module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 106,
"column": 33
} | {
"line": 106,
"column": 50
} | {
"line": 106,
"column": 51
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx : (𝓞 K)ˣ\nw : InfinitePlace K\n⊢ Real.log (w ((algebraMap (𝓞 K) K) ↑x)) = 0 ↔ w ((algebraMap (𝓞 K) K) ↑x) = 1",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NumberField.InfinitePlace.instFunLikeReal",
"Units.val",
"Eq.mpr",
... | [
"K : Type u_1\ninst✝ : Field K\nx : (𝓞 K)ˣ\nw : InfinitePlace K\n⊢ w ((algebraMap (𝓞 K) K) ↑x) = 0 ∨ w ((algebraMap (𝓞 K) K) ↑x) = 1 ∨ w ((algebraMap (𝓞 K) K) ↑x) = -1 ↔\n w ((algebraMap (𝓞 K) K) ↑x) = 1",
"K : Type u_1\ninst✝ : Field K\nx : (𝓞 K)ˣ\nw : InfinitePlace K\n⊢ ¬↑w.mult = 0"
] | Real.log_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 52
} | {
"line": 122,
"column": 6
} | [
{
"pp": "case pos\nK : 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\n⊢ -↑w₀.mult * Real.log (w₀ ((algebraMap (𝓞 K) K) ↑x)) = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NumberFiel... | [
"case pos\nK : 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\n⊢ ∀ x_1 ∈ univ, (logEmbedding K) (Additive.ofMul x) x_1 = 0"
] | rw [← sum_logEmbedding_component, sum_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Three | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 41
} | {
"line": 70,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nu : (𝓞 K)ˣ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nhrank : rank K = 0\nx : ↥(torsion K)\ne : Fin (rank K) → ℤ\nhxu : u = ↑x\nn : ℕ\nhnpos : 0 < n\nhn : ↑x ^ n = 1\n⊢ (algebraMap (𝓞 K) K) (↑u ^ ↑⟨n, hnpos⟩) = 1",
... | [
"K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\nu : (𝓞 K)ˣ\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nhrank : rank K = 0\nx : ↥(torsion K)\ne : Fin (rank K) → ℤ\nhxu : u = ↑x\nn : ℕ\nhnpos : 0 < n\nhn : ↑x ^ n = 1\n⊢ ↑u ^ ↑⟨n, hnpos⟩ = 1"
] | convert! map_one (algebraMap (𝓞 K) K) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 268,
"column": 6
} | {
"line": 268,
"column": 53
} | {
"line": 269,
"column": 6
} | [
{
"pp": "case succ\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw₁ : InfinitePlace K\nB : ℕ\nhB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B\nn : ℕ\nw : InfinitePlace K\nhw : w ≠ w₁\nm : ℕ\nm_ih : n < m → w ((algebraMap (𝓞 K) K) ↑(seq K w₁ hB m)) < w ((algebraMap (𝓞 K) K) ↑(seq K w₁ hB n))\n... | [] | cases eq_or_lt_of_le (Nat.le_of_lt_succ h) with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 730,
"column": 19
} | {
"line": 730,
"column": 33
} | {
"line": 730,
"column": 34
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²¹ : CommRing A\ninst✝²⁰ : Field K\ninst✝¹⁹ : CommRing B\ninst✝¹⁸ : Field L\ninst✝¹⁷ : Algebra A K\ninst✝¹⁶ : Algebra B L\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Algebra A L\ninst✝¹² : IsScalarTower A K L\ninst✝¹¹ : IsScalarTow... | [
"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²¹ : CommRing A\ninst✝²⁰ : Field K\ninst✝¹⁹ : CommRing B\ninst✝¹⁸ : Field L\ninst✝¹⁷ : Algebra A K\ninst✝¹⁶ : Algebra B L\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Algebra A L\ninst✝¹² : IsScalarTower A K L\ninst✝¹¹ : IsScalarTower A B L\nin... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 300,
"column": 2
} | {
"line": 310,
"column": 45
} | {
"line": 311,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw₁ : InfinitePlace K\nB : ℕ\nhB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B\nn m : ℕ\nhnm : n < m\nh : Ideal.span {↑(seq K w₁ hB n)} = Ideal.span {↑(seq K w₁ hB m)}\n⊢ ∃ u, ∀ (w : InfinitePlace K), w ≠ w₁ → Real.log (w ((algebraMap (𝓞 K) K... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw₁ : InfinitePlace K\nB : ℕ\nhB : minkowskiBound K 1 < ↑(convexBodyLTFactor K) * ↑B\n⊢ ∃ n m, n < m ∧ Ideal.span {↑(seq K w₁ hB n)} = Ideal.span {↑(seq K w₁ hB m)}"
] | · have hu := Ideal.span_singleton_eq_span_singleton.mp h
refine ⟨hu.choose, fun w hw ↦ Real.log_neg (pos_at_place hu.choose w) ?_⟩
calc
_ = w (algebraMap (𝓞 K) K (seq K w₁ hB m) * (algebraMap (𝓞 K) K (seq K w₁ hB n))⁻¹) := by
rw [← congr_arg (algebraMap (𝓞 K) K) hu.choose_spec, mul_comm, map_mu... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 334,
"column": 59
} | {
"line": 334,
"column": 75
} | {
"line": 334,
"column": 76
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : Basis { w // w ≠ w₀ } ℝ ({ w // w ≠ w₀ } → ℝ) := Pi.basisFun ℝ { w // w ≠ w₀ }\nv : { w // w ≠ w₀ } → logSpace K := fun w ↦ (logEmbedding K) (Additive.ofMul ⋯.choose)\nw : { w // w ≠ w₀ }\n⊢ 0 < |v w w| - ∑ x ∈ univ.erase w, -v w x",
"ppTer... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : Basis { w // w ≠ w₀ } ℝ ({ w // w ≠ w₀ } → ℝ) := Pi.basisFun ℝ { w // w ≠ w₀ }\nv : { w // w ≠ w₀ } → logSpace K := fun w ↦ (logEmbedding K) (Additive.ofMul ⋯.choose)\nw : { w // w ≠ w₀ }\n⊢ 0 < |v w w| - -∑ x ∈ univ.erase w, v w x",
"K : Type u_1\ninst✝... | sum_neg_distrib, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 343,
"column": 4
} | {
"line": 343,
"column": 39
} | {
"line": 344,
"column": 4
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : Basis { w // w ≠ w₀ } ℝ ({ w // w ≠ w₀ } → ℝ) := Pi.basisFun ℝ { w // w ≠ w₀ }\nv : { w // w ≠ w₀ } → logSpace K := fun w ↦ (logEmbedding K) (Additive.ofMul ⋯.choose)\nw x : { w // w ≠ w₀ }\nhx : x ∈ univ.erase w\n⊢ 0 < ↑(↑x).mul... | [
"case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : Basis { w // w ≠ w₀ } ℝ ({ w // w ≠ w₀ } → ℝ) := Pi.basisFun ℝ { w // w ≠ w₀ }\nv : { w // w ≠ w₀ } → logSpace K := fun w ↦ (logEmbedding K) (Additive.ofMul ⋯.choose)\nw x : { w // w ≠ w₀ }\nhx : x ∈ univ.erase w\n⊢ ↑x ≠ ↑w"
] | · rw [mult]; split_ifs <;> norm_num | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.FactorisationProperties | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 45
} | {
"line": 234,
"column": 2
} | [
{
"pp": "a : ℕ\n⊢ ∃ b ∈ {n | Odd n ∧ n.Abundant}, a < b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.FactorisationProperties.0.Nat.infinite_odd_abundant._proof_1_1",
"instOfNatNat",
"Nat.Abundant",
"Nat",
"OfNat.ofNat"
],
... | [
"a : ℕ\nha : Abundant 945\n⊢ ∃ b ∈ {n | Odd n ∧ n.Abundant}, a < b"
] | have ha : Abundant 945 := by decide +kernel | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic | {
"line": 76,
"column": 59
} | {
"line": 77,
"column": 76
} | {
"line": 79,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\ninst✝¹ : Fintype F\ninst✝ : DecidableEq F\n⊢ quadraticCharFun F 1 = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"False",
"NeZero.one",
"instDecidableTrue",
"if_true",
"MulZeroClass.toMul",
"Fintype.IsSquare.d... | [] | by
simp only [quadraticCharFun, one_ne_zero, IsSquare.one, if_true, if_false] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic | {
"line": 268,
"column": 89
} | {
"line": 278,
"column": 7
} | {
"line": 280,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\n⊢ IsSquare (-1) ↔ Fintype.card F % 4 ≠ 3",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"NegZeroClass.toNeg",
"Nat.instMulZeroClass",
"NeZero.one",
"ZMod.χ₄",
"ZMod.c... | [] | by
classical -- suggested by the linter (instead of `[DecidableEq F]`)
by_cases hF : ringChar F = 2
· simp only [FiniteField.isSquare_of_char_two hF, Ne, true_iff]
exact fun hf ↦
one_ne_zero <|
(Nat.odd_of_mod_four_eq_three hf).symm.trans <| FiniteField.even_card_of_char_two hF
· have h₁ := Fi... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LegendreSymbol.Basic | {
"line": 240,
"column": 8
} | {
"line": 240,
"column": 32
} | {
"line": 240,
"column": 32
