module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Valuation.Archimedean | {
"line": 71,
"column": 63
} | {
"line": 81,
"column": 9
} | {
"line": 83,
"column": 0
} | [
{
"pp": "F : Type u_1\nΓ₀ : Type u_2\nO : Type u_3\ninst✝⁴ : Field F\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : CommRing O\ninst✝¹ : Algebra O F\nv : Valuation F Γ₀\ninst✝ : MulArchimedean ↥(MonoidHom.mrange v)\nhv : v.Integers O\n⊢ IsPrincipalIdealRing O ↔ ¬DenselyOrdered ↑(Set.range ⇑v)",
"ppTe... | [] | by
refine ⟨fun _ ↦ not_denselyOrdered_of_isPrincipalIdealRing hv, fun H ↦ ?_⟩
rcases subsingleton_or_nontrivial (MonoidHom.mrange v)ˣ with hs | _
· have := bijective_algebraMap_of_subsingleton_units_mrange hv
exact .of_surjective _ (RingEquiv.ofBijective _ this).symm.surjective
have : IsDomain O := hv.hom_i... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.Embedding | {
"line": 132,
"column": 6
} | {
"line": 140,
"column": 16
} | {
"line": 141,
"column": 4
} | [
{
"pp": "case mp.refine_2\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\na : ℚ\nh : a.num • 1 < a.den • (x + y)\nk : ℕ\nhk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)\nhk... | [] | have hk' : 1 + (k • a.num • 1 - k • a.den • y) ≤ k • a.den • x - 1 := by
rw [smul_add, smul_sub, smul_add, le_sub_iff_add_le, ← sub_le_iff_le_add] at hk
rw [le_sub_iff_add_le]
convert! hk using 1
abel
have : k • a.num • 1 - k • a.den • y < m • 1 :=
lt_of_lt_of_le (lt_add_of... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Real.Embedding | {
"line": 132,
"column": 6
} | {
"line": 140,
"column": 16
} | {
"line": 141,
"column": 4
} | [
{
"pp": "case mp.refine_2\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\na : ℚ\nh : a.num • 1 < a.den • (x + y)\nk : ℕ\nhk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)\nhk... | [] | have hk' : 1 + (k • a.num • 1 - k • a.den • y) ≤ k • a.den • x - 1 := by
rw [smul_add, smul_sub, smul_add, le_sub_iff_add_le, ← sub_le_iff_le_add] at hk
rw [le_sub_iff_add_le]
convert! hk using 1
abel
have : k • a.num • 1 - k • a.den • y < m • 1 :=
lt_of_lt_of_le (lt_add_of... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 522,
"column": 4
} | {
"line": 525,
"column": 89
} | {
"line": 526,
"column": 2
} | [
{
"pp": "case inr\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : s.Countable\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ x ∈ Set.Ioo (m... | [] | specialize this b₂ a₂
rw [uIcc_comm b₂ a₂, min_comm b₂ a₂, max_comm b₂ a₂] at this
simp only [intervalIntegral.integral_symm b₂ a₂, intervalIntegral.integral_neg]
refine (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_ge h₂))).trans ?_; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 522,
"column": 4
} | {
"line": 525,
"column": 89
} | {
"line": 526,
"column": 2
} | [
{
"pp": "case inr\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₂ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : s.Countable\na₁ b₁ : ℝ\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ x ∈ Set.Ioo (m... | [] | specialize this b₂ a₂
rw [uIcc_comm b₂ a₂, min_comm b₂ a₂, max_comm b₂ a₂] at this
simp only [intervalIntegral.integral_symm b₂ a₂, intervalIntegral.integral_neg]
refine (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_ge h₂))).trans ?_; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 48
} | {
"line": 351,
"column": 2
} | [
{
"pp": "case inl\nM : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\nC : ℝ\nhC : 0 ≤ C\nh : ∀ (i : ι), ‖f i‖ ≤ C\nh✝ : IsEmpty ι\n⊢ ‖∏' (i : ι), f i‖ ≤ C",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
... | [] | simpa only [tprod_empty, norm_one'] using hC | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 48
} | {
"line": 351,
"column": 2
} | [
{
"pp": "case inl\nM : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\nC : ℝ\nhC : 0 ≤ C\nh : ∀ (i : ι), ‖f i‖ ≤ C\nh✝ : IsEmpty ι\n⊢ ‖∏' (i : ι), f i‖ ≤ C",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
... | [] | simpa only [tprod_empty, norm_one'] using hC | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 48
} | {
"line": 351,
"column": 2
} | [
{
"pp": "case inl\nM : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\nC : ℝ\nhC : 0 ≤ C\nh : ∀ (i : ι), ‖f i‖ ≤ C\nh✝ : IsEmpty ι\n⊢ ‖∏' (i : ι), f i‖ ≤ C",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
... | [] | simpa only [tprod_empty, norm_one'] using hC | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 93,
"column": 10
} | {
"line": 93,
"column": 93
} | {
"line": 94,
"column": 10
} | [
{
"pp": "case h\nK : Type u_1\nhK : NormedField K\ninst✝ : IsUltrametricDist K\nU : Set K\nε : ℝ\nhε : ε > 0\nh : Metric.ball 0 ε ⊆ U\nH : ∀ {γ : ℝ≥0}, γ ≠ 0 → ∃ x, x ≠ 0 ∧ (RankLeOne.hom' valuation) (valuation.restrict x) < γ\nx : K\nhx : x ≠ 0\nhxy : (RankLeOne.hom' valuation) (valuation.restrict x) < ⟨ε, ⋯⟩\... | [
"case h\nK : Type u_1\nhK : NormedField K\ninst✝ : IsUltrametricDist K\nU : Set K\nε : ℝ\nhε : ε > 0\nh : Metric.ball 0 ε ⊆ U\nH : ∀ {γ : ℝ≥0}, γ ≠ 0 → ∃ x, x ≠ 0 ∧ (RankLeOne.hom' valuation) (valuation.restrict x) < γ\nx : K\nhx : x ≠ 0\nhxy : (RankLeOne.hom' valuation) (valuation.restrict x) < ⟨ε, ⋯⟩\ny : K\nhy :... | simp only [Units.val_mk0, mem_setOf_eq, restrict_lt_iff, ← NNReal.coe_lt_coe] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 65
} | {
"line": 168,
"column": 6
} | [
{
"pp": "L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\n⊢ uniformity L = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | v.norm (p.1 - p.2) < ε}",
"ppTerm": "?m.163",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
... | [
"L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nthis : Nonempty { ε // ε > 0 }\n⊢ uniformity L = ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | v.norm (p.1 - p.2) < ε}"
] | haveI : Nonempty { ε : ℝ // ε > 0 } := nonempty_Ioi_subtype | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 178,
"column": 71
} | {
"line": 181,
"column": 89
} | {
"line": 183,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : Nontrivial R\nι : Type u_4\nw : σ → ℕ\nf : ι → MvPowerSeries σ R\ns : Finset ι\n⊢ weightedOrder w (∏ i ∈ s, f i) = ∑ i ∈ s, weightedOrder w (f i)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [... | [] | by
induction s using Finset.cons_induction with
| empty => simp
| cons a s ha ih => rw [Finset.sum_cons ha, Finset.prod_cons ha, weightedOrder_mul, ih] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 469,
"column": 38
} | {
"line": 469,
"column": 89
} | {
"line": 470,
"column": 6
} | [
{
"pp": "case pos\nF : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhP : W.Nonsingular P\nhPz : P z = 0\n⊢ toAffine W (-P y • ![0, 1, 0]) = -toAffine W P",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"WeierstrassCurve.Project... | [
