module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.QuasiFinite.Polynomial | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 26
} | {
"line": 72,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nf : R[X] →ₐ[R] S\nhf : Function.Surjective ⇑f\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : Algebra.WeaklyQuasiFiniteAt R P\nH : RingHom.ker f ≤ Ideal.map C (Ideal.under R P)\n⊢ False",
"ppTerm": "?m.49",
"a... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nf : R[X] →ₐ[R] S\nhf : Function.Surjective ⇑f\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : Algebra.WeaklyQuasiFiniteAt R P\nH : RingHom.ker f ≤ Ideal.map C (Ideal.under R P)\nalgInst✝ : Algebra R[X] S := f.toAlgebra\n⊢ False"
... | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 216,
"column": 4
} | {
"line": 217,
"column": 52
} | {
"line": 217,
"column": 53
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\nq : Ideal T\ninst✝¹ : q.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R q\nthis : QuasiFiniteAt S (Ideal.map (I... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\nq : Ideal T\ninst✝¹ : q.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R q\nthis : QuasiFiniteAt S (Ideal.map (Ideal.Quotien... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 239,
"column": 4
} | {
"line": 240,
"column": 63
} | {
"line": 241,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFinit... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFiniteAt p.Residu... | rw [← IsLocalization.AtPrime.map_eq_maximalIdeal p, Ideal.map_le_iff_le_comap,
← Ideal.comap_coe (F := AlgHom _ _ _), Ideal.comap_comap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 47
} | {
"line": 241,
"column": 48
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFinit... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFiniteAt p.Residu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 162,
"column": 4
} | {
"line": 163,
"column": 35
} | {
"line": 164,
"column": 6
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np : FractionRing K[X]\nh : { toFractionRing := p } ≠ 0\nthis : p ≠ 0\n⊢ { toFractionRing := p } * { toFractionRing := p }⁻¹ = 1",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"RatFunc.ofFractionRing.injEq",
"Eq.mpr"... | [
"K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np : FractionRing K[X]\nh : { toFractionRing := p } ≠ 0\nthis : p ≠ 0\n⊢ p * p⁻¹ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 254,
"column": 8
} | {
"line": 254,
"column": 58
} | {
"line": 254,
"column": 59
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFinit... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFiniteAt p.Residu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 330,
"column": 6
} | {
"line": 330,
"column": 32
} | {
"line": 330,
"column": 33
} | [
{
"pp": "K : Type u\ninst✝⁶ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁵ : CommGroupWithZero G₀\ninst✝⁴ : Field L\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]... | [
"K : Type u\ninst✝⁶ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁵ : CommGroupWithZero G₀\ninst✝⁴ : Field L\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰\nf : R⟮X⟯\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 343,
"column": 42
} | {
"line": 343,
"column": 53
} | {
"line": 343,
"column": 54
} | [
{
"pp": "K : Type u\ninst✝⁶ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁵ : CommGroupWithZero G₀\ninst✝⁴ : Field L\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]... | [
"K : Type u\ninst✝⁶ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁵ : CommGroupWithZero G₀\ninst✝⁴ : Field L\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰\np : R[X]\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 360,
"column": 2
} | {
"line": 362,
"column": 45
} | {
"line": 362,
"column": 46
} | [
{
"pp": "case H.H\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰\nhf : Function.Injective ⇑φ\ny✝¹ y✝ : R[X] × ↥R[X]⁰\nh :\n (map φ hφ) { toFractionRing := Localiza... | [
"case H.H\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : FunLike F R[X] S[X]\ninst✝ : MonoidHomClass F R[X] S[X]\nφ : F\nhφ : R[X]⁰ ≤ Submonoid.comap φ S[X]⁰\nhf : Function.Injective ⇑φ\ny✝¹ y✝ : R[X] × ↥R[X]⁰\nh :\n (map φ hφ) { toFractionRing := Localization.mk y✝¹.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.ResidueField.Polynomial | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 15
} | {
"line": 118,
"column": 16
} | [
{
"pp": "case refine_4\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nI : Ideal R\ninst✝⁵ : I.IsPrime\nJ : Ideal R[X]\ninst✝⁴ : J.IsPrime\ninst✝³ : J.LiesOver I\ninst✝² : Algebra (Localization.AtPrime I) (Localization.AtPrime J)\ninst✝¹ : Localization.AtPrime.IsLies... | [
"case refine_4\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nI : Ideal R\ninst✝⁵ : I.IsPrime\nJ : Ideal R[X]\ninst✝⁴ : J.IsPrime\ninst✝³ : J.LiesOver I\ninst✝² : Algebra (Localization.AtPrime I) (Localization.AtPrime J)\ninst✝¹ : Localization.AtPrime.IsLiesOverAlgebra ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 263,
"column": 6
} | {
"line": 263,
"column": 24
} | {
"line": 263,
"column": 25
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\nins... | [
"case refine_1\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 39
} | {
"line": 162,
"column": 4
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyQuasiFinite f\ny : ↥Y\n⊢ IsDiscrete (⇑f ⁻¹' {y})",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryThe... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyQuasiFinite f\ny : ↥Y\n⊢ IsDiscrete (⇑f ⁻¹' {y})"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 39
} | {
"line": 172,
"column": 40
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : LocallyQuasiFinite f\ninst✝ : QuasiCompact f\ny : ↥Y\n⊢ (⇑f ⁻¹' {y}).Finite",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.ConcreteCategory.hom",
"CommRingCat",
... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : LocallyQuasiFinite f\ninst✝ : QuasiCompact f\ny : ↥Y\n⊢ (⇑f ⁻¹' {y}).Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 603,
"column": 2
} | {
"line": 603,
"column": 13
} | {
"line": 603,
"column": 14
} | [
{
