module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 212,
"column": 15
} | {
"line": 212,
"column": 26
} | {
"line": 212,
"column": 27
} | [
{
"pp": "σ : Type u_1\nk : Type u_3\ninst✝ : Field k\nφ : MvPowerSeries σ k\nh : φ⁻¹ = 0\n⊢ constantCoeff φ = 0",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"σ : Type u_1\nk : Type u_3\ninst✝ : Field k\nφ : MvPowerSeries σ k\nh : φ⁻¹ = 0\n⊢ constantCoeff φ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 13
} | {
"line": 280,
"column": 14
} | [
{
"pp": "case inr\nσ : Type u_1\nk : Type u_3\ninst✝ : Field k\nr : k\nhr : r ≠ 0\n⊢ constantCoeff (C r) ≠ 0",
"ppTerm": "?inr✝",
"assigned": true,
"usedConstants": [
"MvPowerSeries",
"RingHom",
"MvPowerSeries.constantCoeff",
"id",
"Ne",
"Field.toSemifield",
... | [
"case inr\nσ : Type u_1\nk : Type u_3\ninst✝ : Field k\nr : k\nhr : r ≠ 0\n⊢ ¬r = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 42
} | {
"line": 80,
"column": 43
} | [
{
"pp": "case neg.e_a.h.inr.hc.refine_1\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ antidiagonal n\nH : j < n\n⊢ (match (i, j) with\n | (a, b) => (single () a, single () b)).2\n PUnit.unit ≤\n (single () n) PUnit.unit",
"ppTerm": "?neg.e_a.h.in... | [
"case neg.e_a.h.inr.hc.refine_1\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ antidiagonal n\nH : j < n\n⊢ j ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 81,
"column": 6
} | {
"line": 81,
"column": 42
} | {
"line": 81,
"column": 43
} | [
{
"pp": "case neg.e_a.h.inr.hc.refine_2\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ antidiagonal n\nH : j < n\nhh :\n single () n ≤\n (match (i, j) with\n | (a, b) => (single () a, single () b)).2\n⊢ n ≤ j",
"ppTerm": "?neg.e_a.h.inr.hc.refine_2✝",
... | [
"case neg.e_a.h.inr.hc.refine_2\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ antidiagonal n\nH : j < n\nhh :\n single () n ≤\n (match (i, j) with\n | (a, b) => (single () a, single () b)).2\n⊢ n ≤ j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 278,
"column": 7
} | {
"line": 283,
"column": 42
} | {
"line": 283,
"column": 42
} | [
{
"pp": "k : Type u_2\ninst✝ : Field k\n⊢ ∀ {x : k⟦X⟧}, x ≠ 0 → ∃ n, Associated (X ^ n) x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"PowerSerie... | [] | by
intro f hf
use f.order.toNat
use Unit_of_divided_by_X_pow_order f
simp only [Unit_of_divided_by_X_pow_order_nonzero hf]
exact X_pow_order_mul_divXPowOrder | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 13
} | {
"line": 63,
"column": 14
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : IsDedekindDomain A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nv : HeightOneSpectrum A\nhv : Finite (A ⧸ v.asIdeal)\n⊢ ¬(Set.range ⇑(valuation K v)).Subsingleton",
"ppTerm": "?m.48",
"assigned": true,
"usedCo... | [
"A : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : IsDedekindDomain A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nv : HeightOneSpectrum A\nhv : Finite (A ⧸ v.asIdeal)\n⊢ (Set.range ⇑(valuation K v)).Nontrivial"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 13
} | {
"line": 72,
"column": 14
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : IsDedekindDomain A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nv : HeightOneSpectrum A\nhv : Finite (A ⧸ v.asIdeal)\n⊢ ¬(Set.range ⇑Valued.v).Subsingleton",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants... | [
"A : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : IsDedekindDomain A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nv : HeightOneSpectrum A\nhv : Finite (A ⧸ v.asIdeal)\n⊢ (Set.range ⇑Valued.v).Nontrivial"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 365,
"column": 29
} | {
"line": 365,
"column": 52
} | {
"line": 365,
"column": 53
} | [
{
"pp": "K : Type u_1\ninst✝⁵ : Field K\nR : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Algebra R K\ninst✝² : IsDedekindDomain R\ninst✝¹ : IsFractionRing R K\nv✝ : HeightOneSpectrum R\ninst✝ : NumberField K\nv : FinitePlace K\nx y : K\n⊢ v (x + y) ≤ v x + v y",
"ppTerm": "?m.25",
"assigned": true,
"use... | [
"K : Type u_1\ninst✝⁵ : Field K\nR : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Algebra R K\ninst✝² : IsDedekindDomain R\ninst✝¹ : IsFractionRing R K\nv✝ : HeightOneSpectrum R\ninst✝ : NumberField K\nv : FinitePlace K\nx y : K\n⊢ ↑v (x + y) ≤ ↑v x + ↑v y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 393,
"column": 6
} | {
"line": 393,
"column": 29
} | {
"line": 393,
"column": 29
} | [
{
"pp": "case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nv₁ v₂ : HeightOneSpectrum (𝓞 K)\nh : v₁ ≠ v₂\nx : 𝓞 K\nhx1 : x ∈ v₁.asIdeal\nhx2 : x ∉ v₂.asIdeal\n⊢ ‖(embedding v₁) ↑x‖ ≠ ‖(embedding v₂) ↑x‖",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Re... | [
"case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nv₁ v₂ : HeightOneSpectrum (𝓞 K)\nh : v₁ ≠ v₂\nx : 𝓞 K\nhx1 : ‖(embedding v₁) ((algebraMap (𝓞 K) K) x)‖ < 1\nhx2 : x ∉ v₂.asIdeal\n⊢ ‖(embedding v₁) ↑x‖ ≠ ‖(embedding v₂) ↑x‖"
] | ← norm_lt_one_iff_mem K | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 22
} | {
"line": 457,
"column": 23
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nv : FinitePlace K\nx y : K\nw : HeightOneSpectrum (𝓞 K)\nhw : place (embedding w) = ↑v\nH : ∀ (x : K), v x = (HeightOneSpectrum.adicAbv K w) x\n⊢ v (x + y) ≤ max (v x) (v y)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nv : FinitePlace K\nx y : K\nw : HeightOneSpectrum (𝓞 K)\nhw : place (embedding w) = ↑v\nH : ∀ (x : K), v x = (HeightOneSpectrum.adicAbv K w) x\n⊢ (HeightOneSpectrum.adicAbv K w) (x + y) ≤ max ((HeightOneSpectrum.adicAbv K w) x) ((HeightOneSpectrum.adicAbv K w... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Geometrically.Connected | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 39
} | {
"line": 92,
"column": 40
} | [
{
"pp": "X S : Scheme\nf : X ⟶ S\ninst✝¹ : GeometricallyConnected f\ninst✝ : ConnectedSpace ↥S\nhf : IsOpenMap ⇑f\n⊢ ConnectedSpace ↥X",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Algebraic... | [
"X S : Scheme\nf : X ⟶ S\ninst✝¹ : GeometricallyConnected f\ninst✝ : ConnectedSpace ↥S\nhf : IsOpenMap ⇑f\n⊢ _root_.IsConnected Set.univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 246,
"column": 4
} | {
"line": 246,
"column": 43
} | {
"line": 246,
"column": 43
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z ≠ W.negY Q * P z\n⊢ W.dblXYZ P =\n W.dblZ P •\n ![W.toAffine.addX (P x / P z) (Q x / Q z) (W.toA... | [
"F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z ≠ W.negY Q * P z\n⊢ W.dblZ P •\n ![W.toAffine.addX (P x / P z) (Q x / Q z) (W.toAffine.slope (P x / P z) (Q x /... | dblXYZ_of_Z_ne_zero hP hQ hPz hQz hx hy | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 11
} | {
"line": 238,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nJ : GrothendieckTopology C\nX : C\nE : J.OneHypercover X\nF : Cᵒᵖ ⥤ Type u_2\nhF : Presieve.IsSheaf J F\nS : Sieve X\nh₁ : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows\nh₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (S... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nJ : GrothendieckTopology C\nX : C\nE : J.OneHypercover X\nF : Cᵒᵖ ⥤ Type u_2\nhF : Presieve.IsSheaf J F\nS : Sieve X\nh₁ : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows\nh₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (Sieve.pullbac... | intro Y f | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 13
