module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Adjoin.Polynomial.Bivariate | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 52
} | {
"line": 47,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nx : A\nhx : Transcendental R x\np : R[X][Y]\n⊢ (algEquivAdjoin hx) (swap p) = (aeval (C ⟨x, ⋯⟩)) p",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Polynomi... | [] | simp [algEquivAdjoin, Bivariate.aveal_eq_map_swap] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.NormTrace | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 60
} | {
"line": 70,
"column": 2
} | [
{
"pp": "case inr\nR : 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 (alge... | [
"case inr\nR : 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 (algebraMapSubmon... | letI := Classical.decEq (Module.Free.ChooseBasisIndex R S) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.RingTheory.Localization.NormTrace | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 60
} | {
"line": 91,
"column": 2
} | [
{
"pp": "case inr\nR : 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 (alge... | [
"case inr\nR : 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 (algebraMapSubmon... | letI := Classical.decEq (Module.Free.ChooseBasisIndex R S) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.RingTheory.Discriminant | {
"line": 197,
"column": 31
} | {
"line": 197,
"column": 44
} | {
"line": 197,
"column": 45
} | [
{
"pp": "case e_a.e_a\nK : Type u\nL : Type v\nE : Type z\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Field E\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra K E\ninst✝² : Module.Finite K L\ninst✝¹ : IsAlgClosed E\npb : PowerBasis K L\ninst✝ : Algebra.IsSeparable K L\ne : Fin pb.dim ≃ (L →ₐ[K] E)\nthis : ∀ (x : Fin p... | [
"case e_a.e_a\nK : Type u\nL : Type v\nE : Type z\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Field E\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra K E\ninst✝² : Module.Finite K L\ninst✝¹ : IsAlgClosed E\npb : PowerBasis K L\ninst✝ : Algebra.IsSeparable K L\ne : Fin pb.dim ≃ (L →ₐ[K] E)\nthis : ∀ (x : Fin pb.dim), ↑x +... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 491,
"column": 44
} | {
"line": 491,
"column": 57
} | {
"line": 491,
"column": 58
} | [
{
"pp": "case h\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nm n : ℕ\nh : m ≤ n\nx : ℤ_[p]\nc : ℕ\nhc : x.appr n - x.appr m = p ^ m * c\n⊢ ↑(p ^ m * c) = ↑p ^ m * ↑c",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semigroup.toM... | [
"case h\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nm n : ℕ\nh : m ≤ n\nx : ℤ_[p]\nc : ℕ\nhc : x.appr n - x.appr m = p ^ m * c\n⊢ ↑(p ^ m) * ↑c = ↑p ^ m * ↑c"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 560,
"column": 2
} | {
"line": 560,
"column": 7
} | {
"line": 561,
"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\nr : R\nε : ℚ\nhε : ε > 0\nk : ℕ\nhk : ↑p ^ (-↑k) < ε\n⊢ ∃ i, ∀ j ≥ i, padicNorm p ((fun n ↦ ↑(n... | [
"case h\nR : 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\nr : R\nε : ℚ\nhε : ε > 0\nk : ℕ\nhk : ↑p ^ (-↑k) < ε\n⊢ ∀ j ≥ k, padicNorm p ((fun n ↦ ↑(nthHom f r... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Discriminant | {
"line": 229,
"column": 2
} | {
"line": 230,
"column": 97
} | {
"line": 231,
"column": 4
} | [
{
"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.refine_1\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 ... | refine prod_bij' (fun i _ ↦ ⟨e i.2, e i.1 pb.gen⟩)
(fun σ hσ ↦ ⟨e.symm (PowerBasis.lift pb σ.2 ?_), e.symm σ.1⟩) ?_ ?_ ?_ ?_ (fun i _ ↦ by simp) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Cyclotomic.Discriminant | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 45
} | {
"line": 102,
"column": 4
} | [
{
"pp": "case e_a\np 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... | [
"case e_a\np 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 + ... | replace H := congr_arg (Algebra.norm K) H | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 231,
"column": 16
} | {
"line": 231,
"column": 31
} | {
"line": 231,
"column": 31
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn : ℕ\npn : xz a1 n * xz a1 n - ↑(d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n\n⊢ ↑(xn a1 n * xn a1 n) - ↑(d a1 * yn a1 n * yn a1 n) = 1",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Pell.xn",
"congrArg",
... | [
"a : ℕ\na1 : 1 < a\nn : ℕ\npn : xz a1 n * xz a1 n - ↑(d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n\n⊢ ↑(xn a1 n) * ↑(xn a1 n) - ↑(d a1 * yn a1 n * yn a1 n) = 1"
] | Int.natCast_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 230,
"column": 75
} | {
"line": 231,
"column": 42
} | {
"line": 232,
"column": 2
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn : ℕ\npn : xz a1 n * xz a1 n - ↑(d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n\n⊢ ↑(xn a1 n * xn a1 n) - ↑(d a1 * yn a1 n * yn a1 n) = 1",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Pell.xn",
"congrArg",
... | [] | by
repeat' rw [Int.natCast_mul]; exact pn | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Dioph | {
"line": 175,
"column": 16
} | {
"line": 175,
"column": 65
} | {
"line": 176,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx✝ : Poly α\n⊢ 0 + x✝ = x✝",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Poly.instAdd",
"Poly.ext",
"Poly",
"congrArg",
"AddMonoid.toAddZeroClass",
"Poly.instFunLike",
"id",
"Int",
"Int.instAddMon... | [] | by ext; simp_rw [add_apply, zero_apply, zero_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 115,
"column": 4
} | {
"line": 129,
"column": 56
} | {
"line": 131,
"column": 0
} | [
{
"pp": "case neg\nξ : 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\nH : ∀ m ∈ D, f m ≠ ↑n\n⊢ ∃ j k, 0 < k ∧ k ≤ ↑n ∧ |(ξ * ↑k - ↑j) * (↑n + 1)| ≤ 1",
"ppTerm": "?neg✝",
"assigned... | [] | have hD : #(Ico (0 : ℤ) n) < #D := by rw [card_Icc, card_Ico]; exact lt_add_one n
have hfu' : ∀ m, f m ≤ n := fun m => lt_add_one_iff.mp (floor_lt.mpr (mod_cast hfu m))
have hwd : ∀ m : ℤ, m ∈ D → f m ∈ Ico (0 : ℤ) n := fun x hx =>
mem_Ico.mpr
⟨floor_nonneg.mpr (mul_nonneg (fract_nonneg (ξ * x)) h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 115,
"column": 4
} | {
"line": 129,
"column": 56
} | {
"line": 131,
"column": 0
} | [
{
"pp": "case neg\nξ : 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\nH : ∀ m ∈ D, f m ≠ ↑n\n⊢ ∃ j k, 0 < k ∧ k ≤ ↑n ∧ |(ξ * ↑k - ↑j) * (↑n + 1)| ≤ 1",
"ppTerm": "?neg✝",
"assigned... | [] | have hD : #(Ico (0 : ℤ) n) < #D := by rw [card_Icc, card_Ico]; exact lt_add_one n
have hfu' : ∀ m, f m ≤ n := fun m => lt_add_one_iff.mp (floor_lt.mpr (mod_cast hfu m))
have hwd : ∀ m : ℤ, m ∈ D → f m ∈ Ico (0 : ℤ) n := fun x hx =>
mem_Ico.mpr
⟨floor_nonneg.mpr (mul_nonneg (fract_nonneg (ξ * x)) h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.DirichletCharacter.Bounds | {
"line": 28,
"column": 2
} | {
"line": 30,
"column": 13
} | {
"line": 32,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝ : NormedField F\nn : ℕ\nχ : DirichletCharacter F n\na : (ZMod n)ˣ\n⊢ ‖χ ↑a‖ = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Units.val",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | [] | refine (pow_eq_one_iff_of_nonneg (norm_nonneg _) (Nat.card_pos (α := (ZMod n)ˣ)).ne').mp ?_
rw [← norm_pow, ← map_pow, ← Units.val_pow_eq_pow_val, pow_card_eq_one', Units.val_one, map_one,
norm_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.DirichletCharacter.Bounds | {
