module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.NumberTheory.NumberField.Discriminant.Different | {
"line": 161,
"column": 59
} | {
"line": 161,
"column": 79
} | [
{
"pp": "L : Type u_3\ninst✝¹ : Field L\ninst✝ : NumberField L\nK₁ K₂ : IntermediateField ℚ L\nh₁ : K₁.LinearDisjoint ↥K₂\nh₂ : K₁ ⊔ K₂ = ⊤\nh₃ :\n IsCoprime (Ideal.map (algebraMap (𝓞 ↥K₁) (𝓞 L)) (differentIdeal ℤ (𝓞 ↥K₁)))\n (Ideal.map (algebraMap (𝓞 ↥K₂) (𝓞 L)) (differentIdeal ℤ (𝓞 ↥K₂)))\nx✝ : Alge... | rwa [isCoprime_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.NumberField.Discriminant.Different | {
"line": 161,
"column": 59
} | {
"line": 161,
"column": 79
} | [
{
"pp": "L : Type u_3\ninst✝¹ : Field L\ninst✝ : NumberField L\nK₁ K₂ : IntermediateField ℚ L\nh₁ : K₁.LinearDisjoint ↥K₂\nh₂ : K₁ ⊔ K₂ = ⊤\nh₃ :\n IsCoprime (Ideal.map (algebraMap (𝓞 ↥K₁) (𝓞 L)) (differentIdeal ℤ (𝓞 ↥K₁)))\n (Ideal.map (algebraMap (𝓞 ↥K₂) (𝓞 L)) (differentIdeal ℤ (𝓞 ↥K₂)))\nx✝ : Alge... | rwa [isCoprime_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Discriminant.Different | {
"line": 161,
"column": 59
} | {
"line": 161,
"column": 79
} | [
{
"pp": "L : Type u_3\ninst✝¹ : Field L\ninst✝ : NumberField L\nK₁ K₂ : IntermediateField ℚ L\nh₁ : K₁.LinearDisjoint ↥K₂\nh₂ : K₁ ⊔ K₂ = ⊤\nh₃ :\n IsCoprime (Ideal.map (algebraMap (𝓞 ↥K₁) (𝓞 L)) (differentIdeal ℤ (𝓞 ↥K₁)))\n (Ideal.map (algebraMap (𝓞 ↥K₂) (𝓞 L)) (differentIdeal ℤ (𝓞 ↥K₂)))\nx✝ : Alge... | rwa [isCoprime_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 12
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝²¹ : CommRing R\ninst✝²⁰ : IsDomain R\nS : Type u_3\ninst✝¹⁹ : CommRing S\ninst✝¹⁸ : IsDomain S\ninst✝¹⁷ : IsIntegrallyClosed R\ninst✝¹⁶ : IsIntegrallyClosed S\ninst✝¹⁵ : Algebra R S\ninst✝¹⁴ : Module.Finite R S\ninst✝¹³ : IsTorsionFree R S\nT : Type u_4\ninst✝¹² : CommRing... | subst hP | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 123,
"column": 55
} | {
"line": 123,
"column": 64
} | [
{
"pp": "case refine_2.h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nh : ∀ (w : InfinitePlace K), w ((algebraMap (𝓞 K) K) ↑x) = 1\nw : { w // w ≠ w₀ }\n⊢ ↑(↑w).mult * 0 = 0 w",
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 156,
"column": 6
} | {
"line": 157,
"column": 62
} | [
{
"pp": "case pos.refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nr : ℝ\nx : (𝓞 K)ˣ\nhr : 0 ≤ r\nh : ‖(logEmbedding K) (Additive.ofMul x)‖ ≤ r\nw : InfinitePlace K\ntool : ∀ (x : ℝ), 0 ≤ x → x ≤ ↑w.mult * x\nhw : w = w₀\nhyp : ↑w₀.mult * |Real.log (w₀ ((algebraMap (𝓞 K) K) ↑x))| ≤ ∑ x_1, |(lo... | refine (sum_le_card_nsmul univ _ _
(fun w _ ↦ logEmbedding_component_le hr h w)).trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 309,
"column": 2
} | {
"line": 310,
"column": 50
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw₁ : InfinitePlace K\nB : ℕ\nhB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B\n⊢ ∃ n m, n < m ∧ Ideal.span {↑(seq K w₁ hB n)} = Ideal.span {↑(seq K w₁ hB m)}",
"usedConstants": [
"Nat.instMulZeroOneClass",
"NonUnitalCommRing.t... | refine Set.Finite.exists_lt_map_eq_of_forall_mem (t := {I : Ideal (𝓞 K) | Ideal.absNorm I ≤ B})
(fun n ↦ ?_) (Ideal.finite_setOf_absNorm_le B) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 859,
"column": 2
} | {
"line": 861,
"column": 29
} | [
{
"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... | have hPbot : P ≠ ⊥ := by
rintro rfl; apply hp'
rwa [Ideal.bot_mul] at ha | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 482,
"column": 52
} | {
"line": 487,
"column": 32
} | [
{
"pp": "p 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))\n⊢ Finite (𝓞 K ⧸ Ideal.span {hζ.toInteger - 1})",
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"Set.fi... | by
have : NumberField K := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K
refine Ideal.finiteQuotientOfFreeOfNeBot _ (fun h ↦ ?_)
simp only [Ideal.span_singleton_eq_bot, sub_eq_zero] at h
exact hζ.ne_one (one_lt_pow₀ hp.1.one_lt (Nat.zero_ne_add_one k).symm)
(RingOfIntegers.ext_iff.1 h) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic | {
"line": 99,
"column": 12
} | {
"line": 99,
"column": 21
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\na b : F\nha : ¬a = 0\nhb : b = 0\n⊢ quadraticCharFun F (a * 0) = quadraticCharFun F a * quadraticCharFun F 0",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.h... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.Basic | {
"line": 194,
"column": 6
} | {
"line": 194,
"column": 20
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℕ\n⊢ legendreSym p ↑a = -1 ↔ ¬IsSquare ↑a",
"usedConstants": [
"legendreSym.eq_neg_one_iff",
"Int.cast",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"ZMod.commRing",
"cong... | eq_neg_one_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.Basic | {
"line": 230,
"column": 30
} | {
"line": 230,
"column": 39
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℤ\nha : ↑a ≠ 0\nx : ZMod p\nhx : x ≠ 0\nhxy : x ^ 2 - ↑a * 0 = 0\n⊢ False",
"usedConstants": [
"Int.cast",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"ZMod.commRing",
"MulZe... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Fermat | {
"line": 167,
"column": 10
} | {
"line": 167,
"column": 13
} | [
{
