module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 153,
"column": 26
} | {
"line": 154,
"column": 96
} | {
"line": 155,
"column": 4
} | [
{
"pp": "L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nx : L\n⊢ v.norm (x - x) = 0",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.to... | [] | by
simp only [sub_self, Valuation.norm, Valuation.map_zero, hv.hom.map_zero, NNReal.coe_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 212,
"column": 15
} | {
"line": 215,
"column": 7
} | {
"line": 215,
"column": 8
} | [
{
"pp": "L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nx y z : L\n⊢ dist x z ≤ max (dist x y) (dist y z)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | [] | by
refine (Valuation.norm_add_le _ (x - y) (y - z)).trans_eq' ?_
simp only [sub_add_sub_cancel]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 122,
"column": 28
} | {
"line": 122,
"column": 44
} | {
"line": 122,
"column": 44
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Ring R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : constantCoeff φ = ↑u\nn : σ →₀ ℕ\nthis✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)\nH : ¬n = 0\nthis : (0, n) ∈ antidiagonal n\ni j : σ →₀ ℕ\nhij : (i, j) ≠ (0, n) ∧ (i, j) ∈ antidiagonal n\n⊢ (coeff (i, j).1) φ * (c... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : Ring R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : constantCoeff φ = ↑u\nn : σ →₀ ℕ\nthis✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)\nH : ¬n = 0\nthis : (0, n) ∈ antidiagonal n\ni j : σ →₀ ℕ\nhij : (i, j) ≠ (0, n) ∧ (i, j).1 + (i, j).2 = n\n⊢ (coeff (i, j).1) φ * (coeff (i, j).... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 114,
"column": 6
} | {
"line": 127,
"column": 16
} | {
"line": 130,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Ring R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : constantCoeff φ = ↑u\nn : σ →₀ ℕ\nthis : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)\nH : ¬n = 0\n⊢ (coeff n) (φ * φ.invOfUnit u) = (coeff n) 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
... | [] | have : ((0 : σ →₀ ℕ), n) ∈ antidiagonal n := by rw [mem_antidiagonal, zero_add]
rw [coeff_one, if_neg H, coeff_mul, ← Finset.insert_erase this,
Finset.sum_insert (Finset.notMem_erase _ _), coeff_zero_eq_constantCoeff_apply, h,
coeff_invOfUnit, if_neg H, neg_mul, mul_neg, Units.mul_inv_cancel_left,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 114,
"column": 6
} | {
"line": 127,
"column": 16
} | {
"line": 130,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Ring R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : constantCoeff φ = ↑u\nn : σ →₀ ℕ\nthis : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)\nH : ¬n = 0\n⊢ (coeff n) (φ * φ.invOfUnit u) = (coeff n) 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
... | [] | have : ((0 : σ →₀ ℕ), n) ∈ antidiagonal n := by rw [mem_antidiagonal, zero_add]
rw [coeff_one, if_neg H, coeff_mul, ← Finset.insert_erase this,
Finset.sum_insert (Finset.notMem_erase _ _), coeff_zero_eq_constantCoeff_apply, h,
coeff_invOfUnit, if_neg H, neg_mul, mul_neg, Units.mul_inv_cancel_left,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 151,
"column": 59
} | {
"line": 151,
"column": 68
} | {
"line": 151,
"column": 69
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP : Fin 3 → R\nhP : W'.Equation P\n⊢ ![0, -W'.dblZ P, 0] = ![-W'.dblZ P * ![0, 1, 0] x, -W'.dblZ P * ![0, 1, 0] y, -W'.dblZ P * ![0, 1, 0] z]",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNe... | [
"R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP : Fin 3 → R\nhP : W'.Equation P\n⊢ ![0, -W'.dblZ P, 0] = ![0, -W'.dblZ P * ![0, 1, 0] y, 0]"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 369,
"column": 2
} | {
"line": 369,
"column": 32
} | {
"line": 370,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nv₁ v₂ : HeightOneSpectrum (𝓞 K)\n⊢ mk v₁ = mk v₂ ↔ v₁ = v₂",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NumberField.instCommRingRingOfIntegers",
"congrArg",
"NumberField.FinitePlace.mk",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nv₁ v₂ : HeightOneSpectrum (𝓞 K)\n⊢ mk v₁ = mk v₂ → v₁ = v₂"
] | refine ⟨?_, fun a ↦ by rw [a]⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Sites.Preserves | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 46
} | {
"line": 93,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nα : Type u_1\ninst✝ : Small.{w, u_1} α\nX : α → C\nc : Cofan X\nhc : IsColimit c\nthis : HasCoproduct X\nh : (Pi.lift fun i ↦ F.map (c.inj i).op) = F.map (Pi.lift fun i ↦ (c.inj i).op) ≫ piComparison F fun i ↦ op (X i)\n⊢ (piComparison F fun x ↦... | [
"C : Type u\ninst✝¹ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nα : Type u_1\ninst✝ : Small.{w, u_1} α\nX : α → C\nc : Cofan X\nhc : IsColimit c\nthis : HasCoproduct X\nh : (Pi.lift fun i ↦ F.map (c.inj i).op) = F.map (Pi.lift fun i ↦ (c.inj i).op) ≫ piComparison F fun i ↦ op (X i)\n⊢ (piComparison F fun x ↦ op (X x)) =... | rw [h, ← Category.assoc, ← Functor.map_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Preserves | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 88
} | {
"line": 109,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nα : Type u_1\ninst✝² : Small.{w, u_1} α\nX : α → C\nc : Cofan X\nhc : IsColimit c\ninst✝¹ : (ofArrows X c.inj).HasPairwisePullbacks\ninst✝ : PreservesLimit (Discrete.functor fun x ↦ op (X x)) F\n⊢ IsSheafFor F (ofArrows X c.inj)",
"ppTerm": ... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nα : Type u_1\ninst✝² : Small.{w, u_1} α\nX : α → C\nc : Cofan X\nhc : IsColimit c\ninst✝¹ : (ofArrows X c.inj).HasPairwisePullbacks\ninst✝ : PreservesLimit (Discrete.functor fun x ↦ op (X x)) F\n⊢ ∀ (y : Equalizer.Presieve.Arrows.FirstObj F X),\n (Concre... | rw [Equalizer.Presieve.Arrows.sheaf_condition, Limits.Types.type_equalizer_iff_unique] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 102,
"column": 48
} | {
"line": 102,
"column": 57
} | {
"line": 102,
"column": 58
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ -P y - W'.a₁ * 0 - W'.a₃ * 0 = -P y",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul.hMul",
... | [
"R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ -P y - 0 - W'.a₃ * 0 = -P y"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 102,
"column": 68
} | {
"line": 102,
"column": 77
} | {
"line": 102,
"column": 78
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ -P y - W'.a₃ * 0 = -P y",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul.hMul",
"MulZeroCl... | [
"R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ -P y - 0 = -P y"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.Basic | {
"line": 470,
"column": 62
} | {
"line": 470,
"column": 91
} | {
"line": 471,
"column": 4
} | [
{
"pp": "R✝ : Type u\nA✝ : Type v\ninst✝¹⁴ : CommRing R✝\ninst✝¹³ : CommRing A✝\ninst✝¹² : Algebra R✝ A✝\nB✝ : Type u_1\nP : Type u_2\nC✝ : Type u_3\ninst✝¹¹ : CommRing B✝\ninst✝¹⁰ : Algebra R✝ B✝\ninst✝⁹ : CommRing C✝\ninst✝⁸ : Algebra R✝ C✝\ninst✝⁷ : CommRing P\ninst✝⁶ : Algebra R✝ P\nR : Type u_4\ninst✝⁵ : C... | [] | by simpa [Algebra.ofId_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Smooth.Basic | {
"line": 462,
"column": 2
} | {
"line": 471,
"column": 87
} | {
"line": 473,
"column": 0
} | [
{
"pp": "R✝ : Type u\nA✝ : Type v\ninst✝¹⁴ : CommRing R✝\ninst✝¹³ : CommRing A✝\ninst✝¹² : Algebra R✝ A✝\nB✝ : Type u_1\nP : Type u_2\nC : Type u_3\ninst✝¹¹ : CommRing B✝\ninst✝¹⁰ : Algebra R✝ B✝\ninst✝⁹ : CommRing C\ninst✝⁸ : Algebra R✝ C\ninst✝⁷ : CommRing P\ninst✝⁶ : Algebra R✝ P\nR : Type u_4\ninst✝⁵ : Comm... | [] | refine .of_comp_surjective fun C _ _ I hI f ↦ ?_