"line": 28,
"column": 2
} | {
"line": 30,
"column": 13
} | {
"line": 32,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝ : NormedField F\nn : ℕ\nχ : DirichletCharacter F n\na : (ZMod n)ˣ\n⊢ ‖χ ↑a‖ = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Units.val",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | [] | refine (pow_eq_one_iff_of_nonneg (norm_nonneg _) (Nat.card_pos (α := (ZMod n)ˣ)).ne').mp ?_
rw [← norm_pow, ← map_pow, ← Units.val_pow_eq_pow_val, pow_card_eq_one', Units.val_one, map_one,
norm_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.MulChar.Duality | {
"line": 46,
"column": 2
} | {
"line": 47,
"column": 81
} | {
"line": 49,
"column": 0
} | [
{
"pp": "case refine_2\nM : Type u_1\nR : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommRing R\na : Mˣ\nx✝ : ∃ φ, φ a ≠ 1\nφ : Mˣ →* Rˣ\nhφ : φ a ≠ 1\n⊢ (ofUnitHom φ) ↑a ≠ 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Units.val",
"_private.Mathlib.NumberTheory.MulChar.D... | [] | · contrapose hφ
simpa only [ofUnitHom_eq, equivToUnitHom_symm_coe, Units.val_eq_one] using hφ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 50
} | {
"line": 257,
"column": 2
} | [
{
"pp": "ξ : ℚ\nf : ℚ → ℤ × ℕ := fun q ↦ (q.num, q.den)\n⊢ {q | |ξ - q| < 1 / ↑q.den ^ 2}.Finite",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Rat.instSub",
"Eq.mpr",
"instHDiv",
"abs",
"congrArg",
"Rat",
"setOf",
"H... | [
"ξ : ℚ\nf : ℚ → ℤ × ℕ := fun q ↦ (q.num, q.den)\ns : Set ℚ := {q | |ξ - q| < 1 / ↑q.den ^ 2}\n⊢ s.Finite"
] | set s := {q : ℚ | |ξ - q| < 1 / (q.den : ℚ) ^ 2} | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.NumberTheory.SmoothNumbers | {
"line": 142,
"column": 2
} | {
"line": 143,
"column": 76
} | {
"line": 144,
"column": 2
} | [
{
"pp": "s : Finset ℕ\nN : ℕ\nhN : ¬Prime N\nm : ℕ\n⊢ m ∈ factoredNumbers (insert N s) ↔ m ∈ factoredNumbers s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"Insert.insert",
"Ne",
"instOfNatNat",
"Finset.instInsert",
"... | [
"s : Finset ℕ\nN : ℕ\nhN : ¬Prime N\nm : ℕ\nhm : m ∈ factoredNumbers (insert N s)\np : ℕ\nhp : p ∈ m.primeFactorsList\n⊢ p ∈ s"
] | refine ⟨fun hm ↦ ⟨hm.1, fun p hp ↦ ?_⟩,
fun hm ↦ ⟨hm.1, fun p hp ↦ Finset.mem_insert_of_mem <| hm.2 p hp⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 767,
"column": 23
} | {
"line": 806,
"column": 94
} | {
"line": 806,
"column": 94
} | [
{
"pp": "a k x y : ℕ\nx✝ : ∃ (a1 : 1 < a), xn a1 k = x ∧ yn a1 k = y\na1 : 1 < a\nhx : xn a1 k = x\nhy : yn a1 k = y\n⊢ 1 < a ∧\n k ≤ y ∧\n (x = 1 ∧ y = 0 ∨\n ∃ u v s t b,\n x * x - (a * a - 1) * y * y = 1 ∧\n u * u - (a * a - 1) * v * v = 1 ∧\n s * s - (b * b - 1... | [] | by
rw [← hx, ← hy]
refine ⟨a1,
(Nat.eq_zero_or_pos k).elim (fun k0 => by rw [k0]; exact ⟨le_rfl, Or.inl ⟨rfl, rfl⟩⟩)
fun kpos => ?_⟩
exact
let x := xn a1 k
let y := yn a1 k
let m := 2 * (k * y)
let u := xn a1 m
let v := yn a1 m
have ky : k ≤ y := yn_ge_n... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 443,
"column": 2
} | {
"line": 443,
"column": 52
} | {
"line": 444,
"column": 2
} | [
{
"pp": "ξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nh : ContfracLegendre.Ass ξ u v\nhξ₀ : 0 < fract ξ\nu' : ℤ := u - ⌊ξ⌋ * v\nhu₀ : 0 < u'\nhuv : u' < v\nhu' : u' = u - ⌊ξ⌋ * v\n⊢ |(fract ξ)⁻¹ - ↑v / (↑u - ↑⌊ξ⌋ * ↑v)| < ((↑u - ↑⌊ξ⌋ * ↑v) * (2 * (↑u - ↑⌊ξ⌋ * ↑v) - 1))⁻¹",
"ppTerm": "?m.138",
"assigned": true,
"used... | [
"ξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nh : ContfracLegendre.Ass ξ u v\nhξ₀ : 0 < fract ξ\nu' : ℤ := u - ⌊ξ⌋ * v\nhu₀ : 0 < u'\nhuv : u' < v\nhu' : u' = u - ⌊ξ⌋ * v\nhu'ℝ : ↑u' = ↑u - ↑⌊ξ⌋ * ↑v\n⊢ |(fract ξ)⁻¹ - ↑v / (↑u - ↑⌊ξ⌋ * ↑v)| < ((↑u - ↑⌊ξ⌋ * ↑v) * (2 * (↑u - ↑⌊ξ⌋ * ↑v) - 1))⁻¹"
] | have hu'ℝ : (u' : ℝ) = u - ⌊ξ⌋ * v := mod_cast hu' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LSeries.Convolution | {
"line": 125,
"column": 4
} | {
"line": 127,
"column": 54
} | {
"line": 128,
"column": 4
} | [
{
"pp": "case inl\nf g : ℕ → ℂ\ns : ℂ\nhS : (fun p ↦ p.1 * p.2) ⁻¹' {0} = {0} ×ˢ univ ∪ univ ×ˢ {0}\n⊢ 0 = ∑' (b : ↑((fun p ↦ p.1 * p.2) ⁻¹' {0})), term f s (↑b).1 * term g s (↑b).2",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"HMul.hMul",
"SProd.sprod"... | [
"case inl\nf g : ℕ → ℂ\ns : ℂ\nhS : (fun p ↦ p.1 * p.2) ⁻¹' {0} = {0} ×ˢ univ ∪ univ ×ˢ {0}\nthis : ∀ (p : ↑((fun p ↦ p.1 * p.2) ⁻¹' {0})), term f s (↑p).1 * term g s (↑p).2 = 0\n⊢ 0 = ∑' (b : ↑((fun p ↦ p.1 * p.2) ⁻¹' {0})), term f s (↑b).1 * term g s (↑b).2"
] | have : ∀ p : (fun p : ℕ × ℕ ↦ p.1 * p.2) ⁻¹' {0}, term f s p.val.1 * term g s p.val.2 = 0 := by
rintro ⟨⟨_, _⟩, hp⟩
rcases hS ▸ hp with ⟨rfl, -⟩ | ⟨-, rfl⟩ <;> simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 924,
"column": 28
} | {
"line": 924,
"column": 41
} | {
"line": 924,
"column": 42
} | [
{
"pp": "m n k : ℕ\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : ℕ\nmt : m < t\nnw : n ≤ w\nkw : k ≤ w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : ℕ\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k ≡ yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) ... | [
"m n k : ℕ\nhk0 : 0 < k\nhn0 : 0 < n\nw t z : ℕ\nmt : m < t\nnw : n ≤ w\nkw : k ≤ w\nhw0 : 0 < w\nhw1 : 1 < w + 1\nj : ℕ\nyj : w * z = yn hw1 j\na1 : 1 < xn hw1 j\ntm : xn a1 k ≡ yn a1 k * (xn hw1 j - n) + m [MOD t]\nta : 2 * xn hw1 j * n = t + (n * n + 1)\nzp : xn hw1 j * xn hw1 j - ((w + 1) * (w + 1) - 1) * (w * ... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 63
} | {
"line": 168,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhsum : Summable f\nhf₀ : f 0 = 0\nε : ℝ\nεpos : 0 < ε\nN₀ : ℕ\nhN₀ : ∀ (s : Finset ℕ), N₀.primesBelow ⊆ s → ‖∑' (m : ℕ), f m - ∑' (m : ↑(factoredNumbers s)), f ↑m‖ < ε\nN : ℕ\nhN : N ≥ N₀\np : ℕ\nhp : p ∈ N₀.primesBelow\n⊢ p ∈... | [] | exact mem_range.mpr <| (lt_of_mem_primesBelow hp).trans_le hN | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 153,
"column": 6
} | {
"line": 154,
"column": 96
} | {
"line": 155,
"column": 6
} | [
{
"pp": "case refine_2.inr.inr.inl\nz τ : ℂ\nhτ✝ : τ.im ≤ 0\nhτ : τ.im = 0\nh : Summable fun x ↦ rexp (-(2 * π * ↑x * z.im))\nhz : z.im = 0\n⊢ False",
"ppTerm": "?refine_2.inr.inr.inl",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"False",
"Real.partialOrder",
... | [
"case refine_2.inr.inr.inr\nz τ : ℂ\nhτ✝ : τ.im ≤ 0\nhτ : τ.im = 0\nh : Summable fun x ↦ rexp (-(2 * π * ↑x * z.im))\nhz : 0 < z.im\n⊢ False"
] | · revert h
simpa only [hz, mul_zero, neg_zero, Real.exp_zero, summable_const_iff] using one_ne_zero | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 56
} | {
"line": 293,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nr x : ℝ\nhx : 1 < x\n⊢ P.f x - P.f₀ = P.f_modif x - 0",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instPow",
"False",
"Real",
"Set.Ioi",
... | [] | simp [f_modif, mem_Ioi.mpr hx, notMem_Ioo_of_ge hx.le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 325,
"column": 6
} | {
"line": 325,
"column": 60
} | {
"line": 325,
"column": 60
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nx : ℝ\nhx : 0 < x\nhx' : x < 1\n⊢ 0 + (P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀) - P.f x + P.f₀ =\n P.f₀ - (P.ε * ↑(x ^ (-P.k))) • P.g₀ + {1}.indicator (fun x ↦ P.f₀ - P.f 1) x",
"ppTerm": "?inl",
"... | [
"case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nx : ℝ\nhx : 0 < x\nhx' : x < 1\n⊢ 0 + (P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀) - P.f x + P.f₀ = P.f₀ - (P.ε * ↑(x ^ (-P.k))) • P.g₀ + 0"