} | {
"line": 78,
"column": 14
} | [
{
"pp": "X : TopCat\nE : precoverage.ZeroHypercover X\n⊢ ∀ (x : ↑X), ∃ i, x ∈ Set.range ⇑(ConcreteCategory.hom (E.f i))",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.PreZeroHypercover.f",
"TopCat.precoverage",
"congrArg",
"CategoryT... | [
"X : TopCat\nE : precoverage.ZeroHypercover X\n⊢ ∀ (x : ↑X), ∃ i y, (ConcreteCategory.hom (E.f i)) y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 13
} | {
"line": 82,
"column": 14
} | [
{
"pp": "X : TopCat\nE : precoverage.ZeroHypercover X\n⊢ ∀ (i : E.I₀), Topology.IsOpenEmbedding ⇑(ConcreteCategory.hom (E.f i))",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"TopCat.precoverage",
"CategoryTheory.ConcreteCategory.hom... | [
"X : TopCat\nE : precoverage.ZeroHypercover X\n⊢ ∀ (i : E.I₀), Topology.IsOpenEmbedding ⇑(ConcreteCategory.hom (E.f i))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 26
} | {
"line": 84,
"column": 0
} | [
{
"pp": "X : TopCat\nE : precoverage.ZeroHypercover X\n⊢ ∀ (i : E.I₀), Topology.IsOpenEmbedding ⇑(ConcreteCategory.hom (E.f i))",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"TopCat.precoverage",
"TopCat.isOpenEmbedding_iff._simp_1"... | [] | simpa using E.mem₀.right | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 26
} | {
"line": 84,
"column": 0
} | [
{
"pp": "X : TopCat\nE : precoverage.ZeroHypercover X\n⊢ ∀ (i : E.I₀), Topology.IsOpenEmbedding ⇑(ConcreteCategory.hom (E.f i))",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"TopCat.precoverage",
"TopCat.isOpenEmbedding_iff._simp_1"... | [] | simpa using E.mem₀.right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 26
} | {
"line": 84,
"column": 0
} | [
{
"pp": "X : TopCat\nE : precoverage.ZeroHypercover X\n⊢ ∀ (i : E.I₀), Topology.IsOpenEmbedding ⇑(ConcreteCategory.hom (E.f i))",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"TopCat.precoverage",
"TopCat.isOpenEmbedding_iff._simp_1"... | [] | simpa using E.mem₀.right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.GluingOneHypercover | {
"line": 60,
"column": 28
} | {
"line": 60,
"column": 39
} | {
"line": 60,
"column": 40
} | [
{
"pp": "D : GlueData\ni₁ i₂ : D.J\nW : Scheme\np₁ : W ⟶ D.U i₁\np₂ : W ⟶ D.U i₂\nfac : p₁ ≫ D.ι i₁ = p₂ ≫ D.ι i₂\nT : Scheme\ng : T ⟶ W\nx✝ : ⊤.arrows g\n⊢ (g ≫ p₁) ≫ D.ι i₁ = (g ≫ p₂) ≫ D.ι i₂",
"ppTerm": "?m.177",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
... | [
"D : GlueData\ni₁ i₂ : D.J\nW : Scheme\np₁ : W ⟶ D.U i₁\np₂ : W ⟶ D.U i₂\nfac : p₁ ≫ D.ι i₁ = p₂ ≫ D.ι i₂\nT : Scheme\ng : T ⟶ W\nx✝ : ⊤.arrows g\n⊢ g ≫ p₁ ≫ D.ι i₁ = g ≫ p₂ ≫ D.ι i₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 127,
"column": 2
} | {
"line": 139,
"column": 57
} | {
"line": 141,
"column": 0
} | [
{
"pp": "⊢ precoverage ≤ Precoverage.comap uliftFunctor precoverage",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.PreZeroHypercover.f",
"_private.Mathlib.Topology.Category.TopCat.GrothendieckTopology.0.TopCat.precoverage_le_comap_uliftFunctor._s... | [] | refine Precoverage.le_of_zeroHypercover fun X E ↦ ?_
refine ⟨?_, ?_⟩
· simp only [Presieve.map_ofArrows, Precoverage.mem_comap_iff,
Types.ofArrows_mem_jointlySurjectivePrecoverage_iff, ConcreteCategory.hom_ofHom,
Set.mem_range, TypeCat.Fun.coe_mk]
intro ⟨x⟩
obtain ⟨i, y, rfl⟩ := exists_mem_zeroH... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 127,
"column": 2
} | {
"line": 139,
"column": 57
} | {
"line": 141,
"column": 0
} | [
{
"pp": "⊢ precoverage ≤ Precoverage.comap uliftFunctor precoverage",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.PreZeroHypercover.f",
"_private.Mathlib.Topology.Category.TopCat.GrothendieckTopology.0.TopCat.precoverage_le_comap_uliftFunctor._s... | [] | refine Precoverage.le_of_zeroHypercover fun X E ↦ ?_
refine ⟨?_, ?_⟩
· simp only [Presieve.map_ofArrows, Precoverage.mem_comap_iff,
Types.ofArrows_mem_jointlySurjectivePrecoverage_iff, ConcreteCategory.hom_ofHom,
Set.mem_range, TypeCat.Fun.coe_mk]
intro ⟨x⟩
obtain ⟨i, y, rfl⟩ := exists_mem_zeroH... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RingHom.Unramified | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 15
} | {
"line": 101,
"column": 16
} | [
{
"pp": "R S : Type u_3\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\ns : Set S\nhs : Ideal.span s = ⊤\nH :\n ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] ↦ FormallyUnramified) ((algebraMap S (Localization.Away ↑r)).comp f)\nalgInst✝ : Algebra R S := f.toAlgebra\nx : PrimeSpectrum S\n⊢ ∃ r ∈ s, x ... | [
"R S : Type u_3\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\ns : Set S\nhs : Ideal.span s = ⊤\nH :\n ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] ↦ FormallyUnramified) ((algebraMap S (Localization.Away ↑r)).comp f)\nalgInst✝ : Algebra R S := f.toAlgebra\nx : PrimeSpectrum S\n⊢ ∃ r ∈ s, r ∉ x.asIdeal"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Locus | {
"line": 155,
"column": 16
} | {
"line": 155,
"column": 27
} | {
"line": 155,
"column": 28
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : FinitePresentation R A\np : Ideal A\ninst✝¹ : p.IsPrime\ninst✝ : IsSmoothAt R p\nf : A\nhxf : { asIdeal := p, isPrime := inst✝¹ } ∈ ↑(basicOpen f)\nhf : ↑(basicOpen f) ⊆ smoothLocus R A\n⊢ f ∉ p",
"... | [
"R : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : FinitePresentation R A\np : Ideal A\ninst✝¹ : p.IsPrime\ninst✝ : IsSmoothAt R p\nf : A\nhxf : { asIdeal := p, isPrime := inst✝¹ } ∈ ↑(basicOpen f)\nhf : ↑(basicOpen f) ⊆ smoothLocus R A\n⊢ f ∉ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.Field | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 15
} | {
"line": 156,
"column": 16
} | [
{
"pp": "case refine_6\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsSeparable K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf : L →ₐ[K] B ⧸ I\ng : (k : L) → ↥K⟮k⟯ →ₐ[K] B\nhg₁ : ∀ (k : L), (fun g ↦ (Ideal.... | [
"case refine_6\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsSeparable K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf : L →ₐ[K] B ⧸ I\ng : (k : L) → ↥K⟮k⟯ →ₐ[K] B\nhg₁ : ∀ (k : L), (fun g ↦ (Ideal.Quotient.mkₐ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.Field | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 31
} | {
"line": 174,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nA : Type u\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra K A\ninst✝² : EssFiniteType K A\ninst✝¹ : FormallyEtale K A\np : Ideal A\ninst✝ : p.IsPrime\nthis✝³ : Module.Finite K A\nthis✝² : IsArtinianRing A\nthis✝¹ : IsReduced ... | [] | exact this ⟨p, inferInstance⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 15
} | {
"line": 275,
"column": 2
} | [
{
"pp": "case tmul.tmul.tmul\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ni... | [] | | tmul x z => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 15
} | {