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) ?_, ?_⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp Tens... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.Basic | {
"line": 462,
"column": 2
} | {
"line": 471,
"column": 87
} | {
"line": 473,
"column": 0
} | [
{
"pp": "R✝ : Type u\nA✝ : Type v\ninst✝¹⁴ : CommRing R✝\ninst✝¹³ : CommRing A✝\ninst✝¹² : Algebra R✝ A✝\nB✝ : Type u_1\nP : Type u_2\nC : Type u_3\ninst✝¹¹ : CommRing B✝\ninst✝¹⁰ : Algebra R✝ B✝\ninst✝⁹ : CommRing C\ninst✝⁸ : Algebra R✝ C\ninst✝⁷ : CommRing P\ninst✝⁶ : Algebra R✝ P\nR : Type u_4\ninst✝⁵ : Comm... | [] | refine .of_comp_surjective fun C _ _ I hI f ↦ ?_
letI := ((algebraMap B C).comp (algebraMap R B)).toAlgebra
haveI : IsScalarTower R B C := IsScalarTower.of_algebraMap_eq' rfl
refine ⟨TensorProduct.productLeftAlgHom (Algebra.ofId B C) ?_, ?_⟩
· exact FormallySmooth.lift I ⟨2, hI⟩ ((f.restrictScalars R).comp Tens... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 431,
"column": 2
} | {
"line": 431,
"column": 80
} | {
"line": 432,
"column": 2
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nι' : Type w₃\nσ : Type w₂\nσ' : Type w₄\nQ : Generators S T ι\nP : Generators R S σ\nQ' : Gen... | [
"R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nι' : Type w₃\nσ : Type w₂\nσ' : Type w₄\nQ : Generators S T ι\nP : Generators R S σ\nQ' : Generators S T ... | simp only [LinearMap.domRestrict_apply, Extension.Cotangent.map_mk, δ_eq_δAux] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.RingHom.Smooth | {
"line": 64,
"column": 2
} | {
"line": 67,
"column": 16
} | {
"line": 69,
"column": 0
} | [
{
"pp": "⊢ IsStableUnderBaseChange @FormallySmooth",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CommRing",
"Algebra.to_smulCommClass",
"RingHom.IsStableUnderBaseChange.mk",
"Algebra.algebraMap",
"congrArg",
"CommSemiring.toSemiring",
... | [] | refine .mk respectsIso ?_
introv H
rw [formallySmooth_algebraMap] at H ⊢
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RingHom.Smooth | {
"line": 64,
"column": 2
} | {
"line": 67,
"column": 16
} | {
"line": 69,
"column": 0
} | [
{
"pp": "⊢ IsStableUnderBaseChange @FormallySmooth",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CommRing",
"Algebra.to_smulCommClass",
"RingHom.IsStableUnderBaseChange.mk",
"Algebra.algebraMap",
"congrArg",
"CommSemiring.toSemiring",
... | [] | refine .mk respectsIso ?_
introv H
rw [formallySmooth_algebraMap] at H ⊢
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Unramified.Locus | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 78
} | {
"line": 140,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : FiniteType R A\np : Ideal A\ninst✝¹ : p.IsPrime\ninst✝ : IsUnramifiedAt R p\n⊢ ∃ f ∉ p, Unramified R (Localization.Away f)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Se... | [
"R : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : FiniteType R A\np : Ideal A\ninst✝¹ : p.IsPrime\ninst✝ : IsUnramifiedAt R p\nf : A\nhfp : f ∉ p\nH : FormallyUnramified R (Localization.Away f)\n⊢ ∃ f ∉ p, Unramified R (Localization.Away f)"
] | obtain ⟨f, hfp, H⟩ := exists_formallyUnramified_of_isUnramifiedAt (R := R) p | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Unramified.Pi | {
"line": 33,
"column": 4
} | {
"line": 34,
"column": 93
} | {
"line": 35,
"column": 2
} | [
{
"pp": "case intro.mp\nR : Type u_1\nI : Type u_2\ninst✝³ : Finite I\nf : I → Type u_3\ninst✝² : CommRing R\ninst✝¹ : (i : I) → CommRing (f i)\ninst✝ : (i : I) → Algebra R (f i)\nval✝ : Fintype I\n⊢ FormallyUnramified R ((i : I) → f i) → ∀ (i : I), FormallyUnramified R (f i)",
"ppTerm": "?intro.mp",
"a... | [] | intro _ i
exact FormallyUnramified.of_surjective (Pi.evalAlgHom R f i) (Function.surjective_eval i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Unramified.Pi | {
"line": 33,
"column": 4
} | {
"line": 34,
"column": 93
} | {
"line": 35,
"column": 2
} | [
{
"pp": "case intro.mp\nR : Type u_1\nI : Type u_2\ninst✝³ : Finite I\nf : I → Type u_3\ninst✝² : CommRing R\ninst✝¹ : (i : I) → CommRing (f i)\ninst✝ : (i : I) → Algebra R (f i)\nval✝ : Fintype I\n⊢ FormallyUnramified R ((i : I) → f i) → ∀ (i : I), FormallyUnramified R (f i)",
"ppTerm": "?intro.mp",
"a... | [] | intro _ i
exact FormallyUnramified.of_surjective (Pi.evalAlgHom R f i) (Function.surjective_eval i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.Pi | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 97
} | {
"line": 77,
"column": 4
} | [
{
"pp": "R : Type u_1\nI : Type u_2\nA : I → Type u_3\ninst✝³ : CommRing R\ninst✝² : (i : I) → CommRing (A i)\ninst✝¹ : (i : I) → Algebra R (A i)\ninst✝ : Finite I\nval✝ : Fintype I\nH : ∀ (i : I), FormallySmooth R (A i)\nB : Type (max u_1 u_2 u_3)\nx✝¹ : CommRing B\nx✝ : Algebra R B\nJ : Ideal B\nhJ : J ^ 2 = ... | [] | rw [← mul_assoc, ← map_mul, mul_sub, mul_one, (he.idem i).eq, sub_self, map_zero, zero_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 386,
"column": 2
} | {
"line": 386,
"column": 45
} | {
"line": 387,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\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 S T\ninst✝¹ : IsScalarTower R S T\nM : Submonoid S\ninst✝ : IsLocalization M T\n⊢ IsLocalizedModule M (map R R S T)",
"ppTerm": "?m.3... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\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 S T\ninst✝¹ : IsScalarTower R S T\nM : Submonoid S\ninst✝ : IsLocalization M T\n⊢ IsBaseChange T (map R R S T)"
] | rw [isLocalizedModule_iff_isBaseChange M T] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 450,
"column": 4
} | {
"line": 450,
"column": 13
} | {
"line": 450,
"column": 14
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ ![0, P y ^ 4, 0] = ![P y ^ 4 * ![0, 1, 0] x, P y ^ 4 * ![0, 1, 0] y, P y ^ 4 * ![0, 1, 0] z]",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Eq.... | [
"R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ ![0, P y ^ 4, 0] = ![0, P y ^ 4 * ![0, 1, 0] y, 0]"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 456,
"column": 43
} | {
"line": 456,
"column": 52
} | {
"line": 456,
"column": 53
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP Q : Fin 3 → F\nhP : W.Equation P\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z = Q y * P z\nhy' : P y * Q z = W.negY Q * P z\n⊢ ![0, W.dblU P, 0] = ![W.dblU P * ![0, 1, 0] x, W.dblU P * ![0, 1, 0] y, W.dblU P * ![0, 1, 0] z]",... | [
"F : Type u\ninst✝ : Field F\nW : Projective F\nP Q : Fin 3 → F\nhP : W.Equation P\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z = Q y * P z\nhy' : P y * Q z = W.negY Q * P z\n⊢ ![0, W.dblU P, 0] = ![0, W.dblU P * ![0, 1, 0] y, 0]"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Etale.Field | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 20
} | {
"line": 142,
"column": 20
} | [
{
"pp": "case refine_1\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsSeparable K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf : L →ₐ[K] B ⧸ I\ng : (k : L) → ↥K⟮k⟯ →ₐ[K] B\nhg₁ : ∀ (k : L), (fun g ↦ (Ideal.... | [
"case refine_1\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsSeparable K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf : L →ₐ[K] B ⧸ I\ng : (k : L) → ↥K⟮k⟯ →ₐ[K] B\nhg₁ : ∀ (k : L), (fun g ↦ (Ideal.Quotient.mkₐ... | change g 1 1 = 1 | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Etale.Field | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 46
} | {
"line": 169,
"column": 2
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nA : Type u\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra K A\ninst✝² : EssFiniteType K A\ninst✝¹ : FormallyEtale K A\np : Ideal A\ninst✝ : p.IsPrime\nthis✝² : Module.Finite K A\nthis✝¹ : IsArtinianRing A\nthis✝ : IsReduced A... | [