] | indicator_of_notMem (mem_singleton_iff.not.mpr hx'.ne) | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 271,
"column": 4
} | {
"line": 271,
"column": 66
} | {
"line": 272,
"column": 2
} | [
{
"pp": "a : UnitAddCircle\nr v : ℝ\nhv : 0 < v\nhv' : (fun x ↦ ‖↑(evenKernel a x) - if a = 0 then 1 else 0‖) =O[atTop] fun x ↦ rexp (-v * x)\n⊢ (fun x ↦ ‖(ofReal ∘ evenKernel a) x - if a = 0 then 1 else 0‖) =O[atTop] fun x ↦ x ^ r",
"ppTerm": "?m.185",
"assigned": true,
"usedConstants": [
"No... | [] | exact hv'.trans (isLittleO_exp_neg_mul_rpow_atTop hv _).isBigO | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 361,
"column": 4
} | {
"line": 363,
"column": 59
} | {
"line": 364,
"column": 2
} | [
{
"pp": "case hg\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nP : WeakFEPair E\ninst✝ : CompleteSpace E\ns : ℂ\nhs : P.k < s.re\nh_re1 : -1 < (s - 1).re\nh_re2 : -1 < (s - ↑P.k - 1).re\n⊢ Integrable (fun x ↦ P.ε • ↑x ^ (s - ↑P.k - 1) • P.g₀) (volume.restrict (Ioc 0 1))",
"ppTerm":... | [] | · refine (Integrable.smul_const ?_ _).smul _
rw [← IntegrableOn, ← intervalIntegrable_iff_integrableOn_Ioc_of_le zero_le_one]
exact intervalIntegral.intervalIntegrable_cpow' h_re2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 545,
"column": 4
} | {
"line": 545,
"column": 27
} | {
"line": 546,
"column": 4
} | [
{
"pp": "case refine_1\na : ℝ\ns : ℂ\nhs : 1 < s.re\nhF :\n ∀ (t : ℝ),\n 0 < t →\n HasSum (fun n ↦ if ↑n + a = 0 then 0 else 1 / 2 * ↑(rexp (-π * (↑n + a) ^ 2 * t)))\n ((↑(evenKernel (↑a) t) - ↑(if ↑a = 0 then 1 else 0)) / 2)\n⊢ Summable fun i ↦ ‖1 / 2‖ * (1 / |↑i + a| ^ s.re)",
"ppTerm": "?... | [
"case refine_1\na : ℝ\ns : ℂ\nhs : 1 < s.re\nhF :\n ∀ (t : ℝ),\n 0 < t →\n HasSum (fun n ↦ if ↑n + a = 0 then 0 else 1 / 2 * ↑(rexp (-π * (↑n + a) ^ 2 * t)))\n ((↑(evenKernel (↑a) t) - ↑(if ↑a = 0 then 1 else 0)) / 2)\n⊢ Summable fun i ↦ 1 / |↑i + a| ^ s.re"
] | apply Summable.mul_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 709,
"column": 36
} | {
"line": 713,
"column": 80
} | {
"line": 715,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\nn : ℕ\n⊢ cosZeta a (-2 * (↑n + 1)) = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"HurwitzZeta.cosZeta.eq_1",
... | [] | by
have : (-2 : ℂ) * (n + 1) ≠ 0 :=
mul_ne_zero (neg_ne_zero.mpr two_ne_zero) (Nat.cast_add_one_ne_zero n)
rw [cosZeta, Function.update_of_ne this,
Gammaℝ_eq_zero_iff.mpr ⟨n + 1, by rw [neg_mul, Nat.cast_add_one]⟩, div_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.FLT.Basic | {
"line": 207,
"column": 6
} | {
"line": 207,
"column": 14
} | {
"line": 208,
"column": 6
} | [
{
"pp": "case pos\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": "?pos✝",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"Distrib.toAdd",
... | [
"case pos\na b c : ℤ\nha : IsUnit a\nhb : IsUnit b\nhc : IsUnit c\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ 0 ↔ FermatLastTheoremFor 0\n⊢ a ^ 0 + b ^ 0 ≠ c ^ 0"
] | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 445,
"column": 4
} | {
"line": 445,
"column": 27
} | {
"line": 446,
"column": 4
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nr : ℤ → ℝ := fun n ↦ ↑n + a\nc : ℤ → ℂ := fun n ↦ 1 / 2\nhF : ∀ (t : ℝ), 0 < t → HasSum (fun n ↦ c n * ↑(r n) * ↑(rexp (-π * r n ^ 2 * t))) (↑(oddKernel (↑a) t) / 2)\n⊢ Summable fun i ↦ ‖1 / 2‖ * (1 / |r i| ^ s.re)",
"ppTerm": "?m.184",
"assigned": true,
"usedCo... | [
"a : ℝ\ns : ℂ\nhs : 1 < s.re\nr : ℤ → ℝ := ⋯\nc : ℤ → ℂ := ⋯\nhF : ∀ (t : ℝ), 0 < t → HasSum (fun n ↦ c n * ↑(r n) * ↑(rexp (-π * r n ^ 2 * t))) (↑(oddKernel (↑a) t) / 2)\n⊢ Summable fun i ↦ 1 / |r i| ^ s.re"
] | apply Summable.mul_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 497,
"column": 89
} | {
"line": 501,
"column": 71
} | {
"line": 503,
"column": 0
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\n⊢ HasSum (fun n ↦ (↑(SignType.sign (↑n + a)) / ↑|↑n + a| ^ s - ↑(SignType.sign (↑n + 1 - a)) / ↑|↑n + 1 - a| ^ s) / 2)\n (hurwitzZetaOdd (↑a) s)",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"HurwitzZeta.hasSum_int_hurwitzZetaOdd",
"Ha... | [] | by
refine (hasSum_int_hurwitzZetaOdd a hs).nat_add_neg_add_one.congr_fun fun n ↦ ?_
rw [Int.cast_neg, Int.cast_add, Int.cast_one, sub_div, sub_eq_add_neg, Int.cast_natCast]
have : -(n + 1) + a = -(n + 1 - a) := by ring_nf
rw [this, Left.sign_neg, abs_neg, SignType.coe_neg, neg_div, neg_div] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Radical.Basic | {
"line": 216,
"column": 2
} | {
"line": 217,
"column": 35
} | {
"line": 218,
"column": 2
} | [
{
"pp": "case mp\nM : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\nha : Irreducible a\nhb : b ≠ 0\n⊢ a ∣ radical b → a ∣ b",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"semigroupDvd",
"S... | [
"case mpr\nM : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\nha : Irreducible a\nhb : b ≠ 0\n⊢ a ∣ b → a ∣ radical b"
] | · intro ha
exact ha.trans radical_dvd_self | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Radical.Basic | {
"line": 251,
"column": 15
} | {
"line": 251,
"column": 45
} | {
"line": 251,
"column": 45
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\nh : radical a = radical b\n⊢ primeFactors a = primeFactors b",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",... | [
"M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\nh : radical a = radical b\n⊢ primeFactors (radical b) = primeFactors b"
] | rw [← primeFactors_radical, h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Radical.Basic | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 73
} | {
"line": 288,
"column": 2
} | [
{
"pp": "case inr\nM : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\nh : normalizedFactors a ≤ normalizedFactors b\nhb₀ : b ≠ 0\nha₀ : a ≠ 0\n⊢ radical a ∣ radical b",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [
"case inr\nM : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na b : M\nh : normalizedFactors a ≤ normalizedFactors b\nhb₀ : b ≠ 0\nha₀ : a ≠ 0\n⊢ normalizedFactors a ⊆ normalizedFactors b"
] | rw [radical_dvd_radical_iff_normalizedFactors_subset_normalizedFactors] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.FLT.Four | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 38
} | {
"line": 240,
"column": 2
} | [
{
"pp": "a b c : ℤ\nh : Minimal a b c\nha2 : a % 2 = 1\nhc : 0 < c\nht : PythagoreanTriple (a ^ 2) (b ^ 2) c\nh2 : (a ^ 2).gcd (b ^ 2) = 1\nha22 : a ^ 2 % 2 = 1\nm n : ℤ\nht1 : a ^ 2 = m ^ 2 - n ^ 2\nht2 : b ^ 2 = 2 * m * n\nht3 : c = m ^ 2 + n ^ 2\nht4 : m.gcd n = 1\nht5 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n... | [
"a b c : ℤ\nh : Minimal a b c\nha2 : a % 2 = 1\nhc : 0 < c\nht : PythagoreanTriple (a ^ 2) (b ^ 2) c\nh2 : (a ^ 2).gcd (b ^ 2) = 1\nha22 : a ^ 2 % 2 = 1\nm n : ℤ\nht1 : a ^ 2 = m ^ 2 - n ^ 2\nht2 : b ^ 2 = 2 * m * n\nht3 : c = m ^ 2 + n ^ 2\nht4 : m.gcd n = 1\nht5 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\nht... | have hk2 : s ^ 2 = k ^ 4 := by grind | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.FLT.MasonStothers | {
"line": 94,
"column": 4
} | {
"line": 101,
"column": 42
} | {
"line": 102,
"column": 0
} | [
{
"pp": "case neg\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\na b c : k[X]\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhab : IsCoprime a b\nhsum : a + b + c = 0\nw : k[X] := a.wronskian b\nwab : w = a.wronskian b\nhbc : IsCoprime b c\nhsum' : b + c + a = 0\nhca : IsCoprime c a\nwbc : w = b.wronskian c\nwca... | [] | left