"line": 163,
"column": 16
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (alg... | [
"R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (algebraMap R S)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 81
} | {
"line": 178,
"column": 82
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (alg... | [
"R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (algebraMap R S)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 26
} | {
"line": 209,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Module.Flat R S\ninst✝⁴ : FinitePresentation R S\np : Ideal R\nq : Ideal S\ninst✝³ : p.IsPrime\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : FormallySmooth p.ResidueField (p.Fiber S)\nRp : Type u_... | [
"R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Module.Flat R S\ninst✝⁴ : FinitePresentation R S\np : Ideal R\nq : Ideal S\ninst✝³ : p.IsPrime\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : FormallySmooth p.ResidueField (p.Fiber S)\nRp : Type u_1 := Localiz... | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 105,
"column": 12
} | {
"line": 105,
"column": 23
} | {
"line": 105,
"column": 24
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\nhf : Function.Injective ⇑f\nk : ℕ\nhk : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range =... | [
"R : Type u\ninst✝⁶ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\nhf : Function.Injective ⇑f\nk : ℕ\nhk : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Functoriality | {
"line": 388,
"column": 4
} | {
"line": 388,
"column": 62
} | {
"line": 388,
"column": 63
} | [
{
"pp": "case h.right\nR : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nh : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)\nx : M\ny : N\nn : ℕ\nhxy : f x ≡ y [SMOD I ^ n • ⊤]\nz : M\nhz... | [
"case h.right\nR : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nh : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)\nx : M\ny : N\nn : ℕ\nhxy : f x ≡ y [SMOD I ^ n • ⊤]\nz : M\nhz : z ∈ I ^ n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.AdicCompletion | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 13
} | {
"line": 99,
"column": 14
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nS : Type u_3\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FormallySmooth R A\nI : Ideal S\ninst✝ : IsAdicComplete I S\nf : A →ₐ[R] S ⧸ I\ng : A →ₐ[R] AdicCompletion I S\nhg : (AlgHom.restrictScalars R (A... | [
"R : Type u_1\nA : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nS : Type u_3\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FormallySmooth R A\nI : Ideal S\ninst✝ : IsAdicComplete I S\nf : A →ₐ[R] S ⧸ I\ng : A →ₐ[R] AdicCompletion I S\nhg : (AlgHom.restrictScalars R (AdicCompletio... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 17
} | {
"line": 191,
"column": 18
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : Type u\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\ng : N →ₗ[R] ... | [
"R : Type u\ninst✝⁸ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : Type u\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nhf : Func... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 452,
"column": 4
} | {
"line": 452,
"column": 29
} | {
"line": 452,
"column": 30
} | [
{
"pp": "case intro.intro\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : PreSubmersivePresentation R S ι σ\nι' : Type u_1\nσ' : Type u_2\ne : ι' ≃ ι\nf : σ' ≃ σ\ninst✝¹ : Finite σ\ninst✝ : Finite σ'\nval✝¹ : Fintype σ\nval✝ : Fintype σ'\n⊢ (a... | [
"case intro.intro\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : PreSubmersivePresentation R S ι σ\nι' : Type u_1\nσ' : Type u_2\ne : ι' ≃ ι\nf : σ' ≃ σ\ninst✝¹ : Finite σ\ninst✝ : Finite σ'\nval✝¹ : Fintype σ\nval✝ : Fintype σ'\n⊢ (aeval P.val) ... | ← AlgHom.mapMatrix_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Extension.Presentation.Core | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 53
} | {
"line": 95,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Presentation R S ι σ\nR₀ : Type u_5\ninst✝¹⁰ : CommRing R₀\ninst✝⁹ : Algebra R₀ R\ninst✝⁸ : Algebra R₀ S\ninst✝⁷ : IsScalarTower R₀ R S\ninst✝⁶ : P.HasCoeffs R₀\nR₁ : Type u_6\... | [
"R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\nP : Presentation R S ι σ\nR₀ : Type u_5\ninst✝¹⁰ : CommRing R₀\ninst✝⁹ : Algebra R₀ R\ninst✝⁸ : Algebra R₀ S\ninst✝⁷ : IsScalarTower R₀ R S\ninst✝⁶ : P.HasCoeffs R₀\nR₁ : Type u_6\ninst✝⁵ : Co... | refine ⟨subset_trans (P.coeffs_subset_range R₀) ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Extension.Presentation.Core | {
"line": 142,
"column": 58
} | {
"line": 145,
"column": 10
} | {
"line": 145,
"column": 10
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP : Presentation R S ι σ\nR₀ : Type u_5\ninst✝⁴ : CommRing R₀\ninst✝³ : Algebra R₀ R\ninst✝² : Algebra R₀ S\ninst✝¹ : IsScalarTower R₀ R S\ninst✝ : P.HasCoeffs R₀\n⊢ ∀ a ∈ Ideal.span ... | [] | by
simp_rw [← RingHom.mem_ker, ← SetLike.le_def, Ideal.span_le]
rintro a ⟨i, rfl⟩
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Presentation.Core | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 78
} | {
"line": 324,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Finite σ\nP : SubmersivePresentation R S ι σ\nR₀ : Type u_5\ninst✝⁴ : CommRing R₀\ninst✝³ : Algebra R₀ R\ninst✝² : Algebra R₀ S\ninst✝¹ : IsScalarTower R₀ R S\ninst✝ : P.HasC... | [
"R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Finite σ\nP : SubmersivePresentation R S ι σ\nR₀ : Type u_5\ninst✝⁴ : CommRing R₀\ninst✝³ : Algebra R₀ R\ninst✝² : Algebra R₀ S\ninst✝¹ : IsScalarTower R₀ R S\ninst✝ : P.HasCoeffs R₀\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.StandardSmoothCotangent | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 15
} | {
"line": 114,
"column": 16
} | [
{
"pp": "case hr\nR : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\nx : ↥P.ker\nhx : ∀ (i : σ), (aeval P.val) ((pderiv (P.map i)) ↑x) = 0\nthis✝ : ↑x ∈ Ideal.span (Set.range P.relation)\nc... | [
"case hr\nR : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\nx : ↥P.ker\nhx : ∀ (i : σ), (aeval P.val) ((pderiv (P.map i)) ↑x) = 0\nthis✝ : ↑x ∈ Ideal.span (Set.range P.relation)\nc : σ →₀ P.Ri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 552,
"column": 39
} | {
"line": 552,
"column": 73
} | {
"line": 552,
"column": 73
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ≠ Q x * P z\n⊢ ?m.162 ≠ ?m.163",
"ppTerm": "?m.166",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg... | [] | by rwa [ne_eq, ← X_eq_iff hPz hQz] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 87
} | {
"line": 211,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nι : Type u_4\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\ng : S\ninst✝ : IsLocalization.Away g T\nP : Generators R S ι\nx : ((localizationAway T g... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\nι : Type u_4\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\ng : S\ninst✝ : IsLocalization.Away g T\nP : Generators R S ι\nx : ((localizationAway T g).comp P).to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 68,
"column": 2
} | {