"K : Type u_1\nL : Type u_2\nA : Type u\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra K A\ninst✝² : EssFiniteType K A\ninst✝¹ : FormallyEtale K A\np : Ideal A\ninst✝ : p.IsPrime\nthis✝² : Module.Finite K A\nthis✝¹ : IsArtinianRing A\nthis✝ : IsReduced A\nthis : Fie... | rw [← Algebra.FormallyEtale.iff_isSeparable] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 121,
"column": 22
} | {
"line": 121,
"column": 33
} | {
"line": 121,
"column": 34
} | [
{
"pp": "case smul\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsS... | [
"case smul\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsScalarTower R... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 150,
"column": 62
} | {
"line": 150,
"column": 71
} | {
"line": 150,
"column": 72
} | [
{
"pp": "case tmul.zero\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ ... | [
"case tmul.zero\nR : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsScalarTo... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 38
} | {
"line": 259,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsScalarTower ... | [
"R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\ninst✝⁷ : CommRing A\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra R B\ninst✝³ : Algebra A B\ninst✝² : Algebra S B\ninst✝¹ : IsScalarTower R A B\ninst✝ : IsScalarTower R S B\nh : A... | refine { __ := e₂, map_smul' := ?_ } | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.AdicCompletion.Functoriality | {
"line": 402,
"column": 18
} | {
"line": 402,
"column": 23
} | {
"line": 402,
"column": 23
} | [
{
"pp": "case right\nR : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nh : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)\nx : M\nn : ℕ\ny : N ⧸ I ^ n • ⊤\ny' : N ⧸ I ^ (n + 1) • ⊤\nhyy' ... | [
"case right\nR : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nh : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)\nx : M\nn : ℕ\ny : N ⧸ I ^ n • ⊤\ny' : N ⧸ I ^ (n + 1) • ⊤\nhyy' : (factor ⋯)... | hx'y0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 787,
"column": 46
} | {
"line": 787,
"column": 55
} | {
"line": 787,
"column": 56
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\n⊢ ![0, addU P Q, 0] = ![addU P Q * ![0, 1, 0] x, addU P Q * ![0, 1, 0] y, addU P Q * ![0, 1, 0] z]",
"ppTerm": "?m.104",
"assigned": tr... | [
"F : Type u\ninst✝ : Field F\nW : Projective F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\n⊢ ![0, addU P Q, 0] = ![0, addU P Q * ![0, 1, 0] y, 0]"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.AsTensorProduct | {
"line": 65,
"column": 8
} | {
"line": 65,
"column": 23
} | {
"line": 65,
"column": 24
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nr x : AdicCompletion I R\n⊢ ∀ (x_1 : M),\n (↑R ((LinearMap.lsmul (AdicCompletion I R) (AdicCompletion I M)) (r • x)) ∘ₗ of I M) x_1 =... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 61
} | {
"line": 201,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Module.Flat R S\ninst✝⁴ : FinitePresentation R S\np : Ideal R\nq : Ideal S\ninst✝³ : p.IsPrime\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : FormallySmooth p.ResidueField (p.Fiber S)\nRp : Type u_... | [
"R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : Module.Flat R S\ninst✝⁴ : FinitePresentation R S\np : Ideal R\nq : Ideal S\ninst✝³ : p.IsPrime\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : FormallySmooth p.ResidueField (p.Fiber S)\nRp : Type u_1 := Localiz... | let Sp := Localization (algebraMapSubmonoid S p.primeCompl) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Smooth.StandardSmoothOfFree | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 94
} | {
"line": 117,
"column": 4
} | [
{
"pp": "case inr\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FinitePresentation R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : IsSmoothAt R p\nthis✝ :\n ∀ (R : Type u_1) {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]\n ... | [
"case inr\nR : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FinitePresentation R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : IsSmoothAt R p\nthis✝ :\n ∀ (R : Type u_1) {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]\n [FinitePre... | refine ⟨g * (IsLocalization.Away.sec g g').1, ?_, .of_algEquiv (e.restrictScalars R).symm⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 74
} | {
"line": 217,
"column": 2
} | [
{
"pp": "n : ℕ\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S\nthis✝ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite\nι σ : Type\nw✝¹ : _root_.Finite σ\nw✝ : _root_.Finite ι\nP : Algebra.Submersi... | [
"n : ℕ\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S\nthis✝¹ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite\nι σ : Type\nw✝¹ : _root_.Finite σ\nw✝ : _root_.Finite ι\nP : Algebra.SubmersivePresentat... | have : IsScalarTower R (MvPolynomial (Fin n) R) S := .to₁₂₄ _ _ P.Ring _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.RingHom.LocallyStandardSmooth | {
"line": 47,
"column": 6
} | {
"line": 48,
"column": 35
} | {
"line": 48,
"column": 35
} | [
{
"pp": "R S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : Locally (fun {R S} [CommRing R] [CommRing S] ↦ IsStandardSmooth) f\n⊢ f.Smooth",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CommRing",
"RingHom.locally_iff_of_localizationSpanT... | [
"R S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : Locally (fun {R S} [CommRing R] [CommRing S] ↦ IsStandardSmooth) f\n⊢ Locally (fun {R S} [CommRing R] [CommRing S] ↦ Smooth) f"
] | ← locally_iff_of_localizationSpanTarget Smooth.propertyIsLocal.respectsIso
Smooth.ofLocalizationSpanTarget | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 146,
"column": 44
} | {
"line": 146,
"column": 64
} | {
"line": 146,
"column": 65
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\nx : ↥D.presLeft.toExtension.ker\n⊢ Extension.Cotangent.mk ⟨↑(AlgHom.id R P.Ring) ↑x, ⋯⟩ = Extension.Cot... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\nx : ↥D.presLeft.toExtension.ker\n⊢ Extension.Cotangent.mk\n ⟨(RingHom.id\n (Presentation.naive (fun... | AlgHom.id_toRingHom, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 49
} | {
"line": 98,
"column": 2
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\nq₁ q₂ : MvPolynomial ι k\ne : F = q₁ * q₂\n... | [
"k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\nq₁ q₂ : MvPolynomial ι k\ne : F = q₁ * q₂\nh₁ : (aeval ... | rw [totalDegree_eq_zero_iff_eq_C.mp this] at ne | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.Smooth | {
"line": 77,
"column": 8
} | {
"line": 77,
"column": 19
} | {
"line": 77,
"column": 20
} | [
{
"pp": "n m : ℕ\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\n⊢ Smooth f ↔ affineLocally (fun {R S} [CommRing R] [CommRing S] ↦ RingHom.Smooth) f",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarr... | [
"n m : ℕ\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\n⊢ (∀ {U : Y.Opens},\n IsAffineOpen U →\n ∀ {V : X.Opens},\n IsAffineOpen V → ∀ (e : V ≤ f ⁻¹ᵁ U), (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V e)).Smooth) ↔\n affineLocally (fun {R S} [CommRing R] [CommRing S] ↦ RingHom.Smooth)... | smooth_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Proper | {
"line": 45,
"column": 92
} | {
"line": 48,
"column": 5
} | {
"line": 50,
"column": 0
} | [
{
"pp": "⊢ @IsProper = @IsSeparated ⊓ @UniversallyClosed ⊓ @LocallyOfFiniteType",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"AlgebraicGeometry.Scheme",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | by
ext X Y f
rw [isProper_iff, ← and_assoc]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 169,
"column": 4
} | {
"line": 174,
"column": 80
} | {
"line": 175,
"column": 2
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\np : ℕ\nhp : Nat.Prime p\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\nthis : ∀ (i : ι), ... | [] | have hF''0' : F''.totalDegree ≠ 0 := by