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℤ\nh : legendreSym p a = -1\nx y : ZMod p\nhxy : x ^ 2 - ↑a * y ^ 2 = 0\nhf : ↑a = 0\n⊢ False",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Int.cast",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"con... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℤ\nh : 0 = -1\nx y : ZMod p\nhxy : x ^ 2 - ↑a * y ^ 2 = 0\nhf : ↑a = 0\n⊢ False"
] | (eq_zero_iff p a).mpr hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Three | {
"line": 249,
"column": 49
} | {
"line": 249,
"column": 62
} | {
"line": 250,
"column": 4
} | [
{
"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\nz : 𝓞 K\nhz : S'.c = λ * z\nx : 𝓞 K\nhx : S'.a ^ 3 - 1 = λ ^ 4 * x\ny : 𝓞 K\nhy : S'.b ^ 3 - 1 = λ ^ 4 * y\n⊢ IsPrimitiveRoot ζ (3 ^ 1)",
"ppTerm": "... | [] | rwa [pow_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.FLT.Three | {
"line": 260,
"column": 49
} | {
"line": 260,
"column": 62
} | {
"line": 261,
"column": 4
} | [
{
"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\nz : 𝓞 K\nhz : S'.c = λ * z\nx : 𝓞 K\nhx : S'.a ^ 3 + 1 = λ ^ 4 * x\ny : 𝓞 K\nhy : S'.b ^ 3 + 1 = λ ^ 4 * y\n⊢ IsPrimitiveRoot ζ (3 ^ 1)",
"ppTerm": "... | [] | rwa [pow_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.FermatPsp | {
"line": 240,
"column": 4
} | {
"line": 240,
"column": 23
} | {
"line": 243,
"column": 2
} | [
{
"pp": "b : ℕ\nb_ge_two : 2 ≤ b\np : ℕ\np_prime : Prime p\np_gt_two : 2 < p\nnot_dvd : ¬p ∣ b * (b ^ 2 - 1)\nA : ℕ := (b ^ p - 1) / (b - 1)\nB : ℕ := (b ^ p + 1) / (b + 1)\nhA : p < A\nhi_A : 1 < A\nhi_B : 1 < B\nhi_b : 0 < b\nhi_bsquared : 0 < b ^ 2 - 1\nhi_bpowtwop : 1 ≤ b ^ (2 * p)\nhi_bpowpsubone : 1 ≤ b ^... | [] | exact mod_cast this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 887,
"column": 19
} | {
"line": 887,
"column": 33
} | {
"line": 887,
"column": 34
} | [
{
"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 ... | [
"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 A\ninst✝² : ... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Fermat | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 63
} | {
"line": 193,
"column": 0
} | [
{
"pp": "n p : ℕ\nhn : 1 < n\nhp : Prime p\nthis : Fact (Prime p)\nhpdvd : ↑2 ^ 2 ^ n + 1 = 0\nhp2 : p ≠ 2\nhp8 : p % 8 = 1\na : ZMod p\nha : 2 = a * a\n⊢ ↑a.val ^ 2 ^ (n + 1) + 1 = 0",
"ppTerm": "?m.261",
"assigned": true,
"usedConstants": [
"Nat.pow_succ'",
"instPowNat",
"Eq.mpr"... | [] | rwa [natCast_val, ZMod.cast_id, pow_succ', pow_mul, sq, ← ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.Ideal.NatInt | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 23
} | {
"line": 63,
"column": 4
} | [
{
"pp": "case refine_1.inl\n⊢ ⊥.IsPrime",
"ppTerm": "?refine_1.inl",
"assigned": true,
"usedConstants": [
"IsDomain.to_noZeroDivisors",
"Nat.instNontrivial",
"Nat.instIsDomain",
"Ideal.isPrime_bot",
"Nat",
"Nat.instSemiring"
],
"usedFVars": [],
"usedGo... | [] | exact isPrime_bot | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.NatInt | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 23
} | {
"line": 63,
"column": 4
} | [
{
"pp": "case refine_1.inl\n⊢ ⊥.IsPrime",
"ppTerm": "?refine_1.inl",
"assigned": true,
"usedConstants": [
"IsDomain.to_noZeroDivisors",
"Nat.instNontrivial",
"Nat.instIsDomain",
"Ideal.isPrime_bot",
"Nat",
"Nat.instSemiring"
],
"usedFVars": [],
"usedGo... | [] | exact isPrime_bot | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.NatInt | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 23
} | {
"line": 63,
"column": 4
} | [
{
"pp": "case refine_1.inl\n⊢ ⊥.IsPrime",
"ppTerm": "?refine_1.inl",
"assigned": true,
"usedConstants": [
"IsDomain.to_noZeroDivisors",
"Nat.instNontrivial",
"Nat.instIsDomain",
"Ideal.isPrime_bot",
"Nat",
"Nat.instSemiring"
],
"usedFVars": [],
"usedGo... | [] | exact isPrime_bot | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Height.Basic | {
"line": 564,
"column": 2
} | {
"line": 564,
"column": 97
} | {
"line": 566,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\n⊢ logHeight₁ x = logHeight ![x, 1]",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"Height.logHeight₁",
"Field.toDivisionRi... | [] | simp only [logHeight₁_eq_log_mulHeight₁, logHeight_eq_log_mulHeight, mulHeight₁_eq_mulHeight x] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Height.Basic | {
"line": 564,
"column": 2
} | {
"line": 564,
"column": 97
} | {
"line": 566,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\n⊢ logHeight₁ x = logHeight ![x, 1]",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"Height.logHeight₁",
"Field.toDivisionRi... | [] | simp only [logHeight₁_eq_log_mulHeight₁, logHeight_eq_log_mulHeight, mulHeight₁_eq_mulHeight x] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Height.Basic | {
"line": 564,
"column": 2
} | {
"line": 564,
"column": 97
} | {
"line": 566,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : AdmissibleAbsValues K\nx : K\n⊢ logHeight₁ x = logHeight ![x, 1]",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"Height.logHeight₁",
"Field.toDivisionRi... | [] | simp only [logHeight₁_eq_log_mulHeight₁, logHeight_eq_log_mulHeight, mulHeight₁_eq_mulHeight x] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.ProductFormula | {
"line": 50,
"column": 42
} | {
"line": 50,
"column": 59
} | {
"line": 51,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : 𝓞 K\nh_x_nezero : x ≠ 0\n⊢ span {x} ≠ 0",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
"Submodule",
"False",
"Semiring.toModule",
"eq_false",
"NumberFi... | [] | simp [h_x_nezero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.NumberField.ProductFormula | {
"line": 88,
"column": 27
} | {
"line": 88,
"column": 43
} | {
"line": 88,
"column": 44
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na b : 𝓞 K\nh_x_nezero : (algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b ≠ 0\nhb : b ≠ 0\nha : a ≠ 0\n⊢ (|↑((Algebra.norm ℤ) a)|⁻¹ / |↑((Algebra.norm ℤ) b)|⁻¹)⁻¹ =\n |↑((Algebra.norm ℚ) ((algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b))|",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na b : 𝓞 K\nh_x_nezero : (algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b ≠ 0\nhb : b ≠ 0\nha : a ≠ 0\n⊢ |↑((Algebra.norm ℤ) a)| / |↑((Algebra.norm ℤ) b)| =\n |↑((Algebra.norm ℚ) ((algebraMap (𝓞 K) K) a / (algebraMap (𝓞 K) K) b))|"
] | inv_inv_div_inv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 194,
"column": 10
} | {
"line": 194,
"column": 24
} | {
"line": 194,
"column": 25
} | [
{
"pp": "case e_a\nn : ℕ\nhn : 0 < n\ns : ℝ\nhs : 1 < s\nhv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x\nthis : 0 ∉ uIcc (↑n) (↑n + 1)\n⊢ ↑n * ((↑n ^ (-s) - (↑n + 1) ^ (-s)) / s) = ↑n * (1 / ↑n ^ s - 1 / (↑n + 1) ^ s) / s",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.in... | [
"case e_a\nn : ℕ\nhn : 0 < n\ns : ℝ\nhs : 1 < s\nhv : ∀ x ∈ uIcc (↑n) (↑n + 1), 0 < x\nthis : 0 ∉ uIcc (↑n) (↑n + 1)\n⊢ ↑n * ((↑n ^ (-s) - (↑n + 1) ^ (-s)) / s) = ↑n * ((1 / ↑n ^ s - 1 / (↑n + 1) ^ s) / s)"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 284,
"column": 8
} | {
"line": 284,
"column": 57
} | {
"line": 284,
"column": 57
} | [
{
"pp": "case h₂\nK : Type u_4\ninst✝² : Field K\nι : Type u_5\nι' : Type u_6\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Finite ι'\np : ι' → MvPolynomial ι K\nh : (fun j ↦ constantCoeff (p j)) ≠ 0\nthis : Nonempty ι'\nv : ↑nonarchAbsVal\nj : ι'\n⊢ ↑v (constantCoeff (p j)) ≤ max (⨆ s, ↑v (coeff (↑s) (p j))) 1",
... | [
"case h₂\nK : Type u_4\ninst✝² : Field K\nι : Type u_5\nι' : Type u_6\ninst✝¹ : AdmissibleAbsValues K\ninst✝ : Finite ι'\np : ι' → MvPolynomial ι K\nh : (fun j ↦ constantCoeff (p j)) ≠ 0\nthis : Nonempty ι'\nv : ↑nonarchAbsVal\nj : ι'\n⊢ ↑v (coeff 0 (p j)) ≤ max (⨆ s, ↑v (coeff (↑s) (p j))) 1"
