module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 816,
"column": 43
} | {
"line": 816,
"column": 54
} | {
"line": 816,
"column": 55
} | [
{
"pp": "m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\nhf' : ¬f = 0\n⊢ ?m.137 < ?m.139",
"ppTerm": "?m.148",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}... | [
"m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\nhf' : ¬f = 0\n⊢ ?m.137 < ?m.139"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 819,
"column": 43
} | {
"line": 819,
"column": 54
} | {
"line": 819,
"column": 55
} | [
{
"pp": "m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\nhg' : ¬g = 0\n⊢ (↑(⟨p, hp⟩, ⟨q, hq⟩).1).degree < ?m.186",
"ppTerm": "?m.195",
"assigned": true,
"usedConstants": [
"WithBot.instPreo... | [
"m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\nhg' : ¬g = 0\n⊢ p.degree < ?m.186"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 700,
"column": 8
} | {
"line": 700,
"column": 59
} | {
"line": 700,
"column": 60
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nP : Ideal R\ninst✝ : P.IsPrime\np : R[X]\nf g : P.ResidueField[X]\nhp : p.Monic\nhf : f.Monic\nhg : g.Monic\nH : map (algebraMap R P.ResidueField) p = f * g\nHpq : IsCoprime f g\n⊢ p.natDegree = f.natDegree + g.natDegree",
"ppTerm": "?m.204",
"assigned": false,
... | [
"R : Type u\ninst✝¹ : CommRing R\nP : Ideal R\ninst✝ : P.IsPrime\np : R[X]\nf g : P.ResidueField[X]\nhp : p.Monic\nhf : f.Monic\nhg : g.Monic\nH : map (algebraMap R P.ResidueField) p = f * g\nHpq : IsCoprime f g\n⊢ p.natDegree = f.natDegree + g.natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 847,
"column": 2
} | {
"line": 847,
"column": 40
} | {
"line": 847,
"column": 41
} | [
{
"pp": "m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\ni : Fin (m + n)\nj : Fin m ⊕ Fin n\n⊢ (LinearMap.toMatrix (((degreeLT.basis R m).prod (degreeLT.basis R n)).reindex finSumFinEquiv)\n (degreeLT.basis R (m + n)))\n (f.sylvesterMap g hf hg) i (... | [
"m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\ni : Fin (m + n)\nj : Fin m ⊕ Fin n\n⊢ (f * ↑(Sum.elim ((fun x ↦ (x, 0)) ∘ ⇑(degreeLT.basis R m)) (Prod.mk 0 ∘ ⇑(degreeLT.basis R n)) j).2).coeff ↑i +\n (g * ↑(Sum.elim ((fun x ↦ (x, 0)) ∘ ⇑(degreeLT.basis R m)) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 878,
"column": 35
} | {
"line": 879,
"column": 68
} | {
"line": 879,
"column": 69
} | [
{
"pp": "m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\nH : m ≠ 0 ∨ n ≠ 0\na✝ : Nontrivial R\n⊢ 1 ∈ R[X]_(?m.85 + ?m.86)",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.degre... | [
"m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\nH : m ≠ 0 ∨ n ≠ 0\na✝ : Nontrivial R\n⊢ ¬?m.85 = 0 ∨ ¬?m.86 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 882,
"column": 23
} | {
"line": 882,
"column": 53
} | {
"line": 882,
"column": 54
} | [
{
"pp": "m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\nH : m ≠ 0 ∨ n ≠ 0\na✝ : Nontrivial R\nX : ↥R[X]_m × ↥R[X]_n := (f.adjSylvester g) ⟨1, ⋯⟩\nthis : ↑((f.sylvesterMap g hf hg) X) = ↑((f.resultant g m n • LinearMap.id) ⟨1, ⋯⟩)\n⊢ (↑X.2).degree < ↑n",
"p... | [
"m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\nH : m ≠ 0 ∨ n ≠ 0\na✝ : Nontrivial R\nX : ↥R[X]_m × ↥R[X]_n := (f.adjSylvester g) ⟨1, ⋯⟩\nthis : ↑((f.sylvesterMap g hf hg) X) = ↑((f.resultant g m n • LinearMap.id) ⟨1, ⋯⟩)\n⊢ (↑X.2).degree < ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 883,
"column": 7
} | {
"line": 883,
"column": 37
} | {
"line": 883,
"column": 38
} | [
{
"pp": "m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\nH : m ≠ 0 ∨ n ≠ 0\na✝ : Nontrivial R\nX : ↥R[X]_m × ↥R[X]_n := (f.adjSylvester g) ⟨1, ⋯⟩\nthis : ↑((f.sylvesterMap g hf hg) X) = ↑((f.resultant g m n • LinearMap.id) ⟨1, ⋯⟩)\n⊢ (↑X.1).degree < ↑m",
"p... | [
"m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\nH : m ≠ 0 ∨ n ≠ 0\na✝ : Nontrivial R\nX : ↥R[X]_m × ↥R[X]_n := (f.adjSylvester g) ⟨1, ⋯⟩\nthis : ↑((f.sylvesterMap g hf hg) X) = ↑((f.resultant g m n • LinearMap.id) ⟨1, ⋯⟩)\n⊢ (↑X.1).degree < ↑m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 883,
"column": 48
} | {
"line": 883,
"column": 78
} | {
"line": 883,
"column": 79
} | [
{
"pp": "m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\nH : m ≠ 0 ∨ n ≠ 0\na✝ : Nontrivial R\nX : ↥R[X]_m × ↥R[X]_n := (f.adjSylvester g) ⟨1, ⋯⟩\nthis : ↑((f.sylvesterMap g hf hg) X) = ↑((f.resultant g m n • LinearMap.id) ⟨1, ⋯⟩)\n⊢ f * ↑X.2 + g * ↑X.1 = C (f.... | [
"m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\nH : m ≠ 0 ∨ n ≠ 0\na✝ : Nontrivial R\nX : ↥R[X]_m × ↥R[X]_n := (f.adjSylvester g) ⟨1, ⋯⟩\nthis : ↑((f.sylvesterMap g hf hg) X) = ↑((f.resultant g m n • LinearMap.id) ⟨1, ⋯⟩)\n⊢ f * ↑X.2 + g * ↑X.1 = C (f.resultant g ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 897,
"column": 28
} | {
"line": 897,
"column": 39
} | {
"line": 897,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.Monic\nhf0 : ¬f.natDegree = 0\nb : R[X]\nthis : f.resultant (b * g) f.natDegree (b.natDegree + g.natDegree) = f.resultant b * f.resultant g\ne : 0 * f + b * g = 1\n⊢ b * g = 1",
"ppTerm": "?m.218",
"assigned": false,
"usedConstants": [],
... | [
"R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.Monic\nhf0 : ¬f.natDegree = 0\nb : R[X]\nthis : f.resultant (b * g) f.natDegree (b.natDegree + g.natDegree) = f.resultant b * f.resultant g\ne : 0 * f + b * g = 1\n⊢ b * g = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 898,
"column": 6
} | {
"line": 898,
"column": 35
} | {
"line": 898,
"column": 36
} | [
{
"pp": "case neg.refine_2.inl\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.Monic\nhf0 : ¬f.natDegree = 0\nb : R[X]\nthis : f.coeff f.natDegree ^ (b.natDegree + g.natDegree) = f.resultant b * f.resultant g\ne : 0 * f + b * g = 1\n⊢ 1 = f.resultant b * f.resultant g",
"ppTerm": "?neg.refine_2.inl✝",... | [
"case neg.refine_2.inl\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.Monic\nhf0 : ¬f.natDegree = 0\nb : R[X]\nthis : f.coeff f.natDegree ^ (b.natDegree + g.natDegree) = f.resultant b * f.resultant g\ne : 0 * f + b * g = 1\n⊢ 1 = f.resultant b * f.resultant g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 900,
"column": 8
} | {
"line": 900,
"column": 37
} | {
"line": 900,
"column": 38
} | [
{
"pp": "case neg.refine_2.inr\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.Monic\nhf0 : ¬f.natDegree = 0\na b : R[X]\ne : a * f + b * g = 1\nthis : f.resultant (C 1) f.natDegree (b.natDegree + g.natDegree) = f.resultant b * f.resultant g\nhb0 : a ≠ 0\n⊢ 1 = f.resultant b * f.resultant g",
"ppTerm"... | [
"case neg.refine_2.inr\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.Monic\nhf0 : ¬f.natDegree = 0\na b : R[X]\ne : a * f + b * g = 1\nthis : f.resultant (C 1) f.natDegree (b.natDegree + g.natDegree) = f.resultant b * f.resultant g\nhb0 : a ≠ 0\n⊢ 1 = f.resultant b * f.resultant g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 893,
"column": 4
} | {
"line": 906,
"column": 11
} | {
"line": 908,
"column": 0
} | [
{
"pp": "case neg.refine_2\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.Monic\nhf0 : ¬f.natDegree = 0\n⊢ IsCoprime f g → IsUnit (f.resultant g)",
"ppTerm": "?neg.refine_2✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"one_pow",
"NonUnitalNonAssocCommRing.toNonUnitalNo... | [] | intro ⟨a, b, e⟩
suffices 1 = f.resultant b * f.resultant g from isUnit_iff_exists_inv'.mpr ⟨_, this.symm⟩
have := resultant_mul_right f b g _ le_rfl
obtain rfl | hb0 := eq_or_ne a 0
· rw [show b * g = 1 by simpa using e, resultant_one_right] at this
simpa [hf.leadingCoeff] using this
· rw [← r... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 893,
"column": 4
} | {
"line": 906,
