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.Topology.JacobsonSpace | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 16
} | [
{
"pp": "case mpr\nX : Type u_1\ninst✝ : TopologicalSpace X\n⊢ (∀ (Z : Set X), Z.Nonempty → IsLocallyClosed Z → (Z ∩ closedPoints X).Nonempty) →\n ∀ {Z : Set X}, IsClosed[inst✝] Z → closure[inst✝] (Z ∩ closedPoints X) = Z",
"usedConstants": [
"closedPoints",
"Set.instInter",
"IsLocallyC... | intro H Z hZ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Spectrum.Prime.Jacobson | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsJacobsonRing R\nx : PrimeSpectrum R\ntfae_1_to_2 : IsOpen {x} → IsClopen {x}\ntfae_2_to_3 : IsClopen {x} → IsClosed {x} ∧ StableUnderGeneralization {x}\nh₁ : IsMax x\nh₂ : StableUnderGeneralization {x}\nthis : {x} = (⋃ p ∈ {p | I... | exact (finite_setOf_isMin R).subset fun x h ↦ h.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.Separated | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 64
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\n𝒱 : (i : 𝒰.I₀) → (pullback f (𝒰.f i)).OpenCover\nhf : Function.Injective ⇑f\nx : ↥(pullback.diagonalObj f)\nH : (pullback.fst f f) x = (pullback.snd f f) x\ni : 𝒰.I₀ := Cover.idx 𝒰 (f ((pullback.fst f f) x))\ny : ↥(𝒰.X i)\nhy : (𝒰.f i) y = f ((pullback.... | obtain ⟨w : (𝒱 i).X j, hy : (𝒱 i).f j w = z⟩ := (𝒱 i).covers z | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.Separated | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 64
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n𝒰 : Y.OpenCover\n𝒱 : (i : 𝒰.I₀) → (pullback f (𝒰.f i)).OpenCover\nx : ↥X\ni : 𝒰.I₀ := Cover.idx 𝒰 (f x)\ny : ↥(𝒰.X i)\nhy : (𝒰.f i) y = f x\nz : ↥(pullback f (𝒰.f i))\nhz₁ : (pullback.fst f (𝒰.f i)) z = x\nhz₂ : (pullback.snd f (𝒰.f i)) z = y\nj : (𝒱 i).I₀ := Cover.... | obtain ⟨w : (𝒱 i).X j, hy : (𝒱 i).f j w = z⟩ := (𝒱 i).covers z | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 155,
"column": 2
} | {
"line": 156,
"column": 55
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsClosedImmersion f\n⊢ IsIso (Scheme.Hom.toImage f)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.Scheme.Hom.image",
"AlgebraicGeometry.Scheme.Hom.imageι",
"CategoryTheory.CategoryStruct.toQuiver",
... | have := @of_comp_isClosedImmersion _ _ _ f.toImage f.imageι inferInstance
(by rw [Scheme.Hom.toImage_imageι]; infer_instance) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed | {
"line": 178,
"column": 43
} | {
"line": 178,
"column": 72
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : UniversallyClosed f\ninst✝ : IsDominant f\n⊢ Set.range ⇑f = Set.univ",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.denseRange",
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Presheafed... | ← f.denseRange.closure_range, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.FinitePresentation | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 68
} | [
{
"pp": "case inr\nX : Scheme\ns : Set ↥X\nhs : IsLocallyConstructible s\nR : CommRingCat\nf : X ⟶ Spec R\nhf : LocallyOfFinitePresentation f\ninst✝ : QuasiCompact f\nthis :\n ∀ {X : Scheme} {s : Set ↥X},\n IsLocallyConstructible s →\n ∀ (R : CommRingCat) (f : X ⟶ Spec R) [hf : LocallyOfFinitePresentat... | rw [← 𝒰.isOpenCover_opensRange.iUnion_inter s, Set.image_iUnion] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.Immersion | {
"line": 128,
"column": 2
} | {
"line": 129,
"column": 62
} | [
{
"pp": "case hP\nX Y Z : Scheme\nf : X ⟶ Y\n⊢ ∀ {α β : Type u_1} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] (f : α → β) {ι : Type u_1}\n (U : ι → TopologicalSpace.Opens β),\n TopologicalSpace.IsOpenCover U →\n Continuous f →\n (IsLocallyClosed (Set.range f) ↔ ∀ (i : ι), IsLocal... | · simp_rw [Set.range_restrictPreimage]
exact fun _ _ _ hU _ ↦ hU.isLocallyClosed_iff_coe_preimage | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.AlgClosed.Basic | {
"line": 35,
"column": 4
} | {
"line": 42,
"column": 18
} | [
{
"pp": "X Y : Scheme\nK : Type u\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : LocallyOfFiniteType f\nx : ↥X\nhx : IsClosed {x}\n⊢ IsIso (Spec.preimage (X.fromSpecResidueField x ≫ f))",
"usedConstants": [
"AlgebraicGeometry.IsIntegralHom",
"CommRingCat.forg... | have : IsFinite (X.fromSpecResidueField x ≫ f) := by
rw [isClosed_singleton_iff_isClosedImmersion] at hx
rw [isFinite_iff_locallyOfFiniteType_of_jacobsonSpace]
infer_instance
rw [ConcreteCategory.isIso_iff_bijective]
refine IsAlgClosed.ringHom_bijective_of_isIntegral _ ?_
rw [← IsIntegralH... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AlgClosed.Basic | {
"line": 35,
"column": 4
} | {
"line": 42,
"column": 18
} | [
{
"pp": "X Y : Scheme\nK : Type u\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\nf : X ⟶ Spec (CommRingCat.of K)\ninst✝ : LocallyOfFiniteType f\nx : ↥X\nhx : IsClosed {x}\n⊢ IsIso (Spec.preimage (X.fromSpecResidueField x ≫ f))",
"usedConstants": [
"AlgebraicGeometry.IsIntegralHom",
"CommRingCat.forg... | have : IsFinite (X.fromSpecResidueField x ≫ f) := by
rw [isClosed_singleton_iff_isClosedImmersion] at hx
rw [isFinite_iff_locallyOfFiniteType_of_jacobsonSpace]
infer_instance
rw [ConcreteCategory.isIso_iff_bijective]
refine IsAlgClosed.ringHom_bijective_of_isIntegral _ ?_
rw [← IsIntegralH... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Noetherian | {
"line": 207,
"column": 2
} | {
"line": 215,
"column": 32
} | [
{
"pp": "X Z : Scheme\ninst✝¹ : IsLocallyNoetherian X\nf : Z ⟶ X\ninst✝ : IsOpenImmersion f\n⊢ QuasiCompact f",
"usedConstants": [
"Iff.mpr",
"TopologicalSpace.Opens.map_coe",
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Pres... | apply quasiCompact_iff_forall_isAffineOpen.mpr
intro U hU
rw [Opens.map_coe, ← Set.preimage_inter_range]
apply f.isOpenEmbedding.isInducing.isCompact_preimage'
· apply (noetherianSpace_set_iff _).mp
