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.NumberTheory.PythagoreanTriples | {
"line": 459,
"column": 62
} | {
"line": 459,
"column": 65
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhyo : y % 2 = 1\nhzpos : 0 < z\nh0 : ¬x = 0\nv : ℚ := ↑x / ↑z\nw : ℚ := ↑y / ↑z\nhq : v ^ 2 + w ^ 2 = 1\nhvz : v ≠ 0\nhw1 : w ≠ -1\nhQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0\nhp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}\nq : ℚ := (circleEquivGen hQ)... | hq2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 464,
"column": 54
} | {
"line": 464,
"column": 57
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhyo : y % 2 = 1\nhzpos : 0 < z\nh0 : ¬x = 0\nv : ℚ := ↑x / ↑z\nw : ℚ := ↑y / ↑z\nhq : v ^ 2 + w ^ 2 = 1\nhvz : v ≠ 0\nhw1 : w ≠ -1\nhQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0\nhp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}\nq : ℚ := (circleEquivGen hQ)... | hq2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 468,
"column": 8
} | {
"line": 468,
"column": 20
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhyo : y % 2 = 1\nhzpos : 0 < z\nh0 : ¬x = 0\nv : ℚ := ↑x / ↑z\nw : ℚ := ↑y / ↑z\nhq : v ^ 2 + w ^ 2 = 1\nhvz : v ≠ 0\nhw1 : w ≠ -1\nhQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0\nhp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}\nq : ℚ := (circleEquivGen hQ)... | Int.gcd_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 111,
"column": 19
} | {
"line": 111,
"column": 33
} | [
{
"pp": "case inl\nk : Type u_1\ninst✝ : Field k\np q r : ℕ\nhp : p ≠ 0\nhq : q ≠ 0\nhr : r ≠ 0\nhineq : q * r + r * p + p * q ≤ p * q * r\nchp : ↑p ≠ 0\nchq : ↑q ≠ 0\nchr : ↑r ≠ 0\na b c : k[X]\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhab : IsCoprime a b\nu v w : k\nhu : u ≠ 0\nhv : v ≠ 0\nhw : w ≠ 0\nheq : C u * ... | natDegree_pow, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.FractionalIdeal.Norm | {
"line": 84,
"column": 15
} | {
"line": 84,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDedekindDomain R\ninst✝⁴ : Free ℤ R\ninst✝³ : Module.Finite ℤ R\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nI : FractionalIdeal R⁰ K\na : ↥R⁰\nI₀ : Ideal R\nh : a • ↑I = Submodule.map (Algebra.linearMap R K) I₀\n⊢ {... | ← absNorm_div_norm_eq_absNorm_div_norm a I₀ h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 507,
"column": 6
} | {
"line": 507,
"column": 18
} | [
{
"pp": "case inr\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhzpos : 0 < z\nh2 : x % 2 = 1 ∧ y % 2 = 0\n⊢ h.IsPrimitiveClassified",
"usedConstants": [
"Int.gcd",
"congrArg",
"Eq.mp",
"instOfNatNat",
"Nat",
"OfNat.ofNat",
"Int.gcd_comm",
"Eq"
... | Int.gcd_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 23
} | [
{
"pp": "case inl\nk : Type u_1\ninst✝ : Field k\np q r : ℕ\na b c : k[X]\nu v w : k\nheq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0\nhp : p ≠ 0\nhq : q ≠ 0\nhr : r ≠ 0\nhineq : q * r + r * p + p * q ≤ p * q * r\nchp : ↑p ≠ 0\nchq : ↑q ≠ 0\nchr : ↑r ≠ 0\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhab : IsCoprime a ... | exact natDegree_C _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.FractionalIdeal | {
"line": 95,
"column": 40
} | {
"line": 95,
"column": 60
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\n⊢ Fintype.card (Free.ChooseBasisIndex ℤ ↥↑↑I) = finrank ℤ (𝓞 K)",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Submodule",
"NumberField.instFreeIntSubtypeMemSubmoduleRingOfIntegersCoeToSu... | RingOfIntegers.rank, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 553,
"column": 31
} | {
"line": 553,
"column": 43
} | [
{
"pp": "case mpr.inr.inl.right\nm n : ℤ\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\n⊢ (2 * m * n).gcd (m ^ 2 - n ^ 2) = 1",
"usedConstants": [
"Int.gcd",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"HSub.hSub",
"id",
"instOfNatNat",
"Int",
... | Int.gcd_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 553,
"column": 31
} | {
"line": 553,
"column": 43
} | [
{
"pp": "case mpr.inr.inr.right\nm n : ℤ\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\n⊢ (2 * m * n).gcd (m ^ 2 - n ^ 2) = 1",
"usedConstants": [
"Int.gcd",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"HSub.hSub",
"id",
"instOfNatNat",
"Int",
... | Int.gcd_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 325,
"column": 68
} | {
"line": 326,
"column": 33
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ∑ w, w.mult = ∑ x, 1",
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congrArg",
"NumberField.InfinitePlace.mult",
"Classical.propDecidable",
"RingHom",
"id",
"NumberField.InfinitePlace.mk",
... | ← Finset.univ.sum_fiberwise
(fun φ => InfinitePlace.mk φ) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 324,
"column": 2
} | {
"line": 328,
"column": 82
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ∑ w, w.mult = finrank ℚ K",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
... | classical
rw [← Embeddings.card K ℂ, Fintype.card, Finset.card_eq_sum_ones, ← Finset.univ.sum_fiberwise
(fun φ => InfinitePlace.mk φ)]
exact Finset.sum_congr rfl
(fun _ _ => by rw [Finset.sum_const, smul_eq_mul, mul_one, card_filter_mk_eq]) | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 324,
"column": 2
} | {
"line": 328,
"column": 82
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ∑ w, w.mult = finrank ℚ K",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
... | classical
