n⊢ completedRiemannZeta s = ↑π ^ (-s / 2) * Gamma (s / 2) * ∑' (n : ℕ), 1 / ↑n ^ s"
] | have := (hasSum_nat_completedCosZeta 0 hs).tsum_eq.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 570,
"column": 4
} | {
"line": 570,
"column": 34
} | {
"line": 571,
"column": 4
} | [
{
"pp": "case inr\nΛ : ℂ → ℂ\nhf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s\nL : ℂ\nh_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)\nclaim : ∀ (t : ℂ), t ≠ 0 → t ≠ 1 → DifferentiableAt ℂ (fun u ↦ Λ u / u.Gammaℝ) t\nclaim2 : Tendsto (fun s ↦ Λ s / s.Gammaℝ) (𝓝[≠] 0) (𝓝 (L / 2))\nhs' : 0 ≠ 1\n⊢ Diff... | [
"case inr\nΛ : ℂ → ℂ\nhf : ∀ (s : ℂ), s ≠ 0 → s ≠ 1 → DifferentiableAt ℂ Λ s\nL : ℂ\nh_lim : Tendsto (fun s ↦ s * Λ s) (𝓝[≠] 0) (𝓝 L)\nclaim : ∀ (t : ℂ), t ≠ 0 → t ≠ 1 → DifferentiableAt ℂ (fun u ↦ Λ u / u.Gammaℝ) t\nclaim2 : Tendsto (fun s ↦ Λ s / s.Gammaℝ) (𝓝[≠] 0) (𝓝 (L / 2))\nhs' : 0 ≠ 1\n⊢ DifferentiableAt... | simp_rw [← claim2.limUnder_eq] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 9
} | {
"line": 226,
"column": 0
} | [
{
"pp": "a x : ℝ\nhx : 0 < x\nh1 : HasSum (fun n ↦ jacobiTheta₂_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂ (↑a * I * ↑x) (I * ↑x))\nh2 : HasSum (fun n ↦ jacobiTheta₂'_term n (↑a * I * ↑x) (I * ↑x)) (jacobiTheta₂' (↑a * I * ↑x) (I * ↑x))\nn : ℤ\n⊢ -↑π * (↑n + ↑a) ^ 2 = -↑π * ↑a ^ 2 + (2 * ↑π * ↑n * ↑a * -1 + ↑... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 40
} | {
"line": 240,
"column": 2
} | [
{
"pp": "case refine_2\na t : ℝ\nht : 0 < t\nh : -2 * ↑π ≠ 0\nn : ℤ\n⊢ 2 * ↑π * I * ↑a * ↑n = 2 * ↑π * I * ↑n * ↑a",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Int.cast",
"Real.pi",
"HMul.hMul",
"Nat.instAtLeastTwoHAddOfNat",
"NonUnitalCommSemiring.toC... | [
"case refine_3\na t : ℝ\nht : 0 < t\nh : -2 * ↑π ≠ 0\nn : ℤ\n⊢ -↑π * ↑n ^ 2 * ↑t = ↑π * I * ↑n ^ 2 * (I * ↑t)"
] | · exact mul_right_comm (2 * π * I) a n | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 733,
"column": 82
} | {
"line": 737,
"column": 77
} | {
"line": 739,
"column": 0
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\n⊢ HasSum (fun n ↦ cexp (2 * ↑π * I * ↑a * ↑n) / ↑|n| ^ s / 2) (cosZeta (↑a) s)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"HasSum.congr_fun",
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialO... | [] | by
rw [cosZeta, Function.update_of_ne (ne_zero_of_one_lt_re hs)]
refine ((hasSum_int_completedCosZeta a hs).div_const (Gammaℝ s)).congr_fun fun n ↦ ?_
rw [mul_div_assoc _ (cexp _), div_right_comm _ (2 : ℂ),
mul_div_cancel_left₀ _ (Gammaℝ_ne_zero_of_re_pos (zero_lt_one.trans hs))] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.FLT.Basic | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 20
} | {
"line": 82,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nm : ℕ\nhm : FermatLastTheoremWith R m\na b c : R\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nk : ℕ\n⊢ a ^ (m * k) + b ^ (m * k) ≠ c ^ (m * k)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
... | [
"R : Type u_1\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nm : ℕ\nhm : FermatLastTheoremWith R m\na b c : R\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nk : ℕ\n⊢ (a ^ k) ^ m + (b ^ k) ^ m ≠ (c ^ k) ^ m"
] | simp_rw [pow_mul'] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 746,
"column": 35
} | {
"line": 746,
"column": 42
} | {
"line": 746,
"column": 42
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis : HasSum (fun n ↦ (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s) (cosZeta (↑a) s)\n⊢ HasSum (fun n ↦ (cexp (2 * ↑π * ↑a * ↑n * I) + cexp (-(2 * ↑π * ↑a * ↑n) * I)) / 2 / ↑n ^ s) (cosZeta (↑a) s)",
"ppTerm": "?m.95",
"assigned": tru... | [
"a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis : HasSum (fun n ↦ (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s) (cosZeta (↑a) s)\n⊢ HasSum (fun n ↦ (cexp (2 * ↑π * ↑a * ↑n * I) + cexp (-(2 * ↑π * ↑a * ↑n * I))) / 2 / ↑n ^ s) (cosZeta (↑a) s)"
] | neg_mul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 154,
"column": 2
} | {
"line": 156,
"column": 93
} | {
"line": 157,
"column": 2
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\n⊢ cexp (∑' (p : Primes), -Complex.log (1 - ↑↑p ^ (-s))) = L 1 s",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Complex.log",
"MulOne.toOne",
"Nat.Prime",
"HMul.hMul",
... | [
"s : ℂ\nhs : 1 < s.re\nx✝ : Primes\n⊢ ↑↑x✝ ^ (-s) = 1 ↑↑x✝ * ↑↑x✝ ^ (-s)"
] | convert!