· convert! noetherianSpace_of_isAffineOpen U hU
apply IsLocallyNoetherian.component_noetherian ⟨U, hU⟩
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Noetherian | {
"line": 207,
"column": 2
} | {
"line": 215,
"column": 32
} | [
{
"pp": "X Z : Scheme\ninst✝¹ : IsLocallyNoetherian X\nf : Z ⟶ X\ninst✝ : IsOpenImmersion f\n⊢ QuasiCompact f",
"usedConstants": [
"Iff.mpr",
"TopologicalSpace.Opens.map_coe",
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Pres... | apply quasiCompact_iff_forall_isAffineOpen.mpr
intro U hU
rw [Opens.map_coe, ← Set.preimage_inter_range]
apply f.isOpenEmbedding.isInducing.isCompact_preimage'
· apply (noetherianSpace_set_iff _).mp
· convert! noetherianSpace_of_isAffineOpen U hU
apply IsLocallyNoetherian.component_noetherian ⟨U, hU⟩
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HopkinsLevitzki | {
"line": 186,
"column": 2
} | {
"line": 191,
"column": 80
} | [
{
"pp": "R : Type u_3\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\n⊢ IsArtinianRing R ↔ IsNilpotent (IsLocalRing.maximalIdeal R)",
"usedConstants": [
"Eq.mpr",
"Submodule",
"False",
"Semiring.toModule",
"IsArtinianRing",
"IsScalarTower.right",... | rw [isArtinianRing_iff_krullDimLE_zero,
Ideal.FG.isNilpotent_iff_le_nilradical (IsNoetherian.noetherian _),
← and_iff_left (a := Ring.KrullDimLE 0 R) ‹IsLocalRing R›,
(Ring.krullDimLE_zero_and_isLocalRing_tfae R).out 0 3 rfl rfl,
IsLocalRing.isMaximal_iff, le_antisymm_iff, and_iff_right]
exact IsLocal... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HopkinsLevitzki | {
"line": 186,
"column": 2
} | {
"line": 191,
"column": 80
} | [
{
"pp": "R : Type u_3\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\n⊢ IsArtinianRing R ↔ IsNilpotent (IsLocalRing.maximalIdeal R)",
"usedConstants": [
"Eq.mpr",
"Submodule",
"False",
"Semiring.toModule",
"IsArtinianRing",
"IsScalarTower.right",... | rw [isArtinianRing_iff_krullDimLE_zero,
Ideal.FG.isNilpotent_iff_le_nilradical (IsNoetherian.noetherian _),
← and_iff_left (a := Ring.KrullDimLE 0 R) ‹IsLocalRing R›,
(Ring.krullDimLE_zero_and_isLocalRing_tfae R).out 0 3 rfl rfl,
IsLocalRing.isMaximal_iff, le_antisymm_iff, and_iff_right]
exact IsLocal... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.SpreadingOut | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 82
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nx : ↥X\ninst✝ : IsOpenImmersion f\nH✝ : Y.IsGermInjectiveAt (f x)\nU : Y.Opens\nhxU : f x ∈ U\nhU : IsAffineOpen U\nhU' : U ≤ Scheme.Hom.opensRange f\nH : Function.Injective ⇑(ConcreteCategory.hom (Y.presheaf.germ U (f x) hxU))\n⊢ X.IsGermInjectiveAt x",
"usedConstants": [
... | obtain ⟨V, hV⟩ := (IsOpenImmersion.affineOpensEquiv f).surjective ⟨⟨U, hU⟩, hU'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 162,
"column": 20
} | {
"line": 162,
"column": 91
} | [
{
"pp": "n : Type v\nS X : Scheme\ninst✝ : X.Over S\nv : n → ↑Γ(X, ⊤)\n⊢ (fun f i ↦ (ConcreteCategory.hom (Scheme.Hom.appTop ↑f)) (coord S i)) ((fun v ↦ ⟨homOfVector (X ↘ S) v, ⋯⟩) v) = v",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry... | ext i; simp [-TopologicalSpace.Opens.map_top, homOfVector_appTop_coord] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 162,
"column": 20
} | {
"line": 162,
"column": 91
} | [
{
"pp": "n : Type v\nS X : Scheme\ninst✝ : X.Over S\nv : n → ↑Γ(X, ⊤)\n⊢ (fun f i ↦ (ConcreteCategory.hom (Scheme.Hom.appTop ↑f)) (coord S i)) ((fun v ↦ ⟨homOfVector (X ↘ S) v, ⋯⟩) v) = v",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry... | ext i; simp [-TopologicalSpace.Opens.map_top, homOfVector_appTop_coord] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 305,
"column": 16
} | {
"line": 305,
"column": 71
} | [
{
"pp": "n : Type v\nR S : CommRingCat\nφ : R ⟶ S\ni✝ : n\n| (ConcreteCategory.hom (Scheme.Hom.app (Spec.map (CommRingCat.ofHom (MvPolynomial.map (CommRingCat.Hom.hom φ)))) ⊤))\n ((ConcreteCategory.hom (Scheme.Hom.app (SpecIso n R).inv ⊤)) (coord (Spec R) i✝))",
"usedConstants": [
"AlgebraicGeometr... | enter [2]; tactic => exact SpecIso_inv_appTop_coord _ _ | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 305,
"column": 16
} | {
"line": 305,
"column": 71
} | [
{
"pp": "n : Type v\nR S : CommRingCat\nφ : R ⟶ S\ni✝ : n\n| (ConcreteCategory.hom (Scheme.Hom.app (Spec.map (CommRingCat.ofHom (MvPolynomial.map (CommRingCat.Hom.hom φ)))) ⊤))\n ((ConcreteCategory.hom (Scheme.Hom.app (SpecIso n R).inv ⊤)) (coord (Spec R) i✝))",
"usedConstants": [
"AlgebraicGeometr... | enter [2]; tactic => exact SpecIso_inv_appTop_coord _ _ | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 396,
"column": 33
} | {
"line": 403,
"column": 87
} | [
{
"pp": "n : Type v\nS X : Scheme\ninst✝¹ : X.Over S\ninst✝ : IsEmpty n\n⊢ isomorphisms Scheme (terminal.from (Spec ℤ[n]))",
"usedConstants": [
"Iff.mpr",
"CommRingCat.forgetReflectIsos",
"Eq.mpr",
"AlgebraicGeometry.Spec",
"Nat.instMulZeroClass",
"AlgebraicGeometry.Sheaf... | by
rw [← terminal.comp_from (Spec.map (CommRingCat.ofHom C))]
apply IsStableUnderComposition.comp_mem
· rw [HasAffineProperty.iff_of_isAffine (P := isomorphisms _), ← isomorphisms,
← arrow_mk_iso_iff (isomorphisms _) (arrowIsoΓSpecOfIsAffine _)]
exact ⟨inferInstance, (ConcreteCategory.isIso_iff_bijectiv... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 397,
"column": 57
} | {
"line": 397,
"column": 97
} | [
{
"pp": "case mp\nX Y S : Scheme\ninst✝¹ : X.Over S\ninst✝ : Y.Over S\nf : X ⤏ Y\nh : IsOver S f\ng : X.PartialMap Y\nhg : PartialMap.IsOver S g\ne : g.toRationalMap = f\n⊢ (g.compHom (Y ↘ S)).toRationalMap = (Hom.toPartialMap (X ↘ S)).toRationalMap",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeome... | PartialMap.isOver_iff_eq_restrict.mp hg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 531,
"column": 2
} | {
"line": 531,
"column": 46
} | [
{