"column": 11
} | {
"line": 908,
"column": 0
} | [
{
"pp": "case neg.refine_2\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.Monic\nhf0 : ¬f.natDegree = 0\n⊢ IsCoprime f g → IsUnit (f.resultant g)",
"ppTerm": "?neg.refine_2✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"one_pow",
"NonUnitalNonAssocCommRing.toNonUnitalNo... | [] | intro ⟨a, b, e⟩
suffices 1 = f.resultant b * f.resultant g from isUnit_iff_exists_inv'.mpr ⟨_, this.symm⟩
have := resultant_mul_right f b g _ le_rfl
obtain rfl | hb0 := eq_or_ne a 0
· rw [show b * g = 1 by simpa using e, resultant_one_right] at this
simpa [hf.leadingCoeff] using this
· rw [← r... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 912,
"column": 4
} | {
"line": 912,
"column": 72
} | {
"line": 913,
"column": 6
} | [
{
"pp": "case inl.inr\nK : Type u_2\ninst✝ : Field K\ng : K[X]\nhg : g ≠ 0\n⊢ resultant 0 g = 0 ↔ (0 ≠ 0 ∨ g ≠ 0) ∧ ¬IsCoprime 0 g",
"ppTerm": "?inl.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"not_exists._simp_1",
... | [
"case inl.inr\nK : Type u_2\ninst✝ : Field K\ng : K[X]\nhg : g ≠ 0\n⊢ (∀ (x : K), ¬C x = g) ↔ ∀ (x : K), ¬x = 0 → ¬C x = g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 946,
"column": 4
} | {
"line": 946,
"column": 34
} | {
"line": 947,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\nhf : f.natDegree = 1\ne : Fin (f.natDegree - 1 + f.natDegree) ≃ Fin 1 := finCongr ⋯\nthis : NeZero (f.natDegree - 1 + f.natDegree)\nj : ℕ\nhj : j < 1\nhi : 0 < 1\n⊢ (Matrix.reindex e e) f.sylvesterDeriv ⟨0, hi⟩ ⟨j, hj⟩ = !![1] ⟨0, hi⟩ ⟨j, hj⟩",
"ppTerm": ... | [
"R : Type u_1\ninst✝ : CommRing R\nf : R[X]\nhf : f.natDegree = 1\ne : Fin (f.natDegree - 1 + f.natDegree) ≃ Fin 1 := finCongr ⋯\nthis : NeZero (f.natDegree - 1 + f.natDegree)\nhi hj : 0 < 1\n⊢ (Matrix.reindex e e) f.sylvesterDeriv ⟨0, hi⟩ ⟨0, hj⟩ = !![1] ⟨0, hi⟩ ⟨0, hj⟩"
] | obtain ⟨rfl⟩ : j = 0 := by lia | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 103,
"column": 13
} | {
"line": 103,
"column": 24
} | {
"line": 103,
"column": 25
} | [
{
"pp": "X S : Scheme\nf : X ⟶ S\ninst✝¹ : LocallyOfFiniteType f\ninst✝ : IsSeparated f\nx : ↥X\ns : ↥S\nh : f x = s\nhx : Scheme.Hom.QuasiFiniteAt f x\nU : TopologicalSpace.Opens ↥S\nhU : U ∈ S.affineOpens\nhxU : f x ∈ ↑U\nV : TopologicalSpace.Opens ↥X\nhV : V ∈ X.affineOpens\nhxV : x ∈ ↑V\nhUV : V ≤ f ⁻¹ᵁ U\n... | [
"X S : Scheme\nf : X ⟶ S\ninst✝¹ : LocallyOfFiniteType f\ninst✝ : IsSeparated f\nx : ↥X\ns : ↥S\nh : f x = s\nhx : Scheme.Hom.QuasiFiniteAt f x\nU : TopologicalSpace.Opens ↥S\nhU : U ∈ S.affineOpens\nhxU : f x ∈ ↑U\nV : TopologicalSpace.Opens ↥X\nhV : V ∈ X.affineOpens\nhxV : x ∈ ↑V\nhUV : V ≤ f ⁻¹ᵁ U\nthis✝⁵ : (Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Cartesian.Grp | {
"line": 330,
"column": 4
} | {
"line": 330,
"column": 51
} | {
"line": 330,
"column": 52
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CartesianMonoidalCategory C\nG : C\ninst✝¹ : GrpObj G\ninst✝ : BraidedCategory C\nheq : GrpObj.commutator G = toUnit (G ⊗ G) ≫ η\nX : C\nf g : X ⟶ G\n⊢ f * g = g * f",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": ... | [
"case refine_2\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : CartesianMonoidalCategory C\nG : C\ninst✝¹ : GrpObj G\ninst✝ : BraidedCategory C\nheq : GrpObj.commutator G = toUnit (G ⊗ G) ≫ η\nX : C\nf g : X ⟶ G\n⊢ f * g = g * f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Group.Abelian | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 15
} | {
"line": 54,
"column": 16
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\ninst✝³ : IsAlgClosed K\nG : Over (Spec (CommRingCat.of K))\ninst✝² : IsProper G.hom\ninst✝¹ : IsIntegral (G ⊗ G).left\ninst✝ : GrpObj G\nS : Scheme := Spec (CommRingCat.of K)\npoint : ↥S := IsLocalRing.closedPoint K\nhpoint : IsClosed {point}\nthis✝⁵ : Nonempty ↥G.left\nth... | [
"K : Type u\ninst✝⁴ : Field K\ninst✝³ : IsAlgClosed K\nG : Over (Spec (CommRingCat.of K))\ninst✝² : IsProper G.hom\ninst✝¹ : IsIntegral (G ⊗ G).left\ninst✝ : GrpObj G\nS : Scheme := Spec (CommRingCat.of K)\npoint : ↥S := IsLocalRing.closedPoint K\nhpoint : IsClosed {point}\nthis✝⁵ : Nonempty ↥G.left\nthis✝⁴ : IsPro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 129,
"column": 61
} | {
"line": 129,
"column": 79
} | {
"line": 129,
"column": 80
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx : ↥X\nhx : QuasiFiniteAt f x\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pullback.snd f fT)... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx : ↥X\nhx : QuasiFiniteAt f x\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pullback.snd f fT)\nhv₂ : (pul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 124,
"column": 8
} | {
"line": 124,
"column": 46
} | {
"line": 124,
"column": 47
} | [
{
"pp": "R : Type u\nS : Type v\nT : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra.FiniteType R T\ninst✝¹ : Algebra.IsIntegral R S\nf : S →ₐ[R] T\ng : S\nhg : Function.Surjective ⇑(awayMapₐ f g)\np : Ideal R\ninst✝ : p.IsPri... | [
"R : Type u\nS : Type v\nT : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra.FiniteType R T\ninst✝¹ : Algebra.IsIntegral R S\nf : S →ₐ[R] T\ng : S\nhg : Function.Surjective ⇑(awayMapₐ f g)\np : Ideal R\ninst✝ : p.IsPrime\nhgp : Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 151,
"column": 10
} | {
"line": 152,
"column": 35
} | {
"line": 152,
"column": 36
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx : ↥X\nhx : QuasiFiniteAt f x\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pullback.snd f fT)... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx : ↥X\nhx : QuasiFiniteAt f x\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pullback.snd f fT)\nhv₂ : (pul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 156,
"column": 8
} | {
"line": 157,
"column": 31
} | {
"line": 157,
"column": 32
} | [
{
"pp": "case a\nX Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx : ↥X\nhx : QuasiFiniteAt f x\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pullback.s... | [
"case a\nX Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx : ↥X\nhx : QuasiFiniteAt f x\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pullback.snd f fT)\nhv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 168,
"column": 77
} | {
"line": 168,
"column": 95
} | {
"line": 168,
"column": 96
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx✝¹ : ↥X\nhx : QuasiFiniteAt f x✝¹\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x✝¹\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pullback.snd... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx✝¹ : ↥X\nhx : QuasiFiniteAt f x✝¹\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x✝¹\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pullback.snd f fT)\nhv₂ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Descent | {
"line": 70,
"column": 8
} | {
"line": 70,
"column": 41
} | {
"line": 70,
"column": 42
} | [
{
"pp": "case refine_2.tmul\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Module.FaithfullyFlat R S\nI : Ideal R\nhI : (Ideal.map (algebraMap R S) I).FG\nf : S ⊗[R] ↥I →ₗ[S] S := ⋯\nhf : Function.Injective ⇑f\ns : S\nx : ↥I\nthis : f (s ⊗ₜ[R] x) = s • f (1 ... | [
"case refine_2.tmul\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Module.FaithfullyFlat R S\nI : Ideal R\nhI : (Ideal.map (algebraMap R S) I).FG\nf : S ⊗[R] ↥I →ₗ[S] S := ↑(AlgebraTensorModule.rid R S S) ∘ₗ (AlgebraTensorModule.lTensor S S) (Submodule.subtype I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 183,
"column": 14
} | {
"line": 183,
"column": 25
} | {
"line": 183,
"column": 26