rw [← Embeddings.card K ℂ, Fintype.card, Finset.card_eq_sum_ones, ← Finset.univ.sum_fiberwise
(fun φ => InfinitePlace.mk φ)]
exact Finset.sum_congr rfl
(fun _ _ => by rw [Finset.sum_const, smul_eq_mul, mul_one, card_filter_mk_eq]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 324,
"column": 2
} | {
"line": 328,
"column": 82
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ∑ w, w.mult = finrank ℚ K",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
... | classical
rw [← Embeddings.card K ℂ, Fintype.card, Finset.card_eq_sum_ones, ← Finset.univ.sum_fiberwise
(fun φ => InfinitePlace.mk φ)]
exact Finset.sum_congr rfl
(fun _ _ => by rw [Finset.sum_const, smul_eq_mul, mul_one, card_filter_mk_eq]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 162,
"column": 2
} | {
"line": 163,
"column": 20
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na : K\nx✝ : a ∈ (fun x ↦ (algebraMap (𝓞 K) K) ↑x) '' ↑(torsion K)\nu : (𝓞 K)ˣ\nh_tors : u ∈ ↑(torsion K)\nh_ua : (fun x ↦ (algebraMap (𝓞 K) K) ↑x) u = a\n⊢ IsIntegral ℤ a",
"usedConstants": [
"Subalgebra.instSetLike",
... | · rw [← h_ua]
exact u.val.prop | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 307,
"column": 23
} | {
"line": 307,
"column": 61
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsUnmixed k φ\nhv : ComplexEmbedding.IsReal (φ.comp (algebraMap k K))\n⊢ ComplexEmbedding.IsReal (mk φ).embedding",
"usedConstants": [
"NumberField.ComplexEmbedding.IsReal",
"congrArg",... | simp [embedding_mk_eq_of_isReal, h hv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 307,
"column": 23
} | {
"line": 307,
"column": 61
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsUnmixed k φ\nhv : ComplexEmbedding.IsReal (φ.comp (algebraMap k K))\n⊢ ComplexEmbedding.IsReal (mk φ).embedding",
"usedConstants": [
"NumberField.ComplexEmbedding.IsReal",
"congrArg",... | simp [embedding_mk_eq_of_isReal, h hv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 307,
"column": 23
} | {
"line": 307,
"column": 61
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsUnmixed k φ\nhv : ComplexEmbedding.IsReal (φ.comp (algebraMap k K))\n⊢ ComplexEmbedding.IsReal (mk φ).embedding",
"usedConstants": [
"NumberField.ComplexEmbedding.IsReal",
"congrArg",... | simp [embedding_mk_eq_of_isReal, h hv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 16
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ rootDiscr K = ↑|discr K| ^ (↑(finrank ℚ K))⁻¹",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instPow",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"abs",... | rw [rootDiscr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 16
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ rootDiscr K = ↑|discr K| ^ (↑(finrank ℚ K))⁻¹",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instPow",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"abs",... | rw [rootDiscr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 16
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ rootDiscr K = ↑|discr K| ^ (↑(finrank ℚ K))⁻¹",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instPow",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"abs",... | rw [rootDiscr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 102,
"column": 67
} | {
"line": 103,
"column": 44
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nf : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=\n (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))\ne : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm\nM : Matrix (index K) (index K) ℝ := (mixedEm... | measure_fundamentalDomain
((latticeBasis K).reindex e.symm), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 175,
"column": 8
} | {
"line": 175,
"column": 20
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nB : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))\nh_le : minkowskiBound K I ≤ volume (convexBodySum K B)\nx✝ : K\n⊢ ↑(FractionalIdeal.absNorm ↑I) *\n (2⁻¹ ^ nrComplexPla... | coe_real_pi, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 158,
"column": 11
} | {
"line": 158,
"column": 23
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\n⊢ ∑ x_1, (canonicalEmbedding K) (((integralBasis K).repr x) x_1 • (integralBasis K) x_1) ∈\n Submodule.span ℚ (Set.range ⇑(latticeBasis K))",
"usedConstants": [
"NumberField.canonicalEmbedding.latticeBasis",
"Finsupp.instF... | map_rat_smul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 214,
"column": 52
} | {
"line": 214,
"column": 81
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ Fintype.card { w // w.IsReal } + ∑ i, finrank ℝ ℂ = finrank ℚ K",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",... | Complex.finrank_real_complex, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 85
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na : 𝓞 K\nh_mem : (algebraMap (𝓞 K) K) a ∈ ↑1\nh_nz : (algebraMap (𝓞 K) K) a ≠ 0\nh_nm :\n ↑|(Algebra.norm ℚ) ((algebraMap (𝓞 K) K) a)| ≤\n ↑(FractionalIdeal.absNorm ↑1) * (4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K).factorial /\n ↑(f... | simp_rw [Units.val_one, FractionalIdeal.absNorm_one, Rat.cast_one, one_mul] at h_nm | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 325,
"column": 27
} | {
"line": 327,
"column": 35
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nw : InfinitePlace K\nx : mixedSpace K\n⊢ 0 ≤ (normAtPlace w) x",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"ZeroHom.funLike",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
... | by