"case pos\nF : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhP : W.Nonsingular P\nhPz : P z = 0\n⊢ toAffine W ![0, 1, 0] = -toAffine W P"
] | toAffine_smul _ (isUnit_Y_of_Z_eq_zero hP hPz).neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes | {
"line": 153,
"column": 31
} | {
"line": 159,
"column": 40
} | {
"line": 161,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\nE : PreOneHypercover X\nF : Cᵒᵖ ⥤ Type u_2\nh : E.IsStronglySheafFor F\n⊢ IsLimit (E.multifork F)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.PreZeroHypercover.f",
"Opposite",
... | [] | by
refine Nonempty.some ?_
rw [Multifork.isLimit_types_iff]
refine ⟨fun s t hst ↦ ?_, fun s ↦ ?_⟩
· exact h.isSheafFor_presieve₀.isSeparatedFor.ext fun _ _ ⟨i⟩ ↦ congr($(hst).val i)
· exact ⟨h.amalgamate s.val fun i j k ↦ s.property ⟨(i, j), k⟩, by
ext; exact map_amalgamate _ _ _ _⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 77
} | {
"line": 409,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : Nontrivial R\n⊢ ¬(3 * 0 ^ 2 = 0 ∧ 1 ^ 2 + W'.a₁ * 0 * 1 - W'.a₂ * 0 ^ 2 = 0)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr... | [] | exact fun h => one_ne_zero <| by linear_combination (norm := ring1) h.right | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.Preserves | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 16
} | {
"line": 144,
"column": 0
} | [
{
"pp": "case neg\nC : Type u\ninst✝⁴ : Category.{v, u} C\nI : C\nF : Cᵒᵖ ⥤ Type w\nhF : IsSheafFor F (ofArrows Empty.elim fun a ↦ Empty.instIsEmpty.elim a)\nhI : IsInitial I\nα : Type u_1\ninst✝³ : Small.{w, u_1} α\nX : α → C\nc : Cofan X\ninst✝² : (ofArrows X c.inj).HasPairwisePullbacks\ninst✝¹ : HasInitial C... | [] | exact i.elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Sites.BigZariski | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 12
} | {
"line": 66,
"column": 4
} | [
{
"pp": "X Y : Scheme\n𝓤 : Y.OpenCover\nhS : Presieve.ofArrows 𝓤.X 𝓤.f ∈ (precoverage IsOpenImmersion).coverings Y\nx : Presieve.FamilyOfElements (yoneda.obj X) (Presieve.ofArrows 𝓤.X 𝓤.f)\nhx : x.Compatible\ni j : 𝓤.I₀\n⊢ pullback.fst (𝓤.f i) (𝓤.f j) ≫ x (𝓤.f i) ⋯ = pullback.snd (𝓤.f i) (𝓤.f j) ≫ x ... | [
"X Y : Scheme\n𝓤 : Y.OpenCover\nhS : Presieve.ofArrows 𝓤.X 𝓤.f ∈ (precoverage IsOpenImmersion).coverings Y\nx : Presieve.FamilyOfElements (yoneda.obj X) (Presieve.ofArrows 𝓤.X 𝓤.f)\nhx : x.Compatible\ni j : 𝓤.I₀\n⊢ (limit.cone (cospan (𝓤.f i) (𝓤.f j))).π.1 WalkingCospan.left ≫ 𝓤.f i =\n (limit.cone (cos... | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Sites.BigZariski | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 16
} | {
"line": 118,
"column": 2
} | [
{
"pp": "case e'_5\nF : Schemeᵒᵖ ⥤ Type v\nι : Type u_1\ninst✝¹ : Small.{u, u_1} ι\ninst✝ : Small.{v, u_1} ι\nhF : Presieve.IsSheaf Scheme.zariskiTopology F\nX : ι → Schemeᵒᵖ\nthis : ∀ (i : ι), Mono (Cofan.inj (Sigma.cocone (Discrete.functor (unop ∘ X))) i)\nY✝ : Scheme\nf : Y✝ ⟶ ⊥_ Scheme\nY : Scheme\ni : Empt... | [] | exact i.elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Etale.Basic | {
"line": 307,
"column": 2
} | {
"line": 307,
"column": 29
} | {
"line": 308,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nT : Type u_3\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.FormallyEtale\nhg : g.FormallyEtale\n⊢ (g.comp f).FormallyEtale",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nT : Type u_3\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.FormallyEtale\nhg : g.FormallyEtale\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T := (g.comp f).toAlgebra\nscalarTowerIn... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.Smooth.Basic | {
"line": 216,
"column": 8
} | {
"line": 218,
"column": 78
} | {
"line": 219,
"column": 8
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_1\nP : Type u_2\nC : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommRing C\ninst✝³ : Algebra R C\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\nP₁ : Extension R A\nP₂ : Extension R A\ninst... | [
"R : Type u\nA : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_1\nP : Type u_2\nC : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommRing C\ninst✝³ : Algebra R C\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\nP₁ : Extension R A\nP₂ : Extension R A\ninst✝ : Formally... | have : ∀ r ∈ P₁.ker, lift r ∈ P₂.infinitesimal.ker :=
fun r hr ↦ (FormallySmooth.liftOfSurjective_apply _
(IsScalarTower.toAlgHom R P₂.infinitesimal.Ring A) _ _ r).trans hr | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 175,
"column": 4
} | {
"line": 179,
"column": 98
} | {
"line": 181,
"column": 0
} | [
{
"pp": "case inl\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\nj i : ι\n⊢ (Q.cotangentSpa... | [] | simp only [Basis.repr_symm_apply, Finsupp.linearCombination_single, Basis.prod_apply,
LinearMap.coe_inl, LinearMap.coe_inr, Sum.elim_inl, Function.comp_apply, one_smul,
Basis.repr_self, Finsupp.single_apply, pderiv_X, Pi.single_apply,
Sum.elim_inr, Function.comp_apply, Basis.baseChange_apply, one_smul... | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 175,
"column": 4
} | {
"line": 179,
"column": 98
} | {
"line": 181,
"column": 0
} | [
{
"pp": "case inr\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\nj : ι\ni : σ\n⊢ (Q.cotange... | [] | simp only [Basis.repr_symm_apply, Finsupp.linearCombination_single, Basis.prod_apply,
LinearMap.coe_inl, LinearMap.coe_inr, Sum.elim_inl, Function.comp_apply, one_smul,
Basis.repr_self, Finsupp.single_apply, pderiv_X, Pi.single_apply,
Sum.elim_inr, Function.comp_apply, Basis.baseChange_apply, one_smul... | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 176,
"column": 49
} | {
"line": 176,
"column": 95
} | {
"line": 176,
"column": 95
} | [
{
"pp": "R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\nl : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))\nhl : l ∘ₗ kerToTensor R P S = LinearMap.id\nσ : S → P\nhσ : ∀ (x : S), (algebraMap P S) (σ x) = x... | [] | by simp [hσ, ← IsScalarTower.algebraMap_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RingHom.Smooth | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 29
} | {
"line": 47,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nT : Type u_3\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.FormallySmooth\nhg : g.FormallySmooth\n⊢ (g.comp f).FormallySmooth",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nT : Type u_3\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.FormallySmooth\nhg : g.FormallySmooth\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T := (g.comp f).toAlgebra\nscalarTower... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.RingHom.Smooth | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 29
} | {
