module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 100
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nht : φ.IsIntegralElem t\nhp : φ p * t ∈ φ.range\na : R := p.leadingCoeff\nR' : Type u_1 := Localization.Away a\nS' : Type u_2 := Localization.Away ((algebraMap R S) a)\nthis✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 71
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nht : φ.IsIntegralElem t\nhp : φ p * t ∈ φ.range\na : R := p.leadingCoeff\nR' : Type u_1 := Localization.Away a\nS' : Type u_2 := Localization.Away ((algebraMap R S) a)\nthis✝¹ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 239,
"column": 39
} | {
"line": 239,
"column": 82
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\nhp : φ p * t ∈ conductor R (φ X)\nalgInst✝ : Algebra R[X] S := φ.toAlgebra\nalgebraizeInst✝ : Module.Finite R[X] S\nthis : IsScalar... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 254,
"column": 25
} | {
"line": 254,
"column": 61
} | [
{
"pp": "case add\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\nhp : φ p * t ∈ conductor R (φ X)\nalgInst✝ : Algebra R[X] S := φ.toAlgebra\nalgebraizeInst✝ : Module.Finite R[X] S\nthis ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 273,
"column": 8
} | {
"line": 273,
"column": 63
} | [
{
"pp": "case inr.h.zero.inr\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\ni : ℕ\nthis : ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.leadingCoeff • t ∈ (conductor R (φ X)).radical\np ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 276,
"column": 8
} | {
"line": 276,
"column": 25
} | [
{
"pp": "case inr.h.succ.inl\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\ni : ℕ\nthis : ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.leadingCoeff • t ∈ (conductor R (φ X)).radical\np ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 281,
"column": 6
} | {
"line": 281,
"column": 40
} | [
{
"pp": "case inr.h.succ.inr\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\ni : ℕ\nthis✝ : ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.leadingCoeff • t ∈ (conductor R (φ X)).radical\np... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 285,
"column": 8
} | {
"line": 285,
"column": 29
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np✝ : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\np : R[X]\ni : ℕ\nhi : i = p.natDegree\nn : ℕ\nhn : (φ p * t) ^ n ∈ conductor R (φ X)\n⊢ φ (p ^ n) * t ^ n ∈ conductor R (φ X)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 302,
"column": 2
} | {
"line": 303,
"column": 27
} | [
{
"pp": "case a\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nhRS : integralClosure R S = ⊥\nx : S\nhx : (aeval x).Finite\nu : S\np : R[X]\ne : (aeval x) p * u ∈ (conductor R x).radical\ni : ℕ\n⊢ (map (algebraMap R (S ⧸ (conductor R x).radical)) p * C ((Ideal.Quotie... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 304,
"column": 10
} | {
"line": 304,
"column": 21
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nhRS : integralClosure R S = ⊥\nx : S\nhx : (aeval x).Finite\nu : S\np : R[X]\ne : (aeval x) p * u ∈ (conductor R x).radical\ni : ℕ\n⊢ (aeval x) p * u ∈ (conductor R ((aeval x) X)).radical",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 334,
"column": 17
} | {
"line": 334,
"column": 40
} | [
{
"pp": "case right\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm : ℕ\nhf : f.natDegree ≤ m\nn : ℕ\ni : Fin (n + 1)\nhb : Fin.natAdd (m + 1) i ≠ Fin.castAdd (n + 1) (Fin.last m)\n⊢ f.sylvester g (m + 1) (n + 1) (Fin.last (m + 1 + n)) (Fin.natAdd (m + 1) i) = 0",
"usedConstants": [
"Eq.mpr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 368,
"column": 2
} | {
"line": 368,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : R[X]\nm n : ℕ\n⊢ resultant 1 g m n = (-1) ^ (m * n) * g.coeff n ^ m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\nm n : ℕ\n⊢ f.resultant 1 m n = f.coeff m ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 435,
"column": 4
} | {
"line": 435,
"column": 15
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : IsReduced S\nx✝ : S\nhx' : (aeval x✝).Finite\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R P\nleft✝ : IsDomain S\nright✝ : FaithfulSMul R S\nthis✝ : IsDomain R\nK : Type u_1 := Fractio... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 418,
"column": 27
} | {
"line": 418,
"column": 51
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 444,
"column": 30
} | {
"line": 444,
"column": 42
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : IsReduced S\nx : S\nhx' : (aeval x).Finite\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R P\nleft✝ : IsDomain S\nright✝ : FaithfulSMul R S\nthis✝ : IsDomain R\nK : Type u_1 := FractionR... | simp [g, S'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 481,
"column": 62
} | {
"line": 481,
"column": 95
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 431,
"column": 39
} | {
"line": 431,
"column": 63