"pp": "K : Type u\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\nL : Type u_1\ninst✝ : Field L\nφ : K[X] →+* L\nhφ : K[X]⁰ ≤ Submonoid.comap φ L⁰\nx : K[X]\n⊢ (liftRingHom φ hφ) ((algebraMap K[X] K⟮X⟯) x) = φ x",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"K : Type u\ninst✝² : CommRing K\ninst✝¹ : IsDomain K\nL : Type u_1\ninst✝ : Field L\nφ : K[X] →+* L\nhφ : K[X]⁰ ≤ Submonoid.comap φ L⁰\nx : K[X]\n⊢ (liftRingHom φ hφ) ((algebraMap K[X] K⟮X⟯) x) = φ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 191,
"column": 58
} | {
"line": 191,
"column": 69
} | {
"line": 191,
"column": 70
} | [
{
"pp": "X : Scheme\nU : X.AffineZariskiSite\ns : Set ↑Γ(X, U.toOpens)\nx : ↥X\nhx : x ∈ U.toOpens\nf : ↑Γ(X, ↑U)\nhfs : f ∈ s\nhxf : x ∈ X.basicOpen f\n⊢ x ∈ (U.basicOpen (f * 1)).toOpens",
"ppTerm": "?m.170",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpac... | [
"X : Scheme\nU : X.AffineZariskiSite\ns : Set ↑Γ(X, U.toOpens)\nx : ↥X\nhx : x ∈ U.toOpens\nf : ↑Γ(X, ↑U)\nhfs : f ∈ s\nhxf : x ∈ X.basicOpen f\n⊢ x ∈ X.basicOpen f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 712,
"column": 35
} | {
"line": 712,
"column": 46
} | {
"line": 712,
"column": 47
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nP : K⟮X⟯ → Prop\nx : K⟮X⟯\nf : ∀ (p q : K[X]), q ≠ 0 → P ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q)\np q : K[X]\nhq : q ≠ 0\n⊢ P (RatFunc.mk p q)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nP : K⟮X⟯ → Prop\nx : K⟮X⟯\nf : ∀ (p q : K[X]), q ≠ 0 → P ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q)\np q : K[X]\nhq : q ≠ 0\n⊢ P ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 912,
"column": 43
} | {
"line": 912,
"column": 81
} | {
"line": 914,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝ : Field K\n⊢ num 1 = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"normalize_one",
"RatFunc.num_div",
"Polynomial.monic_one._simp_1",
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAdd... | [] | convert! num_div (1 : K[X]) 1 <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 912,
"column": 43
} | {
"line": 912,
"column": 81
} | {
"line": 914,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝ : Field K\n⊢ num 1 = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"normalize_one",
"RatFunc.num_div",
"Polynomial.monic_one._simp_1",
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAdd... | [] | convert! num_div (1 : K[X]) 1 <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 912,
"column": 43
} | {
"line": 912,
"column": 81
} | {
"line": 914,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝ : Field K\n⊢ num 1 = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"normalize_one",
"RatFunc.num_div",
"Polynomial.monic_one._simp_1",
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAdd... | [] | convert! num_div (1 : K[X]) 1 <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 922,
"column": 4
} | {
"line": 922,
"column": 71
} | {
"line": 922,
"column": 72
} | [
{
"pp": "K : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ (q / gcd p q).leadingCoeff⁻¹ ≠ 0",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"DivisionCommMonoid.toDiv... | [
"K : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ ¬q / gcd p q = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 928,
"column": 61
} | {
"line": 928,
"column": 72
} | {
"line": 928,
"column": 73
} | [
{
"pp": "K : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ C (q / gcd p q).leadingCoeff⁻¹ * (p / gcd p q) ∣ p",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ C (q / gcd p q).leadingCoeff⁻¹ * (p / gcd p q) ∣ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 971,
"column": 4
} | {
"line": 971,
"column": 71
} | {
"line": 971,
"column": 72
} | [
{
"pp": "case neg\nK : Type u\ninst✝ : Field K\np q : K[X]\nhq : ¬q = 0\n⊢ (q / gcd p q).leadingCoeff⁻¹ ≠ 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"DivisionCommM... | [
"case neg\nK : Type u\ninst✝ : Field K\np q : K[X]\nhq : ¬q = 0\n⊢ ¬q / gcd p q = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 977,
"column": 2
} | {
"line": 977,
"column": 66
} | {
"line": 978,
"column": 2
} | [
{
"pp": "case f\nK : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\n⊢ (algebraMap K[X] K⟮X⟯) ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q).num /\n (algebraMap K[X] K⟮X⟯) ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q).denom =\n (algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q",
"p... | [
"case f\nK : Type u\ninst✝ : Field K\np q : K[X]\nhq : q ≠ 0\nq_div_ne_zero : q / gcd p q ≠ 0\n⊢ (algebraMap K[X] K⟮X⟯) ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q).num /\n (algebraMap K[X] K⟮X⟯) ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q).denom =\n (algebraMap K[X] K⟮X⟯) p / (algebraMap ... | have q_div_ne_zero : q / gcd p q ≠ 0 := right_div_gcd_ne_zero hq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 90,
"column": 35
} | {
"line": 90,
"column": 61
} | {
"line": 90,
"column": 62
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\nx : S\nh : P.HasMap x\nf : S →ₐ[R] T\n⊢ IsUnit ((aeval (f x)) P.g)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstan... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\nx : S\nh : P.HasMap x\nf : S →ₐ[R] T\n⊢ IsUnit (f ((aeval x) P.g))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 96,
"column": 11
} | {
"line": 96,
"column": 28
} | {
"line": 96,
"column": 29
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nx : S\nh : P.HasMap x\np₁ p₂ : R[X]\nn : ℕ\ne : derivative P.f * p₁ + P.f * p₂ = P.g ^ n\n⊢ (aeval x) P.g ^ n = (aeval x) (derivative P.f) * ?m.78",
"ppTerm": "?m.79",
"assigned":... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nx : S\nh : P.HasMap x\np₁ p₂ : R[X]\nn : ℕ\ne : derivative P.f * p₁ + P.f * p₂ = P.g ^ n\n⊢ (aeval x) P.g ^ n = (aeval x) (derivative P.f) * ?m.78"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 1038,
"column": 4
} | {
"line": 1038,
"column": 26
} | {
"line": 1040,
"column": 0
} | [
{
"pp": "case mpr\nK : Type u\ninst✝ : Field K\np : K[X]\nhp : p ≠ 0\nq : K[X]\nhq : q ≠ 0\n⊢ ((algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q).num ∣ p",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"RatFunc.num_div_dvd"
],
"usedFVars": [
"K",
"inst✝",
"... | [] | exact num_div_dvd p hq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 67
} | {