"line": 69,
"column": 65
} | {
"line": 69,
"column": 66
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\ni : σ\n⊢ (Subtype.val ∘ D.f ∘ ⇑b) i ∈ ↑(RingHom.ker (aeval P.val))",
"ppTerm": "?m.6... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\ni : σ\n⊢ (algebraMap P.Ring S) ↑(D.f (b i)) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 243,
"column": 8
} | {
"line": 244,
"column": 48
} | {
"line": 244,
"column": 49
} | [
{
"pp": "n : ℕ\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S\nthis✝¹ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite\nι σ : Type\nw✝¹ : _root_.Finite σ\nw✝ : _root_.Finite ι\nP : Algebra.Submers... | [
"n : ℕ\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S\nthis✝¹ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite\nι σ : Type\nw✝¹ : _root_.Finite σ\nw✝ : _root_.Finite ι\nP : Algebra.SubmersivePresentat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 294,
"column": 41
} | {
"line": 294,
"column": 52
} | {
"line": 294,
"column": 53
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝ : Finite α\nσ : Type u_5\nb₀ : Module.Basis σ S P.toExtension.Cotangent\nh✝ : Subsingleton S\n⊢ Ideal.span (Set.range fun x ↦ 1) = (ofSu... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝ : Finite α\nσ : Type u_5\nb₀ : Module.Basis σ S P.toExtension.Cotangent\nh✝ : Subsingleton S\n⊢ ⊤ = (ofSurjective (fun i ↦ 0) ⋯).ker"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 313,
"column": 65
} | {
"line": 313,
"column": 76
} | {
"line": 313,
"column": 77
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝ : Finite α\nσ : Type u_5\nb₀ : Module.Basis σ S P.toExtension.Cotangent\nh✝ : Nontrivial S\nf : P.toExtension.Cotangent → ↥P.toExtension... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝ : Finite α\nσ : Type u_5\nb₀ : Module.Basis σ S P.toExtension.Cotangent\nh✝ : Nontrivial S\nf : P.toExtension.Cotangent → ↥P.toExtension.ker\nhf : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 747,
"column": 13
} | {
"line": 747,
"column": 64
} | {
"line": 747,
"column": 65
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ≠ Q x * P z\n⊢ W.negY ![W.addX P Q, W.negAddY P Q, W.addZ P Q] / W.addZ P Q =\n W.toAffine.addY (P x / P z) (Q x / Q z) (P y / P ... | [
"F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z ≠ Q x * P z\n⊢ W.toAffine.negY (![W.addX P Q, W.negAddY P Q, W.addZ P Q] x / ![W.addX P Q, W.negAddY P Q, W.addZ P Q] z)\n (![W.addX P Q, W... | negY_of_Z_ne_zero <| addZ_ne_zero_of_X_ne hP hQ hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 94,
"column": 6
} | {
"line": 94,
"column": 27
} | {
"line": 94,
"column": 28
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\nq₁ q₂ : MvPolynomial ι k\ne : F = q₁ * q₂\n... | [
"k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\nq₁ q₂ : MvPolynomial ι k\ne : F = q₁ * q₂\nthis : ∀ (q₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 13
} | {
"line": 100,
"column": 14
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\nq₁ q₂ : MvPolynomial ι k\ne : F = q₁ * q₂\n... | [
"k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\nq₁ q₂ : MvPolynomial ι k\ne : F = q₁ * q₂\nh₁ : (aeval ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 21
} | {
"line": 114,
"column": 22
} | [
{
"pp": "case refine_3\nk : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nσ : ι →₀ ℕ\nhσ : σ ∈ F.support\ni : ι\nhσi : σ i ≠ 0\nH✝ : (F.to... | [
"case refine_3\nk : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nσ : ι →₀ ℕ\nhσ : σ ∈ F.support\ni : ι\nhσi : σ i ≠ 0\nH✝ : (F.toPolynomialAd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 62
} | {
"line": 228,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nA : Type u\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : Smooth A B\nP : Presentation A B (Fin (Presentation.ofFinitePresentationVars A B))\n (Fin (Presentation.ofFinitePresentationRels A B)) :=\n Presen... | [
"R : Type u_1\ninst✝⁵ : CommRing R\nA : Type u\nB : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : Smooth A B\nP : Presentation A B (Fin (Presentation.ofFinitePresentationVars A B))\n (Fin (Presentation.ofFinitePresentationRels A B)) :=\n Presentation.ofFin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Smooth | {
"line": 238,
"column": 78
} | {
"line": 241,
"column": 62
} | {
"line": 243,
"column": 0
} | [
{
"pp": "n m : ℕ\nX Y : Scheme\nf : X ⟶ Y\nhf : Smooth f\n⊢ LocallyOfFinitePresentation f",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"RingHom.FinitePresentation",
"CommRing",
"AlgebraicGeometry.Scheme",
"c... | [] | by
rw [HasRingHomProperty.eq_affineLocally @LocallyOfFinitePresentation]
rw [HasRingHomProperty.eq_affineLocally @Smooth] at hf
exact affineLocally_le (fun hf ↦ hf.finitePresentation) f hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.LocalClosure | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 40
} | {
"line": 67,
"column": 4
} | [
{
"pp": "W P Q : MorphismProperty Scheme\nX✝ Y✝ : Scheme\ninst✝³ : W.IsStableUnderBaseChange\ninst✝² : Scheme.IsJointlySurjectivePreserving W\ninst✝¹ : P.RespectsLeft Q\ninst✝ : Q.IsStableUnderBaseChange\nX Y Z : Scheme\nf : X ⟶ Y\nhf : Q f\ng : Y ⟶ Z\nx✝ : sourceLocalClosure W P g\n𝒰 : Scheme.Cover (Scheme.pr... | [
"W P Q : MorphismProperty Scheme\nX✝ Y✝ : Scheme\ninst✝³ : W.IsStableUnderBaseChange\ninst✝² : Scheme.IsJointlySurjectivePreserving W\ninst✝¹ : P.RespectsLeft Q\ninst✝ : Q.IsStableUnderBaseChange\nX Y Z : Scheme\nf : X ⟶ Y\nhf : Q f\ng : Y ⟶ Z\nx✝ : sourceLocalClosure W P g\n𝒰 : Scheme.Cover (Scheme.precoverage W)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 242,
"column": 8
} | {
"line": 242,
"column": 19
} | {
"line": 242,
"column": 20
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\nn : ι\ninst✝ : ExpChar k p\nha' : IsTranscendenceBasis k fun i ↦ a ↑i\nS : Set (MvPol... | [
"k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\nn : ι\ninst✝ : ExpChar k p\nha' : IsTranscendenceBasis k fun i ↦ a ↑i\nS : Set (MvPolynomial ι k)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.LocalClosure | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 42
} | {
"line": 82,
"column": 6
} | [
{
"pp": "case refine_1\nW P Q : MorphismProperty Scheme\nX Y : Scheme\ninst✝³ : W.IsStableUnderBaseChange\ninst✝² : Scheme.IsJointlySurjectivePreserving W\ninst✝¹ : P.RespectsIso\ninst✝ : P.RespectsLeft IsOpenImmersion\nX✝ Y✝ : Scheme\nf : X✝ ⟶ Y✝\n𝒰 : Scheme.zariskiPrecoverage.ZeroHypercover X✝\nx✝ : sourceLo... | [
"case refine_1\nW P Q : MorphismProperty Scheme\nX Y : Scheme\ninst✝³ : W.IsStableUnderBaseChange\ninst✝² : Scheme.IsJointlySurjectivePreserving W\ninst✝¹ : P.RespectsIso\ninst✝ : P.RespectsLeft IsOpenImmersion\nX✝ Y✝ : Scheme\nf : X✝ ⟶ Y✝\n𝒰 : Scheme.zariskiPrecoverage.ZeroHypercover X✝\nx✝ : sourceLocalClosure I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.LocalClosure | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 82
} | {
"line": 90,
"column": 4
} | [
{
"pp": "W P Q : MorphismProperty Scheme\nX✝ Y✝ : Scheme\ninst✝² : W.IsStableUnderBaseChange\ninst✝¹ : Scheme.IsJointlySurjectivePreserving W\ninst✝ : P.IsStableUnderBaseChange\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nx✝¹ : HasPullback f g\nx✝ : sourceLocalClosure W P g\n𝒰 : Scheme.Cover (Scheme.precoverage W) Y... | [