] | show constantCoeff (p j) = coeff 0 (p j) from rfl | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 302,
"column": 4
} | {
"line": 304,
"column": 82
} | {
"line": 305,
"column": 2
} | [
{
"pp": "case 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\n⊢ (mulHeight fun j ↦ (eval 0) (p j)) ≤ max (mulHeightBound p) 1 * mulHeight 0 ^ N",
... | [] | rcases eq_or_ne (fun j ↦ constantCoeff (p j)) 0 with h | h
· simp [h]
· simpa using le_max_of_le_left <| mulHeight_constantCoeff_le_mulHeightBound h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Height.MvPolynomial | {
"line": 302,
"column": 4
} | {
"line": 304,
"column": 82
} | {
"line": 305,
"column": 2
} | [
{
"pp": "case 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\n⊢ (mulHeight fun j ↦ (eval 0) (p j)) ≤ max (mulHeightBound p) 1 * mulHeight 0 ^ N",
... | [] | rcases eq_or_ne (fun j ↦ constantCoeff (p j)) 0 with h | h
· simp [h]
· simpa using le_max_of_le_left <| mulHeight_constantCoeff_le_mulHeightBound h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 472,
"column": 60
} | {
"line": 472,
"column": 70
} | {
"line": 473,
"column": 8
} | [
{
"pp": "s : ℂ\nhs : s ≠ 1\nhg_an : AnalyticOnNhd ℂ (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) {1}ᶜ\nz : ℂ\nhz : 1 < z.re\n⊢ (starRingEnd ℂ) (∑' (n : ℕ), 1 / ↑n ^ (starRingEnd ℂ) z) = riemannZeta z",
"ppTerm": "?m.192",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nor... | [
"s : ℂ\nhs : s ≠ 1\nhg_an : AnalyticOnNhd ℂ (fun z ↦ (starRingEnd ℂ) (riemannZeta ((starRingEnd ℂ) z))) {1}ᶜ\nz : ℂ\nhz : 1 < z.re\n⊢ ∑' (a : ℕ), (starRingEnd ℂ) (1 / ↑a ^ (starRingEnd ℂ) z) = riemannZeta z"
] | conj_tsum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 601,
"column": 26
} | {
"line": 601,
"column": 38
} | {
"line": 601,
"column": 38
} | [
{
"pp": "s : ℝ\nhs : s > 1\nthis : (riemannZeta ↑s).re = (s - 1)⁻¹ * (riemannZeta₁ ↑s).re\n⊢ Real.log (s - 1)⁻¹ + Real.log (riemannZeta₁ ↑s).re = -Real.log (s - 1) + Real.log (riemannZeta₁ ↑s).re",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"riemannZeta₁",
"R... | [
"s : ℝ\nhs : s > 1\nthis : (riemannZeta ↑s).re = (s - 1)⁻¹ * (riemannZeta₁ ↑s).re\n⊢ -Real.log (s - 1) + Real.log (riemannZeta₁ ↑s).re = -Real.log (s - 1) + Real.log (riemannZeta₁ ↑s).re",
"case hx\ns : ℝ\nhs : s > 1\nthis : (riemannZeta ↑s).re = (s - 1)⁻¹ * (riemannZeta₁ ↑s).re\n⊢ (s - 1)⁻¹ ≠ 0",
"case hy\ns :... | Real.log_inv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.ZetaAsymp | {
"line": 630,
"column": 2
} | {
"line": 633,
"column": 20
} | {
"line": 634,
"column": 2
} | [
{
"pp": "⊢ (fun s ↦ deriv riemannZeta s / riemannZeta s + (s - 1)⁻¹ - ↑γ) =O[𝓝[≠] 1] fun x ↦ x - 1",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"riemannZeta₁",
"instHDiv",
"Semiring.toModule",
"riemannZeta",
"co... | [
"⊢ (fun s ↦ deriv riemannZeta₁ s / riemannZeta₁ s - ↑γ) =O[𝓝 1] fun x ↦ x - 1"
] | suffices (fun s ↦ (deriv riemannZeta₁ s) / (riemannZeta₁ s) - γ) =O[𝓝 1] (· - 1) by
refine (this.mono nhdsWithin_le_nhds).congr' ?_ .rfl
filter_upwards [log_deriv_riemannZeta_eq_neg_inv_sub_add]
simp +contextual | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 266,
"column": 49
} | {
"line": 266,
"column": 52
} | {
"line": 266,
"column": 53
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\nf : R[X]\ninst✝¹ : Algebra R S\nh : IsAdjoinRoot S f\nT : Type u_1\ninst✝ : CommRing T\ni : R →+* T\nx : T\nhx : eval₂ i x f = 0\nz : S\nw y : R[X]\nhy : h.repr z - w = f * y\n⊢ eval₂ i x f * eval₂ i x y = 0",
"ppTerm": "?m.123",
"as... | [
"R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : Ring S\nf : R[X]\ninst✝¹ : Algebra R S\nh : IsAdjoinRoot S f\nT : Type u_1\ninst✝ : CommRing T\ni : R →+* T\nx : T\nhx : eval₂ i x f = 0\nz : S\nw y : R[X]\nhy : h.repr z - w = f * y\n⊢ 0 * eval₂ i x y = 0"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IsAdjoinRoot | {
"line": 555,
"column": 8
} | {
"line": 555,
"column": 20
} | {
"line": 555,
"column": 20
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nx y : S\nhxy : ∀ i < f.natDegree, h.coeff x i = h.coeff y i\ni : Fin f.natDegree\n⊢ h.coeff x ↑i = h.coeff y ↑i",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring S\nf : R[X]\ninst✝ : Algebra R S\nh : IsAdjoinRootMonic S f\nx y : S\nhxy : ∀ i < f.natDegree, h.coeff x i = h.coeff y i\ni : Fin f.natDegree\n⊢ h.coeff y ↑i = h.coeff y ↑i"
] | hxy i i.prop | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 456,
"column": 11
} | {
"line": 456,
"column": 14
} | {
"line": 456,
"column": 15
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nx : ι → ℤ\nv : FinitePlace ℚ\nhx : Finset.univ.gcd x = 1\nhv : IsNonarchimedean fun x ↦ v x\nH : ∀ (n : ℤ), v ↑n ≤ 1\nf : ι → ℤ\nhf : v ↑(Finset.univ.gcd x) = v ↑(∑ a, x a * f a)\n⊢ ⨆ i, v ↑(x i) = 1",
"ppTerm": "?m.182",
"assigned": true,
... | [
"ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nx : ι → ℤ\nv : FinitePlace ℚ\nhx : Finset.univ.gcd x = 1\nhv : IsNonarchimedean fun x ↦ v x\nH : ∀ (n : ℤ), v ↑n ≤ 1\nf : ι → ℤ\nhf : v ↑1 = v ↑(∑ a, x a * f a)\n⊢ ⨆ i, v ↑(x i) = 1"
] | hx, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Height.NumberField | {
"line": 471,
"column": 8
} | {
"line": 471,
"column": 11
} | {
"line": 471,
"column": 12
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nx : ι → ℤ\nhx : x = 0\n⊢ Finset.univ.gcd x ≠ 1",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congrArg",
"Int.instStrongNormalizedGCDMonoid",
"instNormalizedGCDMonoidOfSt... | [
"ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nx : ι → ℤ\nhx : x = 0\n⊢ Finset.univ.gcd 0 ≠ 1"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.KummerDedekind | {
"line": 188,
"column": 4
} | {
"line": 196,
"column": 74
} | {
"line": 197,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nx : S\nI : Ideal R\ninst✝³ : IsDomain R\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDedekindDomain S\ninst✝ : Module.IsTorsionFree R S\nhI : I.IsMaximal\nhI' : I ≠ ⊥\nhx : comap (algebraMap R S) (conductor R x) ⊔... | [] | obtain ⟨y, hy⟩ := this
have h := prod_normalizedFactors (show I.map (algebraMap R S) ≠ 0 by
rwa [← bot_eq_zero, Ne,
map_eq_bot_iff_of_injective (FaithfulSMul.algebraMap_injective R S)])
rw [associated_iff_eq, hy, Multiset.prod_singleton] at h
rw [← h]
exact
irreducible_of_nor... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.KummerDedekind | {
"line": 188,
"column": 4
} | {
"line": 196,
"column": 74
} | {
"line": 197,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nx : S\nI : Ideal R\ninst✝³ : IsDomain R\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDedekindDomain S\ninst✝ : Module.IsTorsionFree R S\nhI : I.IsMaximal\nhI' : I ≠ ⊥\nhx : comap (algebraMap R S) (conductor R x) ⊔... | [] | obtain ⟨y, hy⟩ := this
have h := prod_normalizedFactors (show I.map (algebraMap R S) ≠ 0 by
rwa [← bot_eq_zero, Ne,
map_eq_bot_iff_of_injective (FaithfulSMul.algebraMap_injective R S)])
rw [associated_iff_eq, hy, Multiset.prod_singleton] at h
rw [← h]
exact
irreducible_of_nor... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 56,
"column": 4
} | {
"line": 57,
"column": 80
} | {
"line": 58,
"column": 4
} | [
{
"pp": "k : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\n⊢ ∑' (b : ℕ), ↑(Real.cos (2 * π * x * ↑b)) / ↑b ^ (2 * ↑k) =\n (-1) ^ (k + 1) * (2 * ↑π) ^ (2 * k) / 2 / ↑(2 * k)! *\n Polynomial.eval (↑x) (Polynomial.map (algebraMap ℚ ℂ) (Polynomial.bernoulli (2 * k)))",
"ppTerm": "?m.111",
"assigned": true... | [
