module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.Prime | {
"line": 46,
"column": 6
} | {
"line": 47,
"column": 63
} | [
{
"pp": "case insert.inl\nR : Type u_1\ninst✝² : CommMonoidWithZero R\ninst✝¹ : IsCancelMulZero R\nα : Type u_2\ninst✝ : DecidableEq α\np : α → R\ni : α\ns : Finset α\nhis : i ∉ s\nih :\n ∀ {x y a : R},\n (∀ i ∈ s, Prime (p i)) →\n x * y = a * ∏ i ∈ s, p i →\n ∃ t u b c, t ∪ u = s ∧ Disjoint t u... | exact ⟨insert i t, u, d, c, by rw [insert_union, htus], disjoint_insert_left.2 ⟨hiu, htu⟩, by
simp [hbc, prod_insert hit, mul_comm, mul_left_comm]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 40
} | [
{
"pp": "case refine_2.zero\np : ℕ\nhp : Fact (Nat.Prime p)\ni : ℕ\nhi : i < ((cyclotomic (p ^ (0 + 1)) ℤ).comp (X + 1)).natDegree\n⊢ ((cyclotomic (p ^ (0 + 1)) ℤ).comp (X + 1)).coeff i ∈ ℤ ∙ ↑p",
"usedConstants": [
"Polynomial.instOne",
"congrArg",
"CommSemiring.toSemiring",
"Nat.in... | rw [Nat.zero_add, pow_one] at hi ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 59
} | [
{
"pp": "case refine_1\np k : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nhcycl : IsCyclotomicExtension {p ^ k} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ k)\nx : K\nh : IsIntegral ℤ x\nB : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ\nhint : IsIntegral ℤ B.gen\nthis : ... | obtain ⟨u, n, hun⟩ := discr_prime_pow_eq_unit_mul_pow' hζ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 44
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nh : (logEmbedding K) (Additive.ofMul x) = 0\nw : InfinitePlace K\nhw : w = w₀\nthis : ↑w.mult * Real.log (w ((algebraMap (𝓞 K) K) ↑x)) = 0\n⊢ w ((algebraMap (𝓞 K) K) ↑x) = 1",
"usedConstants": [
"NumberField.InfinitePlace.i... | exact mult_log_place_eq_zero.mp this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 28
} | [
{
"pp": "p : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nx : K\nh : IsIntegral ℤ x\nu : ℤˣ\nn : ℕ\nhcycl : IsCyclotomicExtension {p ^ 0} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ 0)\nB : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ\nhint : IsIntegral ℤ B.gen\nthis✝ : F... | rw [singleton_one ℚ K] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 297,
"column": 4
} | {
"line": 298,
"column": 42
} | [
{
"pp": "case pos\np k : ℕ\nK : Type u\ninst✝² : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝¹ : CharZero K\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhtwo : p = 2\n⊢ Prime (hζ.toInteger - 1)",
"usedConstants": [
"IsPrimitiveRoot.zeta_sub_one_prime_of_two... | subst htwo
apply hζ.zeta_sub_one_prime_of_two_pow | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 297,
"column": 4
} | {
"line": 298,
"column": 42
} | [
{
"pp": "case pos\np k : ℕ\nK : Type u\ninst✝² : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝¹ : CharZero K\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhtwo : p = 2\n⊢ Prime (hζ.toInteger - 1)",
"usedConstants": [
"IsPrimitiveRoot.zeta_sub_one_prime_of_two... | subst htwo
apply hζ.zeta_sub_one_prime_of_two_pow | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 415,
"column": 49
} | {
"line": 418,
"column": 60
} | [
{
"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\nhodd : p ≠ 2\n⊢ Prime ((Algebra.norm ℤ) (hζ.toInteger - 1))",
"usedConstants": [
"Eq.mpr",
"CommRing",
"congrArg",
"CommS... | by
have : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K := by simpa using hcycl
replace hζ : IsPrimitiveRoot ζ (p ^ (0 + 1)) := by simpa using hζ
exact hζ.prime_norm_toInteger_sub_one_of_prime_ne_two hodd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Fintype | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 83
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝² : Ring R\ninst✝¹ : Fintype R\ninst✝ : DecidableEq R\nh✝ : Fintype.card R ≤ 3\nh : Fintype.card R = 3\nthis : Nontrivial R\n⊢ #univ ≤ #{0, 1, -1}",
"usedConstants": [
"Finset.card_univ",
"Eq.mpr",
"NegZeroClass.toNeg",
"le_refl",
"Finset.u... | rw [card_univ, h, card_insert_of_notMem, card_insert_of_notMem, card_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.FLT.Three | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 87
} | [
{
"pp": "case inr.inr\na b c : ℤ\nha : a ≠ 0\nh3a : 3 ∣ a\nHgcd : {a, b, c}.gcd id = 1\nH : ∀ (a b c : ℤ), c ≠ 0 → ¬3 ∣ a → ¬3 ∣ b → 3 ∣ c → IsCoprime a b → a ^ 3 + b ^ 3 ≠ c ^ 3\nHF : a ^ 3 + b ^ 3 + c ^ 3 = 0\nx : ℤ\nh3b : 3 ∣ b\nhx : x = c\n⊢ 3 ∣ id x",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",... | · simpa [hx] using dvd_c_of_prime_of_dvd_a_of_dvd_b_of_FLT Int.prime_three h3a h3b HF | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic | {
"line": 236,
"column": 6
} | {
"line": 236,
"column": 56
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\nhF : ringChar F ≠ 2\na : F\nh₀ : ¬a = 0\ns : Finset F := {x | x ^ 2 = a}.toFinset\nh : ¬IsSquare a\n⊢ ∀ (x : F), x ∉ s",
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"Finset.mem_filter._simp... | simpa [s, isSquare_iff_exists_sq, eq_comm] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.FLT.Three | {
"line": 316,
"column": 6
} | {
"line": 316,
"column": 36
} | [
{
"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\nh1 : ¬λ ^ 2 ∣ S'.a + S'.b\nh2 : ¬λ ^ 2 ∣ S'.a + ↑η * S'.b\nh3 : ¬λ ^ 2 ∣ S'.a + ↑η ^ 2 * S'.b\n⊢ False",
"usedConstants": [
"_private.Mathlib.Numb... | ← emultiplicity_lt_iff_not_dvd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Fermat | {
"line": 65,
"column": 25
} | {
"line": 65,
"column": 43
} | [
{
"pp": "n : ℕ\n⊢ n.fermatNumber = n.fermatNumber - 2 + 2",
"usedConstants": [
"Eq.mpr",
"Nat.fermatNumber",
"congrArg",
"HSub.hSub",
"id",
"instSubNat",
"instOfNatNat",
"instHAdd",
"instHSub",
"Nat.sub_add_cancel",
"HAdd.hAdd",
"Nat",