"pp": "case hnot\na n p : ℕ\nhp : Prime p\nhp2 : p ≠ 2\nhpdvd : p ∣ a ^ 2 ^ n + 1\nthis✝ : Fact (2 < p)\nthis : Fact (Prime p)\nha1 : ↑a ^ 2 ^ n = -1\nha0 : ↑a ≠ 0\n⊢ ¬↑a ^ 2 ^ n = 1",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"MulOne.toOne",
"Monoid.toMulOneClass",
... | ha1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 621,
"column": 2
} | {
"line": 622,
"column": 31
} | [
{
"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\n⊢ S.a + S.b + (S.a + ↑η * S.b) * ↑η + (S.a + (-↑⋯.unit - 1) * S.b) * (-↑⋯.unit - 1) = 0",
"usedConstants": [
"IsPrimitiveRoot.toInteger_isPrimitiveR... | calc _ = S.a+S.b+η^2*S.b-S.a+η^2*S.b+2*η*S.b+S.b := by ring
_ = 0 := by rw [eta_sq]; ring | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.NumberTheory.Fermat | {
"line": 182,
"column": 4
} | {
"line": 182,
"column": 64
} | [
{
"pp": "k n : ℕ\nhn : 1 < n + succ 1\nhp : Prime (k * 2 ^ (n + succ 1 + 1) + 1)\nhpdvd : k * 2 ^ (n + succ 1 + 1) + 1 ∣ (n + succ 1).fermatNumber\nthis : Fact (Prime (k * 2 ^ (n + succ 1 + 1) + 1))\nhp2 : k * 2 ^ (n + succ 1 + 1) + 1 ≠ 2\n⊢ (k * 2 ^ (n + succ 1 + 1) + 1) % 8 = 1",
"usedConstants": [
... | rw [add_assoc, pow_add, ← mul_assoc, ← mod_add_mod, mul_mod] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 754,
"column": 71
} | {
"line": 754,
"column": 90
} | [
{
"pp": "K : Type u\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : NumberField K\nF₁ F₂ : IntermediateField ℚ K\nn₁ n₂ : ℕ\ninst✝³ : NeZero n₁\ninst✝² : NeZero n₂\ninst✝¹ : IsCyclotomicExtension {n₁} ℚ ↥F₁\ninst✝ : IsCyclotomicExtension {n₂} ℚ ↥F₂\nζ₁ : ↥F₁\nhζ₁✝ : IsPrimitiveRoot ζ₁ n₁\nh₁ : ℤ[hζ₁✝.toInteger... | Set.singleton_union | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 666,
"column": 2
} | {
"line": 666,
"column": 23
} | [
{
"pp": "case hcong\nK : 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\nthis : λ ^ 2 ∣ λ ^ 4\nh : λ ^ 2 ∣ ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3\n⊢ ∃ n, λ ^ 2 ∣ ↑S.u₄ - ↑n",
"usedConstants": []
}
] | rcases h with ⟨X, hX⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.FrobeniusNumber | {
"line": 171,
"column": 51
} | {
"line": 171,
"column": 60
} | [
{
"pp": "case pos\ns : Set ℕ\nh0 : ¬setGcd s = 0\nt : Finset ℕ\nhts : ↑t ⊆ s\na : ↥t → ℤ\neq : ∑ i, a i • ↑↑i = ↑(setGcd s)\nx : ℕ\nhxs : x ∈ s\nhx : x ≠ 0\nn : ℕ := x / setGcd s * ∑ i, (-a i).toNat * ↑i\nc : ℕ\nge : n + c ≥ n\ndvd : setGcd s ∣ n + c\nq : ℕ := c / x\nr : ℕ := c % x\nrx : ℕ\nhrx : r = setGcd s *... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.Int | {
"line": 38,
"column": 6
} | {
"line": 38,
"column": 77
} | [
{
"pp": "n : ℕ\nih : padicValRat 2 (harmonic n) = -↑(Nat.log 2 n)\nhn : n ≠ 0\n⊢ padicValRat 2 (harmonic n) ≠ padicValRat 2 (↑(n + 1))⁻¹",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"Rat",
"AddGroupWithOne.toAddMonoidWithOne",
... | rw [ih, padicValRat.inv, padicValRat.of_nat, Ne, neg_inj, Nat.cast_inj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 45
} | [
{
"pp": "n : ℕ\nhn : 0 < n\nx : ℝ\nhx : x ∈ Icc (↑n) (↑n + 1)\n⊢ 0 < x",
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Nat.cast_pos",
"Real.lattice",
"Par... | exact (Nat.cast_pos.mpr hn).trans_le hx.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 37
} | [
{
"pp": "n : ℕ\nhn : 0 < n\nhv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x\n⊢ ∫ (x : ℝ) in ↑n..↑n + 1, 1 / x - ↑n / x ^ 2 =\n (∫ (x : ℝ) in ↑n..↑n + 1, 1 / x) - ↑n * ∫ (x : ℝ) in ↑n..↑n + 1, 1 / x ^ 2",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"MulOne.toOne",
"Real",
... | simp_rw [← mul_one_div (n : ℝ)] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 45
} | [
{
"pp": "n : ℕ\nhn : 0 < n\ns : ℝ\nhs : 1 < s\nx : ℝ\nhx : x ∈ Icc (↑n) (↑n + 1)\n⊢ 0 < x",
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Nat.cast_pos",
"Real.l... | exact (Nat.cast_pos.mpr hn).trans_le hx.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 184,
"column": 4
} | {
"line": 188,
"column": 13
} | [
{
"pp": "case e_a\nN : ℕ\ns : ℝ\nhs : 1 < s\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))",
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZero... | induction N with
| zero => simp
| succ N hN =>
rw [Finset.sum_range_succ, hN, Nat.cast_add_one]
ring_nf | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 184,
"column": 4
} | {
"line": 188,
"column": 13
} | [
{
"pp": "case e_a\nN : ℕ\ns : ℝ\nhs : 1 < s\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))",
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZero... | induction N with
| zero => simp
| succ N hN =>
rw [Finset.sum_range_succ, hN, Nat.cast_add_one]
ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 184,
"column": 4
} | {
"line": 188,
"column": 13
} | [
{
"pp": "case e_a\nN : ℕ\ns : ℝ\nhs : 1 < s\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))",
"usedConstants": [
"CharP.cast_eq_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZero... | induction N with
| zero => simp
| succ N hN =>
rw [Finset.sum_range_succ, hN, Nat.cast_add_one]
ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Height.Basic | {
"line": 424,
"column": 17
} | {
"line": 424,
"column": 80
} | [
{