"case refine_1\nk : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\n⊢ ∑' (b : ℕ), ↑(Real.cos (2 * π * x * ↑b)) / ↑b ^ (2 * ↑k) = ↑(∑' (b : ℕ), 1 / ↑b ^ (2 * k) * Real.cos (2 * π * ↑b * x))",
"case refine_2\nk : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\n⊢ ↑((-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / ↑(2 * k)! *\n Polyn... | refine Eq.trans ?_ <|
(congr_arg ofReal (hasSum_one_div_nat_pow_mul_cos hk hx).tsum_eq).trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 16
} | {
"line": 57,
"column": 0
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\n⊢ term (-f) s = -term f s",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"neg_div",
"Pi.instNeg",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"DivisionCommMonoid.toDivisio... | [] | ext ⟨- | n⟩ <;>
simp [neg_div] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 16
} | {
"line": 57,
"column": 0
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\n⊢ term (-f) s = -term f s",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"neg_div",
"Pi.instNeg",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"DivisionCommMonoid.toDivisio... | [] | ext ⟨- | n⟩ <;>
simp [neg_div] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.Linearity | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 16
} | {
"line": 57,
"column": 0
} | [
{
"pp": "f : ℕ → ℂ\ns : ℂ\n⊢ term (-f) s = -term f s",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"neg_div",
"Pi.instNeg",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"DivisionCommMonoid.toDivisio... | [] | ext ⟨- | n⟩ <;>
simp [neg_div] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 139,
"column": 6
} | {
"line": 139,
"column": 20
} | {
"line": 139,
"column": 21
} | [
{
"pp": "case e_a.e_a\nk : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\nh1 : ∀ (n : ℕ), 2 * ↑k ≠ -↑n\nh2 : 2 * ↑k ≠ 1\nh3 : (2 * ↑k).Gammaℂ ≠ 0\n⊢ Complex.cos (↑π * (2 * ↑k) / 2) * (-1) ^ (k + 1) = -1",
"ppTerm": "?e_a.e_a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"R... | [
"case e_a.e_a\nk : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\nh1 : ∀ (n : ℕ), 2 * ↑k ≠ -↑n\nh2 : 2 * ↑k ≠ 1\nh3 : (2 * ↑k).Gammaℂ ≠ 0\n⊢ Complex.cos (↑π * (2 * ↑k / 2)) * (-1) ^ (k + 1) = -1"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaValues | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 20
} | {
"line": 159,
"column": 21
} | [
{
"pp": "case e_a.e_a\nk : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\nh1 : ∀ (n : ℕ), 2 * ↑k + 1 ≠ -↑n\nh3 : (2 * ↑k + 1).Gammaℂ ≠ 0\n⊢ Complex.sin (↑π * (2 * ↑k + 1) / 2) * (-1) ^ (k + 1) = -1",
"ppTerm": "?e_a.e_a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"Real.p... | [
"case e_a.e_a\nk : ℕ\nx : ℝ\nhk : k ≠ 0\nhx : x ∈ Icc 0 1\nh1 : ∀ (n : ℕ), 2 * ↑k + 1 ≠ -↑n\nh3 : (2 * ↑k + 1).Gammaℂ ≠ 0\n⊢ Complex.sin (↑π * ((2 * ↑k + 1) / 2)) * (-1) ^ (k + 1) = -1"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.ZMod | {
"line": 417,
"column": 2
} | {
"line": 420,
"column": 70
} | {
"line": 422,
"column": 2
} | [
{
"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 * completedCosZeta (toAddCircle x) s = completedLFunction (𝓕 Φ) s",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"ZMod.completedLFunction",
"Iff.... | [
"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"
] | suffices ∑ x, Φ x * cosZeta (toAddCircle x) s = LFunction (𝓕 Φ) s by
simpa only [cosZeta, Function.update_of_ne hs₀, ← mul_div_assoc, ← sum_div,
LFunction_eq_completed_div_gammaFactor_even (dft_even_iff.mpr hΦ) _ (.inl hs₀),
div_left_inj' (Gammaℝ_ne_zero_of_re_pos (zero_lt_one.trans hs))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.NumberTheory.LSeries.Nonvanishing | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 86
} | {
"line": 149,
"column": 0
} | [
{
"pp": "case hL\nN : ℕ\ninst✝ : NeZero N\nB : BadChar N\ns : ℂ\nhs : s ≠ 1\n⊢ B.F =ᶠ[𝓝 s] riemannZeta * LFunction B.χ",
"ppTerm": "?hL",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.toSeminormedCommRing",
"T6Space.toT5Space",
"Semiring... | [] | · filter_upwards [eventually_ne_nhds hs] with t ht using Function.update_of_ne ht .. | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 196,
"column": 42
} | {
"line": 196,
"column": 51
} | {
"line": 196,
"column": 52
} | [
{
"pp": "a : ℤ\ne b x✝ : ℕ\nih : J(a ^ x✝ | b) = J(a | b) ^ x✝\n⊢ J(a ^ x✝ * a | b) = J(a | b) ^ x✝ * J(a | b)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"id",
"MulOne.toMul",
"Int",
... | [
"a : ℤ\ne b x✝ : ℕ\nih : J(a ^ x✝ | b) = J(a | b) ^ x✝\n⊢ J(a ^ x✝ | b) * J(a | b) = J(a | b) ^ x✝ * J(a | b)"
] | mul_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 239,
"column": 66
} | {
"line": 239,
"column": 75
} | {
"line": 239,
"column": 76
} | [
{
"pp": "case cons\nn✝ : ℕ\nn : ℤ\nl' : List ℤ\nih : J(l'.prod | n✝) = (List.map (fun a ↦ J(a | n✝)) l').prod\n⊢ J(n * l'.prod | n✝) = J(n | n✝) * (List.map (fun a ↦ J(a | n✝)) l').prod",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
... | [
"case cons\nn✝ : ℕ\nn : ℤ\nl' : List ℤ\nih : J(l'.prod | n✝) = (List.map (fun a ↦ J(a | n✝)) l').prod\n⊢ J(n | n✝) * J(l'.prod | n✝) = J(n | n✝) * (List.map (fun a ↦ J(a | n✝)) l').prod"
] | mul_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 284,
"column": 2
} | {
"line": 285,
"column": 36
} | {
"line": 287,
"column": 0
} | [
{
"pp": "a : ℤ\np : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ J(a | p) = -1 ↔ ¬IsSquare ↑a",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"legendreSym.eq_neg_one_iff",
"Int.cast",
"Eq.mpr",
"congrArg",
"jacobiSym.legendreSym.to_jacobiSym",
"Field.toDivisionRing",... | [] | rw [← legendreSym.to_jacobiSym]
exact legendreSym.eq_neg_one_iff p | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 284,
"column": 2
} | {
"line": 285,
"column": 36
} | {
"line": 287,
"column": 0
} | [
{
"pp": "a : ℤ\np : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ J(a | p) = -1 ↔ ¬IsSquare ↑a",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"legendreSym.eq_neg_one_iff",
"Int.cast",
"Eq.mpr",
"congrArg",
"jacobiSym.legendreSym.to_jacobiSym",
"Field.toDivisionRing",... | [] | rw [← legendreSym.to_jacobiSym]
exact legendreSym.eq_neg_one_iff p | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.GaussEisensteinLemmas | {
"line": 118,
"column": 6
} | {
"line": 118,
"column": 86
} | {
"line": 119,
"column": 4
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp2✝ : Fact (p % 2 = 1)\na : ℕ\nhap : ↑a ≠ 0\nhp2 : ↑p = ↑1\n⊢ ↑(∑ x ∈ Ico 1 (p / 2).succ, (a * x % p + p * (a * x / p))) =\n ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x % p) + ↑(∑ x ∈ Ico 1 (p / 2).succ, a * x / p)",
"ppTerm": "?m.417",
"assigned": true,
"usedCo... | [] | simp [sum_add_distrib, ← mul_sum, Nat.cast_add, Nat.cast_mul, Nat.cast_sum, hp2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 320,
"column": 26
} | {
"line": 320,
"column": 35
} | {
"line": 320,
"column": 36
} | [
{
"pp": "a : ℤ\nb : ℕ\nhb : Odd b\n⊢ J(-1 * a | b) = χ₄ ↑b * J(a | b)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"ZMod.χ₄",
"CommRing.toNonUnitalCommRing",
... | [
"a : ℤ\nb : ℕ\nhb : Odd b\n⊢ J(-1 | b) * J(a | b) = χ₄ ↑b * J(a | b)"
] | mul_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.SumCoeff | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 61
} | {
"line": 221,
"column": 2
} | [
{
"pp": "f : ℕ → ℂ\nl : ℂ\nhlim : Tendsto (fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) atTop (𝓝 l)\nε : ℝ\nhε : 0 < ε\nt : ℝ\nht : 1 ≤ t\n⊢ ((fun n ↦ (∑ k ∈ Icc 1 n, f k) / ↑n) ∘ fun x ↦ ⌊x⌋₊) t * ↑(↑⌊t⌋₊ / t) = (∑ k ∈ Icc 1 ⌊t⌋₊, f k) / ↑t",
"ppTerm": "?m.150",
"assigned": true,
