module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Localization.NormTrace | {
"line": 56,
"column": 2
} | {
"line": 58,
"column": 89
} | {
"line": 60,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝¹¹ : CommRing Rₘ\ninst✝¹⁰ : Algebra R Rₘ\ninst✝⁹ : CommRing Sₘ\ninst✝⁸ : Algebra S Sₘ\nM : Submonoid R\ninst✝⁷ : IsLocalization M Rₘ\ninst✝⁶ : IsLocalization (algebraMapSubm... | [] | ext i j
simp only [Matrix.map_apply, RingHom.mapMatrix_apply, leftMulMatrix_eq_repr_mul, ← map_mul,
Basis.localizationLocalization_apply, Basis.localizationLocalization_repr_algebraMap] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 346,
"column": 2
} | {
"line": 347,
"column": 51
} | {
"line": 348,
"column": 2
} | [
{
"pp": "case pos\np : ℕ\ninst✝³ : Fact (Nat.Prime p)\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx✝ : ∀ (i : ℕ), HasEnoughRootsOfUnity L (p ^ i)\nζ : ℕ → L\nhζ : ∀ (i : ℕ), IsPrimitiveRoot (ζ i) (p ^ i)\n⊢ ContinuousAt (⇑((Units.coeHom ℤ_[p]).comp ((cyclotomicCharacter... | [
"case pos\np : ℕ\ninst✝³ : Fact (Nat.Prime p)\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx✝ : ∀ (i : ℕ), HasEnoughRootsOfUnity L (p ^ i)\nζ : ℕ → L\nhζ : ∀ (i : ℕ), IsPrimitiveRoot (ζ i) (p ^ i)\n⊢ ∀ (ib : ℝ),\n 0 < ib →\n ∃ ia ∈ galGroupBasis K L,\n ∀ x ∈ i... | rw [ContinuousAt, map_one, (galGroupBasis K L).nhds_one_hasBasis.tendsto_iff
(Metric.nhds_basis_ball (α := ℤ_[p]) (x := 1))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 580,
"column": 2
} | {
"line": 580,
"column": 20
} | {
"line": 581,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : NonAssocSemiring R\np : ℕ\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nhp_prime : Fact (Nat.Prime p)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1\nε : ℚ\nhε : ε > 0\n⊢ ∃ i, ∀ j ≥ i, padicNorm p (↑(nthHomSeq f_compat 1 - 1) j) < ε",
"ppTer... | [
"R : Type u_1\ninst✝ : NonAssocSemiring R\np : ℕ\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nhp_prime : Fact (Nat.Prime p)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), (ZMod.castHom ⋯ (ZMod (p ^ k1))).comp (f k2) = f k1\nε : ℚ\nhε : 0 < ε\n⊢ ∃ i, ∀ j ≥ i, padicNorm p (↑(nthHomSeq f_compat 1 - 1) j) < ε"
] | change _ < _ at hε | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Discriminant | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 45
} | {
"line": 229,
"column": 2
} | [
{
"pp": "case e_a\nK : Type u\nL : Type v\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Module.Finite K L\npb : PowerBasis K L\ninst✝ : Algebra.IsSeparable K L\nE : Type v := AlgebraicClosure L\nthis : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)\ne : Fin pb.d... | [
"case e_a\nK : Type u\nL : Type v\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Module.Finite K L\npb : PowerBasis K L\ninst✝ : Algebra.IsSeparable K L\nE : Type v := AlgebraicClosure L\nthis : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)\ne : Fin pb.dim ≃ (L →ₐ[K... | rw [Finset.prod_sigma', Finset.prod_sigma'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.Discriminant | {
"line": 93,
"column": 6
} | {
"line": 93,
"column": 94
} | {
"line": 94,
"column": 2
} | [
{
"pp": "p k : ℕ\nK : Type u\nL : Type v\nζ : L\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhp : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhk : p ^ (k + 1) ≠ 2\nhne : NeZero ↑(p ^ (k ... | [] | exact one_le_mul (one_le_pow _ _ hp.1.pos) (succ_le_iff.2 <| tsub_pos_of_lt hp.1.one_lt) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 106,
"column": 6
} | {
"line": 107,
"column": 76
} | {
"line": 108,
"column": 6
} | [
{
"pp": "ξ : ℝ\nn : ℕ\nn_pos : 0 < n\nf : ℤ → ℤ := fun m ↦ ⌊fract (ξ * ↑m) * (↑n + 1)⌋\nhn : 0 < ↑n + 1\nhfu : ∀ (m : ℤ), fract (ξ * ↑m) * (↑n + 1) < ↑n + 1\nD : Finset ℤ := Icc 0 ↑n\nm : ℤ\nhm : m ∈ D\nhf : f m = ↑n\nhf' : ↑↑n ≤ fract (ξ * ↑m) * (↑n + 1)\n⊢ 0 < m",
"ppTerm": "?m.227",
"assigned": true,... | [
"ξ : ℝ\nn : ℕ\nn_pos : 0 < n\nf : ℤ → ℤ := fun m ↦ ⌊fract (ξ * ↑m) * (↑n + 1)⌋\nhn : 0 < ↑n + 1\nhfu : ∀ (m : ℤ), fract (ξ * ↑m) * (↑n + 1) < ↑n + 1\nD : Finset ℤ := Icc 0 ↑n\nm : ℤ\nhm : m ∈ D\nhf : f m = ↑n\nhf' : ↑↑n ≤ fract (ξ * ↑m) * (↑n + 1)\nhf₀ : f 0 = 0\n⊢ 0 < m"
] | have hf₀ : f 0 = 0 := by
simp only [f, cast_zero, mul_zero, fract_zero, zero_mul, floor_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 43
} | {
"line": 478,
"column": 2
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn : ℕ\nthis :\n { re := ↑(xn a1 (n + 2)), im := ↑(yn a1 (n + 2)) } + { re := ↑(xn a1 n), im := ↑(yn a1 n) } =\n ↑(2 * a) * { re := ↑(xn a1 (n + 1)), im := ↑(yn a1 (n + 1)) }\n⊢ xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1) ∧ yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)",
... | [
"a : ℕ\na1 : 1 < a\nn : ℕ\nthis :\n { re := ↑(xn a1 (n + 2)), im := ↑(yn a1 (n + 2)) } + { re := ↑(xn a1 n), im := ↑(yn a1 n) } =\n { re := ↑(2 * a) * ↑(xn a1 (n + 1)), im := ↑(2 * a) * ↑(yn a1 (n + 1)) }\n⊢ xn a1 (n + 2) + xn a1 n = 2 * a * xn a1 (n + 1) ∧ yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1)"
] | rw [Zsqrtd.nsmul_val (2 * a : ℕ)] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 562,
"column": 36
} | {
"line": 562,
"column": 62
} | {
"line": 562,
"column": 62
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn j : ℕ\nh : j ≤ n\nh1 : xz a1 n ∣ ↑(d a1) * yz a1 n * yz a1 (n - j) + xz a1 j\n⊢ ↑(xn a1 n) ∣ ↑(d a1 * yn a1 n * yn a1 (n - j) + xn a1 j)",
"ppTerm": "?m.147",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dv... | [] | by simpa [xz, yz] using h1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 747,
"column": 10
} | {
"line": 747,
"column": 13
} | {
"line": 747,
"column": 14
} | [
{
"pp": "d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\nh : x * x = d * y * y\ng : ℕ := x.gcd y\ngpos : g > 0\nm n : ℕ\nco : m.Coprime n\nhx : x = m * g\nhy : y = n * g\n⊢ False",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Nat.Coprime",
"HMul.hMul",
"congrArg",
"Eq.mp",
... | [
"d : ℕ\ndnsq : Nonsquare d\nx y : ℕ\ng : ℕ := x.gcd y\ngpos : g > 0\nm : ℕ\nh : m * g * (m * g) = d * y * y\nn : ℕ\nco : m.Coprime n\nhx : x = m * g\nhy : y = n * g\n⊢ False"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MulChar.Duality | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 45
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case pos\nM : Type u_1\nR : Type u_2\ninst✝⁴ : CommMonoid M\ninst✝³ : CommRing R\ninst✝² : Finite M\ninst✝¹ : HasEnoughRootsOfUnity R (Monoid.exponent Mˣ)\ninst✝ : Nontrivial R\na : M\nhu : IsUnit a\nha : hu.unit = 1\n⊢ a = 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"... | [] | rw [← hu.unit_spec, ha, Units.val_eq_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 389,
"column": 2
} | {
"line": 389,
"column": 26
} | {
"line": 390,
"column": 2
} | [
{
"pp": "ξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nh : ContfracLegendre.Ass ξ u v\nhv₀ hv₁ : 0 < ↑v\nhv₂ : 0 < 2 * ↑v - 1\n⊢ 0 < fract ξ",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Int.cast",
"Real",
"instHDiv",
"HMul.hMul",
"Real.lattice",
"Real.instZero",
... | [
"ξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nhv₀ hv₁ : 0 < ↑v\nhv₂ : 0 < 2 * ↑v - 1\nhcop : IsCoprime u v\nleft✝ : v = 1 → -(1 / 2) < ξ - ↑u\nh : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹\n⊢ 0 < fract ξ"
] | obtain ⟨hcop, _, h⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 405,
"column": 2
} | {
"line": 405,
"column": 26
} | {
"line": 406,
"column": 2
} | [
{
"pp": "ξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nh : ContfracLegendre.Ass ξ u v\n⊢ 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Int.cast",