"pp": "X Y Z S : Scheme\nsX : X ⟶ S\nsY : Y ⟶ S\ninst✝¹ : IsReduced X\ninst✝ : Y.IsSeparated\nf : X ⤏ Y\nx y : f.openCoverDomain.I₀\ng : f.openCoverDomain.I₀ → X.PartialMap Y := fun x ↦ Exists.choose ⋯\nhg₁ : ∀ (x : f.openCoverDomain.I₀), (g x).toRationalMap = f\nhg₂ : ∀ (x : f.openCoverDomain.I₀), (g x).doma... | change _ ≫ _ ≫ (g x).hom = _ ≫ _ ≫ (g y).hom | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicGeometry.Cover.Directed | {
"line": 167,
"column": 4
} | {
"line": 174,
"column": 70
} | [
{
"pp": "P : MorphismProperty Scheme\nX : Scheme\n𝒰✝ : Cover (precoverage P) X\ninst✝⁴ : Category.{v_1, ?u.23800} 𝒰✝.I₀\ninst✝³ : 𝒰✝.LocallyDirected\ninst✝² : P.IsStableUnderBaseChange\n𝒰 : Cover (precoverage P) X\ninst✝¹ : Category.{v_2, ?u.28653} 𝒰.I₀\ninst✝ : 𝒰.LocallyDirected\nY : Scheme\nf : Y ⟶ X\ni... | have (k : 𝒰.I₀) (hki : k ⟶ i) (hkj : k ⟶ j) :
(pullback.lift
(pullback.map f (𝒰.f k) f (𝒰.f i) (𝟙 Y) (𝒰.trans hki) (𝟙 X) (by simp) (by simp))
(pullback.map f (𝒰.f k) f (𝒰.f j) (𝟙 Y) (𝒰.trans hkj) (𝟙 X) (by simp) (by simp))
(by simp)) =
pullback.map _ _ _ _ (�... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 305,
"column": 48
} | {
"line": 305,
"column": 70
} | [
{
"pp": "R₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span ... | by simpa [hi] using hc | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ColimitsOver | {
"line": 207,
"column": 4
} | {
"line": 209,
"column": 14
} | [
{
"pp": "case refine_2.h.h\nP : MorphismProperty Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : P.IsMultiplicative\nS : Scheme\nJ : Type u_1\ninst✝⁶ : Category.{v_1, u_1} J\nD : J ⥤ P.Over ⊤ S\n𝒰 : S.OpenCover\ninst✝⁵ : Category.{v_2, ?u.98729} 𝒰.I₀\ninst✝⁴ : LocallyDirected 𝒰\nd : ColimitGluingData D ... | have : (𝒱 a).f i ≫ (D.map f).left =
((D ⋙ Over.pullback _ _ (𝒰.f i)).map f).left ≫ (𝒱 b).f i := by
simp [𝒱] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 419,
"column": 10
} | {
"line": 419,
"column": 40
} | [
{
"pp": "R₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : R →ₐ[R₀] R ⧸ Ideal.span ... | unfold coeffSubmodule powBound | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 426,
"column": 6
} | {
"line": 426,
"column": 21
} | [
{
"pp": "case refine_2.refine_3\nR₀ : Type u_1\ninst✝² : CommRing R₀\nn : ℕ\nR : Type u_6\ninst✝¹ : CommRing R\ninst✝ : Algebra R₀ R\nc : R\ni : Fin n\ne : InductionObj R n\nhi : c = (e.val i).leadingCoeff\nhc : c ≠ 0\nq₁ : R →ₐ[R₀] Localization.Away c := IsScalarTower.toAlgHom R₀ R (Localization.Away c)\nq₂ : ... | unfold powBound | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 306,
"column": 2
} | {
"line": 327,
"column": 18
} | [
{
"pp": "I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\n⊢ TopologicalSpace.Opens.IsBasis {x | ∃ i V, ∃ (_ : IsAffineOpen V), c.π.app i ⁻¹ᵁ V = x}",
"usedConstants": [
"AlgebraicGeometr... | refine TopologicalSpace.Opens.isBasis_iff_nbhd.mpr fun {U x} hxU ↦ ?_
obtain ⟨i⟩ := IsCofiltered.nonempty (C := I)
obtain ⟨_, ⟨V, hV : IsAffineOpen V, rfl⟩, hxV, -⟩ :=
(D.obj i).isBasis_affineOpens.exists_subset_of_mem_open (Set.mem_univ (c.π.app i x)) isOpen_univ
have (j : _) : IsAffine ((opensDiagram D i V)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 306,
"column": 2
} | {
"line": 327,
"column": 18
} | [
{
"pp": "I : Type u\ninst✝² : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝¹ : IsCofiltered I\ninst✝ : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\n⊢ TopologicalSpace.Opens.IsBasis {x | ∃ i V, ∃ (_ : IsAffineOpen V), c.π.app i ⁻¹ᵁ V = x}",
"usedConstants": [
"AlgebraicGeometr... | refine TopologicalSpace.Opens.isBasis_iff_nbhd.mpr fun {U x} hxU ↦ ?_
obtain ⟨i⟩ := IsCofiltered.nonempty (C := I)
obtain ⟨_, ⟨V, hV : IsAffineOpen V, rfl⟩, hxV, -⟩ :=
(D.obj i).isBasis_affineOpens.exists_subset_of_mem_open (Set.mem_univ (c.π.app i x)) isOpen_univ
have (j : _) : IsAffine ((opensDiagram D i V)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 71
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : CharP R 3\n⊢ W.c₄ ^ 3 = W.c₆ ^ 2",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | linear_combination -W.c_relation + 576 * W.Δ * CharP.cast_eq_zero R 3 | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 71
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : CharP R 3\n⊢ W.c₄ ^ 3 = W.c₆ ^ 2",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | linear_combination -W.c_relation + 576 * W.Δ * CharP.cast_eq_zero R 3 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 71
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : CharP R 3\n⊢ W.c₄ ^ 3 = W.c₆ ^ 2",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | linear_combination -W.c_relation + 576 * W.Δ * CharP.cast_eq_zero R 3 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 49
} | [
{
"pp": "case a\nR : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : CharP R 2\n⊢ 4 = 0",
"usedConstants": [
"CharP.cast_eq_zero",
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.Ring.Common.neg_zero",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne"... | · linear_combination 2 * CharP.cast_eq_zero R 2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 448,
"column": 6
} | {
"line": 449,
"column": 68
} | [
{
"pp": "I : Type u\ninst✝⁷ : Category.{u, u} I\ninst✝⁶ : IsCofiltered I\ni j : I\nR : CommRingCat\ninst✝⁵ : IsAffine (Spec R)\nS : CommRingCat\ninst✝⁴ : IsAffine (Spec S)\nφ : R ⟶ S\ninst✝³ : LocallyOfFiniteType (Spec.map φ)\nD : I ⥤ CommRingCatᵒᵖ\nc : Cone (D ⋙ Scheme.Spec)\nhc : IsLimit c\ninst✝² : ∀ (i : I)... | simp only [coconeLeftOpOfCone_pt, Functor.const_obj_obj,
Functor.leftOp_obj, coconeLeftOpOfCone_ι_app, Spec.map_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 517,
"column": 12
} | {
"line": 517,
"column": 76
} | [
{