-- use `abc_subcall` three times, using the symmetry in `a, b, c`
refine ⟨?_, ?_, ?_⟩
· rw [mul_rotate] at abc_dr_dvd_w ⊢
apply abc_subcall wbc <;> assumption
· rw [← mul_rotate] at abc_dr_dvd_w ⊢
apply abc_subcall wca <;> assumption
· apply abc_subcall wab <;> assumption | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FLT.MasonStothers | {
"line": 94,
"column": 4
} | {
"line": 101,
"column": 42
} | {
"line": 102,
"column": 0
} | [
{
"pp": "case neg\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\na b c : k[X]\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhab : IsCoprime a b\nhsum : a + b + c = 0\nw : k[X] := a.wronskian b\nwab : w = a.wronskian b\nhbc : IsCoprime b c\nhsum' : b + c + a = 0\nhca : IsCoprime c a\nwbc : w = b.wronskian c\nwca... | [] | left
-- use `abc_subcall` three times, using the symmetry in `a, b, c`
refine ⟨?_, ?_, ?_⟩
· rw [mul_rotate] at abc_dr_dvd_w ⊢
apply abc_subcall wbc <;> assumption
· rw [← mul_rotate] at abc_dr_dvd_w ⊢
apply abc_subcall wca <;> assumption
· apply abc_subcall wab <;> assumption | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 30
} | {
"line": 87,
"column": 2
} | [
{
"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 ... | [
"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 v * b ^ q + ... | have hCu := C_ne_zero.mpr hu | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.FractionalIdeal.Norm | {
"line": 74,
"column": 43
} | {
"line": 74,
"column": 56
} | {
"line": 74,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDedekindDomain R\ninst✝⁴ : Free ℤ R\ninst✝³ : Module.Finite ℤ R\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nI J : FractionalIdeal R⁰ K\n⊢ ↑(Ideal.absNorm I.num * Ideal.absNorm J.num) / ↑|(Algebra.norm ℤ) ↑I.den * (A... | [
"R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDedekindDomain R\ninst✝⁴ : Free ℤ R\ninst✝³ : Module.Finite ℤ R\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nI J : FractionalIdeal R⁰ K\n⊢ ↑(Ideal.absNorm I.num) * ↑(Ideal.absNorm J.num) / ↑|(Algebra.norm ℤ) ↑I.den * (Algebra.no... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 48
} | {
"line": 157,
"column": 2
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\np q r : ℕ\na b c : k[X]\nu v w : k\nheq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0\nhp : p ≠ 0\nhq : q ≠ 0\nhr : r ≠ 0\nhineq : q * r + r * p + p * q ≤ p * q * r\nchp : ↑p ≠ 0\nchq : ↑q ≠ 0\nchr : ↑r ≠ 0\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhab : IsCoprime a b\nhu : u ... | [
"case inl\nk : Type u_1\ninst✝ : Field k\np q r : ℕ\na b c : k[X]\nu v w : k\nheq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0\nhp : p ≠ 0\nhq : q ≠ 0\nhr : r ≠ 0\nhineq : q * r + r * p + p * q ≤ p * q * r\nchp : ↑p ≠ 0\nchq : ↑q ≠ 0\nchr : ↑r ≠ 0\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nhab : IsCoprime a b\nhu : u ≠ ... | rcases eq_or_ne (ringChar k) 0 with ch0 | chn0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Instances.Complex | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 23
} | {
"line": 45,
"column": 2
} | [
{
"pp": "K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal ⊆ closure (range (ofReal ∘ Rat.cast))",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Set.range_comp",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"Real.instRatCast",
... | [
"K : Subfield ℂ\nhc : IsClosed ↑K\n⊢ range ofReal ⊆ closure (ofReal '' range Rat.cast)"
] | nth_rw 1 [range_comp] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 326,
"column": 43
} | {
"line": 326,
"column": 67
} | {
"line": 326,
"column": 68
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ∑ w, w.mult = #Finset.univ",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"Finset.card_eq_sum_ones",
"congrArg",
"NumberField.InfinitePlace.mult",
"RingHom",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ∑ w, w.mult = ∑ x, 1"
] | Finset.card_eq_sum_ones, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 617,
"column": 4
} | {
"line": 617,
"column": 15
} | {
"line": 618,
"column": 4
} | [
{
"pp": "case mp\nx y z : ℤ\nh : PythagoreanTriple x y z\nk m n : ℤ\nH : (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨ x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧ m.gcd n = 1\n⊢ ∃ k m n,\n (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨ x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧\n (z = k ... | [
"case h\nx y z : ℤ\nh : PythagoreanTriple x y z\nk m n : ℤ\nH : (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨ x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧ m.gcd n = 1\n⊢ (x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨ x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧\n (z = k * (m ^ 2 + n ^ 2) ∨ z = -k * ... | use k, m, n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 256,
"column": 31
} | {
"line": 256,
"column": 63
} | {
"line": 256,
"column": 63
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nhn : 3 ≤ n\nchn : ↑n ≠ 0\na b c a' b' : k[X]\nd : k[X] := gcd a b\nhb : d ≠ 0 ∧ b' ≠ 0\nha : d ≠ 0 ∧ a' ≠ 0\neq_a : a = a' * d\neq_b : b = b' * d\nhd : d ≠ 0\nc' : k[X]\nheq : (a' ^ n + b' ^ n) * d ^ n = c' ^ n * d ^ n\nhc : d ≠ 0 ∧ c' ≠ 0\neq_c : c = c' * d\n⊢ IsU... | [
"k : Type u_1\ninst✝ : Field k\nn : ℕ\nhn : 3 ≤ n\nchn : ↑n ≠ 0\na b c a' b' : k[X]\nd : k[X] := gcd a b\nhb : d ≠ 0 ∧ b' ≠ 0\nha : d ≠ 0 ∧ a' ≠ 0\neq_a : a = a' * d\neq_b : b = b' * d\nhd : d ≠ 0\nc' : k[X]\nheq : a' ^ n + b' ^ n = c' ^ n\nhc : d ≠ 0 ∧ c' ≠ 0\neq_c : c = c' * d\n⊢ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'... | mul_left_inj' (pow_ne_zero n hd) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 447,
"column": 22
} | {
"line": 447,
"column": 46
} | {
"line": 447,
"column": 47
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : ∀ (w : { w // w.IsComplex }), #{φ | mkComplex φ = w} = 2\n⊢ #Finset.univ = 2 * nrComplexPlaces K",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"HMul.hMul",
"Finset... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : ∀ (w : { w // w.IsComplex }), #{φ | mkComplex φ = w} = 2\n⊢ ∑ x, 1 = 2 * nrComplexPlaces K"
] | Finset.card_eq_sum_ones, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 448,
"column": 73
} | {
"line": 448,
"column": 97
} | {
"line": 449,
"column": 6
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : ∀ (w : { w // w.IsComplex }), #{φ | mkComplex φ = w} = 2\n⊢ ∑ x, 2 = 2 * #Finset.univ",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Finset.univ",
"Finset.card_eq_sum_one... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : ∀ (w : { w // w.IsComplex }), #{φ | mkComplex φ = w} = 2\n⊢ ∑ x, 2 = 2 * ∑ x, 1"
] | Finset.card_eq_sum_ones, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 241,
"column": 2
} | {
"line": 270,
"column": 38
} | {
"line": 271,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nhn : 3 ≤ n\nchn : ↑n ≠ 0\n⊢ FermatLastTheoremWith' k[X] n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.C"... | [] | classical
rw [FermatLastTheoremWith']
intro a b c ha hb hc heq
obtain ⟨a', eq_a⟩ := gcd_dvd_left a b
obtain ⟨b', eq_b⟩ := gcd_dvd_right a b
set d := gcd a b
have hd : d ≠ 0 := gcd_ne_zero_of_left ha
rw [eq_a, eq_b, mul_pow, mul_pow, ← mul_add] at heq
obtain ⟨c', eq_c⟩ : ∃ c', c = d * c' :=
(IsIntegr... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 241,
"column": 2
} | {
"line": 270,
"column": 38
} | {
"line": 271,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nhn : 3 ≤ n\nchn : ↑n ≠ 0\n⊢ FermatLastTheoremWith' k[X] n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.C"... | [] | classical
