module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.NumberField.FractionalIdeal | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 13
} | {
"line": 114,
"column": 14
} | [
{
"pp": "case e_6\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\ne : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ Free.ChooseBasisIndex ℤ ↥↑↑I\nx✝ : Free.ChooseBasisIndex ℤ (𝓞 K)\n⊢ ((basisOfFractionalIdeal K I).reindex e.symm) x✝ = (Subtype.val ∘ ⇑((fractionalIdealBasis K ↑I).re... | [
"case e_6\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\ne : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ Free.ChooseBasisIndex ℤ ↥↑↑I\nx✝ : Free.ChooseBasisIndex ℤ (𝓞 K)\n⊢ (basisOfFractionalIdeal K I) (e x✝) = ↑((fractionalIdealBasis K ↑I) (e x✝))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 514,
"column": 45
} | {
"line": 514,
"column": 89
} | {
"line": 514,
"column": 90
} | [
{
"pp": "x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhz : z ≤ 0\n⊢ PythagoreanTriple x y (-z)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
... | [
"x y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhz : z ≤ 0\n⊢ x * x + y * y = z * z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 543,
"column": 6
} | {
"line": 543,
"column": 17
} | {
"line": 543,
"column": 18
} | [
{
"pp": "case h.inl\nz m n : ℤ\nh : PythagoreanTriple (m ^ 2 - n ^ 2) (2 * m * n) z ∧ (m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\nthis : z ^ 2 = (m ^ 2 + n ^ 2) ^ 2\n⊢ z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)",
"ppTerm": "?h.inl",
"assigned"... | [
"case h.inl\nz m n : ℤ\nh : PythagoreanTriple (m ^ 2 - n ^ 2) (2 * m * n) z ∧ (m ^ 2 - n ^ 2).gcd (2 * m * n) = 1\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\nthis : z ^ 2 = (m ^ 2 + n ^ 2) ^ 2\n⊢ z = m ^ 2 + n ^ 2 ∨ z = -n ^ 2 + -m ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 548,
"column": 6
} | {
"line": 548,
"column": 17
} | {
"line": 548,
"column": 18
} | [
{
"pp": "case h.inr\nz m n : ℤ\nh : PythagoreanTriple (2 * m * n) (m ^ 2 - n ^ 2) z ∧ (2 * m * n).gcd (m ^ 2 - n ^ 2) = 1\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\nthis : z ^ 2 = (m ^ 2 + n ^ 2) ^ 2\n⊢ z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)",
"ppTerm": "?h.inr",
"assigned"... | [
"case h.inr\nz m n : ℤ\nh : PythagoreanTriple (2 * m * n) (m ^ 2 - n ^ 2) z ∧ (2 * m * n).gcd (m ^ 2 - n ^ 2) = 1\nco : m.gcd n = 1\npp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\nthis : z ^ 2 = (m ^ 2 + n ^ 2) ^ 2\n⊢ z = m ^ 2 + n ^ 2 ∨ z = -n ^ 2 + -m ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 119,
"column": 17
} | {
"line": 119,
"column": 28
} | {
"line": 119,
"column": 29
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx✝¹ x✝ : InfinitePlace K\nh : x✝¹.embedding = x✝.embedding\n⊢ x✝¹ = x✝",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝ : Field K\nx✝¹ x✝ : InfinitePlace K\nh : x✝¹.embedding = x✝.embedding\n⊢ x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 55
} | {
"line": 141,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\n⊢ ∑ w, ↑w.mult * Real.log (w ((algebraMap (𝓞 K) K) ↑x)) = 0",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\n⊢ ∑ w, ↑w.mult * Real.log (w ((algebraMap (𝓞 K) K) ↑x)) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Units.Basic | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 59
} | {
"line": 209,
"column": 4
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : (𝓞 K)ˣ\nh : ζ ^ torsionOrder K = 1\n⊢ ζ ∈ CommGroup.torsion (𝓞 K)ˣ",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"NumberField.instCommRingRingOfIntegers",
... | [
"case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : (𝓞 K)ˣ\nh : ζ ^ torsionOrder K = 1\n⊢ ∃ n, 0 < n ∧ ζ ^ n = 1"
] | rw [CommGroup.mem_torsion, isOfFinOrder_iff_pow_eq_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 623,
"column": 6
} | {
"line": 623,
"column": 17
} | {
"line": 623,
"column": 18
} | [
{
"pp": "case h.inl\nz k m n : ℤ\nright✝ : m.gcd n = 1\nh : PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) z\nthis : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2\n⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)",
"ppTerm": "?h.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"No... | [
"case h.inl\nz k m n : ℤ\nright✝ : m.gcd n = 1\nh : PythagoreanTriple (k * (m ^ 2 - n ^ 2)) (k * (2 * m * n)) z\nthis : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2\n⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -(k * (m ^ 2 + n ^ 2))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 628,
"column": 6
} | {
"line": 628,
"column": 17
} | {
"line": 628,
"column": 18
} | [
{
"pp": "case h.inr\nz k m n : ℤ\nright✝ : m.gcd n = 1\nh : PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) z\nthis : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2\n⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)",
"ppTerm": "?h.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"No... | [
"case h.inr\nz k m n : ℤ\nright✝ : m.gcd n = 1\nh : PythagoreanTriple (k * (2 * m * n)) (k * (m ^ 2 - n ^ 2)) z\nthis : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2\n⊢ z = k * (m ^ 2 + n ^ 2) ∨ z = -(k * (m ^ 2 + n ^ 2))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 565,
"column": 2
} | {
"line": 565,
"column": 30
} | {
"line": 567,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx : ℚ\nval✝ : AbsoluteValue K ℝ\nproperty✝ : ∃ φ, place φ = val✝\n⊢ ⟨val✝, property✝⟩ ↑x = ‖x‖",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRi... | [] | aesop (add simp [coe_apply]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 564,
"column": 2
} | {
"line": 565,
"column": 30
} | {
"line": 567,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nx : ℚ\n⊢ v ↑x = ‖x‖",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.partialOrder",
"Real",
... | [] | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 564,
"column": 2
} | {
"line": 565,
"column": 30
} | {
"line": 567,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nx : ℚ\n⊢ v ↑x = ‖x‖",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.partialOrder",
"Real",
... | [] | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 570,
"column": 2
} | {
"line": 570,
"column": 30
} | {
"line": 572,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nn : ℕ\nval✝ : AbsoluteValue K ℝ\nproperty✝ : ∃ φ, place φ = val✝\n⊢ ⟨val✝, property✝⟩ ↑n = ↑n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"NonAssocSemiring.toAddCommMonoi... | [] | aesop (add simp [coe_apply]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 569,
"column": 2
} | {
"line": 570,
"column": 30
} | {
"line": 572,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nn : ℕ\n⊢ v ↑n = ↑n",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Rea... | [] | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 569,