... | Nat.sub_add_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.NatInt | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 22
} | [
{
"pp": "case refine_2\nP : Ideal ℕ\nh : P.IsPrime\nh0 : ∃ x ∈ P, x ∉ ⊥\nhsp : ¬∃ p, Nat.Prime p ∧ P = span {p}\nn : ℕ\nhn : n ∈ maximalIdeal ℕ\n⊢ n ∈ P",
"usedConstants": [
"instDecidableNot",
"Semiring.toModule",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Classical.propDec... | let p := Nat.find h0 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Ideal.NatInt | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 66
} | [
{
"pp": "⊢ ringKrullDim ℕ = 2",
"usedConstants": [
"instCompleteLatticeWithBot",
"WithBot",
"Preorder.toLT",
"instCompleteLinearOrderENat",
"ChainCompletePartialOrder.instOfCompleteLattice",
"ENat.instNatCast",
"instLinearOrderENat",
"CompletelyDistribLattice.... | refine le_antisymm (iSup_le fun s ↦ le_of_not_gt fun hs ↦ ?_) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.FermatPsp | {
"line": 102,
"column": 51
} | {
"line": 111,
"column": 35
} | [
{
"pp": "n b : ℕ\nh : 1 ≤ b\n⊢ n.ProbablePrime b ↔ b ^ (n - 1) ≡ 1 [MOD n]",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"IsOrderedRing.toPosMulMono",
"congrArg",
"Nat.instMonoid",
"Nat.cast_sub._simp_1",
"AddGroupWithOne.toAddMonoidWithOne",
"Nat.instZeroLEOneCla... | by
have : 1 ≤ b ^ (n - 1) := one_le_pow₀ h
-- For exact mod_cast
rw [Nat.ModEq.comm]
constructor
· intro h₁
apply Nat.modEq_of_dvd
exact mod_cast h₁
· intro h₁
exact mod_cast Nat.ModEq.dvd h₁ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.FermatPsp | {
"line": 171,
"column": 54
} | {
"line": 171,
"column": 78
} | [
{
"pp": "b p : ℕ\nhb : 0 < b\nhp : 1 ≤ p\nhi_bsquared : 1 ≤ b ^ 2\n⊢ (b * b ^ (p - 1) - b * 1) * (b ^ p + b) = b * (b ^ (p - 1) - 1) * (b ^ p + b)",
"usedConstants": [
"Eq.mpr",
"Nat.mul_sub_left_distrib",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"HSub.hSub",
"id"... | Nat.mul_sub_left_distrib | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 672,
"column": 4
} | {
"line": 673,
"column": 35
} | [
{
"pp": "case hcong.inl.inl\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\nX : 𝓞 K\nhX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X\nY : 𝓞 K\nhY : S.Y ^ 3 - 1 = λ ^ 2 * Y\nZ :... | rw [show λ ^ 2 * (X - Y - S.u₄ * Z) = λ ^ 2 * X - λ ^ 2 * Y - S.u₄ * (λ ^ 2 * Z) by ring,
← hX, ← hY, ← hZ, ← formula2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.FLT.Three | {
"line": 680,
"column": 4
} | {
"line": 681,
"column": 35
} | [
{
"pp": "case hcong.inr.inl\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\nX : 𝓞 K\nhX : ↑S.u₅ * (λ ^ (S.multiplicity - 1) * S.X) ^ 3 = λ ^ 2 * X\nY : 𝓞 K\nhY : S.Y ^ 3 + 1 = λ ^ 2 * Y\nZ :... | rw [show λ ^ 2 * (X - Y - S.u₄ * Z) = λ ^ 2 * X - λ ^ 2 * Y - S.u₄ * (λ ^ 2 * Z) by ring,
← hX, ← hY, ← hZ, ← formula2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 300,
"column": 4
} | {
"line": 300,
"column": 44
} | [
{
"pp": "case h.e_f.h\nthis : Tendsto (fun s ↦ ∑' (n : ℕ), 1 / (↑n + 1) ^ s - 1 / (s - 1)) (𝓝[>] 1) (𝓝 γ)\ns : ℝ\nhs : s ∈ Ioi 1\nn : ℕ\n⊢ 1 / ↑((↑n + 1) ^ s) = 1 / (↑n + 1) ^ ↑s",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.instPow",
"Real.partialOrder",
"Re... | rw [Complex.ofReal_cpow (by positivity)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.RootsOfUnity.Lemmas | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 30
} | [
{
"pp": "case refine_1.hy\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nk n : ℕ\nμ : R\nm : ℕ\nhn : k < m + k + 1\nhμ : IsPrimitiveRoot μ (m + k + 1)\nhdvd : ∀ (k : ℕ), ∃ z ∈ ℤ[μ], μ ^ k - 1 = z * (μ - 1)\nZ : ℕ → R := fun k ↦ Classical.choose ⋯\nZdef : ∀ (k : ℕ), Z k ∈ ℤ[μ] ∧ μ ^ k - 1 = Z k * (μ - 1... | apply Subalgebra.sub_mem | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Height.Basic | {
"line": 829,
"column": 4
} | {
"line": 829,
"column": 44
} | [
{
"pp": "case refine_1.h\nR : Type u_1\nS : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : CommSemiring S\ninst✝² : LinearOrder S\ninst✝¹ : IsOrderedRing S\ninst✝ : CharZero S\nv : AbsoluteValue R S\nι : Type u_3\ns : Finset ι\nhs : s.Nonempty\nx : ι → R\ni : ι\nhi : i ∈ s\n⊢ v (x i) ≤ ∏ i ∈ s, max (v (x i)) 1",
"... | exact le_prod_max_one hi fun i ↦ v (x i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Height.Basic | {
"line": 902,
"column": 6
} | {
"line": 902,
"column": 51
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx y : K\n⊢ mulHeight₁ (x + y) ≤ 2 ^ totalWeight K * mulHeight₁ x * mulHeight₁ y",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Com... | show x + y = Finset.univ.sum ![x, y] by simp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 918,
"column": 2
} | {
"line": 919,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx y : K\n⊢ mulHeight₁ (x - y) ≤ 2 ^ totalWeight K * mulHeight₁ x * mulHeight₁ y",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
... | rw [sub_eq_add_neg, ← mulHeight₁_neg y]
exact mulHeight₁_add_le x (-y) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Height.Basic | {
"line": 918,
"column": 2
} | {
"line": 919,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx y : K\n⊢ mulHeight₁ (x - y) ≤ 2 ^ totalWeight K * mulHeight₁ x * mulHeight₁ y",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
... | rw [sub_eq_add_neg, ← mulHeight₁_neg y]
exact mulHeight₁_add_le x (-y) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 15
} | [
{
"pp": "F : 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⊢ jacobiSum χ ψ = ∑ x, χ x + ∑ x, ψ x - ↑(Fintype.card F) + ∑ x ∈ univ \\ {0, 1}, (χ x - 1) * (ψ (1 - x) - 1)",
"usedConstants": [
"Eq.mpr... | jacobiSum | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 141,
"column": 6
} | {
"line": 141,
"column": 15
} | [
{