modOne_eq_one (R := ℂ) ▸
DirichletCharacter.LSeries_eulerProduct_exp_log (1 : DirichletCharacter ℂ 1) hs using 7 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 769,
"column": 2
} | {
"line": 769,
"column": 9
} | {
"line": 771,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ -↑n\nhs' : a ≠ 0 ∨ s ≠ 1\nthis : hurwitzZetaEven a (1 - s) = completedHurwitzZetaEven a (1 - s) * (1 - s).Gammaℝ⁻¹\nA : ℂ\n⊢ completedCosZeta a s * (A * s.Gammaℝ⁻¹) = A * (completedCosZeta a s / s.Gammaℝ)",
"ppTerm": "?m.151",
"assigned": true,
... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 782,
"column": 2
} | {
"line": 782,
"column": 9
} | {
"line": 784,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nthis : cosZeta a (1 - s) = completedCosZeta a (1 - s) * (1 - s).Gammaℝ⁻¹\nA : ℂ\n⊢ completedHurwitzZetaEven a s * (A * s.Gammaℝ⁻¹) = A * (completedHurwitzZetaEven a s / s.Gammaℝ)",
"ppTerm": "?m.175",
"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.LSeries.HurwitzZetaOdd | {
"line": 430,
"column": 13
} | {
"line": 430,
"column": 20
} | {
"line": 430,
"column": 20
} | [
{
"pp": "case inr.e_a.e_a.e_a\na : ℝ\ns : ℂ\nhs : 1 < s.re\nthis :\n HasSum\n (fun n ↦\n ((s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) +\n (s + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /\n ↑n ^ s /\n 2)\n (completedSinZeta (↑a) s)\nn :... | [
"case inr.e_a.e_a.e_a\na : ℝ\ns : ℂ\nhs : 1 < s.re\nthis :\n HasSum\n (fun n ↦\n ((s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑n) +\n (s + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /\n ↑n ^ s /\n 2)\n (completedSinZeta (↑a) s)\nn : ℕ\nh : n ≠ ... | neg_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 439,
"column": 2
} | {
"line": 442,
"column": 56
} | {
"line": 443,
"column": 2
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nr : ℤ → ℝ := fun n ↦ ↑n + a\nc : ℤ → ℂ := fun n ↦ 1 / 2\n⊢ HasSum (fun n ↦ (s + 1).Gammaℝ * ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2) (completedHurwitzZetaOdd (↑a) s)",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"HasSum.c... | [
"a : ℝ\ns : ℂ\nhs : 1 < s.re\nr : ℤ → ℝ := fun n ↦ ↑n + a\nc : ℤ → ℂ := fun n ↦ 1 / 2\nhF : ∀ (t : ℝ), 0 < t → HasSum (fun n ↦ c n * ↑(r n) * ↑(rexp (-π * r n ^ 2 * t))) (↑(oddKernel (↑a) t) / 2)\n⊢ HasSum (fun n ↦ (s + 1).Gammaℝ * ↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s / 2) (completedHurwitzZetaOdd (↑a) s)"
] | have hF t (ht : 0 < t) : HasSum (fun n ↦ c n * r n * rexp (-π * (r n) ^ 2 * t))
(oddKernel a t / 2) := by
refine ((hasSum_ofReal.mpr (hasSum_int_oddKernel a ht)).div_const 2).congr_fun fun n ↦ ?_
simp [r, c, push_cast, div_mul_eq_mul_div, -one_div] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 139,
"column": 8
} | {
"line": 139,
"column": 19
} | {
"line": 140,
"column": 8
} | [
{
"pp": "z x0 y0 : ℤ\nhx : (x0 * 2 + 1) % 2 = 1\nhy : (y0 * 2 + 1) % 2 = 1\nh : PythagoreanTriple (x0 * 2 + 1) (y0 * 2 + 1) z\nhc : (x0 * 2 + 1).gcd (y0 * 2 + 1) = 1\n⊢ z * z = 4 * (x0 * x0 + x0 + y0 * y0 + y0) + 2",
"ppTerm": "?m.297",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMu... | [
"z x0 y0 : ℤ\nhx : (x0 * 2 + 1) % 2 = 1\nhy : (y0 * 2 + 1) % 2 = 1\nh : PythagoreanTriple (x0 * 2 + 1) (y0 * 2 + 1) z\nhc : (x0 * 2 + 1).gcd (y0 * 2 + 1) = 1\n⊢ (x0 * 2 + 1) * (x0 * 2 + 1) + (y0 * 2 + 1) * (y0 * 2 + 1) = 4 * (x0 * x0 + x0 + y0 * y0 + y0) + 2"
] | rw [← h.eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.FLT.Four | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 25
} | {
"line": 110,
"column": 2
} | [
{
"pp": "a b c : ℤ\nh2 : ∀ (a1 b1 c1 : ℤ), Fermat42 a1 b1 c1 → c.natAbs ≤ c1.natAbs\nha : a ≠ 0\nhb : b ≠ 0\nheq : a ^ 4 + b ^ 4 = c ^ 2\n⊢ c ^ 2 = (-c) ^ 2",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCom... | [] | exact (neg_sq c).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 500,
