module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 347,
"column": 2
} | {
"line": 352,
"column": 50
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ Bornology.IsBounded (convexBodySum K B)",
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Prod.seminormedAddGroup",
"Nu... | 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
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ Bornology.IsBounded (convexBodySum K B)",
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Prod.seminormedAddGroup",
"Nu... | 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
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ Bornology.IsBounded (convexBodySum K B)",
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Prod.seminormedAddGroup",
"Nu... | 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.InfinitePlace.Ramification | {
"line": 274,
"column": 7
} | {
"line": 274,
"column": 18
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"IsSelfAdjoint",
"NumberField.ComplexEmbedding.conjugate",
"NumberF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 281,
"column": 26
} | {
"line": 281,
"column": 63
} | [
{
"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))",
"usedConstants": [
"Eq.mpr",
"NumberField.ComplexEmbedding.conjugate",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 13
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"IsSelfAdjoint",
"NumberField.Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 304,
"column": 26
} | {
"line": 304,
"column": 63
} | [
{
"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))",
"usedConstants": [
"Eq.mpr",
"NumberField.ComplexEmbedding.conjugate",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 308,
"column": 23
} | {
"line": 308,
"column": 34
} | [
{
"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",
"usedConstants": [
"Algebra.algebraMap",
"NumberField.InfinitePlace... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 337,
"column": 62
} | {
"line": 337,
"column": 73
} | [
{
"pp": "k : Type u_1\ninst✝³ : Field k\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nφ : K →+* ℂ\nH : ∀ (σ : Gal(K/k)), ComplexEmbedding.IsConj φ σ → σ = 1\nhφ : ¬ComplexEmbedding.IsConj φ 1 ∧ ComplexEmbedding.IsReal (φ.comp (algebraMap k K))\nthis✝¹ : Algebra k ℂ := (φ.comp (alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 345,
"column": 61
} | {
"line": 350,
"column": 83
} | [
{
"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",
"usedConstants": [
"or_congr",
"Eq.mpr",
"AlgEquiv.instEquivLike",
"instHSMul",
"InvOneClass.toOne"... | 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": 488,
"column": 19
} | {
"line": 488,
"column": 30
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 505,
"column": 19
} | {
"line": 505,
"column": 30
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 485,
"column": 4
} | {
"line": 489,
"column": 34
} | [
{
"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": 605,
"column": 2
} | {
"line": 605,
"column": 30
} | [
{
"pp": "case h\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\nk✝ : K\n⊢ (w.comap (algebraMap K L)) k✝ = v k✝",
"usedConstants": [
"NumberField.InfinitePlace.instFunLikeReal",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 558,
"column": 23
} | {
"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
} | [
{
"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
} | [
{
"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.InfinitePlace.Ramification | {
"line": 617,
"column": 14
} | {
"line": 617,
"column": 25
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 641,
"column": 8
} | {
"line": 641,
"column": 43
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 705,
"column": 20
} | {
"line": 705,
"column": 31
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 706,
"column": 20
} | {
"line": 706,
"column": 31
} | [
{
"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,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 736,
"column": 4
} | {
"line": 736,
"column": 45
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 737,
"column": 4
} | {
"line": 737,
"column": 45