contrapose hF''0
rw [totalDegree_eq_zero_iff_eq_C.mp hF''0, aeval_C, map_eq_zero] at hF''
rw [totalDegree_eq_zero_iff_eq_C.mp hF''0, hF'', map_zero]
replace this := hpm.trans ((HF F'' hF''0 hF'').trans_eq (one_mul _).symm)
exact hp.one_lt.not_ge ((mul_le... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 169,
"column": 4
} | {
"line": 174,
"column": 80
} | {
"line": 175,
"column": 2
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\np : ℕ\nhp : Nat.Prime p\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\nthis : ∀ (i : ι), ... | [] | have hF''0' : F''.totalDegree ≠ 0 := by
contrapose hF''0
rw [totalDegree_eq_zero_iff_eq_C.mp hF''0, aeval_C, map_eq_zero] at hF''
rw [totalDegree_eq_zero_iff_eq_C.mp hF''0, hF'', map_zero]
replace this := hpm.trans ((HF F'' hF''0 hF'').trans_eq (one_mul _).symm)
exact hp.one_lt.not_ge ((mul_le... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified | {
"line": 145,
"column": 2
} | {
"line": 146,
"column": 56
} | {
"line": 147,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf✝ : X ⟶ Y\nS R : CommRingCat\nf : R ⟶ S\ninst✝¹ : (CommRingCat.Hom.hom f).FormallyUnramified\ninst✝ : (CommRingCat.Hom.hom f).FiniteType\nalgInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom f).toAlgebra\nalgebraizeInst✝¹ : Algebra.FormallyUnramified ↑R ↑S\nalgebraizeInst✝ : Algebra.FiniteTy... | [
"X Y : Scheme\nf✝ : X ⟶ Y\nS R : CommRingCat\nf : R ⟶ S\ninst✝¹ : (CommRingCat.Hom.hom f).FormallyUnramified\ninst✝ : (CommRingCat.Hom.hom f).FiniteType\nalgInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom f).toAlgebra\nalgebraizeInst✝¹ : Algebra.FormallyUnramified ↑R ↑S\nalgebraizeInst✝ : Algebra.FiniteType ↑R ↑S\n⊢ ... | rw [show f = CommRingCat.ofHom (algebraMap R S) from rfl, diagonal_SpecMap R S,
cancel_right_of_respectsIso (P := @IsOpenImmersion)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.Descent | {
"line": 70,
"column": 2
} | {
"line": 87,
"column": 57
} | {
"line": 88,
"column": 2
} | [
{
"pp": "case inr\nP P' : MorphismProperty Scheme\ninst✝² : IsZariskiLocalAtTarget P\ninst✝¹ : P'.IsStableUnderBaseChange\nH : ∀ {R : CommRingCat} {X Y : Scheme} (f : X ⟶ Spec R) (g : Y ⟶ Spec R), P' f → P (pullback.fst f g) → P g\nX Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : HasPullback f g\nh : P' f\nhf : P ... | [
"P P' : MorphismProperty Scheme\ninst✝² : IsZariskiLocalAtTarget P\ninst✝¹ : P'.IsStableUnderBaseChange\nH : ∀ {R : CommRingCat} {X Y : Scheme} (f : X ⟶ Spec R) (g : Y ⟶ Spec R), P' f → P (pullback.fst f g) → P g\nX Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : HasPullback f g\nh : P' f\nhf : P (pullback.fst f g)\nhZ... | · rw [IsZariskiLocalAtTarget.iff_of_openCover (P := P) Z.affineCover]
intro i
let ι := Z.affineCover.f i
let e : pullback (pullback.snd f ι) (pullback.snd g ι) ≅
pullback (pullback.fst f g) (pullback.fst f ι) :=
pullbackLeftPullbackSndIso f ι (pullback.snd g ι) ≪≫
pullback.congrHom rfl... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Conductor | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 38
} | {
"line": 96,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p ∈ comap (algebraMap R S) (conductor R x)\nl : R →₀ S\nH : l ∈ Finsupp.supported S S ↑I\nH' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z\n⊢ (algebraMap R S) p * z ∈ ⇑(alg... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p ∈ comap (algebraMap R S) (conductor R x)\nl : R →₀ S\nH : l ∈ Finsupp.supported S S ↑I\nH' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z\n⊢ (l.sum fun a c ↦ c • (algebraMap R S) a * ... | rw [← H', mul_comm, Finsupp.sum_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Conductor | {
"line": 136,
"column": 18
} | {
"line": 136,
"column": 24
} | {
"line": 136,
"column": 24
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤\nh_alg : Function.Injective ⇑(algebraMap (↥R[x]) S)\nz : S\nhz : z ∈ R[x]\nhy : ⟨z, hz⟩ ∈ comap (algebraMap (↥R[x]) S) (Ideal.map (algebraMa... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\nhx : comap (algebraMap R S) (conductor R x) ⊔ I = ⊤\nh_alg : Function.Injective ⇑(algebraMap (↥R[x]) S)\nz : S\nhz : z ∈ R[x]\nhy : ⟨z, hz⟩ ∈ comap (algebraMap (↥R[x]) S) (Ideal.map (algebraMap R S) I)\np... | ← temp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 63
} | {
"line": 204,
"column": 2
} | [
{
"pp": "case pos\nK : Type u\ninst✝⁴ : CommRing K\nR : Type u_1\ninst✝³ : IsDomain K\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R K[X]\ninst✝ : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\nthis : SMulZeroClass R (FractionRing K[X]) := inferInstance\nhq : q = 0\n⊢ RatFunc.mk (c • p) q = c • RatFunc.mk p q",... | [
"case neg\nK : Type u\ninst✝⁴ : CommRing K\nR : Type u_1\ninst✝³ : IsDomain K\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R K[X]\ninst✝ : IsScalarTower R K[X] K[X]\nc : R\np q : K[X]\nthis : SMulZeroClass R (FractionRing K[X]) := inferInstance\nhq : ¬q = 0\n⊢ RatFunc.mk (c • p) q = c • RatFunc.mk p q"
] | · rw [hq, mk_zero, mk_zero, ← ofFractionRing_smul, smul_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 208,
"column": 63
} | {
"line": 208,
"column": 74
} | {
"line": 208,
"column": 75
} | [
{
"pp": "K : Type u\ninst✝⁴ : CommRing K\nR : Type u_1\ninst✝³ : IsDomain K\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R K[X]\ninst✝ : IsScalarTower R K[X] K[X]\nc : R\np : K[X]\nq✝ : K⟮X⟯\nq r : K[X]\nx✝ : r ≠ 0\n⊢ RatFunc.mk ((c • p) • q) r = c • p • RatFunc.mk q r",
"ppTerm": "?m.45",
"assigned": ... | [
"K : Type u\ninst✝⁴ : CommRing K\nR : Type u_1\ninst✝³ : IsDomain K\ninst✝² : Monoid R\ninst✝¹ : DistribMulAction R K[X]\ninst✝ : IsScalarTower R K[X] K[X]\nc : R\np : K[X]\nq✝ : K⟮X⟯\nq r : K[X]\nx✝ : r ≠ 0\n⊢ RatFunc.mk (c • p • q) r = c • p • RatFunc.mk q r"
] | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 201,
"column": 2
} | {
"line": 202,
"column": 46
} | {
"line": 203,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal S\ninst✝⁴ : p.IsPrime\ninst✝³ : QuasiFiniteAt R (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.under R p))) p)\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nq : Ideal (A ⊗... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal S\ninst✝⁴ : p.IsPrime\ninst✝³ : QuasiFiniteAt R (Ideal.map (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.under R p))) p)\nA : Type u_4\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nq : Ideal (A ⊗[R] S)\ninst... | refine .of_surjectiveOnStalks (q.map φ.toRingHom) e.symm.toAlgHom
e.symm.toRingEquiv.surjectiveOnStalks _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 505,
"column": 19
} | {
"line": 509,
"column": 61
} | {
"line": 510,
"column": 2
} | [
{
"pp": "K : Type u\ninst✝³ : CommRing K\ninst✝² : IsDomain K\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R K[X]\nc : R\nx : K⟮X⟯\n⊢ c • x =\n { toFun := fun x ↦ RatFunc.mk ((algebraMap R K[X]) x) 1, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯,\n map_add' := ⋯ }\n c *\n x",
... | [] | by
induction x using RatFunc.induction_on' with | _ p q hq
rw [RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk, mk_one', ← mk_smul,
mk_def_of_ne (c • p) hq, mk_def_of_ne p hq, ← ofFractionRing_mul,
IsLocalization.mul_mk'_eq_mk'_of_mul, Algebra.smul_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 211,
"column": 91
} | {
"line": 254,
"column": 38
} | {
"line": 256,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nhf : ∀ (x : ↥Y), LocallyQuasiFinite (Hom.fiberToSpecResidueField f x)\n⊢ LocallyQuasiFinite f",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.IsImmersion.instOfIsClosedImmersion",
"CategoryTheory.Limits.pullbackSymmetry",
... | [] | by
change id _ -- avoid typeclass synthesis from getting stuck on the wlog hypothesis.