"column": 2
} | {
"line": 570,
"column": 30
} | {
"line": 572,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nn : ℕ\n⊢ v ↑n = ↑n",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Rea... | [] | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 575,
"column": 2
} | {
"line": 575,
"column": 30
} | {
"line": 577,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nz : ℤ\nval✝ : AbsoluteValue K ℝ\nproperty✝ : ∃ φ, place φ = val✝\n⊢ ⟨val✝, property✝⟩ ↑z = ‖z‖",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"NumberField.InfinitePlace.... | [] | aesop (add simp [coe_apply]) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 574,
"column": 2
} | {
"line": 575,
"column": 30
} | {
"line": 577,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nz : ℤ\n⊢ v ↑z = ‖z‖",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRi... | [] | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 574,
"column": 2
} | {
"line": 575,
"column": 30
} | {
"line": 577,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nz : ℤ\n⊢ v ↑z = ‖z‖",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Int.cast",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
"RingHom.instRi... | [] | rcases v with ⟨_, _⟩
aesop (add simp [coe_apply]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 581,
"column": 7
} | {
"line": 581,
"column": 18
} | {
"line": 581,
"column": 19
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv w : InfinitePlace K\nt : ℝ\nh : (fun x ↦ w x) ^ t = ⇑v\nn : ℕ\nhn : 1 < n\n⊢ ↑n ^ t = ↑n ^ 1",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"congrArg",
"id",
"Nat.cast",
"... | [
"K : Type u_1\ninst✝ : Field K\nv w : InfinitePlace K\nt : ℝ\nh : (fun x ↦ w x) ^ t = ⇑v\nn : ℕ\nhn : 1 < n\n⊢ ↑n ^ t = ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 588,
"column": 27
} | {
"line": 588,
"column": 76
} | {
"line": 588,
"column": 77
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv w : InfinitePlace K\nh✝ : (↑w).IsEquiv ↑v\nt : ℝ\nleft✝ : 0 < t\nh : (fun x ↦ ↑w x ^ t) = ⇑↑v\nk : K\n⊢ w k = v k",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"NumberField.InfinitePlace.instFunLikeReal",
"Real",
"id",
"Nu... | [
"K : Type u_1\ninst✝ : Field K\nv w : InfinitePlace K\nh✝ : (↑w).IsEquiv ↑v\nt : ℝ\nleft✝ : 0 < t\nh : (fun x ↦ ↑w x ^ t) = ⇑↑v\nk : K\n⊢ ↑w k = ↑v k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | {
"line": 190,
"column": 4
} | {
"line": 191,
"column": 77
} | {
"line": 192,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝² : Field K\ninst✝¹ : CharZero K\ninst✝ : Algebra.IsAlgebraic ℚ K\nh : maximalRealSubfield K = ⊤\n⊢ IsTotallyReal K",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Algebra.IsIntegral.tower_top",
"Eq.mpr",
"Subfield.toDivisionRing",
"le_r... | [] | have : Algebra.IsIntegral (⊤ : Subfield K) K := Algebra.IsIntegral.tower_top ℚ
rw [← isTotallyReal_top_iff, isTotallyReal_iff_le_maximalRealSubfield, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | {
"line": 190,
"column": 4
} | {
"line": 191,
"column": 77
} | {
"line": 192,
"column": 2
} | [
{
"pp": "K : Type u_2\ninst✝² : Field K\ninst✝¹ : CharZero K\ninst✝ : Algebra.IsAlgebraic ℚ K\nh : maximalRealSubfield K = ⊤\n⊢ IsTotallyReal K",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Algebra.IsIntegral.tower_top",
"Eq.mpr",
"Subfield.toDivisionRing",
"le_r... | [] | have : Algebra.IsIntegral (⊤ : Subfield K) K := Algebra.IsIntegral.tower_top ℚ
rw [← isTotallyReal_top_iff, isTotallyReal_iff_le_maximalRealSubfield, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 25
} | {
"line": 167,
"column": 26
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\nx : K\nx✝ :\n (∀ (a : InfinitePlace K), a.IsReal → a x < ↑(f a)) ∧\n ∀ (a : InfinitePlace K) (b : a.IsComplex),\n a.embedding x ∈ if ⟨a, b⟩ = w₀ then {x | |x.re| < 1 ∧ |x.im| < ↑(f a) ^ 2} else ... | [
"case refine_2\nK : Type u_1\ninst✝ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\nx : K\nx✝ :\n (∀ (a : InfinitePlace K), a.IsReal → a x < ↑(f a)) ∧\n ∀ (a : InfinitePlace K) (b : a.IsComplex),\n a.embedding x ∈ if ⟨a, b⟩ = w₀ then {x | |x.re| < 1 ∧ |x.im| < ↑(f a) ^ 2} else ball 0 ↑(f a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 229,
"column": 8
} | {
"line": 229,
"column": 35
} | {
"line": 229,
"column": 36
} | [
{
"pp": "case e_a.e_a\nK : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\ninst✝ : NumberField K\nvol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2\n⊢ volume (if w₀ = w₀ then {x | |x.re| < 1 ∧ |x.im| < ↑(f ↑w₀) ^ 2} else ball 0 ↑(f ↑w₀)) = 4 * ↑(f ↑w... | [
"case e_a.e_a\nK : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\ninst✝ : NumberField K\nvol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2\n⊢ volume {x | |x.re| < 1 ∧ |x.im| < ↑(f ↑w₀) ^ 2} = 4 * ↑(f ↑w₀) ^ 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 274,
"column": 7
} | {
"line": 274,
"column": 18
} | {
"line": 274,
"column": 19
} | [
{
"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 : IsRamified k w\n⊢ ¬ComplexEmbedding.IsReal (conjugate w.embedding)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsSelfAdjoint",
"NumberFiel... | [
"k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nw : InfinitePlace K\nh : IsRamified k w\n⊢ ¬IsSelfAdjoint w.embedding"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 281,
"column": 26
} | {
"line": 281,
"column": 63
} | {
"line": 281,
"column": 64
} | [
{
"pp": "case refine_1.inr\nk : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsRamified k (mk φ)\nhr : (mk φ).embedding = conjugate φ\n⊢ IsMixed k (star (star φ))",
"ppTerm": "?refine_1.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case refine_1.inr\nk : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsRamified k (mk φ)\nhr : (mk φ).embedding = conjugate φ\n⊢ IsMixed k (conjugate (mk φ).embedding)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 13
} | {
"line": 297,
"column": 14
} | [
{
"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 : (w.comap (algebraMap k K)).IsReal\n⊢ ComplexEmbedding.IsReal (conjugate w.embedding)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"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 : (w.comap (algebraMap k K)).IsReal\n⊢ IsSelfAdjoint w.embedding"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 304,
"column": 26
} | {
"line": 304,
"column": 63
} | {
"line": 304,
"column": 64
} | [
{
"pp": "case refine_1.inr\nk : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsUnramified k (mk φ)\nhr : (mk φ).embedding = conjugate φ\n⊢ IsUnmixed k (star (star φ))",
"ppTerm": "?refine_1.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"case refine_1.inr\nk : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsUnramified k (mk φ)\nhr : (mk φ).embedding = conjugate φ\n⊢ IsUnmixed k (conjugate (mk φ).embedding)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 308,