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 747,
"column": 4
} | {
"line": 747,
"column": 43
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"_privat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 599,
"column": 20
} | {
"line": 599,
"column": 31
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 773,
"column": 50
} | {
"line": 773,
"column": 61
} | [
{
"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",
"usedConstants": [
"Set.Finite.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 101,
"column": 20
} | {
"line": 105,
"column": 100
} | [
{
"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.RamificationInertia.Galois | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 30
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 86,
"column": 8
} | {
"line": 86,
"column": 13
} | [
{
"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... | h_nz, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Galois | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 13
} | [
{
"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.IsMaximal\nP : Ideal S\ninst✝² : P.LiesOver p\ninst✝¹ : P.IsMaximal\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 36
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℂ ((K →+* ℂ) → ℂ)",
"usedConstants": [
"Pi.Function.module",
"Semiring.toModule",
"Pi.addCommMonoid",
"RingHom",
"Finite.of_fintype",
"Field.toSemifield",
"NonUn... | 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.ClassNumber | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 43
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 21
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nh :\n ∀ ⦃I : ↥(Ideal (𝓞 K))⁰⦄,\n (↑I).IsPrime →\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 : ↑(a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 17
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : { w // w.IsReal }\nA : AffineSubspace ℝ (mixedSpace K) :=\n { carrier := {x | x.1 w = 0}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }.toAffineSubspace\nh : A = ⊤\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 68
} | [
{
"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 ^ (span {↑p}).inertiaDeg P ≤\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 68
} | [
{
"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 ^ (span {↑p}).inertiaDeg P ≤\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.ClassNumber | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 94
} | [
{
"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 ^ (span {↑p}).inertiaDeg P ≤\n ⌊(4 / π... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 592,
"column": 52
} | {
"line": 592,
"column": 73
} | [
{
"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... | Fintype.sum_sum_type, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 596,
"column": 32
} | {
"line": 596,
"column": 52
} | [
{
"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 ... | Equiv.prodComm_symm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 640,
"column": 12
} | {
"line": 640,
"column": 57
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ Function.Injective fun r ↦ r • 1",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 423,
"column": 12
} | {
"line": 423,
"column": 64
} | [
{
"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 | ... | minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 423,
"column": 8
} | {
"line": 424,
"column": 59
} | [
{
"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.RingTheory.FractionalIdeal.Extended | {
"line": 87,
"column": 46
} | {
"line": 87,
"column": 78
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 423,
"column": 8
} | {
"line": 424,
"column": 59
} | [
{
"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.NumberField.Discriminant.Basic | {
"line": 423,
"column": 8
} | {
"line": 424,
"column": 59
} | [
{
"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.RingTheory.FractionalIdeal.Extended | {
"line": 106,
"column": 45
} | {
"line": 106,
"column": 56
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 874,
"column": 2
} | {
"line": 874,
"column": 36