"Real",
"instHDiv",
"HMul.hMul",
"Real.lattice",
"Int.floor",
"abs",
"Real.in... | [
"ξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nhcop : IsCoprime u v\nleft✝ : v = 1 → -(1 / 2) < ξ - ↑u\nh : |ξ - ↑u / ↑v| < (↑v * (2 * ↑v - 1))⁻¹\n⊢ 0 < u - ⌊ξ⌋ * v ∧ u - ⌊ξ⌋ * v < v"
] | obtain ⟨hcop, _, h⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 435,
"column": 48
} | {
"line": 435,
"column": 75
} | {
"line": 438,
"column": 0
} | [
{
"pp": "case refine_1.inl\nξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nhcop : IsCoprime u v\nleft✝ : v = 1 → -(1 / 2) < ξ - ↑u\nh : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))\nhv₀ : 0 < ↑v\nhv₀' : 0 < 2 * ↑v - 1\nhv₁ : 0 < 2 * v - 1\nhu₀ : 0 ≤ u - ⌊ξ⌋ * v\nhu₁ : u - ⌊ξ⌋ ... | [] | linarith only [hv, huv_cop] | Mathlib.Tactic._aux_Mathlib_Tactic_Linarith_Frontend___elabRules_Mathlib_Tactic_linarith_1 | Mathlib.Tactic.linarith |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 435,
"column": 48
} | {
"line": 435,
"column": 75
} | {
"line": 438,
"column": 0
} | [
{
"pp": "case refine_1.inr\nξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nhcop : IsCoprime u v\nleft✝ : v = 1 → -(1 / 2) < ξ - ↑u\nh : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))\nhv₀ : 0 < ↑v\nhv₀' : 0 < 2 * ↑v - 1\nhv₁ : 0 < 2 * v - 1\nhu₀ : 0 ≤ u - ⌊ξ⌋ * v\nhu₁ : u - ⌊ξ⌋ ... | [] | linarith only [hv, huv_cop] | Mathlib.Tactic._aux_Mathlib_Tactic_Linarith_Frontend___elabRules_Mathlib_Tactic_linarith_1 | Mathlib.Tactic.linarith |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 435,
"column": 48
} | {
"line": 435,
"column": 75
} | {
"line": 438,
"column": 0
} | [
{
"pp": "case refine_2.inl\nξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nhcop : IsCoprime u v\nleft✝ : v = 1 → -(1 / 2) < ξ - ↑u\nh : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))\nhv₀ : 0 < ↑v\nhv₀' : 0 < 2 * ↑v - 1\nhv₁ : 0 < 2 * v - 1\nhu₀ : 0 ≤ u - ⌊ξ⌋ * v\nhu₁ : u - ⌊ξ⌋ ... | [] | linarith only [hv, huv_cop] | Mathlib.Tactic._aux_Mathlib_Tactic_Linarith_Frontend___elabRules_Mathlib_Tactic_linarith_1 | Mathlib.Tactic.linarith |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 435,
"column": 48
} | {
"line": 435,
"column": 75
} | {
"line": 438,
"column": 0
} | [
{
"pp": "case refine_2.inr\nξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nhcop : IsCoprime u v\nleft✝ : v = 1 → -(1 / 2) < ξ - ↑u\nh : -1 + ξ * (↑v * (2 * ↑v - 1)) < ↑u * (2 * ↑v - 1) ∧ ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))\nhv₀ : 0 < ↑v\nhv₀' : 0 < 2 * ↑v - 1\nhv₁ : 0 < 2 * v - 1\nhu₀ : 0 ≤ u - ⌊ξ⌋ * v\nhu₁ : u - ⌊ξ⌋ ... | [] | linarith only [hv, huv_cop] | Mathlib.Tactic._aux_Mathlib_Tactic_Linarith_Frontend___elabRules_Mathlib_Tactic_linarith_1 | Mathlib.Tactic.linarith |
Mathlib.NumberTheory.SmoothNumbers | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 75
} | {
"line": 96,
"column": 0
} | [
{
"pp": "s : Finset ℕ\nm : ℕ\nhm : m ≠ 0 ∧ ∀ p ∈ m.primeFactorsList, p ∈ s\n⊢ m.primeFactors ⊆ s",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"Ne",
"instOfNatNat",
"List",
"List.instMembership",
"And.right",
"Fi... | [] | exact fun n hn ↦ hm.2 n (mem_primeFactors_iff_mem_primeFactorsList.mp hn) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 108,
"column": 8
} | {
"line": 108,
"column": 48
} | {
"line": 110,
"column": 8
} | [
{
"pp": "case pos.right\nR : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : ∀ {p : ℕ}, Nat.Prime p → Summable fun n ↦ ‖f (p ^ n)‖\np : ℕ\ns : Finset ℕ\nhp : p ∉ s\nih : (Summable fun m ↦ ‖f ↑m‖) ∧ HasSum (fu... | [
"case pos.right\nR : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : ∀ {p : ℕ}, Nat.Prime p → Summable fun n ↦ ‖f (p ^ n)‖\np : ℕ\ns : Finset ℕ\nhp : p ∉ s\nih : (Summable fun m ↦ ‖f ↑m‖) ∧ HasSum (fun m ↦ f ↑m) ... | apply (hsum hpp).of_norm.hasSum.mul ih.2 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.LSeries.Deriv | {
"line": 49,
"column": 63
} | {
"line": 49,
"column": 77
} | {
"line": 49,
"column": 78
} | [
{
"pp": "case inr\nf : ℕ → ℂ\nn : ℕ\ns : ℂ\nhn : n ≠ 0\n⊢ HasDerivAt (fun z ↦ f n / ↑n ^ z) (f n * -log ↑n / ↑n ^ s) s",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr"... | [
"case inr\nf : ℕ → ℂ\nn : ℕ\ns : ℂ\nhn : n ≠ 0\n⊢ HasDerivAt (fun z ↦ f n / ↑n ^ z) (f n * (-log ↑n / ↑n ^ s)) s"
] | mul_div_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 87,
"column": 86
} | {
"line": 105,
"column": 82
} | {
"line": 107,
"column": 0
} | [
{
"pp": "k : ℕ\na t : ℝ\nht : 0 < t\n⊢ Summable (f_nat k a t)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Norm.norm",
"Int.cast",
"Mathlib.Tactic.Ring... | [] | by
have : Summable fun n : ℕ ↦ n ^ k * exp (-π * (n + a) ^ 2 * t) := by
refine (((summable_pow_mul_jacobiTheta₂_term_bound (|a| * t) ht k).mul_right
(rexp (-π * a ^ 2 * t))).comp_injective Nat.cast_injective).of_norm_bounded (fun n ↦ ?_)
simp_rw [mul_assoc, Function.comp_apply, ← Real.exp_add, norm_mul,... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 35,
"column": 61
} | {
"line": 35,
"column": 75
} | {
"line": 35,
"column": 76
} | [
{
"pp": "case e'_5\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\np : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhp : ∀ (i : ι), a i = 0 ∨ 0 < p i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-p i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / p i ^ s.re\ni : ι\n⊢ a i * Complex.Gamma s / ↑(p i) ^ s = ∫ (a_1 : ... | [
"case e'_5\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\np : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhp : ∀ (i : ι), a i = 0 ∨ 0 < p i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-p i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / p i ^ s.re\ni : ι\n⊢ a i * (Complex.Gamma s / ↑(p i) ^ s) = ∫ (a_1 : ℝ) in Ioi ... | mul_div_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 198,
"column": 15
} | {
"line": 198,
"column": 29
} | {
"line": 198,
"column": 30
} | [
{
"pp": "case inr.refine_2\na : ℝ\nha : 0 ≤ a\naux' : (fun t ↦ ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x ↦ 1\nha' : 0 < a\n⊢ (fun t ↦ a * rexp (-π * a ^ 2 * t) / (1 - rexp (-π * t))) =O[atTop] fun t ↦ rexp (-(π * a ^ 2) * t)",
"ppTerm": "?inr.refine_2",
"assigned": true,
"usedConstants": [
... | [
"case inr.refine_2\na : ℝ\nha : 0 ≤ a\naux' : (fun t ↦ ((1 - rexp (-π * t)) ^ 2)⁻¹) =O[atTop] fun x ↦ 1\nha' : 0 < a\n⊢ (fun t ↦ a * (rexp (-π * a ^ 2 * t) / (1 - rexp (-π * t)))) =O[atTop] fun t ↦ rexp (-(π * a ^ 2) * t)"
] | mul_div_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 57,
"column": 50
} | {
"line": 57,
"column": 64
} | {
"line": 57,
"column": 65
} | [
{
"pp": "case e'_5\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\np : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhp : ∀ (i : ι), a i = 0 ∨ 0 < p i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-p i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / p i ^ s.re\ni : ι\n⊢ ‖‖a i‖ * ∫ (a : ℝ) in Ioi 0, ‖↑a ^ (s - 1) * ↑... | [
"case e'_5\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\np : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhp : ∀ (i : ι), a i = 0 ∨ 0 < p i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-p i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / p i ^ s.re\ni : ι\n⊢ ‖‖a i‖ * ∫ (a : ℝ) in Ioi 0, ‖↑a ^ (s - 1) * ↑(rexp (-p i ... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 82,
"column": 13
} | {
"line": 82,
"column": 27
} | {
"line": 82,
"column": 28
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\ni : ι\nthis : a i / ↑(π... | [
"ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\ni : ι\nthis : a i / ↑(π * q i) ^ s ... | mul_div_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 285,
"column": 22
} | {
"line": 285,
"column": 64
} | {
"line": 287,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nx : ℝ\nhx : 0 < x\nhx' : x < 1\nthis : 1 < 1 / x\n⊢ P.ε * ↑(x ^ P.k) * 1 = P.ε * ↑(x ^ P.k) * 1",
"ppTerm": "?m.356",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWit... | [] | field [(rpow_pos_of_pos hx P.k).ne', P.hε] | Mathlib.Tactic.FieldSimp._aux_Mathlib_Tactic_Field___elabRules_Mathlib_Tactic_FieldSimp_field_1 | Mathlib.Tactic.FieldSimp.field |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 285,
"column": 22
} | {
"line": 285,
"column": 64
} | {
"line": 287,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nx : ℝ\nhx : 0 < x\nhx' : x < 1\nthis : 1 < 1 / x\n⊢ -1 = P.ε * ↑(x ^ P.k) * -(P.ε⁻¹ * (↑(x ^ P.k))⁻¹ * 1)",
"ppTerm": "?m.357",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_... | [] | field [(rpow_pos_of_pos hx P.k).ne', P.hε] | Mathlib.Tactic.FieldSimp._aux_Mathlib_Tactic_Field___elabRules_Mathlib_Tactic_FieldSimp_field_1 | Mathlib.Tactic.FieldSimp.field |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 312,
"column": 31
} | {
"line": 312,
"column": 45
} | {
"line": 312,
"column": 46
} | [
{
"pp": "case e_a\na : UnitAddCircle\ns : ℂ\n⊢ completedHurwitzZetaEven₀ a s - (if a = 0 then 1 else 0) * (1 / (s / 2)) / 2 =\n completedHurwitzZetaEven₀ a s - (if a = 0 then 1 else 0) / s",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAss... | [
"case e_a\na : UnitAddCircle\ns : ℂ\n⊢ completedHurwitzZetaEven₀ a s - (if a = 0 then 1 else 0) * (1 / (s / 2) / 2) =\n completedHurwitzZetaEven₀ a s - (if a = 0 then 1 else 0) / s"
] | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 398,
"column": 4
} | {
"line": 400,
"column": 16
} | {
"line": 402,
"column": 0
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\ns : ℂ\nhs : s ≠ 0 ∨ P.f₀ = 0\nhs' : s ≠ ↑P.k ∨ P.g₀ = 0\n⊢ DifferentiableAt ℂ (fun s ↦ (P.ε / (↑P.k - s)) • P.g₀) s",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Nor... | [] | rcases hs' with hs' | hs'
· fun_prop (disch := grind)
· simp [hs'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 398,
"column": 4
} | {
"line": 400,
"column": 16
} | {
"line": 402,
"column": 0
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\ns : ℂ\nhs : s ≠ 0 ∨ P.f₀ = 0\nhs' : s ≠ ↑P.k ∨ P.g₀ = 0\n⊢ DifferentiableAt ℂ (fun s ↦ (P.ε / (↑P.k - s)) • P.g₀) s",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Nor... | [] | rcases hs' with hs' | hs'
· fun_prop (disch := grind)
· simp [hs'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 337,
"column": 39
} | {
"line": 337,
"column": 53
} | {
"line": 337,
"column": 54
} | [
{
"pp": "case e_a\na : UnitAddCircle\ns : ℂ\n⊢ (if a = 0 then 1 else 0) * (1 / (1 / 2 - s / 2)) / 2 = (if a = 0 then 1 else 0) / (1 - s)",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"i... | [
"case e_a\na : UnitAddCircle\ns : ℂ\n⊢ (if a = 0 then 1 else 0) * (1 / (1 / 2 - s / 2) / 2) = (if a = 0 then 1 else 0) / (1 - s)"
] | mul_div_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 456,
"column": 73
} | {
"line": 456,
"column": 83
} | {
"line": 456,
"column": 84
} | [
{
"pp": "z τ : ℂ\n⊢ -(starRingEnd ℂ) (∑' (n : ℤ), jacobiTheta₂'_term n z τ) =\n ∑' (n : ℤ), jacobiTheta₂'_term n (-(starRingEnd ℂ) z) (-(starRingEnd ℂ) τ)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRin... | [
"z τ : ℂ\n⊢ -∑' (a : ℤ), (starRingEnd ℂ) (jacobiTheta₂'_term a z τ) =\n ∑' (n : ℤ), jacobiTheta₂'_term n (-(starRingEnd ℂ) z) (-(starRingEnd ℂ) τ)"
] | conj_tsum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 32
} | {
"line": 82,
"column": 0
} | [
{
"pp": "z τ J J' : ℂ\n⊢ cexp (↑π * I * (z + 1) ^ 2 * τ + -↑π * I * (τ + 2 * (z * τ))) * (J' - 2 * ↑π * I * J + (z * J + J) * (2 * ↑π * I)) =\n cexp (↑π * I * z ^ 2 * τ) * (J' + z * J * (2 * ↑π * I))",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_p... | [] | congrm (cexp ?_ * ?_) <;> ring | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 503,
"column": 4
} | {
"line": 507,
"column": 84
} | {
"line": 508,
"column": 2
} | [
{
"pp": "case refine_1\na : ℝ\ns : ℂ\nhs : 1 < s.re\nc : ℤ → ℂ := fun n ↦ cexp (2 * ↑π * I * ↑a * ↑n) / 2\nhF :\n ∀ (t : ℝ),\n 0 < t → HasSum (fun n ↦ if ↑n = 0 then 0 else c n * ↑(rexp (-π * ↑n ^ 2 * t))) ((↑(cosKernel (↑a) t) - 1) / 2)\n⊢ Summable fun i ↦ ‖c i‖ / |↑i| ^ s.re",
"ppTerm": "?refine_1",
... | [] | apply (((summable_one_div_int_add_rpow 0 s.re).mpr hs).div_const 2).of_norm_bounded
intro i
simp only [c, (by { push_cast; ring } : 2 * π * I * a * i = ↑(2 * π * a * i) * I), norm_div,
RCLike.norm_ofNat, Complex.norm_exp_ofReal_mul_I, add_zero, norm_one,
norm_of_nonneg (by positivity : 0 ≤ |(i : ℝ)|... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 503,
"column": 4
} | {
"line": 507,
"column": 84
} | {
"line": 508,
"column": 2
} | [
{
"pp": "case refine_1\na : ℝ\ns : ℂ\nhs : 1 < s.re\nc : ℤ → ℂ := fun n ↦ cexp (2 * ↑π * I * ↑a * ↑n) / 2\nhF :\n ∀ (t : ℝ),\n 0 < t → HasSum (fun n ↦ if ↑n = 0 then 0 else c n * ↑(rexp (-π * ↑n ^ 2 * t))) ((↑(cosKernel (↑a) t) - 1) / 2)\n⊢ Summable fun i ↦ ‖c i‖ / |↑i| ^ s.re",
"ppTerm": "?refine_1",
... | [] | apply (((summable_one_div_int_add_rpow 0 s.re).mpr hs).div_const 2).of_norm_bounded
intro i
simp only [c, (by { push_cast; ring } : 2 * π * I * a * i = ↑(2 * π * a * i) * I), norm_div,
RCLike.norm_ofNat, Complex.norm_exp_ofReal_mul_I, add_zero, norm_one,
norm_of_nonneg (by positivity : 0 ≤ |(i : ℝ)|... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZeta | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 24
} | {
"line": 180,
"column": 25
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nhs' : ∀ (n : ℕ), s ≠ -↑n\n⊢ 2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * Complex.cos (↑π * s / 2) * hurwitzZetaEven a s +\n I * (2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * Complex.sin (↑π * s / 2) * hurwitzZetaOdd a s) =\n (2 * ↑π) ^ (-s) * Complex.... | [
"a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\nhs' : ∀ (n : ℕ), s ≠ -↑n\n⊢ 2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * Complex.cos (↑π * s / 2) * hurwitzZetaEven a s +\n I * (2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * Complex.sin (↑π * s / 2) * hurwitzZetaOdd a s) =\n (2 * ↑π) ^ (-s) * Complex.Gamma s *\n ... | hurwitzZetaEven_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 35
} | {
"line": 116,
"column": 2
} | [
{
"pp": "s : ℂ\nN : ℕ\nχ : DirichletCharacter ℂ N\nhs : 1 < s.re\n⊢ HasProd (fun p ↦ (1 - χ ↑↑p * ↑↑p ^ (-s))⁻¹) (L (fun n ↦ χ ↑n) s)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Nat.instMulZeroOneClass",
"Nat.Pri... | [
"s : ℂ\nN : ℕ\nχ : DirichletCharacter ℂ N\nhs : 1 < s.re\n⊢ HasProd (fun p ↦ (1 - χ ↑↑p * ↑↑p ^ (-s))⁻¹) (∑' (n : ℕ), (dirichletSummandHom χ ⋯) n)"
] | rw [← tsum_dirichletSummand χ hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 35
} | {
"line": 132,
"column": 2
} | [
{
"pp": "s : ℂ\nN : ℕ\nχ : DirichletCharacter ℂ N\nhs : 1 < s.re\n⊢ Tendsto (fun n ↦ ∏ p ∈ n.primesBelow, (1 - χ ↑p * ↑p ^ (-s))⁻¹) atTop (𝓝 (L (fun n ↦ χ ↑n) s))",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Nat.instMu... | [
"s : ℂ\nN : ℕ\nχ : DirichletCharacter ℂ N\nhs : 1 < s.re\n⊢ Tendsto (fun n ↦ ∏ p ∈ n.primesBelow, (1 - χ ↑p * ↑p ^ (-s))⁻¹) atTop (𝓝 (∑' (n : ℕ), (dirichletSummandHom χ ⋯) n))"
] | rw [← tsum_dirichletSummand χ hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.FLT.Basic | {
"line": 129,