rw [FermatLastTheoremWith']
intro a b c ha hb hc heq
obtain ⟨a', eq_a⟩ := gcd_dvd_left a b
obtain ⟨b', eq_b⟩ := gcd_dvd_right a b
set d := gcd a b
have hd : d ≠ 0 := gcd_ne_zero_of_left ha
rw [eq_a, eq_b, mul_pow, mul_pow, ← mul_add] at heq
obtain ⟨c', eq_c⟩ : ∃ c', c = d * c' :=
(IsIntegr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 241,
"column": 2
} | {
"line": 270,
"column": 38
} | {
"line": 271,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nhn : 3 ≤ n\nchn : ↑n ≠ 0\n⊢ FermatLastTheoremWith' k[X] n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.C"... | [] | classical
rw [FermatLastTheoremWith']
intro a b c ha hb hc heq
obtain ⟨a', eq_a⟩ := gcd_dvd_left a b
obtain ⟨b', eq_b⟩ := gcd_dvd_right a b
set d := gcd a b
have hd : d ≠ 0 := gcd_ne_zero_of_left ha
rw [eq_a, eq_b, mul_pow, mul_pow, ← mul_add] at heq
obtain ⟨c', eq_c⟩ : ∃ c', c = d * c' :=
(IsIntegr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | {
"line": 184,
"column": 4
} | {
"line": 185,
"column": 60
} | {
"line": 187,
"column": 0
} | [
{
"pp": "case inr\nF : Type u_1\ninst✝⁴ : Field F\nK : Type u_2\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : Algebra.IsAlgebraic ℚ K\nι : Type u_3\nk : ι → Subfield K\ninst✝ : ∀ (i : ι), IsTotallyReal ↥(k i)\ni : ι\n⊢ IsTotallyReal ↥(⨆ i, k i)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants... | [] | rw [isTotallyReal_iff_le_maximalRealSubfield, iSup_le_iff]
exact fun i ↦ IsTotallyReal.le_maximalRealSubfield (k i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | {
"line": 184,
"column": 4
} | {
"line": 185,
"column": 60
} | {
"line": 187,
"column": 0
} | [
{
"pp": "case inr\nF : Type u_1\ninst✝⁴ : Field F\nK : Type u_2\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : Algebra.IsAlgebraic ℚ K\nι : Type u_3\nk : ι → Subfield K\ninst✝ : ∀ (i : ι), IsTotallyReal ↥(k i)\ni : ι\n⊢ IsTotallyReal ↥(⨆ i, k i)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants... | [] | rw [isTotallyReal_iff_le_maximalRealSubfield, iSup_le_iff]
exact fun i ↦ IsTotallyReal.le_maximalRealSubfield (k i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | {
"line": 184,
"column": 2
} | {
"line": 185,
"column": 60
} | {
"line": 187,
"column": 0
} | [
{
"pp": "case inr\nF : Type u_1\ninst✝⁴ : Field F\nK : Type u_2\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : Algebra.IsAlgebraic ℚ K\nι : Type u_3\nk : ι → Subfield K\ninst✝ : ∀ (i : ι), IsTotallyReal ↥(k i)\ni : ι\n⊢ IsTotallyReal ↥(⨆ i, k i)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants... | [] | · rw [isTotallyReal_iff_le_maximalRealSubfield, iSup_le_iff]
exact fun i ↦ IsTotallyReal.le_maximalRealSubfield (k i) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 44
} | {
"line": 77,
"column": 2
} | [
{
"pp": "k : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nhφ : w.IsReal\n⊢ ComplexEmbedding.IsReal (w.embedding.comp f)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"NumberField.ComplexEmbedding.IsReal",
"congrArg",
"Numb... | [
"k : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nhφ : ComplexEmbedding.IsReal w.embedding\n⊢ ComplexEmbedding.IsReal (w.embedding.comp f)"
] | rw [← mk_embedding w, isReal_mk_iff] at hφ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 296,
"column": 11
} | {
"line": 296,
"column": 26
} | {
"line": 296,
"column": 27
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nw : InfinitePlace K\nh : IsUnramified k w\nhw : ComplexEmbedding.IsReal ((conjugate w.embedding).comp (algebraMap k K))\n⊢ ComplexEmbedding.IsReal (conjugate w.embedding)",
"ppTerm": "?m.30",
"assigned": true,
... | [
"k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nw : InfinitePlace K\nh : IsUnramified k w\nhw : ComplexEmbedding.IsReal (conjugate (w.embedding.comp (algebraMap k K)))\n⊢ ComplexEmbedding.IsReal (conjugate w.embedding)"
] | conjugate_comp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 505,
"column": 4
} | {
"line": 508,
"column": 48
} | {
"line": 509,
"column": 4
} | [
{
"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⊢ #({a ∉ {w | IsUnramified k w} | a.comap (algebraMap k K) = (f... | [
"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 = Nat.card Gal(K/k) / 2"
] | · congr; ext w'
rw [mem_filter, compl_filter, mem_filter_univ, @eq_comm _ (comap w' _), Set.mem_toFinset,
mem_orbit_iff, and_iff_right_iff_imp]
intro e; rwa [← isUnramifiedIn_comap, ← e] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 299,
"column": 6
} | {
"line": 299,
"column": 85
} | {
"line": 300,
"column": 6
} | [
{
"pp": "case pos.a\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nh_zero : (commMap K) x = 0\nh_mem : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))\nφ : K →+* ℂ\nhφ : ComplexEmbedding.IsReal φ\n⊢ (x φ).re = Complex.re 0",
"ppTerm": "?pos.a✝",
"assigned": true,
... | [
"case pos.a\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nh_zero : (commMap K) x = 0\nh_mem : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))\nφ : K →+* ℂ\nhφ : ComplexEmbedding.IsReal φ\n⊢ ((commMap K) x).1 ⟨InfinitePlace.mk φ, ⋯⟩ = Complex.re 0"
] | rw [← embedding_mk_eq_of_isReal hφ, ← commMap_apply_of_isReal K x ⟨φ, hφ, rfl⟩] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 187,
"column": 57
} | {
"line": 187,
"column": 70
} | {
"line": 187,
"column": 71
} | [
{
"pp": "case e_a.e_a.e_a.e_a.e_a\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nB : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))\nh_le : minkowskiBound K I ≤ volume (convexBodySum K B)\nx✝ : K\n⊢ ↑(nrRealPlaces K) + ↑(2 * nrComp... | [
"case e_a.e_a.e_a.e_a.e_a\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nB : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))\nh_le : minkowskiBound K I ≤ volume (convexBodySum K B)\nx✝ : K\n⊢ ↑(nrRealPlaces K) + ↑2 * ↑(nrComplexPlaces K... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 560,
"column": 6
} | {
"line": 568,
"column": 21
} | {
"line": 569,
"column": 4
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw₀ : InfinitePlace K\nhw₀ : w₀.IsComplex\nB : ℝ≥0\nhB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B\nthis : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)\na : 𝓞 K\nh_nz : a ≠... | [] | rw [if_pos rfl] at h_le₀
dsimp only at h_le₀
rw [h_eq, ← norm_embedding_eq, Real.lt_sqrt (norm_nonneg _), ← Complex.re_add_im
(embedding w₀ _), Complex.norm_add_mul_I, Real.sq_sqrt (by positivity)]
refine add_lt_add ?_ ?_
· rw [← sq_abs, sq_lt_one_iff₀ (abs_nonneg _)]
exact h_le₀... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 560,
"column": 6
} | {
"line": 568,
"column": 21
} | {
"line": 569,
"column": 4
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw₀ : InfinitePlace K\nhw₀ : w₀.IsComplex\nB : ℝ≥0\nhB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B\nthis : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)\na : 𝓞 K\nh_nz : a ≠... | [] | rw [if_pos rfl] at h_le₀
dsimp only at h_le₀
rw [h_eq, ← norm_embedding_eq, Real.lt_sqrt (norm_nonneg _), ← Complex.re_add_im
(embedding w₀ _), Complex.norm_add_mul_I, Real.sq_sqrt (by positivity)]
refine add_lt_add ?_ ?_
· rw [← sq_abs, sq_lt_one_iff₀ (abs_nonneg _)]
exact h_le₀... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 603,
"column": 2
} | {
"line": 603,
"column": 55
} | {