"pp": "case pos\nR✝ : Type u_2\ninst✝³ : CommRing R✝\nn : ℕ\nR : Type u_2\ninst✝² : CommRing R\nc : InductionObj R n\ni j : Fin n\nhi : (c.val i).Monic\nhle : (c.val i).degree ≤ (c.val j).degree\nhne : i ≠ j\nH :\n ∀ {R₀ : Type u_1} [inst : CommRing R₀] [inst_1 : Algebra R₀ R],\n Statement R₀ R n { val :=... | · subst hkj; gcongr; exact (degree_modByMonic_le _ hi).trans hle | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 54
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\n⊢ (C • W).c₄ = ↑C.u⁻¹ ^ 4 * W.c₄",
"usedConstants": [
"WeierstrassCurve.VariableChange.r",
"Units.val",
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"congrArg... | simp only [c₄, variableChange_b₂, variableChange_b₄] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 289,
"column": 2
} | {
"line": 290,
"column": 10
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW : Affine R\nS : Type s\ninst✝ : CommRing S\nf : R →+* S\n⊢ (W.map f).polynomialY = Polynomial.map (mapRingHom f) W.polynomialY",
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Poly... | simp only [polynomialY]
map_simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Basic | {
"line": 289,
"column": 2
} | {
"line": 290,
"column": 10
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW : Affine R\nS : Type s\ninst✝ : CommRing S\nf : R →+* S\n⊢ (W.map f).polynomialY = Polynomial.map (mapRingHom f) W.polynomialY",
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Poly... | simp only [polynomialY]
map_simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 175,
"column": 31
} | {
"line": 176,
"column": 34
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Affine F\ninst✝ : DecidableEq F\nx₁ x₂ y₁ y₂ : F\nhx : x₁ = x₂\nhy : y₁ = W.negY x₂ y₂\n⊢ W.slope x₁ x₂ y₁ y₂ = 0",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"WeierstrassCurve.Affine.slope._proof_2",
"AddGroupWithOne.toAddG... | by
rw [slope, if_pos hx, if_pos hy] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 446,
"column": 4
} | {
"line": 446,
"column": 85
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat iX\nhUS : IsAffineOpen US\nhUX : IsAffineOpen UX\nhUT : IsCompact ... | rw [← mono_comp_iff_of_isIso (pushoutSymmetry _ _).hom]; convert! this; cat_disch | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 446,
"column": 4
} | {
"line": 446,
"column": 85
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat iX\nhUS : IsAffineOpen US\nhUX : IsAffineOpen UX\nhUT : IsCompact ... | rw [← mono_comp_iff_of_isIso (pushoutSymmetry _ _).hom]; convert! this; cat_disch | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 487,
"column": 4
} | {
"line": 487,
"column": 85
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat f\nhUS : IsAffineOpen US\nhUT : IsCompact ↑UT\nhUX : IsCompact ↑UX... | rw [← mono_comp_iff_of_isIso (pushoutSymmetry _ _).hom]; convert! this; cat_disch | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 487,
"column": 4
} | {
"line": 487,
"column": 85
} | [
{
"pp": "X Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\ninst✝ : Flat f\nhUS : IsAffineOpen US\nhUT : IsCompact ↑UT\nhUX : IsCompact ↑UX... | rw [← mono_comp_iff_of_isIso (pushoutSymmetry _ _).hom]; convert! this; cat_disch | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 881,
"column": 2
} | {
"line": 882,
"column": 62
} | [
{
"pp": "I : Type u\ninst✝⁴ : Category.{u, u} I\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝³ : IsCofiltered I\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ns : ↑Γ(c.pt, ⊤)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.obj i)\ninst✝ : ∀ (i : I), QuasiSeparatedSpace ↥(D.obj i)\nthis : CompactSpace ↥c.p... | have hjS {x y} (hx : x ∈ σ) (hy : y ∈ σ) : j x y ∈ S :=
Finset.subset_union_right (Finset.mem_image₂_of_mem hx hy) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 160,
"column": 2
} | {
"line": 168,
"column": 9
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\n⊢ 2 * ↑(expDegree n) = ↑n ^ 2 - if Even n then 4 else 1",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Me... | rcases n.even_or_odd' with ⟨n, rfl | rfl⟩
· rcases n with _ | n
· contradiction
push_cast [expDegree, show (2 * (n + 1)) ^ 2 = 2 * (2 * n * (n + 2)) + 4 by ring1, even_two_mul,
Nat.add_sub_cancel, Nat.mul_div_cancel_left _ two_pos]
ring1
· push_cast [expDegree, show (2 * n + 1) ^ 2 = 2 * (2 * n * ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 160,
"column": 2
} | {
"line": 168,
"column": 9
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\n⊢ 2 * ↑(expDegree n) = ↑n ^ 2 - if Even n then 4 else 1",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Me... | rcases n.even_or_odd' with ⟨n, rfl | rfl⟩
· rcases n with _ | n
· contradiction
push_cast [expDegree, show (2 * (n + 1)) ^ 2 = 2 * (2 * n * (n + 2)) + 4 by ring1, even_two_mul,
Nat.add_sub_cancel, Nat.mul_div_cancel_left _ two_pos]
ring1
· push_cast [expDegree, show (2 * n + 1) ^ 2 = 2 * (2 * n * ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 535,
"column": 4
} | {
"line": 535,
"column": 41
} | [
{
"pp": "F : Type u_2\ninst✝³ : Field F\nW : WeierstrassCurve F\ninst✝² : W.IsElliptic\ninst✝¹ : W.IsCharTwoJNeZeroNF\ninst✝ : CharP F 2\n⊢ W.c₄ ^ 3 / W.Δ = 1 / W.a₆",
"usedConstants": [
"Eq.mpr",
"WeierstrassCurve.Δ",
"instHDiv",
"DivisionCommMonoid.toDivisionMonoid",
"congrAr... | c₄_of_isCharTwoJNeZeroNF_of_char_two, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 201,
"column": 90
} | {
"line": 231,
"column": 82
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℕ\n⊢ (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n)",
"usedConstants": [
"WeierstrassCurve.coeff_Ψ₂Sq",
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"apply_ite₂"... | by
let dm {m n p q} : _ → _ → (p * q : R[X]).natDegree ≤ m + n := natDegree_mul_le_of_le
let dp {m n p} : _ → (p ^ n : R[X]).natDegree ≤ n * m := natDegree_pow_le_of_le n
let cm {m n p q} : _ → _ → (p * q : R[X]).coeff (m + n) = _ := coeff_mul_add_eq_of_natDegree_le