"usedConstants": [
"E... | [] | simp [div_mul_div_cancel₀ (show (⌊t⌋₊ : ℂ) ≠ 0 by simpa)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 413,
"column": 54
} | {
"line": 413,
"column": 63
} | {
"line": 413,
"column": 64
} | [
{
"pp": "a✝ b : ℕ\nha : Odd a✝\nhb : Odd b\na x y : ℕ\n⊢ qrSign x a * qrSign y a * J(↑x * ↑y | a) = qrSign x a * J(↑x | a) * (qrSign y a * J(↑y | a))",
"ppTerm": "?m.226",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"... | [
"a✝ b : ℕ\nha : Odd a✝\nhb : Odd b\na x y : ℕ\n⊢ qrSign x a * qrSign y a * (J(↑x | a) * J(↑y | a)) = qrSign x a * J(↑x | a) * (qrSign y a * J(↑y | a))"
] | mul_left, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 466,
"column": 20
} | {
"line": 466,
"column": 29
} | {
"line": 466,
"column": 30
} | [
{
"pp": "case inr\na b : ℕ\nhb : Odd b\nha₀ : a ≠ 0\nhb' : Odd (b % (4 * a))\ne a' : ℕ\nha₁' : ¬2 ∣ a'\nha₂ : a = 2 ^ e * a'\nha₁ : Odd a'\n⊢ J(↑(2 ^ e) * ↑a' | b) = J(↑(2 ^ e) * ↑a' | b % (4 * a))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAdd... | [
"case inr\na b : ℕ\nhb : Odd b\nha₀ : a ≠ 0\nhb' : Odd (b % (4 * a))\ne a' : ℕ\nha₁' : ¬2 ∣ a'\nha₂ : a = 2 ^ e * a'\nha₁ : Odd a'\n⊢ J(↑(2 ^ e) | b) * J(↑a' | b) = J(↑(2 ^ e) * ↑a' | b % (4 * a))"
] | mul_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol | {
"line": 466,
"column": 30
} | {
"line": 466,
"column": 39
} | {
"line": 466,
"column": 40
} | [
{
"pp": "case inr\na b : ℕ\nhb : Odd b\nha₀ : a ≠ 0\nhb' : Odd (b % (4 * a))\ne a' : ℕ\nha₁' : ¬2 ∣ a'\nha₂ : a = 2 ^ e * a'\nha₁ : Odd a'\n⊢ J(↑(2 ^ e) | b) * J(↑a' | b) = J(↑(2 ^ e) * ↑a' | b % (4 * a))",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemirin... | [
"case inr\na b : ℕ\nhb : Odd b\nha₀ : a ≠ 0\nhb' : Odd (b % (4 * a))\ne a' : ℕ\nha₁' : ¬2 ∣ a'\nha₂ : a = 2 ^ e * a'\nha₁ : Odd a'\n⊢ J(↑(2 ^ e) | b) * J(↑a' | b) = J(↑(2 ^ e) | b % (4 * a)) * J(↑a' | b % (4 * a))"
] | mul_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LucasLehmer | {
"line": 56,
"column": 13
} | {
"line": 58,
"column": 54
} | {
"line": 60,
"column": 0
} | [
{
"pp": "p : ℕ\n⊢ Odd (mersenne (p + 1)) ↔ p + 1 ≠ 0",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"Nat.instMulZeroClass",
"Nat.instIsOrderedAddMonoid",
"IsOrderedRing.toPosMulMono",
"Nat.instOne",
"congrA... | [] | by
simpa using! Nat.Even.sub_odd (one_le_pow₀ one_le_two)
(even_two.pow_of_ne_zero p.succ_ne_zero) odd_one | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Valued.LocallyCompact | {
"line": 310,
"column": 30
} | {
"line": 310,
"column": 63
} | {
"line": 310,
"column": 64
} | [
{
"pp": "K : Type u_3\nΓ : Type u_4\ninst✝¹ : Field K\ninst✝ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K Γ\n⊢ v.IsNontrivial ↔ ¬IsLocalRing.maximalIdeal ↥v.integer ≤ ⊥",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearOrderedCommGroupWithZero.toLinearOrde... | [
"K : Type u_3\nΓ : Type u_4\ninst✝¹ : Field K\ninst✝ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K Γ\n⊢ (∃ x, x ≠ 0 ∧ v x < 1) ↔ ¬IsLocalRing.maximalIdeal ↥v.integer ≤ ⊥"
] | v.isNontrivial_iff_exists_lt_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.Order.ArchimedeanDiscrete | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 12
} | {
"line": 45,
"column": 13
} | [
{
"pp": "case inr.mp\nG✝ : Type u_1\ninst✝⁶ : CommGroup G✝\ninst✝⁵ : TopologicalSpace G✝\nG : Type u_1\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : TopologicalSpace G\ninst✝ : OrderTopology G\ng : G\nha✝ : 1 ≤ g\nha : 1 < g\nn : ℤ\n⊢ g ^ (-1) < g ^ n → g ^ n < g ^ 1 → ⟨g ^... | [
"case inr.mp\nG✝ : Type u_1\ninst✝⁶ : CommGroup G✝\ninst✝⁵ : TopologicalSpace G✝\nG : Type u_1\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : TopologicalSpace G\ninst✝ : OrderTopology G\ng : G\nha✝ : 1 ≤ g\nha : 1 < g\nn : ℤ\nhn : g ^ (-1) < g ^ n\n⊢ g ^ n < g ^ 1 → ⟨g ^ n, ⋯⟩ =... | intro hn | Lean.Elab.Tactic.evalIntro | null |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 179,
"column": 26
} | {
"line": 180,
"column": 60
} | {
"line": 180,
"column": 60
} | [
{
"pp": "𝒢 : Subgroup (GL (Fin 2) ℝ)\ninst✝ : 𝒢.IsArithmetic\ng : SL(2, ℤ)\n⊢ OnePoint.map Rat.cast ((mapGL ℚ) g⁻¹ • ∞) = (mapGL ℝ) g⁻¹ • ∞",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants": [
"OnePoint.instGLAction",
"Matrix.SpecialLinearGroup.map_mapGL",
"Eq.mpr",
"N... | [] | by rw [← Rat.coe_castHom, OnePoint.map_smul, OnePoint.map_infty,
← (Rat.castHom ℝ).algebraMap_toAlgebra, map_mapGL] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.Cusps | {
"line": 302,
"column": 2
} | {
"line": 303,
"column": 18
} | {
"line": 304,
"column": 2
} | [
{
"pp": "Γ : Subgroup SL(2, ℤ)\nhΓ : ModularGroup.T ∈ Γ\nx : ℝ\n⊢ x ∈ (map (mapGL ℝ) Γ).strictPeriods ↔ x ∈ AddSubgroup.zmultiples 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"Real",
"Matrix.SpecialLinearGroup",
"MonoidHom.instFunL... | [
"Γ : Subgroup SL(2, ℤ)\nhΓ : ModularGroup.T ∈ Γ\nx : ℝ\n⊢ (∃ x_1 ∈ Γ, (algebraMap ℤ ℝ).mapMatrix ↑x_1 = ↑(upperRightHom x)) ↔ x ∈ AddSubgroup.zmultiples 1"
] | simp only [mem_strictPeriods_iff, Subgroup.mem_map, Units.ext_iff, mapGL_coe_matrix,
map_apply_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ModularForms.Identities | {
"line": 49,
"column": 6
} | {
"line": 49,
"column": 21
} | {
"line": 49,
"column": 22
} | [
{
"pp": "f : ℍ → ℂ\nk : ℤ\nz : ℍ\n⊢ (f ∣[k] ModularGroup.S) z = f { coe := (-↑z)⁻¹, coe_im_pos := ⋯ } * ↑z ^ (-k)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"MonoidHom.instFunLike",
"UpperH... | [
"f : ℍ → ℂ\nk : ℤ\nz : ℍ\n⊢ f (ModularGroup.S • z) * denom (toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) ModularGroup.S)) ↑z ^ (-k) =\n f { coe := (-↑z)⁻¹, coe_im_pos := ⋯ } * ↑z ^ (-k)"
] | SL_slash_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.Basic | {
"line": 694,
"column": 4
} | {
"line": 697,
"column": 40
} | {
"line": 698,
"column": 2
} | [
{
"pp": "k : ℤ\nΓ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\ninst✝ : ModularFormClass F Γ k\ng : GL (Fin 2) ℝ\nc : OnePoint ℝ\nhc : IsCusp c (toConjAct g⁻¹ • Γ)\nγ : GL (Fin 2) ℝ\nhγ : γ • ∞ = c\n⊢ IsBoundedAtImInfty ((SlashInvariantForm.translate f g).toFun ∣[k] γ)",
"ppTerm": ... | [] | rw [SlashInvariantForm.toFun_eq_coe, SlashInvariantForm.coe_translate,
← SlashAction.slash_mul, ← isBoundedAt_infty_iff, ← OnePoint.IsBoundedAt.smul_iff]
apply ModularFormClass.bdd_at_cusps f
simpa [mul_smul, hγ] using hc.smul g | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.Basic | {
"line": 694,
"column": 4
} | {
"line": 697,
"column": 40
} | {
"line": 698,
"column": 2
} | [
{
"pp": "k : ℤ\nΓ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\ninst✝ : ModularFormClass F Γ k\ng : GL (Fin 2) ℝ\nc : OnePoint ℝ\nhc : IsCusp c (toConjAct g⁻¹ • Γ)\nγ : GL (Fin 2) ℝ\nhγ : γ • ∞ = c\n⊢ IsBoundedAtImInfty ((SlashInvariantForm.translate f g).toFun ∣[k] γ)",
"ppTerm": ... | [] | rw [SlashInvariantForm.toFun_eq_coe, SlashInvariantForm.coe_translate,
← SlashAction.slash_mul, ← isBoundedAt_infty_iff, ← OnePoint.IsBoundedAt.smul_iff]
apply ModularFormClass.bdd_at_cusps f
simpa [mul_smul, hγ] using hc.smul g | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.Basic | {
"line": 749,
"column": 12
} | {
"line": 749,
"column": 27
} | {
"line": 749,
"column": 27
} | [
{
"pp": "k : ℤ\nF : Type u_1\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : FunLike F ℍ ℂ\nf : F\ninst✝¹ : Γ.IsArithmetic\ng : SL(2, ℤ)\ninst✝ : ModularFormClass F Γ k\n⊢ toGL ((map (Int.castRingHom ℝ)) g) • ∞ = (mapGL ℝ) g • ∞",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"OnePoint.instGL... | [] | by simp [mapGL] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.Basic | {