"column": 23
} | {
"line": 308,
"column": 34
} | {
"line": 308,
"column": 35
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsUnmixed k φ\nhv : ¬ComplexEmbedding.IsReal (φ.comp (algebraMap k K))\n⊢ ((mk φ).comap (algebraMap k K)).IsComplex",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"Algebra... | [
"k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nh : IsUnmixed k φ\nhv : ¬ComplexEmbedding.IsReal (φ.comp (algebraMap k K))\n⊢ (mk (φ.comp (algebraMap k K))).IsComplex"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 298,
"column": 6
} | {
"line": 298,
"column": 30
} | {
"line": 298,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\n⊢ ‖x‖ ≤ convexBodySumFun x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NumberField.mixedEmbedding.norm_eq_sup'_normAtPlace",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\n⊢ (Finset.univ.sup' ⋯ fun w ↦ (normAtPlace w) x) ≤ convexBodySumFun x"
] | norm_eq_sup'_normAtPlace | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 346,
"column": 61
} | {
"line": 351,
"column": 83
} | {
"line": 353,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nφ : K →+* ℂ\nσ : Gal(K/k)\n⊢ σ ∈ Stab (mk φ) ↔ σ = 1 ∨ ComplexEmbedding.IsConj φ σ",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"or_congr",
"Eq.mpr",
"AlgEquiv.instEquivLike",... | [] | by
simp only [MulAction.mem_stabilizer_iff, smul_mk, mk_eq_iff]
rw [← ComplexEmbedding.isConj_symm, ComplexEmbedding.conjugate, star_eq_iff_star_eq]
refine or_congr ⟨fun H ↦ ?_, fun H ↦ H ▸ rfl⟩ Iff.rfl
exact congr_arg AlgEquiv.symm
(AlgEquiv.ext (g := AlgEquiv.refl) fun x ↦ φ.injective (RingHom.congr_fun H... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 347,
"column": 2
} | {
"line": 352,
"column": 50
} | {
"line": 354,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ Bornology.IsBounded (convexBodySum K B)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing"... | [] | classical
refine Metric.isBounded_iff.mpr ⟨B + B, fun x hx y hy => ?_⟩
simp_rw [dist_eq_norm]
refine le_trans (norm_sub_le x y) (add_le_add ?_ ?_)
· exact le_trans (norm_le_convexBodySumFun x) hx
· exact le_trans (norm_le_convexBodySumFun y) hy | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 347,
"column": 2
} | {
"line": 352,
"column": 50
} | {
"line": 354,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ Bornology.IsBounded (convexBodySum K B)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing"... | [] | classical
refine Metric.isBounded_iff.mpr ⟨B + B, fun x hx y hy => ?_⟩
simp_rw [dist_eq_norm]
refine le_trans (norm_sub_le x y) (add_le_add ?_ ?_)
· exact le_trans (norm_le_convexBodySumFun x) hx
· exact le_trans (norm_le_convexBodySumFun y) hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 347,
"column": 2
} | {
"line": 352,
"column": 50
} | {
"line": 354,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ Bornology.IsBounded (convexBodySum K B)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing"... | [] | classical
refine Metric.isBounded_iff.mpr ⟨B + B, fun x hx y hy => ?_⟩
simp_rw [dist_eq_norm]
refine le_trans (norm_sub_le x y) (add_le_add ?_ ?_)
· exact le_trans (norm_le_convexBodySumFun x) hx
· exact le_trans (norm_le_convexBodySumFun y) hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 36
} | {
"line": 119,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℂ ((K →+* ℂ) → ℂ)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
"Semiring.toModule",
"Pi.addCommMonoid",
"RingHom",
"Finite.of_fint... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : Basis (K →+* ℂ) ℂ ((K →+* ℂ) → ℂ) := Pi.basisFun ℂ (K →+* ℂ)\n⊢ Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℂ ((K →+* ℂ) → ℂ)"
] | let B := Pi.basisFun ℂ (K →+* ℂ) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 486,
"column": 4
} | {
"line": 490,
"column": 34
} | {
"line": 491,
"column": 2
} | [
{
"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 = Nat.card Gal(K/k)",
... | [] | · rw [Nat.card_eq_fintype_card,
← MulAction.card_orbit_mul_card_stabilizer_eq_card_group _ w,
← Nat.card_eq_fintype_card (α := Stab w), card_stabilizer, if_pos,
mul_one, Set.toFinset_card]
rwa [← isUnramifiedIn_comap] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 618,
"column": 14
} | {
"line": 618,
"column": 25
} | {
"line": 618,
"column": 26
} | [
{
"pp": "case inr\nK : 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\nhr : (mk (w.embedding.comp (algebraMap K L))).embedding = conjugate (w.embedding.comp (algebraMap K L))\n⊢ w.embedding.comp (algebraMap K L... | [
"case inr\nK : 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\nhr : (mk (w.embedding.comp (algebraMap K L))).embedding = conjugate (w.embedding.comp (algebraMap K L))\n⊢ w.embedding.comp (algebraMap K L) = v.embedd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 642,
"column": 8
} | {
"line": 642,
"column": 43
} | {
"line": 642,
"column": 44
} | [
{
"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": [
... | [
"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⊢ v.IsReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 706,
"column": 20
} | {
"line": 706,
"column": 31
} | {
"line": 706,
"column": 32
} | [
{
"pp": "case inl\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : InfinitePlace K\nψ : L →+* ℂ\nh : ψ ∈ mixedEmbeddingsOver L v.embedding\nhl : (mk ψ).embedding = ψ\n⊢ ψ ∈ Sum.elim embedding (conjugate ∘ embedding) '' Set.sumEquiv.symm (ramifiedPlacesOver L v, ramifiedP... | [
"case inl\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : InfinitePlace K\nψ : L →+* ℂ\nh : ψ ∈ mixedEmbeddingsOver L v.embedding\nhl : (mk ψ).embedding = ψ\n⊢ (∃ a ∈ ramifiedPlacesOver L v, a.embedding = ψ) ∨ ∃ b ∈ ramifiedPlacesOver L v, conjugate b.embedding = ψ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 707,
"column": 20
} | {
"line": 707,
"column": 31
} | {
"line": 707,
"column": 32
} | [
{
"pp": "case inr\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : InfinitePlace K\nψ : L →+* ℂ\nh : ψ ∈ mixedEmbeddingsOver L v.embedding\nhr : (mk ψ).embedding = conjugate ψ\n⊢ ψ ∈ Sum.elim embedding (conjugate ∘ embedding) '' Set.sumEquiv.symm (ramifiedPlacesOver L v,... | [
"case inr\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : InfinitePlace K\nψ : L →+* ℂ\nh : ψ ∈ mixedEmbeddingsOver L v.embedding\nhr : (mk ψ).embedding = conjugate ψ\n⊢ (∃ a ∈ ramifiedPlacesOver L v, a.embedding = ψ) ∨ ∃ b ∈ ramifiedPlacesOver L v, conjugate b.embedding = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 737,
"column": 4
} | {
"line": 737,
"column": 45
} | {
"line": 737,
"column": 46
} | [
{