"pp": "case h.inr\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\ne : ↑(support x) ⊕ ↑(support x)ᶜ ≃ ι := Equiv.Set.sumCompl (support x)\nval : ↑(support x)ᶜ\n⊢ x (e (Sum.inr val)) = Sum.elim (fun i ↦ x ↑i) 0 (Sum.inr val)",
"usedConstants": [
... | exact notMem_support.mp <| (Set.mem_compl_iff _ _ ).mp val.prop | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Height.Basic | {
"line": 424,
"column": 17
} | {
"line": 424,
"column": 80
} | [
{
"pp": "case h.inr\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\ne : ↑(support x) ⊕ ↑(support x)ᶜ ≃ ι := Equiv.Set.sumCompl (support x)\nval : ↑(support x)ᶜ\n⊢ x (e (Sum.inr val)) = Sum.elim (fun i ↦ x ↑i) 0 (Sum.inr val)",
"usedConstants": [
... | exact notMem_support.mp <| (Set.mem_compl_iff _ _ ).mp val.prop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Height.Basic | {
"line": 424,
"column": 17
} | {
"line": 424,
"column": 80
} | [
{
"pp": "case h.inr\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\ne : ↑(support x) ⊕ ↑(support x)ᶜ ≃ ι := Equiv.Set.sumCompl (support x)\nval : ↑(support x)ᶜ\n⊢ x (e (Sum.inr val)) = Sum.elim (fun i ↦ x ↑i) 0 (Sum.inr val)",
"usedConstants": [
... | exact notMem_support.mp <| (Set.mem_compl_iff _ _ ).mp val.prop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 156,
"column": 26
} | {
"line": 157,
"column": 96
} | [
{
"pp": "L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nx : L\n⊢ v.norm (x - x) = 0",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"Real... | by
simp only [sub_self, Valuation.norm, Valuation.map_zero, hv.hom.map_zero, NNReal.coe_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 215,
"column": 15
} | {
"line": 218,
"column": 7
} | [
{
"pp": "L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nx y z : L\n⊢ dist x z ≤ max (dist x y) (dist y z)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"... | by
refine (Valuation.norm_add_le _ (x - y) (y - z)).trans_eq' ?_
simp only [sub_add_sub_cancel]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RootsOfUnity.Lemmas | {
"line": 46,
"column": 52
} | {
"line": 46,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nμ : R\nhμ : IsPrimitiveRoot μ (n + 1)\n⊢ (-1) ^ n = ∏ k ∈ range n, -1",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
"... | prod_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 115,
"column": 81
} | {
"line": 120,
"column": 8
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ totalWeight K = ∑ v, v.mult",
"usedConstants": [
"Multiset.sum",
"Eq.mpr",
"MulOne.toOne",
"Real.partialOrder",
"Real",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"Finset.univ",
"Mul... | by
simp only [totalWeight]
convert sum_archAbsVal_eq (fun _ ↦ (1 : ℕ))
· rw [← Multiset.sum_map_toList, ← Fin.sum_univ_fun_getElem, ← Multiset.length_toList,
Fin.sum_const, Multiset.length_toList, smul_eq_mul, mul_one]
· simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 93,
"column": 70
} | {
"line": 93,
"column": 79
} | [
{
"pp": "case e_a.e_a\nF : Type u_1\nR : Type u_2\ninst✝⁴ : CommRing F\ninst✝³ : Nontrivial F\ninst✝² : Fintype F\ninst✝¹ : DecidableEq F\ninst✝ : CommRing R\nχ ψ : MulChar F R\n⊢ 0 = (χ 0 - 1) * 0 + 0 * (ψ (1 - 1) - 1)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRin... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 173,
"column": 63
} | {
"line": 173,
"column": 72
} | [
{
"pp": "F : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : Fintype F\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nχ φ : MulChar F R\nh : χ * φ ≠ 1\nψ : AddChar F R\n⊢ gaussSum (χ * φ) ψ * jacobiSum χ φ = ∑ x ∈ univ \\ {0}, ∑ x_1, χ x_1 * φ (x - x_1) * ψ x + φ (-1) * 0",
"usedConstants": [
"Eq.mpr",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 74
} | [
{
"pp": "F : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : Fintype F\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nχ φ : MulChar F R\nh : χ * φ ≠ 1\nψ : AddChar F R\nt : F\nht : ¬t = 0\n⊢ (∑ x, χ x * φ (t - x)) * ψ t = (∑ y, χ (t * y) * φ (t - t * y)) * ψ t",
"usedConstants": [
"NonUnitalCommRing.toN... | exact congrArg (· * ψ t) (Equiv.sum_comp (Equiv.mulLeft₀ t ht) _).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 218,
"column": 8
} | {
"line": 218,
"column": 25
} | [
{
"pp": "F : Type u_1\nF' : Type u_2\ninst✝² : Fintype F\ninst✝¹ : Field F\ninst✝ : Field F'\nh : ringChar F' ≠ ringChar F\nχ φ : MulChar F F'\nhχ : χ ≠ 1\nhφ : φ ≠ 1\nhχφ : χ * φ ≠ 1\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nψ : PrimitiveAddChar F F' := FiniteField.primitiveC... | ← ringHomComp_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 66
} | [
{
"pp": "case a\nF : Type u_1\nF' : Type u_2\ninst✝² : Fintype F\ninst✝¹ : Field F\ninst✝ : Field F'\nh : ringChar F' ≠ ringChar F\nχ φ : MulChar F F'\nhχ : χ ≠ 1\nhφ : φ ≠ 1\nhχφ : χ * φ ≠ 1\nn : ℕ+\nhp : Nat.Prime (ringChar F)\nhc : Fintype.card F = ringChar F ^ ↑n\nψ : PrimitiveAddChar F F' := FiniteField.pr... | have Hφ : φ' ≠ 1 := (MulChar.ringHomComp_ne_one_iff hinj).mpr hφ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 189,
"column": 6
} | {
"line": 189,
"column": 14
} | [
{
"pp": "K : Type u_4\ninst✝¹ : Field K\nι : Type u_5\ninst✝ : Finite ι\nv : AbsoluteValue K ℝ\np : MvPolynomial ι K\nN : ℕ\nhp : p.IsHomogeneous N\nx : ι → K\n⊢ v ((eval x) p) ≤ (Finsupp.sum p fun x c ↦ v c) * (⨆ i, v (x i)) ^ N",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunL... | eval_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 262,