wlog hY : ∃ R, Y = Spec R
· refine (IsZariskiLocalAtTarget.iff_of_openCover Y.affineCover).mpr fun i ↦
this (f := pullback.snd _ _) (fun x ↦ ?_) ⟨_, rfl⟩
have (x : Y) : IsLocallyArtinian (f.fiber x) :=
.of_local... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 228,
"column": 4
} | {
"line": 238,
"column": 7
} | {
"line": 240,
"column": 0
} | [
{
"pp": "X : Scheme\nU V : (directedCover X).I₀\nx : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))\n⊢ ∃ k hki hkj y, (pullback.lift (X.homOfLE ⋯) (X.homOfLE ⋯) ⋯) y = x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mp... | [] | let a := (pullback.fst _ _ ≫ U.1.ι) x
have haU : a ∈ U.1 := (pullback.fst U.1.ι V.1.ι x).2
have haV : a ∈ V.1 := by unfold a; rw [pullback.condition]; exact (pullback.snd U.1.ι V.1.ι x).2
obtain ⟨f, g, e, hxf⟩ := exists_basicOpen_le_affine_inter U.2 V.2 _ ⟨haU, haV⟩
refine ⟨U.basicOpen f, homOfLE (U.bas... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 228,
"column": 4
} | {
"line": 238,
"column": 7
} | {
"line": 240,
"column": 0
} | [
{
"pp": "X : Scheme\nU V : (directedCover X).I₀\nx : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))\n⊢ ∃ k hki hkj y, (pullback.lift (X.homOfLE ⋯) (X.homOfLE ⋯) ⋯) y = x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Eq.mp... | [] | let a := (pullback.fst _ _ ≫ U.1.ι) x
have haU : a ∈ U.1 := (pullback.fst U.1.ι V.1.ι x).2
have haV : a ∈ V.1 := by unfold a; rw [pullback.condition]; exact (pullback.snd U.1.ι V.1.ι x).2
obtain ⟨f, g, e, hxf⟩ := exists_basicOpen_le_affine_inter U.2 V.2 _ ⟨haU, haV⟩
refine ⟨U.basicOpen f, homOfLE (U.bas... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 401,
"column": 50
} | {
"line": 401,
"column": 73
} | {
"line": 401,
"column": 74
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyOfFiniteType f\nthis :\n ∀ {X Y : Scheme} {f : X ⟶ Y} [LocallyOfFiniteType f] {x : ↥X},\n (∃ R, Y = Spec R) → (QuasiFiniteAt f x ↔ IsOpen {⟨x, ⋯⟩})\nhY : ¬∃ R, Y = Spec R\ni : Y.affineCover.I₀\nx : ↥(pullback f (Y.affineCover.f i))\nhy : (Y.affineCover.f i) (... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyOfFiniteType f\nthis :\n ∀ {X Y : Scheme} {f : X ⟶ Y} [LocallyOfFiniteType f] {x : ↥X},\n (∃ R, Y = Spec R) → (QuasiFiniteAt f x ↔ IsOpen {⟨x, ⋯⟩})\nhY : ¬∃ R, Y = Spec R\ni : Y.affineCover.I₀\nx : ↥(pullback f (Y.affineCover.f i))\nhy : (Y.affineCover.f i) ((pullback.sn... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 60
} | {
"line": 209,
"column": 4
} | [
{
"pp": "case refine_3\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\n⊢ ∀ (a : AdjoinRoot P.f) (b : a ∈ Submonoid.powers ((AdjoinRoot.mk P.f) P.g)),\n IsUnit ((AdjoinRoot.liftAlgHom ... | [
"case refine_3\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\n⊢ Submonoid.powers ((AdjoinRoot.mk P.f) P.g) ≤\n Submonoid.comap (AdjoinRoot.liftAlgHom P.f (Algebra.ofId R P.Ring) P.X ⋯) (... | change Submonoid.powers _ ≤ (IsUnit.submonoid _).comap _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 60
} | {
"line": 226,
"column": 4
} | [
{
"pp": "case refine_3\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\n⊢ ∀ (a : R[X]) (b : a ∈ Submonoid.powers P.g), IsUnit ((aeval P.X) ↑⟨a, b⟩)",
"ppTerm": "?refine_3",
"assig... | [
"case refine_3\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\n⊢ Submonoid.powers P.g ≤ Submonoid.comap (aeval P.X) (IsUnit.submonoid P.Ring)"
] | change Submonoid.powers _ ≤ (IsUnit.submonoid _).comap _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 22
} | {
"line": 69,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\nT : Type u_4\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nH : Function.Bijective ⇑(toIntegralClosure R S... | [
"R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra R B\nT : Type u_4\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nH : Function.Bijective ⇑(toIntegralClosure R S B)\nH' : Fu... | convert! e.bijective | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 74,
"column": 34
} | {
"line": 74,
"column": 78
} | {
"line": 76,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU V : (TopologicalSpace.Opens ↥Y)ᵒᵖ\ni : U ⟶ V\nx : ↑(Y.presheaf.obj U)\n⊢ (CommRingCat.Hom.hom\n (Y.presheaf.map i ≫\n CommRingCat.ofHom\n (algebraMap ↑Γ(Y, Opposite.unop V) ↥(integralClosure ↑Γ(Y, Opposite.unop V) ↑Γ(X, f ⁻¹ᵁ Opposite.unop V)))))\n ... | [] | exact Subtype.ext congr($(f.naturality i) x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 605,
"column": 8
} | {
"line": 605,
"column": 67
} | {
"line": 606,
"column": 8
} | [
{
"pp": "case refine_1\nn : 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 : CommRi... | [
"case refine_1\nn : 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... | simp only [Subalgebra.restrictScalars_top, Algebra.map_top] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 250,
"column": 8
} | {
"line": 250,
"column": 17
} | {
"line": 250,
"column": 18
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝ : CommRing R\nf p : R[X]\nm n : ℕ\nhp : p.natDegree + m ≤ n\nhf : f.natDegree ≤ m\ng : R[X]\nH : 0 ≤ n - m + 1\n⊢ f.resultant (g + f * 0) m n = f.resultant g m n",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
... | [
"case zero\nR : Type u_1\ninst✝ : CommRing R\nf p : R[X]\nm n : ℕ\nhp : p.natDegree + m ≤ n\nhf : f.natDegree ≤ m\ng : R[X]\nH : 0 ≤ n - m + 1\n⊢ f.resultant (g + 0) m n = f.resultant g m n"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 252,
"column": 31
} | {
"line": 252,
"column": 39
} | {
"line": 252,
"column": 40
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\nf p : R[X]\nm n : ℕ\nhp : p.natDegree + m ≤ n\nhf : f.natDegree ≤ m\nk : ℕ\nIH :\n ∀ (g : R[X]),\n k ≤ n - m + 1 → f.resultant (g + f * ∑ n ∈ Finset.range k, (monomial n) (p.coeff n)) m n = f.resultant g m n\ng : R[X]\nH : k + 1 ≤ n - m + 1\n⊢ f.resultan... | [
"case succ\nR : Type u_1\ninst✝ : CommRing R\nf p : R[X]\nm n : ℕ\nhp : p.natDegree + m ≤ n\nhf : f.natDegree ≤ m\nk : ℕ\nIH :\n ∀ (g : R[X]),\n k ≤ n - m + 1 → f.resultant (g + f * ∑ n ∈ Finset.range k, (monomial n) (p.coeff n)) m n = f.resultant g m n\ng : R[X]\nH : k + 1 ≤ n - m + 1\n⊢ f.resultant (g + (f * ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 54
} | {
"line": 397,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\nm : ℕ\nr : R\nhf : f.natDegree ≤ m\n⊢ f.resultant (X - C r) m 1 = (-1) ^ m * eval r f",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.eval",
"NegZeroClass.toNeg",
"HMul.... | [
"R : Type u_1\ninst✝ : CommRing R\nf : R[X]\nm : ℕ\nr : R\nhf : f.natDegree ≤ m\n⊢ (-1) ^ (m * 1) * eval r f = (-1) ^ m * eval r f"
] | rw [resultant_comm, resultant_X_sub_C_left _ _ _ hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 451,
"column": 4
} | {
"line": 453,
"column": 52
} | {
"line": 454,
"column": 4
} | [
{
"pp": "case refine_3\nX 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 : (norma... | [
"case refine_3\nX 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 : (normalizationOpen... | have h₁ : f ⁻¹ᵁ U.1 ≤ f₀ ⁻¹ᵁ g ⁻¹ᵁ U.1 := by
simp only [← Scheme.Hom.comp_preimage, f₀, Category.assoc,