"pp": "case pos\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : InfinitePlace K\nw : InfinitePlace L\nleft✝ : (↑w).LiesOver ↑v\nhw : IsUnramified K w\nh : ComplexEmbedding.LiesOver w.embedding v.embedding\n⊢ v.embeddingConjugateIte w ∈ unmixedEmbeddingsOver L v.embedd... | [
"case pos\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : InfinitePlace K\nw : InfinitePlace L\nleft✝ : (↑w).LiesOver ↑v\nhw : IsUnramified K w\nh : ComplexEmbedding.LiesOver w.embedding v.embedding\n⊢ w.embedding ∈ unmixedEmbeddingsOver L v.embedding"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 738,
"column": 4
} | {
"line": 738,
"column": 45
} | {
"line": 739,
"column": 6
} | [
{
"pp": "case neg\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : InfinitePlace K\nw : InfinitePlace L\nleft✝ : (↑w).LiesOver ↑v\nhw : IsUnramified K w\nh : ¬ComplexEmbedding.LiesOver w.embedding v.embedding\n⊢ v.embeddingConjugateIte w ∈ unmixedEmbeddingsOver L v.embed... | [
"case neg\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : InfinitePlace K\nw : InfinitePlace L\nleft✝ : (↑w).LiesOver ↑v\nhw : IsUnramified K w\nh : ¬ComplexEmbedding.LiesOver w.embedding v.embedding\n⊢ conjugate w.embedding ∈ unmixedEmbeddingsOver L v.embedding"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 748,
"column": 4
} | {
"line": 748,
"column": 43
} | {
"line": 748,
"column": 44
} | [
{
"pp": "case inr\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : InfinitePlace K\nψ : L →+* ℂ\nh : ψ ∈ unmixedEmbeddingsOver L v.embedding\nhψ : (mk ψ).embedding = conjugate ψ\n⊢ v.embeddingConjugateIte (mk ψ) = ψ",
"ppTerm": "?inr",
"assigned": true,
"used... | [
"case inr\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : InfinitePlace K\nψ : L →+* ℂ\nh : ψ ∈ unmixedEmbeddingsOver L v.embedding\nhψ : (mk ψ).embedding = conjugate ψ\n⊢ ComplexEmbedding.LiesOver (conjugate ψ) v.embedding → conjugate ψ = ψ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 774,
"column": 50
} | {
"line": 774,
"column": 61
} | {
"line": 774,
"column": 62
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nv : InfinitePlace K\ninst✝¹ : NumberField K\ninst✝ : NumberField L\nthis : Algebra K ℂ := v.embedding.toAlgebra\nσ : L →ₐ[K] ℂ\nx✝ : σ ∈ ↑univ\n⊢ σ.toRingHom ∈ ↑⋯.toFinset",
"ppTerm": "?m.122",
"assigned": tru... | [
"K : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nv : InfinitePlace K\ninst✝¹ : NumberField K\ninst✝ : NumberField L\nthis : Algebra K ℂ := v.embedding.toAlgebra\nσ : L →ₐ[K] ℂ\nx✝ : σ ∈ ↑univ\n⊢ ComplexEmbedding.LiesOver (↑σ) v.embedding"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 488,
"column": 19
} | {
"line": 488,
"column": 30
} | {
"line": 488,
"column": 31
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n (fundamentalDomain (fr... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nh : minkowskiBound K I < volume (convexBodyLT K f)\nh_fund :\n IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n (fundamentalDomain (fractionalIdea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 17
} | {
"line": 253,
"column": 18
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : { w // w.IsReal }\nA : AffineSubspace ℝ (mixedSpace K) := ↑{ carrier := {x | x.1 w = 0}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nh : A = ⊤\n⊢ False",
"ppTerm": "?m.66",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : { w // w.IsReal }\nA : AffineSubspace ℝ (mixedSpace K) := ↑{ carrier := {x | x.1 w = 0}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nh : A = ⊤\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 505,
"column": 19
} | {
"line": 505,
"column": 30
} | {
"line": 505,
"column": 31
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nw₀ : { w // w.IsComplex }\nh : minkowskiBound K I < volume (convexBodyLT' K f w₀)\nh_fund :\n IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgro... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nf : InfinitePlace K → ℝ≥0\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nw₀ : { w // w.IsComplex }\nh : minkowskiBound K I < volume (convexBodyLT' K f w₀)\nh_fund :\n IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)\n (fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 558,
"column": 23
} | {
"line": 558,
"column": 57
} | {
"line": 558,
"column": 57
} | [
{
"pp": "K : 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 ≠ 0\nh_le :... | [] | convert! if_neg h_ne ▸ h_le w h_ne | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 558,
"column": 23
} | {
"line": 558,
"column": 57
} | {
"line": 558,
"column": 57
} | [
{
"pp": "K : 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 ≠ 0\nh_le :... | [] | convert! if_neg h_ne ▸ h_le w h_ne | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 558,
"column": 23
} | {
"line": 558,
"column": 57
} | {
"line": 558,
"column": 57
} | [
{
"pp": "K : 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 ≠ 0\nh_le :... | [] | convert! if_neg h_ne ▸ h_le w h_ne | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 599,
"column": 20
} | {
"line": 599,
"column": 31
} | {
"line": 599,
"column": 32
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 593,
"column": 52
} | {
"line": 593,
"column": 73
} | {
"line": 594,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nc : index K\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) c) =\n ∑ x_1,\n fromBlocks (diagonal fun x ↦ 1... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nc : index K\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) c) =\n ∑ a₁,\n fromBlocks (diagonal fun x ↦ 1) 0 0\n ... | Fintype.sum_sum_type, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 597,
"column": 32
} | {
"line": 597,
"column": 52
} | {
"line": 597,
"column": 53
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nc : index K\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) c) =\n ∑ x_1,\n fromBlocks 1 0 0\n ... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nc : index K\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) c) =\n ∑ x_1,\n fromBlocks 1 0 0\n ((blockDia... | Equiv.prodComm_symm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Invariant.Galois | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 17
} | {
"line": 53,
"column": 4
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Field K\ninst✝¹² : Field L\ninst✝¹¹ : Algebra A K\ninst✝¹⁰ : Algebra B L\ninst✝⁹ : IsFractionRing A K\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : Algebra A B\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra ... | [
"A : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Field K\ninst✝¹² : Field L\ninst✝¹¹ : Algebra A K\ninst✝¹⁰ : Algebra B L\ninst✝⁹ : IsFractionRing A K\ninst✝⁸ : IsFractionRing B L\ninst✝⁷ : Algebra A B\ninst✝⁶ : Algebra K L\ninst✝⁵ : Algebra A L\ninst✝⁴ ... | rintro ⟨g, -⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.IntegralClosure.IntegralRestrict | {