"column": 43
} | {
"line": 500,
"column": 50
} | {
"line": 501,
"column": 2
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nn : ℕ\n⊢ -(↑n + 1) + a = -(↑n + 1 - a)",
"ppTerm": "?m.150",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 308,
"column": 6
} | {
"line": 308,
"column": 21
} | {
"line": 309,
"column": 4
} | [
{
"pp": "m n : ℤ\nh : m.gcd n = 1\nhm : m % 2 = 0\nhn : n % 2 = 1\nH : ¬(m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1\np : ℕ\nhp : Nat.Prime p\nhp1 : ↑p ∣ m ^ 2 - n ^ 2\nhp2 : ↑p ∣ m ^ 2 + n ^ 2\nh2m : ↑p ∣ 2 * m ^ 2\nh2n : ↑p ∣ 2 * n ^ 2\nhmc : p = 2 ∨ p ∣ m.natAbs\nhnc : p = 2 ∨ p ∣ n.natAbs\nh2 : p = 2\nh3 : (m ^ ... | [] | rwa [h2] at hp2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 36
} | {
"line": 55,
"column": 2
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\np q r : ℕ\na b c : k[X]\nu v w : k\nhp : p ≠ 0\nhq : q ≠ 0\nhr : r ≠ 0\nhu : u ≠ 0\nhv : v ≠ 0\nhw : w ≠ 0\nheq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0\nhab : IsCoprime (C u * a ^ p) (C v * b ^ q)\nhCu : IsUnit (C u)\nhCv : IsUnit (C v)\nhCw : IsUnit (C w)\n⊢ IsCopr... | [
"k : Type u_1\ninst✝ : Field k\np q r : ℕ\na b c : k[X]\nu v w : k\nhp : p ≠ 0\nhq : q ≠ 0\nhr : r ≠ 0\nhu : u ≠ 0\nhv : v ≠ 0\nhw : w ≠ 0\nheq : -(C u * a ^ p + C v * b ^ q) = C w * c ^ r\nhab : IsCoprime (C u * a ^ p) (C v * b ^ q)\nhCu : IsUnit (C u)\nhCv : IsUnit (C v)\nhCw : IsUnit (C w)\n⊢ IsCoprime b c"
] | rw [add_eq_zero_iff_neg_eq] at heq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 182,
"column": 6
} | {
"line": 183,
"column": 44
} | {
"line": 184,
"column": 6
} | [
{
"pp": "case inr.hi\nk : Type u_1\ninst✝ : Field k\np q r : ℕ\nu v w : k\nhp : p ≠ 0\nhq : q ≠ 0\nhr : r ≠ 0\nhineq : q * r + r * p + p * q ≤ p * q * r\nchp : ↑p ≠ 0\nchq : ↑q ≠ 0\nchr : ↑r ≠ 0\nhu : u ≠ 0\nhv : v ≠ 0\nhw : w ≠ 0\nd : ℕ\nih_d :\n ∀ m ≤ d,\n ∀ {a b c : k[X]},\n C u * a ^ p + C v * b ^ ... | [
"case inr.hi\nk : Type u_1\ninst✝ : Field k\np q r : ℕ\nu v w : k\nhp : p ≠ 0\nhq : q ≠ 0\nhr : r ≠ 0\nhineq : q * r + r * p + p * q ≤ p * q * r\nchp : ↑p ≠ 0\nchq : ↑q ≠ 0\nchr : ↑r ≠ 0\nhu : u ≠ 0\nhv : v ≠ 0\nhw : w ≠ 0\nd : ℕ\nih_d :\n ∀ m ≤ d,\n ∀ {a b c : k[X]},\n C u * a ^ p + C v * b ^ q + C w * c ... | suffices hca : ca.natDegree = 0 by
rw [← eq_d, eq_deg_a, hca, zero_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 498,
"column": 12
} | {
"line": 498,
"column": 37
} | {
"line": 498,
"column": 38
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhyo : y % 2 = 1\nhzpos : 0 < z\nh0 : ¬x = 0\nv : ℚ := ↑x / ↑z\nw : ℚ := ↑y / ↑z\nhq : v ^ 2 + w ^ 2 = 1\nhvz : v ≠ 0\nhw1 : w ≠ -1\nhQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0\nhp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}\nq : ℚ := (circleEquivGen hQ)... | [
"x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhyo : y % 2 = 1\nhzpos : 0 < z\nh0 : ¬x = 0\nv : ℚ := ↑x / ↑z\nw : ℚ := ↑y / ↑z\nhq : v ^ 2 + w ^ 2 = 1\nhvz : v ≠ 0\nhw1 : w ≠ -1\nhQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0\nhp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}\nq : ℚ := (circleEquivGen hQ).symm ⟨(v, w... | Int.ediv_mul_cancel h1.1, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 254,
"column": 20
} | {
"line": 254,
"column": 44
} | {
"line": 254,
"column": 44
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh : Odd (Module.finrank ℚ K)\nthis : Fintype ↥(torsion K) := Fintype.ofFinite ↥(torsion K)\n⊢ Nat.card ↥(torsion K) = 2",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NumberField.instCommRingRingOfInteg... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh : Odd (Module.finrank ℚ K)\nthis : Fintype ↥(torsion K) := Fintype.ofFinite ↥(torsion K)\n⊢ Fintype.card ↥(torsion K) = 2"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 445,