"line": 105,
"column": 2
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.Smooth\nhg : g.Smooth\n⊢ (g.comp f).Smooth",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"CommSemiring.toSem... | [
"R : Type u_3\nS : Type u_4\nT : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.Smooth\nhg : g.Smooth\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T := (g.comp f).toAlgebra\nscalarTowerInst✝ : IsScalar... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.RingHom.Unramified | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 29
} | {
"line": 46,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nT : Type u_3\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nh : (g.comp f).FormallyUnramified\n⊢ g.FormallyUnramified",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"CommSemiri... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nT : Type u_3\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nh : (g.comp f).FormallyUnramified\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T := (g.comp f).toAlgebra\nscalarTowerInst✝ : IsS... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.RingHom.Unramified | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 29
} | {
"line": 52,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nT : Type u_3\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.FormallyUnramified\nhg : g.FormallyUnramified\n⊢ (g.comp f).FormallyUnramified",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.al... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nT : Type u_3\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.FormallyUnramified\nhg : g.FormallyUnramified\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T := (g.comp f).toAlgebra\nsca... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.RingHom.Unramified | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 29
} | {
"line": 126,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nT : Type u_3\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.FormallyUnramified\nh : (g.comp f).FormallyEtale\n⊢ g.FormallyEtale",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\nT : Type u_3\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.FormallyUnramified\nh : (g.comp f).FormallyEtale\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T := (g.comp f).toAlgebra\n... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 292,
"column": 19
} | {
"line": 292,
"column": 50
} | {
"line": 292,
"column": 50
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z = Q y * P z\nhy' : P y * Q z = W'.negY Q * P z\n⊢ ((eval P) W'.polynomialX ^ 2 - W'.a₁ * (eval P) W'.polynomialX * ... | [
"R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z = Q y * P z\nhy' : P y * Q z = W'.negY Q * P z\n⊢ ((eval P) W'.polynomialX ^ 2 - W'.a₁ * (eval P) W'.polynomialX * P z * (W'.ne... | Y_eq_negY_of_Y_eq hQz hx hy hy' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 373,
"column": 22
} | {
"line": 373,
"column": 53
} | {
"line": 373,
"column": 53
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z = Q y * P z\nhy' : P y * Q z = W'.negY Q * P z\n⊢ -(eval P) W'.polynomialX *\n ((eval P) W'.polynomialX ^ 2 - W'.a₁ * (eval... | [
"R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z = Q y * P z\nhy' : P y * Q z = W'.negY Q * P z\n⊢ -(eval P) W'.polynomialX *\n ((eval P) W'.polynomialX ^ 2 - W'.a₁ * (eval P) W'.polyn... | Y_eq_negY_of_Y_eq hQz hx hy hy' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Unramified.Finite | {
"line": 270,
"column": 18
} | {
"line": 270,
"column": 48
} | {
"line": 271,
"column": 4
} | [
{
"pp": "case tmul\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : FormallyUnramified R S\ninst✝ : EssFiniteType R S\nx : M\nr s : S\n⊢ (_root_.Ten... | [] | simp [mul_smul, smul_comm r s] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Unramified.Finite | {
"line": 270,
"column": 18
} | {
"line": 270,
"column": 48
} | {
"line": 271,
"column": 4
} | [
{
"pp": "case tmul\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : FormallyUnramified R S\ninst✝ : EssFiniteType R S\nx : M\nr s : S\n⊢ (_root_.Ten... | [] | simp [mul_smul, smul_comm r s] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Unramified.Finite | {
"line": 270,
"column": 18
} | {
"line": 270,
"column": 48
} | {
"line": 271,
"column": 4
} | [
{
"pp": "case tmul\nR : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : FormallyUnramified R S\ninst✝ : EssFiniteType R S\nx : M\nr s : S\n⊢ (_root_.Ten... | [] | simp [mul_smul, smul_comm r s] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Etale.Field | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 37
} | {
"line": 154,
"column": 4
} | [
{
"pp": "case refine_5\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_5\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... | change g _ (algebraMap K _ r) = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 99,
"column": 41
} | {
"line": 105,
"column": 79
} | {
"line": 107,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R A\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra A B\ninst✝³ : Algebra S B\ninst✝² : IsScalarTower R A B\ninst✝¹ : IsScalarTowe... | [] | by
apply IsScalarTower.of_algebraMap_smul
intro r x
change (Algebra.pushoutDesc B (Algebra.lsmul R (A := S) S (S ⊗[R] Ω[A⁄R]))
(Algebra.lsmul R (A := A) _ _) (LinearMap.ext <| smul_comm · ·)
(algebraMap A B r)) • x = r • x
simp only [Algebra.pushoutDesc_right, Module.End.smul_def, Algebra.lsmul_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 43
} | {
"line": 121,
"column": 2
} | [
{
"pp": "case zero\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\ninst✝ : IsS... | [] | simp only [zero_smul, map_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 43
} | {
"line": 121,
"column": 2
} | [
{
"pp": "case zero\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\ninst✝ : IsS... | [] | simp only [zero_smul, map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 120,
"column": 12
} | {
"line": 120,
"column": 43
} | {
"line": 121,
"column": 2
} | [
{
"pp": "case zero\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\ninst✝ : IsS... | [] | simp only [zero_smul, map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdicCompletion.Algebra | {
"line": 334,
"column": 19
} | {
"line": 334,
"column": 34
} | {
"line": 334,
"column": 35
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nI : Ideal R\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : AdicCompletion I R\nx : AdicCompletion I M\nn : ℕ\n⊢ ↑(r • s) n • ↑x n = ↑(r • s • x) n",
"ppTerm": "?m.46",
"assigned": true,
"usedConsta... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nI : Ideal R\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : AdicCompletion I R\nx : AdicCompletion I M\nn : ℕ\n⊢ (r • ↑s n) • ↑x n = ↑(r • s • x) n"