"line": 756,
"column": 12
} | {
"line": 756,
"column": 27
} | {
"line": 756,
"column": 27
} | [
{
"pp": "k : ℤ\nF : Type u_1\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : FunLike F ℍ ℂ\nf : F\ninst✝¹ : Γ.IsArithmetic\ng : SL(2, ℤ)\ninst✝ : CuspFormClass F Γ k\n⊢ toGL ((map (Int.castRingHom ℝ)) g) • ∞ = (mapGL ℝ) g • ∞",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"OnePoint.instGLAct... | [] | by simp [mapGL] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 231,
"column": 73
} | {
"line": 237,
"column": 41
} | {
"line": 239,
"column": 0
} | [
{
"pp": "h : ℝ\nf : ℍ → ℂ\nhh : 0 < h\nc : ℕ → ℂ\nhf : ∀ (τ : ℍ), HasSum (fun m ↦ c m • 𝕢 h ↑τ ^ m) (f τ)\nq : ℂ\nhq : ‖q‖ < 1\nhq1 : q ≠ 0\n⊢ HasSum (fun m ↦ c m • q ^ m) (cuspFunction h f q)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"UpperHalfPlane.ofComplex",
"NormedC... | [] | by
have h1 := Periodic.im_invQParam_pos_of_norm_lt_one hh hq hq1
let τ : ℍ := ⟨Periodic.invQParam h q, h1⟩
have h2 := (Periodic.cuspFunction_eq_of_nonzero h (f ∘ ofComplex) hq1)
have : cuspFunction h f q = f τ := by simpa [UpperHalfPlane.ofComplex_apply_of_im_pos h1]
using! h2
grind [hf τ, Periodic.qParam... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Modular | {
"line": 251,
"column": 91
} | {
"line": 267,
"column": 67
} | {
"line": 269,
"column": 0
} | [
{
"pp": "z : ℍ\np : Fin 2 → ℤ\nhp : IsCoprime (p 0) (p 1)\n⊢ Tendsto (fun g ↦ |(↑g • z).re|) cofinite atTop",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Int.cast",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"G... | [] | by
suffices
Tendsto (fun g : (fun g : SL(2, ℤ) => g 1) ⁻¹' {p} => ((g : SL(2, ℤ)) • z).re) cofinite
(cocompact ℝ)
by exact tendsto_norm_cocompact_atTop.comp this
have : ((p 0 : ℝ) ^ 2 + (p 1 : ℝ) ^ 2)⁻¹ ≠ 0 := by
apply inv_ne_zero
exact mod_cast hp.sq_add_sq_ne_zero
let f := Homeomorph.mulRi... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.LevelOne.Basic | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 62
} | {
"line": 94,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nk : ℤ\ninst✝ : ModularFormClass F (Matrix.SpecialLinearGroup.mapGL ℝ).range k\nhk : k ≤ 0\nf : F\nz : ℍ\n⊢ f z = Function.const ℍ (UpperHalfPlane.cuspFunction 1 (⇑f) 0) z",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"... | [
"F : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nk : ℤ\ninst✝ : ModularFormClass F (Matrix.SpecialLinearGroup.mapGL ℝ).range k\nhk : k ≤ 0\nf : F\nz : ℍ\nhQ : 𝕢 1 ↑z ∈ Metric.ball 0 1\n⊢ f z = Function.const ℍ (UpperHalfPlane.cuspFunction 1 (⇑f) 0) z"
] | have hQ : 𝕢 1 z ∈ (Metric.ball 0 1) := by
simpa using (norm_qParam_lt_iff zero_lt_one 0 z.1).mpr z.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs | {
"line": 96,
"column": 2
} | {
"line": 111,
"column": 28
} | {
"line": 113,
"column": 0
} | [
{
"pp": "r : ℕ\ninst✝ : NeZero r\n⊢ BijOn (divIntMap ↑r) (gammaSet 1 r 0) (gammaSet 1 1 0)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Int.cast",
"Int.gcd",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.cast_natCast",
"Int.instDiv",
... | [] | refine ⟨?_, ?_, ?_⟩
· intro x hx
simp only [divIntMap, mem_gammaSet_one] at *
exact finGcdMap_div _ hx.2
· intro x hx v hv hv2
ext i
exact (Int.ediv_left_inj (gammaSet_div_gcd hx i) (gammaSet_div_gcd hv i)).mp
(congr_fun hv2 i)
· intro x hx
use r • x
simp only [nsmul_eq_mul, divIntMa... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs | {
"line": 96,
"column": 2
} | {
"line": 111,
"column": 28
} | {
"line": 113,
"column": 0
} | [
{
"pp": "r : ℕ\ninst✝ : NeZero r\n⊢ BijOn (divIntMap ↑r) (gammaSet 1 r 0) (gammaSet 1 1 0)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Int.cast",
"Int.gcd",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.cast_natCast",
"Int.instDiv",
... | [] | refine ⟨?_, ?_, ?_⟩
· intro x hx
simp only [divIntMap, mem_gammaSet_one] at *
exact finGcdMap_div _ hx.2
· intro x hx v hv hv2
ext i
exact (Int.ediv_left_inj (gammaSet_div_gcd hx i) (gammaSet_div_gcd hv i)).mp
(congr_fun hv2 i)
· intro x hx
use r • x
simp only [nsmul_eq_mul, divIntMa... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs | {
"line": 210,
"column": 11
} | {
"line": 210,
"column": 26
} | {
"line": 210,
"column": 27
} | [
{
"pp": "N : ℕ\na : Fin 2 → ZMod N\nk : ℤ\nγ : SL(2, ℤ)\nz : ℍ\n⊢ (eisensteinSeries a k ∣[k] γ) z = eisensteinSeries (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ)) k z",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"M... | [
"N : ℕ\na : Fin 2 → ZMod N\nk : ℤ\nγ : SL(2, ℤ)\nz : ℍ\n⊢ eisensteinSeries a k (γ • z) *\n denom (SpecialLinearGroup.toGL ((SpecialLinearGroup.map (Int.castRingHom ℝ)) γ)) ↑z ^ (-k) =\n eisensteinSeries (a ᵥ* ↑((SpecialLinearGroup.map (Int.castRingHom (ZMod N))) γ)) k z"
] | SL_slash_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.Defs | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 42
} | {
"line": 222,
"column": 4
} | [
{
"pp": "N r : ℕ\na : Fin 2 → ZMod N\ninst✝ : NeZero r\nk : ℤ\nA : GL (Fin 2) ℝ\nhA : A ∈ Subgroup.map (SpecialLinearGroup.mapGL ℝ) Γ(N)\n⊢ eisensteinSeries a k ∣[k] A = eisensteinSeries a k",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Real",
"Matrix.SpecialLinearGroup",
... | [
"N r : ℕ\na : Fin 2 → ZMod N\ninst✝ : NeZero r\nk : ℤ\nA : SL(2, ℤ)\nhA : A ∈ Γ(N)\n⊢ eisensteinSeries a k ∣[k] (SpecialLinearGroup.mapGL ℝ) A = eisensteinSeries a k"
] | obtain ⟨A, (hA : A ∈ Γ(N)), rfl⟩ := hA | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 119,
"column": 6
} | {
"line": 120,
"column": 58
} | {
"line": 121,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t\nΓ : Subgroup SL(2, ℤ)\ninst✝ : Γ.FiniteIndex\nhf_inv : ∀ g ∈... | [] | rw [← Quotient.eq_iff_equiv, Quotient.eq, QuotientGroup.leftRel_apply] at hgh
exact ⟨g⁻¹ * h, hgh, (mul_inv_cancel_left g h).symm⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.Bounds | {
"line": 119,
"column": 6
} | {
"line": 120,
"column": 58
} | {
"line": 121,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\nf : ℍ → E\nhf_cont : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nt : ℝ\nht : 0 ≤ t\nhf_infinity : ∀ (g : SL(2, ℤ)), (fun τ ↦ f (g • τ)) =O[atImInfty] fun z ↦ z.im ^ t\nΓ : Subgroup SL(2, ℤ)\ninst✝ : Γ.FiniteIndex\nhf_inv : ∀ g ∈... | [] | rw [← Quotient.eq_iff_equiv, Quotient.eq, QuotientGroup.leftRel_apply] at hgh
exact ⟨g⁻¹ * h, hgh, (mul_inv_cancel_left g h).symm⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.UniformConvergence | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 97
} | {
"line": 65,
"column": 2
} | [
{
"pp": "k : ℤ\nN : ℕ\nhk : 3 ≤ k\na : Fin 2 → ZMod N\n⊢ TendstoLocallyUniformlyOn (fun s ↦ (fun z ↦ ∑ x ∈ s, eisSummand k (↑x) z) ∘ ↑ofComplex)\n (⇑(eisensteinSeriesSIF a k) ∘ ↑ofComplex) Filter.atTop (UpperHalfPlane.coe '' Set.univ)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
... | [
"k : ℤ\nN : ℕ\nhk : 3 ≤ k\na : Fin 2 → ZMod N\n⊢ TendstoLocallyUniformlyOn (fun n z ↦ ∑ x ∈ n, eisSummand k (↑x) z) (⇑(eisensteinSeriesSIF a k)) Filter.atTop ⊤",
"k : ℤ\nN : ℕ\nhk : 3 ≤ k\na : Fin 2 → ZMod N\n⊢ MapsTo (↑(IsOpenEmbedding.toOpenPartialHomeomorph UpperHalfPlane.coe isOpenEmbedding_coe).symm)\n (I... | apply TendstoLocallyUniformlyOn.comp (s := ⊤) _ _ _ (OpenPartialHomeomorph.continuousOn_symm _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.ModularForms.QExpansion | {
"line": 538,
"column": 4
} | {
"line": 538,
"column": 37
} | {
"line": 539,
"column": 4
} | [
{
"pp": "h : ℝ\nm : ℕ\nq : ℂ\n⊢ cuspFunction h 1 q = 1 q",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Complex.instZero",
"UpperHalfPlane.cuspFunction",