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx : ↥X\nhx : QuasiFiniteAt f x\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pullback.snd f fT)... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx : ↥X\nhx : QuasiFiniteAt f x\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pullback.snd f fT)\nhv₂ : (pul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 242,
"column": 6
} | {
"line": 243,
"column": 42
} | {
"line": 243,
"column": 43
} | [
{
"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 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Group.Abelian | {
"line": 104,
"column": 48
} | {
"line": 104,
"column": 59
} | {
"line": 104,
"column": 60
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\ninst✝³ : IsAlgClosed K\nG : Over (Spec (CommRingCat.of K))\ninst✝² : IsProper G.hom\ninst✝¹ : IsIntegral (G ⊗ G).left\ninst✝ : GrpObj G\nS : Scheme := Spec (CommRingCat.of K)\npoint : ↥S := IsLocalRing.closedPoint K\nhpoint : IsClosed {point}\nthis✝⁹ : Nonempty ↥G.left\nth... | [
"K : Type u\ninst✝⁴ : Field K\ninst✝³ : IsAlgClosed K\nG : Over (Spec (CommRingCat.of K))\ninst✝² : IsProper G.hom\ninst✝¹ : IsIntegral (G ⊗ G).left\ninst✝ : GrpObj G\nS : Scheme := Spec (CommRingCat.of K)\npoint : ↥S := IsLocalRing.closedPoint K\nhpoint : IsClosed {point}\nthis✝⁹ : Nonempty ↥G.left\nthis✝⁸ : IsPro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 263,
"column": 6
} | {
"line": 263,
"column": 70
} | {
"line": 264,
"column": 6
} | [
{
"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 :... | simp only [opensRange_homOfLE, image_preimage_eq_opensRange_inf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 437,
"column": 2
} | {
"line": 437,
"column": 62
} | {
"line": 437,
"column": 63
} | [
{
"pp": "X Y S : Scheme\nf : X ⟶ Y\ng : Y ⟶ S\ns : ↥S\nH : (⇑f '' ⇑(f ≫ g) ⁻¹' {s}).Finite\ninst✝² : IsProper (f ≫ g)\ninst✝¹ : IsSeparated g\ninst✝ : LocallyOfFiniteType g\nthis✝ : IsProper f\nthis : IsProper (Scheme.Hom.imageι f ≫ g)\nx : ↥X\nhx : (Scheme.Hom.toImage f) x ∈ ⇑(Scheme.Hom.imageι f ≫ g) ⁻¹' {s}\... | [
"X Y S : Scheme\nf : X ⟶ Y\ng : Y ⟶ S\ns : ↥S\nH : (⇑f '' ⇑(f ≫ g) ⁻¹' {s}).Finite\ninst✝² : IsProper (f ≫ g)\ninst✝¹ : IsSeparated g\ninst✝ : LocallyOfFiniteType g\nthis✝ : IsProper f\nthis : IsProper (Scheme.Hom.imageι f ≫ g)\nx : ↥X\nhx : (Scheme.Hom.toImage f) x ∈ ⇑(Scheme.Hom.imageι f ≫ g) ⁻¹' {s}\n⊢ (f ≫ g) x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Group.Smooth | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 47
} | {
"line": 59,
"column": 48
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\nG : Scheme\nf : G ⟶ Spec (CommRingCat.of K)\ninst✝³ : LocallyOfFiniteType f\ninst✝² : GrpObj (Over.mk f)\ninst✝¹ : IsReduced G\ninst✝ : IsAlgClosed K\nthis✝ : JacobsonSpace ↥G\nthis : Nonempty ↥G\nH : (↑(Scheme.Hom.smoothLocus f))ᶜ.Nonempty\nx : ↥G\nhx : x ∈ (↑(Scheme.Hom.... | [
"K : Type u\ninst✝⁴ : Field K\nG : Scheme\nf : G ⟶ Spec (CommRingCat.of K)\ninst✝³ : LocallyOfFiniteType f\ninst✝² : GrpObj (Over.mk f)\ninst✝¹ : IsReduced G\ninst✝ : IsAlgClosed K\nthis✝ : JacobsonSpace ↥G\nthis : Nonempty ↥G\nH : (↑(Scheme.Hom.smoothLocus f))ᶜ.Nonempty\nx : ↥G\nhx : x ∈ (↑(Scheme.Hom.smoothLocus ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 272,
"column": 4
} | {
"line": 285,
"column": 8
} | {
"line": 286,
"column": 2
} | [
{
"pp": "case refine_1\nR : 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' : Idea... | [] | have : (P'.1.comap φ.toRingHom).LiesOver P := inferInstanceAs ((P'.1.comap φ).LiesOver P)
apply Ideal.eq_of_comap_eq_comap_of_bijective_residueFieldMap hP
simp only [Ideal.comap_comap, AlgHom.toRingHom_eq_coe,
← @AlgHom.coe_restrictScalars R R', ← AlgHom.comp_toRingHom,
Algebra.TensorProduct.map_res... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 272,
"column": 4
} | {
"line": 285,
"column": 8
} | {
"line": 286,
"column": 2
} | [
{
"pp": "case refine_1\nR : 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' : Idea... | [] | have : (P'.1.comap φ.toRingHom).LiesOver P := inferInstanceAs ((P'.1.comap φ).LiesOver P)
apply Ideal.eq_of_comap_eq_comap_of_bijective_residueFieldMap hP
simp only [Ideal.comap_comap, AlgHom.toRingHom_eq_coe,
← @AlgHom.coe_restrictScalars R R', ← AlgHom.comp_toRingHom,
Algebra.TensorProduct.map_res... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.Functor.StrictPseudofunctor | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 15
} | {
"line": 117,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictPseudofunctorCore B C\na✝ b✝ c✝ : B\nf : a✝ ⟶ b✝\nx✝¹ x✝ : b✝ ⟶ c✝\nη : x✝¹ ⟶ x✝\n⊢ S.map₂ (f ◁ η) = (eqToIso ⋯).hom ≫ S.map f ◁ S.map₂ η ≫ (eqToIso ⋯).inv",
"ppTerm": "?m.107",
... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictPseudofunctorCore B C\na✝ b✝ c✝ : B\nf : a✝ ⟶ b✝\nx✝¹ x✝ : b✝ ⟶ c✝\nη : x✝¹ ⟶ x✝\n⊢ S.map₂ (f ◁ η) = eqToHom ⋯ ≫ S.map f ◁ S.map₂ η ≫ eqToHom ⋯"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictPseudofunctor | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 15
} | {
"line": 119,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictPseudofunctorCore B C\na✝ b✝ c✝ : B\nf✝ g✝ : a✝ ⟶ b✝\nη : f✝ ⟶ g✝\nf : b✝ ⟶ c✝\n⊢ S.map₂ (η ▷ f) = (eqToIso ⋯).hom ≫ S.map₂ η ▷ S.map f ≫ (eqToIso ⋯).inv",
"ppTerm": "?m.123",
"a... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictPseudofunctorCore B C\na✝ b✝ c✝ : B\nf✝ g✝ : a✝ ⟶ b✝\nη : f✝ ⟶ g✝\nf : b✝ ⟶ c✝\n⊢ S.map₂ (η ▷ f) = eqToHom ⋯ ≫ S.map₂ η ▷ S.map f ≫ eqToHom ⋯"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictPseudofunctor | {
"line": 115,
"column": 4
} | {
"line": 115,
"column": 15
} | {
"line": 115,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictPseudofunctorCore B C\na✝ b✝ c✝ d✝ : B\nf : a✝ ⟶ b✝\ng : b✝ ⟶ c✝\nh : c✝ ⟶ d✝\n⊢ S.map₂ (α_ f g h).hom =\n (eqToIso ⋯).hom ≫\n (eqToIso ⋯).hom ▷ S.map h ≫ (α_ (S.map f) (S.map g)... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictPseudofunctorCore B C\na✝ b✝ c✝ d✝ : B\nf : a✝ ⟶ b✝\ng : b✝ ⟶ c✝\nh : c✝ ⟶ d✝\n⊢ S.map₂ (α_ f g h).hom = eqToHom ⋯ ≫ (α_ (S.map f) (S.map g) (S.map h)).hom ≫ eqToHom ⋯"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictPseudofunctor | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 15
} | {
"line": 111,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictPseudofunctorCore B C\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ S.map₂ (λ_ f).hom = (eqToIso ⋯).hom ≫ (eqToIso ⋯).hom ▷ S.map f ≫ (λ_ (S.map f)).hom",
"ppTerm": "?m.154",
"assigned": true,
... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictPseudofunctorCore B C\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ S.map₂ (λ_ f).hom = eqToHom ⋯ ≫ (λ_ (S.map f)).hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictPseudofunctor | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 15
} | {
"line": 113,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictPseudofunctorCore B C\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ S.map₂ (ρ_ f).hom = (eqToIso ⋯).hom ≫ S.map f ◁ (eqToIso ⋯).hom ≫ (ρ_ (S.map f)).hom",
"ppTerm": "?m.167",
"assigned": true,
... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictPseudofunctorCore B C\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ S.map₂ (ρ_ f).hom = eqToHom ⋯ ≫ (ρ_ (S.map f)).hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictPseudofunctor | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 15
} | {
"line": 168,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝⁴ : Bicategory B\nC : Type u₂\ninst✝³ : Bicategory C\nD : Type u₃\ninst✝² : Bicategory D\ninst✝¹ : Strict B\ninst✝ : Strict C\nS : StrictPseudofunctorPreCore B C\na✝ b✝ c✝ : B\nf : a✝ ⟶ b✝\nx✝¹ x✝ : b✝ ⟶ c✝\nη : x✝¹ ⟶ x✝\n⊢ S.map₂ (f ◁ η) = (eqToIso ⋯).hom ≫ S.map f ◁ S.map₂ η ≫ (eqTo... | [