] | val_smul_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.Algebra | {
"line": 334,
"column": 35
} | {
"line": 334,
"column": 50
} | {
"line": 334,
"column": 51
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nI : Ideal R\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : AdicCompletion I R\nx : AdicCompletion I M\nn : ℕ\n⊢ (r • ↑s n) • ↑x n = ↑(r • s • x) n",
"ppTerm": "?m.60",
"assigned": true,
"usedConsta... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nI : Ideal R\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr : R\ns : AdicCompletion I R\nx : AdicCompletion I M\nn : ℕ\n⊢ (r • ↑s n) • ↑x n = r • ↑(s • x) n"
] | val_smul_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 44
} | {
"line": 102,
"column": 2
} | [
{
"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)... | rw [← Submodule.comap_map_eq_of_injective hf (I ^ n • ⊤ : Submodule R M),
Submodule.map_smul'', Submodule.map_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 157,
"column": 2
} | {
"line": 158,
"column": 44
} | {
"line": 159,
"column": 2
} | [
{
"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)... | obtain ⟨n, f₀, hf₀⟩ := Algebra.FiniteType.iff_quotient_mvPolynomial''.mp
(inferInstance : Algebra.FiniteType R P) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 85
} | {
"line": 175,
"column": 4
} | [
{
"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)... | refine le_antisymm ?_ (Ideal.map_le_iff_le_comap.mpr fun x hx ↦ by simp_all [fP]) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 70
} | {
"line": 86,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Function.Exact... | [
"R : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Function.Exact ⇑f ⇑g\nι : ... | simp only [codisjoint_iff, Set.sup_eq_union, Set.top_eq_univ] at hab | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.StandardSmoothCotangent | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 12
} | {
"line": 94,
"column": 2
} | [
{
"pp": "R : 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.... | [] | apply hx | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Smooth.StandardSmoothOfFree | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 22
} | {
"line": 136,
"column": 2
} | [
{
"pp": "R✝ : Type u_1\nS✝ : Type u_2\ninst✝⁸ : CommRing R✝\ninst✝⁷ : CommRing S✝\ninst✝⁶ : Algebra R✝ S✝\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FinitePresentation R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : IsSmoothAt R p\nh : Smooth R S\nκ : Typ... | [] | simp [l₂, e, hb] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.StandardSmoothOfFree | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 22
} | {
"line": 136,
"column": 2
} | [
{
"pp": "R✝ : Type u_1\nS✝ : Type u_2\ninst✝⁸ : CommRing R✝\ninst✝⁷ : CommRing S✝\ninst✝⁶ : Algebra R✝ S✝\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FinitePresentation R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : IsSmoothAt R p\nh : Smooth R S\nκ : Typ... | [] | simp [l₂, e, hb] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.StandardSmoothOfFree | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 22
} | {
"line": 136,
"column": 2
} | [
{
"pp": "R✝ : Type u_1\nS✝ : Type u_2\ninst✝⁸ : CommRing R✝\ninst✝⁷ : CommRing S✝\ninst✝⁶ : Algebra R✝ S✝\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FinitePresentation R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : IsSmoothAt R p\nh : Smooth R S\nκ : Typ... | [] | simp [l₂, e, hb] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 35
} | {
"line": 86,
"column": 0
} | [
{
"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\n⊢ (compLocalizationAwayAlg... | [] | simp [compLocalizationAwayAlgHom] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 35
} | {
"line": 86,
"column": 0
} | [
{
"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\n⊢ (compLocalizationAwayAlg... | [] | simp [compLocalizationAwayAlgHom] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 35
} | {
"line": 86,
"column": 0
} | [
{
"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\n⊢ (compLocalizationAwayAlg... | [] | simp [compLocalizationAwayAlgHom] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 141,
"column": 6
} | {
"line": 141,
"column": 79
} | {
"line": 141,
"column": 79
} | [
{
"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\nQ : Generators S T Unit :=... | [
"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\nQ : Generators S T Unit := localizatio... | IsLocalization.map_eq_zero_iff (Submonoid.powers f) (Localization.Away f) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 131,
"column": 2
} | {
"line": 133,
"column": 52
} | {
"line": 135,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\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\nx : D.presLeft.Ring\nhx : x ∈ D.presLeft.ker\n⊢ x ∈ P.ker",
"ppTerm": "?m.53",
"assigned": true... | [] | simpa only [toExtension_commRing, toExtension_Ring, RingHom.mem_ker,
toExtension_algebra₂, algebraMap_apply, Ideal.Quotient.algebraMap_eq,
map_zero] using! (algebraMap D.T S).congr_arg hx | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 31
} | {
"line": 114,
"column": 4
} | [
{
"pp": "X Y Z : Scheme\nf✝ : X ⟶ Y\ng✝ : Y ⟶ Z\ninst✝ : FormallyUnramified (f✝ ≫ g✝)\nR S T : Type u_1\nx✝² : CommRing R\nx✝¹ : CommRing S\nx✝ : CommRing T\nf : R →+* S\ng : S →+* T\nH : (g.comp f).FormallyUnramified\n⊢ g.FormallyUnramified",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
... | [
"X Y Z : Scheme\nf✝ : X ⟶ Y\ng✝ : Y ⟶ Z\ninst✝ : FormallyUnramified (f✝ ≫ g✝)\nR S T : Type u_1\nx✝² : CommRing R\nx✝¹ : CommRing S\nx✝ : CommRing T\nf : R →+* S\ng : S →+* T\nH : (g.comp f).FormallyUnramified\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T :=... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.Unramified.LocalRing | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 23
} | {
"line": 226,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal R\ninst✝² : p.IsPrime\nq : Ideal S\ninst✝¹ : q.IsPrime\nhRS : (RingHom.ker (algebraMap R S)).FG\ninst✝ : q.LiesOver p\nH : Function.Injective ⇑(localRingHom p q (algebraMap R S) ⋯)\n⊢ ∃ r ∉ p, ∀ (r' : ... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal R\ninst✝² : p.IsPrime\nq : Ideal S\ninst✝¹ : q.IsPrime\ninst✝ : q.LiesOver p\nH : Function.Injective ⇑(localRingHom p q (algebraMap R S) ⋯)\ns : Finset R\nhs : Ideal.span ↑s = RingHom.ker (algebraMap R S)\n⊢ ∃ r ∉... | obtain ⟨s, hs⟩ := hRS | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.RingHom.OpenImmersion | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 29
} | {
