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.AlgebraicGeometry.AffineSpace | {
"line": 191,
"column": 10
} | {
"line": 199,
"column": 14
} | [
{
"pp": "case e.hf.hC\nn : Type v\nS X : Scheme\ninst✝¹ : X.Over S\ninst✝ : IsAffine S\n⊢ (CommRingCat.Hom.hom\n (CommRingCat.ofHom (eval₂Hom (CommRingCat.Hom.hom (Scheme.Hom.appTop (𝔸(n; S) ↘ S))) (coord S)) ≫\n Scheme.Hom.app\n (homOfVector (Spec.map (CommRingCat.ofHom C) ≫ S... | change _ = (CommRingCat.ofHom C ≫ _).hom
rw [CommRingCat.hom_comp, RingHom.comp_assoc, CommRingCat.hom_ofHom, eval₂Hom_comp_C,
← CommRingCat.hom_comp, ← CommRingCat.hom_ext_iff,
← cancel_mono (Scheme.ΓSpecIso _).hom]
rw [← Scheme.Hom.comp_appTop, homOfVector_over, Scheme.Hom.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 191,
"column": 10
} | {
"line": 199,
"column": 14
} | [
{
"pp": "case e.hf.hC\nn : Type v\nS X : Scheme\ninst✝¹ : X.Over S\ninst✝ : IsAffine S\n⊢ (CommRingCat.Hom.hom\n (CommRingCat.ofHom (eval₂Hom (CommRingCat.Hom.hom (Scheme.Hom.appTop (𝔸(n; S) ↘ S))) (coord S)) ≫\n Scheme.Hom.app\n (homOfVector (Spec.map (CommRingCat.ofHom C) ≫ S... | change _ = (CommRingCat.ofHom C ≫ _).hom
rw [CommRingCat.hom_comp, RingHom.comp_assoc, CommRingCat.hom_ofHom, eval₂Hom_comp_C,
← CommRingCat.hom_comp, ← CommRingCat.hom_ext_iff,
← cancel_mono (Scheme.ΓSpecIso _).hom]
rw [← Scheme.Hom.comp_appTop, homOfVector_over, Scheme.Hom.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Cover.Directed | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 38
} | [
{
"pp": "P : MorphismProperty Scheme\nX : Scheme\n𝒰 : Cover (precoverage P) X\ninst✝³ : Category.{v_1, ?u.9584} 𝒰.I₀\ninst✝² : 𝒰.LocallyDirected\ninst✝¹ : P.IsStableUnderBaseChange\ninst✝ : P.HasOfPostcompProperty P\ni j : 𝒰.I₀\n⊢ { I₀ := (k : 𝒰.I₀) × (k ⟶ i) × (k ⟶ j), X := fun k ↦ 𝒰.X k.fst,\n f ... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Birational.RationalMap | {
"line": 512,
"column": 4
} | {
"line": 512,
"column": 38
} | [
{
"pp": "X Y Z S : Scheme\nsX : X ⟶ S\nsY : Y ⟶ S\nf : X ⤏ Y\n⊢ { I₀ := ↑{x | ∃ g, ∃ (_ : g.toRationalMap = f), g.domain = x}, X := fun U ↦ ↑↑U,\n f := fun U ↦ X.homOfLE ⋯ }.presieve₀ ∈\n (precoverage IsOpenImmersion).coverings ↑f.domain",
"usedConstants": [
"CategoryTheory.PreZeroHypercover... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective | {
"line": 61,
"column": 4
} | {
"line": 64,
"column": 7
} | [
{
"pp": "case a\n⊢ @UniversallyInjective ≤ diagonal @Surjective",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.UniversallyInjective",
"AlgebraicGeometry.PresheafedSpace.carrier",
"AlgebraicGeometry.Scheme.Pu... | intro X Y f hf
refine ⟨fun x ↦ ⟨pullback.fst f f x, hf.1 _ _ _ (IsPullback.of_hasPullback f f) ?_⟩⟩
rw [← Scheme.Hom.comp_apply, pullback.diagonal_fst]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyInjective | {
"line": 61,
"column": 4
} | {
"line": 64,
"column": 7
} | [
{
"pp": "case a\n⊢ @UniversallyInjective ≤ diagonal @Surjective",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.UniversallyInjective",
"AlgebraicGeometry.PresheafedSpace.carrier",
"AlgebraicGeometry.Scheme.Pu... | intro X Y f hf
refine ⟨fun x ↦ ⟨pullback.fst f f x, hf.1 _ _ _ (IsPullback.of_hasPullback f f) ?_⟩⟩
rw [← Scheme.Hom.comp_apply, pullback.diagonal_fst]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Cover.Directed | {
"line": 265,
"column": 19
} | {
"line": 265,
"column": 45
} | [
{
"pp": "P : MorphismProperty Scheme\nX : Scheme\n𝒰 : X.OpenCover\ninst✝ : Preorder 𝒰.I₀\nhle : ∀ {i j : 𝒰.I₀}, i ≤ j ↔ Hom.opensRange (𝒰.f i) ≤ Hom.opensRange (𝒰.f j)\nH : TopologicalSpace.Opens.IsBasis (Set.range fun i ↦ Hom.opensRange (𝒰.f i))\ni : 𝒰.I₀\n⊢ IsOpenImmersion.lift (𝒰.f i) (𝒰.f i) ⋯ = 𝟙... | rw [← cancel_mono (𝒰.f i)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Cover.Directed | {
"line": 308,
"column": 4
} | {
"line": 308,
"column": 38
} | [
{
"pp": "P : MorphismProperty Scheme\nX : Scheme\n⊢ { I₀ := ↑X.affineOpens, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.presieve₀ ∈ (precoverage IsOpenImmersion).coverings X",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.mk",
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"CategoryTheory.... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Cover.Over | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 38
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\ninst✝⁵ : P.IsStableUnderBaseChange\ninst✝⁴ : IsJointlySurjectivePreserving P\nX W : Scheme\n𝒰 : Cover (precoverage P) X\nf : W ⟶ X\ninst✝³ : W.Over S\ninst✝² : X.Over S\ninst✝¹ : Cover.Over S 𝒰\ninst✝ : Hom.IsOver f S\n⊢ { I₀ := 𝒰.I₀, X := fun x ↦ (pullback (... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Cover.Over | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 38