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 444,
"column": 46
} | {
"line": 444,
"column": 83
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 499,
"column": 8
} | {
"line": 499,
"column": 38
} | [
{
"pp": "case h.e'_5\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nx : S\nhx : R[x] = ⊤\nH : integralClosure R S = ⊥\nH₀ : Functi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 504,
"column": 4
} | {
"line": 504,
"column": 34
} | [
{
"pp": "case neg\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nx : S\nhx : R[x] = ⊤\nH : integralClosure R S = ⊥\nH₀ : Function.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 524,
"column": 71
} | {
"line": 524,
"column": 82
} | [
{
"pp": "R✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : R[X] →ₐ[R] S\nhf : f.Finite\nH : integralClosure R S = ⊥\nJ : Ideal S :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 538,
"column": 45
} | {
"line": 538,
"column": 90
} | [
{
"pp": "R✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : R[X] →ₐ[R] S\nH : integralClosure R S = ⊥\nhf : ¬conductor R (f X) ≤ p\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 550,
"column": 19
} | {
"line": 550,
"column": 30
} | [
{
"pp": "R✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : R[X] →ₐ[R] S\nH : integralClosure R S = ⊥\nhf : ¬conductor R (f X) ≤ p\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 551,
"column": 4
} | {
"line": 551,
"column": 33
} | [
{
"pp": "case refine_2\nR✝ S✝ : Type u\ninst✝⁷ : CommRing R✝\ninst✝⁶ : CommRing S✝\ninst✝⁵ : Algebra R✝ S✝\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : R[X] →ₐ[R] S\nH : integralClosure R S = ⊥\nhf : ¬conduct... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 487,
"column": 4
} | {
"line": 487,
"column": 77
} | [
{
"pp": "case inr.a\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nn : ℕ\nhg : g.natDegree ≤ n\nhf : f.Splits\nthis✝ :\n ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] (f g : R[X]) (n : ℕ),\n g.natDegree ≤ n →\n f.Splits →\n IsField R → f.resultant g f.natDegree ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 495,
"column": 4
} | {
"line": 496,
"column": 27
} | [
{
"pp": "case neg.inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nn : ℕ\nhg : g.natDegree ≤ n\nhf : f.Splits\nhR : IsField R\nhf0 : ¬f = 0\nthis✝ :\n ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] (f g : R[X]) (n : ℕ),\n g.natDegree ≤ n →\n f.Splits →\n IsFiel... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 509,
"column": 4
} | {
"line": 509,
"column": 93
} | [
{
"pp": "case neg.inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nn : ℕ\nhg : g.natDegree ≤ n\nhf : f.Splits\nhR : IsField R\nhfm : f.Monic\nhg0 : ¬g = 0\nthis✝ :\n ∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] (f g : R[X]) (n : ℕ),\n g.natDegree ≤ n →\n f.Splits →\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 537,
"column": 98
} | {
"line": 550,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np : R[X]\nP : {R : Type u} → [inst : CommRing R] → R[X] → Prop\nSplits : ∀ (R : Type u) [inst : Field R] (p : R[X]), p.Splits → P p\ninjective :\n ∀ (R S : Type u) [inst : CommRing R] [inst_1 : CommRing S] (φ : R →+* S),\n Function.Injective ⇑φ → ∀ (p : R[X]), P (map... | by
wlog hR : IsDomain R generalizing R
· exact surjective _ _ (MvPolynomial.eval₂Hom (algebraMap ℤ R) id)
(fun x ↦ ⟨.X x, by simp [MvPolynomial.eval₂Hom]⟩) p
(fun _ ↦ this _ inferInstance)
wlog hR : IsField R generalizing R
· exact injective _ _ _ (FaithfulSMul.algebraMap_injective R (FractionRing R... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 569,
"column": 4
} | {
"line": 569,
"column": 85
} | [
{
"pp": "case injective.a\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR SatisfiesM : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing SatisfiesM\nφ : R →+* SatisfiesM\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g₁ g₂ : SatisfiesM[X]),\n (map φ f).resultant (g₁ * g₂) (map φ f).natDegree (g₁.natDegree + g₂.natD... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 596,
"column": 4
} | {
"line": 596,
"column": 93
} | [
{
"pp": "case injective.a\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH : (map φ f).resultant (map φ f) = 0 ^ (map φ f).natDegree\n⊢ φ (f.resultant f) = φ (0 ^ f.natDegree)",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 615,
"column": 69
} | {
"line": 615,
"column": 80
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nH : IsCoprime f g\nhf : f ≠ 0\nhg : g ≠ 0\nb : R[X]\ne : 0 * f + b * g = 1\n⊢ b * ?m.152 = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 618,
"column": 69
} | {
"line": 618,
"column": 80
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nH : IsCoprime f g\nhf : f ≠ 0\nhg : g ≠ 0\na : R[X]\nha : a ≠ 0\ne : a * f + 0 * g = 1\n⊢ a * ?m.222 = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 636,
"column": 2
} | {
"line": 636,
"column": 21