"column": 75
} | {
"line": 262,
"column": 84
} | [
{
"pp": "F : 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\n⊢ (χ 0 - 1) * 0 = 0 * (μ - 1) ^ 2",
"usedConstants": [
"Eq.mpr",
"NonUn... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 377,
"column": 6
} | {
"line": 377,
"column": 32
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\ng : R[X]\n⊢ h.map (g %ₘ f) = h.map g",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"AlgHom.algHomClass",
"IsAdjoinRootMonic.toIsAdjoinRoot",
"A... | ← RingHom.sub_mem_ker_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 406,
"column": 79
} | {
"line": 406,
"column": 91
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nn : ℕ\nhdeg : n < f.natDegree\na✝ : Nontrivial R\n⊢ (X ^ n).degree < f.degree",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Preorder.t... | degree_X_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 303,
"column": 4
} | {
"line": 304,
"column": 78
} | [
{
"pp": "case pos\nF : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype F\nn : ℕ\nhn : 2 < n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhμ : IsPrimitiveRoot μ n\nq : ℕ\nhq : Fintype.card F = n * q + 1\nz₁ : R\nhz₁ : z₁ ∈ ℤ[μ]\nHz₁ : ↑n = z₁ * (μ... | refine ⟨0, Subalgebra.zero_mem _, ?_⟩
rw [jacobiSum_comm, hψ₀, jacobiSum_one_nontrivial hχ₀, zero_mul, add_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 303,
"column": 4
} | {
"line": 304,
"column": 78
} | [
{
"pp": "case pos\nF : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype F\nn : ℕ\nhn : 2 < n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhμ : IsPrimitiveRoot μ n\nq : ℕ\nhq : Fintype.card F = n * q + 1\nz₁ : R\nhz₁ : z₁ ∈ ℤ[μ]\nHz₁ : ↑n = z₁ * (μ... | refine ⟨0, Subalgebra.zero_mem _, ?_⟩
rw [jacobiSum_comm, hψ₀, jacobiSum_one_nontrivial hχ₀, zero_mul, add_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 284,
"column": 59
} | {
"line": 284,
"column": 71
} | [
{
"pp": "K : Type u_4\ninst✝² : Field K\nι : Type u_5\nι' : Type u_6\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Finite ι'\np : ι' → MvPolynomial ι K\nh : (fun j ↦ constantCoeff (p j)) ≠ 0\nthis : Nonempty ι'\nv : ↑nonarchAbsVal\nj : ι'\nh₀ : coeff 0 (p j) ≠ 0\n⊢ 0 ∈ (p j).support",
"usedConstants": [
"F... | by simp [h₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 293,
"column": 2
} | {
"line": 316,
"column": 10
} | [
{
"pp": "F : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype F\nn : ℕ\nhn : 2 < n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhn' : n ∣ Fintype.card F - 1\nhμ : IsPrimitiveRoot μ n\n⊢ ∃ z ∈ ℤ[μ], jacobiSum χ ψ = -1 + z * (μ - 1) ^ 2",
"usedC... | obtain ⟨q, hq⟩ := hn'
rw [Nat.sub_eq_iff_eq_add NeZero.one_le] at hq
obtain ⟨z₁, hz₁, Hz₁⟩ := hμ.self_sub_one_pow_dvd_order hn
by_cases hχ₀ : χ = 1 <;> by_cases hψ₀ : ψ = 1
· rw [hχ₀, hψ₀, jacobiSum_one_one]
refine ⟨q * z₁, Subalgebra.mul_mem _ (Subalgebra.natCast_mem _ q) hz₁, ?_⟩
rw [hq, Nat.cast_add,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 293,
"column": 2
} | {
"line": 316,
"column": 10
} | [
{
"pp": "F : Type u_1\nR : Type u_2\ninst✝³ : Field F\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype F\nn : ℕ\nhn : 2 < n\nχ ψ : MulChar F R\nμ : R\nhχ : χ ^ n = 1\nhψ : ψ ^ n = 1\nhn' : n ∣ Fintype.card F - 1\nhμ : IsPrimitiveRoot μ n\n⊢ ∃ z ∈ ℤ[μ], jacobiSum χ ψ = -1 + z * (μ - 1) ^ 2",
"usedC... | obtain ⟨q, hq⟩ := hn'
rw [Nat.sub_eq_iff_eq_add NeZero.one_le] at hq
obtain ⟨z₁, hz₁, Hz₁⟩ := hμ.self_sub_one_pow_dvd_order hn
by_cases hχ₀ : χ = 1 <;> by_cases hψ₀ : ψ = 1
· rw [hχ₀, hψ₀, jacobiSum_one_one]
refine ⟨q * z₁, Subalgebra.mul_mem _ (Subalgebra.natCast_mem _ q) hz₁, ?_⟩
rw [hq, Nat.cast_add,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 308,
"column": 34
} | {
"line": 308,
"column": 43
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhΦ : Function.Even Φ\ns : ℂ\nthis : ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s = 0\n⊢ ↑N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s + ↑N ^ (-s) * 0 =\n ↑N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s",
"used... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 306,
"column": 97
} | {
"line": 310,
"column": 43
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhΦ : Function.Even Φ\ns : ℂ\n⊢ completedLFunction Φ s = ↑N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s",
"usedConstants": [
"ZMod.completedLFunction",
"Eq.mpr",
"Function.Even.mul_odd",
"NegZeroClass.toNeg",
"... | by
suffices ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s = 0 by
rw [completedLFunction, this, mul_zero, add_zero]
refine (hΦ.mul_odd fun j ↦ ?_).sum_eq_zero
rw [map_neg, completedHurwitzZetaOdd_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 315,
"column": 34
} | {
"line": 315,
"column": 43
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhΦ : Function.Odd Φ\ns : ℂ\nthis : ∑ j, Φ j * completedHurwitzZetaEven (toAddCircle j) s = 0\n⊢ ↑N ^ (-s) * 0 + ↑N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s =\n ↑N ^ (-s) * ∑ j, Φ j * completedHurwitzZetaOdd (toAddCircle j) s",
"usedCo... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 190,