"B : Type u₁\ninst✝⁴ : Bicategory B\nC : Type u₂\ninst✝³ : Bicategory C\nD : Type u₃\ninst✝² : Bicategory D\ninst✝¹ : Strict B\ninst✝ : Strict C\nS : StrictPseudofunctorPreCore B C\na✝ b✝ c✝ : B\nf : a✝ ⟶ b✝\nx✝¹ x✝ : b✝ ⟶ c✝\nη : x✝¹ ⟶ x✝\n⊢ S.map₂ (f ◁ η) = eqToHom ⋯ ≫ S.map f ◁ S.map₂ η ≫ eqToHom ⋯"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictPseudofunctor | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 15
} | {
"line": 170,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝⁴ : Bicategory B\nC : Type u₂\ninst✝³ : Bicategory C\nD : Type u₃\ninst✝² : Bicategory D\ninst✝¹ : Strict B\ninst✝ : Strict C\nS : StrictPseudofunctorPreCore B C\na✝ b✝ c✝ : B\nf✝ g✝ : a✝ ⟶ b✝\nη : f✝ ⟶ g✝\nf : b✝ ⟶ c✝\n⊢ S.map₂ (η ▷ f) = (eqToIso ⋯).hom ≫ S.map₂ η ▷ S.map f ≫ (eqToIs... | [
"B : Type u₁\ninst✝⁴ : Bicategory B\nC : Type u₂\ninst✝³ : Bicategory C\nD : Type u₃\ninst✝² : Bicategory D\ninst✝¹ : Strict B\ninst✝ : Strict C\nS : StrictPseudofunctorPreCore B C\na✝ b✝ c✝ : B\nf✝ g✝ : a✝ ⟶ b✝\nη : f✝ ⟶ g✝\nf : b✝ ⟶ c✝\n⊢ S.map₂ (η ▷ f) = eqToHom ⋯ ≫ S.map₂ η ▷ S.map f ≫ eqToHom ⋯"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictlyUnitary | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 15
} | {
"line": 134,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryLaxFunctorCore B C\na✝ b✝ c✝ d✝ : B\nf : a✝ ⟶ b✝\ng : b✝ ⟶ c✝\nh : c✝ ⟶ d✝\n⊢ S.mapComp f g ▷ S.map h ≫ S.mapComp (f ≫ g) h ≫ S.map₂ (α_ f g h).hom =\n (α_ (S.map f) (S.map g... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryLaxFunctorCore B C\na✝ b✝ c✝ d✝ : B\nf : a✝ ⟶ b✝\ng : b✝ ⟶ c✝\nh : c✝ ⟶ d✝\n⊢ S.mapComp f g ▷ S.map h ≫ S.mapComp (f ≫ g) h ≫ S.map₂ (α_ f g h).hom =\n (α_ (S.map f) (S.map g) (S.map h))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictlyUnitary | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 15
} | {
"line": 130,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryLaxFunctorCore B C\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ S.map₂ (λ_ f).inv = (λ_ (S.map f)).inv ≫ eqToHom ⋯ ▷ S.map f ≫ S.mapComp (𝟙 a✝) f",
"ppTerm": "?m.112",
"assigned": true,
... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryLaxFunctorCore B C\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ S.map₂ (λ_ f).inv = (λ_ (S.map f)).inv ≫ eqToHom ⋯ ≫ S.mapComp (𝟙 a✝) f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictlyUnitary | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 15
} | {
"line": 132,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryLaxFunctorCore B C\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ S.map₂ (ρ_ f).inv = (ρ_ (S.map f)).inv ≫ S.map f ◁ eqToHom ⋯ ≫ S.mapComp f (𝟙 b✝)",
"ppTerm": "?m.125",
"assigned": true,
... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryLaxFunctorCore B C\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ S.map₂ (ρ_ f).inv = (ρ_ (S.map f)).inv ≫ eqToHom ⋯ ≫ S.mapComp f (𝟙 b✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictlyUnitary | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 15
} | {
"line": 302,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryPseudofunctorCore B C\na✝ b✝ c✝ : B\nf : a✝ ⟶ b✝\nx✝¹ x✝ : b✝ ⟶ c✝\nη : x✝¹ ⟶ x✝\n⊢ S.map₂ (f ◁ η) = (S.mapComp f x✝¹).hom ≫ S.map f ◁ S.map₂ η ≫ (S.mapComp f x✝).inv",
"ppT... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryPseudofunctorCore B C\na✝ b✝ c✝ : B\nf : a✝ ⟶ b✝\nx✝¹ x✝ : b✝ ⟶ c✝\nη : x✝¹ ⟶ x✝\n⊢ S.map₂ (f ◁ η) = (S.mapComp f x✝¹).hom ≫ S.map f ◁ S.map₂ η ≫ (S.mapComp f x✝).inv"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictlyUnitary | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 15
} | {
"line": 304,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryPseudofunctorCore B C\na✝ b✝ c✝ : B\nf✝ g✝ : a✝ ⟶ b✝\nη : f✝ ⟶ g✝\nf : b✝ ⟶ c✝\n⊢ S.map₂ (η ▷ f) = (S.mapComp f✝ f).hom ≫ S.map₂ η ▷ S.map f ≫ (S.mapComp g✝ f).inv",
"ppTerm... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryPseudofunctorCore B C\na✝ b✝ c✝ : B\nf✝ g✝ : a✝ ⟶ b✝\nη : f✝ ⟶ g✝\nf : b✝ ⟶ c✝\n⊢ S.map₂ (η ▷ f) = (S.mapComp f✝ f).hom ≫ S.map₂ η ▷ S.map f ≫ (S.mapComp g✝ f).inv"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictlyUnitary | {
"line": 300,
"column": 4
} | {
"line": 300,
"column": 15
} | {
"line": 300,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryPseudofunctorCore B C\na✝ b✝ c✝ d✝ : B\nf : a✝ ⟶ b✝\ng : b✝ ⟶ c✝\nh : c✝ ⟶ d✝\n⊢ S.map₂ (α_ f g h).hom =\n (S.mapComp (f ≫ g) h).hom ≫\n (S.mapComp f g).hom ▷ S.map h ≫\... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryPseudofunctorCore B C\na✝ b✝ c✝ d✝ : B\nf : a✝ ⟶ b✝\ng : b✝ ⟶ c✝\nh : c✝ ⟶ d✝\n⊢ S.map₂ (α_ f g h).hom =\n (S.mapComp (f ≫ g) h).hom ≫\n (S.mapComp f g).hom ▷ S.map h ≫\n (α_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictlyUnitary | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 15
} | {
"line": 296,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryPseudofunctorCore B C\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ S.map₂ (λ_ f).hom = (S.mapComp (𝟙 a✝) f).hom ≫ (eqToIso ⋯).hom ▷ S.map f ≫ (λ_ (S.map f)).hom",
"ppTerm": "?m.135",
"as... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryPseudofunctorCore B C\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ S.map₂ (λ_ f).hom = (S.mapComp (𝟙 a✝) f).hom ≫ eqToHom ⋯ ≫ (λ_ (S.map f)).hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Functor.StrictlyUnitary | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 15
} | {
"line": 298,
"column": 16
} | [
{
"pp": "B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryPseudofunctorCore B C\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ S.map₂ (ρ_ f).hom = (S.mapComp f (𝟙 b✝)).hom ≫ S.map f ◁ (eqToIso ⋯).hom ≫ (ρ_ (S.map f)).hom",
"ppTerm": "?m.148",
"as... | [
"B : Type u₁\ninst✝² : Bicategory B\nC : Type u₂\ninst✝¹ : Bicategory C\nD : Type u₃\ninst✝ : Bicategory D\nS : StrictlyUnitaryPseudofunctorCore B C\na✝ b✝ : B\nf : a✝ ⟶ b✝\n⊢ S.map₂ (ρ_ f).hom = (S.mapComp f (𝟙 b✝)).hom ≫ eqToHom ⋯ ≫ (ρ_ (S.map f)).hom"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 395,
"column": 87
} | {
"line": 396,
"column": 53
} | {
"line": 396,
"column": 53
} | [
{
"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\nR' : Type u\nw✝⁶ : CommRing R'\nw✝⁵ : Algebra R R'\nw✝⁴ : Etale R R'\nP : ... | [] | by
rwa [Ideal.disjoint_powers_iff_notMem_of_isPrime] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.Adjunction.Adj | {
"line": 145,
"column": 8
} | {
"line": 145,
"column": 19
} | {
"line": 145,
"column": 20
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d : Adj B\nα : a ⟶ b\n⊢ (conjugateEquiv α.adj (𝟙 a ≫ α).adj) (λ_ α.l).hom = (ρ_ α.r).symm.hom",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
"CategoryTheory... | [
"B : Type u\ninst✝ : Bicategory B\na b c d : Adj B\nα : a ⟶ b\n⊢ (conjugateEquiv α.adj ((Adjunction.id a.obj).comp α.adj)) (λ_ α.l).hom = (ρ_ α.r).inv"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Adjunction.Adj | {
"line": 145,
"column": 5
} | {
"line": 145,
"column": 72
} | {
"line": 145,
"column": 72
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d : Adj B\nα : a ⟶ b\n⊢ (conjugateEquiv α.adj (𝟙 a ≫ α).adj) (λ_ α.l).hom = (ρ_ α.r).symm.hom",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"CategoryTheory.Bicategory.Adjunction.comp",
"Equiv.instEquivLike",
"CategoryThe... | [] | by simpa using conjugateEquiv_id_comp_right_apply α.adj α.adj (𝟙 _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Bicategory.Adjunction.Adj | {
"line": 152,
"column": 8
} | {
"line": 152,
"column": 19
} | {
"line": 152,
"column": 20
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d : Adj B\nα : a ⟶ b\n⊢ (conjugateEquiv α.adj (α ≫ 𝟙 b).adj) (ρ_ α.l).hom = (λ_ α.r).symm.hom",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