let cp {m n p} : _ → (p ^ m : R[X]).coeff (m ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 25
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℕ\ndm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=\n fun {m n} {p q} ↦ natDegree_mul_le_of_le\ndp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegr... | rcases n with _ | _ | n | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 830,
"column": 2
} | {
"line": 830,
"column": 41
} | [
{
"pp": "R : Type r\nF : Type u\nK : Type v\nL : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : Field F\ninst✝¹¹ : Field K\ninst✝¹⁰ : Field L\nW' : Affine R\ninst✝⁹ : DecidableEq F\ninst✝⁸ : DecidableEq K\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R F\ninst✝⁵ : Algebra R K\ninst✝⁴ : Algebra R L\ninst✝³ : Algebra F K\ni... | convert! map_map (Algebra.ofId F K) f P | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.FieldTheory.IsSepClosed | {
"line": 203,
"column": 2
} | {
"line": 211,
"column": 75
} | [
{
"pp": "k : Type u\ninst✝⁴ : Field k\nK : Type v\ninst✝³ : Field K\ninst✝² : IsSepClosed k\ninst✝¹ : Algebra k K\ninst✝ : Algebra.IsSeparable k K\n⊢ Function.Surjective ⇑(algebraMap k K)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Polynomial.C",
"NegZeroClass.toNeg",
"NonA... | refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (Algebra.IsSeparable.isIntegral k x)
have hsep : IsSeparable k x := Algebra.IsSeparable.isSeparable k x
have h : (minpoly k x).degree = 1 :=
degree_eq_one_of_irreducible k (minpoly.irreducible (Algebra.IsSep... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IsSepClosed | {
"line": 203,
"column": 2
} | {
"line": 211,
"column": 75
} | [
{
"pp": "k : Type u\ninst✝⁴ : Field k\nK : Type v\ninst✝³ : Field K\ninst✝² : IsSepClosed k\ninst✝¹ : Algebra k K\ninst✝ : Algebra.IsSeparable k K\n⊢ Function.Surjective ⇑(algebraMap k K)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Polynomial.C",
"NegZeroClass.toNeg",
"NonA... | refine fun x => ⟨-(minpoly k x).coeff 0, ?_⟩
have hq : (minpoly k x).leadingCoeff = 1 := minpoly.monic (Algebra.IsSeparable.isIntegral k x)
have hsep : IsSeparable k x := Algebra.IsSeparable.isSeparable k x
have h : (minpoly k x).degree = 1 :=
degree_eq_one_of_irreducible k (minpoly.irreducible (Algebra.IsSep... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Galois.Basic | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 27
} | [
{
"pp": "case mpr\nF : Type u_1\nE : Type u_3\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\n⊢ IsGalois F E → IsGalois (↥⊥) E",
"usedConstants": [
"IsGalois.tower_top_intermediateField",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"IntermediateField",
"O... | intro h; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Galois.Basic | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 27
} | [
{
"pp": "case mpr\nF : Type u_1\nE : Type u_3\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\n⊢ IsGalois F E → IsGalois (↥⊥) E",
"usedConstants": [
"IsGalois.tower_top_intermediateField",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"IntermediateField",
"O... | intro h; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Galois.Basic | {
"line": 215,
"column": 2
} | {
"line": 216,
"column": 92
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nH : Subgroup Gal(E/F)\nx : E\n⊢ x ∈ fixedField H ↔ ∀ f ∈ H, f x = x",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"congrArg",
"_private.Mathlib.FieldTheory.Gal... | change x ∈ MulAction.fixedPoints H E ↔ _
simp only [MulAction.mem_fixedPoints, Subtype.forall, Subgroup.mk_smul, AlgEquiv.smul_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Galois.Basic | {
"line": 215,
"column": 2
} | {
"line": 216,
"column": 92
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nH : Subgroup Gal(E/F)\nx : E\n⊢ x ∈ fixedField H ↔ ∀ f ∈ H, f x = x",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"congrArg",
"_private.Mathlib.FieldTheory.Gal... | change x ∈ MulAction.fixedPoints H E ↔ _
simp only [MulAction.mem_fixedPoints, Subtype.forall, Subgroup.mk_smul, AlgEquiv.smul_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 90
} | [
{
"pp": "case of_j_eq_zero.of_j_eq_zero\nF : Type u_1\ninst✝⁶ : Field F\ninst✝⁵ : IsSepClosed F\nE E' : WeierstrassCurve F\ninst✝⁴ : E.IsElliptic\ninst✝³ : E'.IsElliptic\ninst✝² : CharP F 2\nheq : E.j = E'.j\nC : VariableChange F\ninst✝¹ : (C • E).IsCharTwoJEqZeroNF\nC' : VariableChange F\ninst✝ : (C' • E').IsC... | obtain ⟨C'', hC⟩ := exists_variableChange_of_char_two_of_j_eq_zero (C • E) (C' • E') | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 125,
"column": 9
} | {
"line": 125,
"column": 28
} | [
{
"pp": "case h\nR : Type r\nP : Fin 3 → R\nn : Fin (Nat.succ 0).succ.succ\n⊢ ![P x, P y, P z] n = P n",
"usedConstants": [
"Fintype.elems",
"Nat.le_refl",
"HEq.refl",
"Finset",
"List.Mem.tail",
"False.elim",
"noConfusion_of_Nat",
"Membership.mem",
"Fin.... | fin_cases n <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 153,
"column": 20
} | {
"line": 153,
"column": 78
} | [
{
"pp": "R : Type r\nF : Type u\ninst✝⁵ : CommRing R\ninst✝⁴ : Field F\nW' : Jacobian R\nW : Jacobian F\nS : Type s\ninst✝³ : CommRing S\nA : Type u\ninst✝² : CommRing A\nB : Type v\ninst✝¹ : CommRing B\nK : Type v\ninst✝ : Field K\nx✝² x✝¹ : R\nx✝ : Fin 3 → R\n⊢ (x✝² * x✝¹) • x✝ = x✝² • x✝¹ • x✝",
"usedCon... | by simp only [smul_fin3, mul_pow, mul_assoc, fin3_def_ext] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 433,
"column": 2
} | {
"line": 436,
"column": 98
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Jacobian R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Nonsingular P\nhPz : P z = 0\n⊢ P x ≠ 0",
"usedConstants": [
"ne_self_iff_false",
"False",
"Nat.instMulZeroClass",
"HMul.hMul",
"OfNat.ofNat_ne_zero",
"pow_eq_zero_... | intro hPx