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
split_ifs <;> exact norm_nonneg _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 237,
"column": 33
} | {
"line": 237,
"column": 46
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsTotallyComplex K\nh : 0 < ↑(finrank ℚ K)\n⊢ ↑(finrank ℚ K) ^ (2 * finrank ℚ K) / ((4 / π) ^ finrank ℚ K * ↑(finrank ℚ K).factorial ^ 2) ≤ ↑|discr K|",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instPow",
"... | Real.rpow_two | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 593,
"column": 54
} | {
"line": 593,
"column": 66
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nc : index K\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) c) =\n ∑ x_1,\n fromBlocks 1 0 0\n ... | sum_product, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 361,
"column": 59
} | {
"line": 361,
"column": 75
} | [
{
"pp": "case inl.a\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nN : ℕ\nhK : |discr K| ≤ ↑N\nh_nz : N ≠ 0\nh₂ : 1 < 3 * π / 4\nh✝ : 1 < finrank ℚ K\nh : Real.logb (3 * π / 4) (9 / 4 * ↑N) < ↑(finrank ℚ K)\n⊢ ↑↑N ≤ ↑N",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 645,
"column": 44
} | {
"line": 645,
"column": 64
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : LinearIndependent ℝ (⇑(commMap K) ∘ ⇑(canonicalEmbedding.latticeBasis K))\n⊢ finrank ℤ (𝓞 K) = finrank ℝ (mixedSpace K)",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"Pi.Function.module",
"In... | RingOfIntegers.rank, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 646,
"column": 26
} | {
"line": 646,
"column": 55
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : LinearIndependent ℝ (⇑(commMap K) ∘ ⇑(canonicalEmbedding.latticeBasis K))\n⊢ finrank ℚ K = Fintype.card { w // w.IsReal } + ∑ i, finrank ℝ ℂ",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProdu... | Complex.finrank_real_complex, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 692,
"column": 11
} | {
"line": 692,
"column": 23
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\n⊢ ∑ x_1, (mixedEmbedding K) (((integralBasis K).repr x) x_1 • (integralBasis K) x_1) ∈\n Submodule.span ℚ (Set.range ⇑(latticeBasis K))",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Finsupp.instFunLike",
... | map_rat_smul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 22
} | [
{
"pp": "case refine_1\nA : Type u_1\ninst✝⁷ : CommRing A\nB : Type u_2\ninst✝⁶ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁵ : CommRing K\ninst✝⁴ : Algebra A K\ninst✝³ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝² : CommRing L\ninst✝¹ : Algebra B L\ninst✝ : IsLocalization N L\n... | rintro ⟨b, _, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.LocalRing.Quotient | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 40
} | [
{
"pp": "case a\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : IsLocalRing R\ninst✝¹ : Module.Finite R S\ninst✝ : Free R S\nthis : Module.Finite (R ⧸ p) (S ⧸ pS)\n⊢ finrank (R ⧸ p) (S ⧸ pS) ≤ finrank R S",
"usedConstants": [
"NonUnitalCommRing.toN... | let b := Module.Free.chooseBasis R S | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 844,
"column": 2
} | {
"line": 844,
"column": 13
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ (stdOrthonormalBasis K).toBasis.map ↑(toMixed K) = stdBasis K",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Pi.Function.module",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 844,
"column": 2
} | {
"line": 844,
"column": 13
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ (stdOrthonormalBasis K).toBasis.map ↑(toMixed K) = stdBasis K",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Pi.Function.module",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 844,
"column": 2
} | {
"line": 844,
"column": 13
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ (stdOrthonormalBasis K).toBasis.map ↑(toMixed K) = stdBasis K",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Pi.Function.module",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 234,
"column": 4
} | {
"line": 234,
"column": 65
} | [
{
"pp": "case refine_2\nA : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : IsDomain A\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : IsDomain B\ninst✝¹⁴ : Algebra A B\ninst✝¹³ : IsTorsionFree A B\ninst✝¹² : Field K\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra A K\ninst✝⁹ : Algebra B L\ninst✝⁸ : ... | exact IsIntegral.tower_bot_of_field <| isIntegral_trans _ hx₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 266,
"column": 32
} | {
"line": 266,
"column": 60
} | [
{
"pp": "case pos\nA : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : IsDomain A\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : IsDomain B\ninst✝¹⁴ : Algebra A B\ninst✝¹³ : IsTorsionFree A B\ninst✝¹² : Field K\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra A K\ninst✝⁹ : Algebra B L\ninst✝⁸ : IsFra... | extendedHom_eq_zero_iff L B, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 268,
"column": 23
} | {
"line": 268,
"column": 51
} | [
{
"pp": "case pos\nA : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : IsDomain A\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : IsDomain B\ninst✝¹⁴ : Algebra A B\ninst✝¹³ : IsTorsionFree A B\ninst✝¹² : Field K\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra A K\ninst✝⁹ : Algebra B L\ninst✝⁸ : IsFra... | extendedHom_eq_zero_iff L B, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.PID | {
"line": 246,
"column": 30