"line": 124,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.IsStandardOpenImmersion\nhg : g.IsStandardOpenImmersion\n⊢ (g.comp f).IsStandardOpenImmersion",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\ng : S →+* T\nhf : f.IsStandardOpenImmersion\nhg : g.IsStandardOpenImmersion\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T := (g.comp f).toAl... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.RingHom.QuasiFinite | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 29
} | {
"line": 124,
"column": 2
} | [
{
"pp": "R : Type u_6\nS : Type u_7\nT : Type u_8\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\nhf : f.IsIntegral\ng : S →+* T\nhg✝ : g.IsStandardOpenImmersion\nhg : (g.comp f).FiniteType\n⊢ (g.comp f).QuasiFinite",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": ... | [
"R : Type u_6\nS : Type u_7\nT : Type u_8\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\nhf : f.IsIntegral\ng : S →+* T\nhg✝ : g.IsStandardOpenImmersion\nhg : (g.comp f).FiniteType\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Algebra R T :=... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.FieldTheory.RatFunc.Defs | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 24
} | {
"line": 166,
"column": 25
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np q : K[X]\nhq : q ≠ 0\n⊢ RatFunc.mk p q = { toFractionRing := Localization.mk p ⟨q, ⋯⟩ }",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"RatFunc.mk_def_of_ne",
"Iff.mpr",
"Eq.mpr",
"Localization.mk",
... | [
"K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np q : K[X]\nhq : q ≠ 0\n⊢ { toFractionRing := IsLocalization.mk' (FractionRing K[X]) p ⟨q, ⋯⟩ } = { toFractionRing := Localization.mk p ⟨q, ⋯⟩ }"
] | mk_def_of_ne _ hq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.RatFunc.Defs | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 24
} | {
"line": 174,
"column": 25
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np q p' q' : K[X]\nhq : q ≠ 0\nhq' : q' ≠ 0\n⊢ RatFunc.mk p q = RatFunc.mk p' q' ↔ p * q' = p' * q",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"RatFunc.mk_def_of_ne",
"Iff.mpr",
"Eq.mpr",
"IsDomain.to_... | [
"K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np q p' q' : K[X]\nhq : q ≠ 0\nhq' : q' ≠ 0\n⊢ { toFractionRing := IsLocalization.mk' (FractionRing K[X]) p ⟨q, ⋯⟩ } = RatFunc.mk p' q' ↔ p * q' = p' * q"
] | mk_def_of_ne _ hq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Conductor | {
"line": 172,
"column": 6
} | {
"line": 172,
"column": 59
} | {
"line": 172,
"column": 59
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤\nh_alg : Function.Injective ⇑(algebraMap (↥R[x]) S)\nf : ↥R[x] ⧸ Ideal.map (algebraMap R ↥R[x]) I →+* S ⧸ Ideal.map (algebraM... | [
"case refine_1\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤\nh_alg : Function.Injective ⇑(algebraMap (↥R[x]) S)\nf : ↥R[x] ⧸ Ideal.map (algebraMap R ↥R[x]) I →+* S ⧸ Ideal.map (algebraMap R S) I :=... | comap_map_eq_map_adjoin_of_coprime_conductor hx h_alg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 18
} | {
"line": 360,
"column": 2
} | [
{
"pp": "R : 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\n⊢ Function.Injective ⇑(map φ hφ)",
"ppTerm": "?m.34",
"assigned": true,
... | [
"R : 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\nx y : FractionRing R[X]\nh : (map φ hφ) { toFractionRing := x } = (map φ hφ) { toFractionRin... | rintro ⟨x⟩ ⟨y⟩ h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 399,
"column": 6
} | {
"line": 399,
"column": 82
} | {
"line": 400,
"column": 6
} | [
{
"pp": "case inl\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : FunLike F R[X] S[X]\nφ : R[X] →*₀ G₀\nhφ : R[X]⁰ ≤ Submonoid.comap φ G₀⁰\nf : R⟮X⟯\np q p'... | [
"case inr\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : FunLike F R[X] S[X]\nφ : R[X] →*₀ G₀\nhφ : R[X]⁰ ≤ Submonoid.comap φ G₀⁰\nf : R⟮X⟯\np q p' q' : R[X]\n... | · rw [Subsingleton.elim p q, Subsingleton.elim p' q, Subsingleton.elim q' q] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 126,
"column": 2
} | {
"line": 127,
"column": 77
} | {
"line": 129,
"column": 0
} | [
{
"pp": "X : Scheme\nU : X.AffineZariskiSite\nS : Sieve U\nx : ↥X\nhxU : x ∈ (toOpensFunctor X).obj U\n⊢ (∃ V f, S.arrows f ∧ x ∈ V.toOpens) → ∃ U_1 f, (Sieve.functorPushforward (toOpensFunctor X) S).arrows f ∧ x ∈ U_1",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"AlgebraicGeometr... | [] | · rintro ⟨W, g, hg, hxW⟩
exact ⟨W.toOpens, homOfLE (toOpens_mono g.le), ⟨W, g, 𝟙 _, hg, rfl⟩, hxW⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 1059,
"column": 2
} | {
"line": 1059,
"column": 81
} | {
"line": 1060,
"column": 2
} | [
{
"pp": "case neg\nK : Type u\ninst✝ : Field K\nx y : K⟮X⟯\nhx : ¬x = 0\nhy : ¬y = 0\n⊢ ∃ q, q ≠ 0 ∧ x * y = (algebraMap K[X] K⟮X⟯) (x.num * y.num) / (algebraMap K[X] K⟮X⟯) q",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"IsDomain.to_noZeroDivisors",
"instHDiv",
"RatFun... | [
"case neg\nK : Type u\ninst✝ : Field K\nx y : K⟮X⟯\nhx : ¬x = 0\nhy : ¬y = 0\n⊢ x * y = (algebraMap K[X] K⟮X⟯) (x.num * y.num) / (algebraMap K[X] K⟮X⟯) (x.denom * y.denom)"
] | refine ⟨x.denom * y.denom, mul_ne_zero (denom_ne_zero x) (denom_ne_zero y), ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 20
} | {
"line": 143,
"column": 2
} | [
{
"pp": "case inr\nR : 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\nH : ∃ x, R[x] = ⊤\nQ : Ideal S\ninst✝¹ : Q.IsPrime\ninst✝ : IsUnramifiedAt R Q\nh✝ : Nontrivial S\nthis : Nontrivial R\n⊢ ∃ f ∉ Q, HasStandardEtaleSurjectionOn R f",
"pp... | [
"case inr\nR : 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\nH : ∃ x, R[x] = ⊤\nQ : Ideal S\ninst✝¹ : Q.IsPrime\ninst✝ : IsUnramifiedAt R Q\nh✝ : Nontrivial S\nthis : Nontrivial R\nP : Ideal R := Ideal.under R Q\n⊢ ∃ f ∉ Q, HasStandardEtaleSurjec... | let P := Q.under R | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 146,
"column": 79
} | {
"line": 175,
"column": 65
} | {
"line": 177,
"column": 0
} | [
{
"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\n⊢ IsIntegral R t",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants... | [] | by
let St := Localization.Away t
let t' : St := IsLocalization.Away.invSelf t
have ht't : t' * algebraMap S St t = 1 := by rw [mul_comm, IsLocalization.Away.mul_invSelf]
let R₁ := Algebra.adjoin R {t'}
let R₂ := Algebra.adjoin R₁ {algebraMap S St (φ X)}
letI : Algebra R₁ R₂ := R₂.algebra
letI : Algebra R₂... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 204,
"column": 6
} | {
"line": 204,
"column": 44
} | {
"line": 204,
"column": 45
} | [
{
"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) (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✝¹ : Nontrivial R\nP : Ideal R := Ideal.under R Q\nthis✝ : Algebra (Localization.AtPrime P) (Localization.AtPri... | ← aeval_map_algebraMap P.ResidueField, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 230,