"column": 6
} | {
"line": 129,
"column": 31
} | {
"line": 130,
"column": 2
} | [
{
"pp": "case inr.inr.inr.inr\nn : ℕ\nh : FermatLastTheoremWith ℕ n\na b c : ℤ\nha✝ : a ≠ 0\nhb✝ : b ≠ 0\nhc✝ : c ≠ 0\nhabc : a ^ n + b ^ n = c ^ n\nhn : Odd n\nha : 0 < a\nhb : 0 < b\nhc : 0 < c\n⊢ |a| ^ n + |b| ^ n = |c| ^ n",
"ppTerm": "?inr.inr.inr.inr",
"assigned": true,
"usedConstants": [
... | [] | simp only [abs_of_pos, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.FLT.Basic | {
"line": 209,
"column": 10
} | {
"line": 209,
"column": 30
} | {
"line": 209,
"column": 31
} | [
{
"pp": "case neg\nn : ℕ\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n\na b c : ℤ\nha : IsUnit a\nhb : IsUnit b\nhc : IsUnit c\nhn : ¬n = 0\n⊢ a ^ n + b ^ n ≠ c ^ n",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
... | [
"case neg\nn : ℕ\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n\na b c : ℤ\nha : IsUnit (a ^ n)\nhb : IsUnit (b ^ n)\nhc : IsUnit (c ^ n)\nhn : ¬n = 0\n⊢ a ^ n + b ^ n ≠ c ^ n"
] | ← isUnit_pow_iff hn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Wronskian | {
"line": 114,
"column": 42
} | {
"line": 114,
"column": 73
} | {
"line": 114,
"column": 73
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\na : R[X]\nha : a ≠ 0\nhw : a.wronskian 0 ≠ 0\n⊢ False",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp",
"Ne",
"Polynomial",
"CommRing.toCommSemiring",
"Polynom... | [
"R : Type u_1\ninst✝ : CommRing R\na : R[X]\nha : a ≠ 0\nhw : 0 ≠ 0\n⊢ False"
] | rw [wronskian_zero_right] at hw | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Radical.Basic | {
"line": 137,
"column": 2
} | {
"line": 144,
"column": 65
} | {
"line": 146,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\nhc : IsRelPrime a b\n⊢ primeFactors (a * b) = (primeFactors a).disjUnion (primeFactors b) ⋯",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | obtain rfl | ha := eq_or_ne a 0
· rw [isRelPrime_zero_left] at hc
simp only [zero_mul, primeFactors_zero, Finset.empty_disjUnion, primeFactors_of_isUnit hc]
obtain rfl | hb := eq_or_ne b 0
· rw [isRelPrime_zero_right] at hc
simp only [mul_zero, primeFactors_zero, primeFactors_of_isUnit hc, Finset.disjUnio... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Radical.Basic | {
"line": 137,
"column": 2
} | {
"line": 144,
"column": 65
} | {
"line": 146,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\nhc : IsRelPrime a b\n⊢ primeFactors (a * b) = (primeFactors a).disjUnion (primeFactors b) ⋯",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | obtain rfl | ha := eq_or_ne a 0
· rw [isRelPrime_zero_left] at hc
simp only [zero_mul, primeFactors_zero, Finset.empty_disjUnion, primeFactors_of_isUnit hc]
obtain rfl | hb := eq_or_ne b 0
· rw [isRelPrime_zero_right] at hc
simp only [mul_zero, primeFactors_zero, primeFactors_of_isUnit hc, Finset.disjUnio... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 83,
"column": 2
} | {
"line": 130,
"column": 13
} | {
"line": 133,
"column": 0
} | [
{
"pp": "k : 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 * a ^ p + C ... | [] | have hbc : IsCoprime b c := by apply rot_coprime heq <;> assumption
have hca : IsCoprime c a := by
rw [add_rotate] at heq; apply rot_coprime heq <;> assumption
have hCu := C_ne_zero.mpr hu
have hCv := C_ne_zero.mpr hv
have hCw := C_ne_zero.mpr hw
have hap := pow_ne_zero p ha
have hbq := pow_ne_zero q hb... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 83,
"column": 2
} | {
"line": 130,
"column": 13
} | {
"line": 133,
"column": 0
} | [
{
"pp": "k : 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 * a ^ p + C ... | [] | have hbc : IsCoprime b c := by apply rot_coprime heq <;> assumption
have hca : IsCoprime c a := by
rw [add_rotate] at heq; apply rot_coprime heq <;> assumption
have hCu := C_ne_zero.mpr hu
have hCv := C_ne_zero.mpr hv
have hCw := C_ne_zero.mpr hw
have hap := pow_ne_zero p ha
have hbq := pow_ne_zero q hb... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.Complex | {
"line": 32,
"column": 8
} | {
"line": 32,
"column": 25
} | {
"line": 32,
"column": 26
} | [
{
"pp": "K : Subfield ℂ\nhc : IsClosed ↑K\nthis : range ofReal ⊆ ↑K\n⊢ K = ofRealHom.fieldRange ∨ K = ⊤",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real",
"congrArg",
"Membership.mem",
"Field.toDivisionRing",
"Eq.mp",
"LE.le",
"Subfield.instSe... | [
"K : Subfield ℂ\nhc : IsClosed ↑K\nthis : ∀ (y : ℝ), ↑y ∈ ↑K\n⊢ K = ofRealHom.fieldRange ∨ K = ⊤"
] | range_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FractionalIdeal.Norm | {
"line": 128,
"column": 56
} | {
"line": 128,
"column": 70
} | {
"line": 128,
"column": 71
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : IsDedekindDomain R\ninst✝⁹ : Free ℤ R\ninst✝⁸ : Module.Finite ℤ R\nK : Type u_2\ninst✝⁷ : CommRing K\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K\ninst✝³ : Algebra ℚ K\ninst✝² : IsDomain K\... | [
"R : Type u_1\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : IsDedekindDomain R\ninst✝⁹ : Free ℤ R\ninst✝⁸ : Module.Finite ℤ R\nK : Type u_2\ninst✝⁷ : CommRing K\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K\ninst✝³ : Algebra ℚ K\ninst✝² : IsDomain K\nι : Type u_... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 521,
"column": 4
} | {
"line": 523,
"column": 17
} | {
"line": 524,
"column": 2
} | [
{
"pp": "case pos\nx y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\n⊢ h.IsClassified",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"one_pow",
"Int.gcd",
"MulOne.toOne",
"False",
"HMul.hMul",
"Int.gcd_zero",
... | [] | obtain ⟨hx, hy⟩ := Int.gcd_eq_zero_iff.mp h0
use 0, 1, 0
simp [hx, hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 521,
"column": 4
} | {
"line": 523,
"column": 17
} | {
"line": 524,
"column": 2
} | [
{
"pp": "case pos\nx y z : ℤ\nh : PythagoreanTriple x y z\nh0 : x.gcd y = 0\n⊢ h.IsClassified",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"one_pow",
"Int.gcd",
"MulOne.toOne",
"False",
"HMul.hMul",
"Int.gcd_zero",
... | [] | obtain ⟨hx, hy⟩ := Int.gcd_eq_zero_iff.mp h0
use 0, 1, 0
simp [hx, hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Norm | {
"line": 143,
"column": 48
} | {
"line": 143,
"column": 62
} | {
"line": 143,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDedekindDomain R\ninst✝⁷ : Free ℤ R\ninst✝⁶ : Module.Finite ℤ R\nK : Type u_2\ninst✝⁵ : CommRing K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K\ninst✝¹ : Algebra ℚ K\ninst✝ : Module.Finite ... | [
"R : Type u_1\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDedekindDomain R\ninst✝⁷ : Free ℤ R\ninst✝⁶ : Module.Finite ℤ R\nK : Type u_2\ninst✝⁵ : CommRing K\ninst✝⁴ : Algebra R K\ninst✝³ : IsFractionRing R K\ninst✝² : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K\ninst✝¹ : Algebra ℚ K\ninst✝ : Module.Finite ℚ K\nx : K\n... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 158,
"column": 83
} | {
"line": 158,
"column": 86
} | {
"line": 159,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nh : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1\nn : ℕ\nhn : 0 < n\nhx : (algebraMap (𝓞 K) K) ↑x ^ n = 1\n⊢ (algebraMap (𝓞 K) K) ↑x ^ n = ↑↑1",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"Units... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nh : ∀ (φ : K →+* ℂ), ‖φ ((algebraMap (𝓞 K) K) ↑x)‖ = 1\nn : ℕ\nhn : 0 < n\nhx : (algebraMap (𝓞 K) K) ↑x ^ n = 1\n⊢ 1 = ↑↑1"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 57
} | {