"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 : MulChar F R\nhχ : χ ≠ 1\n⊢ jacobiSum χ χ⁻¹ = -χ (-1)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMu... | jacobiSum | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 140,
"column": 2
} | {
"line": 161,
"column": 40
} | [
{
"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 : MulChar F R\nhχ : χ ≠ 1\n⊢ jacobiSum χ χ⁻¹ = -χ (-1)",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Finset.mem_... | classical
rw [jacobiSum]
conv => enter [1, 2, x]; rw [MulChar.inv_apply', ← map_mul, ← div_eq_mul_inv]
rw [sum_eq_sum_diff_singleton_add (mem_univ (1 : F)), sub_self, div_zero, χ.map_zero, add_zero]
have : ∑ x ∈ univ \ {1}, χ (x / (1 - x)) = ∑ x ∈ univ \ {-1}, χ x := by
refine sum_bij' (fun a _ ↦ a / (1 - a... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 140,
"column": 2
} | {
"line": 161,
"column": 40
} | [
{
"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 : MulChar F R\nhχ : χ ≠ 1\n⊢ jacobiSum χ χ⁻¹ = -χ (-1)",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Finset.mem_... | classical
rw [jacobiSum]
conv => enter [1, 2, x]; rw [MulChar.inv_apply', ← map_mul, ← div_eq_mul_inv]
rw [sum_eq_sum_diff_singleton_add (mem_univ (1 : F)), sub_self, div_zero, χ.map_zero, add_zero]
have : ∑ x ∈ univ \ {1}, χ (x / (1 - x)) = ∑ x ∈ univ \ {-1}, χ x := by
refine sum_bij' (fun a _ ↦ a / (1 - a... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 140,
"column": 2
} | {
"line": 161,
"column": 40
} | [
{
"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 : MulChar F R\nhχ : χ ≠ 1\n⊢ jacobiSum χ χ⁻¹ = -χ (-1)",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Finset.mem_... | classical
rw [jacobiSum]
conv => enter [1, 2, x]; rw [MulChar.inv_apply', ← map_mul, ← div_eq_mul_inv]
rw [sum_eq_sum_diff_singleton_add (mem_univ (1 : F)), sub_self, div_zero, χ.map_zero, add_zero]
have : ∑ x ∈ univ \ {1}, χ (x / (1 - x)) = ∑ x ∈ univ \ {-1}, χ x := by
refine sum_bij' (fun a _ ↦ a / (1 - a... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 71
} | [
{
"pp": "case inr.h₂\nK : Type u_4\ninst✝¹ : Field K\nι : Type u_5\ninst✝ : Finite ι\nv : AbsoluteValue K ℝ\nhv : IsNonarchimedean ⇑v\np : MvPolynomial ι K\nN : ℕ\nhp : p.IsHomogeneous N\nx : ι → K\nhp₀ : p ≠ 0\ns : ι →₀ ℕ\nhs₁ : s ∈ p.support\nhs₂ : v (∑ d ∈ p.support, coeff d p * ∏ i ∈ d.support, x i ^ d i) ≤... | rw [hp.degree_eq_sum_deg_support hs₁, ← Finset.prod_pow_eq_pow_sum] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 93
} | [
{
"pp": "K : Type u_4\ninst✝¹ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝ : AdmissibleAbsValues K\n⊢ max ((Multiset.map (fun x ↦ ⨆ j, 0) archAbsVal).prod * ∏ᶠ (v : ↑nonarchAbsVal), ⨆ j, 1) 1 = 1",
"usedConstants": [
"Multiset.prod_replicate",
"Eq.mpr",
"MulOne.toOne",
"Real.partial... | simp only [Real.iSup_const_zero, Multiset.map_const', Multiset.prod_replicate, zero_pow_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 24
} | [
{
"pp": "case inr.inl\nK : Type u_4\ninst✝³ : Field K\nι : Type u_5\nι' : Type u_6\ninst✝² : AdmissibleAbsValues K\ninst✝¹ : Finite ι'\ninst✝ : Finite ι\nN : ℕ\np : ι' → MvPolynomial ι K\nhp : ∀ (i : ι'), (p i).IsHomogeneous N\nx : ι → K\nhx : x ≠ 0\nh₀ : (fun j ↦ (eval x) (p j)) = 0\n⊢ (mulHeight fun j ↦ (eval... | grw [← le_max_right] | Mathlib.Tactic._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_grwSeq_1 | Mathlib.Tactic.grwSeq |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 46
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\nf : R[X]\ninst✝¹ : Algebra R S\nh : IsAdjoinRootMonic S f\ninst✝ : Nontrivial S\n⊢ 0 < f.natDegree",
"usedConstants": [
"Ring.toNonAssocRing",
"CommSemiring.toSemiring",
"Module.Basis.index_nonempty",
"Algebra.toM... | rcases h.basis.index_nonempty with ⟨⟨i, hi⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 441,
"column": 4
} | {
"line": 442,
"column": 70
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhΦ : Function.Odd Φ\ns : ℂ\nhs : 1 < s.re\nthis : ∑ x, Φ x * sinZeta (toAddCircle x) s = I * LFunction (𝓕 Φ) s\nhs' : 0 < (s + 1).re\n⊢ ∑ x, Φ x * completedSinZeta (toAddCircle x) s = I * completedLFunction (𝓕 Φ) s",
"usedConstants": [
"ZMod.complete... | simpa only [sinZeta, ← mul_div_assoc, ← sum_div, div_left_inj' (Gammaℝ_ne_zero_of_re_pos hs'),
LFunction_eq_completed_div_gammaFactor_odd (dft_odd_iff.mpr hΦ)] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 449,
"column": 4
} | {
"line": 449,
"column": 21
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhΦ : Function.Odd Φ\ns : ℂ\nhs : 1 < s.re\n⊢ (∑ x, Φ x * expZeta (toAddCircle x) s) / (2 * I) - (∑ x, Φ x * expZeta (toAddCircle (-x)) s) / (2 * I) =\n (∑ x, Φ (-x) * expZeta (toAddCircle (-x)) s) / ?m.227 - ?m.229",
"usedConstants": [
"NegZeroClass... | congrm ?_ / _ - _ | Mathlib.Tactic._aux_Mathlib_Tactic_CongrM___elabRules_Mathlib_Tactic_congrM_1 | Mathlib.Tactic.congrM |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 62
} | [
{
"pp": "case h.e'_2\n⊢ riemannZeta 2 = ↑(∑' (b : ℕ), 1 / ↑b ^ 2)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Nat.instIsOrderedAddMonoid",
"riemannZeta",
"AddMonoid... | rw [← Nat.cast_two, zeta_nat_eq_tsum_of_gt_one one_lt_two] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 74
} | [
{
"pp": "case inl\n⊢ riemannZeta (-↑0) = -↑(bernoulli' (0 + 1)) / (↑0 + 1)",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"NegZeroClass.toNeg",
"neg_div",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"InvOneClass.toOne",
"riemannZeta",
"... | rw [Nat.cast_zero, neg_zero, riemannZeta_zero, zero_add, zero_add, div_one,
bernoulli'_one, Rat.cast_div, Rat.cast_one, Rat.cast_ofNat, neg_div] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 74
} | [
{
"pp": "case inl\n⊢ riemannZeta (-↑0) = -↑(bernoulli' (0 + 1)) / (↑0 + 1)",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"NegZeroClass.toNeg",