"Ne",
"Or.casesOn",
"Pi.instOne",
"UpperHalfPlane",
"Eq.ndrec",
"One.toOfNat1",
... | [
"case inl\nh : ℝ\nm : ℕ\n⊢ cuspFunction h 1 0 = 1 0",
"case inr\nh : ℝ\nm : ℕ\nq : ℂ\nhq : q ≠ 0\n⊢ cuspFunction h 1 q = 1 q"
] | rcases eq_or_ne q 0 with rfl | hq | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.ModularForms.DedekindEta | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 74
} | {
"line": 113,
"column": 2
} | [
{
"pp": "z : ℂ\nhz : z ∈ ℍₒ\n⊢ ∏' (n : ℕ), (1 - eta_q n z) ≠ 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"ModularForm.eta_q",
"NormedDivisionRing.toNormMulClass",
"Complex.instNormedField",
"CommCStarAlgebra.toNormedCommRing",
"NormedDivisionRing.to_norm... | [
"case refine_1\nz : ℂ\nhz : z ∈ ℍₒ\n⊢ ∀ (i : ℕ), 1 + -eta_q i z ≠ 0",
"case refine_2\nz : ℂ\nhz : z ∈ ℍₒ\n⊢ Summable fun x ↦ ‖-eta_q x z‖"
] | refine tprod_one_add_ne_zero_of_summable (f := fun n ↦ -eta_q n z) ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 129,
"column": 6
} | {
"line": 129,
"column": 21
} | {
"line": 129,
"column": 22
} | [
{
"pp": "z : ℍ\n⊢ (Δ ∣[12] ModularGroup.T) z = Δ z",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"MonoidHom.instFunLike",
"UpperHalfPlane.SLAction",
"HMul.hMul",
"UpperHalfPlane.co... | [
"z : ℍ\n⊢ Δ (ModularGroup.T • z) *\n denom (Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) ModularGroup.T)) ↑z ^\n (-12) =\n Δ z"
] | SL_slash_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.Discriminant | {
"line": 146,
"column": 8
} | {
"line": 146,
"column": 23
} | {
"line": 146,
"column": 24
} | [
{
"pp": "z : ℍ\nthis : η (-(↑z)⁻¹) ^ 24 * (↑z ^ 12)⁻¹ = η ↑z ^ 24\n⊢ (Δ ∣[12] ModularGroup.S) z = Δ z",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"MonoidHom.instFunLike",
"UpperHalfPlane.SLA... | [
"z : ℍ\nthis : η (-(↑z)⁻¹) ^ 24 * (↑z ^ 12)⁻¹ = η ↑z ^ 24\n⊢ Δ (ModularGroup.S • z) *\n denom (Matrix.SpecialLinearGroup.toGL ((Matrix.SpecialLinearGroup.map (Int.castRingHom ℝ)) ModularGroup.S)) ↑z ^\n (-12) =\n Δ z"
] | SL_slash_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Modular | {
"line": 750,
"column": 2
} | {
"line": 770,
"column": 9
} | {
"line": 771,
"column": 2
} | [
{
"pp": "case mp\ng : SL(2, ℤ)\n⊢ g • ρ = ρ → g ∈ {1, -1, S * T, -(S * T), T⁻¹ * S, -(T⁻¹ * S)}",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCom... | [
"case mpr\ng : SL(2, ℤ)\n⊢ g ∈ {1, -1, S * T, -(S * T), T⁻¹ * S, -(T⁻¹ * S)} → g • ρ = ρ"
] | · intro hg
have neS : g ≠ S ∧ g ≠ -S := by
have : S • ρ ≠ ρ := by
rw [ne_eq, UpperHalfPlane.ext_iff, modular_S_smul, coe_mk, Complex.ext_iff]
norm_num [ρ, ← pow_two, div_pow]
grind [SL_neg_smul]
have neT : g ≠ T ∧ g ≠ -T ∧ g ≠ T⁻¹ ∧ g ≠ -T⁻¹ := by
have : T • ρ ≠ ρ := by
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Modular | {
"line": 831,
"column": 2
} | {
"line": 831,
"column": 51
} | {
"line": 833,
"column": 0
} | [
{
"pp": "τ : ℍ\nhτ : τ ∈ 𝒟\n⊢ ↑τ ∈ {z | 0 < z.im ∧ 1 ≤ ‖z‖ ∧ |z.re| ≤ 1 / 2}",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"Norm.norm",
"ModularGroup.fd._proof_1",
"Complex.one_le_normSq_iff",
"Real.instLE",
"Real",
"UpperHalfPlane.im_pos",
"in... | [] | exact ⟨τ.im_pos, one_le_normSq_iff.mp hτ.1, hτ.2⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Modular | {
"line": 927,
"column": 4
} | {
"line": 927,
"column": 71
} | {
"line": 928,
"column": 2
} | [
{
"pp": "ho1 : interior 𝒟 ⊆ UpperHalfPlane.re ⁻¹' interior (UpperHalfPlane.re '' 𝒟)\nho2 : interior 𝒟 ⊆ (fun τ ↦ ‖↑τ‖) ⁻¹' interior ((fun τ ↦ ‖↑τ‖) '' 𝒟)\nx : ℍ\nhx : x ∈ interior 𝒟\nξ : ℍ\nhξ : ξ ∈ 𝒟\n⊢ ξ ∈ (fun τ ↦ ‖↑τ‖) ⁻¹' Set.Ici 1",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": ... | [] | simpa [Set.mem_preimage, Set.mem_Ici, one_le_normSq_iff] using hξ.1 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.ModularForms.ProperlyDiscontinuous | {
"line": 26,
"column": 2
} | {
"line": 26,
"column": 85
} | {
"line": 27,
"column": 2
} | [
{
"pp": "𝒮ℒ' : Subgroup SL(2, ℝ) := (SpecialLinearGroup.map (Int.castRingHom ℝ)).range\nthis : ∀ {K L : Set ℍ}, IsCompact K → IsCompact L → {g | g ∈ 𝒮ℒ' ∧ (g • K ∩ L).Nonempty}.Finite\n⊢ ∀ {K L : Set ℍ},\n IsCompact K → IsCompact L → {g | g ∈ (SpecialLinearGroup.mapGL ℝ).range ∧ (g • K ∩ L).Nonempty}.Finit... | [
"𝒮ℒ' : Subgroup SL(2, ℝ) := (SpecialLinearGroup.map (Int.castRingHom ℝ)).range\nthis : ∀ {K L : Set ℍ}, IsCompact K → IsCompact L → {g | g ∈ 𝒮ℒ' ∧ (g • K ∩ L).Nonempty}.Finite\nK L : Set ℍ\nhK : IsCompact K\nhL : IsCompact L\ng : GL (Fin 2) ℝ\n⊢ g ∈ {g | g ∈ (SpecialLinearGroup.mapGL ℝ).range ∧ (g • K ∩ L).Nonemp... | refine fun K L hK hL ↦ ((this hK hL).map SpecialLinearGroup.toGL).subset fun g ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula | {
"line": 251,
"column": 49
} | {
"line": 251,
"column": 61
} | {
"line": 252,
"column": 2
} | [
{
"pp": "case h\nk : ℕ\nihn :\n ∀ m < k,\n Even m →\n Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑m) =\n ↑(if m ≡ 2 [MOD 12] then m / 12 else m / 12 + 1)\nhk2 : Even k\n⊢ Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑k) =\n ↑(if k ≡ 2 [MOD 12]... | [] | | h k ihn => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 57,
"column": 4
} | {
"line": 59,
"column": 45
} | {
"line": 60,
"column": 4
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\np x : R\nn : ℕ\n⊢ p ^ 2 ∣ (x + p) ^ (n + 1) - x ^ (n + 1 - 1) * p * ↑(n + 1) - x ^ (n + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemiring.toAddCommMonoidWit... | [
"case succ\nR : Type u_1\ninst✝ : CommRing R\np x : R\nn : ℕ\n⊢ p ^ 2 ∣\n ∑ m ∈ range n, x ^ m * p ^ (n + 1 - m) * ↑((n + 1).choose m) + x ^ n * p * (↑n + 1) + x ^ (n + 1) -\n x ^ n * p * (↑n + 1) -\n x ^ (n + 1)"
] | simp only [add_pow, sum_range_succ, add_tsub_cancel_left, pow_one, Nat.choose_succ_self_right,
Nat.cast_succ, tsub_self, pow_zero, mul_one, Nat.choose_self, Nat.cast_zero, zero_add,
Nat.succ_sub_succ_eq_sub, Nat.sub_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Multiplicity | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 25
} | {
"line": 254,
"column": 2
} | [
{
"pp": "x : ℤ\nhx : ∃ k, x = 2 * k + 1\n⊢ x ^ 2 % 4 = 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"Exists",
"Distrib.toAdd",
"instHMod",
"AddMonoidWithOne.toNatCast",
"Odd._proof_1",
... | [
"w✝ : ℤ\n⊢ (2 * w✝ + 1) ^ 2 % 4 = 1"
] | rcases hx with ⟨_, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.RestrictedProduct.Basic | {
"line": 434,
"column": 18
} | {
"line": 436,
"column": 27
} | {
"line": 438,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\n𝓕 𝓖 : Filter ι\nι₁ : Type u_3\nι₂ : Type u_4\nR₁ : ι₁ → Type u_5\nR₂ : ι₂ → Type u_6\n𝓕₁ : Filter ι₁\n𝓕₂ : Filter ι₂\nA₁ : (i : ι₁) → Set (R₁ i)\nA₂ : (i : ι₂) → Set (R₂ i)\nS₁ : ι₁ → Type u_7\nS₂ : ι₂ → Type u_8\ninst✝⁵ : (i : ι₁) → SetLike ... | [] | by
ext i
exact map_mul (φ i) _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.RestrictedProduct.Basic | {
"line": 536,
"column": 2
} | {
"line": 536,
"column": 36
} | {
"line": 536,
"column": 36
} | [
{
"pp": "ι : Type u_1\nS : ι → Type u_3\nG : ι → Type u_4\ninst✝⁴ : (i : ι) → SetLike (S i) (G i)\nA : (i : ι) → S i\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → MulZeroClass (G i)\ninst✝¹ : ∀ (i : ι), ZeroMemClass (S i) (G i)\ninst✝ : ∀ (i : ι), MulMemClass (S i) (G i)\ni : ι\nr : G i\nx : Πʳ (i : ι), [G i, ↑(A... | [
"case inl\nι : Type u_1\nS : ι → Type u_3\nG : ι → Type u_4\ninst✝⁴ : (i : ι) → SetLike (S i) (G i)\nA : (i : ι) → S i\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → MulZeroClass (G i)\ninst✝¹ : ∀ (i : ι), ZeroMemClass (S i) (G i)\ninst✝ : ∀ (i : ι), MulMemClass (S i) (G i)\ni : ι\nr : G i\nx : Πʳ (i : ι), [G i, ↑(A i... | rcases eq_or_ne i j with rfl | hne | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.RestrictedProduct.Basic | {