"line": 194,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nφ : K →+* ℂ\nh : ComplexEmbedding.IsReal φ\nthis : (mk φ).embedding = φ ∨ (mk φ).embedding = ComplexEmbedding.conjugate φ\n⊢ (mk φ).embedding = φ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"NumberField.ComplexEmbedding.conjugate",
"N... | [] | rwa [ComplexEmbedding.isReal_iff.mp h, or_self] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 45
} | {
"line": 255,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh : Odd (Module.finrank ℚ K)\nthis : Fintype ↥(torsion K) := Fintype.ofFinite ↥(torsion K)\n⊢ torsionOrder K = 2",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NumberField.instCommRingRingOfIntegers",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh : Odd (Module.finrank ℚ K)\nthis : Fintype ↥(torsion K) := Fintype.ofFinite ↥(torsion K)\n⊢ Fintype.card ↥(torsion K) = 2"
] | rw [torsionOrder, Nat.card_eq_fintype_card] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 362,
"column": 35
} | {
"line": 362,
"column": 70
} | {
"line": 362,
"column": 70
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\nx✝ : K →+* ℂ\n⊢ ‖x✝ x‖ = ‖(RingHom.equivRatAlgHom x✝) x‖",
"ppTerm": "?m.170",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"Equiv.instEquivLike",
"congrArg",
"AlgHom",
"AlgHom.fun... | [] | simp [RingHom.equivRatAlgHom_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 362,
"column": 35
} | {
"line": 362,
"column": 70
} | {
"line": 362,
"column": 70
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\nx✝ : K →+* ℂ\n⊢ ‖x✝ x‖ = ‖(RingHom.equivRatAlgHom x✝) x‖",
"ppTerm": "?m.170",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"Equiv.instEquivLike",
"congrArg",
"AlgHom",
"AlgHom.fun... | [] | simp [RingHom.equivRatAlgHom_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 362,
"column": 35
} | {
"line": 362,
"column": 70
} | {
"line": 362,
"column": 70
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\nx✝ : K →+* ℂ\n⊢ ‖x✝ x‖ = ‖(RingHom.equivRatAlgHom x✝) x‖",
"ppTerm": "?m.170",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"Equiv.instEquivLike",
"congrArg",
"AlgHom",
"AlgHom.fun... | [] | simp [RingHom.equivRatAlgHom_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | {
"line": 127,
"column": 48
} | {
"line": 127,
"column": 51
} | {
"line": 127,
"column": 52
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nK : Type u_2\ninst✝ : Field K\na✝ b✝ : K\nhx : a✝ ∈ {x | ∀ (φ : K →+* ℂ), star (φ x) = φ x}\nhy : b✝ ∈ {x | ∀ (φ : K →+* ℂ), star (φ x) = φ x}\nx✝ : K →+* ℂ\n⊢ star (x✝ b✝) * star (x✝ a✝) = x✝ a✝ * x✝ b✝",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants":... | [
"F : Type u_1\ninst✝¹ : Field F\nK : Type u_2\ninst✝ : Field K\na✝ b✝ : K\nhx : a✝ ∈ {x | ∀ (φ : K →+* ℂ), star (φ x) = φ x}\nhy : b✝ ∈ {x | ∀ (φ : K →+* ℂ), star (φ x) = φ x}\nx✝ : K →+* ℂ\n⊢ star (x✝ b✝) * x✝ a✝ = x✝ a✝ * x✝ b✝"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 39
} | {
"line": 135,
"column": 0
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\ninst✝ : NumberField K\nw₁ : InfinitePlace K\nB : ℝ≥0\nhf : ∀ (w : InfinitePlace K), w ≠ w₁ → f w ≠ 0\nS : ℝ≥0 := ∏ w ∈ Finset.univ.erase w₁, f w ^ w.mult\n⊢ ↑w₁.mult ≠ 0",
"ppTerm": "?refine_2✝",
"assigned": true,
"us... | [] | · rw [mult]; split_ifs <;> norm_num | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | {
"line": 129,
"column": 48
} | {
"line": 129,
"column": 51
} | {
"line": 129,
"column": 52
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\nK : Type u_2\ninst✝ : Field K\na✝ b✝ : K\nhx : a✝ ∈ {x | ∀ (φ : K →+* ℂ), star (φ x) = φ x}\nhy : b✝ ∈ {x | ∀ (φ : K →+* ℂ), star (φ x) = φ x}\nx✝ : K →+* ℂ\n⊢ star (x✝ a✝) + star (x✝ b✝) = x✝ a✝ + x✝ b✝",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants":... | [
"F : Type u_1\ninst✝¹ : Field F\nK : Type u_2\ninst✝ : Field K\na✝ b✝ : K\nhx : a✝ ∈ {x | ∀ (φ : K →+* ℂ), star (φ x) = φ x}\nhy : b✝ ∈ {x | ∀ (φ : K →+* ℂ), star (φ x) = φ x}\nx✝ : K →+* ℂ\n⊢ x✝ a✝ + star (x✝ b✝) = x✝ a✝ + x✝ b✝"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 25
} | {
"line": 204,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nw : InfinitePlace K\n⊢ (w.comap (algebraMap k K)).mult ≤ w.mult",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"NumberField.InfinitePlace.mult_comap_le",
"... | [] | exact mult_comap_le _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 45
} | {
"line": 335,
"column": 2
} | [
{
"pp": "k : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nφ : K →+* ℂ\nH : ∀ (σ : Gal(K/k)), ComplexEmbedding.IsConj φ σ → σ = 1\nhφ : ¬ComplexEmbedding.IsConj φ 1 ∧ ComplexEmbedding.IsReal (φ.comp (algebraMap k K))\n⊢ False",
"ppTerm": "?m.111",
... | [
"k : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nφ : K →+* ℂ\nH : ∀ (σ : Gal(K/k)), ComplexEmbedding.IsConj φ σ → σ = 1\nhφ : ¬ComplexEmbedding.IsConj φ 1 ∧ ComplexEmbedding.IsReal (φ.comp (algebraMap k K))\nthis : Algebra k ℂ := (φ.comp (algebraMap k K)).... | letI := (φ.comp (algebraMap k K)).toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 72,
"column": 45
} | {
"line": 72,
"column": 48
} | {
"line": 72,
"column": 49
} | [
{
"pp": "case refine_3\nK : Type u_1\ninst✝ : Field K\nx : (K →+* ℂ) → ℂ\nφ : K →+* ℂ\nhx✝ : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))\nx✝³ x✝² : (K →+* ℂ) → ℂ\nx✝¹ : x✝³ ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))\nx✝ : x✝² ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))\nhx : ... | [
"case refine_3\nK : Type u_1\ninst✝ : Field K\nx : (K →+* ℂ) → ℂ\nφ : K →+* ℂ\nhx✝ : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))\nx✝³ x✝² : (K →+* ℂ) → ℂ\nx✝¹ : x✝³ ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))\nx✝ : x✝² ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))\nhx : (starRingEnd... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 411,
"column": 61
} | {
"line": 414,
"column": 55
} | {
"line": 416,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra k K\nw : InfinitePlace K\ninst✝ : IsGalois k K\n⊢ Nat.card ↥(Stab w) = if IsUnramified k w then 1 else 2",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NumberField.InfinitePlac... | [] | by
split
· rwa [← isUnramified_iff_card_stabilizer_eq_one]
· rwa [← not_isUnramified_iff_card_stabilizer_eq_two] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 477,
"column": 55
} | {
"line": 477,
"column": 69
} | {
"line": 477,
"column": 70
} | [
{
"pp": "k : Type u_1\ninst✝⁵ : Field k\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra k K\ninst✝² : NumberField K\ninst✝¹ : NumberField k\ninst✝ : IsGalois k K\n⊢ ∑ b with IsUnramifiedIn K b, #({a ∈ {w | IsUnramified k w} | a.comap (algebraMap k K) = b}) =\n #{w | IsUnramifiedIn K w} * Nat.card Gal(K/k)"... | [
"k : Type u_1\ninst✝⁵ : Field k\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra k K\ninst✝² : NumberField K\ninst✝¹ : NumberField k\ninst✝ : IsGalois k K\n⊢ ∑ b with IsUnramifiedIn K b, #({a ∈ {w | IsUnramified k w} | a.comap (algebraMap k K) = b}) =\n #{w | IsUnramifiedIn K w} • Nat.card Gal(K/k)",
"k : Typ... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 500,
"column": 69
} | {
"line": 500,
"column": 83
} | {
"line": 500,
"column": 84
} | [
{
"pp": "k : Type u_1\ninst✝⁵ : Field k\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra k K\ninst✝² : NumberField K\ninst✝¹ : NumberField k\ninst✝ : IsGalois k K\n⊢ ∑ b ∈ {w | IsUnramifiedIn K w}ᶜ, #({a ∉ {w | IsUnramified k w} | a.comap (algebraMap k K) = b}) =\n #{w | IsUnramifiedIn K w}ᶜ * (Nat.card Gal... | [