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nH : IsCoprime f g\nhf : f ≠ 0\ne : f.resultant g = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 648,
"column": 51
} | {
"line": 648,
"column": 86
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nι : Type u_3\nf : ι → R[X]\ng : R[X]\nn : ℕ\nhn : g.natDegree ≤ n\na : ι\ns : Finset ι\nhas : a ∉ s\nIH :\n ∏ i ∈ s, (f i).leadingCoeff ≠ 0 →\n (∏ i ∈ s, f i).resultant g (∏ i ∈ s, f i).natDegree n = ∏ i ∈ s, (f i).resultant g (f i).natDegree n\nhf : ∏ i ∈ insert a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 710,
"column": 35
} | {
"line": 710,
"column": 46
} | [
{
"pp": "R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g : S[X]) (r : S),\n map φ f ≠ 0 →\n g ≠ 0 →\n ((map φ f).scaleRoots r).resultant (g.scaleRoots r) (map φ f).natDegree g.natDegree =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 711,
"column": 10
} | {
"line": 711,
"column": 21
} | [
{
"pp": "R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g : S[X]) (r : S),\n map φ f ≠ 0 →\n g ≠ 0 →\n ((map φ f).scaleRoots r).resultant (g.scaleRoots r) (map φ f).natDegree g.natDegree =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 714,
"column": 6
} | {
"line": 714,
"column": 52
} | [
{
"pp": "case inr.inr.injective.a\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g : S[X]) (r : S),\n map φ f ≠ 0 →\n g ≠ 0 →\n ((map φ f).scaleRoots r).resultant (g.scaleRoots r) (map φ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 626,
"column": 6
} | {
"line": 627,
"column": 47
} | [
{
"pp": "n : ℕ\nIH :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]\n [WeaklyQuasiFiniteAt R p] (f : MvPolynomial (Fin n) R →ₐ[R] S), f.Finite → ZariskisMainProperty R p\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 649,
"column": 15
} | {
"line": 650,
"column": 9
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FiniteType R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] S\nhf : Function.Surjective ⇑f\nthis : Small.{u, v} S\nr : Shrink.{u, v} S\nhr : r ∉... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 13
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU : ↑Y.affineOpens\n⊢ fromNormalization f ⁻¹ᵁ ↑U = opensRange ((normalizationOpenCover f).f U)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 201,
"column": 10
} | {
"line": 201,
"column": 21
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU : ↑Y.affineOpens\n⊢ Set.range ⇑(fromNormalization f ⁻¹ᵁ ↑U).ι = Set.range ⇑((normalizationOpenCover f).f U)",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.fromNormalization",
"Eq.mpr",
"AlgebraicGeom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 220,
"column": 8
} | {
"line": 220,
"column": 19
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU : Y.Opens\nhU : IsAffineOpen U\n⊢ (normalizationOpenCover f).f ⟨U, hU⟩ ''ᵁ ⊤ = fromNormalization f ⁻¹ᵁ U",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"AlgebraicGeometry.Scheme.Hom.fromNorm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 695,
"column": 2
} | {
"line": 695,
"column": 66
} | [
{
"pp": "case refine_3\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FiniteType R S\np : Ideal S\nH : ZariskisMainProperty R p\ns : Finset S\nhs : adjoin R ↑s = ⊤\nr : S\nhrp : r ∉ p\nhr : IsIntegral R r\nm : S → ℕ\nhm : ∀ (x : S), IsIntegral R (r ^ m x * x... | obtain ⟨y, hy : Localization.awayMap _ _ _ = _⟩ := this ⟨x, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 720,
"column": 6
} | {
"line": 720,
"column": 44
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FiniteType R S\np : Ideal S\ninst✝ : p.IsPrime\nH✝ : ZariskisMainProperty R p\nS' : Subalgebra R S\nhS' : (Subalgebra.toSubmodule S').FG\nr : ↥S'\nhrp : ↑r ∉ p\nH : Function.Bijective ⇑(Localization.awa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 724,
"column": 30
} | {
"line": 724,
"column": 41
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FiniteType R S\np : Ideal S\ninst✝ : p.IsPrime\nH✝ : ZariskisMainProperty R p\nS' : Subalgebra R S\nhS' : (Subalgebra.toSubmodule S').FG\nr : ↥S'\nhrp : ↑r ∉ p\nH : Function.Bijective ⇑(Localization.awa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 782,
"column": 4
} | {
"line": 782,
"column": 50
} | [
{
"pp": "case injective.a\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH : ∀ (g : S[X]) (r : S), ((taylor r) (map φ f)).resultant ((taylor r) g) = (map φ f).resultant g\ng : R[X]\nr : R\nthis : (map φ ((taylor ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 810,
"column": 4
} | {
"line": 810,
"column": 32
} | [
{
"pp": "m n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\n⊢ f * ↑(⟨p, hp⟩, ⟨q, hq⟩).2 + g * ↑(⟨p, hp⟩, ⟨q, hq⟩).1 ∈ R[X]_(m + n)",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"P... | rw [Polynomial.mem_degreeLT] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 814,
"column": 43
} | {
"line": 814,
"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\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\nhf' : ¬f = 0\n⊢ ?m.137 < ?m.139",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 817,