"column": 58
} | {
"line": 190,
"column": 73
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nH :\n (fun s ↦ (s - 1) * LFunctionTrivChar N s) =ᶠ[𝓝[≠] 1] fun s ↦\n (∏ p ∈ N.primeFactors, (1 - ↑p ^ (-s))) * ((s - 1) * riemannZeta s)\n| (∏ p ∈ N.primeFactors, (1 - (↑p)⁻¹)) * 1",
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"Complex.commRing",
... | enter [1, 2, p] | Lean.Elab.Tactic.Conv.evalEnter | Lean.Parser.Tactic.Conv.enter |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 18
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nχ : DirichletCharacter ℂ N\nhχ : χ.IsPrimitive\ns : ℂ\nhN : N ≠ 1\nh_sum : ∑ j, χ j = 0\nε : ℂ := I ^ if χ.Even then 0 else 1\n⊢ ↑N ^ (s - 1) * χ (-1) / ε * ZMod.completedLFunction (𝓕 ⇑χ) s =\n ?m.245 * ZMod.completedLFunction (fun j ↦ χ⁻¹ (-1) * gaussSum χ stdAddChar * χ⁻¹... | congr 2 with j | Batteries.Tactic._aux_Batteries_Tactic_Congr___macroRules_Batteries_Tactic_congrConfigWith_1 | Batteries.Tactic.congrConfigWith |
Mathlib.NumberTheory.ZetaValues | {
"line": 174,
"column": 88
} | {
"line": 174,
"column": 97
} | [
{
"pp": "m : ℕ\nm0 : m ≠ 0\nx : ℝ\nm0' : ↑m ≠ 0\nf : ℕ → ℝ → ℝ := fun k x ↦ bernoulliFun k (↑m * x) - ↑m ^ k / ↑m * ∑ i ∈ Finset.range m, bernoulliFun k (x + ↑i / ↑m)\nk : ℕ\nh : ∀ (x : ℝ), f k x = 0\nd : ∀ (x : ℝ), HasDerivAt (f (k + 1)) 0 x\nc : ℝ\nfc : ∀ (x : ℝ), f (k + 1) x = c\n⊢ ((↑m)⁻¹ • ∫ (x : ℝ) in ↑m ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 252,
"column": 54
} | {
"line": 252,
"column": 63
} | [
{
"pp": "case inl\nk : ℕ\nhk : k ≠ 0\n⊢ 0 = -↑k ! / (2 * ↑π * I * 0) ^ k",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
"AddGroupWithOne.... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 173,
"column": 6
} | {
"line": 173,
"column": 20
} | [
{
"pp": "x : ℝ\nhx : x ∈ Icc 0 1\nk : ℕ\nhk : k.succ ≠ 0\n⊢ hurwitzZetaOdd (↑x) (1 - 2 * ↑k.succ) = 0",
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.t... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 26
} | [
{
"pp": "case e_a\nm n : ℕ\nz : ℂ\nx : ℝ\nHn : 0 ≤ (↑n + 1)⁻¹\n⊢ (↑n + 1) ^ ↑x / ↑m ^ ↑x = ((↑m / (↑n + 1)) ^ ↑x)⁻¹",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Complex.instPow",
"Complex.in... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 43
} | [
{
"pp": "f : ℕ → ℂ\nn : ℕ\nh : ∀ m ≤ n, f m = 0\nha : abscissaOfAbsConv f < ⊤\n⊢ Tendsto (fun x ↦ (↑n + 1) ^ ↑x * LSeries f ↑x) atTop (nhds (f (n + 1)))",
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"EReal",
"instTopEReal",
"exists_between",
"instPartia... | obtain ⟨y, hay, hyt⟩ := exists_between ha | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.ZetaValues | {
"line": 346,
"column": 22
} | {
"line": 346,
"column": 30
} | [
{
"pp": "case h.e'_5.h\nk : ℕ\nhk : 2 ≤ k\nx : ℝ\nhx : x ∈ Icc 0 1\nn : ℕ\n⊢ 1 / ↑n ^ k * ((fourier ↑n) ↑x + (-1) ^ k * (fourier (-↑n)) ↑x) =\n 1 / ↑↑n ^ k * (fourier ↑n) ↑x + 1 / (-↑↑n) ^ k * (fourier (-↑n)) ↑x",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Distrib.leftDistribClass",
... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 346,
"column": 4
} | {
"line": 346,
"column": 43
} | [
{
"pp": "case h.e'_5.h\nk : ℕ\nhk : 2 ≤ k\nx : ℝ\nhx : x ∈ Icc 0 1\nn : ℕ\n⊢ 1 / ↑n ^ k * ((fourier ↑n) ↑x + (-1) ^ k * (fourier (-↑n)) ↑x) =\n 1 / ↑↑n ^ k * (fourier ↑n) ↑x + 1 / ↑(-↑n) ^ k * (fourier (-↑n)) ↑x",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Distr... | rw [Int.cast_neg, mul_add, ← mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ZetaValues | {
"line": 439,
"column": 16
} | {
"line": 439,
"column": 25
} | [
{
"pp": "case h.e'_5.h\nk : ℕ\nhk : k ≠ 0\nn : ℕ\n⊢ 1 / ↑n ^ (2 * k) = 1 / ↑n ^ (2 * k) * Real.cos (2 * π * ↑n * 0)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 363,
"column": 42
} | {
"line": 363,
"column": 51
} | [
{
"pp": "case inl\nN : ℕ\nχ : DirichletCharacter ℂ N\ninst✝ : NeZero N\nt : ℝ\nh : χ ^ 2 ≠ 1 ∨ t ≠ 0\nHz : LFunction χ (1 + I * ↑t) = 0\nhz₁ : t ≠ 0 ∨ χ ≠ 1\nhz₂ : 2 * t ≠ 0 ∨ χ ^ 2 ≠ 1\n⊢ (1 / 0) ^ 3 * 0 * 1 = 0",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonA... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nv₁ v₂ : HeightOneSpectrum (𝓞 K)\n⊢ mk v₁ = mk v₂ ↔ v₁ = v₂",
"usedConstants": [
"Eq.mpr",
"NumberField.instCommRingRingOfIntegers",
"congrArg",
"NumberField.FinitePlace.mk",
"IsDedekindDomain.HeightOneSpectrum",
... | refine ⟨?_, fun a ↦ by rw [a]⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 67,
"column": 70
} | {
"line": 67,
"column": 81
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℤ\nhap : ↑a ≠ 0\n⊢ ↑a ^ (p / 2) * ↑(p / 2)! = ↑((∏ x ∈ Ico 1 (p / 2).succ, a) * ↑(p / 2)!)",
"usedConstants": [
"Int.instCommMonoid",
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHDiv",
"NonUnitalCom... | prod_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 61
} | [
{
"pp": "m₁ m₂ n : ℕ\n⊢ qrSign (m₁ * m₂) n = qrSign m₁ n * qrSign m₂ n",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"ZMod.χ₄",
"qrSign",
"ZMod.commRing",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiri... | simp_rw [qrSign, Nat.cast_mul, map_mul, jacobiSym.mul_left] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 61
} | [
{
"pp": "m₁ m₂ n : ℕ\n⊢ qrSign (m₁ * m₂) n = qrSign m₁ n * qrSign m₂ n",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"ZMod.χ₄",