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ IsZLattice ℝ (euclidean.integerLattice K)",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
"WithLp.instProdNormedAddCommGr... | simp_rw [euclidean.integerLattice] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 230,
"column": 4
} | {
"line": 231,
"column": 11
} | [
{
"pp": "case refine_1\nA : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁸ : CommRing A\ninst✝¹⁷ : IsDomain A\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : IsDomain B\ninst✝¹⁴ : Algebra A B\ninst✝¹³ : IsTorsionFree A B\ninst✝¹² : Field K\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra A K\ninst✝⁹ : Algebra B L\ninst✝⁸ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1112,
"column": 2
} | {
"line": 1112,
"column": 17
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : InfinitePlace K\nA : AffineSubspace ℝ (realSpace K) :=\n { carrier := {x | x w = 0}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }.toAffineSubspace\nh : A = ⊤\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1156,
"column": 41
} | {
"line": 1158,
"column": 35
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx : mixedSpace K\nw : { w // w.IsComplex }\n⊢ normAtComplexPlaces x ↑w = ‖x.2 w‖",
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"NumberField.InfinitePlace.IsComplex",
"MonoidWithZeroHom.funLike",
... | 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": 1215,
"column": 4
} | {
"line": 1216,
"column": 53
} | [
{
"pp": "case h.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",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.lattice",
"abs",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1217,
"column": 4
} | {
"line": 1218,
"column": 56
} | [
{
"pp": "case h.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",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"NumberF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 472,
"column": 12
} | {
"line": 472,
"column": 64
} | [
{
"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✝ :... | minpoly.isIntegrallyClosed_eq_field_fractions' ℚ hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 472,
"column": 8
} | {
"line": 473,
"column": 59
} | [
{
"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": 472,
"column": 8
} | {
"line": 473,
"column": 59
} | [
{
"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": 472,
"column": 8
} | {
"line": 473,
"column": 59
} | [
{
"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.RingTheory.Ideal.Int | {
"line": 99,
"column": 47
} | {
"line": 99,
"column": 58
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Int | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 31
} | [
{
"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",
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Submodule.Quotient.instZeroQuotient",
"Dvd.dvd",
"Semiring.toModule",... | Quotient.eq_zero_iff_mem, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.LinearDisjoint | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 55
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝³⁴ : CommRing A\ninst✝³³ : Field K\ninst✝³² : Algebra A K\ninst✝³¹ : IsFractionRing A K\ninst✝³⁰ : CommRing B\ninst✝²⁹ : Field L\ninst✝²⁸ : Algebra B L\ninst✝²⁷ : Algebra A L\ninst✝²⁶ : Algebra K L\ninst✝²⁵ : FiniteDimensional K L\ninst✝²⁴ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Instances | {
"line": 127,
"column": 21
} | {
"line": 127,
"column": 32
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 63,
"column": 67
} | {
"line": 63,
"column": 78
} | [
{
"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 = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.NormalClosure | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 13
} | [
{
"pp": "case a\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsDomain S\ninst✝³ : Algebra R S\ninst✝² : Module.IsTorsionFree R S\ninst✝¹ : Module.Finite R S\ninst✝ : PerfectField K\nx✝ : K\n⊢ ((algebraMap K (FractionRing T)).comp ↑(FractionRing.algEquiv R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Discriminant.Different | {
"line": 52,
"column": 10
} | {
"line": 52,
"column": 21