"neg_div",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"InvOneClass.toOne",
"riemannZeta",
"... | rw [Nat.cast_zero, neg_zero, riemannZeta_zero, zero_add, zero_add, div_one,
bernoulli'_one, Rat.cast_div, Rat.cast_one, Rat.cast_ofNat, neg_div] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 74
} | [
{
"pp": "case inl\n⊢ riemannZeta (-↑0) = -↑(bernoulli' (0 + 1)) / (↑0 + 1)",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"NegZeroClass.toNeg",
"neg_div",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"InvOneClass.toOne",
"riemannZeta",
"... | rw [Nat.cast_zero, neg_zero, riemannZeta_zero, zero_add, zero_add, div_one,
bernoulli'_one, Rat.cast_div, Rat.cast_one, Rat.cast_ofNat, neg_div] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.Injectivity | {
"line": 157,
"column": 12
} | {
"line": 157,
"column": 39
} | [
{
"pp": "f : ℕ → ℂ\nh : ¬abscissaOfAbsConv f = ⊤\nH : (fun x ↦ LSeries f ↑x) =ᶠ[atTop] 0\nF : ℕ → ℂ := fun n ↦ if n = 0 then 0 else f n\nhF₀ : F 0 = 0\nhF : ∀ {n : ℕ}, n ≠ 0 → F n = f n\nha : ¬abscissaOfAbsConv F = ⊤\nh' : ∀ (x : ℝ), LSeries F ↑x = LSeries f ↑x\nn : ℕ\n⊢ (fun x ↦ ↑n ^ ↑x * LSeries f ↑x) =ᶠ[atTo... | eventuallyEq_iff_exists_mem | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ZetaValues | {
"line": 443,
"column": 19
} | {
"line": 443,
"column": 37
} | [
{
"pp": "k : ℕ\nhk : k ≠ 0\n⊢ 2 ^ (2 * k - 1 + 1) = 2 ^ (2 * k)",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"HSub.hSub",
"id",
"instSubNat",
"instMulNat",
"instOfNatNat",
"Monoid.toPow",
... | Nat.sub_add_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.PrimesInAP | {
"line": 289,
"column": 2
} | {
"line": 290,
"column": 79
} | [
{
"pp": "q : ℕ\na : ZMod q\ninst✝ : NeZero q\nha : IsUnit a\ns : ℂ\nhs : 1 < s.re\n⊢ ↑q.totient * L (fun n ↦ ↑(residueClass a n)) s = ∑ x, x a⁻¹ * L ((fun x_1 ↦ x ↑x_1) * fun n ↦ ↑(Λ n)) s",
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"DirichletCharacter.fintype",
"Eq.mpr",
... | simp_rw [← LSeries_smul,
← LSeries_sum <| fun χ _ ↦ (LSeriesSummable_twist_vonMangoldt χ hs).smul _] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 74
} | [
{
"pp": "case inl.inl\np : ℕ\nh : Fact (Nat.Prime p)\na : ℤ\nha0 : ↑a ≠ 0\nhp : Odd p\nthis : ↑(legendreSym p a) = ↑((-1) ^ #({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val}))\nh✝¹ : legendreSym p a = 1\nh✝ : (-1) ^ #({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val}) = 1\n⊢ legendreSym p a = (-1) ^ #({x ∈ Ico ... | simp_all [ne_neg_self hp one_ne_zero, (ne_neg_self hp one_ne_zero).symm] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 74
} | [
{
"pp": "case inl.inr\np : ℕ\nh : Fact (Nat.Prime p)\na : ℤ\nha0 : ↑a ≠ 0\nhp : Odd p\nthis : ↑(legendreSym p a) = ↑((-1) ^ #({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val}))\nh✝¹ : legendreSym p a = 1\nh✝ : (-1) ^ #({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val}) = -1\n⊢ legendreSym p a = (-1) ^ #({x ∈ Ico... | simp_all [ne_neg_self hp one_ne_zero, (ne_neg_self hp one_ne_zero).symm] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 74
} | [
{
"pp": "case inr.inl\np : ℕ\nh : Fact (Nat.Prime p)\na : ℤ\nha0 : ↑a ≠ 0\nhp : Odd p\nthis : ↑(legendreSym p a) = ↑((-1) ^ #({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val}))\nh✝¹ : legendreSym p a = -1\nh✝ : (-1) ^ #({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val}) = 1\n⊢ legendreSym p a = (-1) ^ #({x ∈ Ico... | simp_all [ne_neg_self hp one_ne_zero, (ne_neg_self hp one_ne_zero).symm] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 74
} | [
{
"pp": "case inr.inr\np : ℕ\nh : Fact (Nat.Prime p)\na : ℤ\nha0 : ↑a ≠ 0\nhp : Odd p\nthis : ↑(legendreSym p a) = ↑((-1) ^ #({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val}))\nh✝¹ : legendreSym p a = -1\nh✝ : (-1) ^ #({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑a * ↑x).val}) = -1\n⊢ legendreSym p a = (-1) ^ #({x ∈ Ic... | simp_all [ne_neg_self hp one_ne_zero, (ne_neg_self hp one_ne_zero).symm] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 388,
"column": 4
} | {
"line": 389,
"column": 62
} | [
{
"pp": "case neg\nN : ℕ\nχ : DirichletCharacter ℂ N\ninst✝ : NeZero N\ns : ℂ\nhs : s.re = 1\nhχs : χ ≠ 1 ∨ s ≠ 1\nh : ¬(χ ^ 2 = 1 ∧ s = 1)\n⊢ LFunction χ s ≠ 0",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNo... | have hs' : s = 1 + I * s.im := by
conv_lhs => rw [← re_add_im s, hs, ofReal_one, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 33
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : p ≠ 2\na : ℕ\nha1 : a % 2 = 1\nha0 : ↑a ≠ 0\nhp' : Fact (p % 2 = 1)\nha0' : ↑↑a ≠ 0\n⊢ legendreSym p ↑a = (-1) ^ ∑ x ∈ Ico 1 (p / 2).succ, x * a / p",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instHDiv",
"HMul.hMul",
"c... | neg_one_pow_eq_pow_mod_two, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 209,
"column": 55
} | {
"line": 209,
"column": 82
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : p ≠ 2\na : ℕ\nha1 : a % 2 = 1\nha0 : ↑a ≠ 0\nhp' : Fact (p % 2 = 1)\nha0' : ↑↑a ≠ 0\n⊢ (-1) ^ #({x ∈ Ico 1 (p / 2).succ | p / 2 < (↑↑a * ↑x).val}) = (-1) ^ ((∑ x ∈ Ico 1 (p / 2).succ, x * a / p) % 2)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
... | neg_one_pow_eq_pow_mod_two, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 172,
"column": 4
} | {
"line": 173,
"column": 21
} | [
{
"pp": "case inl\na : ℤ\nb : ℕ\nh : a.gcd ↑b = 1\nhb : b = 0\n⊢ J(a | b) ≠ 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Int.instNeZeroOfNatOfNat",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"Ne",
"instOfNatNat",
"Int",
"A... | rw [hb, zero_right]
exact one_ne_zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 172,
"column": 4
} | {
"line": 173,
"column": 21
} | [
{