"column": 43
} | {
"line": 817,
"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\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",
"usedConstants": [
"WithBot.instPreorder",
"Polynomial.degreeLT",
"Sub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 845,
"column": 2
} | {
"line": 845,
"column": 40
} | [
{
"pp": "case a\nm 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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 15
} | [
{
"pp": "case h.e'_3\nX Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\ninst✝ : IsIntegral X\n⊢ ⊤ = closure (⇑(toNormalization f) '' Set.univ)",
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 876,
"column": 35
} | {
"line": 877,
"column": 68
} | [
{
"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)",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.degreeLT",
"Nat.instCanonicallyOrderedAdd",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 880,
"column": 23
} | {
"line": 880,
"column": 53
} | [
{
"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",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 881,
"column": 7
} | {
"line": 881,
"column": 37
} | [
{
"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",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 881,
"column": 48
} | {
"line": 881,
"column": 78
} | [
{
"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.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 895,
"column": 28
} | {
"line": 895,
"column": 39
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 896,
"column": 6
} | {
"line": 896,
"column": 35
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 898,
"column": 8
} | {
"line": 898,
"column": 37
} | [
{
"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",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 891,
"column": 4
} | {
"line": 904,
"column": 11
} | [
{
"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)",
"usedConstants": [
"Iff.mpr",
"one_pow",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Polynomial.C",
"NonA... | 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": 891,
"column": 4
} | {
"line": 904,
"column": 11
} | [
{
"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)",
"usedConstants": [
"Iff.mpr",
"one_pow",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Polynomial.C",
"NonA... | 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": 910,
"column": 4
} | {
"line": 910,
"column": 72
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"not_exists._simp_1",
"False",
"IsDomain.to_noZeroDivisors",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 944,
"column": 4
} | {
"line": 944,
"column": 34
} | [
{
"pp": "case a\nR : 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⟩",
"u... | obtain ⟨rfl⟩ : j = 0 := by lia | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 463,
"column": 6
} | {
"line": 463,
"column": 49
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ f₂ : normalization f ⟶ T\ng : T ⟶ Y\ninst✝ : IsAffineHom g\nH₁ : toNormalization f ≫ f₁ = toNormalization f ≫ f₂\nhf₁ : f₁ ≫ g = fromNormalization f\nhf₂ : f₂ ≫ g = fromNormalization f\nU : (normalizationOpenCov... | rw [this, f.toNormalization_app_preimage U] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 13
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing R'\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R R'\ninst✝³ : Algebra R S\np : Ideal R\nq : Ideal R'\ninst✝² : p.IsPrime\ninst✝¹ : q.IsPrime\ninst✝ : q.LiesOver p\nH : Function.Bijective ⇑(ResidueField.mapₐ p q (Algebra.ofId R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 123,
"column": 8
} | {
"line": 123,
"column": 46
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 477,
"column": 6
} | {
"line": 477,
"column": 49
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ f₂ : normalization f ⟶ T\ng : T ⟶ Y\ninst✝ : IsAffineHom g\nH₁ : toNormalization f ≫ f₁ = toNormalization f ≫ f₂\nhf₁ : f₁ ≫ g = fromNormalization f\nhf₂ : f₂ ≫ g = fromNormalization f\nU : (normalizationOpenCov... | rw [this, f.toNormalization_app_preimage U] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 159,
"column": 4
} | {
"line": 161,
"column": 37
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nq : Ideal S\ninst✝³ : q.IsPrime\ninst✝² : q.LiesOver p\ninst✝¹ : FiniteType R S\ninst✝ : QuasiFiniteAt R q\ns₁ : S\nhs₁q : s₁ ∉ q\nhs₁ : ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q → q... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 15
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nq : Ideal S\ninst✝³ : q.IsPrime\ninst✝² : q.LiesOver p\ninst✝¹ : FiniteType R S\ninst✝ : QuasiFiniteAt R q\ns₁ : S\nhs₁q : s₁ ∉ q\nhs₁ : ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q → q... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 195,
"column": 6
} | {
"line": 195,
"column": 58