"column": 2
} | {
"line": 449,
"column": 61
} | {
"line": 450,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ card { φ // ¬ComplexEmbedding.IsReal φ } = 2 * nrComplexPlaces K",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mul_sum",
"instDecidableNot",
"instHSMul",
"HMul.hMul",
"... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ∀ (w : { w // w.IsComplex }), #{φ | mkComplex φ = w} = 2"
] | suffices ∀ w : { w : InfinitePlace K // IsComplex w },
#{φ : {φ //¬ ComplexEmbedding.IsReal φ} | mkComplex φ = w} = 2 by
rw [Fintype.card, Finset.card_eq_sum_ones, ← Finset.sum_fiberwise _ (fun φ => mkComplex φ)]
simp_rw [Finset.sum_const, this, smul_eq_mul, mul_one, Fintype.card, Finset.card_eq_sum_ones,
... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 127,
"column": 4
} | {
"line": 133,
"column": 39
} | {
"line": 135,
"column": 0
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\ninst✝ : NumberField K\nw₁ : InfinitePlace K\nB : ℝ≥0\nhf : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0\nS : ℝ≥0 := ∏ w ∈ Finset.univ.erase w₁, f w ^ w.mult\n⊢ ∏ w, Function.update f w₁ ((B * S⁻¹) ^ (↑w₁.mult)⁻¹) w ^ w.mult = B",
... | [] | rw [← Finset.mul_prod_erase Finset.univ _ (Finset.mem_univ w₁), Function.update_self,
Finset.prod_congr rfl fun w hw => by rw [Function.update_of_ne (Finset.ne_of_mem_erase hw)],
← NNReal.rpow_natCast, ← NNReal.rpow_mul, inv_mul_cancel₀, NNReal.rpow_one, mul_assoc,
inv_mul_cancel₀, mul_one]
· rw [... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 127,
"column": 4
} | {
"line": 133,
"column": 39
} | {
"line": 135,
"column": 0
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\ninst✝ : NumberField K\nw₁ : InfinitePlace K\nB : ℝ≥0\nhf : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0\nS : ℝ≥0 := ∏ w ∈ Finset.univ.erase w₁, f w ^ w.mult\n⊢ ∏ w, Function.update f w₁ ((B * S⁻¹) ^ (↑w₁.mult)⁻¹) w ^ w.mult = B",
... | [] | rw [← Finset.mul_prod_erase Finset.univ _ (Finset.mem_univ w₁), Function.update_self,
Finset.prod_congr rfl fun w hw => by rw [Function.update_of_ne (Finset.ne_of_mem_erase hw)],
← NNReal.rpow_natCast, ← NNReal.rpow_mul, inv_mul_cancel₀, NNReal.rpow_one, mul_assoc,
inv_mul_cancel₀, mul_one]
· rw [... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 167,
"column": 82
} | {
"line": 174,
"column": 20
} | {
"line": 176,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝⁴ : Field k\nK : Type u_2\ninst✝³ : Field K\nF : Type u_3\ninst✝² : Field F\ninst✝¹ : Algebra k K\nσ : Gal(K/k)\nw : InfinitePlace K\ninst✝ : IsGalois k K\n⊢ Quotient (MulAction.orbitRel Gal(K/k) (InfinitePlace K)) ≃ InfinitePlace k",
"ppTerm": "?m.22",
"assigned": true,
... | [] | by
refine Equiv.ofBijective (Quotient.lift (comap · (algebraMap k K))
fun _ _ e ↦ (mem_orbit_iff.mp e).symm) ⟨?_, ?_⟩
· rintro ⟨w⟩ ⟨w'⟩ e
exact Quotient.sound (mem_orbit_iff.mpr e.symm)
· intro w
obtain ⟨w', hw⟩ := comap_surjective (K := K) w
exact ⟨⟦w'⟧, hw⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 62
} | {
"line": 71,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝ : Field K\nx : (K →+* ℂ) → ℂ\nφ : K →+* ℂ\nhx : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))\n⊢ ∀ x ∈ Set.range ⇑(canonicalEmbedding K), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)",
"ppTerm": "?refine_1",
"assigned": true,
"usedCo... | [] | rintro _ ⟨x, rfl⟩
rw [apply_at, apply_at, ComplexEmbedding.conjugate_coe_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 62
} | {
"line": 71,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝ : Field K\nx : (K →+* ℂ) → ℂ\nφ : K →+* ℂ\nhx : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))\n⊢ ∀ x ∈ Set.range ⇑(canonicalEmbedding K), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)",
"ppTerm": "?refine_1",
"assigned": true,