} | {
"line": 252,
"column": 39
} | [
{
"pp": "R : Type u_2\ninst✝² : CommRing R\ninst✝¹ : IsDedekindDomain R\ninst✝ : UniqueFactorizationMonoid R\n⊢ IsPrincipalIdealRing R",
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
"Iff.mpr",
"Eq.mpr",
"Submodule",
"IsDedekindDomain.toIsDomain",
"False",
"Is... | by
refine .of_prime_ne_bot fun P hp hp₀ ↦ ?_
obtain ⟨x, hx₁, hx₂⟩ := hp.exists_mem_prime_of_ne_bot hp₀
suffices Ideal.span {x} = P from this ▸ inferInstance
have := (Ideal.span_singleton_prime hx₂.ne_zero).mpr hx₂
exact (Ring.DimensionLeOne.prime_le_prime_iff_eq (by aesop)).mp <|
P.span_singleton_le_iff_m... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Int | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\ninst✝² : Algebra.IsIntegral ℤ R\nP : Ideal R\ninst✝¹ : P.IsPrime\ninst✝ : NeZero P\nthis : under ℤ P = ⊥\n⊢ P = ⊥",
"usedConstants": [
"Int.instNontrivial",
"Ideal.eq_bot_of_comap_eq_bot",
"Int",
"Int.instCommRing",
... | exact eq_bot_of_comap_eq_bot this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 408,
"column": 29
} | {
"line": 408,
"column": 36
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDomain R\nS : Type u_3\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : IsIntegrallyClosed R\ninst✝⁸ : IsIntegrallyClosed S\ninst✝⁷ : Algebra R S\ninst✝⁶ : Module.Finite R S\ninst✝⁵ : IsTorsionFree R S\ninst✝⁴ : IsDedekindDomain R\ninst✝³ :... | bot_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 412,
"column": 2
} | {
"line": 420,
"column": 46
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDomain R\nS : Type u_3\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : IsIntegrallyClosed R\ninst✝⁸ : IsIntegrallyClosed S\ninst✝⁷ : Algebra R S\ninst✝⁶ : Module.Finite R S\ninst✝⁵ : IsTorsionFree R S\ninst✝⁴ : IsDedekindDomain R\ninst✝³ :... | have h₀ : ∀ Q ∈ (p.primesOver S).toFinset,
relNorm R Q ^ ramificationIdx p Q = p ^ ((p.ramificationIdxIn S) * s) := by
intro Q hQ
rw [Set.mem_toFinset] at hQ
have : Q.IsPrime := hQ.1
have : Q.LiesOver p := hQ.2
rw [← ramificationIdxIn_eq_ramificationIdx p Q G]
obtain ⟨σ, rfl⟩ := Ideal.exis... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 444,
"column": 4
} | {
"line": 444,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDomain R\nS : Type u_3\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : IsIntegrallyClosed R\ninst✝⁸ : IsIntegrallyClosed S\ninst✝⁷ : Algebra R S\ninst✝⁶ : Module.Finite R S\ninst✝⁵ : IsTorsionFree R S\ninst✝⁴ : IsDedekindDomain R\ninst✝³ : IsDedekin... | inertiaDeg_algebra_tower p P Q, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 27
} | [
{
"pp": "case neg.hi.convert_2\nR : Type u\nK : Type v\nL : Type z\np : R\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra R L\ninst✝⁴ : Algebra R K\ninst✝³ : IsScalarTower R K L\ninst✝² : IsDomain R\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\nB : ... | choose! g hg using hind | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___macroRules_Mathlib_Tactic_Choose_tacticChoose!___Using__1 | Mathlib.Tactic.Choose.tacticChoose!___Using_ |
Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral | {
"line": 321,
"column": 4
} | {
"line": 322,
"column": 39
} | [
{
"pp": "case neg.hi.convert_2\nR : Type u\nK : Type v\nL : Type z\np : R\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra R L\ninst✝⁴ : Algebra R K\ninst✝³ : IsScalarTower R K L\ninst✝² : IsDomain R\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\nB : ... | rw [aeval_eq_sum_range, Hj, range_add, sum_union (disjoint_range_addLeftEmbedding _ _),
sum_congr rfl hg, add_comm] at hQ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 719,
"column": 4
} | {
"line": 720,
"column": 37
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²¹ : CommRing A\ninst✝²⁰ : Field K\ninst✝¹⁹ : CommRing B\ninst✝¹⁸ : Field L\ninst✝¹⁷ : Algebra A K\ninst✝¹⁶ : Algebra B L\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Algebra A L\ninst✝¹² : IsScalarTower A K L\ninst✝¹¹ : IsScalarTow... | simp only [FractionalIdeal.mem_coeIdeal, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂] at hx | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 340,
"column": 64
} | {
"line": 343,
"column": 85
} | [
{
"pp": "k : ℕ\nK : Type u\ninst✝² : Field K\nζ : K\ninst✝¹ : CharZero K\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\n⊢ (Algebra.norm ℤ) (hζ.toInteger ^ 2 ^ k - 1) = (-2) ^ 2 ^ k",
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractio... | by
have : NumberField K := IsCyclotomicExtension.numberField {2 ^ (k + 1)} ℚ K
rw [Algebra.norm_eq_iff ℤ (Sₘ := K) (Rₘ := ℚ) le_rfl]
simp [hζ.norm_pow_sub_one_two (cyclotomic.irreducible_rat (pow_pos (by decide) _))] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 876,
"column": 4
} | {
"line": 877,
"column": 37
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra A B\ninst✝⁸ : IsDomain A\ninst✝⁷ : IsDedekindDomain A\ninst✝⁶ : IsDedekindDomain B\ninst✝⁵ : IsTorsionFree A B\ninst✝⁴ : Module.Finite A B\ninst✝³ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal ... | simp only [FractionalIdeal.mem_coeIdeal, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂] at hx | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 898,
"column": 2
} | {
"line": 899,
"column": 45
} | [
{
"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 hx' : (e (Ideal.Quotient.mk _ x)).2 = 0 := by
simpa [e, Ideal.Quotient.eq_zero_iff_mem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 459,