"line": 170,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nI : Ideal S\nhI : I ^ 2 = ⊥\nx : S\nhx : P.HasMap ((Ideal.Quotient.mk I) x)\nhf : (aeval x) P.f ∈ I\na : S\nha : (aeval x) P.g * a - 1 ∈ I\np₁ p₂ : R[X]\nn : ℕ\ne : (aeval ... | [] | rw [Polynomial.aeval_add_of_sq_eq_zero _ _ _ (by grind)]; grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 67
} | {
"line": 170,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nI : Ideal S\nhI : I ^ 2 = ⊥\nx : S\nhx : P.HasMap ((Ideal.Quotient.mk I) x)\nhf : (aeval x) P.f ∈ I\na : S\nha : (aeval x) P.g * a - 1 ∈ I\np₁ p₂ : R[X]\nn : ℕ\ne : (aeval ... | [] | rw [Polynomial.aeval_add_of_sq_eq_zero _ _ _ (by grind)]; grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 189,
"column": 52
} | {
"line": 189,
"column": 63
} | {
"line": 189,
"column": 64
} | [
{
"pp": "R : Type u_1\nS✝ : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\ninst✝² : CommRing T\ninst✝¹ : Algebra R S✝\ninst✝ : Algebra R T\nP : StandardEtalePair R\nS : Type u_1\nx✝¹ : CommRing S\nx✝ : Algebra R S\nI : Ideal S\nhI : I ^ 2 = ⊥\nx : S\nhx : P.HasMap ((Ideal.Quotient.mk I) x)\n... | [
"R : Type u_1\nS✝ : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S✝\ninst✝² : CommRing T\ninst✝¹ : Algebra R S✝\ninst✝ : Algebra R T\nP : StandardEtalePair R\nS : Type u_1\nx✝¹ : CommRing S\nx✝ : Algebra R S\nI : Ideal S\nhI : I ^ 2 = ⊥\nx : S\nhx : P.HasMap ((Ideal.Quotient.mk I) x)\nε : S\nH : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 31
} | {
"line": 222,
"column": 32
} | [
{
"pp": "case refine_3\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\n⊢ Ideal.span {(algebraMap R[X] (Localization.Away P.g)) P.f} ≤ RingHom.ker (IsLocalization.Away.liftAlgHom P.g ⋯)",... | [
"case refine_3\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\n⊢ (aeval P.X) P.f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 88
} | {
"line": 330,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePresentation R S\nx : Localization.Away ((AdjoinRoot.mk P.f) P.g)\nn : ℕ\np : R[X]\ne :\n x *\n (algebraMap (AdjoinRoot P.f) (Localization.Away ((AdjoinRoot.mk P.f) P.g)))\n ↑((Adjoi... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePresentation R S\nx : Localization.Away ((AdjoinRoot.mk P.f) P.g)\nn : ℕ\np : R[X]\ne :\n x *\n (algebraMap (AdjoinRoot P.f) (Localization.Away ((AdjoinRoot.mk P.f) P.g)))\n ↑((AdjoinRoot.mk P.f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 359,
"column": 6
} | {
"line": 359,
"column": 44
} | {
"line": 359,
"column": 45
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP✝ : StandardEtalePair R\nP : StandardEtalePresentation R S\nalgInst✝ : Algebra R (P.map (algebraMap R T)).Ring :=\n ((algebraMap T (P.map (algebraMap R T... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP✝ : StandardEtalePair R\nP : StandardEtalePresentation R S\nalgInst✝ : Algebra R (P.map (algebraMap R T)).Ring :=\n ((algebraMap T (P.map (algebraMap R T)).Ring).com... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 379,
"column": 27
} | {
"line": 379,
"column": 70
} | {
"line": 379,
"column": 71
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP✝¹ : StandardEtalePair R\nP✝ : StandardEtalePresentation R S\nP : StandardEtalePair R\n⊢ Function.Bijective ⇑(P.lift P.X ⋯)",
"ppTerm": "?m.30",
"... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP✝¹ : StandardEtalePair R\nP✝ : StandardEtalePresentation R S\nP : StandardEtalePair R\n⊢ Function.Bijective _root_.id"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 15
} | {
"line": 121,
"column": 16
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nP : Ideal R\ninst✝⁵ : P.IsPrime\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Q.LiesOver P\ninst✝² : IsUnramifiedAt R Q\nx : S\np : R[X]\ninst✝¹ : Algebra (Localization.AtPrime P) (Localization.AtPrime Q)\ninst✝ : ... | [
"R : Type u_1\nS : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nP : Ideal R\ninst✝⁵ : P.IsPrime\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Q.LiesOver P\ninst✝² : IsUnramifiedAt R Q\nx : S\np : R[X]\ninst✝¹ : Algebra (Localization.AtPrime P) (Localization.AtPrime Q)\ninst✝ : Localization... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 422,
"column": 21
} | {
"line": 424,
"column": 74
} | {
"line": 424,
"column": 75
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsStandardEtale R S\nSₛ : Type u_4\ninst✝⁴ : CommRing Sₛ\ninst✝³ : Algebra S Sₛ\ninst✝² : Algebra R Sₛ\ninst✝¹ : IsScalarTower R S Sₛ\ns : S\ninst✝ : IsLocalization.Away s Sₛ\nP : StandardEtalePresentat... | [] | by
simp [IsScalarTower.algebraMap_apply R S' (Localization.Away _),
- AlgEquiv.symm_toRingEquiv, IsScalarTower.algebraMap_eq R S Sₛ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 155,
"column": 27
} | {
"line": 155,
"column": 38
} | {
"line": 155,
"column": 39
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : Module.Finite R S\nQ : Ideal S\ninst✝¹ : Q.IsPrime\ninst✝ : IsUnramifiedAt R Q\nh✝ : Nontrivial S\nthis✝ : Nontrivial R\nP✝ : Ideal R := Ideal.under R Q\nthis : Algebra (Localization.AtPrime P✝) (Locali... | [
"R : Type u_1\nS : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : Module.Finite R S\nQ : Ideal S\ninst✝¹ : Q.IsPrime\ninst✝ : IsUnramifiedAt R Q\nh✝ : Nontrivial S\nthis✝ : Nontrivial R\nP✝ : Ideal R := Ideal.under R Q\nthis : Algebra (Localization.AtPrime P✝) (Localization.AtPri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 15
} | {
"line": 160,
"column": 16
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : Module.Finite R S\nQ : Ideal S\ninst✝¹ : Q.IsPrime\ninst✝ : IsUnramifiedAt R Q\nh✝ : Nontrivial S\nthis✝¹ : Nontrivial R\nP✝ : Ideal R := Ideal.under R Q\nthis✝ : Algebra (Localization.AtPrime P✝) (Loca... | [
"R : Type u_1\nS : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : Module.Finite R S\nQ : Ideal S\ninst✝¹ : Q.IsPrime\ninst✝ : IsUnramifiedAt R Q\nh✝ : Nontrivial S\nthis✝¹ : Nontrivial R\nP✝ : Ideal R := Ideal.under R Q\nthis✝ : Algebra (Localization.AtPrime P✝) (Localization.AtP... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 230,
"column": 28
} | {
"line": 230,
"column": 48
} | {
"line": 230,
"column": 49
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOv... | [
"R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOverAlgebra (I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 15
} | {
"line": 207,
"column": 16
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\ny : B\nhy : IsIntegral R y\nh✝ : Nontrivial B\nhf : Monic 1\nhf' : ∀ (i : ℕ), IsIntegral R (coeff 1 i)\nhfx ... | [