simp only [nonsingular_of_Z_eq_zero hPz, equation_of_Z_eq_zero hPz, hPx, mul_zero, zero_mul,
zero_pow <| OfNat.ofNat_ne_zero _, ne_self_iff_false, or_false, false_or] at hP
rwa [pow_eq_zero_iff two_ne_zero, hP.left, eq_self, true_and, mul_zero, ne_self_iff_false] at hP | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 433,
"column": 2
} | {
"line": 436,
"column": 98
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Jacobian R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Nonsingular P\nhPz : P z = 0\n⊢ P x ≠ 0",
"usedConstants": [
"ne_self_iff_false",
"False",
"Nat.instMulZeroClass",
"HMul.hMul",
"OfNat.ofNat_ne_zero",
"pow_eq_zero_... | intro hPx
simp only [nonsingular_of_Z_eq_zero hPz, equation_of_Z_eq_zero hPz, hPx, mul_zero, zero_mul,
zero_pow <| OfNat.ofNat_ne_zero _, ne_self_iff_false, or_false, false_or] at hP
rwa [pow_eq_zero_iff two_ne_zero, hP.left, eq_self, true_and, mul_zero, ne_self_iff_false] at hP | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 104,
"column": 31
} | {
"line": 104,
"column": 60
} | [
{
"pp": "case a.a\nF : Type u\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ W.negY P / P z ^ 3 + (-W.a₁ * P x / P z ^ 2 * 1 + W.a₃ * (P z ^ 3 / P z ^ 3)) -\n (W.toAffine.negY (P x / P z ^ 2) (P y / P z ^ 3) + (-W.a₁ * P x / P z ^ 2 * (P z / P z) + W.a₃ * 1)) =\n 0",
"usedConstant... | rw [negY, Affine.negY]; ring1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 104,
"column": 31
} | {
"line": 104,
"column": 60
} | [
{
"pp": "case a.a\nF : Type u\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ W.negY P / P z ^ 3 + (-W.a₁ * P x / P z ^ 2 * 1 + W.a₃ * (P z ^ 3 / P z ^ 3)) -\n (W.toAffine.negY (P x / P z ^ 2) (P y / P z ^ 3) + (-W.a₁ * P x / P z ^ 2 * (P z / P z) + W.a₃ * 1)) =\n 0",
"usedConstant... | rw [negY, Affine.negY]; ring1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 473,
"column": 6
} | {
"line": 474,
"column": 62
} | [
{
"pp": "case neg\nF : Type u\ninst✝ : Field F\nW : Jacobian F\nP : Fin 3 → F\nu : F\nhu : IsUnit u\nhP : W.Nonsingular P\nhPz : ¬P z = 0\n⊢ toAffine W (u • P) = toAffine W P",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"WeierstrassCurve.Jacobian.instSMulForallFinOfNatNat._proof_1",
"We... | rw [toAffine_of_Z_ne_zero ((nonsingular_smul P hu).mpr hP) <| mul_ne_zero hu.ne_zero hPz,
toAffine_of_Z_ne_zero hP hPz, Affine.Point.some.injEq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 287,
"column": 2
} | {
"line": 288,
"column": 7
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nu : R\n⊢ W'.negDblY (u • P) = (u ^ 4) ^ 3 * W'.negDblY P",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemir... | simp only [negDblY, dblU_smul, dblX_smul, negY_smul, smul_fin3_ext]
ring1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 287,
"column": 2
} | {
"line": 288,
"column": 7
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\nP : Fin 3 → R\nu : R\n⊢ W'.negDblY (u • P) = (u ^ 4) ^ 3 * W'.negDblY P",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemir... | simp only [negDblY, dblU_smul, dblX_smul, negY_smul, smul_fin3_ext]
ring1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 249,
"column": 79
} | {
"line": 250,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx y z : R\nhx : 0 <ᵥ x\n⊢ x * y ≤ᵥ x * z ↔ y ≤ᵥ z",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"ValuativeRel.mul_vle_... | by
simp [mul_comm x, hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 545,
"column": 4
} | {
"line": 545,
"column": 49
} | [
{
"pp": "case mk\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\nx y z : R\nb c : ValueGroupWithZero R\nhbc : b ≤ c\na₁ : R\na₂ : ↥(posSubmonoid R)\nhab : ValueGroupWithZero.mk a₁ a₂ ≤ b\n⊢ ValueGroupWithZero.mk a₁ a₂ ≤ c",
"usedConstants": [
"CommSemiring.toSemiring",
"Membership.me... | induction b using ValueGroupWithZero.ind with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1000,
"column": 2
} | {
"line": 1003,
"column": 47
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : ValuativeRel R\n⊢ Nontrivial (ValueGroupWithZero R)ˣ → IsNontrivial R",
"usedConstants": [
"Nontrivial",
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"False",
"congrArg",
"Exists",
"Unit... | · rintro ⟨r, s, h⟩
rcases eq_or_ne r 1 with rfl | hr
· exact ⟨s.val, by simp, by simpa using h.symm⟩
· exact ⟨r.val, by simp, by simpa using hr⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.BilinearForm.DualLattice | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 20
} | [
{
"pp": "R : Type u_1\nS : Type u_3\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : Field S\ninst✝⁴ : AddCommGroup M\ninst✝³ : Algebra R S\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\nB : BilinForm S M\nN₁ N₂ : Submodule R M\n⊢ (∀ x ∈ N₁, ∀ y ∈ N₂, ∃ y_1, (algebraMap R S) y_1 = (B y) x) ... | exact forall₂_comm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 35
} | [
{
"pp": "case inr.inl\nα : Type u_1\nIsBase : Set α → Prop\nB₁ B₂ : Set α\nexch : ExchangeProperty IsBase\nhB₁ : IsBase B₁\nhB₂ : IsBase B₂\nhinf : (B₂ \\ B₁).encard = ⊤\n⊢ (B₁ \\ B₂).encard ≤ (B₂ \\ B₁).encard",
"usedConstants": [
"Set.encard",
"LE.le.trans_eq",
"instTopENat",
"inst... | · exact le_top.trans_eq hinf.symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 29
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\ns : S\nhs : IsIntegral R s\np : R[X]\nhp : (Polynomial.aeval s) p = 0\nK : Type u_1 := FractionR... | apply dvd_mul_of_dvd_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 988,
"column": 11
} | {
"line": 988,
"column": 64
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\ne : α\nhI : M.Indep I\n⊢ M.IsBasis I (insert e I) ↔ M.Dep (insert e I) ∨ e ∈ I",
"usedConstants": [
"Eq.mpr",
"Matroid.Dep",
"congrArg",
"Membership.mem",
"id",
"Insert.insert",
"SDiff.sdiff",
"Iff",