"column": 6
} | {
"line": 230,
"column": 52
} | {
"line": 231,
"column": 4
} | [
{
"pp": "case pos\nR : 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.AtPrim... | [] | exact ⟨1, by simp, by simpa [p, hx0] using hx⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 230,
"column": 6
} | {
"line": 230,
"column": 52
} | {
"line": 231,
"column": 4
} | [
{
"pp": "case pos\nR : 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.AtPrim... | [] | exact ⟨1, by simp, by simpa [p, hx0] using hx⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 230,
"column": 6
} | {
"line": 230,
"column": 52
} | {
"line": 231,
"column": 4
} | [
{
"pp": "case pos\nR : 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.AtPrim... | [] | exact ⟨1, by simp, by simpa [p, hx0] using hx⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 45
} | {
"line": 289,
"column": 2
} | [
{
"pp": "case neg\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\np : R[X]\ni : ℕ\nhi : i = p.natDegree\nn : ℕ\nhn : (φ p * t) ^ n ∈ conductor R (φ X)\nk : ℕ\nhk : (p ^ n).leadingCoeff ^... | [
"case neg\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\np : R[X]\ni : ℕ\nhi : i = p.natDegree\nn : ℕ\nhn : (φ p * t) ^ n ∈ conductor R (φ X)\nk : ℕ\nhk : p.leadingCoeff ^ (n * k) • t ^ n ∈... | rw [leadingCoeff_pow' hpn, ← pow_mul] at hk | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 478,
"column": 32
} | {
"line": 478,
"column": 73
} | {
"line": 478,
"column": 73
} | [
{
"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\nx : S\nhx : R[x] = ⊤\nH : integralClosure R S = ⊥\n⊢ (aeval x).range = ⊤",
... | [
"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\nx : S\nhx : R[x] = ⊤\nH : integralClosure R S = ⊥\n⊢ R[x] = ⊤"
] | ← Algebra.adjoin_singleton_eq_range_aeval | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 494,
"column": 88
} | {
"line": 495,
"column": 89
} | {
"line": 496,
"column": 6
} | [
{
"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\nx : S\nhx : R[x] = ⊤\nH : integralClosure R S = ⊥\nH₀ : Function.Surjective... | [] | by
simpa [eraseLead_coeff, show n ≠ f.natDegree by rintro rfl; exact hfn (by simpa)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 71,
"column": 37
} | {
"line": 71,
"column": 53
} | {
"line": 73,
"column": 0
} | [
{
"pp": "case left\nR : Type u_1\ninst✝ : Semiring R\nf g : R[X]\nm n : ℕ\ni : Fin (m + n)\ni✝ : Fin m\n⊢ f.sylvester g m n i (Fin.castAdd n i✝) =\n (Matrix.reindex (finCongr ⋯) (finSumFinEquiv.symm.trans ((Equiv.sumComm (Fin n) (Fin m)).trans finSumFinEquiv)))\n (g.sylvester f n m) i (Fin.castAdd n i✝)... | [] | simp [sylvester] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 71,
"column": 37
} | {
"line": 71,
"column": 53
} | {
"line": 73,
"column": 0
} | [
{
"pp": "case right\nR : Type u_1\ninst✝ : Semiring R\nf g : R[X]\nm n : ℕ\ni : Fin (m + n)\ni✝ : Fin n\n⊢ f.sylvester g m n i (Fin.natAdd m i✝) =\n (Matrix.reindex (finCongr ⋯) (finSumFinEquiv.symm.trans ((Equiv.sumComm (Fin n) (Fin m)).trans finSumFinEquiv)))\n (g.sylvester f n m) i (Fin.natAdd m i✝)"... | [] | simp [sylvester] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 640,
"column": 2
} | {
"line": 650,
"column": 48
} | {
"line": 652,
"column": 0
} | [
{
"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\n⊢ ZariskisMainProperty R p",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgE... | [] | obtain ⟨n, f, hf⟩ := Algebra.FiniteType.iff_quotient_mvPolynomial''.mp ‹_›
have : Small.{u} S := small_of_surjective hf
have := ZariskisMainProperty.of_algHom_mvPolynomial (p.comap (Shrink.algEquiv R S).toRingHom)
((Shrink.algEquiv R S).symm.toAlgHom.comp f)
(.of_surjective _ <| (Shrink.algEquiv R S).symm.s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 640,
"column": 2
} | {
"line": 650,
"column": 48
} | {
"line": 652,
"column": 0
} | [
{
"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\n⊢ ZariskisMainProperty R p",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgE... | [] | obtain ⟨n, f, hf⟩ := Algebra.FiniteType.iff_quotient_mvPolynomial''.mp ‹_›
have : Small.{u} S := small_of_surjective hf
have := ZariskisMainProperty.of_algHom_mvPolynomial (p.comap (Shrink.algEquiv R S).toRingHom)
((Shrink.algEquiv R S).symm.toAlgHom.comp f)
(.of_surjective _ <| (Shrink.algEquiv R S).symm.s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 306,
"column": 6
} | {
"line": 306,
"column": 36
} | {
"line": 307,
"column": 6
} | [
{
"pp": "case succ.e_a.e_a\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm : ℕ\nhf : f.natDegree ≤ m\nn : ℕ\n⊢ (-1) ^ (m + 1 + n + m) = (-1) ^ (n + 1)",
"ppTerm": "?succ.e_a.e_a✝",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"HMul.hMul",
"CommSemiring.toSemiring",
... | [
"R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm : ℕ\nhf : f.natDegree ≤ m\nn : ℕ\n⊢ (-1) ^ (m + 1 + n + m) = (-1) ^ (2 * m + (n + 1))",
"R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm : ℕ\nhf : f.natDegree ≤ m\nn : ℕ\n⊢ (-1) ^ (2 * m + (n + 1)) = (-1) ^ (n + 1)"
] | trans (-1) ^ (2 * m + (n + 1)) | Batteries.Tactic._aux_Batteries_Tactic_Trans___elabRules_Batteries_Tactic_tacticTrans____1 | Batteries.Tactic.tacticTrans___ |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 208,
"column": 6
} | {
"line": 209,
"column": 75
} | {
"line": 210,
"column": 2
} | [
{
"pp": "case a.convert_2.C.mul_X\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nf : MvPolynomial (Fin m ⊕ Fin k) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R :=\n eval₂Hom (↑(universalFactorizationMap R n m k hn)) (Sum.elim (fun x ↦ X x ⊗ₜ[R] 1) fun x ↦ 1 ⊗ₜ[R... | [] | simp only [map_mul]
exact Ideal.mul_sub_mul_mem _ IH (Ideal.subset_span ⟨i, by simp [f]⟩) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 208,
"column": 6
} | {
"line": 209,
"column": 75
} | {
"line": 210,
"column": 2
} | [
{
"pp": "case a.convert_2.C.mul_X\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nf : MvPolynomial (Fin m ⊕ Fin k) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R :=\n eval₂Hom (↑(universalFactorizationMap R n m k hn)) (Sum.elim (fun x ↦ X x ⊗ₜ[R] 1) fun x ↦ 1 ⊗ₜ[R... | [] | simp only [map_mul]
exact Ideal.mul_sub_mul_mem _ IH (Ideal.subset_span ⟨i, by simp [f]⟩) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 247,
"column": 2
} | {
"line": 264,
"column": 14
} | {
"line": 266,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\ni : Fin m\nj : Fin n\n⊢ (pderiv (Sum.inl i)) ((tensorEquivSum R (Fin m) (Fin k) R) ((universalFactorizationMap R n m k hn) (X j))) =\n if ↑j < ↑i then 0 else if h : ↑j - ↑i < k then X (Sum.inr ⟨↑j - ↑i, h⟩) else if ↑j - ↑i = k then 1 else ... | [] | classical