"R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\ny : B\nhy : IsIntegral R y\nh✝ : Nontrivial B\nhf : Monic 1\nhf' : ∀ (i : ℕ), IsIntegral R (coeff 1 i)\nhfx : RingHom.ke... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 66
} | {
"line": 208,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\ny : B\nhy : IsIntegral R y\nh✝ : Nontrivial B\nhf : Monic 1\nhf' : ∀ (i : ℕ), IsIntegral R (coeff 1 i)\nhfx ... | [] | simpa using (RingHom.ker_ne_top φ.toRingHom).symm.trans_eq hfx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 248,
"column": 4
} | {
"line": 248,
"column": 55
} | {
"line": 248,
"column": 56
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOv... | [
"R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOverAlgebra (I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 29
} | {
"line": 223,
"column": 6
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRing... | [
"case refine_2\nR : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRingHom = Ideal.... | · exact isIntegral_zero | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 46
} | {
"line": 250,
"column": 47
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOv... | [
"R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOverAlgebra (I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 223,
"column": 8
} | {
"line": 223,
"column": 19
} | {
"line": 223,
"column": 20
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRing... | [
"case refine_2\nR : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRingHom = Ideal.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 68
} | {
"line": 227,
"column": 69
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRingHom = Ideal.spa... | [
"R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRingHom = Ideal.span {f}\nh✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 15
} | {
"line": 120,
"column": 16
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\np : Ideal T\ninst✝ : p.IsPrime\ns : S\nhsp : s ∉ Ideal.under S p\nhs : IsIntegral R s\n... | [
"case refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\np : Ideal T\ninst✝ : p.IsPrime\ns : S\nhsp : s ∉ Ideal.under S p\nhs : IsIntegral R s\nHs : ∀ (x : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 258,
"column": 6
} | {
"line": 258,
"column": 17
} | {
"line": 258,
"column": 18
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRingHom = Ideal.spa... | [
"R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRingHom = Ideal.span {f}\nh✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 19
} | {
"line": 259,
"column": 20
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRingHom = Ideal.spa... | [
"R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRingHom = Ideal.span {f}\nh✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 31
} | {
"line": 266,
"column": 32
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOv... | [
"R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOverAlgebra (I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 13
} | {
"line": 174,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np r : R[X]\nht : φ.IsIntegralElem t\nhpm : p.Monic\nhpr : r.natDegree < p.natDegree\nhp : φ p * t = φ r\nSt : Type u_2 := Localization.Away t\nt' : St := IsLocalization.Away.invSelf t\nht... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np r : R[X]\nht : φ.IsIntegralElem t\nhpm : p.Monic\nhpr : r.natDegree < p.natDegree\nhp : φ p * t = φ r\nSt : Type u_2 := Localization.Away t\nt' : St := IsLocalization.Away.invSelf t\nht't : t' * (a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 272,
"column": 6
} | {
"line": 272,
"column": 88
} | {
"line": 272,
"column": 89
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRing... | [
"case refine_2\nR : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRingHom = Ideal.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 72
} | {
"line": 217,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nht : φ.IsIntegralElem t\nhp : φ p * t ∈ φ.range\na : R := p.leadingCoeff\nR' : Type u_1 := Localization.Away a\nS' : Type u_2 := Localization.Away ((algebraMap R S) a)\nthis✝ : ... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nht : φ.IsIntegralElem t\nhp : φ p * t ∈ φ.range\na : R := p.leadingCoeff\nR' : Type u_1 := Localization.Away a\nS' : Type u_2 := Localization.Away ((algebraMap R S) a)\nthis✝ : Algebra R' S... | generalize IsLocalization.integerNormalization (.powers a) q = q' at e | Lean.Elab.Tactic.evalGeneralize | Lean.Parser.Tactic.generalize |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 100
} | {
"line": 220,
"column": 8
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nht : φ.IsIntegralElem t\nhp : φ p * t ∈ φ.range\na : R := p.leadingCoeff\nR' : Type u_1 := Localization.Away a\nS' : Type u_2 := Localization.Away ((algebraMap R S) a)\nthis✝ : ... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nht : φ.IsIntegralElem t\nhp : φ p * t ∈ φ.range\na : R := p.leadingCoeff\nR' : Type u_1 := Localization.Away a\nS' : Type u_2 := Localization.Away ((algebraMap R S) a)\nthis✝ : Algebra R' S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 71
} | {
"line": 222,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nht : φ.IsIntegralElem t\nhp : φ p * t ∈ φ.range\na : R := p.leadingCoeff\nR' : Type u_1 := Localization.Away a\nS' : Type u_2 := Localization.Away ((algebraMap R S) a)\nthis✝¹ :... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nht : φ.IsIntegralElem t\nhp : φ p * t ∈ φ.range\na : R := p.leadingCoeff\nR' : Type u_1 := Localization.Away a\nS' : Type u_2 := Localization.Away ((algebraMap R S) a)\nthis✝¹ : Algebra R' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 239,
"column": 39
} | {
"line": 239,
"column": 82
} | {
"line": 239,
"column": 83
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\nhp : φ p * t ∈ conductor R (φ X)\nalgInst✝ : Algebra R[X] S := φ.toAlgebra\nalgebraizeInst✝ : Module.Finite R[X] S\nthis : IsScalar... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\nhp : φ p * t ∈ conductor R (φ X)\nalgInst✝ : Algebra R[X] S := φ.toAlgebra\nalgebraizeInst✝ : Module.Finite R[X] S\nthis : IsScalarTower R R[X]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 254,
"column": 25
} | {
"line": 254,
"column": 61