"line": 604,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nB : ℝ\nh : minkowskiBound K I ≤ volume (convexBodySum K B)\nhB : 0 ≤ B\nh1 : 0 < (↑(finrank ℚ K))⁻¹\nh2 : 0 ≤ B / ↑(finrank ℚ K)\nh_fund :\n IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasi... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nB : ℝ\nh : minkowskiBound K I ≤ volume (convexBodySum K B)\nhB : 0 ≤ B\nh1 : 0 < (↑(finrank ℚ K))⁻¹\nh2 : 0 ≤ B / ↑(finrank ℚ K)\nh_fund :\n IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toA... | refine le_trans ?_ ((convexBodySum_mem K B).mp h_mem) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 26
} | {
"line": 100,
"column": 4
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁶ : CommRing A\ninst✝¹⁵ : CommRing B\ninst✝¹⁴ : Algebra A B\np : Ideal A\nG : Type u_3\ninst✝¹³ : Group G\ninst✝¹² : MulSemiringAction G B\ninst✝¹¹ : SMulCommClass G A B\nK : Type u_4\nL : Type u_5\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra A K\ninst✝⁷ : Is... | [
"A : Type u_1\nB : Type u_2\ninst✝¹⁶ : CommRing A\ninst✝¹⁵ : CommRing B\ninst✝¹⁴ : Algebra A B\np : Ideal A\nG : Type u_3\ninst✝¹³ : Group G\ninst✝¹² : MulSemiringAction G B\ninst✝¹¹ : SMulCommClass G A B\nK : Type u_4\nL : Type u_5\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra A K\ninst✝⁷ : IsFractionRing... | change map _ Q.1 = Q.1 | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 196,
"column": 2
} | {
"line": 197,
"column": 63
} | {
"line": 199,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁹ : CommRing A\ninst✝¹⁸ : CommRing B\ninst✝¹⁷ : Algebra A B\np : Ideal A\nP : Ideal B\nhPp : P.IsPrime\nhp : P.LiesOver p\nG : Type u_3\ninst✝¹⁶ : Group G\ninst✝¹⁵ : Finite G\ninst✝¹⁴ : MulSemiringAction G B\ninst✝¹³ : IsGaloisGroup G A B\nC : Type u_4\ninst✝¹² : CommR... | [] | rw [inertiaDegIn_eq_inertiaDeg p P G, inertiaDegIn_eq_inertiaDeg p Q GAC,
inertiaDegIn_eq_inertiaDeg P Q GBC, ← inertiaDeg_tower P Q] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1037,
"column": 6
} | {
"line": 1038,
"column": 16
} | {
"line": 1039,
"column": 4
} | [
{
"pp": "case refine_1.inl\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w ↦ ‖x.1 w‖, x.2) ∈ A\ns : Set { w // w.IsReal }\nx : mixedSpace K\nhx : x ∈ A\nright✝ : x ∈ {x | ∀ (w : { w // w.IsReal }), 0 < x.1 w}\n⊢ (negAt s) x ∈ A",
"ppTerm": "?refine_1.inl",
... | [] | simp_rw +singlePass [hA, negAt_apply_norm_isReal, negAt_apply_snd]
rwa [← hA] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1037,
"column": 6
} | {
"line": 1038,
"column": 16
} | {
"line": 1039,
"column": 4
} | [
{
"pp": "case refine_1.inl\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w ↦ ‖x.1 w‖, x.2) ∈ A\ns : Set { w // w.IsReal }\nx : mixedSpace K\nhx : x ∈ A\nright✝ : x ∈ {x | ∀ (w : { w // w.IsReal }), 0 < x.1 w}\n⊢ (negAt s) x ∈ A",
"ppTerm": "?refine_1.inl",
... | [] | simp_rw +singlePass [hA, negAt_apply_norm_isReal, negAt_apply_snd]
rwa [← hA] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 32
} | {
"line": 68,
"column": 2
} | [
{
"pp": "case neg.refine_1\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : HasFiniteQuotients R\nI : Ideal R\nhI : ¬I = 0\nx : R\nhx₁ : x ∈ I\nhx₂ : x ≠ 0\nthis : Finite (R ⧸ Ideal.span {x})\n⊢ (Submodule.map (Submodule.mkQ (Ideal.span {x})) I).FG",
"ppTerm": "?neg.refine_1✝",
"assigned": true,
"usedCon... | [] | exact Submodule.FG.of_finite | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 79
} | {
"line": 181,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nA : Type u_1\ninst✝¹¹ : CommRing A\nB : Type u_2\ninst✝¹⁰ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁹ : CommRing K\ninst✝⁸ : Algebra A K\ninst✝⁷ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝⁶ : CommRing L\ninst✝⁵ : Algebra B L\ninst✝⁴ : IsLocali... | [] | exact (mem_extended_iff L hf I _).2 <| Submodule.subset_span ⟨x, hx, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 79
} | {
"line": 181,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nA : Type u_1\ninst✝¹¹ : CommRing A\nB : Type u_2\ninst✝¹⁰ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁹ : CommRing K\ninst✝⁸ : Algebra A K\ninst✝⁷ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝⁶ : CommRing L\ninst✝⁵ : Algebra B L\ninst✝⁴ : IsLocali... | [] | exact (mem_extended_iff L hf I _).2 <| Submodule.subset_span ⟨x, hx, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 79
} | {
"line": 181,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nA : Type u_1\ninst✝¹¹ : CommRing A\nB : Type u_2\ninst✝¹⁰ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁹ : CommRing K\ninst✝⁸ : Algebra A K\ninst✝⁷ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝⁶ : CommRing L\ninst✝⁵ : Algebra B L\ninst✝⁴ : IsLocali... | [] | exact (mem_extended_iff L hf I _).2 <| Submodule.subset_span ⟨x, hx, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LocalRing.Quotient | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 53
} | {
"line": 122,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\nI : Ideal R\ninst✝ : IsArtinianRing (R ⧸ I)\nhI : ¬I = ⊤\nthis✝¹ : Nontrivial (R ⧸ I)\nthis✝ : IsLocalRing (R ⧸ I)\nthis : IsLocalHom (Ideal.Quotient.mk I)\nn : ℕ\nhn : ⊥.jacobson ^ n = 0\nx : R ⧸ I\n⊢ x ∈ Ideal.map (Ideal.Quotient.mk I) p ↔ x ... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\nI : Ideal R\ninst✝ : IsArtinianRing (R ⧸ I)\nhI : ¬I = ⊤\nthis✝¹ : Nontrivial (R ⧸ I)\nthis✝ : IsLocalRing (R ⧸ I)\nthis : IsLocalHom (Ideal.Quotient.mk I)\nn : ℕ\nhn : ⊥.jacobson ^ n = 0\nx : R\n⊢ (Ideal.Quotient.mk I) x ∈ Ideal.map (Ideal.Quotient.mk I) ... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.DedekindDomain.PID | {
"line": 161,
"column": 4
} | {
"line": 164,
"column": 67
} | {
"line": 164,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nh : {I | I.IsMaximal}.Finite\nI : Ideal R\n⊢ Submodule.IsPrincipal I",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"FractionRing.field",
"IsDedekindDomain.toIsDomain",
"Semiring.toModule",
"Or... | [] | obtain rfl | hI := eq_or_ne I ⊥
· exact bot_isPrincipal
apply Ideal.IsPrincipal.of_finite_maximals_of_isUnit h
exact .of_mul_eq_one _ (FractionalIdeal.coe_ideal_mul_inv I hI) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.PID | {
"line": 161,
"column": 4
} | {
"line": 164,
"column": 67
} | {
"line": 164,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nh : {I | I.IsMaximal}.Finite\nI : Ideal R\n⊢ Submodule.IsPrincipal I",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"FractionRing.field",
"IsDedekindDomain.toIsDomain",
"Semiring.toModule",
"Or... | [] | obtain rfl | hI := eq_or_ne I ⊥
· exact bot_isPrincipal
apply Ideal.IsPrincipal.of_finite_maximals_of_isUnit h
exact .of_mul_eq_one _ (FractionalIdeal.coe_ideal_mul_inv I hI) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Int | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 62
} | {
"line": 64,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nI✝ I : Ideal R\n⊢ I.LiesOver (span {↑(absNorm (under ℤ I))})",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"Nat.instMulZeroOneClass",
"RingHom.instRingHomClass",
"Ideal.absNorm",
... | [] | rw [liesOver_iff, under_def, Int.ideal_span_absNorm_eq_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Int | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 62
} | {
"line": 64,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nI✝ I : Ideal R\n⊢ I.LiesOver (span {↑(absNorm (under ℤ I))})",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"Nat.instMulZeroOneClass",
"RingHom.instRingHomClass",
"Ideal.absNorm",