"pp": "case inl\na : ℤ\nb : ℕ\nh : a.gcd ↑b = 1\nhb : b = 0\n⊢ J(a | b) ≠ 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Int.instNeZeroOfNatOfNat",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"Ne",
"instOfNatNat",
"Int",
"A... | rw [hb, zero_right]
exact one_ne_zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 120,
"column": 4
} | {
"line": 121,
"column": 64
} | [
{
"pp": "case succ\nK : Type u_1\nΓ₀ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\ninst✝¹ : Valued K Γ₀\ninst✝ : IsDiscreteValuationRing ↥𝒪[K]\nh : Finite 𝓀[K]\nn : ℕ\nthis : 𝓂[K] ^ (n + 1) ≤ 𝓂[K] ^ n\nih : Finite ((↥𝒪[K] ⧸ 𝓂[K] ^ (n + 1)) ⧸ Ideal.map (Ideal.Quotient.mk (𝓂[K] ... | suffices Finite (Ideal.map (Ideal.Quotient.mk (𝓂[K] ^ (n + 1))) (𝓂[K] ^ n)) from
.of_ideal_quotient (.map (Ideal.Quotient.mk _) (𝓂[K] ^ n)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 528,
"column": 2
} | {
"line": 528,
"column": 60
} | [
{
"pp": "a b : ℕ\nflip : Bool\nha0 : a > 0\nhb2 : b % 2 = 1\nhb1 : b > 1\n⊢ fastJacobiSymAux a b flip ha0 = if flip = true then -J(↑a | b) else J(↑a | b)",
"usedConstants": [
"Int.instAddCommGroup",
"Nat.gcd",
"Iff.mpr",
"Int.gcd",
"Eq.mpr",
"instDecidableNot",
"Int... | induction a using Nat.strongRecOn generalizing b flip with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.SimpleRing.Field | {
"line": 45,
"column": 32
} | {
"line": 45,
"column": 40
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : IsSimpleRing A\nx : A\nhx1✝ : x ∈ Subring.center A\nhx1 : ∀ (g : A), g * x = x * g\nhx2 : x ≠ 0\nI : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯\ny : A\nhy : x * y = 1\na : A\n⊢ x * y * a * y = y * x * a * y",
"usedConstants": [
"Eq.mp... | rw [hx1] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.SimpleRing.Field | {
"line": 45,
"column": 32
} | {
"line": 45,
"column": 40
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : IsSimpleRing A\nx : A\nhx1✝ : x ∈ Subring.center A\nhx1 : ∀ (g : A), g * x = x * g\nhx2 : x ≠ 0\nI : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯\ny : A\nhy : x * y = 1\na : A\n⊢ x * y * a * y = y * x * a * y",
"usedConstants": [
"Eq.mp... | rw [hx1] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.SimpleRing.Field | {
"line": 45,
"column": 32
} | {
"line": 45,
"column": 40
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : IsSimpleRing A\nx : A\nhx1✝ : x ∈ Subring.center A\nhx1 : ∀ (g : A), g * x = x * g\nhx2 : x ≠ 0\nI : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯\ny : A\nhy : x * y = 1\na : A\n⊢ x * y * a * y = y * x * a * y",
"usedConstants": [
"Eq.mp... | rw [hx1] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SimpleRing.Field | {
"line": 47,
"column": 32
} | {
"line": 47,
"column": 40
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : IsSimpleRing A\nx : A\nhx1✝ : x ∈ Subring.center A\nhx1 : ∀ (g : A), g * x = x * g\nhx2 : x ≠ 0\nI : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯\ny : A\nhy : x * y = 1\na : A\n⊢ y * (x * a) * y = y * (a * x) * y",
"usedConstants": [
"E... | rw [hx1] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.SimpleRing.Field | {
"line": 47,
"column": 32
} | {
"line": 47,
"column": 40
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : IsSimpleRing A\nx : A\nhx1✝ : x ∈ Subring.center A\nhx1 : ∀ (g : A), g * x = x * g\nhx2 : x ≠ 0\nI : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯\ny : A\nhy : x * y = 1\na : A\n⊢ y * (x * a) * y = y * (a * x) * y",
"usedConstants": [
"E... | rw [hx1] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.SimpleRing.Field | {
"line": 47,
"column": 32
} | {
"line": 47,
"column": 40
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : IsSimpleRing A\nx : A\nhx1✝ : x ∈ Subring.center A\nhx1 : ∀ (g : A), g * x = x * g\nhx2 : x ≠ 0\nI : TwoSidedIdeal A := mk' (Set.range fun x_1 ↦ x * x_1) ⋯ ⋯ ⋯ ⋯ ⋯\ny : A\nhy : x * y = 1\na : A\n⊢ y * (x * a) * y = y * (a * x) * y",
"usedConstants": [
"E... | rw [hx1] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 324,
"column": 4
} | {
"line": 328,
"column": 16
} | [
{
"pp": "case refine_1\nK : Type u_1\nΓ₀ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\ninst✝¹ : Valued K Γ₀\ninst✝ : v.RankOne\nh : CompactSpace ↥𝒪[K]\n⊢ CompleteSpace ↥𝒪[K] ∧ IsDiscreteValuationRing ↥𝒪[K] ∧ Finite 𝓀[K]",
"usedConstants": [
"TotallyBounded",
"IsCo... | have : IsDiscreteValuationRing 𝒪[K] := isDiscreteValuationRing_of_compactSpace
refine ⟨complete_of_compact, by assumption, ?_⟩
rw [← isCompact_univ_iff, isCompact_iff_totallyBounded_isComplete,
totallyBounded_iff_finite_residueField] at h
exact h.left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 324,
"column": 4
} | {
"line": 328,
"column": 16
} | [
{
"pp": "case refine_1\nK : Type u_1\nΓ₀ : Type u_2\ninst✝³ : Field K\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\ninst✝¹ : Valued K Γ₀\ninst✝ : v.RankOne\nh : CompactSpace ↥𝒪[K]\n⊢ CompleteSpace ↥𝒪[K] ∧ IsDiscreteValuationRing ↥𝒪[K] ∧ Finite 𝓀[K]",
"usedConstants": [
"TotallyBounded",
"IsCo... | have : IsDiscreteValuationRing 𝒪[K] := isDiscreteValuationRing_of_compactSpace
refine ⟨complete_of_compact, by assumption, ?_⟩
rw [← isCompact_univ_iff, isCompact_iff_totallyBounded_isComplete,
totallyBounded_iff_finite_residueField] at h
exact h.left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LucasLehmer | {
"line": 199,
"column": 53
} | {
"line": 212,
"column": 8
} | [
{
"pp": "p : ℕ\nw : 1 < p\n⊢ lucasLehmerResidue p = 0 ↔ sMod p (p - 2) = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.instMulZeroClass",
"Preorder.toLT",
"Dvd.dvd",
"LinearOrderedC... | by
dsimp [lucasLehmerResidue]
rw [sZMod_eq_sMod p]
constructor
· -- We want to use that fact that `0 ≤ s_mod p (p-2) < 2^p - 1`
-- and `lucas_lehmer_residue p = 0 → 2^p - 1 ∣ s_mod p (p-2)`.