trans ∑ x ∈ Finset.antidiagonal ↑j,
if h : x.2 < k then if x.1 < m ∧ x.1 = ↑i then X (Sum.inr ⟨x.2, h⟩) else 0
else if x.2 = k ∧ x.1 < m ∧ x.1 = ↑i then 1 else 0
· simp [universalFactorizationMap, mapEquivMonic, Polynomial.coeff_mul, coeff_freeMonic,
apply_dite, apply_ite, ← Algebra.TensorPr... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 247,
"column": 2
} | {
"line": 264,
"column": 14
} | {
"line": 266,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\ni : Fin m\nj : Fin n\n⊢ (pderiv (Sum.inl i)) ((tensorEquivSum R (Fin m) (Fin k) R) ((universalFactorizationMap R n m k hn) (X j))) =\n if ↑j < ↑i then 0 else if h : ↑j - ↑i < k then X (Sum.inr ⟨↑j - ↑i, h⟩) else if ↑j - ↑i = k then 1 else ... | [] | classical
trans ∑ x ∈ Finset.antidiagonal ↑j,
if h : x.2 < k then if x.1 < m ∧ x.1 = ↑i then X (Sum.inr ⟨x.2, h⟩) else 0
else if x.2 = k ∧ x.1 < m ∧ x.1 = ↑i then 1 else 0
· simp [universalFactorizationMap, mapEquivMonic, Polynomial.coeff_mul, coeff_freeMonic,
apply_dite, apply_ite, ← Algebra.TensorPr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 247,
"column": 2
} | {
"line": 264,
"column": 14
} | {
"line": 266,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\ni : Fin m\nj : Fin n\n⊢ (pderiv (Sum.inl i)) ((tensorEquivSum R (Fin m) (Fin k) R) ((universalFactorizationMap R n m k hn) (X j))) =\n if ↑j < ↑i then 0 else if h : ↑j - ↑i < k then X (Sum.inr ⟨↑j - ↑i, h⟩) else if ↑j - ↑i = k then 1 else ... | [] | classical
trans ∑ x ∈ Finset.antidiagonal ↑j,
if h : x.2 < k then if x.1 < m ∧ x.1 = ↑i then X (Sum.inr ⟨x.2, h⟩) else 0
else if x.2 = k ∧ x.1 < m ∧ x.1 = ↑i then 1 else 0
· simp [universalFactorizationMap, mapEquivMonic, Polynomial.coeff_mul, coeff_freeMonic,
apply_dite, apply_ite, ← Algebra.TensorPr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 259,
"column": 32
} | {
"line": 259,
"column": 70
} | {
"line": 259,
"column": 71
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FiniteType R S\np : Ideal R\ninst✝³ : p.IsPrime\nq : Ideal S\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : QuasiFiniteAt R q\ns : S\nhsq : s ∉ q\nhRs : IsIntegral R s\nhs : ∀ (q' : Ideal S), q'.IsPrim... | [
"R : Type u\nS : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FiniteType R S\np : Ideal R\ninst✝³ : p.IsPrime\nq : Ideal S\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : QuasiFiniteAt R q\ns : S\nhsq : s ∉ q\nhRs : IsIntegral R s\nhs : ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q →... | ← aeval_map_algebraMap P.ResidueField, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 266,
"column": 4
} | {
"line": 267,
"column": 22
} | {
"line": 268,
"column": 4
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FiniteType R S\np : Ideal R\ninst✝³ : p.IsPrime\nq : Ideal S\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : QuasiFiniteAt R q\ns : S\nhsq : s ∉ q\nhRs : IsIntegral R s\nhs : ∀ (q' : Ideal S), q'.IsPrim... | [
"R : Type u\nS : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FiniteType R S\np : Ideal R\ninst✝³ : p.IsPrime\nq : Ideal S\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : QuasiFiniteAt R q\ns : S\nhsq : s ∉ q\nhRs : IsIntegral R s\nhs : ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q →... | rw [← map_mul, eq_sub_iff_add_eq'.mpr hcd, map_sub, Submodule.sub_mem_iff_left _ H,
map_one] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 663,
"column": 2
} | {
"line": 665,
"column": 77
} | {
"line": 666,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nP : Ideal R\ninst✝ : P.IsPrime\nf : MonicDegreeEq P.ResidueField m\ng : MonicDegreeEq P.ResidueField k\nH : map (algebraMap R P.ResidueField) ↑p = ↑f * ↑g\nHpq : IsCoprime ↑f ↑g\nφ : 𝓡' →ₐ[R] P.ResidueField := (homEqu... | [
"R : Type u_1\ninst✝¹ : CommRing R\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nP : Ideal R\ninst✝ : P.IsPrime\nf : MonicDegreeEq P.ResidueField m\ng : MonicDegreeEq P.ResidueField k\nH : map (algebraMap R P.ResidueField) ↑p = ↑f * ↑g\nHpq : IsCoprime ↑f ↑g\nφ : 𝓡' →ₐ[R] P.ResidueField := (homEquiv P.Residue... | let e : P.ResidueField ≃ₐ[R] Q.ResidueField :=
.ofAlgHom φi φ' (AlgHom.ext fun x ↦ φ'.injective <|
show (φ'.comp φi) (φ' x) = AlgHom.id R _ (φ' x) by congr; ext) (by ext) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 272,
"column": 2
} | {
"line": 277,
"column": 76
} | {
"line": 278,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nV : { x // QuasiFiniteAt f x } → (normalization f).Opens\nhxV✝ : ∀ (x : { x // QuasiFiniteAt f x }), (toNormalization f) ↑x ∈ V x\nhV : ∀ (x : { x // QuasiFiniteAt f x }), IsIso (toNormalization f ∣... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nV : { x // QuasiFiniteAt f x } → (normalization f).Opens\nhxV✝ : ∀ (x : { x // QuasiFiniteAt f x }), (toNormalization f) ↑x ∈ V x\nhV : ∀ (x : { x // QuasiFiniteAt f x }), IsIso (toNormalization f ∣_ V x)\n𝒰 :... | have : (f.appLE U W H).hom.QuasiFinite := by
have : (f.appLE U W H).hom.FiniteType := f.finiteType_appLE hU hr H
rw [← H', CommRingCat.hom_comp, RingHom.finiteType_respectsIso.cancel_right_isIso] at this
rw [← H', CommRingCat.hom_comp, RingHom.QuasiFinite.respectsIso.cancel_right_isIso]
exact .of_isInte... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 283,
"column": 18
} | {
"line": 283,
"column": 34
} | {
"line": 283,
"column": 35
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nV : { x // QuasiFiniteAt f x } → (normalization f).Opens\nhxV✝ : ∀ (x : { x // QuasiFiniteAt f x }), (toNormalization f) ↑x ∈ V x\nhV : ∀ (x : { x // QuasiFiniteAt f x }), IsIso (toNormalization f ∣... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nV : { x // QuasiFiniteAt f x } → (normalization f).Opens\nhxV✝ : ∀ (x : { x // QuasiFiniteAt f x }), (toNormalization f) ↑x ∈ V x\nhV : ∀ (x : { x // QuasiFiniteAt f x }), IsIso (toNormalization f ∣_ V x)\n𝒰 :... | appLE_map_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 24
} | {
"line": 113,
"column": 24
} | [
{
"pp": "X S : Scheme\nf : X ⟶ S\ninst✝² : IsAffine S\ninst✝¹ : Flat f\ninst✝ : IsFinite f\ns : ↥S\n⊢ IsAffine.finrank (pullback.snd f (𝟙 S)) s = IsAffine.finrank f ((𝟙 S) s)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"Al... | [
"X S : Scheme\nf : X ⟶ S\ninst✝² : IsAffine S\ninst✝¹ : Flat f\ninst✝ : IsFinite f\ns : ↥S\n⊢ IsAffine.finrank f ((𝟙 S) s) = IsAffine.finrank f ((𝟙 S) s)"
] | IsAffine.finrank_snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 36
} | {
"line": 556,
"column": 0
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\nc d : B\nl₁ l₂ l₃ : c ⟶ d\nr₁ r₂ r₃ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nadj₃ : l₃ ⊣ r₃\nα : r₁ ⟶ r₂\nβ : r₂ ⟶ r₃\n⊢ (conjugateEquiv adj₁ adj₂) ((conjugateEquiv adj₁ adj₂).symm α) ≫\n (conjugateEquiv adj₂ adj₃) ((conjugateEquiv adj₂ adj₃).symm β) =\n α ≫ β"... | [] | simp only [Equiv.apply_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 629,