hf₁, toNormalization_fromNormalization]; rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 579,
"column": 6
} | {
"line": 579,
"column": 47
} | {
"line": 579,
"column": 47
} | [
{
"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\nΔ : 𝓡 := (presentation m k hn p).jacobian\nhΔ : IsUnit ((algebraMap 𝓡 (Localization.Away Δ)) Δ)\nP : Al... | [] | simp [Algebra.Presentation.dimension, hn] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 767,
"column": 4
} | {
"line": 779,
"column": 71
} | {
"line": 780,
"column": 2
} | [
{
"pp": "case Splits\nR✝ : Type u_1\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : Field R\nf : R[X]\nhf' : f.Splits\ng : R[X]\nr : R\n⊢ ((taylor r) f).resultant ((taylor r) g) = f.resultant g",
"ppTerm": "?Splits",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Polynomial.taylor_eva... | [] | induction hf' using Submonoid.closure_induction with
| mem x h =>
obtain (⟨s, rfl⟩ | ⟨s, rfl⟩) := h
· rw [taylor_C]; simp
· nontriviality R
rw [map_add, taylor_X, taylor_C, add_assoc, ← map_add]
simp [-map_add, taylor_eval]
| one => simp
| mul x y hx hy hx' hy' =>
by_... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 767,
"column": 4
} | {
"line": 779,
"column": 71
} | {
"line": 780,
"column": 2
} | [
{
"pp": "case Splits\nR✝ : Type u_1\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : Field R\nf : R[X]\nhf' : f.Splits\ng : R[X]\nr : R\n⊢ ((taylor r) f).resultant ((taylor r) g) = f.resultant g",
"ppTerm": "?Splits",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Polynomial.taylor_eva... | [] | induction hf' using Submonoid.closure_induction with
| mem x h =>
obtain (⟨s, rfl⟩ | ⟨s, rfl⟩) := h
· rw [taylor_C]; simp
· nontriviality R
rw [map_add, taylor_X, taylor_C, add_assoc, ← map_add]
simp [-map_add, taylor_eval]
| one => simp
| mul x y hx hy hx' hy' =>
by_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 767,
"column": 4
} | {
"line": 779,
"column": 71
} | {
"line": 780,
"column": 2
} | [
{
"pp": "case Splits\nR✝ : Type u_1\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : Field R\nf : R[X]\nhf' : f.Splits\ng : R[X]\nr : R\n⊢ ((taylor r) f).resultant ((taylor r) g) = f.resultant g",
"ppTerm": "?Splits",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Polynomial.taylor_eva... | [] | induction hf' using Submonoid.closure_induction with
| mem x h =>
obtain (⟨s, rfl⟩ | ⟨s, rfl⟩) := h
· rw [taylor_C]; simp
· nontriviality R
rw [map_add, taylor_X, taylor_C, add_assoc, ← map_add]
simp [-map_add, taylor_eval]
| one => simp
| mul x y hx hy hx' hy' =>
by_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Group.Abelian | {
"line": 89,
"column": 56
} | {
"line": 89,
"column": 79
} | {
"line": 90,
"column": 6
} | [
{
"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✝¹⁰ : Nonempty ↥G.left\nt... | [
"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✝¹⁰ : Nonempty ↥G.left\nthis✝⁹ : IsPr... | ← Scheme.Hom.comp_base, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 49
} | {
"line": 227,
"column": 0
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : B\nf g f' : a ⟶ b\nη θ : f ≅ g\nη_f : 𝟙 a ≫ f ≅ f'\nη_g : 𝟙 a ≫ g ≅ f'\nh_η : 𝟙 a ◁ η ≪≫ η_g = η_f\nh_θ : 𝟙 a ◁ θ ≪≫ η_g = η_f\n⊢ (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom = θ.hom",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
... | [] | simp [← reassoc_of% (congrArg Iso.hom h_θ)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 49
} | {
"line": 227,
"column": 0
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : B\nf g f' : a ⟶ b\nη θ : f ≅ g\nη_f : 𝟙 a ≫ f ≅ f'\nη_g : 𝟙 a ≫ g ≅ f'\nh_η : 𝟙 a ◁ η ≪≫ η_g = η_f\nh_θ : 𝟙 a ◁ θ ≪≫ η_g = η_f\n⊢ (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom = θ.hom",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
... | [] | simp [← reassoc_of% (congrArg Iso.hom h_θ)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 49
} | {
"line": 227,
"column": 0
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : B\nf g f' : a ⟶ b\nη θ : f ≅ g\nη_f : 𝟙 a ≫ f ≅ f'\nη_g : 𝟙 a ≫ g ≅ f'\nh_η : 𝟙 a ◁ η ≪≫ η_g = η_f\nh_θ : 𝟙 a ◁ θ ≪≫ η_g = η_f\n⊢ (λ_ f).inv ≫ η_f.hom ≫ η_g.inv ≫ (λ_ g).hom = θ.hom",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
... | [] | simp [← reassoc_of% (congrArg Iso.hom h_θ)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 43
} | {
"line": 206,
"column": 6
} | [
{
"pp": "case refine_1.inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : Flat f\ninst✝ : IsFinite f\nh : 1 ≤ finrank f\nthis : ∀ {X Y : Scheme} (f : X ⟶ Y) [Flat f] [IsFinite f], 1 ≤ finrank f → (∃ R, Y = Spec R) → Surjective f\nhY : ¬∃ R, Y = Spec R\ni : Y.affineCover.toPreZeroHypercover.1\n⊢ Surjective (Cover.pullbackHo... | [
"case refine_1.inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : Flat f\ninst✝ : IsFinite f\nh : 1 ≤ finrank f\nthis : ∀ {X Y : Scheme} (f : X ⟶ Y) [Flat f] [IsFinite f], 1 ≤ finrank f → (∃ R, Y = Spec R) → Surjective f\nhY : ¬∃ R, Y = Spec R\ni : Y.affineCover.toPreZeroHypercover.1\n⊢ Surjective (pullback.snd f (Y.affineCove... | dsimp only [Scheme.Cover.pullbackHom] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 316,
"column": 8
} | {
"line": 316,
"column": 50
} | {
"line": 316,
"column": 50
} | [
{
"pp": "case h₁\nA : 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\nt : Set (ProjectiveSpectrum 𝒜)\nx : ProjectiveSpectrum 𝒜\nhx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus 𝒜 fs)\nht : t ... | [
"case h₁\nA : 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\nt : Set (ProjectiveSpectrum 𝒜)\nx : ProjectiveSpectrum 𝒜\nhx : x ∈ zeroLocus 𝒜 ↑(vanishingIdeal t)\nfs : Set A\nht' : IsClosed (zeroLocus 𝒜 fs)\nht : fs ⊆ ↑(vanishi... | subset_zeroLocus_iff_subset_vanishingIdeal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.FunLike.Graded | {
"line": 75,
"column": 25
} | {
"line": 75,
"column": 60
} | {
"line": 77,
"column": 0
} | [
{
"pp": "E : Type u_1\nA : Type u_2\nB : Type u_3\nσ : Type u_4\nτ : Type u_5\nι : Type u_6\ninst✝³ : SetLike σ A\ninst✝² : SetLike τ B\n𝒜 : ι → σ\nℬ : ι → τ\ninst✝¹ : EquivLike E A B\ninst✝ : GradedEquivLike E 𝒜 ℬ\ne : E\ni : ι\nx✝ : ↥(ℬ i)\n⊢ ↑(subtypeMap e i ((fun y ↦ ⟨EquivLike.inv e ↑y, ⋯⟩) x✝)) = ↑x✝",
... | [] | exact EquivLike.apply_inv_apply e _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 637,
"column": 61
} | {
"line": 637,
"column": 70
} | {
"line": 637,
"column": 71
} | [
{
"pp": "case pos\nι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubgroupClass σ A\ninst✝² : AddCommMonoid ι\ninst✝¹ : DecidableEq ι\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ι\nhf : f ∈ 𝒜 m\nz : Away 𝒜 f\nk : ℕ\nhk : f ^ k = den z\nk' : ℕ\nhk' : k ≤ k'\... | [
"case pos\nι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubgroupClass σ A\ninst✝² : AddCommMonoid ι\ninst✝¹ : DecidableEq ι\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ι\nhf : f ∈ 𝒜 m\nz : Away 𝒜 f\nk : ℕ\nhk : f ^ k = den z\nk' : ℕ\nhk' : k ≤ k'\nhfk : f ^ k... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 258,
"column": 8
} | {
"line": 258,
"column": 64
} | {
"line": 259,
"column": 8
} | [
{