intro h
apply Int.eq_zero_of_dvd_of_nonneg_of_lt _ _
(by simpa [ZMod.intCast_zmod_eq_zero_iff_dvd] usin... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LucasLehmer | {
"line": 637,
"column": 4
} | {
"line": 637,
"column": 60
} | [
{
"pp": "case succ\np : ℕ\nhp : 2 ≤ p\nn✝ : ℕ\na✝ : ↑(sModNat (2 ^ p - 1) n✝) = sMod p n✝\n⊢ ↑(sModNat (2 ^ p - 1) (n✝ + 1)) = sMod p (n✝ + 1)",
"usedConstants": [
"Nat.instMonoid",
"_private.Mathlib.NumberTheory.LucasLehmer.0.LucasLehmer.norm_num_ext.sModNat_eq_sMod._proof_1_8",
"instOfNa... | have : 2 ^ 2 ≤ 2 ^ p := Nat.pow_le_pow_right (by lia) hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 44
} | [
{
"pp": "case neg\nn : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nR : Type u_2\ninst✝ : Ring R\n𝒢 : Subgroup (GL n R)\nhG : -1 ∉ 𝒢\n⊢ 𝒢.relIndex 𝒢.adjoinNegOne ≠ 0",
"usedConstants": [
"False",
"congrArg",
"Matrix",
"Nat.instAtLeastTwoHAddOfNat",
"Subgroup.adjoin... | simp [𝒢.relindex_adjoinNegOne_eq_two hG] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 44
} | [
{
"pp": "case neg\nn : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nR : Type u_2\ninst✝ : Ring R\n𝒢 : Subgroup (GL n R)\nhG : -1 ∉ 𝒢\n⊢ 𝒢.relIndex 𝒢.adjoinNegOne ≠ 0",
"usedConstants": [
"False",
"congrArg",
"Matrix",
"Nat.instAtLeastTwoHAddOfNat",
"Subgroup.adjoin... | simp [𝒢.relindex_adjoinNegOne_eq_two hG] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.ArithmeticSubgroups | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 44
} | [
{
"pp": "case neg\nn : Type u_1\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\nR : Type u_2\ninst✝ : Ring R\n𝒢 : Subgroup (GL n R)\nhG : -1 ∉ 𝒢\n⊢ 𝒢.relIndex 𝒢.adjoinNegOne ≠ 0",
"usedConstants": [
"False",
"congrArg",
"Matrix",
"Nat.instAtLeastTwoHAddOfNat",
"Subgroup.adjoin... | simp [𝒢.relindex_adjoinNegOne_eq_two hG] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactification.OnePoint.ProjectiveLine | {
"line": 179,
"column": 4
} | {
"line": 179,
"column": 80
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : DecidableEq K\nc : K\ng : GL (Fin 2) K\nh : ↑g 1 0 * c + ↑g 1 1 = 0\n⊢ ↑g 1 0 * c ^ 2 + (↑g 1 1 - ↑g 0 0) * c - ↑g 0 1 = 0 ↔ ∞ = ↑c",
"usedConstants": [
"Units.val",
"HMul.hMul",
"OnePoint.infty",
"CommSemiring.toSemiring",
... | refine ⟨fun hg ↦ (g.det_ne_zero ?_).elim, fun hg ↦ (infty_ne_coe _ hg).elim⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ModularForms.Identities | {
"line": 29,
"column": 24
} | {
"line": 35,
"column": 72
} | [
{
"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\n⊢ f (h +ᵥ τ) = f τ",
"usedConstants": [
"UpperHalfPlane.glAction",
"Real.instIsOrderedRing",
"Units.val",
"Eq.mpr",... | by
rw [← congr_fun (slash_action_eqn f _ <| Γ.mem_strictPeriods_iff.mp hH) τ]
suffices GeneralLinearGroup.upperRightHom h • τ = h +ᵥ τ by
simp_rw [slash_def, this]
simp [σ, denom, GeneralLinearGroup.val_det_apply, denom]
ext
simp [σ, num, denom, coe_vadd, UpperHalfPlane.coe_smul, num, add_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 69
} | [
{
"pp": "case right\nR : Type u_1\ninst✝ : Ring R\n𝒢 : Subgroup (GL (Fin 2) R)\n⊢ 𝒢.periods.relIndex 𝒢.strictPeriods ≠ 0",
"usedConstants": [
"Iff.mpr",
"False",
"Nat.instMulZeroClass",
"Nat.instOne",
"AddGroupWithOne.toAddGroup",
"congrArg",
"PartialOrder.toPreo... | · simp [AddSubgroup.relIndex_eq_one.mpr 𝒢.strictPeriods_le_periods] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 287,
"column": 2
} | {
"line": 288,
"column": 88
} | [
{
"pp": "h : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nhfper : Periodic (f ∘ ↑ofComplex) ↑h\nhfhol : MDiff f\nhfbdd : IsBoundedAtImInfty f\nn : ℕ\nR : ℝ\nhR : 0 < R\nhR' : R < 1\n⊢ (PowerSeries.coeff n) (qExpansion h f) =\n (2 * ↑π * Complex.I)⁻¹ * ∮ (z : ℂ) in C(0, R), cuspFunction h f z / z ^ (n + 1)",
"usedConstants"... | have := ((differentiableOn_cuspFunction_ball hh hfper hfhol hfbdd).mono
(Metric.closedBall_subset_ball hR')).circleIntegral_one_div_sub_center_pow_smul hR n | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ModularForms.LevelOne.Basic | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 31
} | [
{
"pp": "case inl\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\nhf : ⇑f = Function.const ℍ (UpperHalfPlane.cuspFunction 1 (⇑f) 0)\ninst✝ : ModularFormClass F (Matrix.SpecialLinearGroup.mapGL ℝ).range 0\nhk : 0 < 0\n⊢ ⇑f = 0",
"usedConstants": [
"lt_irrefl",
"instConditionallyCompleteLinearOrder"... | · exact (lt_irrefl _ hk).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Modular | {
"line": 279,
"column": 2
} | {
"line": 290,
"column": 41
} | [
{
"pp": "z : ℍ\n⊢ ∃ g, ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"UpperHalfPlane.im_pos",
"ModularGroup.bottom_row_coprime"... | let s : Set (Fin 2 → ℤ) := {cd | IsCoprime (cd 0) (cd 1)}
have hs : s.Nonempty := ⟨![1, 1], isCoprime_one_left⟩
obtain ⟨p, hp_coprime, hp⟩ :=
Filter.Tendsto.exists_within_forall_le hs (tendsto_normSq_coprime_pair z)
obtain ⟨g, -, hg⟩ := bottom_row_surj hp_coprime
refine ⟨g, fun g' => ?_⟩
rw [ModularGroup.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Modular | {
"line": 279,
"column": 2
} | {
"line": 290,
"column": 41
} | [
{
"pp": "z : ℍ\n⊢ ∃ g, ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"UpperHalfPlane.im_pos",
"ModularGroup.bottom_row_coprime"... | let s : Set (Fin 2 → ℤ) := {cd | IsCoprime (cd 0) (cd 1)}
have hs : s.Nonempty := ⟨![1, 1], isCoprime_one_left⟩
obtain ⟨p, hp_coprime, hp⟩ :=
Filter.Tendsto.exists_within_forall_le hs (tendsto_normSq_coprime_pair z)
obtain ⟨g, -, hg⟩ := bottom_row_surj hp_coprime
refine ⟨g, fun g' => ?_⟩
rw [ModularGroup.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs | {
"line": 174,
"column": 6
} | {
"line": 175,
"column": 72
} | [
{
"pp": "N r : ℕ\na : Fin 2 → ZMod N\ninst✝ : NeZero r\nγ : SL(2, ℤ)\nv : ↑(gammaSet N r (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ)))\nthis :\n ↑v ᵥ* ↑γ⁻¹ ∈\n gammaSet N r\n (a ᵥ*\n ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ *\n (SpecialLinearGroup.map... | simpa only [SpecialLinearGroup.map_apply_coe, RingHom.mapMatrix_apply, Int.coe_castRingHom,