"qrSign",
"ZMod.commRing",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiri... | simp_rw [qrSign, Nat.cast_mul, map_mul, jacobiSym.mul_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 61
} | [
{
"pp": "m₁ m₂ n : ℕ\n⊢ qrSign (m₁ * m₂) n = qrSign m₁ n * qrSign m₂ n",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"ZMod.χ₄",
"qrSign",
"ZMod.commRing",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiri... | simp_rw [qrSign, Nat.cast_mul, map_mul, jacobiSym.mul_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 407,
"column": 8
} | {
"line": 407,
"column": 81
} | [
{
"pp": "a✝ b : ℕ\nha : Odd a✝\nhb : Odd b\na x y : ℕ\n⊢ qrSign (x * y) a * J(↑(x * y) | a) = qrSign x a * J(↑x | a) * (qrSign y a * J(↑y | a))",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"qrSign",
"mul_mul_mul_comm",
"congrArg",... | simp_rw [qrSign.mul_left x y a, Nat.cast_mul, mul_left, mul_mul_mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 407,
"column": 8
} | {
"line": 407,
"column": 81
} | [
{
"pp": "a✝ b : ℕ\nha : Odd a✝\nhb : Odd b\na x y : ℕ\n⊢ qrSign (x * y) a * J(↑(x * y) | a) = qrSign x a * J(↑x | a) * (qrSign y a * J(↑y | a))",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"qrSign",
"mul_mul_mul_comm",
"congrArg",... | simp_rw [qrSign.mul_left x y a, Nat.cast_mul, mul_left, mul_mul_mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 407,
"column": 8
} | {
"line": 407,
"column": 81
} | [
{
"pp": "a✝ b : ℕ\nha : Odd a✝\nhb : Odd b\na x y : ℕ\n⊢ qrSign (x * y) a * J(↑(x * y) | a) = qrSign x a * J(↑x | a) * (qrSign y a * J(↑y | a))",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"qrSign",
"mul_mul_mul_comm",
"congrArg",... | simp_rw [qrSign.mul_left x y a, Nat.cast_mul, mul_left, mul_mul_mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 455,
"column": 8
} | {
"line": 455,
"column": 17
} | [
{
"pp": "case inl\nb : ℕ\nhb : Odd b\n⊢ J(↑0 | b) = J(↑0 | b % (4 * 0))",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"id",
"Nat.instMod",
"instHMod",
"instMulNat",
"instOfNatNat",
"Int"... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.DiscreteValuativeRel | {
"line": 67,
"column": 10
} | {
"line": 67,
"column": 47
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : ValuativeRel R\nv : Valuation R (WithZero (Multiplicative ℤ))\ninst✝ : v.Compatible\nthis : IsRankLeOne R\nh : Nontrivial (ValueGroupWithZero R)ˣ\nH : Nonempty (ValueGroupWithZero R ≃*o WithZero (Multiplicative ℤ))\n⊢ IsDiscrete R",
"usedConstan... | nonempty_orderIso_withZeroMul_int_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 27
} | [
{
"pp": "f : 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[atTop] fun n ↦... | rw [← integral_const_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 238,
"column": 8
} | {
"line": 238,
"column": 40
} | [
{
"pp": "case refine_3\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)\nt : ℝ\nht : t ∈ Set.Ioi T\n⊢ ‖S t - l * ↑t‖ * t ^ (... | have ht' : 0 < t := hT₀.trans ht | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 167,
"column": 4
} | {
"line": 169,
"column": 23
} | [
{
"pp": "case mpr\nK : Type u_1\nΓ₀ : Type u_2\ninst✝⁴ : Field K\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : Valued K Γ₀\ninst✝¹ : v.RankOne\ninst✝ : IsDiscreteValuationRing ↥𝒪[K]\nH : Finite 𝓀[K]\nε : ℝ\nεpos : ε > 0\np : ↥𝒪[K]\nhp : Irreducible p\nhp' : v ↑p < 1\nn : ℕ\nhn : ‖↑p‖ ^ n < ε\nhF : Fi... | have : {y : 𝒪[K] | v (y : K) ≤ v (p : K) ^ n} = Metric.closedBall 0 (‖p‖ ^ n) := by
ext
simp [← norm_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 328,
"column": 10
} | {
"line": 328,
"column": 18
} | [
{
"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... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LucasLehmer | {
"line": 509,
"column": 6
} | {
"line": 509,
"column": 14
} | [
{
"pp": "case h\np' : ℕ\nk : ℤ\nh : ω ^ 2 ^ p' * (ω ^ 2 ^ p' + ωb ^ 2 ^ p') = ω ^ 2 ^ p' * ↑((2 ^ (p' + 2) - 1) * k)\nt : 2 ^ p' + 2 ^ p' = 2 ^ (p' + 1)\n⊢ ω ^ 2 ^ (p' + 1) = ↑k * ↑(mersenne (p' + 2)) * ω ^ 2 ^ p' - 1",
"usedConstants": [
"PNat.val",
"Distrib.leftDistribClass",
"Int.cast",... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 200,
"column": 3
} | {
"line": 200,
"column": 59
} | [
{
"pp": "n : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nR : Type u_2\ninst✝ : Ring R\n𝒢 : Subgroup (GL n R)\n⊢ 𝒢.adjoinNegOne.relIndex 𝒢 ≠ 0",
"usedConstants": [
"Iff.mpr",
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
"congrArg",
"Matrix",
"PartialO... | by simp [Subgroup.relIndex_eq_one.mpr 𝒢.le_adjoinNegOne] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 96
} | [
{
"pp": "n : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nR : Type u_3\ninst✝ : CommRing R\n𝒢 : Subgroup (GL n R)\n⊢ 𝒢.adjoinNegOne.HasDetPlusMinusOne ↔ 𝒢.HasDetPlusMinusOne",
"usedConstants": [
"Subgroup.HasDetPlusMinusOne",
"CommSemiring.toSemiring",
"Matrix",
"Subgroup... | refine ⟨fun _ ↦ ⟨fun {g} hg ↦ HasDetPlusMinusOne.det_eq (𝒢.le_adjoinNegOne hg)⟩, fun _ ↦ ⟨?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ModularForms.SlashActions | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 73
} | [
{