"line": 544,
"column": 2
} | {
"line": 544,
"column": 36
} | {
"line": 544,
"column": 36
} | [
{
"pp": "ι : Type u_1\nS : ι → Type u_3\nG : ι → Type u_4\ninst✝⁴ : (i : ι) → SetLike (S i) (G i)\nA : (i : ι) → S i\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → MulZeroClass (G i)\ninst✝¹ : ∀ (i : ι), ZeroMemClass (S i) (G i)\ninst✝ : ∀ (i : ι), MulMemClass (S i) (G i)\ni : ι\nr : G i\nx : Πʳ (i : ι), [G i, ↑(A... | [
"case inl\nι : Type u_1\nS : ι → Type u_3\nG : ι → Type u_4\ninst✝⁴ : (i : ι) → SetLike (S i) (G i)\nA : (i : ι) → S i\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → MulZeroClass (G i)\ninst✝¹ : ∀ (i : ι), ZeroMemClass (S i) (G i)\ninst✝ : ∀ (i : ι), MulMemClass (S i) (G i)\ni : ι\nr : G i\nx : Πʳ (i : ι), [G i, ↑(A i... | rcases eq_or_ne i j with rfl | hne | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.RingTheory.DedekindDomain.FiniteAdeleRing | {
"line": 158,
"column": 2
} | {
"line": 161,
"column": 28
} | {
"line": 163,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\na : FiniteAdeleRing R K\n⊢ IsUnit a ↔\n (∀ (v : HeightOneSpectrum R), a v ≠ 0) ∧ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, Valued.v (a v) = 1",
... | [] | rw [RestrictedProduct.isUnit_iff]
simp only [isUnit_iff_ne_zero, adicCompletionIntegers.isUnit_iff_valued_eq_one, exists_prop,
Filter.eventually_cofinite, not_and_or, Set.setOf_or]
simpa using! fun _ _ ↦ a.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.FiniteAdeleRing | {
"line": 158,
"column": 2
} | {
"line": 161,
"column": 28
} | {
"line": 163,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\na : FiniteAdeleRing R K\n⊢ IsUnit a ↔\n (∀ (v : HeightOneSpectrum R), a v ≠ 0) ∧ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, Valued.v (a v) = 1",
... | [] | rw [RestrictedProduct.isUnit_iff]
simp only [isUnit_iff_ne_zero, adicCompletionIntegers.isUnit_iff_valued_eq_one, exists_prop,
Filter.eventually_cofinite, not_and_or, Set.setOf_or]
simpa using! fun _ _ ↦ a.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Multiplicity | {
"line": 345,
"column": 4
} | {
"line": 352,
"column": 7
} | {
"line": 353,
"column": 2
} | [
{
"pp": "case inl\nx y : ℕ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nn : ℕ\nhn : Even n\nhyx : y ≤ x\n⊢ emultiplicity 2 (x ^ n - y ^ n) + 1 = emultiplicity 2 (x + y) + emultiplicity 2 (x - y) + emultiplicity 2 n",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
... | [] | iterate 3 rw [← Int.natCast_emultiplicity]
simp only [Int.ofNat_sub hyx, Int.ofNat_sub (pow_le_pow_left' hyx _), Int.natCast_add,
Int.natCast_pow]
rw [← Int.natCast_dvd_natCast] at hx
rw [← Int.natCast_dvd_natCast, Int.ofNat_sub hyx] at hxy
convert! Int.two_pow_sub_pow hxy hx hn using 2
rw [← ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Multiplicity | {
"line": 345,
"column": 4
} | {
"line": 352,
"column": 7
} | {
"line": 353,
"column": 2
} | [
{
"pp": "case inl\nx y : ℕ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nn : ℕ\nhn : Even n\nhyx : y ≤ x\n⊢ emultiplicity 2 (x ^ n - y ^ n) + 1 = emultiplicity 2 (x + y) + emultiplicity 2 (x - y) + emultiplicity 2 n",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
... | [] | iterate 3 rw [← Int.natCast_emultiplicity]
simp only [Int.ofNat_sub hyx, Int.ofNat_sub (pow_le_pow_left' hyx _), Int.natCast_add,
Int.natCast_pow]
rw [← Int.natCast_dvd_natCast] at hx
rw [← Int.natCast_dvd_natCast, Int.ofNat_sub hyx] at hxy
convert! Int.two_pow_sub_pow hxy hx hn using 2
rw [← ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 667,
"column": 6
} | {
"line": 667,
"column": 9
} | {
"line": 667,
"column": 9
} | [
{
"pp": "ι₁ : Type u_3\nι₂ : Type u_4\nR₁ : ι₁ → Type u_5\nR₂ : ι₂ → Type u_6\ninst✝¹ : (i : ι₁) → TopologicalSpace (R₁ i)\ninst✝ : (i : ι₂) → TopologicalSpace (R₂ i)\n𝓕₁ : Filter ι₁\n𝓕₂ : Filter ι₂\nA₁ : (i : ι₁) → Set (R₁ i)\nA₂ : (i : ι₂) → Set (R₂ i)\nf : ι₂ → ι₁\nhf : Tendsto f 𝓕₂ 𝓕₁\nφ : (j : ι₂) → R₁... | [
"ι₁ : Type u_3\nι₂ : Type u_4\nR₁ : ι₁ → Type u_5\nR₂ : ι₂ → Type u_6\ninst✝¹ : (i : ι₁) → TopologicalSpace (R₁ i)\ninst✝ : (i : ι₂) → TopologicalSpace (R₂ i)\n𝓕₁ : Filter ι₁\n𝓕₂ : Filter ι₂\nA₁ : (i : ι₁) → Set (R₁ i)\nA₂ : (i : ι₂) → Set (R₂ i)\nf : ι₂ → ι₁\nhf : Tendsto f 𝓕₂ 𝓕₁\nφ : (j : ι₂) → R₁ (f j) → R₂ ... | key | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 118,
"column": 39
} | {
"line": 119,
"column": 43
} | {
"line": 119,
"column": 44
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nthis :\n (Subgroup.map (QuotientGroup.mk' (torsion K)) (Subgroup.closure (Set.range u))).index =\n (Subgroup.closure (Set.range u) ⊔ torsion K).index\n⊢ (Subgroup.toAddSubgroup (Subgroup.map (QuotientGroup.mk' (torsi... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nthis :\n (Subgroup.map (QuotientGroup.mk' (torsion K)) (Subgroup.closure (Set.range u))).index =\n (Subgroup.closure (Set.range u) ⊔ torsion K).index\n⊢ (AddSubgroup.map (↑(logEmbeddingEquiv K).toAddEquiv)\n (Subgroup.t... | ← AddSubgroup.index_map_equiv
_ (logEmbeddingEquiv K).toAddEquiv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 130,
"column": 20
} | {
"line": 130,
"column": 23
} | {
"line": 130,
"column": 24
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nhx : mixedEmbedding.norm x = 1\nw : { w // w ≠ w₀ }\n⊢ ↑(↑w).mult * (Real.log ((normAtPlace ↑w) x) - Real.log (mixedEmbedding.norm x) * (↑(finrank ℚ K))⁻¹) =\n ↑(↑w).mult * Real.log ((normAtPlace ↑w) x)",
"ppTerm": "?m.39",... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nhx : mixedEmbedding.norm x = 1\nw : { w // w ≠ w₀ }\n⊢ ↑(↑w).mult * (Real.log ((normAtPlace ↑w) x) - Real.log 1 * (↑(finrank ℚ K))⁻¹) =\n ↑(↑w).mult * Real.log ((normAtPlace ↑w) x)"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 191,
"column": 2
} | {
"line": 193,
"column": 40
} | {
"line": 195,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : Algebra.IsIntegral ℚ K\n⊢ Subgroup.zpowers (complexConj K) = ⊤",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Subgroup.instFiniteSubtypeMem",
"Eq.mpr",
"IsGalois.card_aut_eq_finr... | [] | refine Subgroup.eq_top_of_card_eq _ ?_
rw [Nat.card_zpowers, orderOf_complexConj, IsGalois.card_aut_eq_finrank,
IsQuadraticExtension.finrank_eq_two] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 191,
"column": 2
} | {
"line": 193,
"column": 40
} | {
"line": 195,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : Algebra.IsIntegral ℚ K\n⊢ Subgroup.zpowers (complexConj K) = ⊤",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Subgroup.instFiniteSubtypeMem",
"Eq.mpr",
"IsGalois.card_aut_eq_finr... | [] | refine Subgroup.eq_top_of_card_eq _ ?_
rw [Nat.card_zpowers, orderOf_complexConj, IsGalois.card_aut_eq_finrank,
IsQuadraticExtension.finrank_eq_two] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 209,
"column": 2
} | {
"line": 217,
"column": 27
} | {
"line": 219,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : Algebra.IsIntegral ℚ K\nx : 𝓞 K\n⊢ (complexConj K) ↑x = ↑x ↔ ∃ y, (algebraMap (𝓞 ↥K⁺) K) y = ↑x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
... | [] | rw [complexConj_eq_self_iff]
refine ⟨fun h ↦ ?_, fun ⟨y, hy⟩ ↦ ?_⟩
· have : IsIntegral ℤ (⟨x, h⟩ : K⁺) :=
(isIntegral_algebraMap_iff (FaithfulSMul.algebraMap_injective K⁺ K)).mp x.isIntegral_coe
refine ⟨⟨⟨x, h⟩, this⟩, ?_⟩
rw [IsScalarTower.algebraMap_apply (𝓞 K⁺) K⁺, RingOfIntegers.map_mk]
rfl
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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