map_inv, mul_inv_cancel, SpecialLinearGroup.coe_one, vecMul_one] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.Modular | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 23
} | [
{
"pp": "g : SL(2, ℤ)\nhc : ↑g 1 0 = 0\n⊢ ∃ n, ∀ (z : ℍ), g • z = T ^ n • z",
"usedConstants": [
"Matrix",
"instDecidableEqFin",
"AddGroupWithOne.toAddMonoidWithOne",
"Matrix.SpecialLinearGroup.det_coe",
"instOfNatNat",
"Int",
"Fin.fintype",
"AddMonoidWithOne.... | have had := g.det_coe | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 83,
"column": 2
} | {
"line": 84,
"column": 38
} | [
{
"pp": "case ha\nE : Type u_1\ninst✝ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : f =O[atImInfty] fun z ↦ z.im ^ t\nhf_inv : ∀ (g : SL(2, ℤ)) (τ : ℍ), f (g • τ) = f τ\nF : ℝ\nτ : ℍ\ng : SL(2, ℤ)\nhg : g •... | · rw [← div_le_iff₀ (by positivity)] at hF𝒟
exact le_trans (by positivity) hF𝒟 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Modular | {
"line": 446,
"column": 4
} | {
"line": 446,
"column": 15
} | [
{
"pp": "case right\nz : ℍ\ng₀ : SL(2, ℤ)\nhg₀ : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g₀ • z).im\ng : SL(2, ℤ)\nhg : ↑g 1 = ↑g₀ 1\nhg' : ∀ (g' : SL(2, ℤ)), ↑g 1 = ↑g' 1 → |(g • z).re| ≤ |(g' • z).re|\nhg₀' : ∀ (g' : SL(2, ℤ)), (g' • z).im ≤ (g • z).im\n⊢ |(g • z).re| ≤ 1 / 2",
"usedConstants": [
"Eq.mpr"... | rw [abs_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 175,
"column": 34
} | {
"line": 175,
"column": 45
} | [
{
"pp": "k : ℤ\nhk : 0 ≤ k\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝⁴ : Γ.IsArithmetic\nF : Type u_2\nF' : Type u_3\nf : F\nf' : F'\ninst✝³ : FunLike F ℍ ℂ\ninst✝² : FunLike F' ℍ ℂ\ninst✝¹ : ModularFormClass F Γ k\ninst✝ : ModularFormClass F' Γ k\nC : ℝ\nτ : ℍ\n| ‖petersson k (⇑f) (⇑f') τ‖",
"usedConstants": [
... | ← norm_norm | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 190,
"column": 34
} | {
"line": 190,
"column": 45
} | [
{
"pp": "k : ℤ\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝⁴ : Γ.IsArithmetic\nF : Type u_2\nF' : Type u_3\nf : F\nf' : F'\ninst✝³ : FunLike F ℍ ℂ\ninst✝² : FunLike F' ℍ ℂ\ninst✝¹ : CuspFormClass F Γ k\ninst✝ : ModularFormClass F' Γ k\nC : ℝ\nτ : ℍ\n| ‖petersson k (⇑f) (⇑f') τ‖",
"usedConstants": [
"Norm.norm"... | ← norm_norm | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.NumberTheory.TsumDivisorsAntidiagonal | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 78
} | [
{
"pp": "x : ↥(divisorsAntidiagonal 1)\nh : (↑x).1 * (↑x).2 = 1 ∧ 1 ≠ 0\n⊢ divisorsAntidiagonalFactors 1 x = (1, 1)",
"usedConstants": [
"MulOne.toOne",
"False",
"Nat.instMulZeroClass",
"HMul.hMul",
"Nat.divisorsAntidiagonal",
"Nat.instOne",
"and_true",
"Monoi... | simp only [mul_eq_one, ne_eq, one_ne_zero, not_false_eq_true, and_true] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Modular | {
"line": 667,
"column": 40
} | {
"line": 667,
"column": 66
} | [
{
"pp": "g✝¹ : SL(2, ℤ)\nz✝¹ : ℍ\ng✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhden : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ ≤ 1\nhc : 0 ≤ ↑g 1 0\n⊢ ↑g 1 0 = 0 ∨ ↑g 1 0 = 1",
"usedConstants": [
"_private.Mathlib.NumberTheory.Modular.0.ModularGro... | grind [abs_c_le_one hz hg] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 136,
"column": 6
} | {
"line": 136,
"column": 93
} | [
{
"pp": "case e_f.h\nz : ℍ\nt : ℂ := ∑' (m : ℤ) (n : ℤ), G2Term z ![m, n]\na : ℤ\n⊢ ∑' (n : ℤ), eisSummand 2 ![a, n] z =\n ∑' (b : ℤ), (G2Term z ![(a, b).1, (a, b).2] + (1 / (↑a * ↑z + ↑b) - 1 / (↑a * ↑z + ↑b + 1)))",
"usedConstants": [
"_private.Mathlib.NumberTheory.ModularForms.EisensteinSeries.E... | exact tsum_congr (fun b ↦ by simp [eisSummand, G2Term, aux_identity z a b, zpow_ofNat]) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Transform | {
"line": 156,
"column": 2
} | {
"line": 160,
"column": 73
} | [
{
"pp": "case hf\nz : ℍ\n⊢ Summable (fun n ↦ ∑' (m : ℤ), 1 / (↑m * ↑z + ↑n) ^ 2) (symmetricIco ℤ)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Int.instIsStrictOrderedRing",
"Real",
"DivInvMonoid.toInv",
"in... | · apply HasSum.summable (a := (z.1 ^ 2)⁻¹ * G2 (S • z))
rw [hasSum_symmetricIco_int_iff]
apply (tendsto_double_sum_S_act z).congr (fun x ↦ ?_)
rw [Summable.tsum_finsetSum (fun i hi ↦ ?_)]
simpa using linear_left_summable (ne_zero z) i (k := 2) (by norm_num) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.Derivative | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 52
} | [
{
"pp": "case h\nF G : ℍ → ℂ\nhF : MDiff F\nhG : MDiff G\nz : ℍ\nhFz : DifferentiableAt ℂ (F ∘ ↑ofComplex) ↑z\nhGz : DifferentiableAt ℂ (G ∘ ↑ofComplex) ↑z\n⊢ D (F - G) z = (D F - D G) z",
"usedConstants": [
"HSub.hSub",
"id",
"instHSub",
"UpperHalfPlane",
"Pi.instSub",
"... | simp only [normalizedDerivOfComplex, Pi.sub_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.E2.Summable | {
"line": 239,
"column": 4
} | {
"line": 239,
"column": 21
} | [
{
"pp": "case h\nz : ℍ\nN : ℕ+\nh2 :\n ↑π * I -\n 2 * ↑π * I *\n (cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ 0 +\n ∑' (n : ℕ+), cexp (2 * ↑π * I * ↑{ coe := -↑↑N / ↑z, coe_im_pos := ⋯ }) ^ ↑n) -\n ↑z / -↑↑N =\n ↑π * I -\n 2 * ↑π * I *\n (cexp... | field [ne_zero z] | Mathlib.Tactic.FieldSimp._aux_Mathlib_Tactic_Field___elabRules_Mathlib_Tactic_FieldSimp_field_1 | Mathlib.Tactic.FieldSimp.field |
Mathlib.NumberTheory.Modular | {
"line": 712,
"column": 40
} | {
"line": 712,
"column": 66
} | [
{
"pp": "g✝ : SL(2, ℤ)\nz✝ : ℍ\ng : SL(2, ℤ)\nz : ℍ\nhz : z ∈ 𝒟\nhg : g • z ∈ 𝒟\nhim : ‖denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) g)) ↑z‖ = 1\nhc : 0 ≤ ↑g 1 0\n⊢ ↑g 1 0 = 0 ∨ ↑g 1 0 = 1",
"usedConstants": [
"_private.Mathlib.NumberTheory.Modular.0.ModularGroup.cases_of_mem_fd_smul_me... | grind [abs_c_le_one hz hg] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 96
} | [
{
"pp": "case hf.hg\n⊢ DifferentiableOn ℂ η upperHalfPlaneSet",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Semiring.toModule",
"PseudoMetricSpace.toUniformSpace",
"DifferentiableAt.differentiableWithinAt",
"Membership.mem",
"NormedField.toField",