"column": 51
} | {
"line": 459,
"column": 67
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ finrank ℤ (Additive ((𝓞 K)ˣ ⧸ torsion K)) = Fintype.card (Fin (rank K))",
"usedConstants": [
"Eq.mpr",
"NumberField.instCommRingRingOfIntegers",
"congrArg",
"CommSemiring.toSemiring",
"Additive",
"NumberFi... | rank_modTorsion, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic | {
"line": 103,
"column": 2
} | {
"line": 105,
"column": 58
} | [
{
"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\nhab : a * b ≠ 0\nhF : ringChar F = 2\n⊢ quadraticCharFun F (a * b) = quadraticCharFun F a * quadraticCharFun F b",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"... | · -- case `ringChar F = 2`
rw [quadraticCharFun_eq_one_of_char_two hF ha, quadraticCharFun_eq_one_of_char_two hF hb,
quadraticCharFun_eq_one_of_char_two hF hab, mul_one] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LegendreSymbol.Basic | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 23
} | [
{
"pp": "case h.e'_3.h.e'_6.h.e'_5\np : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℤ\nhc : ringChar (ZMod p) ≠ 2\n⊢ p = Fintype.card (ZMod p)",
"usedConstants": [
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Nat.instOne",
"ZMod.fintype",
"Fintype.card",
... | exact (card p).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.LegendreSymbol.Basic | {
"line": 228,
"column": 2
} | {
"line": 232,
"column": 49
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℤ\nha : ↑a ≠ 0\nx y : ZMod p\nhx : x ≠ 0\nhxy : x ^ 2 - ↑a * y ^ 2 = 0\n⊢ legendreSym p a = 1",
"usedConstants": [
"Int.cast",
"False",
"Nat.instMulZeroClass",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"MulZeroClass.toMul",... | haveI hy : y ≠ 0 := by
rintro rfl
rw [zero_pow two_ne_zero, mul_zero, sub_zero, sq_eq_zero_iff] at hxy
exact hx hxy
exact eq_one_of_sq_sub_mul_sq_eq_zero ha hy hxy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.Basic | {
"line": 228,
"column": 2
} | {
"line": 232,
"column": 49
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℤ\nha : ↑a ≠ 0\nx y : ZMod p\nhx : x ≠ 0\nhxy : x ^ 2 - ↑a * y ^ 2 = 0\n⊢ legendreSym p a = 1",
"usedConstants": [
"Int.cast",
"False",
"Nat.instMulZeroClass",
"IsDomain.to_noZeroDivisors",
"HMul.hMul",
"MulZeroClass.toMul",... | haveI hy : y ≠ 0 := by
rintro rfl
rw [zero_pow two_ne_zero, mul_zero, sub_zero, sq_eq_zero_iff] at hxy
exact hx hxy
exact eq_one_of_sq_sub_mul_sq_eq_zero ha hy hxy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 739,
"column": 4
} | {
"line": 739,
"column": 78
} | [
{
"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] ... | rw [Int.isCoprime_iff_nat_coprime, natAbs_discr n₁ F₁, natAbs_discr n₂ F₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LegendreSymbol.Basic | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 56
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℤ\nh : legendreSym p a = -1\nx y : ℤ\nhxy : ↑p ∣ x ^ 2 - a * y ^ 2\n⊢ ↑p ∣ x ∧ ↑p ∣ y",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring",
"HS... | simp_rw [← ZMod.intCast_zmod_eq_zero_iff_dvd] at hxy ⊢ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 89
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : p ≠ 2\nthis : -2 = ↑(-2)\n⊢ legendreSym p (-2) = χ₈' ↑p",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"Nat.instMulZeroClass",
"legendreSym._proof_1",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
... | rw [legendreSym, ← this, quadraticChar_neg_two ((ringChar_zmod_n p).substr hp), card p] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity | {
"line": 81,
"column": 2
} | {
"line": 83,
"column": 5
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : p ≠ 2\n⊢ IsSquare (-2) ↔ p % 8 = 1 ∨ p % 8 = 3",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NegZeroClass.toNeg",
"Nat.instMulZeroClass",
"Nat.Prime",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Nat.instOne",
... | rw [FiniteField.isSquare_neg_two_iff, card p]
have h₁ := (Prime.mod_two_eq_one_iff_ne_two Fact.out).mpr hp
lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity | {
"line": 81,
"column": 2
} | {
"line": 83,
"column": 5
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : p ≠ 2\n⊢ IsSquare (-2) ↔ p % 8 = 1 ∨ p % 8 = 3",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NegZeroClass.toNeg",
"Nat.instMulZeroClass",
"Nat.Prime",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Nat.instOne",
... | rw [FiniteField.isSquare_neg_two_iff, card p]
have h₁ := (Prime.mod_two_eq_one_iff_ne_two Fact.out).mpr hp
lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum | {
"line": 69,
"column": 4
} | {
"line": 72,
"column": 7
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ¬ringChar F = 2\n⊢ IsSquare (-2) ↔ Fintype.card F % 8 ≠ 5 ∧ Fintype.card F % 8 ≠ 7",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ZMod.commRin... | have h := FiniteField.odd_card_of_char_ne_two hF
rw [← quadraticChar_one_iff_isSquare (neg_ne_zero.mpr (Ring.two_ne_zero hF)),
quadraticChar_neg_two hF, χ₈'_nat_eq_if_mod_eight]
lia | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum | {
"line": 69,
"column": 4
} | {
"line": 72,
"column": 7
} | [
{
"pp": "case neg\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ¬ringChar F = 2\n⊢ IsSquare (-2) ↔ Fintype.card F % 8 ≠ 5 ∧ Fintype.card F % 8 ≠ 7",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ZMod.commRin... | have h := FiniteField.odd_card_of_char_ne_two hF
rw [← quadraticChar_one_iff_isSquare (neg_ne_zero.mpr (Ring.two_ne_zero hF)),