"Set.... | hI.isBasis_iff_forall_insert_dep (subset_insert _ _), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Dual | {
"line": 62,
"column": 19
} | {
"line": 62,
"column": 36
} | [
{
"pp": "α : Type u_1\nM✝ : Matroid α\nI✝ B✝ X✝ : Set α\nM : Matroid α\nI X : Set α\nhIE : I ⊆ M.E\nB : Set α\nhB : M.IsBase B\nhIB : Disjoint I B\nhI_not_max : ¬Maximal (fun I ↦ I ⊆ M.E ∧ ∃ B, M.IsBase B ∧ Disjoint I B) I\nhX_max : Maximal (fun I ↦ I ⊆ M.E ∧ ∃ B, M.IsBase B ∧ Disjoint I B) X\nhXE : X ⊆ M.E\nB'... | union_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Dual | {
"line": 89,
"column": 10
} | {
"line": 89,
"column": 27
} | [
{
"pp": "α : Type u_1\nM✝ : Matroid α\nI B✝ X✝ : Set α\nM : Matroid α\nX I' : Set α\nhI'E : I' ⊆ M.E\nB : Set α\nhB : M.IsBase B\nhI'B : Disjoint I' B\nhI'X : I' ⊆ X\nB' : Set α\nhB' : M.IsBase B'\nhI : M.IsBasis (B' \\ X) (M.E \\ X)\nhIB' : B' \\ X ⊆ B'\nhB'IB : B' ⊆ B' \\ X ∪ B\nJ : Set α\nhJE : J ⊆ M.E\nB'' ... | union_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Restrict | {
"line": 95,
"column": 10
} | {
"line": 95,
"column": 27
} | [
{
"pp": "case h\nα : Type u_1\nM✝ : Matroid α\nR✝ I✝ X Y : Set α\nM : Matroid α\nR I : Set α\nhI : M.Indep I\nhIY : I ⊆ R\nhIn : ¬M.IsBasis I (R ∩ M.E)\nB' : Set α\nhB' : M.IsBase B'\nhI' : M✶.IsBasis (M.E \\ (B' ∪ R ∩ M.E)) (M.E \\ (R ∩ M.E))\nB : Set α\nhB : M.IsBase B\nhIB : I ⊆ B\nhBIB' : B ⊆ I ∪ B'\n⊢ B ∪ ... | union_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Restrict | {
"line": 445,
"column": 2
} | {
"line": 445,
"column": 37
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X J : Set α\ne : α\nhI : M.IsBasis I X\nhJ : M.IsBasis J X\nhIJ : I \\ J = {e}\n⊢ ∃ f ∈ J \\ I, J = insert f I \\ {e}",
"usedConstants": [
"Matroid.IsBasis.subset_ground",
"congrArg",
"Matroid.IsBase",
"Eq.mp",
"propext",
"Matroid.i... | rw [← isBase_restrict_iff] at hI hJ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 279,
"column": 6
} | {
"line": 279,
"column": 95
} | [
{
"pp": "case neg\nα : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug :\n ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_compact : ∀ (I : Set α), (∀... | exact hI₀ f ⟨Or.elim (hJss hfJ) (fun hfe ↦ (heJ <| hfe ▸ hfJ).elim) (by aesop), hfI₀⟩ hfi | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Map | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 21
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nE : Set β\n⊢ (loopyOn E).comap f = loopyOn (f ⁻¹' E)",
"usedConstants": [
"Eq.mpr",
"Matroid.loopyOn",
"congrArg",
"Matroid.E",
"Matroid.Indep",
"id",
"HasSubset.Subset",
"And",
"Set.preimage",
"prope... | rw [eq_loopyOn_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Map | {
"line": 669,
"column": 59
} | {
"line": 670,
"column": 56
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set ↑M.E\n⊢ (M.restrictSubtype M.E).IsBasis I X ↔ M.IsBasis (Subtype.val '' I) (Subtype.val '' X)",
"usedConstants": [
"Eq.mpr",
"Matroid.restrictSubtype_isBasis_iff",
"congrArg",
"Matroid.E",
"Set.univ",
"Iff.rfl",
"Matro... | by
rw [restrictSubtype_isBasis_iff, isBasis'_iff_isBasis] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\n⊢ (M.map f hf).loops = f '' M.loops",
"usedConstants": [
"congrArg",
"Matroid.map_closure_eq",
"True",
"eq_self",
"Set.instEmptyCollection",
"Matroid.closure",
"of_eq_true",
"Set.... | simp [loops] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\n⊢ (M.map f hf).loops = f '' M.loops",
"usedConstants": [
"congrArg",
"Matroid.map_closure_eq",
"True",
"eq_self",
"Set.instEmptyCollection",
"Matroid.closure",
"of_eq_true",
"Set.... | simp [loops] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\n⊢ (M.map f hf).loops = f '' M.loops",
"usedConstants": [
"congrArg",
"Matroid.map_closure_eq",
"True",
"eq_self",
"Set.instEmptyCollection",
"Matroid.closure",
"of_eq_true",
"Set.... | simp [loops] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 356,
"column": 6
} | {
"line": 356,
"column": 24
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne f : α\nhe : M.IsNonloop e\nhef : e ∈ M.closure {f}\n⊢ f ∈ M.closure {e}",
"usedConstants": [
"Eq.mpr",
"Set.union_empty",
"congrArg",
"Membership.mem",
"Set.instUnion",
"Eq.mp",
"Set.instSingletonSet",
"id",
"Set.i... | ← union_empty {_}, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 376,
"column": 55
} | {
"line": 376,
"column": 96
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne f : α\nhe : M.IsNonloop e\nhef : e ∈ M.closure {f}\n⊢ M.closure {e} = M.closure (insert e (M.closure {f}))",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.closure_insert_closure_eq_closure_insert",
"Set.instSingletonSet",
"id",
... | closure_insert_closure_eq_closure_insert, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 378,
"column": 4
} | {
"line": 378,
"column": 45
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne f : α\nhe : M.IsNonloop e\nhef : e ∈ M.closure {f}\n⊢ M.closure (insert f (M.closure {e})) = M.closure {e, f}",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.closure_insert_closure_eq_closure_insert",
"Set.instSingletonSet",
"id",
... | closure_insert_closure_eq_closure_insert, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 68
} | [
{
"pp": "case neg\nα : Type u_1\nM : Matroid α\ne : α\ninst✝ : M✶.RankPos\nhe : e ∉ M.E\nC : Set α\nhC : M.IsCircuit C\n⊢ M.IsColoop e ↔ ∀ (C : Set α), M.IsCircuit C → e ∈ M.E \\ C",
"usedConstants": [
"Matroid.E",
"Matroid.IsColoop.mem_ground",
"Membership.mem",
"Matroid.IsColoop",
... | exact iff_of_false (fun h ↦ he h.mem_ground) fun h ↦ he (h C hC).1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 566,
"column": 2
} | {
"line": 569,
"column": 68
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\ninst✝ : M✶.RankPos\n⊢ M.IsColoop e ↔ ∀ (C : Set α), M.IsCircuit C → e ∈ M.E \\ C",
"usedConstants": [
"congrArg",
"Matroid.E",
"Matroid.IsColoop.mem_ground",