} | {
"line": 254,
"column": 62
} | [
{
"pp": "case add\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\nhp : φ p * t ∈ conductor R (φ X)\nalgInst✝ : Algebra R[X] S := φ.toAlgebra\nalgebraizeInst✝ : Module.Finite R[X] S\nthis ... | [
"case add\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\nhp : φ p * t ∈ conductor R (φ X)\nalgInst✝ : Algebra R[X] S := φ.toAlgebra\nalgebraizeInst✝ : Module.Finite R[X] S\nthis : IsScalarTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 273,
"column": 8
} | {
"line": 273,
"column": 63
} | {
"line": 273,
"column": 64
} | [
{
"pp": "case inr.h.zero.inr\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\ni : ℕ\nthis : ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.leadingCoeff • t ∈ (conductor R (φ X)).radical\np ... | [
"case inr.h.zero.inr\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\ni : ℕ\nthis : ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.leadingCoeff • t ∈ (conductor R (φ X)).radical\np : R[X]\nhp :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 276,
"column": 8
} | {
"line": 276,
"column": 25
} | {
"line": 276,
"column": 26
} | [
{
"pp": "case inr.h.succ.inl\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\ni : ℕ\nthis : ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.leadingCoeff • t ∈ (conductor R (φ X)).radical\np ... | [
"case inr.h.succ.inl\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\ni : ℕ\nthis : ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.leadingCoeff • t ∈ (conductor R (φ X)).radical\np : R[X]\nhp :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 281,
"column": 6
} | {
"line": 281,
"column": 40
} | {
"line": 282,
"column": 8
} | [
{
"pp": "case inr.h.succ.inr\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\ni : ℕ\nthis✝ : ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.leadingCoeff • t ∈ (conductor R (φ X)).radical\np... | [
"case inr.h.succ.inr\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\ni : ℕ\nthis✝ : ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.leadingCoeff • t ∈ (conductor R (φ X)).radical\np : R[X]\nhp ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 349,
"column": 6
} | {
"line": 350,
"column": 91
} | {
"line": 350,
"column": 92
} | [
{
"pp": "case inr.refine_1\nR : Type u_1\nS : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nQ : Ideal S\ninst✝² : Q.IsPrime\ninst✝¹ : FiniteType R S\ninst✝ : IsUnramifiedAt R Q\nthis :\n ∀ {R : Type u_1} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]... | [
"case inr.refine_1\nR : Type u_1\nS : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nQ : Ideal S\ninst✝² : Q.IsPrime\ninst✝¹ : FiniteType R S\ninst✝ : IsUnramifiedAt R Q\nthis :\n ∀ {R : Type u_1} {S : Type u_3} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (Q : Ideal ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 285,
"column": 8
} | {
"line": 285,
"column": 29
} | {
"line": 285,
"column": 30
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np✝ : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\np : R[X]\ni : ℕ\nhi : i = p.natDegree\nn : ℕ\nhn : (φ p * t) ^ n ∈ conductor R (φ X)\n⊢ φ (p ^ n) * t ^ n ∈ conductor R (φ X)",
... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np✝ : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\np : R[X]\ni : ℕ\nhi : i = p.natDegree\nn : ℕ\nhn : (φ p * t) ^ n ∈ conductor R (φ X)\n⊢ φ p ^ n * t ^ n ∈ conductor R (φ X)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 302,
"column": 2
} | {
"line": 303,
"column": 27
} | {
"line": 303,
"column": 28
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nhRS : integralClosure R S = ⊥\nx : S\nhx : (aeval x).Finite\nu : S\np : R[X]\ne : (aeval x) p * u ∈ (conductor R x).radical\ni : ℕ\n⊢ (map (algebraMap R (S ⧸ (conductor R x).radical)) p * C ((Ideal.Quotient.mk (c... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nhRS : integralClosure R S = ⊥\nx : S\nhx : (aeval x).Finite\nu : S\np : R[X]\ne : (aeval x) p * u ∈ (conductor R x).radical\ni : ℕ\n⊢ (algebraMap R S) (p.coeff i) * u ∈ (conductor R x).radical"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 304,
"column": 10
} | {
"line": 304,
"column": 21
} | {
"line": 304,
"column": 22
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nhRS : integralClosure R S = ⊥\nx : S\nhx : (aeval x).Finite\nu : S\np : R[X]\ne : (aeval x) p * u ∈ (conductor R x).radical\ni : ℕ\n⊢ (aeval x) p * u ∈ (conductor R ((aeval x) X)).radical",
"ppTerm": "?m.128"... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nhRS : integralClosure R S = ⊥\nx : S\nhx : (aeval x).Finite\nu : S\np : R[X]\ne : (aeval x) p * u ∈ (conductor R x).radical\ni : ℕ\n⊢ (aeval x) p * u ∈ (conductor R x).radical"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 404,
"column": 27
} | {
"line": 406,
"column": 16
} | {
"line": 406,
"column": 17
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal S\ninst✝² : p.IsPrime\ninst✝¹ : FinitePresentation R S\ninst✝ : IsSmoothAt R p\nf : S\nhfp : f ∉ p\nH✝ : IsStandardSmooth R (Localization.Away f)\nn : ℕ\nφ : MvPolynomial (Fin n) R →+* Localization.Awa... | [
"R : Type u_1\nS : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal S\ninst✝² : p.IsPrime\ninst✝¹ : FinitePresentation R S\ninst✝ : IsSmoothAt R p\nf : S\nhfp : f ∉ p\nH✝ : IsStandardSmooth R (Localization.Away f)\nn : ℕ\nφ : MvPolynomial (Fin n) R →+* Localization.Away f\nhgC : φ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 53
} | {
"line": 73,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf g : R[X]\nm n : ℕ\ni j : Fin (m + n)\n⊢ f.sylvester g m n i j =\n (Matrix.reindex (finCongr ⋯) (finSumFinEquiv.symm.trans ((Equiv.sumComm (Fin n) (Fin m)).trans finSumFinEquiv)))\n (g.sylvester f n m) i j",
"ppTerm": "?m.41",
"assigned": true,
"us... | [] | induction j using Fin.addCases <;> simp [sylvester] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 435,
"column": 4
} | {
"line": 435,
"column": 15
} | {
"line": 435,
"column": 16
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : IsReduced S\nx✝ : S\nhx' : (aeval x✝).Finite\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R P\nleft✝ : IsDomain S\nright✝ : FaithfulSMul R S\nthis✝ : IsDomain R\nK : Type u_1 := Fractio... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : IsReduced S\nx✝ : S\nhx' : (aeval x✝).Finite\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R P\nleft✝ : IsDomain S\nright✝ : FaithfulSMul R S\nthis✝ : IsDomain R\nK : Type u_1 := FractionRing R\nL :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 444,