"CategoryTheory... | [
"B : Type u\ninst✝ : Bicategory B\na b c d : Adj B\nα : a ⟶ b\n⊢ (conjugateEquiv α.adj (α.adj.comp (Adjunction.id b.obj))) (ρ_ α.l).hom = (λ_ α.r).inv"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Adjunction.Cat | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 15
} | {
"line": 54,
"column": 16
} | [
{
"pp": "C✝ D✝ E : Type u\ninst✝² : Category.{v, u} C✝\ninst✝¹ : Category.{v, u} D✝\ninst✝ : Category.{v, u} E\nF✝ : C✝ ⥤ D✝\nG✝ : D✝ ⥤ C✝\nadj✝ : F✝ ⊣ G✝\nF' : D✝ ⥤ E\nG' : E ⥤ D✝\nadj' : F' ⊣ G'\nC D : Cat\nF : C ⟶ D\nG : D ⟶ C\nadj : F ⊣ G\nX : ↑C\n⊢ F.toFunctor.map (adj.unit.toNatTrans.app X) ≫ adj.counit.t... | [
"C✝ D✝ E : Type u\ninst✝² : Category.{v, u} C✝\ninst✝¹ : Category.{v, u} D✝\ninst✝ : Category.{v, u} E\nF✝ : C✝ ⥤ D✝\nG✝ : D✝ ⥤ C✝\nadj✝ : F✝ ⊣ G✝\nF' : D✝ ⥤ E\nG' : E ⥤ D✝\nadj' : F' ⊣ G'\nC D : Cat\nF : C ⟶ D\nG : D ⟶ C\nadj : F ⊣ G\nX : ↑C\n⊢ F.toFunctor.map (adj.unit.toNatTrans.app X) ≫ adj.counit.toNatTrans.ap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Adjunction.Cat | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 15
} | {
"line": 56,
"column": 16
} | [
{
"pp": "C✝ D✝ E : Type u\ninst✝² : Category.{v, u} C✝\ninst✝¹ : Category.{v, u} D✝\ninst✝ : Category.{v, u} E\nF✝ : C✝ ⥤ D✝\nG✝ : D✝ ⥤ C✝\nadj✝ : F✝ ⊣ G✝\nF' : D✝ ⥤ E\nG' : E ⥤ D✝\nadj' : F' ⊣ G'\nC D : Cat\nF : C ⟶ D\nG : D ⟶ C\nadj : F ⊣ G\nX : ↑D\n⊢ adj.unit.toNatTrans.app (G.toFunctor.obj X) ≫ G.toFunctor.... | [
"C✝ D✝ E : Type u\ninst✝² : Category.{v, u} C✝\ninst✝¹ : Category.{v, u} D✝\ninst✝ : Category.{v, u} E\nF✝ : C✝ ⥤ D✝\nG✝ : D✝ ⥤ C✝\nadj✝ : F✝ ⊣ G✝\nF' : D✝ ⥤ E\nG' : E ⥤ D✝\nadj' : F' ⊣ G'\nC D : Cat\nF : C ⟶ D\nG : D ⟶ C\nadj : F ⊣ G\nX : ↑D\n⊢ adj.unit.toNatTrans.app (G.toFunctor.obj X) ≫ G.toFunctor.map (adj.cou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 574,
"column": 42
} | {
"line": 574,
"column": 63
} | {
"line": 574,
"column": 64
} | [
{
"pp": "R : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\nthis✝² : IsArtinianRing (p.ResidueField ⊗[R] S)\nhpSfin : (p.primesOver S).Finite\nn✝ : ℕ\nIH :\n ∀ m < n✝ + 1,\n ∀ {R : Type u} {S : Type (max... | [
"R : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\nthis✝² : IsArtinianRing (p.ResidueField ⊗[R] S)\nhpSfin : (p.primesOver S).Finite\nn✝ : ℕ\nIH :\n ∀ m < n✝ + 1,\n ∀ {R : Type u} {S : Type (max u v)} [inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 583,
"column": 6
} | {
"line": 584,
"column": 47
} | {
"line": 584,
"column": 48
} | [
{
"pp": "R : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\nthis✝⁴ : IsArtinianRing (p.ResidueField ⊗[R] S)\nhpSfin : (p.primesOver S).Finite\nn✝ : ℕ\nIH :\n ∀ m < n✝ + 1,\n ∀ {R : Type u} {S : Type (max... | [
"R : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\nthis✝⁴ : IsArtinianRing (p.ResidueField ⊗[R] S)\nhpSfin : (p.primesOver S).Finite\nn✝ : ℕ\nIH :\n ∀ m < n✝ + 1,\n ∀ {R : Type u} {S : Type (max u v)} [inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 599,
"column": 4
} | {
"line": 599,
"column": 72
} | {
"line": 599,
"column": 73
} | [
{
"pp": "case h.succ\nR : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\nthis✝⁵ : IsArtinianRing (p.ResidueField ⊗[R] S)\nhpSfin : (p.primesOver S).Finite\nn✝ : ℕ\nIH :\n ∀ m < n✝ + 1,\n ∀ {R : Type u} {... | [
"case h.succ\nR : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\nthis✝⁵ : IsArtinianRing (p.ResidueField ⊗[R] S)\nhpSfin : (p.primesOver S).Finite\nn✝ : ℕ\nIH :\n ∀ m < n✝ + 1,\n ∀ {R : Type u} {S : Type (ma... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 594,
"column": 7
} | {
"line": 594,
"column": 48
} | {
"line": 594,
"column": 49
} | [
{
"pp": "B : Type u\ninst✝¹ : Bicategory B\nc d : B\nl₁ l₂ : c ⟶ d\nr₁ r₂ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nα : l₂ ⟶ l₁\ninst✝ : IsIso ((conjugateEquiv adj₁ adj₂) α)\nthis : IsIso ((conjugateEquiv adj₁ adj₂).symm ((conjugateEquiv adj₁ adj₂) α))\n⊢ IsIso α",
"ppTerm": "?m.62",
"assigned": false,
... | [
"B : Type u\ninst✝¹ : Bicategory B\nc d : B\nl₁ l₂ : c ⟶ d\nr₁ r₂ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nα : l₂ ⟶ l₁\ninst✝ : IsIso ((conjugateEquiv adj₁ adj₂) α)\nthis : IsIso ((conjugateEquiv adj₁ adj₂).symm ((conjugateEquiv adj₁ adj₂) α))\n⊢ IsIso α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 604,
"column": 7
} | {
"line": 604,
"column": 48
} | {
"line": 604,
"column": 49
} | [
{
"pp": "B : Type u\ninst✝¹ : Bicategory B\nc d : B\nl₁ l₂ : c ⟶ d\nr₁ r₂ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nα : r₁ ⟶ r₂\ninst✝ : IsIso ((conjugateEquiv adj₁ adj₂).symm α)\nthis : IsIso ((conjugateEquiv adj₁ adj₂) ((conjugateEquiv adj₁ adj₂).symm α))\n⊢ IsIso α",
"ppTerm": "?m.64",
"assigned": fal... | [
"B : Type u\ninst✝¹ : Bicategory B\nc d : B\nl₁ l₂ : c ⟶ d\nr₁ r₂ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nα : r₁ ⟶ r₂\ninst✝ : IsIso ((conjugateEquiv adj₁ adj₂).symm α)\nthis : IsIso ((conjugateEquiv adj₁ adj₂) ((conjugateEquiv adj₁ adj₂).symm α))\n⊢ IsIso α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 632,
"column": 13
} | {
"line": 632,
"column": 24
} | {
"line": 632,
"column": 25
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\nR' : Type u\nw✝⁶ : CommRing R'\nw✝⁵ : Algebra R R'\nw✝⁴ : Etale R R'\nP : Ideal R'\nw✝³ : P.IsPrime\nw✝² : P.LiesOver p\nn : ℕ\ne : Fin (n + 1) → R' ⊗[R] U... | [
"R : Type u\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\nR' : Type u\nw✝⁶ : CommRing R'\nw✝⁵ : Algebra R R'\nw✝⁴ : Etale R R'\nP : Ideal R'\nw✝³ : P.IsPrime\nw✝² : P.LiesOver p\nn : ℕ\ne : Fin (n + 1) → R' ⊗[R] ULift.{u, v} ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.ModuleFinitePresentation | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 28
} | {
"line": 60,
"column": 4
} | [
{
"pp": "case insert\nR : Type u\nS : Type u_1\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Module.Finite R S\na : S\ns : Finset S\nhas : a ∉ s\nS' : Type u\nw✝⁴ : CommRing S'\nw✝³ : Algebra R S'\nw✝² : Module.Finite R S'\nw✝¹ : Free R S'\nw✝ : Algebra.FinitePresentation R S'\nf : S'... | [
"case insert\nR : Type u\nS : Type u_1\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Module.Finite R S\na : S\ns : Finset S\nhas : a ∉ s\nS' : Type u\nw✝⁴ : CommRing S'\nw✝³ : Algebra R S'\nw✝² : Module.Finite R S'\nw✝¹ : Free R S'\nw✝ : Algebra.FinitePresentation R S'\nf : S' →ₐ[R] S\nhs... | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 60
} | {
"line": 134,
"column": 0
} | [
{
"pp": "case hy\nR S : CommRingCat\nf : R ⟶ S\nhf₁✝ : IsFinite (Spec.map f)\nhf₂✝ : Flat (Spec.map f)\nhf₁ : (CommRingCat.Hom.hom (appTop (Spec.map f))).Finite\nhf₂ : (CommRingCat.Hom.hom (appTop (Spec.map f))).Flat\nx✝ : ↥(Spec R)\nthis : f = (ΓSpecIso R).inv ≫ appTop (Spec.map f) ≫ (ΓSpecIso S).hom\n⊢ x✝ = P... | [] | · simp [isoSpec_Spec_hom, SpecMap_ΓSpecIso_hom, ← AlgebraicGeometry.Spec.map_apply,
← Scheme.Hom.comp_apply, toSpecΓ_SpecMap_ΓSpecIso_inv] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.KrullDimension.NonZeroDivisors | {
"line": 55,
"column": 19
} | {
"line": 55,
"column": 36
} | {
"line": 55,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nr : R\nhr : r ∈ R⁰\nhr' : ¬Ideal.span {r} = ⊤\nthis✝¹ : Nonempty ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))\nthis✝ : Nontrivial (R ⧸ Ideal.span {r})\nthis : Nontrivial R\nl : LTSeries ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))\np : Ideal R\nhp : p ∈ ⊥.minimalPrimes... | [