"line": 94,
"column": 4
} | {
"line": 96,
"column": 45
} | {
"line": 96,
"column": 46
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u_3\nL₂ : Type u_4\nL₃ : Type u_5\nB : Type u_6\nB₂ : Type u_7\nB₃ : Type u_8\ninst✝³¹ : CommRing A\ninst✝³⁰ : CommRing B\ninst✝²⁹ : CommRing B₂\ninst✝²⁸ : CommRing B₃\ninst✝²⁷ : Algebra A B\ninst✝²⁶ : Algebra A B₂\ninst✝²⁵ : Algebra A B₃\ninst✝²⁴ : Field K\ninst✝²³... | [
"A : Type u_1\nK : Type u_2\nL : Type u_3\nL₂ : Type u_4\nL₃ : Type u_5\nB : Type u_6\nB₂ : Type u_7\nB₃ : Type u_8\ninst✝³¹ : CommRing A\ninst✝³⁰ : CommRing B\ninst✝²⁹ : CommRing B₂\ninst✝²⁸ : CommRing B₃\ninst✝²⁷ : Algebra A B\ninst✝²⁶ : Algebra A B₂\ninst✝²⁵ : Algebra A B₃\ninst✝²⁴ : Field K\ninst✝²³ : Field L\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 605,
"column": 4
} | {
"line": 618,
"column": 27
} | {
"line": 620,
"column": 0
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nc : { w // w.IsComplex } × Fin 2\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) (Sum.inr c)) =\n ∑ x... | [] | rcases c with ⟨w, j⟩
fin_cases j
· simp only [Fin.zero_eta, Fin.isValue, stdBasis_apply_isComplex_fst, re_eq_add_conj,
mul_neg, fromBlocks_apply₂₁, Matrix.zero_apply, zero_mul, sum_const_zero,
fromBlocks_apply₂₂, submatrix_apply, Prod.swap_prod_mk, blockDiagonal_apply, of_apply,
cons_val... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 605,
"column": 4
} | {
"line": 618,
"column": 27
} | {
"line": 620,
"column": 0
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nc : { w // w.IsComplex } × Fin 2\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) (Sum.inr c)) =\n ∑ x... | [] | rcases c with ⟨w, j⟩
fin_cases j
· simp only [Fin.zero_eta, Fin.isValue, stdBasis_apply_isComplex_fst, re_eq_add_conj,
mul_neg, fromBlocks_apply₂₁, Matrix.zero_apply, zero_mul, sum_const_zero,
fromBlocks_apply₂₂, submatrix_apply, Prod.swap_prod_mk, blockDiagonal_apply, of_apply,
cons_val... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 641,
"column": 12
} | {
"line": 641,
"column": 57
} | {
"line": 641,
"column": 58
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ Function.Injective fun r ↦ r • 1",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ Function.Injective fun r ↦ ↑r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Invariant.Galois | {
"line": 111,
"column": 67
} | {
"line": 111,
"column": 78
} | {
"line": 111,
"column": 79
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁹ : CommRing A\ninst✝¹⁸ : CommRing B\ninst✝¹⁷ : Algebra A B\nG : Type u_3\ninst✝¹⁶ : Finite G\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MulSemiringAction G B\ninst✝¹³ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝¹² : Q.LiesOver P\ninst✝¹¹ : P.IsPrime\ninst✝¹⁰ : Q.Is... | [
"A : Type u_1\nB : Type u_2\ninst✝¹⁹ : CommRing A\ninst✝¹⁸ : CommRing B\ninst✝¹⁷ : Algebra A B\nG : Type u_3\ninst✝¹⁶ : Finite G\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MulSemiringAction G B\ninst✝¹³ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝¹² : Q.LiesOver P\ninst✝¹¹ : P.IsPrime\ninst✝¹⁰ : Q.IsPrime\nK : T... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Invariant.Galois | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 47
} | {
"line": 117,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁹ : CommRing A\ninst✝¹⁸ : CommRing B\ninst✝¹⁷ : Algebra A B\nG : Type u_3\ninst✝¹⁶ : Finite G\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MulSemiringAction G B\ninst✝¹³ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝¹² : Q.LiesOver P\ninst✝¹¹ : P.IsPrime\ninst✝¹⁰ : Q.Is... | [] | exact MulSemiringAction.splits_charpoly G b | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.IntegralClosure.IntegralRestrict | {
"line": 384,
"column": 27
} | {
"line": 384,
"column": 38
} | {
"line": 384,
"column": 39
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u_3\nL₂ : Type u_4\nL₃ : Type u_5\nB : Type u_6\nB₂ : Type u_7\nB₃ : Type u_8\ninst✝⁴² : CommRing A\ninst✝⁴¹ : CommRing B\ninst✝⁴⁰ : CommRing B₂\ninst✝³⁹ : CommRing B₃\ninst✝³⁸ : Algebra A B\ninst✝³⁷ : Algebra A B₂\ninst✝³⁶ : Algebra A B₃\ninst✝³⁵ : Field K\ninst✝³⁴... | [
"A : Type u_1\nK : Type u_2\nL : Type u_3\nL₂ : Type u_4\nL₃ : Type u_5\nB : Type u_6\nB₂ : Type u_7\nB₃ : Type u_8\ninst✝⁴² : CommRing A\ninst✝⁴¹ : CommRing B\ninst✝⁴⁰ : CommRing B₂\ninst✝³⁹ : CommRing B₃\ninst✝³⁸ : Algebra A B\ninst✝³⁷ : Algebra A B₂\ninst✝³⁶ : Algebra A B₃\ninst✝³⁵ : Field K\ninst✝³⁴ : Field L\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 875,
"column": 2
} | {
"line": 875,
"column": 36
} | {
"line": 876,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ IsZLattice ℝ (euclidean.integerLattice K)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ IsZLattice ℝ (ZLattice.comap ℝ (mixedEmbedding.integerLattice K) ↑↑(toMixed K))"
] | simp_rw [euclidean.integerLattice] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 424,
"column": 12
} | {
"line": 424,
"column": 64
} | {
"line": 424,
"column": 65
} | [
{
"pp": "case refine_2.refine_1\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := rankOfDiscrBdd N\nB : ℝ≥0 := boundOfDiscBdd N\nC : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ : ⟨K, hK₀⟩ ∈ {K | ... | [
"case refine_2.refine_1\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := rankOfDiscrBdd N\nB : ℝ≥0 := boundOfDiscBdd N\nC : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsRea... | minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 424,
"column": 8
} | {
"line": 425,
"column": 59
} | {
"line": 426,
"column": 6
} | [
{
"pp": "case refine_2.refine_1\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := rankOfDiscrBdd N\nB : ℝ≥0 := boundOfDiscBdd N\nC : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ : ⟨K, hK₀⟩ ∈ {K | ... | [] | rw [minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx, coeff_map, eq_intCast,
Int.norm_cast_rat, Int.norm_eq_abs, Int.cast_abs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 424,
"column": 8
} | {
"line": 425,
"column": 59
} | {
"line": 426,
"column": 6
} | [
{
"pp": "case refine_2.refine_1\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := rankOfDiscrBdd N\nB : ℝ≥0 := boundOfDiscBdd N\nC : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ : ⟨K, hK₀⟩ ∈ {K | ... | [] | rw [minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx, coeff_map, eq_intCast,
Int.norm_cast_rat, Int.norm_eq_abs, Int.cast_abs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 101,
"column": 20
} | {
"line": 105,
"column": 100
} | {
"line": 107,
"column": 0
} | [
{
"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... | [] | by
apply Subtype.val_inj.mp
change map _ Q.1 = map _ (map _ Q.1)
rw [map_mul]