"pp": "case h\nk : ℤ\nA : GL (Fin 2) ℝ\nf : ℍ → ℂ\nc : ℂ\nτ : ℍ\n⊢ ((c • f) ∣[k] A) τ = ((σ A) c • f ∣[k] A) τ",
"usedConstants": [
"UpperHalfPlane.glAction",
"Units.val",
"NormedCommRing.toSeminormedCommRing",
"ContinuousAlgEquivClass.toAlgEquivClass",
"Semigroup.toMul",
... | simp only [slash_apply, Pi.smul_apply, smul_eq_mul, map_mul, mul_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Modular | {
"line": 151,
"column": 35
} | {
"line": 151,
"column": 43
} | [
{
"pp": "case h.h.«1»\nz : ℍ\nthis✝ : Module ℝ (Fin 2 → ℝ) := Pi.normedSpace.toModule\nπ₀ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 0\nπ₁ : (Fin 2 → ℝ) →ₗ[ℝ] ℝ := LinearMap.proj 1\nf : (Fin 2 → ℝ) →ₗ[ℝ] ℂ := π₀.smulRight ↑z + π₁.smulRight 1\nf_def : ⇑f = fun p ↦ ↑(p 0) * ↑z + ↑(p 1)\nthis : (fun p ↦ normSq (↑(p 0... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.Identities | {
"line": 32,
"column": 13
} | {
"line": 32,
"column": 23
} | [
{
"pp": "Γ : Subgroup (GL (Fin 2) ℝ)\nk : ℤ\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\ninst✝ : SlashInvariantFormClass F Γ k\nf : F\nτ : ℍ\nh : ℝ\nhH : h ∈ Γ.strictPeriods\nthis : GeneralLinearGroup.upperRightHom h • τ = h +ᵥ τ\n⊢ f (h +ᵥ τ) = (⇑f ∣[k] GeneralLinearGroup.upperRightHom h) τ",
"usedConstants": [
... | slash_def, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.Identities | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 72
} | [
{
"pp": "case coe\nΓ : Subgroup (GL (Fin 2) ℝ)\nk : ℤ\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\ninst✝ : SlashInvariantFormClass F Γ k\nf : F\nτ : ℍ\nh : ℝ\nhH : h ∈ Γ.strictPeriods\n⊢ ↑(GeneralLinearGroup.upperRightHom h • τ) = ↑(h +ᵥ τ)",
"usedConstants": [
"UpperHalfPlane.glAction",
"Units.val",
... | simp [σ, num, denom, coe_vadd, UpperHalfPlane.coe_smul, num, add_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 52
} | [
{
"pp": "case coe\nK : Type u_1\ninst✝⁶ : Field K\ninst✝⁵ : DecidableEq K\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsPrincipalIdealRing A\nq : K\ng : SL(2, A)\nhg0 : ↑g 0 0 = IsFractionRing.num A q\nhg1 : ↑g 1 0 = ↑(IsFractionRing.den A ... | exact ⟨g, by simp [hg0, hg1, smul_infty_eq_ite]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Modular | {
"line": 349,
"column": 14
} | {
"line": 349,
"column": 49
} | [
{
"pp": "g : SL(2, ℤ)\nhc : ↑g 1 0 = 1\nhg : ↑g 0 1 = ↑g 0 0 * ↑g 1 1 - 1\n| ↑g",
"usedConstants": [
"Equiv.instEquivLike",
"congrArg",
"Matrix",
"Matrix.of",
"instDecidableEqFin",
"AddGroupWithOne.toAddMonoidWithOne",
"Matrix.eta_fin_two",
"Fin.instOfNat",
... | rw [(g : Matrix _ _ ℤ).eta_fin_two] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.NumberTheory.Modular | {
"line": 349,
"column": 14
} | {
"line": 349,
"column": 49
} | [
{
"pp": "g : SL(2, ℤ)\nhc : ↑g 1 0 = 1\nhg : ↑g 0 1 = ↑g 0 0 * ↑g 1 1 - 1\n| ↑g",
"usedConstants": [
"Equiv.instEquivLike",
"congrArg",
"Matrix",
"Matrix.of",
"instDecidableEqFin",
"AddGroupWithOne.toAddMonoidWithOne",
"Matrix.eta_fin_two",
"Fin.instOfNat",
... | rw [(g : Matrix _ _ ℤ).eta_fin_two] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 349,
"column": 14
} | {
"line": 349,
"column": 49
} | [
{
"pp": "g : SL(2, ℤ)\nhc : ↑g 1 0 = 1\nhg : ↑g 0 1 = ↑g 0 0 * ↑g 1 1 - 1\n| ↑g",
"usedConstants": [
"Equiv.instEquivLike",
"congrArg",
"Matrix",
"Matrix.of",
"instDecidableEqFin",
"AddGroupWithOne.toAddMonoidWithOne",
"Matrix.eta_fin_two",
"Fin.instOfNat",
... | rw [(g : Matrix _ _ ℤ).eta_fin_two] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.NumberTheory.Modular | {
"line": 394,
"column": 2
} | {
"line": 395,
"column": 42
} | [
{
"pp": "τ : ℍ\nh : τ ∈ 𝒟\n⊢ 3 ≤ 4 * τ.im ^ 2",
"usedConstants": [
"Mathlib.Tactic.Ring.mul_pp_pf_overlap",
"Iff.mpr",
"Real.instIsOrderedRing",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.single_pow",
"ModularGroup.fd._pro... | have : 1 ≤ τ.re * τ.re + τ.im * τ.im := by simpa [Complex.normSq_apply] using h.1
cases abs_cases τ.re <;> nlinarith [h.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 394,
"column": 2
} | {
"line": 395,
"column": 42
} | [
{
"pp": "τ : ℍ\nh : τ ∈ 𝒟\n⊢ 3 ≤ 4 * τ.im ^ 2",
"usedConstants": [
"Mathlib.Tactic.Ring.mul_pp_pf_overlap",
"Iff.mpr",
"Real.instIsOrderedRing",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.single_pow",
"ModularGroup.fd._pro... | have : 1 ≤ τ.re * τ.re + τ.im * τ.im := by simpa [Complex.normSq_apply] using h.1
cases abs_cases τ.re <;> nlinarith [h.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Modular | {
"line": 473,
"column": 2
} | {
"line": 473,
"column": 31
} | [
{
"pp": "g : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟ᵒ\nhg : g • z ∈ 𝒟ᵒ\nc' : ℤ := ↑g 1 0\nc : ℝ := ↑c'\nhc : c ≠ 0\nh₁ : 3 * 3 * c ^ 4 < 4 * (g • z).im ^ 2 * (4 * z.im ^ 2) * c ^ 4\nh₂ : (c * z.im) ^ 4 / normSq (denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z) ^ 2 ≤ 1\n⊢ 9 * c ^ 4 < 16",
"usedConstan... | let nsq := normSq (denom g z) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.Modular | {
"line": 692,
"column": 81
} | {
"line": 695,
"column": 66
} | [
{
"pp": "τ : ℍ\n⊢ ∃ γ, 1 / 2 ≤ (γ • τ).im",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Mathlib.Tactic.Ring.mul_pp_pf_overlap",
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.single_pow",
"instNeZer... | by
obtain ⟨γ, hγ⟩ := exists_smul_mem_fd τ