"column": 4
} | {
"line": 629,
"column": 38
} | {
"line": 631,
"column": 0
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\nc d : B\nl₁ l₂ : c ⟶ d\nr₁ r₂ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nα : r₁ ≅ r₂\n⊢ (fun α ↦\n { hom := (conjugateEquiv adj₁ adj₂) α.hom, inv := (conjugateEquiv adj₂ adj₁) α.inv, hom_inv_id := ⋯,\n inv_hom_id := ⋯ })\n ((fun β ↦\n { hom ... | [] | simp only [Equiv.apply_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 662,
"column": 2
} | {
"line": 662,
"column": 36
} | {
"line": 664,
"column": 0
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d : B\nf₁ : a ⟶ c\nu₁ : c ⟶ a\nf₂ : b ⟶ d\nu₂ : d ⟶ b\nl₁ : a ⟶ b\nr₁ : b ⟶ a\nl₂ : c ⟶ d\nr₂ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nadj₃ : f₁ ⊣ u₁\nadj₄ : f₂ ⊣ u₂\nα : u₂ ≫ r₁ ⟶ r₂ ≫ u₁\n⊢ (mateEquiv adj₄ adj₃) ((mateEquiv adj₁ adj₂) ((mateEquiv adj₁ adj₂).sym... | [] | simp only [Equiv.apply_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 330,
"column": 4
} | {
"line": 330,
"column": 78
} | {
"line": 332,
"column": 0
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\n𝒜 : ι → σ\nx : Submonoid A\nα : Type u_4\ninst✝² : SMul α A\ninst✝¹ : IsScalarTower α A A\ninst✝ : SMulMemClass σ α A\nm : α\nc1 c2 : NumDenSameDeg 𝒜 x\nh : Localization.mk ↑c1.num ⟨↑c1.den, ⋯⟩ = Localization.mk ↑c2.... | [] | convert! congr_arg (fun z : at x => m • z) h <;> rw [Localization.smul_mk] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 725,
"column": 14
} | {
"line": 725,
"column": 51
} | {
"line": 727,
"column": 0
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝¹³ : CommRing A\ninst✝¹² : SetLike σ A\ninst✝¹¹ : AddSubgroupClass σ A\ninst✝¹⁰ : AddCommMonoid ι\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → σ\ninst✝⁸ : GradedRing 𝒜\nB : Type u_4\nτ : Type u_5\ninst✝⁷ : CommRing B\ninst✝⁶ : SetLike τ B\ninst✝⁵ : AddSubgroupClass ... | [] | rintro _ ⟨n, rfl⟩; exact ⟨n, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 725,
"column": 14
} | {
"line": 725,
"column": 51
} | {
"line": 727,
"column": 0
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝¹³ : CommRing A\ninst✝¹² : SetLike σ A\ninst✝¹¹ : AddSubgroupClass σ A\ninst✝¹⁰ : AddCommMonoid ι\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → σ\ninst✝⁸ : GradedRing 𝒜\nB : Type u_4\nτ : Type u_5\ninst✝⁷ : CommRing B\ninst✝⁶ : SetLike τ B\ninst✝⁵ : AddSubgroupClass ... | [] | rintro _ ⟨n, rfl⟩; exact ⟨n, by simp⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 54
} | {
"line": 106,
"column": 55
} | [
{
"pp": "case e'_1\nσ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nι : Type u_2\nf : ι → A\nhfn : ∀ (i : ι), ∃ n, f i ∈ 𝒜 n\nhf : Algebra.adjoin (↥(𝒜 0)) (Set.range f) = ⊤\nx : A\nhx : x ∈ (HomogeneousIdeal.irrelevant 𝒜).... | [
"case e'_1\nσ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nι : Type u_2\nf : ι → A\nhfn : ∀ (i : ι), ∃ n, f i ∈ 𝒜 n\nhf : Algebra.adjoin (↥(𝒜 0)) (Set.range f) = ⊤\nx : A\nhx : x ∈ (HomogeneousIdeal.irrelevant 𝒜).toIdeal\n⊢ x... | (HomogeneousIdeal.mem_irrelevant_iff _ _).mp hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 16
} | {
"line": 131,
"column": 4
} | [
{
"pp": "σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nι : Type u_2\nf : ι → A\nhfn : ∀ (i : ι), ∃ n, f i ∈ 𝒜 n\nhf : Algebra.adjoin (↥(𝒜 0)) (Set.range f) = ⊤\nx✝ x y : A\nhx : x ∈ Algebra.adjoin (↥(𝒜 0)) (Set.range f)\... | [
"σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nι : Type u_2\nf : ι → A\nhfn : ∀ (i : ι), ∃ n, f i ∈ 𝒜 n\nhf : Algebra.adjoin (↥(𝒜 0)) (Set.range f) = ⊤\nx✝ x y : A\nhx : x ∈ Algebra.adjoin (↥(𝒜 0)) (Set.range f)\nhy : y ∈ Al... | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Maps | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 26
} | {
"line": 46,
"column": 6
} | [
{
"pp": "case smul\nA : Type u_1\nB : Type u_2\nC : Type u_3\nσ : Type u_4\nτ : Type u_5\nω : Type u_6\nι : Type u_7\nF : Type u_8\nG : Type u_9\ninst✝¹³ : Semiring A\ninst✝¹² : Semiring B\ninst✝¹¹ : Semiring C\ninst✝¹⁰ : SetLike σ A\ninst✝⁹ : SetLike τ B\ninst✝⁸ : SetLike ω C\ninst✝⁷ : AddSubmonoidClass σ A\ni... | [] | | smul a₁ a₂ ha₂ ih => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Maps | {
"line": 37,
"column": 4
} | {
"line": 47,
"column": 63
} | {
"line": 49,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\nσ : Type u_4\nτ : Type u_5\nω : Type u_6\nι : Type u_7\nF : Type u_8\nG : Type u_9\ninst✝¹³ : Semiring A\ninst✝¹² : Semiring B\ninst✝¹¹ : Semiring C\ninst✝¹⁰ : SetLike σ A\ninst✝⁹ : SetLike τ B\ninst✝⁸ : SetLike ω C\ninst✝⁷ : AddSubmonoidClass σ A\ninst✝⁶ : Add... | [] | rw [Ideal.map] at hb
induction hb using Submodule.span_induction generalizing i with
| zero => simp
| add => simp [*, Ideal.add_mem]
| mem a ha =>
obtain ⟨a, ha, rfl⟩ := ha
rw [← f.map_directSumDecompose]
exact Ideal.mem_map_of_mem _ (I.2 _ ha)
| smul a₁ a₂ ha₂ ih =>
classica... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Maps | {
"line": 37,
"column": 4
} | {
"line": 47,
"column": 63
} | {
"line": 49,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\nσ : Type u_4\nτ : Type u_5\nω : Type u_6\nι : Type u_7\nF : Type u_8\nG : Type u_9\ninst✝¹³ : Semiring A\ninst✝¹² : Semiring B\ninst✝¹¹ : Semiring C\ninst✝¹⁰ : SetLike σ A\ninst✝⁹ : SetLike τ B\ninst✝⁸ : SetLike ω C\ninst✝⁷ : AddSubmonoidClass σ A\ninst✝⁶ : Add... | [] | rw [Ideal.map] at hb
induction hb using Submodule.span_induction generalizing i with
| zero => simp
| add => simp [*, Ideal.add_mem]
| mem a ha =>
obtain ⟨a, ha, rfl⟩ := ha
rw [← f.map_directSumDecompose]
exact Ideal.mem_map_of_mem _ (I.2 _ ha)
| smul a₁ a₂ ha₂ ih =>
classica... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ValuativeCriterion | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 35
} | {
"line": 128,
"column": 4
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nH : Existence f\nx' : ↥X\ny : ↥Y\nh : flip (fun x1 x2 ↦ x1 ⤳ x2) y (f x')\nstalk_y_to_residue_x' : Y.presheaf.stalk y ⟶ X.residueField x' :=\n Y.presheaf.stalkSpecializes h ≫ Scheme.Hom.stalkMap f x' ≫ X.residue x'\nA : ValuationSubring ↑(X.residueField x')\nhA : ∀ (x : ↑(Y.pr... | [
"X Y : Scheme\nf : X ⟶ Y\nH : Existence f\nx' : ↥X\ny : ↥Y\nh : flip (fun x1 x2 ↦ x1 ⤳ x2) y (f x')\nstalk_y_to_residue_x' : Y.presheaf.stalk y ⟶ X.residueField x' :=\n Y.presheaf.stalkSpecializes h ≫ Scheme.Hom.stalkMap f x' ≫ X.residue x'\nA : ValuationSubring ↑(X.residueField x')\nhA : ∀ (x : ↑(Y.presheaf.stalk... | simp_rw [← Spec.map_comp_assoc] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
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