"column": 30
} | {
"line": 444,
"column": 42
} | {
"line": 444,
"column": 42
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : IsReduced S\nx : S\nhx' : (aeval x).Finite\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R P\nleft✝ : IsDomain S\nright✝ : FaithfulSMul R S\nthis✝ : IsDomain R\nK : Type u_1 := FractionR... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : IsReduced S\nx : S\nhx' : (aeval x).Finite\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R P\nleft✝ : IsDomain S\nright✝ : FaithfulSMul R S\nthis✝ : IsDomain R\nK : Type u_1 := FractionRing R\nL : T... | simp [g, S'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 25
} | {
"line": 178,
"column": 26
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nn m : ℕ\nthis : ∀ (i : Fin (m + 1 + n)), f.sylvester 0 (m + 1) n i ⟨0, ⋯⟩ = 0\n⊢ f.resultant 0 (m + 1) n = 0 ^ (m + 1) * f.coeff 0 ^ n",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Nat.i... | [
"case succ\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nn m : ℕ\nthis : ∀ (i : Fin (m + 1 + n)), f.sylvester 0 (m + 1) n i ⟨0, ⋯⟩ = 0\n⊢ (f.sylvester 0 (m + 1) n).det = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 524,
"column": 71
} | {
"line": 524,
"column": 82
} | {
"line": 524,
"column": 83
} | [
{
"pp": "R✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : R[X] →ₐ[R] S\nhf : f.Finite\nH : integralClosure R S = ⊥\nJ : Ideal S :... | [
"R✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : R[X] →ₐ[R] S\nhf : f.Finite\nH : integralClosure R S = ⊥\nJ : Ideal S := (conductor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 538,
"column": 45
} | {
"line": 538,
"column": 90
} | {
"line": 538,
"column": 91
} | [
{
"pp": "R✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : R[X] →ₐ[R] S\nH : integralClosure R S = ⊥\nhf : ¬conductor R (f X) ≤ p\... | [
"R✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : R[X] →ₐ[R] S\nH : integralClosure R S = ⊥\nhf : ¬conductor R (f X) ≤ p\nx : S\nhx :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 550,
"column": 19
} | {
"line": 550,
"column": 30
} | {
"line": 550,
"column": 31
} | [
{
"pp": "R✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : R[X] →ₐ[R] S\nH : integralClosure R S = ⊥\nhf : ¬conductor R (f X) ≤ p\... | [
"R✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : R[X] →ₐ[R] S\nH : integralClosure R S = ⊥\nhf : ¬conductor R (f X) ≤ p\nx : S\nhxp ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 551,
"column": 4
} | {
"line": 551,
"column": 33
} | {
"line": 551,
"column": 34
} | [
{
"pp": "case refine_2\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : R[X] →ₐ[R] S\nH : integralClosure R S = ⊥\nhf : ¬conduct... | [
"case refine_2\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : R[X] →ₐ[R] S\nH : integralClosure R S = ⊥\nhf : ¬conductor R (f X) ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 336,
"column": 17
} | {
"line": 336,
"column": 40
} | {
"line": 336,
"column": 41
} | [
{
"pp": "case right\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm : ℕ\nhf : f.natDegree ≤ m\nn : ℕ\ni : Fin (n + 1)\nhb : Fin.natAdd (m + 1) i ≠ Fin.castAdd (n + 1) (Fin.last m)\n⊢ f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n)) (Fin.natAdd (m + 1) i) = 0",
"ppTerm": "?right",
"assigned": true,
... | [
"case right\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm : ℕ\nhf : f.natDegree ≤ m\nn : ℕ\ni : Fin (n + 1)\nhb : Fin.natAdd (m + 1) i ≠ Fin.castAdd (n + 1) (Fin.last m)\n⊢ ↑i ≤ m + 1 + n → m + 1 + n ≤ ↑i + (m + 1) → f.coeff (m + 1 + n - ↑i) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 203,
"column": 10
} | {
"line": 203,
"column": 21
} | {
"line": 203,
"column": 22
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU : ↑Y.affineOpens\n⊢ Set.range ⇑(fromNormalization f ⁻¹ᵁ ↑U).ι = Set.range ⇑((normalizationOpenCover f).f U)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.fromNormali... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU : ↑Y.affineOpens\n⊢ ⇑(fromNormalization f) ⁻¹' ↑↑U = Set.range ⇑((normalizationOpenCover f).f U)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 420,
"column": 27
} | {
"line": 420,
"column": 51
} | {
"line": 420,
"column": 52
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | [
"n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fun ij ↦ ij.1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 626,
"column": 6
} | {
"line": 627,
"column": 47
} | {
"line": 627,
"column": 48
} | [
{
"pp": "n : ℕ\nIH :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]\n [WeaklyQuasiFiniteAt R p] (f : MvPolynomial (Fin n) R →ₐ[R] S), f.Finite → ZariskisMainProperty R p\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : ... | [
"n : ℕ\nIH :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]\n [WeaklyQuasiFiniteAt R p] (f : MvPolynomial (Fin n) R →ₐ[R] S), f.Finite → ZariskisMainProperty R p\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 649,
"column": 15
} | {
"line": 650,
"column": 9
} | {
"line": 650,
"column": 10
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FiniteType R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] S\nhf : Function.Surjective ⇑f\nthis : Small.{u, v} S\nr : Shrink.{u, v} S\nhr : r ∉... | [
"R : Type u\nS : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FiniteType R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] S\nhf : Function.Surjective ⇑f\nthis : Small.{u, v} S\nr : Shrink.{u, v} S\nhr : r ∉ Ideal.comap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 433,
"column": 39
} | {
"line": 433,
"column": 63
} | {
"line": 433,
"column": 64
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | [
"n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fun ij ↦ ij.1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 695,
"column": 2
} | {
"line": 695,
"column": 66
} | {
"line": 696,
"column": 2
} | [
{
"pp": "case refine_3\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FiniteType R S\np : Ideal S\nH : ZariskisMainProperty R p\ns : Finset S\nhs : adjoin R ↑s = ⊤\nr : S\nhrp : r ∉ p\nhr : IsIntegral R r\nm : S → ℕ\nhm : ∀ (x : S), IsIntegral R (r ^ m x * x... | [