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nq : Ideal S\ninst✝³ : q.IsPrime\ninst✝² : q.LiesOver p\ninst✝¹ : FiniteType R S\ninst✝ : QuasiFiniteAt R q\ns₁ : S\nhs₁q : s₁ ∉ q\nhs₁ : ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q → q... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 196,
"column": 33
} | {
"line": 196,
"column": 83
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nq : Ideal S\ninst✝³ : q.IsPrime\ninst✝² : q.LiesOver p\ninst✝¹ : FiniteType R S\ninst✝ : QuasiFiniteAt R q\ns₁ : S\nhs₁q : s₁ ∉ q\nhs₁ : ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q → q... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 102,
"column": 13
} | {
"line": 102,
"column": 24
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 60
} | [
{
"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'.I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 227,
"column": 4
} | {
"line": 228,
"column": 76
} | [
{
"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'.I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 348,
"column": 6
} | {
"line": 348,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k) ℤ)\nthis : Algebra (MvPo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k) ℤ)\nthis : Algebra (MvPo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 356,
"column": 6
} | {
"line": 356,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k) ℤ)\nthis : Algebra (MvPo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 362,
"column": 4
} | {
"line": 362,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k) ℤ)\nthis : Algebra (MvPo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 371,
"column": 15
} | {
"line": 371,
"column": 26
} | [
{
"pp": "case tmul.refine_1.C\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k) ℤ)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 258,
"column": 4
} | {
"line": 258,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 373,
"column": 24
} | {
"line": 373,
"column": 47
} | [
{
"pp": "case tmul.refine_1.mul_X\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 375,
"column": 15
} | {
"line": 375,
"column": 96
} | [
{
"pp": "case tmul.refine_2.C\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k) ℤ)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 378,
"column": 24
} | {
"line": 378,
"column": 47
} | [
{
"pp": "case tmul.refine_2.mul_X\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 271,
"column": 4
} | {
"line": 284,
"column": 8
} | [
{
"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": 271,
"column": 4
} | {
"line": 284,
"column": 8
} | [
{
"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.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 610,
"column": 4
} | {
"line": 610,
"column": 19
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nq : { q // ↑q.1 * ↑q.2 = map (algebraMap R S) ↑p ∧ IsCoprime ↑q.1 ↑q.2 }\nf : failed to pretty print expr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 128,
"column": 61
} | {
"line": 128,
"column": 79
} | [
{
"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)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 703,
"column": 8
} | {
"line": 703,
"column": 59
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 150,
"column": 10
} | {
"line": 151,
"column": 35
} | [
{
"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)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Monoidal.Cartesian.Grp | {
"line": 325,
"column": 4
} | {
"line": 325,
"column": 51
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Group.Abelian | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 155,
"column": 8
} | {
"line": 156,
"column": 31
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 167,
"column": 77
} | {
"line": 167,
"column": 95
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 394,
"column": 87
} | {
"line": 395,
"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.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 182,
"column": 14
} | {
"line": 182,
"column": 25
} | [
{
"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)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 241,
"column": 6
} | {
"line": 242,
"column": 42
} | [
{
"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 ∣_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 70
} | [
{
"pp": "case a\nX 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 (toNormaliza... | simp only [opensRange_homOfLE, image_preimage_eq_opensRange_inf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Group.Abelian | {
"line": 103,
"column": 48
} | {
"line": 103,
"column": 59
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Descent | {
"line": 70,
"column": 8
} | {
"line": 70,
"column": 41
} | [
{
"pp": "case refine_2.tmul.a\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 (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 62
} | [
{
"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}\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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