"usedCo... | [] | rintro _ ⟨x, rfl⟩
rw [apply_at, apply_at, ComplexEmbedding.conjugate_coe_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 129,
"column": 32
} | {
"line": 129,
"column": 48
} | {
"line": 129,
"column": 48
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : Basis (K →+* ℂ) ℂ ((K →+* ℂ) → ℂ) := Pi.basisFun ℂ (K →+* ℂ)\ne : (K →+* ℂ) ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := Fintype.equivOfCardEq ⋯\nM : Matrix (K →+* ℂ) (K →+* ℂ) ℂ := B.toMatrix fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))\nN ... | [] | { ext : 2; rfl } | Lean.Elab.Tactic.evalTacticSeqBracketed | Lean.Parser.Tactic.tacticSeqBracketed |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 129,
"column": 32
} | {
"line": 129,
"column": 48
} | {
"line": 129,
"column": 48
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : Basis (K →+* ℂ) ℂ ((K →+* ℂ) → ℂ) := Pi.basisFun ℂ (K →+* ℂ)\ne : (K →+* ℂ) ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := Fintype.equivOfCardEq ⋯\nM : Matrix (K →+* ℂ) (K →+* ℂ) ℂ := B.toMatrix fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))\nN ... | [] | { ext : 2; rfl } | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 467,
"column": 22
} | {
"line": 467,
"column": 79
} | {
"line": 467,
"column": 79
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\n⊢ volume (fundamentalDomain (fractionalIdealLatticeBasis K I)) ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Units.val",
"InnerProd... | [] | classical exact ZSpan.measure_fundamentalDomain_ne_zero _ | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 467,
"column": 22
} | {
"line": 467,
"column": 79
} | {
"line": 467,
"column": 79
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\n⊢ volume (fundamentalDomain (fractionalIdealLatticeBasis K I)) ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Units.val",
"InnerProd... | [] | classical exact ZSpan.measure_fundamentalDomain_ne_zero _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 467,
"column": 22
} | {
"line": 467,
"column": 79
} | {
"line": 467,
"column": 79
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\n⊢ volume (fundamentalDomain (fractionalIdealLatticeBasis K I)) ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Units.val",
"InnerProd... | [] | classical exact ZSpan.measure_fundamentalDomain_ne_zero _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 771,
"column": 6
} | {
"line": 771,
"column": 26
} | {
"line": 771,
"column": 27
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nv : InfinitePlace K\ninst✝¹ : NumberField K\ninst✝ : NumberField L\nthis : Algebra K ℂ := v.embedding.toAlgebra\n⊢ (unramifiedPlacesOver L v).ncard + 2 * (ramifiedPlacesOver L v).ncard = finrank K L",
"ppTerm": "?... | [
"K : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nv : InfinitePlace K\ninst✝¹ : NumberField K\ninst✝ : NumberField L\nthis : Algebra K ℂ := v.embedding.toAlgebra\n⊢ (unramifiedPlacesOver L v).ncard + 2 * (ramifiedPlacesOver L v).ncard = Fintype.card (L →ₐ[K] ℂ)"
] | ← AlgHom.card K L ℂ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 192,
"column": 42
} | {
"line": 192,
"column": 54
} | {
"line": 192,
"column": 54
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nB : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))\nh_le : minkowskiBound K I ≤ volume (convexBodySum K B)\nx✝ : K\n⊢ ↑(FractionalIdeal.absNorm ↑I) * 4 ^ ↑(nrComplexPlaces K) *... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nB : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))\nh_le : minkowskiBound K I ≤ volume (convexBodySum K B)\nx✝ : K\n⊢ ↑(FractionalIdeal.absNorm ↑I) * 4 ^ nrComplexPlaces K * √|↑(discr K)| ... | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 199,
"column": 2
} | {
"line": 203,
"column": 12
} | {