"R : Type u_1\ninst✝ : CommRing R\nr : R\nhr : r ∈ R⁰\nhr' : ¬Ideal.span {r} = ⊤\nthis✝¹ : Nonempty ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))\nthis✝ : Nontrivial (R ⧸ Ideal.span {r})\nthis : Nontrivial R\nl : LTSeries ↑(PrimeSpectrum.zeroLocus ↑(Ideal.span {r}))\np : Ideal R\nhp : p ∈ ⊥.minimalPrimes\nhp' : p ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.KrullDimension.NonZeroDivisors | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 54
} | {
"line": 83,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nσ : Type u_3\ninst✝ : Finite σ\n⊢ ringKrullDim R + ↑(Nat.card σ) ≤ ringKrullDim (MvPolynomial σ R)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CharP.cast_eq_zero",
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"... | [] | induction σ using Finite.induction_empty_option with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.OrderOfVanishing.Basic | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 13
} | {
"line": 206,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nx : R\nhx : IsUnit x\n⊢ ord R x = 0",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommRing R\nx : R\nhx : IsUnit x\n⊢ ord R x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 35
} | {
"line": 39,
"column": 36
} | [
{
"pp": "ι : Type u_2\nR : ι → Type u_1\ninst✝ : (i : ι) → CommRing (R i)\ns : Set ((i : ι) → R i)\nhs : s.Finite\nf : (i : ι) → { x // x ∈ s } →₀ R i\nhf : ∀ (i : ι), ((f i).sum fun i_1 a ↦ a * (⇑(Pi.evalRingHom (fun x ↦ R x) i) ∘ Subtype.val) i_1) = 1\nthis : Fintype ↑s\ni : ι\n⊢ (Finsupp.equivFunOnFinite.sym... | [
"ι : Type u_2\nR : ι → Type u_1\ninst✝ : (i : ι) → CommRing (R i)\ns : Set ((i : ι) → R i)\nhs : s.Finite\nf : (i : ι) → { x // x ∈ s } →₀ R i\nhf : ∀ (i : ι), ((f i).sum fun i_1 a ↦ a * (⇑(Pi.evalRingHom (fun x ↦ R x) i) ∘ Subtype.val) i_1) = 1\nthis : Fintype ↑s\ni : ι\n⊢ ∑ x, (f i) x * ↑x i = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 52,
"column": 84
} | {
"line": 52,
"column": 95
} | {
"line": 52,
"column": 96
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nU : (Spec (CommRingCat.of ((i : ι) → ↑(R i)))).Opens\nV : Set ↥(Spec (CommRingCat.of ((i : ι) → ↑(R i))))\nhV : ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ V\nhV' : IsCompact V\nhVU : V ⊆ ↑U\ns : Set ↑(CommRingCat.of ((i : ι) → ↑(R i)))\nhs : U.carrierᶜ = zeroLocus s\nt : ... | [
"ι : Type u\nR : ι → CommRingCat\nU : (Spec (CommRingCat.of ((i : ι) → ↑(R i)))).Opens\nV : Set ↥(Spec (CommRingCat.of ((i : ι) → ↑(R i))))\nhV : ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ V\nhV' : IsCompact V\nhVU : V ⊆ ↑U\ns : Set ↑(CommRingCat.of ((i : ι) → ↑(R i)))\nhs : U.carrierᶜ = zeroLocus s\nt : Finset ↑s\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 72
} | {
"line": 54,
"column": 73
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nU : (Spec (CommRingCat.of ((i : ι) → ↑(R i)))).Opens\nV : Set ↥(Spec (CommRingCat.of ((i : ι) → ↑(R i))))\nhV : ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ V\nhV' : IsCompact V\nhVU : V ⊆ ↑U\ns : Set ↑(CommRingCat.of ((i : ι) → ↑(R i)))\nhs : U.carrierᶜ = zeroLocus s\nt : ... | [
"ι : Type u\nR : ι → CommRingCat\nU : (Spec (CommRingCat.of ((i : ι) → ↑(R i)))).Opens\nV : Set ↥(Spec (CommRingCat.of ((i : ι) → ↑(R i))))\nhV : ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ V\nhV' : IsCompact V\nhVU : V ⊆ ↑U\ns : Set ↑(CommRingCat.of ((i : ι) → ↑(R i)))\nhs : U.carrierᶜ = zeroLocus s\nt : Set ↑(CommRi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.OrderOfVanishing | {
"line": 130,
"column": 2
} | {
"line": 131,
"column": 44
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case neg\nX : Scheme\ninst✝² : IsIntegral X\ninst✝¹ : IsLocallyNoetherian X\nx : ↥X\nU : X.Opens\ninst✝ : Nonempty ↥↑U\nhxU : x ∈ U\na : ↑Γ(X, U)\nha : a ≠ 0\nf : ↑X.functionField\nhx : coheight x = 1\nhf : ¬f = 0\nthis✝ : a • f ≠ 0\nalgInst✝ : Algebra ↑(X.presheaf.obj (Opposite.op U)) ↑(X.presheaf.sta... | [
"case neg\nX : Scheme\ninst✝² : IsIntegral X\ninst✝¹ : IsLocallyNoetherian X\nx : ↥X\nU : X.Opens\ninst✝ : Nonempty ↥↑U\nhxU : x ∈ U\na : ↑Γ(X, U)\nha : a ≠ 0\nf : ↑X.functionField\nhx : coheight x = 1\nhf : ¬f = 0\nthis✝¹ : a • f ≠ 0\nalgInst✝ : Algebra ↑(X.presheaf.obj (Opposite.op U)) ↑(X.presheaf.stalk x) :=\n ... | have : IsScalarTower ↑Γ(X, U) ↑(X.presheaf.stalk x) ↑X.functionField :=
functionField_isScalarTower X U ⟨x, hxU⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 71
} | {
"line": 64,
"column": 72
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nU : (Spec (CommRingCat.of ((i : ι) → ↑(R i)))).Opens\nV : Set ↥(Spec (CommRingCat.of ((i : ι) → ↑(R i))))\nhV : ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ V\nhV' : IsCompact V\nhVU : V ⊆ ↑U\ns : Set ↑(CommRingCat.of ((i : ι) → ↑(R i)))\nhs : U.carrierᶜ = zeroLocus s\nt : ... | [
"ι : Type u\nR : ι → CommRingCat\nU : (Spec (CommRingCat.of ((i : ι) → ↑(R i)))).Opens\nV : Set ↥(Spec (CommRingCat.of ((i : ι) → ↑(R i))))\nhV : ↑(Scheme.Hom.opensRange (sigmaSpec R)) ⊆ V\nhV' : IsCompact V\nhVU : V ⊆ ↑U\ns : Set ↑(CommRingCat.of ((i : ι) → ↑(R i)))\nhs : U.carrierᶜ = zeroLocus s\nt : Set ↑(CommRi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 71
} | {
"line": 73,
"column": 4
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nI : Ideal ((i : ι) → ↑(R i))\nf : (∐ fun i ↦ Spec (R i)) ⟶ Spec (CommRingCat.of (((i : ι) → ↑(R i)) ⧸ I))\nhf : f ≫ Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk I)) = sigmaSpec R\nx : (i : ι) → ↑(R i)\nhx : x ∈ I\ni : ι\n⊢ x i = 0 i",
"ppTerm": "?m.51",
"assig... | [
"ι : Type u\nR : ι → CommRingCat\nI : Ideal ((i : ι) → ↑(R i))\nf : (∐ fun i ↦ Spec (R i)) ⟶ Spec (CommRingCat.of (((i : ι) → ↑(R i)) ⧸ I))\nhf : f ≫ Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk I)) = sigmaSpec R\nx : (i : ι) → ↑(R i)\nhx : x ∈ I\ni : ι\n⊢ x i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 49
} | {
"line": 134,
"column": 50
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nideal_gc : GaloisConnection Ideal.span SetLike.coe\n⊢ GaloisConnection (fun s ↦ zeroLocus 𝒜 s) fun t ↦ ↑(vanishingIdeal t)",
"ppTerm": "?m.52",
"assigned": ... | [
"A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nideal_gc : GaloisConnection Ideal.span SetLike.coe\n⊢ GaloisConnection (fun s ↦ zeroLocus 𝒜 s) fun t ↦ ↑(vanishingIdeal t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 260,
"column": 22
} | {
"line": 260,
"column": 33
} | {
"line": 260,
"column": 34
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf g : A\nx : ProjectiveSpectrum 𝒜\n⊢ x ∈ zeroLocus 𝒜 {f * g} ↔ x ∈ zeroLocus 𝒜 {f} ∪ zeroLocus 𝒜 {g}",
"ppTerm": "?m.49",
"assigned": true,
"usedCons... | [
"A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf g : A\nx : ProjectiveSpectrum 𝒜\n⊢ f * g ∈ x.asHomogeneousIdeal ↔ f ∈ x.asHomogeneousIdeal ∨ g ∈ x.asHomogeneousIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 265,
"column": 22
} | {
"line": 265,
"column": 33
} | {
"line": 265,
"column": 34
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nn : ℕ\nhn : 0 < n\nx : ProjectiveSpectrum 𝒜\n⊢ x ∈ zeroLocus 𝒜 {f ^ n} ↔ x ∈ zeroLocus 𝒜 {f}",
"ppTerm": "?m.42",
"assigned": true,
"usedConsta... | [
"A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nn : ℕ\nhn : 0 < n\nx : ProjectiveSpectrum 𝒜\n⊢ f ^ n ∈ x.asHomogeneousIdeal ↔ f ∈ x.asHomogeneousIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 373,
"column": 35
} | {
"line": 373,
"column": 46
} | {
"line": 373,
"column": 47
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nn : ℕ\nhn : 0 < n\n⊢ ↑(basicOpen 𝒜 (f ^ n)) = ↑(basicOpen 𝒜 f)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nn : ℕ\nhn : 0 < n\n⊢ zeroLocus 𝒜 {f ^ n} = zeroLocus 𝒜 {f}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.FiniteType | {
"line": 37,
"column": 38
} | {
"line": 37,
"column": 49
} | {
"line": 37,
"column": 50
} | [
{