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\ninst✝⁵ : P.IsStableUnderBaseChange\ninst✝⁴ : IsJointlySurjectivePreserving P\nX W : Scheme\n𝒰 : Cover (precoverage P) X\nf : W ⟶ X\ninst✝³ : W.Over S\ninst✝² : X.Over S\ninst✝¹ : Cover.Over S 𝒰\ninst✝ : Hom.IsOver f S\n⊢ { I₀ := 𝒰.I₀, X := fun x ↦ (pullback (... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Cover.Over | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 38
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : IsJointlySurjectivePreserving P\nX W : Scheme\n𝒰 : Cover (precoverage P) X\nf : W ⟶ X\ninst✝⁶ : W.Over S\ninst✝⁵ : X.Over S\ninst✝⁴ : Cover.Over S 𝒰\ninst✝³ : Hom.IsOver f S\nQ : MorphismProperty Scheme\ninst✝² : Q.... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Cover.Over | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 38
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\ninst✝⁸ : P.IsStableUnderBaseChange\ninst✝⁷ : IsJointlySurjectivePreserving P\nX W : Scheme\n𝒰 : Cover (precoverage P) X\nf : W ⟶ X\ninst✝⁶ : W.Over S\ninst✝⁵ : X.Over S\ninst✝⁴ : Cover.Over S 𝒰\ninst✝³ : Hom.IsOver f S\nQ : MorphismProperty Scheme\ninst✝² : Q.... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 75,
"column": 47
} | {
"line": 75,
"column": 55
} | [
{
"pp": "S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝ : (i : ι) → TopologicalSpace (X i)\nU : Set S\nx✝ : IsCompactOpenCovered f U\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier\nhU : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 75,
"column": 47
} | {
"line": 75,
"column": 55
} | [
{
"pp": "S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝ : (i : ι) → TopologicalSpace (X i)\nU : Set S\nx✝ : IsCompactOpenCovered f U\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier\nhU : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.CompactOpenCovered | {
"line": 75,
"column": 47
} | {
"line": 75,
"column": 55
} | [
{
"pp": "S : Type u_1\nι : Type u_2\nX : ι → Type u_3\nf : (i : ι) → X i → S\ninst✝ : (i : ι) → TopologicalSpace (X i)\nU : Set S\nx✝ : IsCompactOpenCovered f U\ns : Set ι\nhs : s.Finite\nV : (i : ι) → i ∈ s → Opens (X i)\nhc : ∀ (i : ι) (h : i ∈ s), IsCompact (V i h).carrier\nhU : ⋃ i, ⋃ (h : i ∈ s), f i '' ↑(... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 54,
"column": 58
} | {
"line": 54,
"column": 66
} | [
{
"pp": "S : Scheme\n𝒰 : PreZeroHypercover S\ninst✝ : QuasiCompactCover 𝒰\nU : S.Opens\nhU : IsCompact ↑U\nUs : Set (TopologicalSpace.Opens ↥S)\nhUs : Us ⊆ S.affineOpens\nhUf : Us.Finite\nhUc : U = sSup Us\n⊢ ⋃ t ∈ SetLike.coe '' Us, t = ↑U",
"usedConstants": [
"Set.iUnion_iUnion_eq_right",
"A... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 54,
"column": 58
} | {
"line": 54,
"column": 66
} | [
{
"pp": "S : Scheme\n𝒰 : PreZeroHypercover S\ninst✝ : QuasiCompactCover 𝒰\nU : S.Opens\nhU : IsCompact ↑U\nUs : Set (TopologicalSpace.Opens ↥S)\nhUs : Us ⊆ S.affineOpens\nhUf : Us.Finite\nhUc : U = sSup Us\n⊢ ⋃ t ∈ SetLike.coe '' Us, t = ↑U",
"usedConstants": [
"Set.iUnion_iUnion_eq_right",
"A... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Cover.QuasiCompact | {
"line": 54,
"column": 58
} | {
"line": 54,
"column": 66
} | [
{
"pp": "S : Scheme\n𝒰 : PreZeroHypercover S\ninst✝ : QuasiCompactCover 𝒰\nU : S.Opens\nhU : IsCompact ↑U\nUs : Set (TopologicalSpace.Opens ↥S)\nhUs : Us ⊆ S.affineOpens\nhUf : Us.Finite\nhUc : U = sSup Us\n⊢ ⋃ t ∈ SetLike.coe '' Us, t = ↑U",
"usedConstants": [
"Set.iUnion_iUnion_eq_right",
"A... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ColimitsOver | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 25
} | [
{
"pp": "P : 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_2} 𝒰.I₀\ninst✝⁴ : LocallyDirected 𝒰\nd : ColimitGluingData D 𝒰\ninst✝³ :\n ∀ {i j :... | simp only [gluedCocone] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Cover.Sigma | {
"line": 36,
"column": 4
} | {
"line": 36,
"column": 38
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\ninst✝¹ : IsZariskiLocalAtSource P\ninst✝ : UnivLE.{v, u}\n𝒰 : Cover (precoverage P) S\n⊢ { I₀ := PUnit.{v + 1}, X := fun x ↦ ∐ 𝒰.X, f := fun x ↦ Sigma.desc 𝒰.f }.presieve₀ ∈ (precoverage P).coverings S",
"usedConstants": [
"CategoryTheory.PreZeroHyp... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.EffectiveEpi.Comp | {
"line": 37,
"column": 4
} | {
"line": 41,
"column": 9
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nα : Type u_2\nB : C\nX Y : α → C\nf : (a : α) → X a ⟶ B\ng : (a : α) → Y a ⟶ X a\ni : (a : α) → X a ⟶ Y a\nhi : ∀ (a : α), i a ≫ g a = 𝟙 (X a)\ninst✝ : EffectiveEpiFamily X f\nW✝ : C\ne : (a : α) → Y a ⟶ W✝\nw : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ Y a₁) (g₂ : ... | simp only [← Category.assoc]
apply w _ _ (g₁ ≫ i a₁) (g₂ ≫ i a₂)
simp only [Category.assoc]
simp only [← Category.assoc, hi]
simpa | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.EffectiveEpi.Comp | {
"line": 37,
"column": 4
} | {
"line": 41,
"column": 9