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Discriminant.Different | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 45
} | [
{
"pp": "case h\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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 282,
"column": 2
} | {
"line": 282,
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 31
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 385,
"column": 2
} | {
"line": 387,
"column": 75
} | [
{
"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... | · 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 |
Mathlib.NumberTheory.NumberField.Discriminant.Different | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 98
} | [
{
"pp": "case refine_3.a\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 ↑↑(differentI... | 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.DedekindDomain.Different | {
"line": 122,
"column": 74
} | {
"line": 122,
"column": 87
} | [
{
"pp": "case neg\nA : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : CommRing B\ninst✝⁷ : Field L\ninst✝⁶ : Algebra A K\ninst✝⁵ : Algebra B L\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra K L\ninst✝² : Algebra A L\ninst✝¹ : IsScalarTower A K L\ninst✝ : IsScalarTowe... | true_implies, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 21
} | [
{
"pp": "case neg\nA : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁰ : CommRing A\ninst✝⁹ : Field K\ninst✝⁸ : CommRing B\ninst✝⁷ : Field L\ninst✝⁶ : Algebra A K\ninst✝⁵ : Algebra B L\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra K L\ninst✝² : Algebra A L\ninst✝¹ : IsScalarTower A K L\ninst✝ : IsScalarTowe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 29
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝¹⁷ : CommRing A\ninst✝¹⁶ : Field K\ninst✝¹⁵ : CommRing B\ninst✝¹⁴ : Field L\ninst✝¹³ : Algebra A K\ninst✝¹² : Algebra B L\ninst✝¹¹ : Algebra A B\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : Algebra A L\ninst✝⁸ : IsScalarTower A K L\ninst✝⁷ : IsScalarTower ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 30
} | [
{
"pp": "case refine_2.succ\np : ℕ\nhp : Fact (Nat.Prime p)\nn i k : ℕ\nhi : p * k < p * (p ^ n * (p - 1))\nhk : i = p * k\nhn : (Polynomial.map (Int.castRingHom (ZMod p)) ((cyclotomic (p ^ (n + 1)) ℤ).comp (X + C 1))).coeff k = 0\n⊢ ((cyclotomic (p ^ (n + 1)) (ZMod p)).comp (X + 1)).coeff k = 0",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Prime | {
"line": 69,
"column": 57
} | {
"line": 69,
"column": 78
} | [
{
"pp": "α : Type u_1\ninst✝ : CommRing α\np : α\nh1 : p ≠ 0\nh2 : ¬IsUnit p\nh3 : ∀ (a b : α), p ∣ a * b → p ∣ a ∨ p ∣ b\n⊢ ∀ (a b : α), -p ∣ a * b → -p ∣ a ∨ -p ∣ b",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"Dvd.dvd",
"NonUnitalCommRing.toNonUnit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 421,
"column": 2
} | {
"line": 421,
"column": 13
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_5\ninst✝³⁸ : CommRing A\ninst✝³⁷ : Field K\ninst✝³⁶ : CommRing B\ninst✝³⁵ : Field L\ninst✝³⁴ : Algebra A K\ninst✝³³ : Algebra B L\ninst✝³² : Algebra A B\ninst✝³¹ : Algebra K L\ninst✝³⁰ : Algebra A L\ninst✝²⁹ : IsScalarTower A K L\ninst✝²⁸ : IsScalarTow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 435,
"column": 2
} | {
"line": 436,
"column": 47
} | [
{
"pp": "case h\nA : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_5\ninst✝⁴⁰ : CommRing A\ninst✝³⁹ : Field K\ninst✝³⁸ : CommRing B\ninst✝³⁷ : Field L\ninst✝³⁶ : Algebra A K\ninst✝³⁵ : Algebra B L\ninst✝³⁴ : Algebra A B\ninst✝³³ : Algebra K L\ninst✝³² : Algebra A L\ninst✝³¹ : IsScalarTower A K L\ninst✝³⁰ : IsS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 489,
"column": 6
} | {
"line": 489,
"column": 16
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : IsDomain A\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : IsTorsionFree A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 491,
"column": 4
} | {
"line": 491,
"column": 70
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝⁹ : CommRing A\ninst✝⁸ : CommRing B\ninst✝⁷ : Algebra A B\ninst✝⁶ : IsDomain A\ninst✝⁵ : IsIntegrallyClosed A\ninst✝⁴ : IsDedekindDomain B\ninst✝³ : IsTorsionFree A B\ninst✝² : Algebra.IsIntegral A B\ninst✝¹ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral | {
"line": 246,
"column": 4
} | {
"line": 246,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 621,
"column": 6
} | {
"line": 621,
"column": 71
} | [
{
"pp": "case neg\nA : 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... | rw [Function.comp_apply, coeff_eq_zero_of_natDegree_lt, mul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 693,
"column": 42
} | {
"line": 693,
"column": 59
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²¹ : CommRing A\ninst✝²⁰ : Field K\ninst✝¹⁹ : CommRing B\ninst✝¹⁸ : Field L\ninst✝¹⁷ : Algebra A K\ninst✝¹⁶ : Algebra B L\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Algebra A L\ninst✝¹² : IsScalarTower A K L\ninst✝¹¹ : IsScalarTow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 715,
"column": 49
} | {
"line": 715,
"column": 60
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²¹ : CommRing A\ninst✝²⁰ : Field K\ninst✝¹⁹ : CommRing B\ninst✝¹⁸ : Field L\ninst✝¹⁷ : Algebra A K\ninst✝¹⁶ : Algebra B L\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Algebra A L\ninst✝¹² : IsScalarTower A K L\ninst✝¹¹ : IsScalarTow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 717,
"column": 59
} | {
"line": 717,
"column": 70
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²¹ : CommRing A\ninst✝²⁰ : Field K\ninst✝¹⁹ : CommRing B\ninst✝¹⁸ : Field L\ninst✝¹⁷ : Algebra A K\ninst✝¹⁶ : Algebra B L\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Algebra A L\ninst✝¹² : IsScalarTower A K L\ninst✝¹¹ : IsScalarTow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Units.DirichletTheorem | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (𝓞 K)ˣ\nh : ∑ i, ↑(↑i).mult * Real.log (↑i ((algebraMap (𝓞 K) K) ↑x)) = -↑w₀.mult * Real.log (w₀ ((algebraMap (𝓞 K) K) ↑x))\n⊢ ∑ w, (logEmbedding K) (Additive.ofMul x) w = -↑w₀.mult * Real.log (w₀ ((algebraMap (𝓞 K) K) ↑x))",
"usedConst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 718,
"column": 74
} | {
"line": 718,
"column": 85
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u\nB : Type u_3\ninst✝²¹ : CommRing A\ninst✝²⁰ : Field K\ninst✝¹⁹ : CommRing B\ninst✝¹⁸ : Field L\ninst✝¹⁷ : Algebra A K\ninst✝¹⁶ : Algebra B L\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : Algebra A L\ninst✝¹² : IsScalarTower A K L\ninst✝¹¹ : IsScalarTow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 99,
"column": 48
} | {
"line": 99,
"column": 59
} | [
{
"pp": "p : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nx : K\nh : IsIntegral ℤ x\nu : ℤˣ\nn : ℕ\nhcycl : IsCyclotomicExtension {p ^ 0} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ 0)\nB : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ\nhint : IsIntegral ℤ B.gen\nthis : Fi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 120,
"column": 56
} | {
"line": 120,
"column": 67
} | [
{
"pp": "p : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nx : K\nh : IsIntegral ℤ x\nu : ℤˣ\nn n✝ : ℕ\nhcycl : IsCyclotomicExtension {p ^ (n✝ + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ (n✝ + 1))\nB : PowerBasis ℚ K := IsPrimitiveRoot.subOnePowerBasis ℚ hζ\nhint : IsIntegral ℤ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 258,
"column": 4
} | {
"line": 258,
"column": 15
} | [
{
"pp": "case refine_1\np k : ℕ\nK : Type u\ninst✝² : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝¹ : CharZero K\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhodd : p ≠ 2\nthis : NumberField K := numberField {p ^ (k + 1)} ℚ K\nh : hζ.toInteger = 1\n⊢ ζ ^ 1 = 1",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 15
} | [
{
"pp": "case refine_1\nk : ℕ\nK : Type u\ninst✝² : Field K\nζ : K\ninst✝¹ : CharZero K\ninst✝ : IsCyclotomicExtension {2 ^ (k + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (2 ^ (k + 1))\nthis : NumberField K\nh : hζ.toInteger = 1\n⊢ ζ ^ 1 = 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Nat.instOn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 284,
"column": 4
} | {
"line": 286,
"column": 11
} | [
{