"pp": "S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\nF : Finset S\nhF : Algebra.adjoin ↥(𝒜 0) ↑F = ⊤\nι₀ : Type (... | [
"S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\nF : Finset S\nhF : Algebra.adjoin ↥(𝒜 0) ↑F = ⊤\nι₀ : Type (max u_1 u_3)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.FiniteType | {
"line": 41,
"column": 53
} | {
"line": 41,
"column": 64
} | {
"line": 41,
"column": 65
} | [
{
"pp": "S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\nF : Finset S\nhF : Algebra.adjoin ↥(𝒜 0) ↑F = ⊤\nι₀ : Type (... | [
"S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\nF : Finset S\nhF : Algebra.adjoin ↥(𝒜 0) ↑F = ⊤\nι₀ : Type (max u_1 u_3)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.FiniteType | {
"line": 47,
"column": 37
} | {
"line": 47,
"column": 48
} | {
"line": 47,
"column": 49
} | [
{
"pp": "S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\ns : Finset S\nh₁ : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤\nn : S → ι\n... | [
"S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\ns : Finset S\nh₁ : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤\nn : S → ι\nhn : ∀ i ∈ s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.FiniteType | {
"line": 53,
"column": 36
} | {
"line": 53,
"column": 53
} | {
"line": 53,
"column": 54
} | [
{
"pp": "S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\ns : Finset S\nh₁ : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤\nn : S → ι\n... | [
"S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\ns : Finset S\nh₁ : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤\nn : S → ι\nhn : ∀ i ∈ s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 271,
"column": 12
} | {
"line": 271,
"column": 78
} | {
"line": 272,
"column": 14
} | [
{
"pp": "R : CommRingCat\nM✝ : ModuleCat ↑R\nM : (Spec (CommRingCat.of ↑R)).Modules\nf g : (↑R)ᵒᵖ\ni : f ⟶ g\nN : ModuleCat ↑(CommRingCat.of ↑R) := (modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)\n⊢ ∃ n, unop f ∣ unop g ^ n",
"ppTerm": "?m.232",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"R : CommRingCat\nM✝ : ModuleCat ↑R\nM : (Spec (CommRingCat.of ↑R)).Modules\nf g : (↑R)ᵒᵖ\ni : f ⟶ g\nN : ModuleCat ↑(CommRingCat.of ↑R) := (modulesSpecToSheaf.obj M).presheaf.obj (op ⊤)\n⊢ ∃ n, unop f ∣ unop g ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 251,
"column": 23
} | {
"line": 251,
"column": 61
} | {
"line": 251,
"column": 62
} | [
{
"pp": "case succ\nι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\nx : Submonoid A\ninst✝² : AddCommMonoid ι\ninst✝¹ : DecidableEq ι\ninst✝ : GradedRing 𝒜\nc : NumDenSameDeg 𝒜 x\nn : ℕ\nih : ↑(GradedMonoid.GMonoid.gnpow n c.den)... | [
"case succ\nι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\nx : Submonoid A\ninst✝² : AddCommMonoid ι\ninst✝¹ : DecidableEq ι\ninst✝ : GradedRing 𝒜\nc : NumDenSameDeg 𝒜 x\nn : ℕ\nih : ↑(GradedMonoid.GMonoid.gnpow n c.den) ∈ x\n⊢ ↑c.d... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 447,
"column": 5
} | {
"line": 447,
"column": 76
} | {
"line": 447,
"column": 76
} | [
{
"pp": "R : CommRingCat\nM : ModuleCat ↑R\ns : Set ↑M\nhs : Submodule.span (↑R) s = ⊤\nt : Set (↑s →₀ ↑R)\nht : Submodule.span (↑R) t = (Finsupp.linearCombination (↑R) Subtype.val).ker\nH₁ :\n Function.Exact ⇑(ConcreteCategory.hom (ModuleCat.ofHom (Finsupp.linearCombination (↑R) Subtype.val)))\n ⇑(Concrete... | [] | by simp [← LinearMap.range_eq_top, Finsupp.range_linearCombination, hs] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 426,
"column": 42
} | {
"line": 426,
"column": 58
} | {
"line": 426,
"column": 59
} | [
{
"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\nX : Scheme\nf✝ : A →+* ↑Γ(X, ⊤)\nx✝ x' : ↑Γ(X, ⊤)\nt t' : A\nd d' : ℕ\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nx : A\nhx : x ... | [
"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\nX : Scheme\nf✝ : A →+* ↑Γ(X, ⊤)\nx✝ x' : ↑Γ(X, ⊤)\nt t' : A\nd d' : ℕ\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nx : A\nhx : x ∈ (Homogeneo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 526,
"column": 4
} | {
"line": 526,
"column": 15
} | {
"line": 526,
"column": 16
} | [
{
"pp": "case refine_2\nR : CommRingCat\nM : (Spec R).Modules\nh : IsLocalizing (modulesSpecToSheaf.obj M)\n⊢ IsIso ((modulesSpecToSheaf.obj M).obj.map (homOfLE ⋯).op)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"CategoryTheory.... | [
"case refine_2\nR : CommRingCat\nM : (Spec R).Modules\nh : IsLocalizing (modulesSpecToSheaf.obj M)\n⊢ IsIso (𝟙 ((modulesSpecToSheaf.obj M).obj.obj (op ⊤)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 449,
"column": 4
} | {
"line": 449,
"column": 70
} | {
"line": 450,
"column": 6
} | [
{
"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\nX : Scheme\nf✝ : A →+* ↑Γ(X, ⊤)\nx✝ x' : ↑Γ(X, ⊤)\nt t' : A\nd d' : ℕ\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nx y : (openCov... | [
"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\nX : Scheme\nf✝ : A →+* ↑Γ(X, ⊤)\nx✝ x' : ↑Γ(X, ⊤)\nt t' : A\nd d' : ℕ\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nx y : (openCoverOfMapIrrel... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 451,
"column": 4
} | {
"line": 451,
"column": 70
} | {
"line": 452,
"column": 6
} | [
{
"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\nX : Scheme\nf✝ : A →+* ↑Γ(X, ⊤)\nx✝ x' : ↑Γ(X, ⊤)\nt t' : A\nd d' : ℕ\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nx y : (openCov... | [
"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\nX : Scheme\nf✝ : A →+* ↑Γ(X, ⊤)\nx✝ x' : ↑Γ(X, ⊤)\nt t' : A\nd d' : ℕ\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nx y : (openCoverOfMapIrrel... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 593,
"column": 4
} | {
"line": 594,
"column": 11
} | {
"line": 594,
"column": 12
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubgroupClass σ A\n𝒜 : ι → σ\nx : Submonoid A\ninst✝³ : AddCommMonoid ι\ninst✝² : DecidableEq ι\ninst✝¹ : GradedRing 𝒜\n𝔭 : Ideal A\ninst✝ : 𝔭.IsPrime\na : AtPrime 𝒜 𝔭\n⊢ IsUnit a ∨ IsUnit (1 - a)",
... | [
"ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : SetLike σ A\ninst✝⁴ : AddSubgroupClass σ A\n𝒜 : ι → σ\nx : Submonoid A\ninst✝³ : AddCommMonoid ι\ninst✝² : DecidableEq ι\ninst✝¹ : GradedRing 𝒜\n𝔭 : Ideal A\ninst✝ : 𝔭.IsPrime\na : AtPrime 𝒜 𝔭\n⊢ IsUnit (val a) ∨ IsUnit (1 - val a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 468,
"column": 47
} | {
"line": 468,
"column": 80
} | {
"line": 469,
"column": 6
} | [
{
"pp": "case a\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\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nr : A\nn : ℕ\nhn : 0 < n\nhr : r ∈ 𝒜 n\ni : (openCoverOfMapIrrel... | [
"case a\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\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nr : A\nn : ℕ\nhn : 0 < n\nhr : r ∈ 𝒜 n\ni : (openCoverOfMapIrrelevantEqTop �... | ← TopologicalSpace.Opens.map_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 470,
"column": 26
} | {
"line": 470,
"column": 59
} | {
"line": 470,
"column": 60
} | [
{
"pp": "case a\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\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nr : A\nn : ℕ\nhn : 0 < n\nhr : r ∈ 𝒜 n\ni : (openCoverOfMapIrrel... | [
"case a\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\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nr : A\nn : ℕ\nhn : 0 < n\nhr : r ∈ 𝒜 n\ni : (openCoverOfMapIrrelevantEqTop �... | ← TopologicalSpace.Opens.map_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 485,
"column": 6
} | {
"line": 485,
"column": 49
} | {
"line": 485,
"column": 50
} | [
{