use γ
nlinarith [three_le_four_mul_im_sq_of_mem_fd hγ, im_pos (γ • τ)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.ConditionalInt | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 13
} | [
{
"pp": "α : Type u_1\nf : ℤ → α\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : ContinuousMul α\na : α\nhf2 : Tendsto (fun N ↦ (f ↑N)⁻¹) atTop (𝓝 1)\nhf : Tendsto ((fun s ↦ ∏ b ∈ s, f b) ∘ fun N ↦ Icc (-↑N) ↑N) atTop (𝓝 a)\n⊢ Tendsto (((fun s ↦ ∏ b ∈ s, f b) ∘ fun N ↦ Ico (-N) N) ∘ Nat.cast / (fu... | simpa [Pi.div_def, fun N : ℕ ↦ prod_Icc_eq_prod_Ico_mul f (show (-N : ℤ) ≤ N by lia)]
using hf2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.MDifferentiable | {
"line": 37,
"column": 4
} | {
"line": 39,
"column": 70
} | [
{
"pp": "case inl.h\nk : ℤ\na : Fin 2 → ℤ\nha : a ≠ 0\n⊢ (∀ x ∈ {z | 0 < z.im}, ↑(a 0) * x + ↑(a 1) ≠ 0) ∨ 0 ≤ -k",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Real",
"Function.comp_ne_zero_iff",
"NonUnitalCommRing.toNonUnitalNonAssocCommRin... | · left
exact fun z hz ↦ linear_ne_zero ⟨z, hz⟩
((comp_ne_zero_iff _ Int.cast_injective Int.cast_zero).mpr ha) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 116,
"column": 83
} | {
"line": 120,
"column": 59
} | [
{
"pp": "E : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t\nΓ : Subgroup SL(2, ℤ)\ninst✝ : Γ.FiniteIndex\nhf_inv : ∀ g ∈ Γ, ∀ (τ : ℍ), f (g • τ) = f τ\nτ : ℍ\ng h : SL(2, ℤ)\nh... | by
obtain ⟨j, hj, hj'⟩ : ∃ j ∈ Γ, h = g * j := by
rw [← Quotient.eq_iff_equiv, Quotient.eq, QuotientGroup.leftRel_apply] at hgh
exact ⟨g⁻¹ * h, hgh, (mul_inv_cancel_left g h).symm⟩
simp [-sl_moeb, hj', mul_smul, hf_inv j⁻¹ (inv_mem hj)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.TsumDivisorsAntidiagonal | {
"line": 70,
"column": 56
} | {
"line": 75,
"column": 97
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : CompleteSpace 𝕜\nk : ℕ\nr : 𝕜\nhr : ‖r‖ < 1\n⊢ Summable fun n ↦ ↑n ^ k * r ^ n / (1 - r ^ n)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Norm... | by
simp only [div_eq_mul_one_div (_ * _ ^ _)]
apply Summable.mul_tendsto_const (c := 1 / (1 - 0))
(by simpa using summable_norm_pow_mul_geometric_of_norm_lt_one k hr)
simpa only [Nat.cofinite_eq_atTop] using
tendsto_const_nhds.div ((tendsto_pow_atTop_nhds_zero_of_norm_lt_one hr).const_sub 1) (by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.TsumDivisorsAntidiagonal | {
"line": 137,
"column": 8
} | {
"line": 137,
"column": 79
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nr : 𝕜\nhr : ‖r‖ < 1\nh1 : ∀ (m : ℕ+), ‖r ^ ↑m‖ < 1\nm : ℕ+\nthis : ∑' (n : ℕ), (r ^ ↑m) ^ n = (1 - r ^ ↑m)⁻¹\n⊢ ∑' (n : ℕ+), r ^ (↑n * ↑m) = (1 - r ^ ↑m)⁻¹ - 1",
"usedConstants": [
"PNat.val",
"NormedCommRing.toNormedRing",
"Norm... | ← tsum_zero_pnat_eq_tsum_nat (summable_geometric_of_norm_lt_one (h1 m)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 26
} | [
{
"pp": "case h\nK : Set ℂ\nhK : K ⊆ ℍₒ\nhcK : IsCompact K\n⊢ HasProdUniformlyOn (fun n a ↦ 1 + -eta_q n a) (fun z ↦ ∏' (n : ℕ), (1 + -eta_q n z)) K",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"ModularForm.eta_q",
"Complex.instNormedAddCommGroup",
"Complex.commRing",
... | by_cases hN : K.Nonempty | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 47
} | [
{
"pp": "case hg\nz : ℍ\n⊢ Summable (fun n ↦ ∑' (m : ℤ), (1 / (↑m * ↑z + ↑n) - 1 / (↑m * ↑z + ↑n + 1))) (symmetricIco ℤ)",
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"instHDiv",
"Real.pi",
"HMul.hMul",
"instConditionallyCompleteLinearOrder",
... | apply HasSum.summable (a := -2 * π * I / z) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 66
} | [
{
"pp": "case hg\nz : ℍ\n⊢ HasSum (fun n ↦ ∑' (m : ℤ), (1 / (↑m * ↑z + ↑n) - 1 / (↑m * ↑z + ↑n + 1))) (-2 * ↑π * I / ↑z) (symmetricIco ℤ)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Set.Ioi",
"instHDiv",
"Real.pi",
"HMul.hMul",... | rw [hasSum_symmetricIco_int_iff, ← tendsto_comp_val_Ioi_atTop] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 31
} | [
{
"pp": "z : ℍ\nd : ℕ+\n⊢ ∑' (m : ℕ+), (1 / (↑↑m * ↑z - ↑↑d) + 1 / (-↑↑m * ↑z + -↑↑d) - 1 / (↑↑m * ↑z + ↑↑d) - 1 / (-↑↑m * ↑z + ↑↑d)) =\n 2 / ↑z * ∑' (m : ℕ+), (1 / (-↑↑d / ↑z - ↑↑m) + 1 / (-↑↑d / ↑z + ↑↑m))",
"usedConstants": [
"PNat.val",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing... | rw [← Summable.tsum_mul_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 51,
"column": 71
} | {
"line": 51,
"column": 80
} | [
{
"pp": "τ : ℍ\nh0 : ↑τ ≠ 0\nh1 : (-I * ↑τ) ^ (1 / 2) ≠ 0\n⊢ jacobiTheta₂ 0 (-↑τ)⁻¹ =\n (-I * ↑τ) ^ (1 / 2) * (1 / (-I * ↑τ) ^ (1 / 2) * cexp (-↑π * I * 0 / ↑τ) * jacobiTheta₂ (0 / ↑τ) (-1 / ↑τ))",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 67,
"column": 4
} | {
"line": 69,
"column": 94
} | [
{
"pp": "case refine_2\nτ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\n⊢ y ^ n ^ 2 ≤ rexp (-π * τ.im) ^ n.natAbs",
"usedConstants": [
"zpow_natCast",
"instPowNat",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.par... | have : n ^ 2 = (n.natAbs ^ 2 :) := by rw [Nat.cast_pow, Int.natAbs_sq]
rw [this, zpow_natCast]
exact pow_le_pow_of_le_one (exp_pos _).le h.le ((sq n.natAbs).symm ▸ n.natAbs.le_mul_self) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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