"line": 205,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ∃ a,\n a ≠ 0 ∧\n ↑|(Algebra.norm ℚ) ↑a| ≤\n (4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K).factorial / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Rat.inst... | [] | obtain ⟨_, h_mem, h_nz, h_nm⟩ := exists_ne_zero_mem_ideal_of_norm_le_mul_sqrt_discr K ↑1
obtain ⟨a, rfl⟩ := (FractionalIdeal.mem_one_iff _).mp h_mem
refine ⟨a, ne_zero_of_map h_nz, ?_⟩
simp_rw [Units.val_one, FractionalIdeal.absNorm_one, Rat.cast_one, one_mul] at h_nm
exact h_nm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 199,
"column": 2
} | {
"line": 203,
"column": 12
} | {
"line": 205,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ∃ a,\n a ≠ 0 ∧\n ↑|(Algebra.norm ℚ) ↑a| ≤\n (4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K).factorial / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Rat.inst... | [] | obtain ⟨_, h_mem, h_nz, h_nm⟩ := exists_ne_zero_mem_ideal_of_norm_le_mul_sqrt_discr K ↑1
obtain ⟨a, rfl⟩ := (FractionalIdeal.mem_one_iff _).mp h_mem
refine ⟨a, ne_zero_of_map h_nz, ?_⟩
simp_rw [Units.val_one, FractionalIdeal.absNorm_one, Rat.cast_one, one_mul] at h_nm
exact h_nm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Invariant.Galois | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 15
} | {
"line": 64,
"column": 0
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Field K\ninst✝¹² : Field L\ninst✝¹¹ : Algebra A K\ninst✝¹⁰ : Algebra B L\ninst✝⁹ : IsFractionRing A K\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : Algebra A B\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra ... | [] | exact ⟨a, hk⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 618,
"column": 6
} | {
"line": 618,
"column": 13
} | {
"line": 618,
"column": 13
} | [
{
"pp": "case inr.«1»\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nw : { w // w.IsComplex }\n⊢ (x (↑w).embedding - (starRingEnd ℂ) (x (↑w).embedding)) / (2 * I) =\n -(2⁻¹ * I * x (↑w).embedding) + 2⁻... | [
"case inr.«1»\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nw : { w // w.IsComplex }\n⊢ x (↑w).embedding * I⁻¹ * (1 / 2) + (starRingEnd ℂ) (x (↑w).embedding) * I⁻¹ * (-1 / 2) =\n x (↑w).embedding * I * (-... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.RamificationInertia.Unramified | {
"line": 58,
"column": 6
} | {
"line": 58,
"column": 14
} | {
"line": 59,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : CommRing T\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Algebra S T\ninst✝⁸ : Algebra R T\ninst✝⁷ : IsScalarTower R S T\np : Ideal S\nP : Ideal T\ninst✝⁶ : P.LiesOver p\ninst✝⁵ : p.IsPrime\ninst✝⁴ : P.IsPrime\ninst✝³ : ... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : CommRing T\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Algebra S T\ninst✝⁸ : Algebra R T\ninst✝⁷ : IsScalarTower R S T\nP : Ideal T\ninst✝⁶ : P.IsPrime\ninst✝⁵ : IsUnramifiedAt R P\ninst✝⁴ : EssFiniteType R S\ninst✝³ : EssFiniteTy... | subst hp | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 67,
"column": 37
} | {
"line": 67,
"column": 52
} | {
"line": 67,
"column": 53
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝⁷ : CommRing A\nB : Type u_2\ninst✝⁶ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁵ : CommRing K\ninst✝⁴ : Algebra A K\ninst✝³ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝² : CommRing L\ninst✝¹ : Algebra B L\ninst✝ : IsLocalization N L\n... | [
"case refine_2\nA : Type u_1\ninst✝⁷ : CommRing A\nB : Type u_2\ninst✝⁶ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁵ : CommRing K\ninst✝⁴ : Algebra A K\ninst✝³ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝² : CommRing L\ninst✝¹ : Algebra B L\ninst✝ : IsLocalization N L\nhf : M ≤ Sub... | mul_comm (f a), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Unramified | {