quadraticChar_neg_two hF, χ₈'_nat_eq_if_mod_eight]
lia | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Fermat | {
"line": 65,
"column": 2
} | {
"line": 66,
"column": 41
} | [
{
"pp": "n : ℕ\n⊢ n.fermatNumber = ∏ k ∈ range n, k.fermatNumber + 2",
"usedConstants": [
"Eq.mpr",
"Nat.fermatNumber",
"congrArg",
"HSub.hSub",
"Nat.prod_fermatNumber",
"le_of_lt",
"id",
"instSubNat",
"instOfNatNat",
"Finset.prod",
"Finset.r... | rw [prod_fermatNumber, Nat.sub_add_cancel]
exact le_of_lt <| two_lt_fermatNumber _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Fermat | {
"line": 65,
"column": 2
} | {
"line": 66,
"column": 41
} | [
{
"pp": "n : ℕ\n⊢ n.fermatNumber = ∏ k ∈ range n, k.fermatNumber + 2",
"usedConstants": [
"Eq.mpr",
"Nat.fermatNumber",
"congrArg",
"HSub.hSub",
"Nat.prod_fermatNumber",
"le_of_lt",
"id",
"instSubNat",
"instOfNatNat",
"Finset.prod",
"Finset.r... | rw [prod_fermatNumber, Nat.sub_add_cancel]
exact le_of_lt <| two_lt_fermatNumber _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FermatPsp | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 39
} | [
{
"pp": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : ¬p ∣ b * (b ^ 2 - 1)\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nhA : p < A\nhi_A : 1 < A\nhi_B : 1 < B\nhi_b : 0 < b\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 ≤ b ^ (2 * p)\nhi_bpowpsubone : 1 ≤ b ^... | rw [Nat.div_mul_cancel hd] at AB_id | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Harmonic.Int | {
"line": 48,
"column": 2
} | {
"line": 49,
"column": 100
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\n⊢ ‖↑(harmonic n)‖ = 2 ^ Nat.log 2 n",
"usedConstants": [
"zpow_natCast",
"Rat.instOfNat",
"Norm.norm",
"Nat.cast_ofNat",
"Eq.mpr",
"Padic.eq_padicNorm",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"DivisionRing.to... | rw [Padic.eq_padicNorm, padicNorm.eq_zpow_of_nonzero (harmonic_pos hn).ne',
padicValRat_two_harmonic, neg_neg, zpow_natCast, Rat.cast_pow, Rat.cast_natCast, Nat.cast_ofNat] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Harmonic.Int | {
"line": 48,
"column": 2
} | {
"line": 49,
"column": 100
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\n⊢ ‖↑(harmonic n)‖ = 2 ^ Nat.log 2 n",
"usedConstants": [
"zpow_natCast",
"Rat.instOfNat",
"Norm.norm",
"Nat.cast_ofNat",
"Eq.mpr",
"Padic.eq_padicNorm",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"DivisionRing.to... | rw [Padic.eq_padicNorm, padicNorm.eq_zpow_of_nonzero (harmonic_pos hn).ne',
padicValRat_two_harmonic, neg_neg, zpow_natCast, Rat.cast_pow, Rat.cast_natCast, Nat.cast_ofNat] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Harmonic.Int | {
"line": 48,
"column": 2
} | {
"line": 49,
"column": 100
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\n⊢ ‖↑(harmonic n)‖ = 2 ^ Nat.log 2 n",
"usedConstants": [
"zpow_natCast",
"Rat.instOfNat",
"Norm.norm",
"Nat.cast_ofNat",
"Eq.mpr",
"Padic.eq_padicNorm",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"DivisionRing.to... | rw [Padic.eq_padicNorm, padicNorm.eq_zpow_of_nonzero (harmonic_pos hn).ne',
padicValRat_two_harmonic, neg_neg, zpow_natCast, Rat.cast_pow, Rat.cast_natCast, Nat.cast_ofNat] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 88,
"column": 6
} | {
"line": 93,
"column": 79
} | [
{
"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": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",... | simp_rw [← mul_one_div (n : ℝ)]
rw [intervalIntegral.integral_sub]
· simp_rw [intervalIntegral.integral_const_mul]
· exact intervalIntegral.intervalIntegrable_one_div (fun x hx ↦ (hv x hx).ne') (by fun_prop)
· exact (intervalIntegral.intervalIntegrable_one_div
(fun x hx ↦ (sq_pos_of_po... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 88,
"column": 6
} | {
"line": 93,
"column": 79
} | [
{
"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": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",... | simp_rw [← mul_one_div (n : ℝ)]
rw [intervalIntegral.integral_sub]
· simp_rw [intervalIntegral.integral_const_mul]
· exact intervalIntegral.intervalIntegrable_one_div (fun x hx ↦ (hv x hx).ne') (by fun_prop)
· exact (intervalIntegral.intervalIntegrable_one_div
(fun x hx ↦ (sq_pos_of_po... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 119,
"column": 4
} | {
"line": 123,
"column": 11
} | [
{
"pp": "case succ\nN : ℕ\nhN : term_sum 1 N = log (↑N + 1) - ↑(harmonic (N + 1)) + 1\n⊢ term_sum 1 (N + 1) = log (↑(N + 1) + 1) - ↑(harmonic (N + 1 + 1)) + 1",
"usedConstants": [
"zero_le",
"NormedCommRing.toNormedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZer... | unfold term_sum at hN ⊢
rw [Finset.sum_range_succ, hN, harmonic_succ (N + 1),
term_one (by positivity : 0 < N + 1)]
push_cast
ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 119,
"column": 4
} | {
"line": 123,
"column": 11
} | [
{
"pp": "case succ\nN : ℕ\nhN : term_sum 1 N = log (↑N + 1) - ↑(harmonic (N + 1)) + 1\n⊢ term_sum 1 (N + 1) = log (↑(N + 1) + 1) - ↑(harmonic (N + 1 + 1)) + 1",
"usedConstants": [
"zero_le",
"NormedCommRing.toNormedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZer... | unfold term_sum at hN ⊢
rw [Finset.sum_range_succ, hN, harmonic_succ (N + 1),
term_one (by positivity : 0 < N + 1)]
push_cast
ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.ProductFormula | {
"line": 52,
"column": 61
} | {
"line": 52,
"column": 77