"_private.Mathlib.Combinatorics.Matroid.Loop.0.Matroid.isColoop_iff_forall_mem_co... | by_cases he : e ∈ M.E
· simp [isColoop_iff_forall_notMem_isCircuit, he]
obtain ⟨C, hC⟩ := M.exists_isCircuit
exact iff_of_false (fun h ↦ he h.mem_ground) fun h ↦ he (h C hC).1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 566,
"column": 2
} | {
"line": 569,
"column": 68
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\ninst✝ : M✶.RankPos\n⊢ M.IsColoop e ↔ ∀ (C : Set α), M.IsCircuit C → e ∈ M.E \\ C",
"usedConstants": [
"congrArg",
"Matroid.E",
"Matroid.IsColoop.mem_ground",
"_private.Mathlib.Combinatorics.Matroid.Loop.0.Matroid.isColoop_iff_forall_mem_co... | by_cases he : e ∈ M.E
· simp [isColoop_iff_forall_notMem_isCircuit, he]
obtain ⟨C, hC⟩ := M.exists_isCircuit
exact iff_of_false (fun h ↦ he h.mem_ground) fun h ↦ he (h C hC).1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 680,
"column": 32
} | {
"line": 682,
"column": 34
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX K : Set α\nhXK : Disjoint X K\nhK : K ⊆ M.coloops\n⊢ Disjoint (M.closure X) K",
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"PartialOrder.t... | by
rwa [disjoint_iff_inter_eq_empty, closure_inter_eq_of_subset_coloops X hK,
← disjoint_iff_inter_eq_empty] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 125,
"column": 16
} | {
"line": 125,
"column": 17
} | [
{
"pp": "α : Type u\nM : Matroid α\nX Y : Set α\nhXY : X ⊆ Y\n⊢ ∀ ⦃I : Set α⦄, M.IsBasis' I X → #↑I ≤ M.cRk Y",
"usedConstants": [
"Set"
]
}
] | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 723,
"column": 4
} | {
"line": 723,
"column": 45
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX : Set α\ne f : α\nhfE : f ∈ M.E\nheE : e ∈ M.E\nI : Set α\nhef : e ∈ M.closure (insert f (M.closure I))\nhe : M.Indep (insert e I) ∧ e ∉ I\nhI : M.IsBasis I (X ∩ M.E)\n⊢ f ∈ M.closure (insert e (M.closure I))",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | closure_insert_closure_eq_closure_insert, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 743,
"column": 59
} | {
"line": 743,
"column": 100
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX : Set α\ne f : α\nhe : e ∈ M.closure (insert f X) \\ M.closure X\nhf : f ∈ M.closure (insert e X) \\ M.closure X\n⊢ M.closure (insert e (M.closure (insert f X))) = M.closure (insert e X)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.closure_ins... | closure_insert_closure_eq_closure_insert, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.MvPolynomial | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 26
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\nf : R[X]\n⊢ IsAlgebraic R f → IsAlgebraic R ((Polynomial.aeval (X i)) f)",
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"IsAlgebraic",
"IsAlgebraic.algHom",
"CommSemiring.toSemiring",
... | exact fun h ↦ h.algHom _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 79,
"column": 2
} | {
"line": 83,
"column": 43
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nx y r s : K\ny_ne : y ≠ 0\nhr : r ≠ 0\nhs : s ≠ 0\nh : v (x - y) < min (v s / v r * (v y * v y)) (v y)\n⊢ v (x⁻¹ - y⁻¹) * v r < v s",
"usedConstants": [
"Units.val",
"Eq.... | have hr' : 0 < v r := by simp [zero_lt_iff, hr]
let γ : Γ₀ˣ := .mk0 (v s / v r) (by simp [hs, hr])
calc
v (x⁻¹ - y⁻¹) * v r < γ * v r := by gcongr; exact Valuation.inversion_estimate v y_ne h
_ = v s := div_mul_cancel₀ _ (by simpa) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 79,
"column": 2
} | {
"line": 83,
"column": 43
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nx y r s : K\ny_ne : y ≠ 0\nhr : r ≠ 0\nhs : s ≠ 0\nh : v (x - y) < min (v s / v r * (v y * v y)) (v y)\n⊢ v (x⁻¹ - y⁻¹) * v r < v s",
"usedConstants": [
"Units.val",
"Eq.... | have hr' : 0 < v r := by simp [zero_lt_iff, hr]
let γ : Γ₀ˣ := .mk0 (v s / v r) (by simp [hs, hr])
calc
v (x⁻¹ - y⁻¹) * v r < γ * v r := by gcongr; exact Valuation.inversion_estimate v y_ne h
_ = v s := div_mul_cancel₀ _ (by simpa) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 179,
"column": 8
} | {
"line": 179,
"column": 28
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nF : Filter K\nhF : Cauchy F\nh0 : 𝓝 0 ⊓ F = ⊥\nU : Set K\nU_in : U ∈ 𝓝 0\nM : Set K\nM_in : M ∈ F\nH : U ∩ M = ∅\nγ₀ : (MonoidWithZeroHom.ValueGroup₀ v)ˣ\nhU : {x | v.restrict x < ↑γ₀} ⊆ U\nx :... | apply le_of_not_gt _ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Polynomial.SeparableDegree | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 65
} | [
{
"pp": "case zero\nF : Type u_1\ninst✝¹ : Field F\nf : F[X]\nirred : Irreducible f\ninst✝ : CharZero F\n⊢ HasSeparableContraction 1 f",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"AlgHom",
"Nat.instMo... | · exact ⟨f, irred.separable, ⟨0, by rw [pow_zero, expand_one]⟩⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 334,
"column": 6
} | {
"line": 334,
"column": 35
} | [
{
"pp": "case ih\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nx✝ y✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nx y : K\n⊢ v.restrict (x * y) = v.restrict x * v.restrict y",
"usedConstants":... | exact Valuation.map_mul _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 390,
"column": 4
} | {
"line": 395,
"column": 18
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\nγ : Γ₀ˣ\nx : hat K\nγ₀' : ValueGroup₀ v := extension x\nhγ₀'_def : γ₀' = extension x\nγ₀ : Γ₀ := extensionValuation x\nhγ₀_def : γ₀ = extensionValuation x\nheq : γ₀ = embedding γ₀'\nthis : γ₀ ≠ 0... | rcases eq_or_ne γ₀ 0 with h | h
· simp only [(Valuation.zero_iff _).mp h, mem_setOf_eq, Valuation.map_zero, Units.zero_lt,
iff_true]
apply subset_closure
exact ⟨0, by simp only [mem_setOf_eq, Valuation.map_zero, Units.zero_lt, true_and]; rfl⟩
· exact this h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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