"W P Q : MorphismProperty Scheme\nX✝ Y✝ : Scheme\ninst✝² : W.IsStableUnderBaseChange\ninst✝¹ : Scheme.IsJointlySurjectivePreserving W\ninst✝ : P.IsStableUnderBaseChange\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nx✝¹ : HasPullback f g\nx✝ : sourceLocalClosure W P g\n𝒰 : Scheme.Cover (Scheme.precoverage W) Y\nhg : ∀ (i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.LocalClosure | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 89
} | {
"line": 102,
"column": 4
} | [
{
"pp": "W P Q : MorphismProperty Scheme\nX✝ Y✝ : Scheme\ninst✝⁴ : W.IsStableUnderBaseChange\ninst✝³ : Scheme.IsJointlySurjectivePreserving W\ninst✝² : W.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderBaseChange\ninst✝ : P.IsStableUnderComposition\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nx✝² : sourceLocalClosu... | [
"W P Q : MorphismProperty Scheme\nX✝ Y✝ : Scheme\ninst✝⁴ : W.IsStableUnderBaseChange\ninst✝³ : Scheme.IsJointlySurjectivePreserving W\ninst✝² : W.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderBaseChange\ninst✝ : P.IsStableUnderComposition\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\nx✝² : sourceLocalClosure W P f\nx✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 76
} | {
"line": 218,
"column": 6
} | [
{
"pp": "X Y Z' Z : Scheme\ni : Z' ⟶ Z\nhi : IsNilpotent (Scheme.Hom.ker i)\ninst✝¹ : IsClosedImmersion i\nf : X ⟶ Y\ninst✝ : FormallyUnramified f\ng₁ g₂ : Z ⟶ X\nhig : i ≫ g₁ = i ≫ g₂\nhgf : g₁ ≫ f = g₂ ≫ f\nthis✝² : IsDominant i\nx : ↥Z\nU : Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : f (g₁ x) ∈ ↑U\nV : Opens ↥X\... | [
"X Y Z' Z : Scheme\ni : Z' ⟶ Z\nhi : IsNilpotent (Scheme.Hom.ker i)\ninst✝¹ : IsClosedImmersion i\nf : X ⟶ Y\ninst✝ : FormallyUnramified f\ng₁ g₂ : Z ⟶ X\nhig : i ≫ g₁ = i ≫ g₂\nhgf : g₁ ≫ f = g₂ ≫ f\nthis✝² : IsDominant i\nx : ↥Z\nU : Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : f (g₁ x) ∈ ↑U\nV : Opens ↥X\nhV : V ∈ X.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified | {
"line": 221,
"column": 17
} | {
"line": 221,
"column": 28
} | {
"line": 221,
"column": 29
} | [
{
"pp": "X Y Z' Z : Scheme\ni : Z' ⟶ Z\ninst✝¹ : IsClosedImmersion i\nf : X ⟶ Y\ninst✝ : FormallyUnramified f\ng₁ g₂ : Z ⟶ X\nhig : i ≫ g₁ = i ≫ g₂\nhgf : g₁ ≫ f = g₂ ≫ f\nthis✝¹ : IsDominant i\nx : ↥Z\nU : Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : f (g₁ x) ∈ ↑U\nV : Opens ↥X\nhV : V ∈ X.affineOpens\nhxV : g₁ x ∈... | [
"X Y Z' Z : Scheme\ni : Z' ⟶ Z\ninst✝¹ : IsClosedImmersion i\nf : X ⟶ Y\ninst✝ : FormallyUnramified f\ng₁ g₂ : Z ⟶ X\nhig : i ≫ g₁ = i ≫ g₂\nhgf : g₁ ≫ f = g₂ ≫ f\nthis✝¹ : IsDominant i\nx : ↥Z\nU : Opens ↥Y\nhU : U ∈ Y.affineOpens\nhxU : f (g₁ x) ∈ ↑U\nV : Opens ↥X\nhV : V ∈ X.affineOpens\nhxV : g₁ x ∈ ↑V\nhVU : V... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Descent | {
"line": 58,
"column": 21
} | {
"line": 58,
"column": 36
} | {
"line": 58,
"column": 37
} | [
{
"pp": "P : MorphismProperty Scheme\nX : Scheme\ninst✝² : CompactSpace ↥X\ninst✝¹ : IsZariskiLocalAtSource P\ninst✝ : P.ContainsIdentities\n𝒰 : X.OpenCover := X.affineCover.finiteSubcover\np : (∐ fun i ↦ 𝒰.X i) ⟶ X := Sigma.desc fun i ↦ 𝒰.f i\ni : (sigmaOpenCover fun i ↦ 𝒰.X i).I₀\n⊢ P ((sigmaOpenCover fun... | [
"P : MorphismProperty Scheme\nX : Scheme\ninst✝² : CompactSpace ↥X\ninst✝¹ : IsZariskiLocalAtSource P\ninst✝ : P.ContainsIdentities\n𝒰 : X.OpenCover := X.affineCover.finiteSubcover\np : (∐ fun i ↦ 𝒰.X i) ⟶ X := Sigma.desc fun i ↦ 𝒰.f i\ni : (sigmaOpenCover fun i ↦ 𝒰.X i).I₀\n⊢ P (𝒰.f i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Descent | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 20
} | {
"line": 130,
"column": 21
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nP : MorphismProperty C\ninst✝³ : (isomorphisms C).DescendsAlong P\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : HasEqualizers C\ninst✝ : HasPullbacks C\nX Y S T : C\nf g : X ⟶ Y\ns : X ⟶ S\nt : Y ⟶ S\nhf : f ≫ t = s\nhg : g ≫ t = s\nv : T ⟶ S\nhv : P v\nH :... | [
"C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nP : MorphismProperty C\ninst✝³ : (isomorphisms C).DescendsAlong P\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : HasEqualizers C\ninst✝ : HasPullbacks C\nX Y S T : C\nf g : X ⟶ Y\ns : X ⟶ S\nt : Y ⟶ S\nhf : f ≫ t = s\nhg : g ≫ t = s\nv : T ⟶ S\nhv : P v\nH : pullback.ma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.Descent | {
"line": 140,
"column": 34
} | {
"line": 140,
"column": 81
} | {
"line": 140,
"column": 82
} | [
{
"pp": "P P' : MorphismProperty Scheme\ninst✝³ : P'.IsStableUnderBaseChange\ninst✝² : P'.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderBaseChange\nH₁ : @IsLocalIso ⊓ @Surjective ≤ P'\ninst✝ : IsZariskiLocalAtTarget P\nH :\n ∀ {R S : CommRingCat} {Y : Scheme} (φ : R ⟶ S) (g : Y ⟶ Spec R),\n P' (Spec.map... | [
"P P' : MorphismProperty Scheme\ninst✝³ : P'.IsStableUnderBaseChange\ninst✝² : P'.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderBaseChange\nH₁ : @IsLocalIso ⊓ @Surjective ≤ P'\ninst✝ : IsZariskiLocalAtTarget P\nH :\n ∀ {R S : CommRingCat} {Y : Scheme} (φ : R ⟶ S) (g : Y ⟶ Spec R),\n P' (Spec.map φ) → P (pul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 43,
"column": 7
} | {
"line": 43,
"column": 61
} | {
"line": 43,
"column": 62
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nx : S\nh : IsStronglyTranscendental R x\ninst✝ : FaithfulSMul R S\np : R[X]\nhp : (aeval x) p = 0\n⊢ ∀ (n : ℕ), p.coeff n = coeff 0 n",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nx : S\nh : IsStronglyTranscendental R x\ninst✝ : FaithfulSMul R S\np : R[X]\nhp : (aeval x) p = 0\n⊢ ∀ (n : ℕ), p.coeff n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 48,
"column": 2
} | {
"line": 49,
"column": 90
} | {
"line": 49,
"column": 91
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : FaithfulSMul R K\nx : K\nh : Transcendental R x\n⊢ IsStronglyTranscendental R x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"GroupWithZe... | [
"R : Type u_1\ninst✝³ : CommRing R\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : FaithfulSMul R K\nx : K\nh : Transcendental R x\n⊢ ∀ (b : R[X]), (aeval x) b = 0 → ∀ (n : ℕ), b.coeff n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 43
} | {
"line": 68,
"column": 44
} | [
{
"pp": "R : 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\nM : Submonoid S\ninst✝¹ : IsLocalization M T\ninst✝ : IsScalarTower R S T\nx : S\nh : IsStronglyTranscendental R x\np : R[X]\nu : S\... | [
"R : 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\nM : Submonoid S\ninst✝¹ : IsLocalization M T\ninst✝ : IsScalarTower R S T\nx : S\nh : IsStronglyTranscendental R x\np : R[X]\nu : S\ns : ↥M\nhp ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 74
} | {
"line": 71,
"column": 6
} | [
{
"pp": "R : 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\nM : Submonoid S\ninst✝¹ : IsLocalization M T\ninst✝ : IsScalarTower R S T\nx : S\nh : IsStronglyTranscendental R x\np : R[X]\nu : S\... | [