"pp": "R : CommRingCat\nM✝ : ModuleCat ↑R\nM : (Spec (CommRingCat.of ↑R)).Modules\nf : (↑R)ᵒᵖ\n⊢ ∀ (a : ↑R) (b : a ∈ Submonoid.powers (unop f)),\n IsUnit\n ((algebraMap (↑R)\n (Module.End ↑R\n ↑((modulesSpecToSheaf.obj M).obj.obj (op ((inducedFunctor PrimeSpectrum.basicOpen).obj (u... | [
"R : CommRingCat\nM✝ : ModuleCat ↑R\nM : (Spec (CommRingCat.of ↑R)).Modules\nf : (↑R)ᵒᵖ\n⊢ Submonoid.powers (unop f) ≤\n Submonoid.comap\n (algebraMap (↑R)\n (Module.End ↑R\n ↑((modulesSpecToSheaf.obj M).obj.obj (op ((inducedFunctor PrimeSpectrum.basicOpen).obj (unop f))))))\n (IsUnit.s... | change Submonoid.powers _ ≤ (IsUnit.submonoid _).comap _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 283,
"column": 2
} | {
"line": 286,
"column": 5
} | {
"line": 288,
"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\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nm' : ℕ\ng : A\ng_deg : g ∈ 𝒜 m'\nhm' : 0 < m'\nx : A\nhx : x = f * g\n⊢ (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' hx).hom ≫ S... | [] | rw [← cancel_mono (awayι 𝒜 g g_deg hm'), ← Limits.pullback.condition,
← pullbackAwayιIso_hom_awayι 𝒜 f_deg hm g_deg hm' hx,
Category.assoc, SpecMap_awayMap_awayι]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 283,
"column": 2
} | {
"line": 286,
"column": 5
} | {
"line": 288,
"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\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nm' : ℕ\ng : A\ng_deg : g ∈ 𝒜 m'\nhm' : 0 < m'\nx : A\nhx : x = f * g\n⊢ (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' hx).hom ≫ S... | [] | rw [← cancel_mono (awayι 𝒜 g g_deg hm'), ← Limits.pullback.condition,
← pullbackAwayιIso_hom_awayι 𝒜 f_deg hm g_deg hm' hx,
Category.assoc, SpecMap_awayMap_awayι]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 267,
"column": 10
} | {
"line": 267,
"column": 66
} | {
"line": 268,
"column": 10
} | [
{
"pp": "case map_unit\nR : 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⊢ ∀ (a : ↑R) (b : a ∈ Submonoid.powers (unop f)),\n IsUnit\n ((algebraMap (↑R)\n (Mo... | [
"case map_unit\nR : 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⊢ Submonoid.powers (unop f) ≤\n Submonoid.comap\n (algebraMap (↑R)\n (Module.End ↑R ↑(((inducedFun... | change Submonoid.powers _ ≤ (IsUnit.submonoid _).comap _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 315,
"column": 6
} | {
"line": 315,
"column": 62
} | {
"line": 316,
"column": 6
} | [
{
"pp": "case map_unit\nR : CommRingCat\nM✝ : ModuleCat ↑R\nM N : (Spec (CommRingCat.of ↑R)).Modules\nf : M ⟶ N\nr : (InducedCategory (TopologicalSpace.Opens ↥(Spec (CommRingCat.of ↑R))) PrimeSpectrum.basicOpen)ᵒᵖ\n⊢ ∀ (a : ↑R) (b : a ∈ Submonoid.powers (unop r)),\n IsUnit\n ((algebraMap (↑R)\n ... | [
"case map_unit\nR : CommRingCat\nM✝ : ModuleCat ↑R\nM N : (Spec (CommRingCat.of ↑R)).Modules\nf : M ⟶ N\nr : (InducedCategory (TopologicalSpace.Opens ↥(Spec (CommRingCat.of ↑R))) PrimeSpectrum.basicOpen)ᵒᵖ\n⊢ Submonoid.powers (unop r) ≤\n Submonoid.comap\n (algebraMap (↑R)\n (Module.End ↑R\n ... | change Submonoid.powers _ ≤ (IsUnit.submonoid _).comap _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 346,
"column": 6
} | {
"line": 346,
"column": 62
} | {
"line": 347,
"column": 6
} | [
{
"pp": "case map_unit\nR : CommRingCat\nM✝ M : ModuleCat ↑R\nr : (InducedCategory (Opens ↥(Spec (CommRingCat.of ↑R))) basicOpen)ᵒᵖ\n⊢ ∀ (a : ↑R) (b : a ∈ Submonoid.powers (unop r)),\n IsUnit\n ((algebraMap (↑R)\n (Module.End ↑R\n ↑(((inducedFunctor basicOpen).op ⋙ (modulesSpecToShea... | [
"case map_unit\nR : CommRingCat\nM✝ M : ModuleCat ↑R\nr : (InducedCategory (Opens ↥(Spec (CommRingCat.of ↑R))) basicOpen)ᵒᵖ\n⊢ Submonoid.powers (unop r) ≤\n Submonoid.comap\n (algebraMap (↑R)\n (Module.End ↑R\n ↑(((inducedFunctor basicOpen).op ⋙ (modulesSpecToSheaf.obj ((tilde.functor R).obj... | change Submonoid.powers _ ≤ (IsUnit.submonoid _).comap _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 1035,
"column": 6
} | {
"line": 1035,
"column": 90
} | {
"line": 1036,
"column": 6
} | [
{
"pp": "case e_y.inr\nA : 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 : ... | [
"case e_y.inr\nA : 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 ∈ �... | rw [← mul_le_mul_iff_of_pos_right hd, ← smul_eq_mul (a := a), ← hai, Finset.sum_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 1039,
"column": 2
} | {
"line": 1039,
"column": 25
} | {
"line": 1040,
"column": 2
} | [
{
"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'... | rw [H, SetLike.mem_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 463,
"column": 6
} | {
"line": 470,
"column": 59
} | {
"line": 471,
"column": 6
} | [
{
"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\nx y : A\nx✝¹ : IsHomogeneousElem 𝒜 x\nx✝ : IsHomogeneousElem 𝒜 y\nhxy : x * y ∈ as... | [
"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\nx y : A\nx✝¹ : IsHomogeneousElem 𝒜 x\nx✝ : IsHomogeneousElem 𝒜 y\nhxy : x * y ∈ asIdeal f_deg ... | · apply q.2.mem_or_mem; convert! hxy (nx + ny) using 1
dsimp
simp_rw [decompose_of_mem_same 𝒜 hnx, decompose_of_mem_same 𝒜 hny,
decompose_of_mem_same 𝒜 (SetLike.GradedMonoid.toGradedMul.mul_mem hnx hny),
mul_pow, pow_add]
simp only [HomogeneousLocalization.ext_iff_val, Hom... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 503,
"column": 50
} | {
"line": 507,
"column": 70
} | {
"line": 509,
"column": 0
} | [
{
"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\nx : ↑↑(Proj.restrict ⋯).toPresheafedSpace\n⊢ FromSpec.toFun f_deg hm ((ConcreteCategory.hom (toSpec 𝒜 f)) x) = x",
"p... | [] | by
refine Subtype.ext <| ProjectiveSpectrum.ext <| HomogeneousIdeal.ext' ?_
intro i z hzi
refine (FromSpec.mem_carrier_iff_of_mem f_deg hm _ _ hzi).trans ?_
exact (ToSpec.mk_mem_carrier _ _).trans (x.1.2.pow_mem_iff_mem m hm) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Sites.Small | {
"line": 109,
"column": 75
} | {
"line": 125,
"column": 50
} | {
"line": 127,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\ninst✝ : P.RespectsIso\n⊢ overGrothendieckTopology P S = (overPretopology P S).toGrothendieck",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
... | [] | by
ext X R
rw [GrothendieckTopology.mem_over_iff]
constructor
· intro hR
obtain ⟨𝒰, hle⟩ := exists_cover_of_mem_grothendieckTopology hR
rw [mem_grothendieckTopology_iff] at hR
letI (i : 𝒰.I₀) : (𝒰.X i).Over S := { hom := 𝒰.f i ≫ X.hom }
letI : 𝒰.Over S :=
{ over := inferInstance
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 603,
"column": 28
} | {
"line": 603,
"column": 96
} | {
"line": 604,
"column": 4
} | [
{
"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\nx✝² x✝¹ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nx✝ : failed to pretty print expression (use 'set_opti... | [] | simp only [map_add, HomogeneousLocalization.val_add, Proj.add_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Elements | {
"line": 93,
"column": 42
} | {
"line": 93,
"column": 73
} | {
"line": 93,
"column": 74