"k : Type u_1\ninst✝⁵ : Field k\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra k K\ninst✝² : NumberField K\ninst✝¹ : NumberField k\ninst✝ : IsGalois k K\n⊢ ∑ b ∈ {w | IsUnramifiedIn K w}ᶜ, #({a ∉ {w | IsUnramified k w} | a.comap (algebraMap k K) = b}) =\n #{w | IsUnramifiedIn K w}ᶜ • (Nat.card Gal(K/k) / 2)",... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 512,
"column": 8
} | {
"line": 512,
"column": 41
} | {
"line": 512,
"column": 42
} | [
{
"pp": "k : Type u_1\ninst✝⁵ : Field k\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra k K\ninst✝² : NumberField K\ninst✝¹ : NumberField k\ninst✝ : IsGalois k K\nw : InfinitePlace K\nhw : ¬IsUnramifiedIn K ((fun x ↦ x.comap (algebraMap k K)) w)\n⊢ #(MulAction.orbit Gal(K/k) w).toFinset = Fintype.card ↑(MulAc... | [
"k : Type u_1\ninst✝⁵ : Field k\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra k K\ninst✝² : NumberField K\ninst✝¹ : NumberField k\ninst✝ : IsGalois k K\nw : InfinitePlace K\nhw : ¬IsUnramifiedIn K ((fun x ↦ x.comap (algebraMap k K)) w)\n⊢ #(MulAction.orbit Gal(K/k) w).toFinset = Fintype.card ↑(MulAction.orbit G... | Nat.mul_div_cancel _ zero_lt_two, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 642,
"column": 5
} | {
"line": 642,
"column": 73
} | {
"line": 642,
"column": 73
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nw : InfinitePlace L\nv : InfinitePlace K\ninst✝ : (↑w).LiesOver ↑v\nhw : IsUnramified K w\nhv : v.IsReal\n⊢ ¬(w.comap (algebraMap K L)).IsComplex",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
... | [] | by simpa [LiesOver.comap_eq w v] using not_isComplex_iff_isReal.2 hv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 20
} | {
"line": 396,
"column": 21
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nthis : univ = image (fun w ↦ ↑w) univ ∪ image (fun w ↦ ↑w) univ\n⊢ ‖x‖₊ =\n max (univ.sup ((fun w ↦ NNReal.mk ((normAtPlace w) x) ⋯) ∘ fun w ↦ ↑w))\n (univ.sup ((fun w ↦ NNReal.mk ((normAtPlace w) x) ⋯) ∘ fun w ↦ ↑w))",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nthis : univ = image (fun w ↦ ↑w) univ ∪ image (fun w ↦ ↑w) univ\n⊢ max ‖x.1‖₊ ‖x.2‖₊ =\n max (univ.sup ((fun w ↦ NNReal.mk ((normAtPlace w) x) ⋯) ∘ fun w ↦ ↑w))\n (univ.sup ((fun w ↦ NNReal.mk ((normAtPlace w) x) ⋯) ∘ fun w ↦ ↑w))"
] | Prod.nnnorm_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 408,
"column": 4
} | {
"line": 408,
"column": 31
} | {
"line": 408,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\n⊢ ↑(univ.sup' ⋯ fun w ↦ NNReal.mk ((normAtPlace w) x) ⋯) = univ.sup' ⋯ fun w ↦ (normAtPlace w) x",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NumberField.instNonemptyInfinitePlaceOfR... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\n⊢ (OrderHom.Subtype.val fun r ↦ 0 ≤ r) (univ.sup' ⋯ fun w ↦ NNReal.mk ((normAtPlace w) x) ⋯) =\n univ.sup' ⋯ fun w ↦ (normAtPlace w) x"
] | ← OrderHom.Subtype.val_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 702,
"column": 2
} | {
"line": 703,
"column": 43
} | {
"line": 704,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\ni : ChooseBasisIndex ℤ (𝓞 K)\n⊢ ((Basis.restrictScalars ℚ (latticeBasis K)).repr ⟨(mixedEmbedding K) x, ⋯⟩) i = ((integralBasis K).repr x) i",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceR... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\ni : ChooseBasisIndex ℤ (𝓞 K)\nf : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=\n LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (mixedEmbedding K).toRatAlgHom.toLinearMap ⋯\n⊢ ((Basis.restrictScalars ℚ (latticeB... | let f := (mixedEmbedding K).toRatAlgHom.toLinearMap.codRestrict _
(fun x ↦ mem_rat_span_latticeBasis K x) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 383,
"column": 4
} | {
"line": 383,
"column": 20
} | {
"line": 384,
"column": 2
} | [
{
"pp": "case ha\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nN : ℕ\nhK : |discr K| ≤ ↑N\nthis : boundOfDiscBdd N - 1 < boundOfDiscBdd N\n⊢ 1 ≤ 2",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
"ENNReal.instIsOrderedRing",
"ENNReal.instAddCommMonoid",
"IsOrderedR... | [] | exact one_le_two | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 383,
"column": 4
} | {
"line": 383,
"column": 20
} | {
"line": 384,
"column": 2
} | [
{
"pp": "case ha\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nN : ℕ\nhK : |discr K| ≤ ↑N\nthis : boundOfDiscBdd N - 1 < boundOfDiscBdd N\n⊢ 1 ≤ 2",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
"ENNReal.instIsOrderedRing",
"ENNReal.instAddCommMonoid",
"IsOrderedR... | [] | exact one_le_two | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 383,
"column": 4
} | {
"line": 383,
"column": 20
} | {
"line": 384,
"column": 2
} | [
{
"pp": "case ha\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nN : ℕ\nhK : |discr K| ≤ ↑N\nthis : boundOfDiscBdd N - 1 < boundOfDiscBdd N\n⊢ 1 ≤ 2",
"ppTerm": "?ha",
"assigned": true,
"usedConstants": [
"ENNReal.instIsOrderedRing",
"ENNReal.instAddCommMonoid",
"IsOrderedR... | [] | exact one_le_two | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 939,
"column": 4
} | {
"line": 939,
"column": 69
} | {
"line": 940,
"column": 2
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝ : Field K\ns : Set { w // w.IsReal }\nx : mixedSpace K\nw : InfinitePlace K\nhw : w.IsReal\n⊢ (normAtPlace w) ((negAt s) x) = (normAtPlace w) x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Norm.norm",... | [] | simp_rw [normAtPlace_apply_of_isReal hw, negAt_apply_norm_isReal] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 939,
"column": 4
} | {
"line": 939,
"column": 69
} | {
"line": 940,
"column": 2
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝ : Field K\ns : Set { w // w.IsReal }\nx : mixedSpace K\nw : InfinitePlace K\nhw : w.IsReal\n⊢ (normAtPlace w) ((negAt s) x) = (normAtPlace w) x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Norm.norm",... | [] | simp_rw [normAtPlace_apply_of_isReal hw, negAt_apply_norm_isReal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 939,
"column": 4
} | {
"line": 939,
"column": 69
} | {
"line": 940,
"column": 2
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝ : Field K\ns : Set { w // w.IsReal }\nx : mixedSpace K\nw : InfinitePlace K\nhw : w.IsReal\n⊢ (normAtPlace w) ((negAt s) x) = (normAtPlace w) x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Norm.norm",... | [] | simp_rw [normAtPlace_apply_of_isReal hw, negAt_apply_norm_isReal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 158,
"column": 6
} | {
"line": 158,
"column": 24
} | {
"line": 158,
"column": 25
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\np : Ideal A\nP : Ideal B\nhPp : P.IsPrime\nhp : P.LiesOver p\nG : Type u_3\ninst✝³ : Group G\ninst✝² : Finite G\ninst✝¹ : MulSemiringAction G B\ninst✝ : IsGaloisGroup G A B\nh : ∃ P, P.IsPrime ∧ P.LiesOver p\nle... | [
"A : Type u_1\nB : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra A B\np : Ideal A\nP : Ideal B\nhPp : P.IsPrime\nhp : P.LiesOver p\nG : Type u_3\ninst✝³ : Group G\ninst✝² : Finite G\ninst✝¹ : MulSemiringAction G B\ninst✝ : IsGaloisGroup G A B\nh : ∃ P, P.IsPrime ∧ P.LiesOver p\nleft✝ : h.choo... | ramificationIdxIn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 20
} | {
"line": 225,
"column": 21
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : IsDomain A\np : Ideal A\ninst✝⁹ : p.IsPrime\nB : Type u_2\ninst✝⁸ : CommRing B\ninst✝⁷ : IsDomain B\ninst✝⁶ : Algebra A B\ninst✝⁵ : Module.Finite A B\ninst✝⁴ : Flat A B\nG : Type u_3\ninst✝³ : Group G\ninst✝² : Finite G\ninst✝¹ : MulSemiringAction G B\ninst... | [