"R : 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\nM : Submonoid S\ninst✝¹ : IsLocalization M T\ninst✝ : IsScalarTower R S T\nx : S\nh : IsStronglyTranscendental R x\np : R[X]\nu : S\ns : ↥M\nhp ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 42
} | {
"line": 82,
"column": 43
} | [
{
"pp": "R : 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\nM : Submonoid R\ninst✝¹ : IsLocalization M S\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental R x\nt : T\np : S[X]\... | [
"R : 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\nM : Submonoid R\ninst✝¹ : IsLocalization M S\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental R x\nt : T\np : S[X]\nhp : (aeval... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 43
} | {
"line": 86,
"column": 44
} | [
{
"pp": "R : 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\nM : Submonoid R\ninst✝¹ : IsLocalization M S\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental R x\nt : T\np : S[X]\... | [
"R : 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\nM : Submonoid R\ninst✝¹ : IsLocalization M S\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental R x\nt : T\np : S[X]\nhp : (aeval... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Conductor | {
"line": 35,
"column": 25
} | {
"line": 35,
"column": 51
} | {
"line": 35,
"column": 52
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx a✝ b✝ : S\nha : a✝ ∈ {a | ∀ (b : S), a * b ∈ R[x]}\nhb : b✝ ∈ {a | ∀ (b : S), a * b ∈ R[x]}\nc : S\n⊢ (a✝ + b✝) * c ∈ R[x]",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Subalgebra... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx a✝ b✝ : S\nha : a✝ ∈ {a | ∀ (b : S), a * b ∈ R[x]}\nhb : b✝ ∈ {a | ∀ (b : S), a * b ∈ R[x]}\nc : S\n⊢ a✝ * c + b✝ * c ∈ R[x]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Conductor | {
"line": 36,
"column": 27
} | {
"line": 36,
"column": 83
} | {
"line": 36,
"column": 84
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx c a : S\nha : a ∈ {a | ∀ (b : S), a * b ∈ R[x]}\nb : S\n⊢ c • a * b ∈ R[x]",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Semigroup.... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx c a : S\nha : a ∈ {a | ∀ (b : S), a * b ∈ R[x]}\nb : S\n⊢ c * (a * b) ∈ R[x]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 54
} | {
"line": 92,
"column": 55
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental S x\nt : T\np : R[X]\nhp : (aeval x) p * t = 0\n⊢ map (algebraMap R... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental S x\nt : T\np : R[X]\nhp : (aeval x) p * t = 0\n⊢ map (algebraMap R T) p * C t ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 54
} | {
"line": 99,
"column": 55
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental R x\nH : Function.Surjective ⇑(algebraMap R S)\nt : T\np : R[X]\nhp... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental R x\nH : Function.Surjective ⇑(algebraMap R S)\nt : T\np : R[X]\nhp : (aeval x)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 99,
"column": 65
} | {
"line": 99,
"column": 76
} | {
"line": 99,
"column": 77
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental R x\nH : Function.Surjective ⇑(algebraMap R S)\nt : T\np : R[X]\nhp... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx : T\nh : IsStronglyTranscendental R x\nH : Function.Surjective ⇑(algebraMap R S)\nt : T\np : R[X]\nhp : (aeval x)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.FlatDescent | {
"line": 138,
"column": 2
} | {
"line": 139,
"column": 72
} | {
"line": 140,
"column": 2
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : HasPullback f g\nhf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f\nhg : IsOpenImmersion (pullback.fst f g)\nthis : UniversallyOpen g\nU : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }\nf' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι\ng' : Y ⟶ ↑U := IsOpe... | [
"X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : HasPullback f g\nhf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f\nhg : IsOpenImmersion (pullback.fst f g)\nthis✝ : UniversallyOpen g\nU : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }\nf' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι\ng' : Y ⟶ ↑U := IsOpenImmersion.... | have : Surjective g' := ⟨fun ⟨x, ⟨y, hy⟩⟩ ↦
⟨y, by apply U.ι.injective; simp [← Scheme.Hom.comp_apply, g', hy]⟩⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.FlatDescent | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 30
} | {
"line": 143,
"column": 4
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : HasPullback f g\nhf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f\nhg : IsOpenImmersion (pullback.fst f g)\nthis✝ : UniversallyOpen g\nU : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }\nf' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι\ng' : Y ⟶ ↑U := IsOp... | [
"X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : HasPullback f g\nhf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f\nhg : IsOpenImmersion (pullback.fst f g)\nthis✝ : UniversallyOpen g\nU : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }\nf' : pullback f U.ι ⟶ ↑U := pullback.snd f U.ι\ng' : Y ⟶ ↑U := IsOpenImmersion.... | refine ⟨?_, inferInstance⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 29
} | {
"line": 139,
"column": 30
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsReduced S\nx : S\nhx : IsStronglyTranscendental R x\nq : Ideal S\nhq : q ∈ minimalPrimes S\nu : S\np : R[X]\ne : (aeval x) p * u ∈ q\nthis✝¹ : q.IsPrime\nthis✝ : Ring.KrullDimLE 0 (Localization.AtPrime... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsReduced S\nx : S\nhx : IsStronglyTranscendental R x\nq : Ideal S\nhq : q ∈ minimalPrimes S\nu : S\np : R[X]\ne : (aeval x) p * u ∈ q\nthis✝¹ : q.IsPrime\nthis✝ : Ring.KrullDimLE 0 (Localization.AtPrime q)\nthis : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 138,
"column": 2
} | {
"line": 139,
"column": 89
} | {
"line": 140,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsReduced S\nx : S\nhx : IsStronglyTranscendental R x\nq : Ideal S\nhq : q ∈ minimalPrimes S\nu : S\np : R[X]\ne : (aeval x) p * u ∈ q\nthis✝¹ : q.IsPrime\nthis✝ : Ring.KrullDimLE 0 (Localization.AtPrime... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsReduced S\nx : S\nhx : IsStronglyTranscendental R x\nq : Ideal S\nhq : q ∈ minimalPrimes S\nu : S\np : R[X]\ne : (aeval x) p * u ∈ q\nthis✝² : q.IsPrime\nthis✝¹ : Ring.KrullDimLE 0 (Localization.AtPrime q)\nthis✝ ... | have : algebraMap R S (p.coeff i) * u * m = 0 := by
simpa [← mul_assoc] using congr(($(hx (u * m) p (by linear_combination hm))).coeff i) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 44,
"column": 35
} | {
"line": 44,
"column": 46
} | {
"line": 44,
"column": 47
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nP x y : R[X]\nhx✝ : x ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhy✝ : y ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhx : ∀ (i : ℕ), x.coeff i ∈ I ^ i\nhy : ∀ (i : ℕ), y.coeff i ∈ I ^ i\ni : ℕ\nx✝ : ℕ × ℕ\nj₁ j₂ : ℕ\nhj : (j₁, j₂) ∈ Finset.antidiago... | [