"pp": "case h.e'_3.a\nK : Type u\ninst✝² : Field K\nζ : K\ninst✝¹ : CharZero K\nthis : NumberField K\ninst✝ : IsCyclotomicExtension {2 ^ (0 + 1)} ℚ K\nhζ : IsPrimitiveRoot ζ (2 ^ (0 + 1))\n⊢ (Algebra.norm ℚ) ((algebraMap (𝓞 K) K) (hζ.toInteger - 1)) = -2",
"usedConstants": [
"Eq.mpr",
"NonAss... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 799,
"column": 6
} | {
"line": 799,
"column": 95
} | [
{
"pp": "case neg.refine_1.smul\nA : Type u_1\nB : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra A B\ninst✝⁵ : IsDomain A\ninst✝⁴ : IsDedekindDomain A\ninst✝³ : IsDedekindDomain B\ninst✝² : IsTorsionFree A B\ninst✝¹ : Module.Finite A B\ninst✝ : Algebra.IsSeparable (FractionRing A) (Fracti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 59
} | [
{
"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))\n⊢ Prime hζ.subOneIntegralPowerBasisOfPrimePow.gen",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 309,
"column": 55
} | {
"line": 310,
"column": 81
} | [
{
"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))\n⊢ Prime hζ.subOneIntegralPowerBasisOfPrimePow.gen",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
... | by
simpa only [subOneIntegralPowerBasisOfPrimePow_gen] using hζ.zeta_sub_one_prime | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 822,
"column": 4
} | {
"line": 822,
"column": 35
} | [
{
"pp": "A : Type u_1\nB : Type u_3\ninst✝¹² : CommRing A\ninst✝¹¹ : CommRing B\ninst✝¹⁰ : Algebra A B\ninst✝⁹ : IsDomain A\ninst✝⁸ : IsDedekindDomain A\ninst✝⁷ : IsDedekindDomain B\ninst✝⁶ : IsTorsionFree A B\ninst✝⁵ : Module.Finite A B\ninst✝⁴ : Algebra.IsSeparable (FractionRing A) (FractionRing B)\np : Ideal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 38
} | [
{
"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\n⊢ (Algebra.norm ℤ) (hζ.toInteger - 1) = ↑p",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 381,
"column": 2
} | {
"line": 381,
"column": 13
} | [
{
"pp": "K : Type u\ninst✝² : Field K\nζ : K\ninst✝¹ : CharZero K\ninst✝ : IsCyclotomicExtension {2} ℚ K\nhζ✝ : IsPrimitiveRoot ζ 2\nhζ : IsPrimitiveRoot ζ (2 ^ (0 + 1))\n⊢ (Algebra.norm ℤ) (hζ✝.toInteger - 1) = -2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Different | {
"line": 827,
"column": 4
} | {
"line": 827,
"column": 51
} | [
{
"pp": "case refine_1\nA : Type u_1\nB : Type u_3\ninst✝¹² : CommRing A\ninst✝¹¹ : CommRing B\ninst✝¹⁰ : Algebra A B\ninst✝⁹ : IsDomain A\ninst✝⁸ : IsDedekindDomain A\ninst✝⁷ : IsDedekindDomain B\ninst✝⁶ : IsTorsionFree A B\ninst✝⁵ : Module.Finite A B\ninst✝⁴ : Algebra.IsSeparable (FractionRing A) (FractionRin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 387,
"column": 55
} | {
"line": 387,
"column": 66
} | [
{
"pp": "p : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nhcycl : IsCyclotomicExtension {p} ℚ K\nhζ : IsPrimitiveRoot ζ p\nh : p ≠ 2\n⊢ IsCyclotomicExtension {p ^ (0 + 1)} ℚ K",
"usedConstants": [
"Eq.mpr",
"CommRing",
"congrArg",
"CommSemiring... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 388,
"column": 53
} | {
"line": 388,
"column": 64
} | [
{
"pp": "p : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nhcycl : IsCyclotomicExtension {p} ℚ K\nhζ : IsPrimitiveRoot ζ p\nh : p ≠ 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K\n⊢ IsPrimitiveRoot ζ (p ^ (0 + 1))",
"usedConstants": [
"Eq.mpr",
"congrArg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 416,
"column": 55
} | {
"line": 416,
"column": 66
} | [
{
"pp": "p : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nhcycl : IsCyclotomicExtension {p} ℚ K\nhζ : IsPrimitiveRoot ζ p\nhodd : p ≠ 2\n⊢ IsCyclotomicExtension {p ^ (0 + 1)} ℚ K",
"usedConstants": [
"Eq.mpr",
"CommRing",
"congrArg",
"CommSemir... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Basic | {
"line": 417,
"column": 53
} | {
"line": 417,
"column": 64
} | [
{
"pp": "p : ℕ\nK : Type u\ninst✝¹ : Field K\nζ : K\nhp : Fact (Nat.Prime p)\ninst✝ : CharZero K\nhcycl : IsCyclotomicExtension {p} ℚ K\nhζ : IsPrimitiveRoot ζ p\nhodd : p ≠ 2\nthis : IsCyclotomicExtension {p ^ (0 + 1)} ℚ K\n⊢ IsPrimitiveRoot ζ (p ^ (0 + 1))",
"usedConstants": [
"Eq.mpr",
"congr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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