exact (Q.1.map_map ((galRestrict A K L B) τ).toRingHom ((galRestrict A K L B) σ).toRingHom).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 424,
"column": 8
} | {
"line": 425,
"column": 59
} | {
"line": 426,
"column": 6
} | [
{
"pp": "case refine_2.refine_1\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := rankOfDiscrBdd N\nB : ℝ≥0 := boundOfDiscBdd N\nC : ℕ := ⌈max B 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ : ⟨K, hK₀⟩ ∈ {K | ... | [] | rw [minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx, coeff_map, eq_intCast,
Int.norm_cast_rat, Int.norm_eq_abs, Int.cast_abs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 69
} | {
"line": 251,
"column": 70
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝²¹ : CommRing A\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Algebra A B\ninst✝¹⁸ : FaithfulSMul A B\np : Ideal A\nP : Ideal B\ninst✝¹⁷ : P.IsPrime\ninst✝¹⁶ : P.LiesOver p\nG : Type u_3\ninst✝¹⁵ : Group G\ninst✝¹⁴ : Finite G\ninst✝¹³ : MulSemiringAction G B\ninst✝¹² : IsGaloisGroup... | [
"A : Type u_1\nB : Type u_2\ninst✝²¹ : CommRing A\ninst✝²⁰ : CommRing B\ninst✝¹⁹ : Algebra A B\ninst✝¹⁸ : FaithfulSMul A B\np : Ideal A\nP : Ideal B\ninst✝¹⁷ : P.IsPrime\ninst✝¹⁶ : P.LiesOver p\nG : Type u_3\ninst✝¹⁵ : Group G\ninst✝¹⁴ : Finite G\ninst✝¹³ : MulSemiringAction G B\ninst✝¹² : IsGaloisGroup G A B\nC : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 265,
"column": 41
} | {
"line": 265,
"column": 72
} | {
"line": 265,
"column": 73
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁹ : CommRing A\ninst✝¹⁸ : CommRing B\ninst✝¹⁷ : Algebra A B\ninst✝¹⁶ : FaithfulSMul A B\np : Ideal A\nP : Ideal B\ninst✝¹⁵ : P.IsPrime\ninst✝¹⁴ : P.LiesOver p\nG : Type u_3\ninst✝¹³ : Group G\ninst✝¹² : Finite G\ninst✝¹¹ : MulSemiringAction G B\ninst✝¹⁰ : IsGaloisGroup... | [
"A : Type u_1\nB : Type u_2\ninst✝¹⁹ : CommRing A\ninst✝¹⁸ : CommRing B\ninst✝¹⁷ : Algebra A B\ninst✝¹⁶ : FaithfulSMul A B\np : Ideal A\nP : Ideal B\ninst✝¹⁵ : P.IsPrime\ninst✝¹⁴ : P.LiesOver p\nG : Type u_3\ninst✝¹³ : Group G\ninst✝¹² : Finite G\ninst✝¹¹ : MulSemiringAction G B\ninst✝¹⁰ : IsGaloisGroup G A B\nC : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1113,
"column": 2
} | {
"line": 1113,
"column": 17
} | {
"line": 1113,
"column": 18
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : InfinitePlace K\nA : AffineSubspace ℝ (realSpace K) := ↑{ carrier := {x | x w = 0}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nh : A = ⊤\n⊢ False",
"ppTerm": "?m.57",
"assigned": false,
"usedConstants": [],
"usedFVars": []... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : InfinitePlace K\nA : AffineSubspace ℝ (realSpace K) := ↑{ carrier := {x | x w = 0}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nh : A = ⊤\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1157,
"column": 41
} | {
"line": 1159,
"column": 35
} | {
"line": 1161,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx : mixedSpace K\nw : { w // w.IsComplex }\n⊢ normAtComplexPlaces x ↑w = ‖x.2 w‖",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"NumberField.InfinitePlace.IsCo... | [] | by
rw [normAtComplexPlaces, dif_neg (not_isReal_iff_isComplex.mpr w.prop),
normAtPlace_apply_of_isComplex] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1216,
"column": 4
} | {
"line": 1217,
"column": 53
} | {
"line": 1217,
"column": 54
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝ : Field K\nx y : mixedSpace K\nh : normAtComplexPlaces x = normAtComplexPlaces y\nw : InfinitePlace K\nhw : w.IsReal\n⊢ normAtAllPlaces x w = normAtAllPlaces y w",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.... | [
"case inl\nK : Type u_1\ninst✝ : Field K\nx y : mixedSpace K\nh : normAtComplexPlaces x = normAtComplexPlaces y\nw : InfinitePlace K\nhw : w.IsReal\n⊢ |x.1 ⟨w, hw⟩| = |y.1 ⟨w, hw⟩|"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1218,
"column": 4
} | {
"line": 1219,
"column": 56
} | {
"line": 1219,
"column": 57
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝ : Field K\nx y : mixedSpace K\nh : normAtComplexPlaces x = normAtComplexPlaces y\nw : InfinitePlace K\nhw : w.IsComplex\n⊢ normAtAllPlaces x w = normAtAllPlaces y w",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
... | [
"case inr\nK : Type u_1\ninst✝ : Field K\nx y : mixedSpace K\nh : normAtComplexPlaces x = normAtComplexPlaces y\nw : InfinitePlace K\nhw : w.IsComplex\n⊢ ‖x.2 ⟨w, hw⟩‖ = ‖y.2 ⟨w, hw⟩‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 86,
"column": 8
} | {
"line": 86,
"column": 13
} | {
"line": 86,
"column": 14
} | [
{
"pp": "K : 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_nm :\n ↑|(Algebra.norm ℚ) ((Algebra.linearMap (𝓞 K) K) a)| ≤\n ↑(FractionalIdeal.absNorm ↑((FractionalIdeal.mk0 K) J)) * (4 / π) ^ nrComplexPla... | [
"K : 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_nm :\n ↑|(Algebra.norm ℚ) ((Algebra.linearMap (𝓞 K) K) a)| ≤\n ↑(FractionalIdeal.absNorm ↑((FractionalIdeal.mk0 K) J)) * (4 / π) ^ nrComplexPlaces K * ↑(fi... | h_nz, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 42
} | {
"line": 87,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : HasFiniteQuotients R\nx : R\nhx : x ≠ 0\nthis✝ : Finite (R ⧸ Ideal.span {x})\nthis : {I | Ideal.comap (Ideal.Quotient.mk (Ideal.span {x})) ⊥ ≤ I}.Finite\n⊢ {I | x ∈ I}.Finite",
"ppTerm": "?m.67",
"assigned": false,
"usedConstants": [],
"usedFVa... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : HasFiniteQuotients R\nx : R\nhx : x ≠ 0\nthis✝ : Finite (R ⧸ Ideal.span {x})\nthis : {I | Ideal.comap (Ideal.Quotient.mk (Ideal.span {x})) ⊥ ≤ I}.Finite\n⊢ {I | x ∈ I}.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients | {
"line": 165,
"column": 27
} | {
"line": 165,
"column": 38
} | {
"line": 165,
"column": 39
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nhI : ℤ ∙ n ≠ ⊥\n⊢ n ≠ 0",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"Int",
"Zero.toOfNat0",
"OfNat.ofNat",
"MulZeroClass.toZero",
"Int.instSemiring",
"instMulZeroClassOfSemirin... | [
"R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nhI : ℤ ∙ n ≠ ⊥\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 13
} | {
"line": 314,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nG : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Group G\ninst✝⁶ : MulSemiringAction G S\ninst✝⁵ : IsGaloisGroup G R S\ninst✝⁴ : Finite G\np : Ideal R\ninst✝³ : p.IsPrime\nP : Ideal S\ninst✝² : P.LiesOver p\ninst✝¹ : P.IsPrime\ninst✝ : ... | [
"R : Type u_1\nS : Type u_2\nG : Type u_3\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : Group G\ninst✝⁶ : MulSemiringAction G S\ninst✝⁵ : IsGaloisGroup G R S\ninst✝⁴ : Finite G\np : Ideal R\ninst✝³ : p.IsPrime\nP : Ideal S\ninst✝² : P.LiesOver p\ninst✝¹ : P.IsPrime\ninst✝ : PerfectField... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 43