"R : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nP x y : R[X]\nhx✝ : x ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhy✝ : y ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhx : ∀ (i : ℕ), x.coeff i ∈ I ^ i\nhy : ∀ (i : ℕ), y.coeff i ∈ I ^ i\ni : ℕ\nx✝ : ℕ × ℕ\nj₁ j₂ : ℕ\nhj : (j₁, j₂) ∈ Finset.antidiagonal i\n⊢ j₁ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 36,
"column": 6
} | {
"line": 36,
"column": 57
} | {
"line": 36,
"column": 58
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni✝ : ℕ\nhi✝ : i✝ ≠ q.natDegree\nP : Ideal R\nhPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P\nhP : P.IsPrime\ni : ℕ\nhi : i < p.natDegree\n⊢ (map (Ideal.Quotient.mk P) p).coeff i = (X ^ p.natDegre... | [
"case inl\nR : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni✝ : ℕ\nhi✝ : i✝ ≠ q.natDegree\nP : Ideal R\nhPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P\nhP : P.IsPrime\ni : ℕ\nhi : i < p.natDegree\n⊢ p.coeff i ∈ P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 46
} | {
"line": 44,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni : ℕ\nhi : i ≠ q.natDegree\nP : Ideal R\nhPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P\nhP : P.IsPrime\nthis : map (Ideal.Quotient.mk P) p = X ^ p.natDegree\nj : ℕ\nhj : j ≤ p.natDegree\na : (R ⧸ P)[X]ˣ\... | [
"R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni : ℕ\nhi : i ≠ q.natDegree\nP : Ideal R\nhPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P\nhP : P.IsPrime\nthis : map (Ideal.Quotient.mk P) p = X ^ p.natDegree\nj : ℕ\nhj : j ≤ p.natDegree\na : (R ⧸ P)[X]ˣ\nr : R ⧸ P\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 50
} | {
"line": 45,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni : ℕ\nhi : i ≠ q.natDegree\nP : Ideal R\nhPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P\nhP : P.IsPrime\nthis : map (Ideal.Quotient.mk P) p = X ^ p.natDegree\na : (R ⧸ P)[X]ˣ\nr : R ⧸ P\nhr : IsUnit r\ne ... | [
"R : Type u_1\ninst✝ : CommRing R\np q : R[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ p\ni : ℕ\nhi : i ≠ q.natDegree\nP : Ideal R\nhPJ : Ideal.span {x | ∃ i < p.natDegree, p.coeff i = x} ≤ P\nhP : P.IsPrime\nthis : map (Ideal.Quotient.mk P) p = X ^ p.natDegree\na : (R ⧸ P)[X]ˣ\nr : R ⧸ P\nhr : IsUnit r\ne : C r = ↑a⁻¹... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 62,
"column": 4
} | {
"line": 63,
"column": 74
} | {
"line": 63,
"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 : IsDomain S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : Algebra.IsIntegral R S\ninst✝² : IsIntegrallyClosed R\nthis : IsDomain R\np : Ideal R\ninst✝¹ : p.IsPrime\nQ : Ideal S\ninst✝ : Q.IsPrime\nhpQ : p < Ideal... | [
"R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsDomain S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : Algebra.IsIntegral R S\ninst✝² : IsIntegrallyClosed R\nthis : IsDomain R\np : Ideal R\ninst✝¹ : p.IsPrime\nQ : Ideal S\ninst✝ : Q.IsPrime\nhpQ : p < Ideal.under R Q\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 82,
"column": 25
} | {
"line": 82,
"column": 46
} | {
"line": 82,
"column": 47
} | [
{
"pp": "case add\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsIntegral R S\nI : Ideal R\nx✝ : S\nA : Subalgebra R R[X] := Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nthis : Algebra R[X] S[X] := algebra R S\nx y : S\nhx✝ : x ∈ Submodule.span S (... | [
"case add\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsIntegral R S\nI : Ideal R\nx✝ : S\nA : Subalgebra R R[X] := Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nthis : Algebra R[X] S[X] := algebra R S\nx y : S\nhx✝ : x ∈ Submodule.span S (⇑(algebraMap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 15
} | {
"line": 85,
"column": 16
} | [
{
"pp": "case mem\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsIntegral R S\nI : Ideal R\nx✝ : S\nA : Subalgebra R R[X] := Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nthis : Algebra R[X] S[X] := algebra R S\nx : R\nhx : x ∈ ↑I\n⊢ IsIntegral (↥(A... | [
"case mem\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsIntegral R S\nI : Ideal R\nx✝ : S\nA : Subalgebra R R[X] := Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nthis : Algebra R[X] S[X] := algebra R S\nx : R\nhx : x ∈ ↑I\n⊢ IsIntegral (↥(Algebra.adjoi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 59
} | {
"line": 71,
"column": 60
} | [
{
"pp": "case inl\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsDomain S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : Algebra.IsIntegral R S\ninst✝² : IsIntegrallyClosed R\nthis : IsDomain R\np : Ideal R\ninst✝¹ : p.IsPrime\nQ : Ideal S\ninst✝ : Q.IsPrime\nhpQ :... | [
"case inl\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsDomain S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : Algebra.IsIntegral R S\ninst✝² : IsIntegrallyClosed R\nthis : IsDomain R\np : Ideal R\ninst✝¹ : p.IsPrime\nQ : Ideal S\ninst✝ : Q.IsPrime\nhpQ : p < Ideal.u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.GoingDown | {
"line": 75,
"column": 6
} | {
"line": 75,
"column": 77
} | {
"line": 76,
"column": 8
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsDomain S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : Algebra.IsIntegral R S\ninst✝² : IsIntegrallyClosed R\nthis✝ : IsDomain R\np : Ideal R\ninst✝¹ : p.IsPrime\nQ : Ideal S\ninst✝ : Q.IsPrime\nhpQ : p < Idea... | [
"R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : IsDomain S\ninst✝⁴ : FaithfulSMul R S\ninst✝³ : Algebra.IsIntegral R S\ninst✝² : IsIntegrallyClosed R\nthis✝ : IsDomain R\np : Ideal R\ninst✝¹ : p.IsPrime\nQ : Ideal S\ninst✝ : Q.IsPrime\nhpQ : p < Ideal.under R Q\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.RatFunc.Defs | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 59
} | {
"line": 219,
"column": 8
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nP : K⟮X⟯ → Prop\nx : FractionRing K[X]\nf : ∀ (p q : K[X]), q ≠ 0 → P (RatFunc.mk p q)\nx✝ : K[X] × ↥K[X]⁰\np : K[X]\nq : ↥K[X]⁰\n⊢ P { toFractionRing := Localization.mk (p, q).1 (p, q).2 }",
"ppTerm": "?m.29",
"assigned": true,
"usedCons... | [
"K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nP : K⟮X⟯ → Prop\nx : FractionRing K[X]\nf : ∀ (p q : K[X]), q ≠ 0 → P (RatFunc.mk p q)\nx✝ : K[X] × ↥K[X]⁰\np : K[X]\nq : ↥K[X]⁰\n⊢ P { toFractionRing := IsLocalization.mk' (Localization K[X]⁰) p q }"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.IntegralClosure.IsIntegral.AlmostIntegral | {
"line": 101,
"column": 6
} | {
"line": 102,
"column": 43
} | {
"line": 102,
"column": 44
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ns : S\ninst✝ : IsNoetherianRing R\nH' : R⁰ ≤ Submonoid.comap (algebraMap R S) S⁰\nr : R\nhr : r ∈ R⁰\nhr' : ∀ (n : ℕ), r • s ^ n ∈ (algebraMap R S).range\nn : ℕ\na : R\nha : (algebraMap R S) a = r • s ^ n\nthis ... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ns : S\ninst✝ : IsNoetherianRing R\nH' : R⁰ ≤ Submonoid.comap (algebraMap R S) S⁰\nr : R\nhr : r ∈ R⁰\nhr' : ∀ (n : ℕ), r • s ^ n ∈ (algebraMap R S).range\nn : ℕ\na : R\nha : (algebraMap R S) a = r • s ^ n\nthis :\n (algebr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Conductor | {