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nA : C ⥤ Type w\nI : Type u₁\ninst✝³ : Category.{v₁, u₁} I\ninst✝² : Small.{w, u₁} I\nF : I ⥤ A.Elements\ninst✝¹ : HasLimitsOfShape I C\ninst✝ : PreservesLimitsOfShape I A\ni i' : I\nf : i ⟶ i'\n⊢ ↑(((Functor.const I).obj ⟨limit (F ⋙ π A), liftedConeElement F⟩).ma... | [] | simpa using! (limit.w _ _).symm | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 58
} | {
"line": 308,
"column": 2
} | [
{
"pp": "case refine_1.refine_2\nC₀ : Type u₀\nC : Type u\ninst✝² : Category.{v₀, u₀} C₀\ninst✝¹ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\ninst✝ : IsDenseSubsite J₀ J F\nX : C\ndata : F.OneHypercoverDenseData J₀ J X\nX₀ : C₀\nf : F.obj X₀ ⟶ X\nthis✝ : F.IsCoverDe... | [] | · rw [w₁, assoc, ← reassoc_of% fac, hb.some.fac_assoc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Point.Basic | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 28
} | {
"line": 232,
"column": 2
} | [
{
"pp": "case mp\nC : Type u\ninst✝⁶ : Category.{v, u} C\nJ : GrothendieckTopology C\nΦ : J.Point\nA : Type u'\ninst✝⁵ : Category.{v', u'} A\ninst✝⁴ : HasColimitsOfSize.{w, w, v', u'} A\nFC : A → A → Type u_1\nCC : A → Type w'\ninst✝³ : (X Y : A) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝² : ConcreteCategory A FC\... | [] | exact ⟨Y, f, y, hf, hf'⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 144,
"column": 18
} | {
"line": 144,
"column": 40
} | {
"line": 144,
"column": 41
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) A\nx : Finset A\nhx : Algebra.adjoin ↥(𝒜 0) ↑x = ⊤\nd : (i : A) → i ∈ x → ℕ\nhd : ∀ (i : A) (a : i ∈ x), d i a ≠ 0\nhxd : ∀ (i ... | [
"σ : Type u_1\nA : Type u_2\ninst✝⁴ : CommRing A\ninst✝³ : SetLike σ A\ninst✝² : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) A\nx : Finset A\nhx : Algebra.adjoin ↥(𝒜 0) ↑x = ⊤\nd : (i : A) → i ∈ x → ℕ\nhd : ∀ (i : A) (a : i ∈ x), d i a ≠ 0\nhxd : ∀ (i : A) (a : i ... | Proj.awayι_toSpecZero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 582,
"column": 4
} | {
"line": 583,
"column": 71
} | {
"line": 584,
"column": 4
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize... | [
"C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize.{w, w, v', ... | refine Presheaf.IsSheaf.hom_ext G₀.property
⟨_, cover_lift F J₀ _ (J.pullback_stable a (data Y).mem₀)⟩ _ _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 77
} | {
"line": 74,
"column": 2
} | [
{
"pp": "S : Scheme\nX : Scheme\nE : PreZeroHypercover X\nF : (i : E.I₀) → PreZeroHypercover (E.X i)\nhE : qcCoverFamily.property E\nhF : ∀ (i : E.I₀), qcCoverFamily.property (F i)\n⊢ qcCoverFamily.property (E.bind F)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CategoryTheory.Pr... | [
"S : Scheme\nX : Scheme\nE : PreZeroHypercover X\nF : (i : E.I₀) → PreZeroHypercover (E.X i)\nhE : QuasiCompactCover E\nhF : ∀ (i : E.I₀), QuasiCompactCover (F i)\n⊢ QuasiCompactCover (E.bind F)"
] | simp only [qcCoverFamily_property, Scheme.quasiCompactCover_iff] at hE hF ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 77
} | {
"line": 79,
"column": 2
} | [
{
"pp": "S : Scheme\nX : Scheme\nE F : PreZeroHypercover X\nhE : qcCoverFamily.property E\nhF : qcCoverFamily.property F\n⊢ qcCoverFamily.property (E.sum F)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.QuasiCompactCover",
"AlgebraicGeometry... | [
"S : Scheme\nX : Scheme\nE F : PreZeroHypercover X\nhE : QuasiCompactCover E\nhF : QuasiCompactCover F\n⊢ QuasiCompactCover (E.sum F)"
] | simp only [qcCoverFamily_property, Scheme.quasiCompactCover_iff] at hE hF ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 59
} | {
"line": 104,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : Surjective f\ninst✝ : QuasiCompact f\nE : Cover (precoverage ⊤) Y := cover f trivial\n⊢ qcCoverFamily.property (cover f trivial).toPreZeroHypercover",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProp... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : Surjective f\ninst✝ : QuasiCompact f\nE : Cover (precoverage ⊤) Y := cover f trivial\n⊢ QuasiCompactCover (cover f trivial).toPreZeroHypercover"
] | simp only [qcCoverFamily_property, quasiCompactCover_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.EffectiveEpimorphic | {
"line": 163,
"column": 4
} | {
"line": 166,
"column": 58
} | {
"line": 167,
"column": 2
} | [
{
"pp": "case mp\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : Y ⟶ X\n⊢ (Presieve.singleton f).EffectiveEpimorphic → EffectiveEpi f",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"CategoryTheory.Over",
"CategoryTheory.Sieve.generateSingleton_eq",
"congrArg",
"... | [] | intro (h : Nonempty _)
rw [Sieve.generateSingleton_eq] at h
constructor
apply Nonempty.map (effectiveEpiStructOfIsColimit _) h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.EffectiveEpimorphic | {
"line": 163,
"column": 4
} | {
"line": 166,
"column": 58
} | {
"line": 167,
"column": 2
} | [
{
"pp": "case mp\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : Y ⟶ X\n⊢ (Presieve.singleton f).EffectiveEpimorphic → EffectiveEpi f",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"CategoryTheory.Over",
"CategoryTheory.Sieve.generateSingleton_eq",
"congrArg",
"... | [] | intro (h : Nonempty _)
rw [Sieve.generateSingleton_eq] at h
constructor
apply Nonempty.map (effectiveEpiStructOfIsColimit _) h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 768,
"column": 2
} | {
"line": 769,
"column": 12
} | {
"line": 770,
"column": 2
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize... | [
"C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize.{w, w, v', ... | rw [← cancel_mono (presheafObjObjIso data G₀ ((data X).X i)).inv, assoc, Iso.hom_inv_id,
comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 43
} | {
"line": 86,
"column": 4
} | [
{
"pp": "P : MorphismProperty Scheme\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\nF : Schemeᵒᵖ ⥤ Type u_1\ninst✝ : IsZariskiLocalAtSource P\nx✝ :\n Presieve.IsSheaf zariskiTopology F ∧\n ∀ {R S : CommRingCat} (f : R ⟶ S),\n P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsShea... | [
"P : MorphismProperty Scheme\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\nF : Schemeᵒᵖ ⥤ Type u_1\ninst✝ : IsZariskiLocalAtSource P\nx✝ :\n Presieve.IsSheaf zariskiTopology F ∧\n ∀ {R S : CommRingCat} (f : R ⟶ S),\n P (Spec.map f) → Surjective (Spec.map f) → Presieve.IsSheafFor F (Pres... | obtain ⟨φ, hφ⟩ := Spec.map_surjective f | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Sites.ConstantSheaf | {
"line": 61,
"column": 2
} | {
"line": 65,
"column": 70
} | {
"line": 67,
"column": 0
} | [
{
"pp": "T : Type v\ninst✝ : TopologicalSpace T\n⊢ Presheaf.IsSheaf Scheme.zariskiTopology (continuousMapPresheaf T)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"ULift.topologicalSpace",
"CategoryTheory.Presieve.IsSheaf",
"CategoryTheory.Functor.op",
"Eq.mpr",
... | [] | rw [Presheaf.isSheaf_of_iso_iff (continuousMapPresheafIsoUlift T)]
apply Scheme.forgetToTop.op_comp_isSheaf_of_isSheaf _ TopCat.grothendieckTopology
apply TopCat.uliftFunctor.op_comp_isSheaf_of_isSheaf _ TopCat.grothendieckTopology
rw [isSheaf_iff_isSheaf_of_type]
exact GrothendieckTopology.Subcanonical.isSheaf... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.