module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 856,
"column": 2
} | {
"line": 857,
"column": 34
} | {
"line": 858,
"column": 2
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j k : ι\nhi : c.prev j = i\nhk : c.next j = k\ninst✝¹ : K.HasHomology j\ninst✝ : (K.sc' i j k).HasHomology\n⊢ ShortComplex.cyclesMap ((natIsoSc' C c i j k hi hk).ho... | [
"C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex C c\ni j k : ι\nhi : c.prev j = i\nhk : c.next j = k\ninst✝¹ : K.HasHomology j\ninst✝ : (K.sc' i j k).HasHomology\n⊢ (K.sc j).iCycles = K.iCycles j"
] | simp only [ShortComplex.cyclesMap_i, shortComplexFunctor_obj_X₂, shortComplexFunctor'_obj_X₂,
natIsoSc'_hom_app_τ₂, comp_id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 865,
"column": 2
} | {
"line": 866,
"column": 16
} | {
"line": 868,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\n⊢ Mono (kernelSequence f).f",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Mono",... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 865,
"column": 2
} | {
"line": 866,
"column": 16
} | {
"line": 868,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\n⊢ Mono (kernelSequence f).f",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Mono",... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 870,
"column": 2
} | {
"line": 871,
"column": 16
} | {
"line": 873,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\n⊢ Epi (cokernelSequence f).g",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Short... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 870,
"column": 2
} | {
"line": 871,
"column": 16
} | {
"line": 873,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\n⊢ Epi (cokernelSequence f).g",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory.Short... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Action.Basic | {
"line": 62,
"column": 61
} | {
"line": 62,
"column": 87
} | {
"line": 62,
"column": 87
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝ : Group G\nA : Action V G\ng : G\n⊢ A.ρ (g⁻¹ * g) = 𝟙 A.V",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"MonoidHom.instFunLike",
"inv_... | [] | rw [inv_mul_cancel, ρ_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Action.Basic | {
"line": 62,
"column": 61
} | {
"line": 62,
"column": 87
} | {
"line": 62,
"column": 87
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝ : Group G\nA : Action V G\ng : G\n⊢ A.ρ (g⁻¹ * g) = 𝟙 A.V",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"MonoidHom.instFunLike",
"inv_... | [] | rw [inv_mul_cancel, ρ_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Action.Basic | {
"line": 62,
"column": 61
} | {
"line": 62,
"column": 87
} | {
"line": 62,
"column": 87
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Category.{v_1, u_1} V\nG : Type u_2\ninst✝ : Group G\nA : Action V G\ng : G\n⊢ A.ρ (g⁻¹ * g) = 𝟙 A.V",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"MonoidHom.instFunLike",
"inv_... | [] | rw [inv_mul_cancel, ρ_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.IsBounded | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 84
} | {
"line": 185,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder β\ninst✝ : IsDirectedOrder β\nf : α → β\nb : β\nhb : ∀ᶠ (x : β) in map f cofinite, (fun x1 x2 ↦ x1 ≤ x2) x b\nthis : Nonempty β\n⊢ BddAbove (range f)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image_univ... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder β\ninst✝ : IsDirectedOrder β\nf : α → β\nb : β\nhb : ∀ᶠ (x : β) in map f cofinite, (fun x1 x2 ↦ x1 ≤ x2) x b\nthis : Nonempty β\n⊢ BddAbove (f '' {x | f x ≤ b}) ∧ BddAbove (f '' {x | f x ≤ b}ᶜ)"
] | rw [← image_univ, ← union_compl_self { x | f x ≤ b }, image_union, bddAbove_union] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Homotopy | {
"line": 840,
"column": 16
} | {