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nα : Type u_2\nB : C\nX Y : α → C\nf : (a : α) → X a ⟶ B\ng : (a : α) → Y a ⟶ X a\ni : (a : α) → X a ⟶ Y a\nhi : ∀ (a : α), i a ≫ g a = 𝟙 (X a)\ninst✝ : EffectiveEpiFamily X f\nW✝ : C\ne : (a : α) → Y a ⟶ W✝\nw : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ Y a₁) (g₂ : ... | simp only [← Category.assoc]
apply w _ _ (g₁ ≫ i a₁) (g₂ ≫ i a₂)
simp only [Category.assoc]
simp only [← Category.assoc, hi]
simpa | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.EffectiveEpi.Preserves | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 44
} | [
{
"pp": "case refine_1\nC✝ : Type u_1\ninst✝⁶ : Category.{v_1, u_1} C✝\nD✝ : Type u_2\ninst✝⁵ : Category.{v_2, u_2} D✝\nC : Type u_3\nD : Type u_4\ninst✝⁴ : Category.{v_3, u_3} C\ninst✝³ : Category.{v_4, u_4} D\nX Y : C\nf : X ⟶ Y\ninst✝² : EffectiveEpi f\nF : C ⥤ D\ninst✝¹ : F.PreservesEffectiveEpis\ninst✝ : P... | · simp [← Functor.map_comp, c.condition] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 313,
"column": 85
} | {
"line": 313,
"column": 93
} | [
{
"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\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV✝ : ∀ (... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 313,
"column": 85
} | {
"line": 313,
"column": 93
} | [
{
"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\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV✝ : ∀ (... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 313,
"column": 85
} | {
"line": 313,
"column": 93
} | [
{
"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\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV✝ : ∀ (... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 323,
"column": 85
} | {
"line": 323,
"column": 93
} | [
{
"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\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV✝ : ∀ (... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 323,
"column": 85
} | {
"line": 323,
"column": 93
} | [
{
"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\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV✝ : ∀ (... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Flat | {
"line": 323,
"column": 85
} | {
"line": 323,
"column": 93
} | [
{
"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\nι : Type u_1\ninst✝ : Finite ι\nVX : ι → X.Opens\nhVU : iSup VX = UX\nhV✝ : ∀ (... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 519,
"column": 10
} | {
"line": 519,
"column": 72
} | [
{
"pp": "R✝ : 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 := Function.... | · gcongr; simpa using (degree_modByMonic_lt _ hi).trans_le hle | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange | {
"line": 296,
"column": 14
} | {
"line": 298,
"column": 56
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\nC : VariableChange R\nA : Type v\ninst✝ : CommRing A\nφ : R →+* A\n⊢ map 1 φ = 1",
"usedConstants": [
"WeierstrassCurve.VariableChange.r",
"Units.val",
"RingHom.instRingHomClass",
"MonoidHom.instMonoidHomClass",
... | by
simp only [one_def, map]
ext <;> simp only [map_one, Units.val_one, map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 547,
"column": 12
} | {
"line": 547,
"column": 40
} | [
{
"pp": "R✝ : 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 := Function.... | refine lt_of_lt_of_le ?_ hle | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 10
} | [
{
"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 = (3 * x₁ ^ 2 + 2 * W.a₂ * x₁ + W.a₄ - W.a₁ * y₁) / (y₁ - W.negY x₁ y₁)",
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"instHDiv",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 10
} | [
{
"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 = (3 * x₁ ^ 2 + 2 * W.a₂ * x₁ + W.a₄ - W.a₁ * y₁) / (y₁ - W.negY x₁ y₁)",
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"instHDiv",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 10
} | [
{
"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 = (3 * x₁ ^ 2 + 2 * W.a₂ * x₁ + W.a₄ - W.a₁ * y₁) / (y₁ - W.negY x₁ y₁)",
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"instHDiv",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Formula | {
"line": 186,
"column": 79
} | {
"line": 187,
"column": 10
} | [
{
"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 = (3 * x₁ ^ 2 + 2 * W.a₂ * x₁ + W.a₄ - W.a₁ * y₁) / (y₁ - W.negY x₁ y₁)",
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"instHDiv",
... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 845,
"column": 6
} | {
"line": 848,
"column": 11
} | [
{
"pp": "case h.e'_3.h\nI : 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✝ : C... | · dsimp [TopCat.Presheaf.restrictOpen, TopCat.Presheaf.restrict]
change _ = (c.pt.presheaf.map (homOfLE _).op ≫ c.pt.presheaf.map (homOfLE _).op) s
rw [← Functor.map_comp]
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.FreeModule.Norm | {
"line": 44,
"column": 81
} | {
"line": 44,
"column": 90