} | {
"line": 114,
"column": 44
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh :\n ∀ ⦃I : ↥(Ideal (𝓞 K))⁰⦄,\n ↑(absNorm ↑I) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|) →\n Submodule.IsPrincipal ↑I\nI : ↥(Ideal (𝓞 K))⁰\nhI : ↑(absNorm ↑I) ≤ (4 / π) ^ nrComplexPl... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh :\n ∀ ⦃I : ↥(Ideal (𝓞 K))⁰⦄,\n ↑(absNorm ↑I) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|) →\n Submodule.IsPrincipal ↑I\nI : ↥(Ideal (𝓞 K))⁰\nhI : ↑(absNorm ↑I) ≤ (4 / π) ^ nrComplexPlaces K * (↑(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 68
} | {
"line": 157,
"column": 69
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh✝ :\n ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,\n Nat.Prime p →\n ∀ P ∈ (span {↑p}).primesOver (𝓞 K),\n p ^ P.inertiaDeg ℤ ≤\n ⌊(4 / ... | [
"case inl\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh✝ :\n ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,\n Nat.Prime p →\n ∀ P ∈ (span {↑p}).primesOver (𝓞 K),\n p ^ P.inertiaDeg ℤ ≤\n ⌊(4 / π) ^ nrCompl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 68
} | {
"line": 157,
"column": 69
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh✝ :\n ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,\n Nat.Prime p →\n ∀ P ∈ (span {↑p}).primesOver (𝓞 K),\n p ^ P.inertiaDeg ℤ ≤\n ⌊(4 / ... | [
"case inr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh✝ :\n ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,\n Nat.Prime p →\n ∀ P ∈ (span {↑p}).primesOver (𝓞 K),\n p ^ P.inertiaDeg ℤ ≤\n ⌊(4 / π) ^ nrCompl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 473,
"column": 12
} | {
"line": 473,
"column": 64
} | {
"line": 473,
"column": 65
} | [
{
"pp": "case refine_2.refine_1\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := rankOfDiscrBdd N\nB : ℝ≥0 := boundOfDiscBdd N\nC : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ :... | [
"case refine_2.refine_1\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := rankOfDiscrBdd N\nB : ℝ≥0 := boundOfDiscBdd N\nC : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ : ⟨K, hK₀⟩ ∈ ... | minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 162,
"column": 43
} | {
"line": 162,
"column": 54
} | {
"line": 162,
"column": 55
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh :\n ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,\n Nat.Prime p →\n ∀ P ∈ (span {↑p}).primesOver (𝓞 K),\n p ^ P.inertiaDeg ℤ ≤\n ⌊(4 / π) ^ nrComp... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh :\n ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,\n Nat.Prime p →\n ∀ P ∈ (span {↑p}).primesOver (𝓞 K),\n p ^ P.inertiaDeg ℤ ≤\n ⌊(4 / π) ^ nrComplexPlaces K ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 473,
"column": 8
} | {
"line": 474,
"column": 59
} | {
"line": 475,
"column": 6
} | [
{
"pp": "case refine_2.refine_1\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := rankOfDiscrBdd N\nB : ℝ≥0 := boundOfDiscBdd N\nC : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ :... | [] | rw [minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx, coeff_map, eq_intCast,
Int.norm_cast_rat, Int.norm_eq_abs, Int.cast_abs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 473,
"column": 8
} | {
"line": 474,
"column": 59
} | {
"line": 475,
"column": 6
} | [
{
"pp": "case refine_2.refine_1\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := rankOfDiscrBdd N\nB : ℝ≥0 := boundOfDiscBdd N\nC : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ :... | [] | rw [minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx, coeff_map, eq_intCast,
Int.norm_cast_rat, Int.norm_eq_abs, Int.cast_abs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 473,
"column": 8
} | {
"line": 474,
"column": 59
} | {
"line": 475,
"column": 6
} | [
{
"pp": "case refine_2.refine_1\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := rankOfDiscrBdd N\nB : ℝ≥0 := boundOfDiscBdd N\nC : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ :... | [] | rw [minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx, coeff_map, eq_intCast,
Int.norm_cast_rat, Int.norm_eq_abs, Int.cast_abs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 69
} | {
"line": 164,
"column": 70
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh :\n ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,\n Nat.Prime p →\n ∀ P ∈ (span {↑p}).primesOver (𝓞 K),\n p ^ P.inertiaDeg ℤ ≤\n ⌊(4 / π) ^ nrComp... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh :\n ∀ p ∈ Finset.Icc 1 ⌊(4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)⌋₊,\n Nat.Prime p →\n ∀ P ∈ (span {↑p}).primesOver (𝓞 K),\n p ^ P.inertiaDeg ℤ ≤\n ⌊(4 / π) ^ nrComplexPlaces K ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Unramified | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 38
} | {
"line": 105,
"column": 39
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsDomain R\ninst✝³ : IsDomain S\ninst✝² : Module.IsTorsionFree R S\ninst✝¹ : CharZero R\ninst✝ : Algebra.IsIntegral R S\n⊢ IsFractionRing S (Localization.AtPrime ⊥)",
"ppTerm": "?m.37",
"assigne... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsDomain R\ninst✝³ : IsDomain S\ninst✝² : Module.IsTorsionFree R S\ninst✝¹ : CharZero R\ninst✝ : Algebra.IsIntegral R S\n⊢ IsFractionRing S (Localization.AtPrime ⊥)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Unramified | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 48
} | {
"line": 122,
"column": 49
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsDomain R\ninst✝³ : IsDomain S\ninst✝² : FaithfulSMul R S\ninst✝¹ : CharZero R\ninst✝ : Algebra.IsIntegral R S\nP : Ideal S\nx✝ : P.IsPrime\nhP : P.LiesOver ⊥\n⊢ IsUnramifiedAt R P",
"ppTerm": "?m.... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : IsDomain R\ninst✝³ : IsDomain S\ninst✝² : FaithfulSMul R S\ninst✝¹ : CharZero R\ninst✝ : Algebra.IsIntegral R S\nP : Ideal S\nx✝ : P.IsPrime\nhP : P.LiesOver ⊥\n⊢ IsUnramifiedAt R ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 91,
"column": 46
} | {
"line": 91,
"column": 78
} | {
"line": 91,
"column": 79
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : CommRing A\nB : Type u_2\ninst✝⁷ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁶ : CommRing K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝³ : CommRing L\ninst✝² : Algebra B L\ninst✝¹ : IsLocalization N L\nhf : M ≤ Submo... | [
"A : Type u_1\ninst✝⁸ : CommRing A\nB : Type u_2\ninst✝⁷ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁶ : CommRing K\ninst✝⁵ : Algebra A K\ninst✝⁴ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝³ : CommRing L\ninst✝² : Algebra B L\ninst✝¹ : IsLocalization N L\nhf : M ≤ Submonoid.comap f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 110,