"A : Type u_1\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : IsDomain A\np : Ideal A\ninst✝⁹ : p.IsPrime\nB : Type u_2\ninst✝⁸ : CommRing B\ninst✝⁷ : IsDomain B\ninst✝⁶ : Algebra A B\ninst✝⁵ : Module.Finite A B\ninst✝⁴ : Flat A B\nG : Type u_3\ninst✝³ : Group G\ninst✝² : Finite G\ninst✝¹ : MulSemiringAction G B\ninst✝ : IsGalois... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 22
} | {
"line": 97,
"column": 2
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : HasFiniteQuotients R\nB : ℕ\nh✝ : Finite R\n⊢ {I | Submodule.cardQuot I ≤ B}.Finite",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"CommSemiring.toSemiring",
"Subtype.finite",
"setOf"... | [
"case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : HasFiniteQuotients R\nB : ℕ\nh✝ : Infinite R\n⊢ {I | Submodule.cardQuot I ≤ B}.Finite"
] | · apply Set.toFinite | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 95,
"column": 83
} | {
"line": 95,
"column": 97
} | {
"line": 96,
"column": 6
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nJ : ↥(Ideal (𝓞 K))⁰\nhJ : ClassGroup.mk0 J = C⁻¹\na : 𝓞 K\nha : a ∈ ↑↑J\nh_nz : (Algebra.linearMap (𝓞 K) K) a ≠ 0\nI₀ : Ideal (𝓞 K)\nh_nm :\n ↑(Submodule.cardQuot ↑J) * ↑(absNorm I₀) ≤\n ↑(absNorm ↑J) *... | [
"case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nJ : ↥(Ideal (𝓞 K))⁰\nhJ : ClassGroup.mk0 J = C⁻¹\na : 𝓞 K\nha : a ∈ ↑↑J\nh_nz : (Algebra.linearMap (𝓞 K) K) a ≠ 0\nI₀ : Ideal (𝓞 K)\nh_nm :\n ↑(Submodule.cardQuot ↑J) * ↑(absNorm I₀) ≤\n ↑(absNorm ↑J) * (4 / π) ^ n... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 321,
"column": 2
} | {
"line": 326,
"column": 56
} | {
"line": 328,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nG : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Group G\ninst✝¹⁰ : MulSemiringAction G S\ninst✝⁹ : IsGaloisGroup G R S\ninst✝⁸ : Finite G\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsDomain S\ninst✝⁵ : Module.Finite R S\ninst✝⁴ : Flat R S\np : ... | [] | have H := ncard_primesOver_mul_card_inertia_mul_finrank (G := G) p P
rw [← inertiaDegIn_eq_inertiaDeg p P G] at H
have h1 : (p.primesOver S).ncard ≠ 0 := by grind [Nat.card_pos]
have h2 : p.inertiaDegIn S ≠ 0 := by grind [Nat.card_pos]
rwa [← ncard_primesOver_mul_ramificationIdxIn_mul_inertiaDegIn p S G,
mu... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 321,
"column": 2
} | {
"line": 326,
"column": 56
} | {
"line": 328,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nG : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Group G\ninst✝¹⁰ : MulSemiringAction G S\ninst✝⁹ : IsGaloisGroup G R S\ninst✝⁸ : Finite G\ninst✝⁷ : IsDomain R\ninst✝⁶ : IsDomain S\ninst✝⁵ : Module.Finite R S\ninst✝⁴ : Flat R S\np : ... | [] | have H := ncard_primesOver_mul_card_inertia_mul_finrank (G := G) p P
rw [← inertiaDegIn_eq_inertiaDeg p P G] at H
have h1 : (p.primesOver S).ncard ≠ 0 := by grind [Nat.card_pos]
have h2 : p.inertiaDegIn S ≠ 0 := by grind [Nat.card_pos]
rwa [← ncard_primesOver_mul_ramificationIdxIn_mul_inertiaDegIn p S G,
mu... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.Unramified | {
"line": 60,
"column": 4
} | {
"line": 61,
"column": 47
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : CommRing T\ninst✝¹⁰ : Algebra R S\ninst✝⁹ : Algebra S T\ninst✝⁸ : Algebra R T\ninst✝⁷ : IsScalarTower R S T\np : Ideal S\nP : Ideal T\ninst✝⁶ : P.LiesOver p\ninst✝⁵ : p.IsPrime\ninst✝⁴ : P.IsPrime\... | [] | rw [IsScalarTower.algebraMap_eq _ S, ← Ideal.map_map, this,
Localization.AtPrime.map_eq_maximalIdeal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 67,
"column": 53
} | {
"line": 67,
"column": 67
} | {
"line": 67,
"column": 68
} | [
{
"pp": "case refine_2\nA : Type u_1\ninst✝⁷ : CommRing A\nB : Type u_2\ninst✝⁶ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁵ : CommRing K\ninst✝⁴ : Algebra A K\ninst✝³ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝² : CommRing L\ninst✝¹ : Algebra B L\ninst✝ : IsLocalization N L\n... | [
"case refine_2\nA : Type u_1\ninst✝⁷ : CommRing A\nB : Type u_2\ninst✝⁶ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁵ : CommRing K\ninst✝⁴ : Algebra A K\ninst✝³ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝² : CommRing L\ninst✝¹ : Algebra B L\ninst✝ : IsLocalization N L\nhf : M ≤ Sub... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 509,
"column": 4
} | {
"line": 509,
"column": 28
} | {
"line": 510,
"column": 4
} | [
{
"pp": "case pos\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nhK₁ : ⟨K, hK₀⟩ ∈ {K | |discr ↥↑K| ≤ ↑N}\nthis✝ : CharZero ↥K\nthis : NumberField ↥K\nw₀ : InfinitePlace ↥K\nhw₀ : w₀.IsReal\n⊢ ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥... | [
"case pos\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nhK₁ : ⟨K, hK₀⟩ ∈ {K | |discr ↥↑K| ≤ ↑N}\nthis✝ : CharZero ↥K\nthis : NumberField ↥K\nw₀ : InfinitePlace ↥K\nhw₀ : w₀.IsReal\n⊢ ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}"
] | apply Set.mem_union_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.DedekindDomain.PID | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 68
} | {
"line": 76,
"column": 4
} | [
{
"pp": "R : Type u_2\nA : Type u_3\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))\nh : J = ⊤\nhJ : ... | [
"R : Type u_2\nA : Type u_3\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nI : (FractionalIdeal S A)ˣ\nv : A\nhv : v ∈ ↑I⁻¹\nhinv : ↑I * ↑I⁻¹ = 1\nJ : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))\nh : J = ⊤\nhJ : IsLocalizati... | rw [← hJ, h, IsLocalization.coeSubmodule_top, Submodule.mem_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.PID | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 20
} | {
"line": 95,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nhS : S ≤ R⁰\nhf : {I | I.IsMaximal}.Finite\nI I' : FractionalIdeal S A\nhinv : I * I' = 1\n⊢ (↑I).IsPrincipal",
"ppTerm": "?m.39",
"assigned": true,
"used... | [
"R : Type u_1\ninst✝³ : CommRing R\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nS : Submonoid R\ninst✝ : IsLocalization S A\nhS : S ≤ R⁰\nhf : {I | I.IsMaximal}.Finite\nI I' : FractionalIdeal S A\nhinv hinv' : I * I' = 1\n⊢ (↑I).IsPrincipal"
] | have hinv' := hinv | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.Instances | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 22
} | {
"line": 77,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : CommRing S\ninst✝¹⁶ : IsDomain R\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Algebra R S\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝¹³ : CommRing Rₘ\ninst✝¹² : CommRing Sₘ\ninst✝¹¹ : Algebra R Rₘ\ninst✝¹⁰ : IsTorsionFree R S\ninst✝⁹ : Algebra.IsSeparable K L\nM... | [
"R : Type u_1\nS : Type u_2\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : CommRing S\ninst✝¹⁶ : IsDomain R\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Algebra R S\nRₘ : Type u_4\nSₘ : Type u_5\ninst✝¹³ : CommRing Rₘ\ninst✝¹² : CommRing Sₘ\ninst✝¹¹ : Algebra R Rₘ\ninst✝¹⁰ : IsTorsionFree R S\ninst✝⁹ : Algebra.IsSeparable K L\nM : Submonoid... | apply ringHom_ext R⁰ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 35
} | {
"line": 163,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\nS : Type u_3\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : IsIntegrallyClosed R\ninst✝³ : IsIntegrallyClosed S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\ninst✝ : IsTorsionFree R S\nI J : Ideal S\n⊢ spanNorm R I * spanNorm R J ≤ spanN... | [
"R : Type u_1\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\nS : Type u_3\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : IsIntegrallyClosed R\ninst✝³ : IsIntegrallyClosed S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\ninst✝ : IsTorsionFree R S\nI J : Ideal S\n⊢ map (Algebra.intNorm R S) I * map (Algebra.intN... | rw [spanNorm, spanNorm, spanNorm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 45