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsPrincipalIdealRing R\ninst✝³ : CommRing S\ninst✝² : IsDomain S\ninst✝¹ : Algebra R S\ninst✝ : Fintype ι\nb : Basis ι R S\nf : S\nhf : f ≠ 0\nhI : ¬span {f} = ⊥\nb' : Basis ι R S := ringBasis b (span {f}) hI\n... | ite_smul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 48
} | [
{
"pp": "R : Type r\nS : Type s\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nW' : Affine R\nf : R →+* S\nhf : Function.Injective ⇑f\ny : W'.CoordinateRing\nhy : (map W' f) y = 0\n⊢ y = 0",
"usedConstants": [
"WeierstrassCurve.Affine.CoordinateRing.exists_smul_basis_eq"
]
}
] | obtain ⟨p, q, rfl⟩ := exists_smul_basis_eq y | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 231,
"column": 48
} | {
"line": 231,
"column": 65
} | [
{
"pp": "case nat.succ.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nn✝ : ℕ\n⊢ preNormEDS b c d (2 * (↑n✝ + 1 + 1) + 1) =\n (preNormEDS b c d (↑n✝ + 1 + 1 + 2) * preNormEDS b c d (↑n✝ + 1 + 1) ^ 3 * if Even (↑n✝ + 1 + 1) then b else 1) -\n preNormEDS b c d (↑n✝ + 1) * preNormEDS b c d (↑n✝ + 1 + 1 + ... | Int.even_add_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.EllipticDivisibilitySequence | {
"line": 239,
"column": 69
} | {
"line": 239,
"column": 86
} | [
{
"pp": "case neg.succ\nR : Type u\ninst✝ : CommRing R\nb c d : R\nih :\n ∀ (n : ℕ),\n preNormEDS b c d (2 * ↑n + 1) =\n (preNormEDS b c d (↑n + 2) * preNormEDS b c d ↑n ^ 3 * if Even ↑n then b else 1) -\n preNormEDS b c d (↑n - 1) * preNormEDS b c d (↑n + 1) ^ 3 * if Even ↑n then 1 else b\nm : ... | Int.even_add_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic | {
"line": 484,
"column": 42
} | {
"line": 484,
"column": 59
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW : WeierstrassCurve R\nn : ℤ\n⊢ (mk W) (C X) * (mk W) (C (W.ΨSq n)) -\n (mk W) (C (W.preΨ (n + 1))) * (mk W) (C (W.preΨ (n - 1))) *\n ((mk W) (if Even (n + 1) then W.ψ₂ else 1) * (mk W) (if Even (n - 1) then W.ψ₂ else 1)) =\n (mk W) (C X) * (mk W) (C (W.Ψ... | Int.even_add_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms | {
"line": 498,
"column": 68
} | {
"line": 500,
"column": 7
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\ninst✝ : W.IsCharTwoJNeZeroNF\n⊢ W.c₆ = -W.b₂ ^ 3 - 864 * W.a₆",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemir... | by
rw [c₆, b₄_of_isCharTwoJNeZeroNF, b₆_of_isCharTwoJNeZeroNF]
ring1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Normal.Basic | {
"line": 51,
"column": 39
} | {
"line": 51,
"column": 47
} | [
{
"pp": "case refine_2\nF : Type u_1\nK : Type u_2\ninst✝³ : Field F\ninst✝² : Field K\ninst✝¹ : Algebra F K\nh : Normal F K\ninst✝ : FiniteDimensional F K\ns : Module.Basis (↑(Module.Basis.ofVectorSpaceIndex F K)) F K := Module.Basis.ofVectorSpace F K\nx : ↑(Module.Basis.ofVectorSpaceIndex F K)\n⊢ (aeval (s x)... | map_prod | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 868,
"column": 14
} | {
"line": 868,
"column": 31
} | [
{
"pp": "case zero\nR : Type r\ninst✝ : CommRing R\nW' : Affine R\n⊢ (-zero).xRep = zero.xRep",
"usedConstants": [
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
"instOfNatNat",
"AddMonoidWithOne.toOne",
"CommRing.toCommSemiring",
"Nat",
"eq_self",... | simp [← zero_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.Affine.Point | {
"line": 868,
"column": 14
} | {
"line": 868,
"column": 31
} | [
{
"pp": "case some\nR : Type r\ninst✝ : CommRing R\nW' : Affine R\nx✝ y✝ : R\nh✝ : W'.Nonsingular x✝ y✝\n⊢ (-some x✝ y✝ h✝).xRep = (some x✝ y✝ h✝).xRep",
"usedConstants": [
"AddGroupWithOne.toAddMonoidWithOne",
"instOfNatNat",
"AddMonoidWithOne.toOne",
"Nat",
"eq_self",
"... | simp [← zero_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.Normal.Basic | {
"line": 176,
"column": 2
} | {
"line": 178,
"column": 37
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝³ : Field F\ninst✝² : Field K\ninst✝¹ : Algebra F K\nE : IntermediateField F K\ninst✝ : Normal F ↥E\nf : ↥E →ₐ[F] K\ng : Gal(↥E/F) := f.restrictNormal' ↥E\n⊢ f.fieldRange = E",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"IntermediateField.isScalarTowe... | rw [← show E.val.comp ↑g = f from DFunLike.ext_iff.mpr (f.restrictNormal_commutes E),
← AlgHom.map_fieldRange, AlgEquiv.fieldRange_eq_top g, ← AlgHom.fieldRange_eq_map,
IntermediateField.fieldRange_val] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1106,
"column": 36
} | {
"line": 1106,
"column": 70
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nX : Scheme\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : IsAffine X\ninst✝² : ∀ (i : I), IsAffine (D.obj i)\na : c.pt ⟶ X\nR : CommRingCat\nt : D ⟶ (Functor.const I).obj (Spec R)\nf✝ : X ⟶ Spec R\ninst✝¹ : LocallyOfFinitePresentat... | simpa [← Iso.comp_inv_eq] using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1106,
"column": 36
} | {
"line": 1106,
"column": 70