"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\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nr : A\nn : ℕ\nhn : 0 < n\nhr : r ∈ 𝒜 n\nx : ↥X\nhx : x ∈ X.basicOpen (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\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\nr : A\nn : ℕ\nhn : 0 < n\nhr : r ∈ 𝒜 n\nx : ↥X\nhx : x ∈ X.basicOpen (f r)\nI : (ope... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 513,
"column": 76
} | {
"line": 525,
"column": 70
} | {
"line": 527,
"column": 0
} | [
{
"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\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nhf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤\n⊢ fromOfGlobalSections 𝒜 f hf ≫ toSpecZero 𝒜 = X.toSpecΓ ≫ Spec.map (Co... | [] | by
refine (openCoverOfMapIrrelevantEqTop 𝒜 f hf).hom_ext _ _ fun x ↦ ?_
simp only [fromOfGlobalSections, toBasicOpenOfGlobalSections, CommRingCat.ofHom_comp,
Category.assoc, Scheme.Cover.ι_glueMorphisms_assoc, basicOpenIsoSpec_inv_ι_assoc,
awayι_toSpecZero, Iso.inv_comp_eq]
simp only [openCoverOfMapIrrel... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 702,
"column": 10
} | {
"line": 703,
"column": 40
} | {
"line": 703,
"column": 41
} | [
{
"pp": "case h\nR : CommRingCat\nM : (Spec R).Modules\nV : (Spec R).Opens\nι : Type u_1\ninst✝ : Finite ι\ng : ι → ↑R\nhg : V = ⨆ i, basicOpen (g i)\nh₁ : ∀ (i : ι), Aux M (basicOpen (g i))\nh₂ : ∀ (i j : ι), Aux M (basicOpen (g i * g j))\nhgle : ∀ (i : ι), basicOpen (g i) ≤ V\nhug : ∀ (i : ι) (m : ℕ), IsUnit ... | [
"case h\nR : CommRingCat\nM : (Spec R).Modules\nV : (Spec R).Opens\nι : Type u_1\ninst✝ : Finite ι\ng : ι → ↑R\nhg : V = ⨆ i, basicOpen (g i)\nh₁ : ∀ (i : ι), Aux M (basicOpen (g i))\nh₂ : ∀ (i j : ι), Aux M (basicOpen (g i * g j))\nhgle : ∀ (i : ι), basicOpen (g i) ≤ V\nhug : ∀ (i : ι) (m : ℕ), IsUnit ((algebraMap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.LocalRing.LocalSubring | {
"line": 114,
"column": 4
} | {
"line": 115,
"column": 73
} | {
"line": 115,
"column": 74
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nK : Type u_3\ninst✝¹ : Field K\nA : Subring K\nP : Ideal ↥A\ninst✝ : P.IsPrime\nx : ↥A\ns : ↥P.primeCompl\ny : ↥A\nt : ↥P.primeCompl\ne :\n (IsLocalization.liftAlgHom ⋯) (IsLocalization.mk' (Localization.AtPrime P) x s) =\n (IsLo... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nK : Type u_3\ninst✝¹ : Field K\nA : Subring K\nP : Ideal ↥A\ninst✝ : P.IsPrime\nx : ↥A\ns : ↥P.primeCompl\ny : ↥A\nt : ↥P.primeCompl\ne :\n (IsLocalization.liftAlgHom ⋯) (IsLocalization.mk' (Localization.AtPrime P) x s) =\n (IsLocalization.l... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 1044,
"column": 40
} | {
"line": 1044,
"column": 78
} | {
"line": 1044,
"column": 79
} | [
{
"pp": "A : Type u_2\nσ : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\nf : A\nd : ℕ\nhf : f ∈ 𝒜 d\nι' : Type u_4\ninst✝ : Fintype ι'\nv : ι' → A\nhx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤\ndv : ι' → ℕ\nhxd : ∀ (i : ι'), v i ∈ 𝒜 ... | [
"A : Type u_2\nσ : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\nf : A\nd : ℕ\nhf : f ∈ 𝒜 d\nι' : Type u_4\ninst✝ : Fintype ι'\nv : ι' → A\nhx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤\ndv : ι' → ℕ\nhxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)\nhxd'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 414,
"column": 8
} | {
"line": 414,
"column": 52
} | {
"line": 414,
"column": 53
} | [
{
"pp": "case neg\nA : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\nx : A\nhx : x ∈ carrier f_deg q\nn : ℕ\na : A\nha : a ∈ 𝒜 n\ni : ℕ\nprodu... | [
"case neg\nA : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\nx : A\nhx : x ∈ carrier f_deg q\nn : ℕ\na : A\nha : a ∈ 𝒜 n\ni : ℕ\nproduct : A⁰_ f :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 431,
"column": 37
} | {
"line": 431,
"column": 93
} | {
"line": 431,
"column": 94
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\ni : ℕ\na : A\nha : a ∈ asIdeal f_deg hm q\nj : ℕ\nh : i = j\n⊢ HomogeneousLocalizati... | [
"A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\ni : ℕ\na : A\nha : a ∈ asIdeal f_deg hm q\nj : ℕ\nh : i = j\n⊢ HomogeneousLocalization.mk { deg ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 75,
"column": 20
} | {
"line": 75,
"column": 66
} | {
"line": 75,
"column": 66
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\nx : K\nhx : IsIntegral (↥R.toSubring) x\nS : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x]\nthis : Algebra.IsIntegral ↥R.toSubring ↥S\nQ : Ideal ↥S.toSubring\nhQ : Q.IsMaximal\ne : Ideal.comap (algebraMap ↥R.toSubring ↥S) Q = maximalId... | [
"K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\nx : K\nhx : IsIntegral (↥R.toSubring) x\nS : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x]\nthis : Algebra.IsIntegral ↥R.toSubring ↥S\nQ : Ideal ↥S.toSubring\nhQ : Q.IsMaximal\ne : Ideal.comap (algebraMap ↥R.toSubring ↥S) Q = maximalIdeal ↥R.toSub... | ← IsLocalization.AtPrime.map_eq_maximalIdeal Q | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 433,
"column": 4
} | {
"line": 434,
"column": 38
} | {
"line": 434,
"column": 39
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\ni : ℕ\na : A\nha : a ∈ asIdeal f_deg hm q\nj : ℕ\nh : ¬i = j\n⊢ HomogeneousLocalizat... | [
"A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\ni : ℕ\na : A\nha : a ∈ asIdeal f_deg hm q\nj : ℕ\nh : ¬i = j\n⊢ HomogeneousLocalization.mk { deg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 1080,
"column": 40
} | {
"line": 1080,
"column": 55
} | {
"line": 1080,
"column": 56
} | [
{
"pp": "A : Type u_2\nσ : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\nf : A\nd : ℕ\nhf : f ∈ 𝒜 d\nι' : Type u_4\ninst✝ : Fintype ι'\nv : ι' → A\nhx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤\ndv : ι' → ℕ\nhxd : ∀ (i : ι'), v i ∈ 𝒜 ... | [
"A : Type u_2\nσ : Type u_3\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\nf : A\nd : ℕ\nhf : f ∈ 𝒜 d\nι' : Type u_4\ninst✝ : Fintype ι'\nv : ι' → A\nhx : Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤\ndv : ι' → ℕ\nhxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)\ns : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 91,
"column": 22
} | {
"line": 91,
"column": 79
} | {
"line": 91,
"column": 80
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : ValuationSubring K\nS : LocalSubring K\nhS : R.toLocalSubring ≤ S\nx : K\nhx : x ∈ S.toSubring\nh : x⁻¹ ∈ R.carrier\nh' : x ∉ R.toLocalSubring.toSubring\nhx0 : x ≠ 0\nthis : IsUnit ((Subring.inclusion ⋯) ⟨x⁻¹, h⟩)\nx' : ↥R.toLocalSubring.toSubring\nhx' : ⟨x⁻¹, h⟩ * x'... | [
"K : Type u_3\ninst✝ : Field K\nR : ValuationSubring K\nS : LocalSubring K\nhS : R.toLocalSubring ≤ S\nx : K\nhx : x ∈ S.toSubring\nh : x⁻¹ ∈ R.carrier\nh' : x ∉ R.toLocalSubring.toSubring\nhx0 : x ≠ 0\nthis : IsUnit ((Subring.inclusion ⋯) ⟨x⁻¹, h⟩)\nx' : ↥R.toLocalSubring.toSubring\nhx' : ⟨x⁻¹, h⟩ * x' = 1\n⊢ ↑x' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 108,
"column": 58
} | {
"line": 108,
"column": 69
} | {
"line": 108,
"column": 70
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\nx : K\nhx : x ∉ R.toSubring\nhx0 : x ≠ 0\nthis✝ : Invertible x := invertibleOfNonzero hx0\nS : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x]\nthis : R.toSubring < S.toSubring\np : Polynomial ↥R.toSubring\nhp : p.leadingCoeff - 1 ∈ maxi... | [
"K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\nx : K\nhx : x ∉ R.toSubring\nhx0 : x ≠ 0\nthis✝ : Invertible x := invertibleOfNonzero hx0\nS : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x]\nthis : R.toSubring < S.toSubring\np : Polynomial ↥R.toSubring\nhp : p.leadingCoeff - 1 ∈ maximalIdeal ↥R.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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