... | [] | rw [liesOver_iff, under_def, Int.ideal_span_absNorm_eq_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Int | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 62
} | {
"line": 64,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nI✝ I : Ideal R\n⊢ I.LiesOver (span {↑(absNorm (under ℤ I))})",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"Nat.instMulZeroOneClass",
"RingHom.instRingHomClass",
"Ideal.absNorm",
... | [] | rw [liesOver_iff, under_def, Int.ideal_span_absNorm_eq_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Different | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 17
} | {
"line": 129,
"column": 18
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : NumberField L\ninst✝ : Algebra K L\n⊢ discr K ^ finrank K L ∣ ↑(Ideal.absNorm (differentIdeal (𝓞 K) (𝓞 L)) * (discr K).natAbs ^ finrank K L)",
"ppTerm": "?m.91",
"assigned": true,
"usedConstan... | [
"K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : NumberField L\ninst✝ : Algebra K L\n⊢ discr K ^ finrank K L ∣ ↑(Ideal.absNorm (differentIdeal (𝓞 K) (𝓞 L))) * ↑((discr K).natAbs ^ finrank K L)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 422,
"column": 4
} | {
"line": 422,
"column": 68
} | {
"line": 423,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDomain R\nS : Type u_3\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : IsIntegrallyClosed R\ninst✝⁸ : IsIntegrallyClosed S\ninst✝⁷ : Algebra R S\ninst✝⁶ : Module.Finite R S\ninst✝⁵ : IsTorsionFree R S\ninst✝⁴ : IsDedekindDomain R\ninst✝³ : IsDedekin... | [
"R : Type u_1\ninst✝¹³ : CommRing R\ninst✝¹² : IsDomain R\nS : Type u_3\ninst✝¹¹ : CommRing S\ninst✝¹⁰ : IsDomain S\ninst✝⁹ : IsIntegrallyClosed R\ninst✝⁸ : IsIntegrallyClosed S\ninst✝⁷ : Algebra R S\ninst✝⁶ : Module.Finite R S\ninst✝⁵ : IsTorsionFree R S\ninst✝⁴ : IsDedekindDomain R\ninst✝³ : IsDedekindDomain S\nP... | obtain ⟨σ, rfl⟩ := Ideal.exists_smul_eq_of_isGaloisGroup p P Q G | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.NumberField.Discriminant.Different | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 70
} | {
"line": 181,
"column": 2
} | [
{
"pp": "K : Type u_1\n𝒪 : Type u_2\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : CommRing 𝒪\ninst✝¹ : Algebra 𝒪 K\ninst✝ : IsIntegralClosure 𝒪 ℤ K\np : ℤ\nhp : Prime p\nthis✝ : IsDomain 𝒪\nthis : IsDedekindDomain 𝒪\n⊢ ¬p ∣ discr K ↔ ∀ (P : Ideal 𝒪) (x : P.IsMaximal), P.LiesOver (Ideal.span {p}) → ... | [
"K : Type u_1\n𝒪 : Type u_2\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : CommRing 𝒪\ninst✝¹ : Algebra 𝒪 K\ninst✝ : IsIntegralClosure 𝒪 ℤ K\np : ℤ\nhp : Prime p\nthis✝¹ : IsDomain 𝒪\nthis✝ : IsDedekindDomain 𝒪\nthis : IsFractionRing 𝒪 K\n⊢ ¬p ∣ discr K ↔ ∀ (P : Ideal 𝒪) (x : P.IsMaximal), P.LiesOver (... | have := IsIntegralClosure.isFractionRing_of_finite_extension ℤ ℚ K 𝒪 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 472,
"column": 8
} | {
"line": 472,
"column": 40
} | {
"line": 472,
"column": 41
} | [
{
"pp": "case neg.refine_3\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_3\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... | relNorm_eq_pow_of_isMaximal Q P, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral | {
"line": 156,
"column": 94
} | {
"line": 163,
"column": 8
} | {
"line": 165,
"column": 2
} | [
{
"pp": "R : Type u\nK : Type v\nL : Type z\np : R\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra R L\ninst✝⁴ : Algebra R K\ninst✝³ : IsScalarTower R K L\ninst✝² : IsDomain R\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\nB : PowerBasis K L\nhp : _r... | [] | by
have hndiv : ¬p ^ 2 ∣ (minpoly R B.gen).coeff 0 := fun h =>
hei.notMem ((span_singleton_pow p 2).symm ▸ Ideal.mem_span_singleton.2 h)
refine hp.dvd_of_pow_dvd_pow_mul_pow_of_square_not_dvd (n := n) (?_ : _ ∣ _) hndiv
convert! (IsUnit.dvd_mul_right ⟨(-1) ^ (n.succ * n), rfl⟩).mpr this using 1
pu... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 530,
"column": 2
} | {
"line": 538,
"column": 40
} | {
"line": 540,
"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))\nhodd : p ≠ 2\nh : hζ.toInteger - 1 ∣ 2\n⊢ False",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
... | [] | have : NumberField K := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K
replace h : hζ.toInteger - 1 ∣ (2 : ℤ) := by simp [h]
rw [← Ideal.norm_dvd_iff, hζ.norm_toInteger_sub_one_of_prime_ne_two hodd] at h
· refine hodd <| (prime_dvd_prime_iff_eq ?_ ?_).1 ?_
· exact Nat.prime_iff.1 hp.1
· exact Nat.pri... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 530,
"column": 2
} | {
"line": 538,
"column": 40
} | {
"line": 540,
"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))\nhodd : p ≠ 2\nh : hζ.toInteger - 1 ∣ 2\n⊢ False",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
... | [] | have : NumberField K := IsCyclotomicExtension.numberField {p ^ (k + 1)} ℚ K
replace h : hζ.toInteger - 1 ∣ (2 : ℤ) := by simp [h]
rw [← Ideal.norm_dvd_iff, hζ.norm_toInteger_sub_one_of_prime_ne_two hodd] at h
· refine hodd <| (prime_dvd_prime_iff_eq ?_ ?_).1 ?_
· exact Nat.prime_iff.1 hp.1
· exact Nat.pri... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral | {
"line": 321,
"column": 6
} | {
"line": 321,
"column": 23
} | {
"line": 321,
"column": 24
} | [
{
"pp": "case neg.hi.convert_2\nR : Type u\nK : Type v\nL : Type z\np : R\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra R L\ninst✝⁴ : Algebra R K\ninst✝³ : IsScalarTower R K L\ninst✝² : IsDomain R\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\nB : ... | [
"case neg.hi.convert_2\nR : Type u\nK : Type v\nL : Type z\np : R\ninst✝⁹ : CommRing R\ninst✝⁸ : Field K\ninst✝⁷ : Field L\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra R L\ninst✝⁴ : Algebra R K\ninst✝³ : IsScalarTower R K L\ninst✝² : IsDomain R\ninst✝¹ : IsFractionRing R K\ninst✝ : IsIntegrallyClosed R\nB : PowerBasis K... | sum_congr rfl hg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.LinearDisjoint | {
"line": 229,
"column": 8
} | {
"line": 229,
"column": 18
} | {
"line": 229,
"column": 18
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝⁵⁴ : CommRing A\ninst✝⁵³ : Field K\ninst✝⁵² : Algebra A K\ninst✝⁵¹ : IsFractionRing A K\ninst✝⁵⁰ : CommRing B\ninst✝⁴⁹ : Field L\ninst✝⁴⁸ : Algebra B L\ninst✝⁴⁷ : Algebra A L\ninst✝⁴⁶ : Algebra K L\ninst✝⁴⁵ : FiniteDimensional K L\ninst✝⁴⁴ : ... | [
"A : Type u_1\nB : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝⁵⁴ : CommRing A\ninst✝⁵³ : Field K\ninst✝⁵² : Algebra A K\ninst✝⁵¹ : IsFractionRing A K\ninst✝⁵⁰ : CommRing B\ninst✝⁴⁹ : Field L\ninst✝⁴⁸ : Algebra B L\ninst✝⁴⁷ : Algebra A L\ninst✝⁴⁶ : Algebra K L\ninst✝⁴⁵ : FiniteDimensional K L\ninst✝⁴⁴ : IsScalarTowe... | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 597,
"column": 2
} | {
"line": 597,
"column": 77
} | {
"line": 598,
"column": 2
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra A K\ninst✝⁷ : Algebra K L\ninst✝⁶ : Algebra A L\ninst✝⁵ : IsScalarTower A K L\ninst✝⁴ : IsDomain A\ninst✝³ : IsFractionRing A K\ninst✝² : FiniteDimensional K L\ninst✝¹ : Algebra.IsSeparab... | [
"A : Type u_1\nK : Type u_2\nL : Type u\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra A K\ninst✝⁷ : Algebra K L\ninst✝⁶ : Algebra A L\ninst✝⁵ : IsScalarTower A K L\ninst✝⁴ : IsDomain A\ninst✝³ : IsFractionRing A K\ninst✝² : FiniteDimensional K L\ninst✝¹ : Algebra.IsSeparable K L\ninst... | have hnondeg : (traceForm K L).Nondegenerate := traceForm_nondegenerate K L | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 87,