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : 𝓞 K\nh_x_nezero : x ≠ 0\nh_span_nezero : span {x} ≠ 0\n⊢ (∏ᶠ (i : HeightOneSpectrum (𝓞 K)), ‖(embedding i) ↑x‖) *\n ↑↑(absNorm (∏ᶠ (v : HeightOneSpectrum (𝓞 K)), v.maxPowDividing (span {x}))) =\n 1",
"usedConstants": [
"Int... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 602,
"column": 8
} | {
"line": 602,
"column": 43
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\nhx : x ≠ 0\nH : x • ![x⁻¹, 1] = ![1, x]\n⊢ mulHeight ![x⁻¹, 1] = mulHeight ![x, 1]",
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"instSMulOfMul",
"DivisionCommMonoid.toDivisionMonoid"... | ← mulHeight_smul_eq_mulHeight _ hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Height.Basic | {
"line": 671,
"column": 2
} | {
"line": 673,
"column": 23
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : AdmissibleAbsValues K\nα : Type u\ninst✝¹ : Fintype α\nι : α → Type v\ninst✝ : ∀ (a : α), Finite (ι a)\nx : (a : α) → ι a → K\nhx : ∀ (a : α), x a ≠ 0\n⊢ (Multiset.map (fun x_1 ↦ ∏ a, ⨆ i, x_1 (x a i)) archAbsVal).prod * ∏ᶠ (v : ↑nonarchAbsVal), ∏ a, ⨆ i, ↑v (x ... | rw [Multiset.prod_map_prod,
finprod_prod_comm _ _ fun b _ ↦ hasFiniteMulSupport_iSup_nonarchAbsVal (hx b),
← prod_mul_distrib] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Height.Basic | {
"line": 756,
"column": 4
} | {
"line": 756,
"column": 34
} | [
{
"pp": "case inr.inr.inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\nhι : Nonempty ι\nhx : x ≠ 0\n⊢ mulHeight (x * 0) ≤ mulHeight x * mulHeight 0",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
... | simpa using one_le_mulHeight x | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.Height.Basic | {
"line": 756,
"column": 4
} | {
"line": 756,
"column": 34
} | [
{
"pp": "case inr.inr.inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\nhι : Nonempty ι\nhx : x ≠ 0\n⊢ mulHeight (x * 0) ≤ mulHeight x * mulHeight 0",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
... | simpa using one_le_mulHeight x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Height.Basic | {
"line": 756,
"column": 4
} | {
"line": 756,
"column": 34
} | [
{
"pp": "case inr.inr.inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : AdmissibleAbsValues K\nι : Type u_2\ninst✝ : Finite ι\nx : ι → K\nhι : Nonempty ι\nhx : x ≠ 0\n⊢ mulHeight (x * 0) ≤ mulHeight x * mulHeight 0",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
... | simpa using one_le_mulHeight x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Height.Basic | {
"line": 925,
"column": 2
} | {
"line": 926,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx y : K\n⊢ logHeight₁ (x - y) ≤ ↑(totalWeight K) * log 2 + logHeight₁ x + logHeight₁ y",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"HMul.hMul",
"Height.logHeight₁_add_l... | rw [sub_eq_add_neg, ← logHeight₁_neg y]
exact logHeight₁_add_le x (-y) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Height.Basic | {
"line": 925,
"column": 2
} | {
"line": 926,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx y : K\n⊢ logHeight₁ (x - y) ≤ ↑(totalWeight K) * log 2 + logHeight₁ x + logHeight₁ y",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"HMul.hMul",
"Height.logHeight₁_add_l... | rw [sub_eq_add_neg, ← logHeight₁_neg y]
exact logHeight₁_add_le x (-y) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 171,
"column": 4
} | {
"line": 172,
"column": 54
} | [
{
"pp": "case h\nF : 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\nx : F\n| χ x * φ (0 - x) * ψ 0",
"usedConstants": [
"NegZeroClass.toNeg",
"MulOne.toOne",
"NonUnitalCommRing.toNonU... | rw [zero_sub, neg_eq_neg_one_mul x, map_mul, mul_left_comm (χ x) (φ (-1)),
← MulChar.mul_apply, ψ.map_zero_eq_one, mul_one] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.NumberTheory.JacobiSum.Basic | {
"line": 305,
"column": 2
} | {
"line": 316,
"column": 10
} | [
{
"pp": "case neg\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... | · classical
rw [jacobiSum_eq_aux, MulChar.sum_eq_zero_of_ne_one hχ₀, MulChar.sum_eq_zero_of_ne_one hψ₀, hq]
have : NeZero n := ⟨by lia⟩
have H := MulChar.exists_apply_sub_one_mul_apply_sub_one hχ hψ hμ
have Hcs x := (H x).choose_spec
refine ⟨-q * z₁ + ∑ x ∈ (univ \ {0, 1} : Finset F), (H x).choose, ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 161,
"column": 22
} | {
"line": 161,
"column": 90
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ZMod N\ns : ℂ\nhs : 1 < s.re\nn : ℕ\nthis : ↑(toCircle ↑(↑j.val * ↑n)) = cexp (2 * ↑π * I * ↑(↑j.val * ↑n) / ↑N)\n| cexp (2 * ↑π * I * ↑(↑j.val * ↑n) / ↑N)",
"usedConstants": [
"Int.cast",
"Int.cast_natCast",
"instHDiv",
"Real.pi",
"HMul.hM... | rw [Int.cast_mul, Int.cast_natCast, Int.cast_natCast, mul_div_assoc] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 161,
"column": 22
} | {
"line": 161,
"column": 90
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ZMod N\ns : ℂ\nhs : 1 < s.re\nn : ℕ\nthis : ↑(toCircle ↑(↑j.val * ↑n)) = cexp (2 * ↑π * I * ↑(↑j.val * ↑n) / ↑N)\n| cexp (2 * ↑π * I * ↑(↑j.val * ↑n) / ↑N)",
"usedConstants": [
"Int.cast",
"Int.cast_natCast",
"instHDiv",
"Real.pi",
"HMul.hM... | rw [Int.cast_mul, Int.cast_natCast, Int.cast_natCast, mul_div_assoc] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 161,
"column": 22
} | {
"line": 161,
"column": 90
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ZMod N\ns : ℂ\nhs : 1 < s.re\nn : ℕ\nthis : ↑(toCircle ↑(↑j.val * ↑n)) = cexp (2 * ↑π * I * ↑(↑j.val * ↑n) / ↑N)\n| cexp (2 * ↑π * I * ↑(↑j.val * ↑n) / ↑N)",
"usedConstants": [
"Int.cast",