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nX : Scheme\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : IsAffine X\ninst✝² : ∀ (i : I), IsAffine (D.obj i)\na : c.pt ⟶ X\nR : CommRingCat\nt : D ⟶ (Functor.const I).obj (Spec R)\nf✝ : X ⟶ Spec R\ninst✝¹ : LocallyOfFinitePresentat... | simpa [← Iso.comp_inv_eq] using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1106,
"column": 36
} | {
"line": 1106,
"column": 70
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nX : Scheme\nD : I ⥤ Scheme\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : IsAffine X\ninst✝² : ∀ (i : I), IsAffine (D.obj i)\na : c.pt ⟶ X\nR : CommRingCat\nt : D ⟶ (Functor.const I).obj (Spec R)\nf✝ : X ⟶ Spec R\ninst✝¹ : LocallyOfFinitePresentat... | simpa [← Iso.comp_inv_eq] using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 113,
"column": 29
} | {
"line": 113,
"column": 32
} | [
{
"pp": "α : Type u_1\nM : Type u_3\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\nm₁ m₂ : M\na : α\nh₁ : a ∈ fixedBy α m₁\nh₂ : a ∈ fixedBy α m₂\n⊢ m₁ • m₂ • a = a",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"id",
"Monoid.toSemigroup",
"HSMul.hSMul",
"Semig... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 149,
"column": 43
} | {
"line": 149,
"column": 55
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ns : Set α\ng : G\n⊢ s = g⁻¹ • s ↔ ∀ (x : α), g • x ∈ s ↔ x ∈ s",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHSMul",
"congrArg",
"Membership.mem",
"id",
"DivInvMonoid.toMonoid"... | Set.ext_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.Normal.Closure | {
"line": 235,
"column": 72
} | {
"line": 236,
"column": 72
} | [
{
"pp": "F : Type u_1\nL : Type u_3\ninst✝³ : Field F\ninst✝² : Field L\ninst✝¹ : Algebra F L\nK : IntermediateField F L\ninst✝ : Normal F ↥K\n⊢ normalClosure F (↥K) L = K",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
"CommRing.toNonUnitalCommRing",... | by
simp only [normalClosure_def, AlgHom.fieldRange_of_normal, iSup_const] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IsSepClosed | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 80
} | [
{
"pp": "case h\nk : Type u\ninst✝¹ : Field k\ninst✝ : IsSepClosed k\nx : k\nn : ℕ\nhn : NeZero ↑n\nhn' : 0 < n\nthis : (X ^ n - C x).degree ≠ 0\nhx : ¬x = 0\nz : k\nhz : (X ^ n - C x).IsRoot z\n⊢ z ^ n = x",
"usedConstants": [
"Polynomial.C",
"Polynomial.eval",
"Polynomial.eval_C",
... | simpa [eval_C, eval_X, eval_pow, eval_sub, IsRoot.def, sub_eq_zero] using hz | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1233,
"column": 29
} | {
"line": 1233,
"column": 41
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.ob... | by simp [𝒱'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.AffineTransitionLimit | {
"line": 1233,
"column": 71
} | {
"line": 1233,
"column": 83
} | [
{
"pp": "I : Type u\ninst✝⁵ : Category.{u, u} I\nS X : Scheme\nD : I ⥤ Scheme\nt : D ⟶ (Functor.const I).obj S\nf : X ⟶ S\nc : Cone D\nhc : IsLimit c\ninst✝⁴ : IsCofiltered I\ninst✝³ : LocallyOfFinitePresentation f\ninst✝² : ∀ {i j : I} (f : i ⟶ j), IsAffineHom (D.map f)\ninst✝¹ : ∀ (i : I), CompactSpace ↥(D.ob... | by simp [𝒱'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Galois.Basic | {
"line": 358,
"column": 24
} | {
"line": 360,
"column": 7
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\nH : Subgroup Gal(E/F)\nK✝ : IntermediateField F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nK L : IntermediateField F E\n⊢ { toFun := ⇑OrderDual.toDual ∘ IntermediateField.fixingSubgroup, invFun := fixedF... | by
rw [← fixedField_fixingSubgroup L, IntermediateField.le_iff_le, fixedField_fixingSubgroup L]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Galois.Basic | {
"line": 556,
"column": 45
} | {
"line": 556,
"column": 81
} | [
{
"pp": "case base\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\np : F[X]\nsp : Polynomial.IsSplittingField F E p\nhp : p.Separable\nhFE : FiniteDimensional F E\nthis : DecidableEq E := Classical.decEq E\ns : Set E := p.rootSet E\nadjoin_root : adjoin F s = ⊤\nP : Interme... | ← IntermediateField.bot_toSubalgebra | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Galois.Basic | {
"line": 692,
"column": 4
} | {
"line": 692,
"column": 12
} | [
{
"pp": "case «0»\nF : Type u_1\nK : Type u_2\ninst✝³ : Field F\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : IsQuadraticExtension F K\nh : Nat.card Gal(K/F) = 0\nthis : 0 ≤ 2\n⊢ IsCyclic Gal(K/F)",
"usedConstants": [
"False",
"congrArg",
"CommSemiring.toSemiring",
"AddCommGroup.t... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.Galois.Basic | {
"line": 692,
"column": 4
} | {
"line": 692,
"column": 12
} | [
{
"pp": "case «0»\nF : Type u_1\nK : Type u_2\ninst✝³ : Field F\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : IsQuadraticExtension F K\nh : Nat.card Gal(K/F) = 0\nthis : 0 ≤ 2\n⊢ IsCyclic Gal(K/F)",
"usedConstants": [
"False",
"congrArg",
"CommSemiring.toSemiring",
"AddCommGroup.t... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Galois.Basic | {