"Co... | exact fun x hx ↦ (differentiableAt_eta_of_mem_upperHalfPlaneSet hx).differentiableWithinAt | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 96
} | [
{
"pp": "case hf.hg\n⊢ DifferentiableOn ℂ η upperHalfPlaneSet",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Semiring.toModule",
"PseudoMetricSpace.toUniformSpace",
"DifferentiableAt.differentiableWithinAt",
"Membership.mem",
"NormedField.toField",
"Co... | exact fun x hx ↦ (differentiableAt_eta_of_mem_upperHalfPlaneSet hx).differentiableWithinAt | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 96
} | [
{
"pp": "case hf.hg\n⊢ DifferentiableOn ℂ η upperHalfPlaneSet",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Semiring.toModule",
"PseudoMetricSpace.toUniformSpace",
"DifferentiableAt.differentiableWithinAt",
"Membership.mem",
"NormedField.toField",
"Co... | exact fun x hx ↦ (differentiableAt_eta_of_mem_upperHalfPlaneSet hx).differentiableWithinAt | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 98,
"column": 2
} | {
"line": 102,
"column": 65
} | [
{
"pp": "case hf\n⊢ DifferentiableOn ℂ (η ∘ fun z ↦ -1 / z) upperHalfPlaneSet",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommCStarAlgebra.toNonUnitalCStarAlgebra",
"instHDiv",
"Semiring.toModule",
"NormedRing.toRin... | · apply DifferentiableOn.comp (t := upperHalfPlaneSet)
· exact fun x hx ↦ (differentiableAt_eta_of_mem_upperHalfPlaneSet hx).differentiableWithinAt
· exact DifferentiableOn.div (by fun_prop) (by fun_prop)
(fun x hx ↦ ne_zero (⟨x, hx⟩ : ℍ))
· exact fun y hy ↦ by grind [im_pnat_div_pos 1 (⟨y, hy⟩ : ℍ)... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 283,
"column": 69
} | {
"line": 283,
"column": 88
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\nHE1 :\n ∑' (v : Fin 2 → ℤ), eisSummand (↑k) v z =\n 2 * riemannZeta ↑k +\n 2 * ((-2 * ↑π * I) ^ k / ↑(k - 1)!) * ∑' (n : ℕ+), ↑((σ (k - 1)) ↑n) * cexp (2 * ↑π * I * ↑z) ^ ↑n\nHE2 : ∑' (v : Fin 2 → ℤ), eisSummand (↑k) v z = riemannZeta ↑k * eisensteinSeri... | by norm_cast; grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 99
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nm : ℕ\nβ : ℂ := -(2 * ↑k / ↑(bernoulli k))\nc : ℕ → ℂ := fun m ↦ if m = 0 then 1 else β * ↑((σ (k - 1)) m)\nτ : ℍ\nhS : Summable fun n ↦ ↑((σ (k - 1)) (n + 1)) * cexp (2 * ↑π * I * ↑τ) ^ (n + 1)\nthis : (E hk) τ = 1 - 2 * ↑k / ↑(bernoulli k) * ∑' (n : ℕ+), ↑((σ (k - 1))... | rw [this, ← tsum_pnat_eq_tsum_succ (f := fun n ↦ (σ (k - 1) n : ℂ) * cexp (2 * π * I * τ) ^ n)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 57,
"column": 4
} | {
"line": 59,
"column": 55
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : SlashInvariantFormClass F 𝒢 k\ninst✝ : 𝒢.IsFiniteRelIndex ℋ\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\n⊢ (let this := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ);\n ∑ q, quotientFunc f q) ∣[k]\n h =\n let this :... | let := Fintype.ofFinite 𝒬
simpa [SlashAction.sum_slash, quotientFunc_smul f hh]
using Equiv.sum_comp (MulAction.toPerm (_ : ℋ)) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 57,
"column": 4
} | {
"line": 59,
"column": 55
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : SlashInvariantFormClass F 𝒢 k\ninst✝ : 𝒢.IsFiniteRelIndex ℋ\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\n⊢ (let this := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ);\n ∑ q, quotientFunc f q) ∣[k]\n h =\n let this :... | let := Fintype.ofFinite 𝒬
simpa [SlashAction.sum_slash, quotientFunc_smul f hh]
using Equiv.sum_comp (MulAction.toPerm (_ : ℋ)) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula | {
"line": 224,
"column": 91
} | {
"line": 241,
"column": 98
} | [
{
"pp": "f : ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 2\n⊢ qExpansion 1 ⇑f = 0",
"usedConstants": [
"EisensteinSeries.E_ne_zero",
"ModularForm",
"Eq.mpr",
"Subgroup.instHasDetOneRangeSpecialLinearGroupGeneralLinearGroupMapGL",
"MonoidHom.range",
"Real.partial... | by
obtain ⟨c4, hc4⟩ : ∃ c4, c4 • E₄ = f.mul f :=
(finrank_eq_one_iff_of_nonzero' E₄ (E_ne_zero _ ⟨2, rfl⟩)).mp
(Module.rank_eq_one_iff_finrank_eq_one.mp levelOne_weight_four_rank_one) _
obtain ⟨c6, hc6⟩ : ∃ c6, c6 • E₆ = (f.mul f).mul f :=
(finrank_eq_one_iff_of_nonzero' E₆ (E_ne_zero _ ⟨3, rfl⟩)).mp
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.InfiniteAdeleRing | {
"line": 122,
"column": 15
} | {
"line": 125,
"column": 96
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : InfiniteAdeleRing K\nhx : ¬IsUnit x\n⊢ ‖x‖ = 0",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Iff.mpr",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Finset.mem_univ",
"Norm.no... | by
rw [Pi.isUnit_iff, not_forall] at hx
obtain ⟨v, hv⟩ := hx
exact Finset.prod_eq_zero_iff.2 ⟨v, Finset.mem_univ v, by simpa [isUnit_iff_ne_zero] using hv⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 59
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\n⊢ Continuous fun x ↦ (fun w ↦ if hw : w.IsReal then x.1 ⟨w, hw⟩ else (x.2 ⟨w, ⋯⟩).1, fun w ↦ (x.2 w).2)",
"usedConstants": [
"Continuous.comp'",
"Real",
"Pi.topologicalSpace",
"Continuous.snd",
"NumberField.InfinitePlace.IsComplex",
... | refine .prodMk (continuous_pi fun w ↦ ?_) (by fun_prop) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 63
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx : realMixedSpace K\nw : { w // w.IsReal }\n⊢ ((homeoRealMixedSpacePolarSpace K) x).1 ↑w = x.1 w",
"usedConstants": [
"Real",
"Pi.topologicalSpace",
"congrArg",
"NumberField.InfinitePlace.IsComplex",
"PseudoMetricSpace.toUniformSpace",
... | simp_rw [homeoRealMixedSpacePolarSpace_apply, dif_pos w.prop] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 63
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx : realMixedSpace K\nw : { w // w.IsReal }\n⊢ ((homeoRealMixedSpacePolarSpace K) x).1 ↑w = x.1 w",
"usedConstants": [
"Real",
"Pi.topologicalSpace",
"congrArg",
"NumberField.InfinitePlace.IsComplex",
"PseudoMetricSpace.toUniformSpace",
... | simp_rw [homeoRealMixedSpacePolarSpace_apply, dif_pos w.prop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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