"column": 15
} | {
"line": 87,
"column": 27
} | {
"line": 88,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ (fun w ↦ ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(Additive.toMul 0)))) = 0",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"NumberField.InfinitePlace.instFunLikeReal",
"Units.val",
"Eq.mpr",
... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Fintype | {
"line": 65,
"column": 2
} | {
"line": 66,
"column": 62
} | {
"line": 68,
"column": 0
} | [
{
"pp": "M₀ : Type u_1\ninst✝² : MonoidWithZero M₀\ninst✝¹ : Nontrivial M₀\ninst✝ : Finite M₀\n⊢ Nat.card M₀ˣ < Nat.card M₀",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Fintype.ofFinite",
"congrArg",
"card_units_lt",
"Classical.propDecidable",
"Units",
... | [] | have : Fintype M₀ := Fintype.ofFinite M₀
simpa only [Fintype.card_eq_nat_card] using card_units_lt M₀ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Fintype | {
"line": 65,
"column": 2
} | {
"line": 66,
"column": 62
} | {
"line": 68,
"column": 0
} | [
{
"pp": "M₀ : Type u_1\ninst✝² : MonoidWithZero M₀\ninst✝¹ : Nontrivial M₀\ninst✝ : Finite M₀\n⊢ Nat.card M₀ˣ < Nat.card M₀",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Fintype.ofFinite",
"congrArg",
"card_units_lt",
"Classical.propDecidable",
"Units",
... | [] | have : Fintype M₀ := Fintype.ofFinite M₀
simpa only [Fintype.card_eq_nat_card] using card_units_lt M₀ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 667,
"column": 51
} | {
"line": 667,
"column": 80
} | {
"line": 667,
"column": 80
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : Field K\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra A K\ninst✝¹³ : Algebra B L\ninst✝¹² : Algebra A B\ninst✝¹¹ : Algebra K L\ninst✝¹⁰ : Algebra A L\ninst✝⁹ : IsScalarTower A K L\ninst✝⁸ : IsScalarTower... | [
"A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : Field K\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra A K\ninst✝¹³ : Algebra B L\ninst✝¹² : Algebra A B\ninst✝¹¹ : Algebra K L\ninst✝¹⁰ : Algebra A L\ninst✝⁹ : IsScalarTower A K L\ninst✝⁸ : IsScalarTower A B L\ninst... | ← AlgHom.map_adjoin_singleton | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Three | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 30
} | {
"line": 102,
"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\nhcong : ∃ n, λ ^ 2 ∣ ↑u - ↑n\n⊢ ∃ n, 3 ∣ ↑u - ↑n",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Int.cast",
"Units.val",
... | [
"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\nn : ℤ\nx : 𝓞 K\nhx : ↑u - ↑n = λ ^ 2 * x\n⊢ ∃ n, 3 ∣ ↑u - ↑n"
] | obtain ⟨n, x, hx⟩ := hcong | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.NumberField.Cyclotomic.Three | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 30
} | {
"line": 111,
"column": 4
} | [
{
"pp": "case «3»\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nhcong : ∃ n, 3 ∣ ↑(-⋯.unit) - ↑n\n⊢ ∃ n, ↑3 ∣ hζ.toInteger - ↑n",
"ppTerm": "?«3»",
"assigned": true,
"usedConstants": [
"IsPrimitiveRoot.toInteger... | [
"case «3»\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nn : ℤ\nx : 𝓞 K\nhx : ↑(-⋯.unit) - ↑n = 3 * x\n⊢ ∃ n, ↑3 ∣ hζ.toInteger - ↑n"
] | obtain ⟨n, x, hx⟩ := hcong | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.NumberField.Cyclotomic.Three | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 30
} | {
"line": 120,
"column": 4
} | [
{
"pp": "case «5»\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nhcong : ∃ n, 3 ∣ ↑(-⋯.unit ^ 2) - ↑n\n⊢ ∃ n, ↑3 ∣ ⋯.toInteger - ↑n",
"ppTerm": "?«5»",
"assigned": true,
"usedConstants": [
"IsPrimitiveRoot.toInte... | [
"case «5»\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nn : ℤ\nx : 𝓞 K\nhx : ↑(-⋯.unit ^ 2) - ↑n = 3 * x\n⊢ ∃ n, ↑3 ∣ ⋯.toInteger - ↑n"
] | obtain ⟨n, x, hx⟩ := hcong | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.NumberField.Cyclotomic.Three | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 62
} | {
"line": 206,
"column": 0
} | [
{
"pp": "case inr.inr\nK : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\nx : 𝓞 K\nh : ¬λ ∣ x\nH : λ ∣ x + 1\n⊢ λ ^ 4 ∣ x ^ 3 + 1",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"IsCyclotomicExtension.Ra... | [] | exact lambda_pow_four_dvd_cube_add_one_of_dvd_add_one hζ H | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 394,
"column": 8
} | {
"line": 396,
"column": 37
} | {
"line": 396,
"column": 37
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ torsion K = (AddMonoidHom.toMultiplicativeRight (logEmbedding K)).ker",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Multiplicative.group",
"Eq.mpr",
"MulOne.toOne",
"AddMonoidHom.toMultiplicativeRi... | [] | ext
rw [MonoidHom.mem_ker, AddMonoidHom.toMultiplicativeRight_apply_apply, ofAdd_eq_one,
← logEmbedding_eq_zero_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 394,
"column": 8
} | {
"line": 396,
"column": 37
} | {
"line": 396,
"column": 37
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ torsion K = (AddMonoidHom.toMultiplicativeRight (logEmbedding K)).ker",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Multiplicative.group",
"Eq.mpr",
"MulOne.toOne",
"AddMonoidHom.toMultiplicativeRi... | [] | ext
rw [MonoidHom.mem_ker, AddMonoidHom.toMultiplicativeRight_apply_apply, ofAdd_eq_one,
← logEmbedding_eq_zero_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.Basic | {
"line": 245,
"column": 4
} | {
"line": 245,
"column": 23
} | {
"line": 246,
"column": 2
} | [
{
"pp": "case inl\np : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℤ\nh : 1 = 0\nx y : ZMod p\nhxy : x ^ 2 - ↑a * y ^ 2 = 0\nha : ↑a ≠ 0\nhf : ¬(x = 0 ∧ y = 0)\nhx : ¬x = 0\n⊢ False",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Int.instNeZeroOfNatOfNat",
"AddGroupWithOne.toAddMonoidWi... | [] | exact one_ne_zero h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.LegendreSymbol.Basic | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 23
} | {
"line": 249,
"column": 0
} | [
{
"pp": "case inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℤ\nh : 1 = 0\nx y : ZMod p\nhxy : x ^ 2 - ↑a * y ^ 2 = 0\nha : ↑a ≠ 0\nhf : ¬(x = 0 ∧ y = 0)\nhy : ¬y = 0\n⊢ False",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Int.instNeZeroOfNatOfNat",
"AddGroupWithOne.toAddMonoidWi... | [] | exact one_ne_zero h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.FLT.Three | {
"line": 168,
"column": 64
} | {
"line": 178,
"column": 26
} | {
"line": 180,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ 3\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {3} ℚ K\n⊢ FermatLastTheoremForThreeGen hζ → FermatLastTheoremFor 3",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"I... | [] | by
intro H
refine fermatLastTheoremThree_of_three_dvd_only_c (fun a b c hc ha hb ⟨x, hx⟩ hcoprime h ↦ ?_)
refine H a b c 1 (by simp [hc]) (fun hdvd ↦ ha ?_) (fun hdvd ↦ hb ?_) ?_ ?_ ?_
· rwa [← Ideal.norm_dvd_iff (hζ.prime_norm_toInteger_sub_one_of_prime_ne_two' (by decide)),
hζ.norm_toInteger_sub_one_of_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.FermatPsp | {
"line": 231,
"column": 42
} | {
"line": 231,
"column": 71
} | {
"line": 231,
"column": 71
} | [
{
"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 ^... | [
"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 ^ (p - 1)\np_... | Nat.even_pow' p_prime.ne_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
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