"line": 692,
"column": 4
} | {
"line": 692,
"column": 12
} | [
{
"pp": "case «0»\nF : Type u_1\nK : Type u_2\ninst✝³ : Field F\ninst✝² : Field K\ninst✝¹ : Algebra F K\ninst✝ : IsQuadraticExtension F K\nh : Nat.card Gal(K/F) = 0\nthis : 0 ≤ 2\n⊢ IsCyclic Gal(K/F)",
"usedConstants": [
"False",
"congrArg",
"CommSemiring.toSemiring",
"AddCommGroup.t... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 286,
"column": 48
} | {
"line": 287,
"column": 58
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Jacobian R\n⊢ W'.Equation ![1, 1, 0]",
"usedConstants": [
"one_pow",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
"_private.Mathlib.AlgebraicGeometr... | by
simp only [equation_of_Z_eq_zero, fin3_def_ext, one_pow] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 455,
"column": 2
} | {
"line": 455,
"column": 58
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Jacobian F\nP Q : Fin 3 → F\nhP : W.Nonsingular P\nhQ : W.Nonsingular Q\nhPz : P z = 0\nhQz : Q z = 0\nhPx : IsUnit (P x)\nhPy : IsUnit (P y)\n⊢ P ≈ Q",
"usedConstants": [
"IsUnit",
"Fin.instOfNat",
"instOfNatNat",
"Field.toSemifield",
... | have hQx : IsUnit <| Q x := isUnit_X_of_Z_eq_zero hQ hQz | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 588,
"column": 2
} | {
"line": 588,
"column": 89
} | [
{
"pp": "R : Type r\ninst✝¹⁰ : CommRing R\nW' : Jacobian R\nS : Type s\ninst✝⁹ : CommRing S\nA : Type u\ninst✝⁸ : CommRing A\nB : Type v\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsSca... | rw [← RingHom.coe_coe, ← map_nonsingular _ hf, AlgHom.toRingHom_eq_coe, map_baseChange] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 588,
"column": 2
} | {
"line": 588,
"column": 89
} | [
{
"pp": "R : Type r\ninst✝¹⁰ : CommRing R\nW' : Jacobian R\nS : Type s\ninst✝⁹ : CommRing S\nA : Type u\ninst✝⁸ : CommRing A\nB : Type v\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsSca... | rw [← RingHom.coe_coe, ← map_nonsingular _ hf, AlgHom.toRingHom_eq_coe, map_baseChange] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic | {
"line": 588,
"column": 2
} | {
"line": 588,
"column": 89
} | [
{
"pp": "R : Type r\ninst✝¹⁰ : CommRing R\nW' : Jacobian R\nS : Type s\ninst✝⁹ : CommRing S\nA : Type u\ninst✝⁸ : CommRing A\nB : Type v\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsSca... | rw [← RingHom.coe_coe, ← map_nonsingular _ hf, AlgHom.toRingHom_eq_coe, map_baseChange] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 277,
"column": 54
} | {
"line": 277,
"column": 81
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq 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 ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ (W.dblU P ^ 2 - W.a₁ * W.dblU P * P z * (P y - W.negY P) - W.a₂ * P z ... | ← (X_eq_iff hPz hQz).mp hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point | {
"line": 558,
"column": 78
} | {
"line": 560,
"column": 23
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Jacobian F\nP : W.Point\n⊢ (-P).toAffineLift = -P.toAffineLift",
"usedConstants": [
"Units.instMulAction",
"CommSemiring.toSemiring",
"WeierstrassCurve.Jacobian.Point.toAffine_neg",
"WeierstrassCurve.Jacobian.instMulActionForallFinOfNatNat",
... | by
rcases P with @⟨⟨_⟩, hP⟩
exact toAffine_neg hP | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 314,
"column": 63
} | {
"line": 314,
"column": 90
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Jacobian F\ninst✝ : DecidableEq 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 ^ 2 = Q x * P z ^ 2\nhy : P y * Q z ^ 3 ≠ W.negY Q * P z ^ 3\n⊢ (-W.dblU P *\n (W.dblU P ^ 2 - W.a₁ * W.dblU P * P z * (P y -... | ← (X_eq_iff hPz hQz).mp hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Nonarchimedean.Bases | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 22
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : TopologicalSpace R\ninst✝ : Nonempty ι\nB : ι → Submodule R M\nhB : SubmodulesBasis B\nm₀ : M\ni : ι\n⊢ ∀ᶠ (x : R) in 𝓝 0, x • ... | exact hB.smul m₀ i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.WithZeroTopology | {
"line": 126,
"column": 2
} | {
"line": 128,
"column": 31
} | [
{
"pp": "Γ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\ns : Set Γ₀\n⊢ IsOpen[topologicalSpace] s ↔ 0 ∉ s ∨ ∃ γ, γ ≠ 0 ∧ Iio γ ⊆ s",
"usedConstants": [
"Pure.pure",
"WithZeroTopology.topologicalSpace",
"Filter.instMembership",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero... | rw [isOpen_iff_mem_nhds, ← and_forall_ne (0 : Γ₀)]
simp +contextual [nhds_of_ne_zero, imp_iff_not_or,
hasBasis_nhds_zero.mem_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.WithZeroTopology | {
"line": 126,
"column": 2
} | {
"line": 128,
"column": 31
} | [
{
"pp": "Γ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\ns : Set Γ₀\n⊢ IsOpen[topologicalSpace] s ↔ 0 ∉ s ∨ ∃ γ, γ ≠ 0 ∧ Iio γ ⊆ s",
"usedConstants": [
"Pure.pure",
"WithZeroTopology.topologicalSpace",