"line": 205,
"column": 39
} | {
"line": 205,
"column": 55
} | {
"line": 205,
"column": 56
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nx : S\nP : Ideal S\ninst✝² : P.IsPrime\nhx : ¬conductor R x ≤ P\ns : Subalgebra R S\nhs : s = R[x]\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\na : S\nha : a ∈ conductor R x\nhaP : a ∉ P\nb : S\n⊢ a ... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nx : S\nP : Ideal S\ninst✝² : P.IsPrime\nhx : ¬conductor R x ≤ P\ns : Subalgebra R S\nhs : s = R[x]\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\na : S\nha : a ∈ conductor R x\nhaP : a ∉ P\nb : S\n⊢ a * b ∈ R[x]"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Conductor | {
"line": 207,
"column": 42
} | {
"line": 207,
"column": 53
} | {
"line": 207,
"column": 54
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nx : S\nP : Ideal S\ninst✝² : P.IsPrime\nhx : ¬conductor R x ≤ P\ns : Subalgebra R S\nhs : s = R[x]\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\na : S\nhaP : a ∉ P\nha : ∀ (b : S), a * b ∈ s\ny : S\nx... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nx : S\nP : Ideal S\ninst✝² : P.IsPrime\nhx : ¬conductor R x ≤ P\ns : Subalgebra R S\nhs : s = R[x]\np : Ideal ↥s\ninst✝¹ : p.IsPrime\ninst✝ : P.LiesOver p\na : S\nhaP : a ∉ P\nha : ∀ (b : S), a * b ∈ s\ny : S\nx✝ : y ∈ ⊤\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 90,
"column": 31
} | {
"line": 90,
"column": 42
} | {
"line": 90,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\np✝ : S[X]\nhp✝ : IsAlmostIntegral R[X] p✝\ni : ℕ\nq : S[X]\np : R[X]\nhp : p ∈ R[X]⁰\nhp' : ∀ (n : ℕ), p • q ^ n ∈ (algebraMap R[X] S[X]).range\n⊢ p.leadingCoeff ∈ ... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\np✝ : S[X]\nhp✝ : IsAlmostIntegral R[X] p✝\ni : ℕ\nq : S[X]\np : R[X]\nhp : p ∈ R[X]⁰\nhp' : ∀ (n : ℕ), p • q ^ n ∈ (algebraMap R[X] S[X]).range\n⊢ ¬p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 17
} | {
"line": 105,
"column": 18
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\ni : ℕ\nH : ∀ {q : S[X]}, IsAlmostIntegral R[X] q → IsAlmostIntegral R q.leadingCoeff\nn : ℕ\nIH : ∀ m < n, ∀ {p : S[X]}, IsAlmostIntegral R[X] p → p.natDe... | [
"case pos\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\ni : ℕ\nH : ∀ {q : S[X]}, IsAlmostIntegral R[X] q → IsAlmostIntegral R q.leadingCoeff\nn : ℕ\nIH : ∀ m < n, ∀ {p : S[X]}, IsAlmostIntegral R[X] p → p.natDegree = m → I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 114,
"column": 20
} | {
"line": 114,
"column": 31
} | {
"line": 114,
"column": 32
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\ni : ℕ\nH : ∀ {q : S[X]}, IsAlmostIntegral R[X] q → IsAlmostIntegral R q.leadingCoeff\nn : ℕ\nIH : ∀ m < n, ∀ {p : S[X]}, IsAlmostIntegral R[X] p → p.natDegree = m →... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\ni : ℕ\nH : ∀ {q : S[X]}, IsAlmostIntegral R[X] q → IsAlmostIntegral R q.leadingCoeff\nn : ℕ\nIH : ∀ m < n, ∀ {p : S[X]}, IsAlmostIntegral R[X] p → p.natDegree = m → IsAlmostInt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 41
} | {
"line": 118,
"column": 4
} | [
{
"pp": "case neg\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\ni : ℕ\nH : ∀ {q : S[X]}, IsAlmostIntegral R[X] q → IsAlmostIntegral R q.leadingCoeff\nn : ℕ\nIH : ∀ m < n, ∀ {p : S[X]}, IsAlmostIntegral R[X] p → p.natDe... | [
"case neg\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\ni : ℕ\nH : ∀ {q : S[X]}, IsAlmostIntegral R[X] q → IsAlmostIntegral R q.leadingCoeff\nn : ℕ\nIH : ∀ m < n, ∀ {p : S[X]}, IsAlmostIntegral R[X] p → p.natDegree = m → I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 137,
"column": 10
} | {
"line": 137,
"column": 21
} | {
"line": 137,
"column": 22
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\nhm₁ : ∀ (i : ℕ... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\nhm₁ : ∀ (i : ℕ), (q.coeff ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 40
} | {
"line": 147,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\nhm₁ : ∀ (i : ℕ... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\nhm₁ : ∀ (i : ℕ), (q.coeff ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 67
} | {
"line": 151,
"column": 68
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\nhm₁ : ∀ (i : ℕ... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\nhm₁ : ∀ (i : ℕ), (q.coeff ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 35
} | {
"line": 155,
"column": 36
} | [
{
"pp": "case inr.inl\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\n... | [
"case inr.inl\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\nhm₁ : ∀ (i :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 177,
"column": 8
} | {
"line": 177,
"column": 48
} | {
"line": 177,
"column": 49
} | [
{
"pp": "R : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nx : A\np : A[X]\nmonic : p.Monic\ndeg : p.natDegree ≠ 0\nhx : IsIntegral R (eval x p)\nhp : ∀ (i : ℕ), IsIntegral R (p.coeff i)\nq : (↥(integralClosure R A))[X]\nhqp : Polynomial.map (algebraMap (↥(integralClosur... | [
"R : Type u_4\nA : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nx : A\np : A[X]\nmonic : p.Monic\ndeg : p.natDegree ≠ 0\nhx : IsIntegral R (eval x p)\nhp : ∀ (i : ℕ), IsIntegral R (p.coeff i)\nq : (↥(integralClosure R A))[X]\nhqp : Polynomial.map (algebraMap (↥(integralClosure R A)) A) q... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Polynomial | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 26
} | {
"line": 45,
"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\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : Algebra.WeaklyQuasiFiniteAt R P\n⊢ Ideal.map C (Ideal.under R P) < Ideal.comap (↑f) P",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants"... | [
"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\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : Algebra.WeaklyQuasiFiniteAt R P\nalgInst✝ : Algebra R[X] S := f.toAlgebra\n⊢ Ideal.map C (Ideal.under R P) < Ideal.comap (↑f) P"
] | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 150,
"column": 8
} | {
"line": 150,
"column": 26
} | {
"line": 150,
"column": 27
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\nh₂ : Ideal.under R P = Ideal.under R Q\ninst✝ : WeaklyQuasiFiniteAt R Q\n⊢ RingHom.ker (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.und... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\nh₂ : Ideal.under R P = Ideal.under R Q\ninst✝ : WeaklyQuasiFiniteAt R Q\n⊢ Ideal.map (algebraMap R S) (Ideal.under R P) ≤ P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 40
} | {
"line": 163,
"column": 41
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\nh₂ : Ideal.under R P = Ideal.under R Q\ninst✝ : WeaklyQuasiFiniteAt R Q\nthis✝¹ : (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ide... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\nh₂ : Ideal.under R P = Ideal.under R Q\ninst✝ : WeaklyQuasiFiniteAt R Q\nthis✝¹ : (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.under R Q... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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