} | {
"line": 396,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : IsDomain R\nS : Type u_3\ninst✝⁹ : CommRing S\ninst✝⁸ : IsDomain S\ninst✝⁷ : IsIntegrallyClosed R\ninst✝⁶ : IsIntegrallyClosed S\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module.Finite R S\ninst✝³ : IsTorsionFree R S\ninst✝² : IsDedekindDomain R\ninst✝¹ : I... | [] | exact ⟨s, by rwa [associated_iff_eq] at hs⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 463,
"column": 35
} | {
"line": 463,
"column": 38
} | {
"line": 463,
"column": 39
} | [
{
"pp": "case neg.refine_1\nR : Type u_1\ninst✝¹⁴ : CommRing R\ninst✝¹³ : IsDomain R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\ninst✝¹⁰ : IsIntegrallyClosed R\ninst✝⁹ : IsIntegrallyClosed S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module.Finite R S\ninst✝⁶ : IsTorsionFree R S\ninst✝⁵ : IsDedekindDomain R... | [
"case neg.refine_1\nR : Type u_1\ninst✝¹⁴ : CommRing R\ninst✝¹³ : IsDomain R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\ninst✝¹⁰ : IsIntegrallyClosed R\ninst✝⁹ : IsIntegrallyClosed S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module.Finite R S\ninst✝⁶ : IsTorsionFree R S\ninst✝⁵ : IsDedekindDomain R\ninst✝⁴ : I... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 481,
"column": 72
} | {
"line": 482,
"column": 95
} | {
"line": 484,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹⁴ : CommRing R\ninst✝¹³ : IsDomain R\nS : Type u_3\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S\ninst✝¹⁰ : IsIntegrallyClosed R\ninst✝⁹ : IsIntegrallyClosed S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Module.Finite R S\ninst✝⁶ : IsTorsionFree R S\ninst✝⁵ : IsDedekindDomain R\ninst✝⁴ : IsDedeki... | [] | by
rw [← absNorm_relNorm ℤ, ← relNorm_relNorm ℤ R, relNorm_algebraMap, absNorm_relNorm, map_pow] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 15
} | {
"line": 358,
"column": 0
} | [
{
"pp": "p k : ℕ\nK : Type u\ninst✝² : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝¹ : CharZero K\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\ns : ℕ\nhs : s ≤ k\nhodd : p ≠ 2\nh : p ^ (k - s + 1) = 2\n⊢ p ^ (k - s).succ = 2 ^ ?m.88",
"ppTerm": "?m.96",
"assig... | [] | rwa [pow_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 531,
"column": 2
} | {
"line": 531,
"column": 55
} | {
"line": 532,
"column": 2
} | [
{
"pp": "p k : ℕ\nK : Type u\ninst✝² : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝¹ : CharZero K\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhodd : p ≠ 2\nh : hζ.toInteger - 1 ∣ 2\nthis : NumberField K\n⊢ False",
"ppTerm": "?m.71",
"assigned": true,
"us... | [
"p k : ℕ\nK : Type u\ninst✝² : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝¹ : CharZero K\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhodd : p ≠ 2\nthis : NumberField K\nh : hζ.toInteger - 1 ∣ ↑2\n⊢ False"
] | replace h : hζ.toInteger - 1 ∣ (2 : ℤ) := by simp [h] | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 34
} | {
"line": 118,
"column": 2
} | [
{
"pp": "case pos\nA : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : CommRing B\ninst✝⁷ : Field L\ninst✝⁶ : Algebra A K\ninst✝⁵ : Algebra B L\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra K L\ninst✝² : Algebra A L\ninst✝¹ : IsScalarTower A K L\ninst✝ : IsScalarTowe... | [] | exact fun _ _ _ _ ↦ h ⟨_, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 570,
"column": 10
} | {
"line": 580,
"column": 69
} | {
"line": 581,
"column": 6
} | [
{
"pp": "case pos.succ\nn : ℕ\nhn✝ : 2 ≤ n\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : K\np : ℕ\nhF : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot ζ n\nthis✝ : NeZero n\nμ : ↥ℚ⟮ζ⟯\nhC : IsCyclotomicExtension {n} ℚ ↥ℚ⟮ζ⟯\nhμ : IsPrimitiveRoot μ n\nhp : ↑p ∣ (Algebra.norm ℤ) (hμ.toInteger - 1) ^ Modul... | [] | rw [← hn'] at hC hμ
refine ⟨q, hq, r + 1, Module.finrank (ℚ⟮ζ⟯) K, r.add_one_ne_zero, hn'.symm, ?_⟩
by_cases hq' : q = 2
· cases r with
| zero =>
rw [← hn', hq', zero_add, pow_one] at hn
exact hn.false.elim
| succ k =>
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 570,
"column": 10
} | {
"line": 580,
"column": 69
} | {
"line": 581,
"column": 6
} | [
{
"pp": "case pos.succ\nn : ℕ\nhn✝ : 2 ≤ n\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : K\np : ℕ\nhF : Fact (Nat.Prime p)\nhζ : IsPrimitiveRoot ζ n\nthis✝ : NeZero n\nμ : ↥ℚ⟮ζ⟯\nhC : IsCyclotomicExtension {n} ℚ ↥ℚ⟮ζ⟯\nhμ : IsPrimitiveRoot μ n\nhp : ↑p ∣ (Algebra.norm ℤ) (hμ.toInteger - 1) ^ Modul... | [] | rw [← hn'] at hC hμ
refine ⟨q, hq, r + 1, Module.finrank (ℚ⟮ζ⟯) K, r.add_one_ne_zero, hn'.symm, ?_⟩
by_cases hq' : q = 2
· cases r with
| zero =>
rw [← hn', hq', zero_add, pow_one] at hn
exact hn.false.elim
| succ k =>
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 649,
"column": 2
} | {
"line": 653,
"column": 20
} | {
"line": 655,
"column": 0
} | [
{
"pp": "p : ℕ\nK : Type u\ninst✝² : Field K\nhp : Fact (Nat.Prime p)\ninst✝¹ : CharZero K\ninst✝ : IsCyclotomicExtension {p} ℚ K\n⊢ NumberField.discr K = (-1) ^ ((p - 1) / 2) * ↑p ^ (p - 2)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Set.fi... | [] | have : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K := by
rw [zero_add, pow_one]
infer_instance
rw [discr_prime_pow_succ p 0 K]
simp [Nat.sub_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 649,
"column": 2
} | {
"line": 653,
"column": 20
} | {
"line": 655,
"column": 0
} | [
{
"pp": "p : ℕ\nK : Type u\ninst✝² : Field K\nhp : Fact (Nat.Prime p)\ninst✝¹ : CharZero K\ninst✝ : IsCyclotomicExtension {p} ℚ K\n⊢ NumberField.discr K = (-1) ^ ((p - 1) / 2) * ↑p ^ (p - 2)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Set.fi... | [] | have : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K := by
rw [zero_add, pow_one]
infer_instance
rw [discr_prime_pow_succ p 0 K]
simp [Nat.sub_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.LinearDisjoint | {
"line": 184,
"column": 58
} | {
"line": 203,
"column": 13
} | {
"line": 205,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝⁵⁵ : CommRing A\ninst✝⁵⁴ : Field K\ninst✝⁵³ : Algebra A K\ninst✝⁵² : IsFractionRing A K\ninst✝⁵¹ : CommRing B\ninst✝⁵⁰ : Field L\ninst✝⁴⁹ : Algebra B L\ninst✝⁴⁸ : Algebra A L\ninst✝⁴⁷ : Algebra K L\ninst✝⁴⁶ : FiniteDimensional K L\ninst✝⁴⁵ : ... | [] | by
classical
have h₂' : F₁ ⊔ F₂ = ⊤ := by
rwa [← sup_toSubalgebra_of_isAlgebraic_right, ← top_toSubalgebra, toSubalgebra_inj] at h₂
have : Finite ι := Module.Finite.finite_basis b
have h_main := congr_arg (Submodule.restrictScalars R₁) <|
congr_arg coeToSubmodule <| (1 : FractionalIdeal B⁰ L).dual_dual ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 769,
"column": 27
} | {
"line": 796,
"column": 86
} | {
"line": 798,
"column": 0
} | [
{
"pp": "n : ℕ\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nhn : NeZero n\nhK : IsCyclotomicExtension {n} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ n\n⊢ ℤ[hζ.toInteger] = ⊤",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Nat.lcm",
"Eq.mpr",
"Nat.Coprime",
"False",
... | [] | by
haveI : NumberField K := IsCyclotomicExtension.numberField {n} ℚ K
induction n using Nat.recOnPrimeCoprime generalizing K hn with
| zero => exact (neZero_zero_iff_false.mp hn).elim
| prime_pow p k hp =>
have : Fact (p.Prime) := ⟨hp⟩
rw [← hζ.integralPowerBasisOfPrimePow.adjoin_gen_eq_top, hζ.integral... | [anonymous] | Lean.Parser.Term.byTactic |
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