"Filter.instMembership",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero... | rw [isOpen_iff_mem_nhds, ← and_forall_ne (0 : Γ₀)]
simp +contextual [nhds_of_ne_zero, imp_iff_not_or,
hasBasis_nhds_zero.mem_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 522,
"column": 13
} | {
"line": 527,
"column": 38
} | [] | y * u * s * z
_ = y * s * (z * u) := by ring
_ ≤ᵥ x * t * (w * v) := by gcongr
_ = x * v * (t * w) := by ring
_ ≤ᵥ z * s * (t * w) := by gcongr
_ = w * t * s * z := by ring | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1124,
"column": 18
} | {
"line": 1124,
"column": 26
} | [
{
"pp": "case pos\nR✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nx✝¹ x✝ : ValueGroupWithZero R\nx y z : R\nhz : 0 <ᵥ z\nw : R\nhw : 0 <ᵥ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1124,
"column": 18
} | {
"line": 1124,
"column": 26
} | [
{
"pp": "case pos\nR✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nx✝¹ x✝ : ValueGroupWithZero R\nx y z : R\nhz : 0 <ᵥ z\nw : R\nhw : 0 <ᵥ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1124,
"column": 18
} | {
"line": 1124,
"column": 26
} | [
{
"pp": "case pos\nR✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nx✝¹ x✝ : ValueGroupWithZero R\nx y z : R\nhz : 0 <ᵥ z\nw : R\nhw : 0 <ᵥ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1124,
"column": 18
} | {
"line": 1124,
"column": 26
} | [
{
"pp": "case neg\nR✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nx✝¹ x✝ : ValueGroupWithZero R\nx y z : R\nhz : 0 <ᵥ z\nw : R\nhw : 0 <ᵥ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1124,
"column": 18
} | {
"line": 1124,
"column": 26
} | [
{
"pp": "case neg\nR✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nx✝¹ x✝ : ValueGroupWithZero R\nx y z : R\nhz : 0 <ᵥ z\nw : R\nhw : 0 <ᵥ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1124,
"column": 18
} | {
"line": 1124,
"column": 26
} | [
{
"pp": "case neg\nR✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nx✝¹ x✝ : ValueGroupWithZero R\nx y z : R\nhz : 0 <ᵥ z\nw : R\nhw : 0 <ᵥ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1124,
"column": 18
} | {
"line": 1124,
"column": 26
} | [
{
"pp": "case pos\nR✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nx✝¹ x✝ : ValueGroupWithZero R\nx y z : R\nhz : 0 <ᵥ z\nw : R\nhw : 0 <ᵥ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1124,
"column": 18
} | {
"line": 1124,
"column": 26
} | [
{
"pp": "case pos\nR✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nx✝¹ x✝ : ValueGroupWithZero R\nx y z : R\nhz : 0 <ᵥ z\nw : R\nhw : 0 <ᵥ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1124,
"column": 18
} | {
"line": 1124,
"column": 26
} | [
{
"pp": "case pos\nR✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nx✝¹ x✝ : ValueGroupWithZero R\nx y z : R\nhz : 0 <ᵥ z\nw : R\nhw : 0 <ᵥ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1124,
"column": 18
} | {
"line": 1124,
"column": 26
} | [
{
"pp": "case neg\nR✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nx✝¹ x✝ : ValueGroupWithZero R\nx y z : R\nhz : 0 <ᵥ z\nw : R\nhw : 0 <ᵥ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1124,
"column": 18
} | {
"line": 1124,
"column": 26
} | [
{
"pp": "case neg\nR✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nx✝¹ x✝ : ValueGroupWithZero R\nx y z : R\nhz : 0 <ᵥ z\nw : R\nhw : 0 <ᵥ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1124,
"column": 18
} | {
"line": 1124,
"column": 26
} | [
{
"pp": "case neg\nR✝ : Type u_1\ninst✝⁵ : CommRing R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\nx✝¹ x✝ : ValueGroupWithZero R\nx y z : R\nhz : 0 <ᵥ z\nw : R\nhw : 0 <ᵥ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1177,
"column": 2
} | {
"line": 1177,
"column": 59
} | [
{
"pp": "R : Type u_2\nΓ : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\na b : ValueGroupWithZero R\nh : a < b\n⊢ (embed v) a < (embed v) b",
"usedConstants": [
"ValuativeRel.exists_valuation_div_valuation_eq... | obtain ⟨a, r, rfl⟩ := exists_valuation_div_valuation_eq a | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 27
} | [
{
"pp": "case inr.inr\nK : Type u\ninst✝² : DivisionRing K\nΓ₀ : Type v\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : MulArchimedean Γ₀\nv : Valuation K Γ₀\nx : K\nhx : v x ≠ 0\nr : Γ₀ˣ\nhr : ↑r ≠ 0\nh : ∀ (x : K), v x ≠ 0 → ↑r < v x\nH : 1 < Units.mk0 (v x) hx\n⊢ v x = 1",
"usedConstants": [
"... | rw [← inv_lt_one'] at H | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 85,
"column": 30
} | {
"line": 85,
"column": 55
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nthis : LinearOrderedCommGroupWithZero (ValueGroup₀ v) := instLinearOrderedCommGroupWithZero\nγ γ₀ : (ValueGroup₀ v)ˣ\nh : γ₀ * γ₀ ≤ γ\nr : R\nr_in : v.restrict r < ↑γ₀\ns : R\ns_in : v.restrict s < ... | gcongr <;> exact zero_le' | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 85,
"column": 30
} | {
"line": 85,
"column": 55