"Int.cast_natCast",
"instHDiv",
"Real.pi",
"HMul.hM... | rw [Int.cast_mul, Int.cast_natCast, Int.cast_natCast, mul_div_assoc] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 346,
"column": 88
} | {
"line": 350,
"column": 6
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\ns : ℂ\n⊢ completedLFunction Φ s = completedLFunction₀ Φ s - ↑N ^ (-s) * Φ 0 / s - (↑N ^ (-s) * ∑ j, Φ j) / (1 - s)",
"usedConstants": [
"ZMod.completedLFunction",
"Eq.mpr",
"Semigroup.toMul",
"Real",
"DivInvMonoid.toInv",
... | by
simp only [completedLFunction, completedHurwitzZetaEven_eq, toAddCircle_eq_zero, div_eq_mul_inv,
ite_mul, one_mul, zero_mul, mul_sub, mul_ite, mul_zero, sum_sub_distrib, Fintype.sum_ite_eq',
← sum_mul, completedLFunction₀, mul_assoc]
abel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 422,
"column": 2
} | {
"line": 423,
"column": 92
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nhΦ : Function.Even Φ\ns : ℂ\nhs : 1 < s.re\nhs₀ : s ≠ 0\nhs₁ : s ≠ 1\n⊢ ∑ x, Φ x * cosZeta (toAddCircle x) s = LFunction (𝓕 Φ) s",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Pi.Function.module",
"GroupWithZero.toMonoid... | simp only [cosZeta_eq, ← mul_div_assoc _ _ (2 : ℂ), mul_add, ← sum_div, sum_add_distrib,
LFunction_dft Φ (.inr hs₁), map_neg, div_eq_iff (two_ne_zero' ℂ), mul_two, add_left_inj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 190,
"column": 10
} | {
"line": 190,
"column": 94
} | [
{
"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| Tendsto (fun s ↦ (∏ p ∈ N.primeFactors, (1 - ↑p ^ (-s))) * ((s - 1) * riemannZeta s)) (𝓝[≠] 1)\n (𝓝 (∏ p ∈ N.primeFactors, (1 - ... | enter [3, 1]; rw [← mul_one <| Finset.prod ..]; enter [1, 2, p]; rw [← cpow_neg_one] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 190,
"column": 10
} | {
"line": 190,
"column": 94
} | [
{
"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| Tendsto (fun s ↦ (∏ p ∈ N.primeFactors, (1 - ↑p ^ (-s))) * ((s - 1) * riemannZeta s)) (𝓝[≠] 1)\n (𝓝 (∏ p ∈ N.primeFactors, (1 - ... | enter [3, 1]; rw [← mul_one <| Finset.prod ..]; enter [1, 2, p]; rw [← cpow_neg_one] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 89
} | [
{
"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) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s * (χ (-1) * χ⁻¹ (-1)) =\n ↑N ^ (s - 1) / ε * gaussSum χ stdAddChar * completed... | rw [← MulChar.mul_apply, mul_inv_cancel, MulChar.one_apply (isUnit_one.neg), mul_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 89
} | [
{
"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) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s * (χ (-1) * χ⁻¹ (-1)) =\n ↑N ^ (s - 1) / ε * gaussSum χ stdAddChar * completed... | rw [← MulChar.mul_apply, mul_inv_cancel, MulChar.one_apply (isUnit_one.neg), mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.DirichletContinuation | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 89
} | [
{
"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) / ε * gaussSum χ stdAddChar * completedLFunction χ⁻¹ s * (χ (-1) * χ⁻¹ (-1)) =\n ↑N ^ (s - 1) / ε * gaussSum χ stdAddChar * completed... | rw [← MulChar.mul_apply, mul_inv_cancel, MulChar.one_apply (isUnit_one.neg), mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 56
} | [
{
"pp": "f g : ℕ → ℂ\ns a b : ℂ\nhf : LSeriesHasSum f s a\nhg : LSeriesHasSum g s b\n⊢ LSeriesHasSum (f - g) s (a - b)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"SummationFilter",
"Co... | simpa [LSeriesHasSum, term_sub] using HasSum.sub hf hg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 56
} | [
{
"pp": "f g : ℕ → ℂ\ns a b : ℂ\nhf : LSeriesHasSum f s a\nhg : LSeriesHasSum g s b\n⊢ LSeriesHasSum (f - g) s (a - b)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"SummationFilter",
"Co... | simpa [LSeriesHasSum, term_sub] using HasSum.sub hf hg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 56
} | [
{
"pp": "f g : ℕ → ℂ\ns a b : ℂ\nhf : LSeriesHasSum f s a\nhg : LSeriesHasSum g s b\n⊢ LSeriesHasSum (f - g) s (a - b)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"SummationFilter",
"Co... | simpa [LSeriesHasSum, term_sub] using HasSum.sub hf hg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 21
} | [
{
"pp": "k : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\nh1 : ∀ (n : ℕ), 2 * ↑k ≠ -↑n\nh2 : 2 * ↑k ≠ 1\n⊢ (2 * ↑π) ^ (-(2 * ↑k)) ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NegZeroClass.toNeg",
"False",
"Real",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
... | simp [pi_ne_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 21
} | [
{
"pp": "k : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\nh1 : ∀ (n : ℕ), 2 * ↑k + 1 ≠ -↑n\n⊢ (2 * ↑π) ^ (-(2 * ↑k + 1)) ≠ 0",
"usedConstants": [
"neg_add_rev",
"NormedCommRing.toNormedRing",
"False",
"Real",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
"Real.inst... | simp [pi_ne_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ZetaValues | {
"line": 177,
"column": 8
} | {
"line": 184,
"column": 71
} | [
{
"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 ^ (k + 1) / ↑m = 0 ∨ ∫ ... | right
rw [intervalIntegral.integral_finsetSum]
· simp only [intervalIntegral.integral_comp_add_right, zero_add, ← one_div, ← add_div,
add_comm (1 : ℝ), ← Nat.cast_add_one]
rw [intervalIntegral.sum_integral_adjacent_intervals]
· simp [div_self m0', integral_bernoulliFun_eq... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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