"line": 108,
"column": 8
} | {
"line": 108,
"column": 46
} | {
"line": 109,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsDomain R\ninst✝³ : IsDomain S\ninst✝² : Module.IsTorsionFree R S\ninst✝¹ : CharZero R\ninst✝ : Algebra.IsIntegral R S\nthis✝ : IsFractionRing S (Localization.AtPrime ⊥)\nthis : Field (Localization.AtP... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsDomain R\ninst✝³ : IsDomain S\ninst✝² : Module.IsTorsionFree R S\ninst✝¹ : CharZero R\ninst✝ : Algebra.IsIntegral R S\nthis✝ : IsFractionRing S (Localization.AtPrime ⊥)\nthis : Field (Localization.AtPrime ⊥) := I... | faithfulSMul_iff_algebraMap_injective, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Quotient.Index | {
"line": 79,
"column": 17
} | {
"line": 79,
"column": 46
} | {
"line": 79,
"column": 46
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Ideal R\nN : Submodule R M\ninst✝ : Finite (R ⧸ I)\ns : Finset M\nhs : span R ↑s = N\nval✝ : Fintype (R ⧸ I)\ne : (↥N ⧸ comap N.subtype (I • N)) ≃ₗ[R] (R ⧸ I) ⊗[R] ↥N :=\n (comap N.subtype (I • N)).quotE... | [
"R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Ideal R\nN : Submodule R M\ninst✝ : Finite (R ⧸ I)\ns : Finset M\nhs : span R ↑s = N\nval✝ : Fintype (R ⧸ I)\ne : (↥N ⧸ comap N.subtype (I • N)) ≃ₗ[R] (R ⧸ I) ⊗[R] ↥N :=\n (comap N.subtype (I • N)).quotEquivOfEq (I ... | rw [← H, LinearMap.mem_range] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Quotient.Index | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 81
} | {
"line": 99,
"column": 4
} | [
{
"pp": "case insert\nR : Type u_1\ninst✝ : CommRing R\nι : Type u_3\nI : ι → Ideal R\na : ι\ns : Finset ι\nhas : a ∉ s\nIH : (∀ i ∈ s, (I i).FG) → (∀ i ∈ s, Finite (R ⧸ I i)) → Finite (R ⧸ ∏ i ∈ s, I i)\nhI : ∀ i ∈ insert a s, (I i).FG\nhI' : ∀ i ∈ insert a s, Finite (R ⧸ I i)\nthis : Finite (R ⧸ I a)\n⊢ Finit... | [
"case insert\nR : Type u_1\ninst✝ : CommRing R\nι : Type u_3\nI : ι → Ideal R\na : ι\ns : Finset ι\nhas : a ∉ s\nIH : (∀ i ∈ s, (I i).FG) → (∀ i ∈ s, Finite (R ⧸ I i)) → Finite (R ⧸ ∏ i ∈ s, I i)\nhI : ∀ i ∈ insert a s, (I i).FG\nhI' : ∀ i ∈ insert a s, Finite (R ⧸ I i)\nthis✝ : Finite (R ⧸ I a)\nthis : Finite (R ⧸... | have := IH (fun i hi ↦ hI _ (by simp [hi])) (fun i hi ↦ hI' _ (by simp [hi])) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 166,
"column": 50
} | {
"line": 166,
"column": 70
} | {
"line": 166,
"column": 70
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : CommRing A\nB : Type u_2\ninst✝¹⁰ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁹ : CommRing K\ninst✝⁸ : Algebra A K\ninst✝⁷ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝⁶ : CommRing L\ninst✝⁵ : Algebra B L\ninst✝⁴ : IsLocalization N L\nhf : M ≤ Sub... | [
"A : Type u_1\ninst✝¹¹ : CommRing A\nB : Type u_2\ninst✝¹⁰ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁹ : CommRing K\ninst✝⁸ : Algebra A K\ninst✝⁷ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝⁶ : CommRing L\ninst✝⁵ : Algebra B L\ninst✝⁴ : IsLocalization N L\nhf : M ≤ Submonoid.comap... | coe_extended_eq_span | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 187,
"column": 38
} | {
"line": 187,
"column": 58
} | {
"line": 187,
"column": 58
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : CommRing A\nB : Type u_2\ninst✝⁶ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁵ : CommRing K\ninst✝⁴ : Algebra A K\ninst✝³ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝² : CommRing L\ninst✝¹ : Algebra B L\ninst✝ : IsLocalization N L\nhf : M ≤ Submon... | [
"A : Type u_1\ninst✝⁷ : CommRing A\nB : Type u_2\ninst✝⁶ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁵ : CommRing K\ninst✝⁴ : Algebra A K\ninst✝³ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝² : CommRing L\ninst✝¹ : Algebra B L\ninst✝ : IsLocalization N L\nhf : M ≤ Submonoid.comap f ... | coe_extended_eq_span | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Int | {