"column": 45
} | {
"line": 110,
"column": 56
} | {
"line": 110,
"column": 57
} | [
{
"pp": "A : 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 ≤ Submon... | [
"A : 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 ≤ Submonoid.comap f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Int | {
"line": 99,
"column": 47
} | {
"line": 99,
"column": 58
} | {
"line": 99,
"column": 59
} | [
{
"pp": "S : Type u_2\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Module.Free ℤ S\nI : Ideal S\nh✝ : Finite (S ⧸ I)\nthis : Fintype (S ⧸ I)\nd : ℕ\nh : ∀ (x : S ⧸ I), (Ideal.Quotient.mk I) ↑d * x = 0\n⊢ (Ideal.Quotient.mk I) ↑d = 0",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants... | [
"S : Type u_2\ninst✝² : CommRing S\ninst✝¹ : IsDedekindDomain S\ninst✝ : Module.Free ℤ S\nI : Ideal S\nh✝ : Finite (S ⧸ I)\nthis : Fintype (S ⧸ I)\nd : ℕ\nh : ∀ (x : S ⧸ I), (Ideal.Quotient.mk I) ↑d * x = 0\n⊢ ↑d = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Int | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 31
} | {
"line": 110,
"column": 32
} | [
{
"pp": "S : Type u_2\ninst✝ : CommRing S\nI : Ideal S\nx : ℕ\n⊢ (Ideal.Quotient.mk I) ↑x = 0 ↔ absNorm (under ℤ I) ∣ x",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Submodule.Quotient.instZeroQuotient"... | [
"S : Type u_2\ninst✝ : CommRing S\nI : Ideal S\nx : ℕ\n⊢ ↑x ∈ I ↔ absNorm (under ℤ I) ∣ x"
] | Quotient.eq_zero_iff_mem, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Instances | {
"line": 128,
"column": 21
} | {
"line": 128,
"column": 32
} | {
"line": 128,
"column": 33
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\nP : Ideal R\ninst✝¹ : P.IsPrime\ninst✝ : FaithfulSMul R S\nx✝ : ↥P.primeCompl\nx : R\nhx : x ∈ P.primeCompl\n⊢ fa... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsDomain S\ninst✝³ : IsDomain T\ninst✝² : Algebra R S\nP : Ideal R\ninst✝¹ : P.IsPrime\ninst✝ : FaithfulSMul R S\nx✝ : ↥P.primeCompl\nx : R\nhx : x ∈ P.primeCompl\n⊢ ¬x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 63,
"column": 67
} | {
"line": 63,
"column": 78
} | {
"line": 63,
"column": 79
} | [
{
"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\nx : R\nhx : x ∈ ⇑(Algebra.intNorm R S) '' ↑⊥\n⊢ x = ... | [
"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\nx : R\nhx : x ∈ ⇑(Algebra.intNorm R S) '' ↑⊥\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Discriminant.Different | {
"line": 52,
"column": 10
} | {
"line": 52,
"column": 21
} | {
"line": 52,
"column": 22
} | [
{
"pp": "K : Type u_1\n𝒪 : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : NumberField K\ninst✝⁵ : CommRing 𝒪\ninst✝⁴ : Algebra 𝒪 K\ninst✝³ : IsFractionRing 𝒪 K\ninst✝² : IsDedekindDomain 𝒪\ninst✝¹ : CharZero 𝒪\ninst✝ : Module.Finite ℤ 𝒪\n⊢ ↑(differentIdeal ℤ 𝒪) ≤ 1⁻¹",
"ppTerm": "?m.147",
"assigned": true... | [
"K : Type u_1\n𝒪 : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : NumberField K\ninst✝⁵ : CommRing 𝒪\ninst✝⁴ : Algebra 𝒪 K\ninst✝³ : IsFractionRing 𝒪 K\ninst✝² : IsDedekindDomain 𝒪\ninst✝¹ : CharZero 𝒪\ninst✝ : Module.Finite ℤ 𝒪\n⊢ ↑(differentIdeal ℤ 𝒪) ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Discriminant.Different | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 45
} | {
"line": 61,
"column": 46
} | [
{
"pp": "K : Type u_1\n𝒪 : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : NumberField K\ninst✝⁵ : CommRing 𝒪\ninst✝⁴ : Algebra 𝒪 K\ninst✝³ : IsFractionRing 𝒪 K\ninst✝² : IsDedekindDomain 𝒪\ninst✝¹ : CharZero 𝒪\ninst✝ : Module.Finite ℤ 𝒪\nthis✝ :\n (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) ≃ₗ[𝒪... | [
"K : Type u_1\n𝒪 : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : NumberField K\ninst✝⁵ : CommRing 𝒪\ninst✝⁴ : Algebra 𝒪 K\ninst✝³ : IsFractionRing 𝒪 K\ninst✝² : IsDedekindDomain 𝒪\ninst✝¹ : CharZero 𝒪\ninst✝ : Module.Finite ℤ 𝒪\nthis✝ :\n (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) ≃ₗ[𝒪]\n ↥↑(↑(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Discriminant.Different | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 98
} | {
"line": 71,
"column": 4
} | [
{
"pp": "case refine_3\nK : Type u_1\n𝒪 : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : NumberField K\ninst✝⁵ : CommRing 𝒪\ninst✝⁴ : Algebra 𝒪 K\ninst✝³ : IsFractionRing 𝒪 K\ninst✝² : IsDedekindDomain 𝒪\ninst✝¹ : CharZero 𝒪\ninst✝ : Module.Finite ℤ 𝒪\nthis✝ :\n (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIde... | [
"case refine_3\nK : Type u_1\n𝒪 : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : NumberField K\ninst✝⁵ : CommRing 𝒪\ninst✝⁴ : Algebra 𝒪 K\ninst✝³ : IsFractionRing 𝒪 K\ninst✝² : IsDedekindDomain 𝒪\ninst✝¹ : CharZero 𝒪\ninst✝ : Module.Finite ℤ 𝒪\nthis✝ :\n (↥↑1 ⧸ Submodule.comap (↑1).subtype ↑↑(differentIdeal ℤ 𝒪)) ≃ₗ... | rw [AddSubgroup.toIntSubmodule_closure, ← LinearMap.BilinForm.dualSubmodule_span_of_basis, hb] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 32
} | {
"line": 284,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\nS : Type u_3\ninst✝⁸ : CommRing S\ninst✝⁷ : IsDomain S\ninst✝⁶ : IsIntegrallyClosed R\ninst✝⁵ : IsIntegrallyClosed S\ninst✝⁴ : Algebra R S\ninst✝³ : Module.Finite R S\ninst✝² : IsTorsionFree R S\ninst✝¹ : IsDedekindDomain R\ninst✝ : IsDedekindDom... | [
"R : Type u_1\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\nS : Type u_3\ninst✝⁸ : CommRing S\ninst✝⁷ : IsDomain S\ninst✝⁶ : IsIntegrallyClosed R\ninst✝⁵ : IsIntegrallyClosed S\ninst✝⁴ : Algebra R S\ninst✝³ : Module.Finite R S\ninst✝² : IsTorsionFree R S\ninst✝¹ : IsDedekindDomain R\ninst✝ : IsDedekindDomain S\n⊢ (re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 31
} | {
"line": 288,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\nS : Type u_3\ninst✝⁸ : CommRing S\ninst✝⁷ : IsDomain S\ninst✝⁶ : IsIntegrallyClosed R\ninst✝⁵ : IsIntegrallyClosed S\ninst✝⁴ : Algebra R S\ninst✝³ : Module.Finite R S\ninst✝² : IsTorsionFree R S\ninst✝¹ : IsDedekindDomain R\ninst✝ : IsDedekindDom... | [
"R : Type u_1\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\nS : Type u_3\ninst✝⁸ : CommRing S\ninst✝⁷ : IsDomain S\ninst✝⁶ : IsIntegrallyClosed R\ninst✝⁵ : IsIntegrallyClosed S\ninst✝⁴ : Algebra R S\ninst✝³ : Module.Finite R S\ninst✝² : IsTorsionFree R S\ninst✝¹ : IsDedekindDomain R\ninst✝ : IsDedekindDomain S\n⊢ (re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 387,
"column": 2
} | {
"line": 389,
"column": 75
} | {
"line": 390,
"column": 2
} | [
{
"pp": "case pos\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... | [
"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 : IsDedekindDom... | · refine ⟨1, ?_⟩
have : P.LiesOver ⊥ := hp ▸ hPp
rw [hp, eq_bot_of_liesOver_bot R P, relNorm_bot, bot_pow (one_ne_zero)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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