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nthis : LinearOrderedCommGroupWithZero (ValueGroup₀ v) := instLinearOrderedCommGroupWithZero\nγ γ₀ : (ValueGroup₀ v)ˣ\nh : γ₀ * γ₀ ≤ γ\nr : R\nr_in : v.restrict r < ↑γ₀\ns : R\ns_in : v.restrict s < ... | gcongr <;> exact zero_le' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 85,
"column": 30
} | {
"line": 85,
"column": 55
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nthis : LinearOrderedCommGroupWithZero (ValueGroup₀ v) := instLinearOrderedCommGroupWithZero\nγ γ₀ : (ValueGroup₀ v)ˣ\nh : γ₀ * γ₀ ≤ γ\nr : R\nr_in : v.restrict r < ↑γ₀\ns : R\ns_in : v.restrict s < ... | gcongr <;> exact zero_le' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 215,
"column": 41
} | {
"line": 215,
"column": 93
} | [
{
"pp": "K : Type u\ninst✝³ : DivisionRing K\nΓ₀ : Type v\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\ninst✝¹ : MulArchimedean Γ₀\ninst✝ : Valued K Γ₀\nr : Γ₀\nhr : r ≠ 0\nh : ∀ (x : K), v x ≠ 0 → r < v x\n⊢ ∀ (x : K), x ≠ 0 → v x = 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Linea... | by simpa using Valued.v.map_eq_one_of_forall_lt hr h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 18
} | [
{
"pp": "R : Type u_2\nS : Type u_1\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\n⊢ minpolyDiv R x * (X - C x) = map (algebraMap R S) (minpoly R x)",
"usedConstants": [
"Polynomial.C",
"HMul.hMul",
"Algebra.algebraMap",
"CommSemiring.toSemiring",
"HSub.... | delta minpolyDiv | Lean.Elab.Tactic.evalDelta | Lean.Parser.Tactic.delta |
Mathlib.FieldTheory.Minpoly.MinpolyDiv | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 18
} | [
{
"pp": "R : Type u_1\nK : Type u_3\nS : Type u_2\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nx : S\nhx : IsIntegral R x\ninst✝⁶ : IsDomain R\ninst✝⁵ : IsIntegrallyClosed R\ninst✝⁴ : IsDomain S\ninst✝³ : Algebra R K\ninst✝² : Algebra K S\ninst✝¹ : IsScalarTower R K S\nins... | delta minpolyDiv | Lean.Elab.Tactic.evalDelta | Lean.Parser.Tactic.delta |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 560,
"column": 2
} | {
"line": 561,
"column": 63
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\n⊢ M.Indep I ↔ ¬M.Dep I ∧ I ⊆ M.E",
"usedConstants": [
"Eq.mpr",
"Matroid.Dep",
"congrArg",
"Matroid.E",
"Matroid.Indep",
"id",
"HasSubset.Subset",
"Matroid.Indep.subset_ground",
"And",
"Iff",
"... | rw [dep_iff, not_and, not_imp_not]
exact ⟨fun h ↦ ⟨fun _ ↦ h, h.subset_ground⟩, fun h ↦ h.1 h.2⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 560,
"column": 2
} | {
"line": 561,
"column": 63
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\n⊢ M.Indep I ↔ ¬M.Dep I ∧ I ⊆ M.E",
"usedConstants": [
"Eq.mpr",
"Matroid.Dep",
"congrArg",
"Matroid.E",
"Matroid.Indep",
"id",
"HasSubset.Subset",
"Matroid.Indep.subset_ground",
"And",
"Iff",
"... | rw [dep_iff, not_and, not_imp_not]
exact ⟨fun h ↦ ⟨fun _ ↦ h, h.subset_ground⟩, fun h ↦ h.1 h.2⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 947,
"column": 2
} | {
"line": 947,
"column": 89
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X Y : Set α\nhI : M.IsBasis I X\nhXY : X ⊆ Y\nhY : Y ⊆ M.E\nJ : Set α\nhJ : M.IsBasis J Y\nhIJ : I ⊆ J\n⊢ J ∩ X ⊆ I",
"usedConstants": [
"Membership.mem",
"Matroid.Indep.subset",
"Insert.insert",
"Set.instInter",
"Inter.inter",
"Set... | exact fun e he ↦ hI.mem_of_insert_indep he.2 (hJ.indep.subset (insert_subset he.1 hIJ)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Dual | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 26
} | [
{
"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\n⊢ ∃ x ∈ X \\ I, i... | have hXE := hX_max.1.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 1087,
"column": 33
} | {
"line": 1087,
"column": 45
} | [
{
"pp": "α : Type u_1\nE : Set α\nhE : E.Finite\nf : Matroid α → Set α × Set (Set α) := fun M ↦ (M.E, {B | M.IsBase B})\nM M' : Matroid α\nhMM' : {B | M.IsBase B} = {B | M'.IsBase B} ∧ M.E = M'.E\n⊢ M = M'",
"usedConstants": [
"congrArg",
"Matroid.E",
"setOf",
"Membership.mem",
... | Set.ext_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Restrict | {
"line": 144,
"column": 27
} | {
"line": 144,
"column": 35
} | [
{
"pp": "α : Type u_1\nM : Matroid α\n⊢ ∀ ⦃I : Set α⦄, I ⊆ (M ↾ M.E).E → ((M ↾ M.E).Indep I ↔ M.Indep I)",
"usedConstants": [
"and_true",
"congrArg",
"Matroid.E",
"Matroid.Indep",
"HasSubset.Subset",
"iff_self",
"And",
"Iff",
"True",
"eq_true",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.Matroid.Minor.Restrict | {
"line": 298,
"column": 6
} | {
"line": 298,
"column": 32
} | [
{
"pp": "α : Type u_1\nR R' : Set α\nM : Matroid α\nh : R ⊆ R'\n⊢ M ↾ R ≤r M ↾ R'",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Matroid.IsRestriction",
"Matroid.restrict_restrict_eq",
"Matroid.restrict",
"Eq.symm",
"Eq",
"Matroid"
]
}
] | ← restrict_restrict_eq M h | Lean.Elab.Tactic.evalRewriteSeq | null |
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