"case refine_3\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FiniteType R S\np : Ideal S\nH : ZariskisMainProperty R p\ns : Finset S\nhs : adjoin R ↑s = ⊤\nr : S\nhrp : r ∉ p\nhr : IsIntegral R r\nm : S → ℕ\nhm : ∀ (x : S), IsIntegral R (r ^ m x * x)\nt : Set S... | obtain ⟨y, hy : Localization.awayMap _ _ _ = _⟩ := this ⟨x, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 723,
"column": 30
} | {
"line": 723,
"column": 41
} | {
"line": 723,
"column": 42
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FiniteType R S\np : Ideal S\ninst✝ : p.IsPrime\nH✝ : ZariskisMainProperty R p\nS' : Subalgebra R S\nhS' : (Subalgebra.toSubmodule S').FG\nr : ↥S'\nhrp : ↑r ∉ p\nH : Function.Bijective ⇑(Localization.awa... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FiniteType R S\np : Ideal S\ninst✝ : p.IsPrime\nH✝ : ZariskisMainProperty R p\nS' : Subalgebra R S\nhS' : (Subalgebra.toSubmodule S').FG\nr : ↥S'\nhrp : ↑r ∉ p\nH : Function.Bijective ⇑(Localization.awayMap S'.val.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 15
} | {
"line": 360,
"column": 16
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\ninst✝ : IsIntegral X\n⊢ ⊤ = closure (⇑(toNormalization f) '' Set.univ)",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"AlgebraicGeometry.SheafedSpace.instTo... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\ninst✝ : IsIntegral X\n⊢ Set.univ = closure (Set.range fun a ↦ (toNormalization f) a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 446,
"column": 46
} | {
"line": 446,
"column": 83
} | {
"line": 446,
"column": 84
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | [
"n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fun ij ↦ ij.1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 489,
"column": 4
} | {
"line": 489,
"column": 77
} | {
"line": 489,
"column": 78
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nn : ℕ\nhg : g.natDegree ≤ n\nhf : f.Splits\nthis✝ :\n ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] (f g : R[X]) (n : ℕ),\n g.natDegree ≤ n →\n f.Splits →\n IsField R → f.resultant g f.natDegree n ... | [
"case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nn : ℕ\nhg : g.natDegree ≤ n\nhf : f.Splits\nthis✝ :\n ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] (f g : R[X]) (n : ℕ),\n g.natDegree ≤ n →\n f.Splits →\n IsField R → f.resultant g f.natDegree n = f.leadingC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 497,
"column": 4
} | {
"line": 498,
"column": 27
} | {
"line": 498,
"column": 28
} | [
{
"pp": "case neg.inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nn : ℕ\nhg : g.natDegree ≤ n\nhf : f.Splits\nhR : IsField R\nhf0 : ¬f = 0\nthis✝ :\n ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] (f g : R[X]) (n : ℕ),\n g.natDegree ≤ n →\n f.Splits →\n IsFiel... | [
"case neg.inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nn : ℕ\nhg : g.natDegree ≤ n\nhf : f.Splits\nhR : IsField R\nhf0 : ¬f = 0\nthis✝ :\n ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] (f g : R[X]) (n : ℕ),\n g.natDegree ≤ n →\n f.Splits →\n IsField R →\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 470,
"column": 6
} | {
"line": 470,
"column": 49
} | {
"line": 471,
"column": 6
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ f₂ : normalization f ⟶ T\ng : T ⟶ Y\ninst✝ : IsAffineHom g\nH₁ : toNormalization f ≫ f₁ = toNormalization f ≫ f₂\nhf₁ : f₁ ≫ g = fromNormalization f\nhf₂ : f₂ ≫ g = fromNormalization f\nU : (normalizationOpenCov... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ f₂ : normalization f ⟶ T\ng : T ⟶ Y\ninst✝ : IsAffineHom g\nH₁ : toNormalization f ≫ f₁ = toNormalization f ≫ f₂\nhf₁ : f₁ ≫ g = fromNormalization f\nhf₂ : f₂ ≫ g = fromNormalization f\nU : (normalizationOpenCover f).I₀\nth... | rw [this, f.toNormalization_app_preimage U] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 511,
"column": 4
} | {
"line": 511,
"column": 93
} | {
"line": 511,
"column": 94
} | [
{
"pp": "case neg.inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nn : ℕ\nhg : g.natDegree ≤ n\nhf : f.Splits\nhR : IsField R\nhfm : f.Monic\nhg0 : ¬g = 0\nthis✝ :\n ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] (f g : R[X]) (n : ℕ),\n g.natDegree ≤ n →\n f.Splits →\... | [
"case neg.inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nn : ℕ\nhg : g.natDegree ≤ n\nhf : f.Splits\nhR : IsField R\nhfm : f.Monic\nhg0 : ¬g = 0\nthis✝ :\n ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] (f g : R[X]) (n : ℕ),\n g.natDegree ≤ n →\n f.Splits →\n IsF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 539,
"column": 98
} | {
"line": 552,
"column": 21
} | {
"line": 554,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np : R[X]\nP : {R : Type u} → [inst : CommRing R] → R[X] → Prop\nSplits : ∀ (R : Type u) [inst : Field R] (p : R[X]), p.Splits → P p\ninjective :\n ∀ (R S : Type u) [inst : CommRing R] [inst_1 : CommRing S] (φ : R →+* S),\n Function.Injective ⇑φ → ∀ (p : R[X]), P (map... | [] | by
wlog hR : IsDomain R generalizing R
· exact surjective _ _ (MvPolynomial.eval₂Hom (algebraMap ℤ R) id)
(fun x ↦ ⟨.X x, by simp [MvPolynomial.eval₂Hom]⟩) p
(fun _ ↦ this _ inferInstance)
wlog hR : IsField R generalizing R
· exact injective _ _ _ (FaithfulSMul.algebraMap_injective R (FractionRing R... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 571,
"column": 4
} | {
"line": 571,
"column": 85
} | {
"line": 571,
"column": 86
} | [
{
"pp": "case injective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR SatisfiesM : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing SatisfiesM\nφ : R →+* SatisfiesM\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g₁ g₂ : SatisfiesM[X]),\n (map φ f).resultant (g₁ * g₂) (map φ f).natDegree (g₁.natDegree + g₂.natDeg... | [
"case injective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR SatisfiesM : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing SatisfiesM\nφ : R →+* SatisfiesM\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g₁ g₂ : SatisfiesM[X]),\n (map φ f).resultant (g₁ * g₂) (map φ f).natDegree (g₁.natDegree + g₂.natDegree) =\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 598,
"column": 4
} | {
"line": 598,
"column": 93
} | {
"line": 598,
"column": 94
} | [
{
"pp": "case injective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH : (map φ f).resultant (map φ f) = 0 ^ (map φ f).natDegree\n⊢ φ (f.resultant f) = φ (0 ^ f.natDegree)",