"line": 100,
"column": 57
} | {
"line": 100,
"column": 81
} | {
"line": 100,
"column": 81
} | [
{
"pp": "case inl\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Module.Free ℤ S\nI : Ideal S\nh✝ : Finite (S ⧸ I)\nthis : Fintype (S ⧸ I)\nh_main : ∀ {d : ℕ}, ↑d ∈ I ↔ ∀ (x : S ⧸ I), d • x = 0\n⊢ absNorm (under ℤ I) ∣ Nat.card (S ⧸ I)",
"ppTerm": "?inl",
"assigned": true,
... | [
"case inl\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Module.Free ℤ S\nI : Ideal S\nh✝ : Finite (S ⧸ I)\nthis : Fintype (S ⧸ I)\nh_main : ∀ {d : ℕ}, ↑d ∈ I ↔ ∀ (x : S ⧸ I), d • x = 0\n⊢ absNorm (under ℤ I) ∣ Fintype.card (S ⧸ I)"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Instances | {
"line": 144,
"column": 26
} | {
"line": 144,
"column": 63
} | {
"line": 144,
"column": 64
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\nP : Ideal R\ninst✝¹ : P.IsPrime\ninst✝ : FaithfulSMul R S\n⊢ (algebraMap Rₚ L).comp (algebraMap R Rₚ) = ((algebra... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\nP : Ideal R\ninst✝¹ : P.IsPrime\ninst✝ : FaithfulSMul R S\n⊢ (algebraMap Rₚ L).comp (algebraMap R Rₚ) = (algebraMap S L).comp... | ← IsScalarTower.algebraMap_eq S Sₚ L, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 50
} | {
"line": 180,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\nS : Type u_3\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : IsIntegrallyClosed R\ninst✝³ : IsIntegrallyClosed S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\ninst✝ : IsTorsionFree R S\neq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I... | [
"case inr.inl\nR : Type u_1\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\nS : Type u_3\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : IsIntegrallyClosed R\ninst✝³ : IsIntegrallyClosed S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\ninst✝ : IsTorsionFree R S\neq_bot_or_top : ∀ (I : Ideal R), I = ⊥ ∨ I = ⊤\nI ... | rcases eq_bot_or_top (spanNorm R J) with hJ | hJ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 11
} | {
"line": 163,
"column": 4
} | [
{
"pp": "R : Type u\nK : Type v\nL : Type z\np : R\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra R L\ninst✝⁴ : Algebra R K\ninst✝³ : IsScalarTower R K L\ninst✝² : IsDomain R\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _r... | [
"R : Type u\nK : Type v\nL : Type z\np : R\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra R L\ninst✝⁴ : Algebra R K\ninst✝³ : IsScalarTower R K L\ninst✝² : IsDomain R\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _root_.Prime p... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 142,
"column": 2
} | {
"line": 143,
"column": 91
} | {
"line": 145,
"column": 0
} | [
{
"pp": "case refine_2\np k : ℕ\nhp : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot (zeta (p ^ k) ℚ (CyclotomicField (p ^ k) ℚ)) (p ^ k)\nx : CyclotomicField (p ^ k) ℚ\n⊢ (∃ y, (algebraMap (CyclotomicRing (p ^ k) ℤ ℚ) (CyclotomicField (p ^ k) ℚ)) y = x) → IsIntegral ℤ x",
"ppTerm": "?refine_2",
"assigned": t... | [] | · rintro ⟨y, rfl⟩
exact IsIntegral.algebraMap ((IsCyclotomicExtension.integral {p ^ k} ℤ _).isIntegral _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 495,
"column": 2
} | {
"line": 495,
"column": 39
} | {
"line": 497,
"column": 0
} | [
{
"pp": "p : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nhcycl : IsCyclotomicExtension {p} ℚ K\nhζ✝ : IsPrimitiveRoot ζ p\nthis : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ (0 + 1))\n⊢ Finite (𝓞 K ⧸ Ideal.span {hζ✝.toInteger - 1})",
"ppTerm... | [] | exact hζ.finite_quotient_span_sub_one | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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