"ppTerm": "?injective",
"ass... | [
"case injective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH : (map φ f).resultant (map φ f) = 0 ^ (map φ f).natDegree\n⊢ φ (f.resultant f) = 0 ^ f.natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 617,
"column": 69
} | {
"line": 617,
"column": 80
} | {
"line": 617,
"column": 81
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nH : IsCoprime f g\nhf : f ≠ 0\nhg : g ≠ 0\nb : R[X]\ne : 0 * f + b * g = 1\n⊢ b * ?m.152 = 1",
"ppTerm": "?m.153",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nH : IsCoprime f g\nhf : f ≠ 0\nhg : g ≠ 0\nb : R[X]\ne : 0 * f + b * g = 1\n⊢ b * ?m.152 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 620,
"column": 69
} | {
"line": 620,
"column": 80
} | {
"line": 620,
"column": 81
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nH : IsCoprime f g\nhf : f ≠ 0\nhg : g ≠ 0\na : R[X]\nha : a ≠ 0\ne : a * f + 0 * g = 1\n⊢ a * ?m.222 = 1",
"ppTerm": "?m.223",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nH : IsCoprime f g\nhf : f ≠ 0\nhg : g ≠ 0\na : R[X]\nha : a ≠ 0\ne : a * f + 0 * g = 1\n⊢ a * ?m.222 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 484,
"column": 6
} | {
"line": 484,
"column": 49
} | {
"line": 485,
"column": 6
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ f₂ : normalization f ⟶ T\ng : T ⟶ Y\ninst✝ : IsAffineHom g\nH₁ : toNormalization f ≫ f₁ = toNormalization f ≫ f₂\nhf₁ : f₁ ≫ g = fromNormalization f\nhf₂ : f₂ ≫ g = fromNormalization f\nU : (normalizationOpenCov... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ f₂ : normalization f ⟶ T\ng : T ⟶ Y\ninst✝ : IsAffineHom g\nH₁ : toNormalization f ≫ f₁ = toNormalization f ≫ f₂\nhf₁ : f₁ ≫ g = fromNormalization f\nhf₂ : f₂ ≫ g = fromNormalization f\nU : (normalizationOpenCover f).I₀\nth... | rw [this, f.toNormalization_app_preimage U] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 638,
"column": 2
} | {
"line": 638,
"column": 21
} | {
"line": 638,
"column": 22
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nH : IsCoprime f g\nhf : f ≠ 0\ne : f.resultant g = 0\n⊢ False",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nH : IsCoprime f g\nhf : f ≠ 0\ne : f.resultant g = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 650,
"column": 51
} | {
"line": 650,
"column": 86
} | {
"line": 650,
"column": 87
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nι : Type u_3\nf : ι → R[X]\ng : R[X]\nn : ℕ\nhn : g.natDegree ≤ n\na : ι\ns : Finset ι\nhas : a ∉ s\nIH :\n ∏ i ∈ s, (f i).leadingCoeff ≠ 0 →\n (∏ i ∈ s, f i).resultant g (∏ i ∈ s, f i).natDegree n = ∏ i ∈ s, (f i).resultant g (f i).natDegree n\nhf : ∏ i ∈ insert a... | [
"R : Type u_1\ninst✝ : CommRing R\nι : Type u_3\nf : ι → R[X]\ng : R[X]\nn : ℕ\nhn : g.natDegree ≤ n\na : ι\ns : Finset ι\nhas : a ∉ s\nIH :\n ∏ i ∈ s, (f i).leadingCoeff ≠ 0 →\n (∏ i ∈ s, f i).resultant g (∏ i ∈ s, f i).natDegree n = ∏ i ∈ s, (f i).resultant g (f i).natDegree n\nhf : ∏ i ∈ insert a s, (f i).le... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 712,
"column": 35
} | {
"line": 712,
"column": 46
} | {
"line": 712,
"column": 47
} | [
{
"pp": "R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g : S[X]) (r : S),\n map φ f ≠ 0 →\n g ≠ 0 →\n ((map φ f).scaleRoots r).resultant (g.scaleRoots r) (map φ f).natDegree g.natDegree =... | [
"R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g : S[X]) (r : S),\n map φ f ≠ 0 →\n g ≠ 0 →\n ((map φ f).scaleRoots r).resultant (g.scaleRoots r) (map φ f).natDegree g.natDegree =\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 713,
"column": 10
} | {
"line": 713,
"column": 21
} | {
"line": 713,
"column": 22
} | [
{
"pp": "R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g : S[X]) (r : S),\n map φ f ≠ 0 →\n g ≠ 0 →\n ((map φ f).scaleRoots r).resultant (g.scaleRoots r) (map φ f).natDegree g.natDegree =... | [
"R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g : S[X]) (r : S),\n map φ f ≠ 0 →\n g ≠ 0 →\n ((map φ f).scaleRoots r).resultant (g.scaleRoots r) (map φ f).natDegree g.natDegree =\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 716,
"column": 6
} | {
"line": 716,
"column": 52
} | {
"line": 716,
"column": 53
} | [
{
"pp": "case inr.inr.injective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g : S[X]) (r : S),\n map φ f ≠ 0 →\n g ≠ 0 →\n ((map φ f).scaleRoots r).resultant (g.scaleRoots r) (map φ f)... | [
"case inr.inr.injective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g : S[X]) (r : S),\n map φ f ≠ 0 →\n g ≠ 0 →\n ((map φ f).scaleRoots r).resultant (g.scaleRoots r) (map φ f).natDegree g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 608,
"column": 4
} | {
"line": 608,
"column": 19
} | {
"line": 608,
"column": 20
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nq : { q // ↑q.1 * ↑q.2 = map (algebraMap R S) ↑p ∧ IsCoprime ↑q.1 ↑q.2 }\nf : failed to pretty print expr... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nq : { q // ↑q.1 * ↑q.2 = map (algebraMap R S) ↑p ∧ IsCoprime ↑q.1 ↑q.2 }\nf : failed to pretty print expression (use ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 784,
"column": 4
} | {
"line": 784,
"column": 50
} | {
"line": 784,
"column": 51
} | [
{
"pp": "case injective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH : ∀ (g : S[X]) (r : S), ((taylor r) (map φ f)).resultant ((taylor r) g) = (map φ f).resultant g\ng : R[X]\nr : R\nthis : (map φ ((taylor r)... | [
"case injective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH : ∀ (g : S[X]) (r : S), ((taylor r) (map φ f)).resultant ((taylor r) g) = (map φ f).resultant g\ng : R[X]\nr : R\nthis : (map φ ((taylor r) f)).resulta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 812,
"column": 4
} | {
"line": 812,
"column": 32
} | {
"line": 813,
"column": 4
} | [
{
"pp": "m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\n⊢ f * ↑(⟨p, hp⟩, ⟨q, hq⟩).2 + g * ↑(⟨p, hp⟩, ⟨q, hq⟩).1 ∈ R[X]_(m + n)",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"Wi... | [
"m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\n⊢ (f * ↑(⟨p, hp⟩, ⟨q, hq⟩).2 + g * ↑(⟨p, hp⟩, ⟨q, hq⟩).1).degree < ↑(m + n)"
] | rw [Polynomial.mem_degreeLT] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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