"line": 840,
"column": 93
} | {
"line": 842,
"column": 0
} | [
{
"pp": "ι✝ : Type u_1\nV : Type u\ninst✝⁶ : Category.{v, u} V\ninst✝⁵ : Preadditive V\nc✝ : ComplexShape ι✝\nC✝ D E : HomologicalComplex V c✝\nf✝ g✝ : C✝ ⟶ D\nh✝ k : D ⟶ E\ni✝ : ι✝\nC : Type u_2\ninst✝⁴ : Category.{v_1, u_2} C\ninst✝³ : Preadditive C\nι : Type ?u.42\nc : ComplexShape ι\ninst✝² : DecidableRel c... | [] | by rw [← homologyMap_comp, h.homotopyInvHomId.homologyMap_eq, homologyMap_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactness.CompactlyCoherentSpace | {
"line": 103,
"column": 2
} | {
"line": 106,
"column": 47
} | {
"line": 108,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactlyCoherentSpace X\ns : Set X\n⊢ IsOpen[inst✝¹] s ↔\n ∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] (f : K → X), Continuous[_, inst✝¹] f → IsOpen (f ⁻¹' s)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
... | [] | refine ⟨fun hs _ _ _ _ hf ↦ hs.preimage hf, fun hs ↦ isOpen_iff |>.mpr ?_⟩
intro K hK
have : CompactSpace K := isCompact_iff_compactSpace.mp hK
exact hs K Subtype.val continuous_subtype_val | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.CompactlyCoherentSpace | {
"line": 103,
"column": 2
} | {
"line": 106,
"column": 47
} | {
"line": 108,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactlyCoherentSpace X\ns : Set X\n⊢ IsOpen[inst✝¹] s ↔\n ∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] (f : K → X), Continuous[_, inst✝¹] f → IsOpen (f ⁻¹' s)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
... | [] | refine ⟨fun hs _ _ _ _ hf ↦ hs.preimage hf, fun hs ↦ isOpen_iff |>.mpr ?_⟩
intro K hK
have : CompactSpace K := isCompact_iff_compactSpace.mp hK
exact hs K Subtype.val continuous_subtype_val | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.CompactlyCoherentSpace | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 45
} | {
"line": 157,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nA : Set (𝐤X)\n⊢ IsOpen[instTopologicalSpace] A ↔\n ∀ (K : Set X), IsCompact K → IsOpen[instTopologicalSpaceSubtype] (K ↓∩ ⇑(CompactCoherentification.mk X) ⁻¹' A)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Eq... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nA : Set (𝐤X)\n⊢ (∀ (i : Set X),\n IsCompact i →\n IsOpen[TopologicalSpace.coinduced Subtype.val\n { IsOpen := instTopologicalSpace._aux_1 i, isOpen_univ := ⋯, isOpen_inter := ⋯, isOpen_sUnion := ⋯ }]\n (⇑(CompactCoherentification.mk X) ⁻¹... | simp_rw [isOpen_coinduced, isOpen_iSup_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Compactness.CompactlyCoherentSpace | {
"line": 242,
"column": 68
} | {
"line": 243,
"column": 42
} | {
"line": 245,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nK : Set X\n⊢ IsCompact (⇑(CompactCoherentification.mk X) '' K) ↔ IsCompact K",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"CompactCoherentification.mk",
"congrArg",
"Iff.rfl"... | [] | by
rw [isCompact_iff, Equiv.preimage_image] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 196,
"column": 76
} | {
"line": 198,
"column": 86
} | {
"line": 200,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf : β → α\na : α\ng : γ → α\ne : ↑(mulSupport f) ≃ ↑(mulSupport g)\nhe : ∀ (x : ↑(mulSupport f)), g ↑(e x) = f ↑x\n⊢ HasProd f a ↔ HasProd g a",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants":... | [] | by
have : (g ∘ (↑)) ∘ e = f ∘ (↑) := funext he
rw [← hasProd_subtype_mulSupport, ← this, e.hasProd_iff, hasProd_subtype_mulSupport] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 33
} | {
"line": 325,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf g : β → α\na b : α\nL : SummationFilter β\ninst✝ : ContinuousMul α\nhf : HasProd f a L\nhg : HasProd g b L\n⊢ HasProd (fun b ↦ f b * g b) (a * b) L",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf g : β → α\na b : α\nL : SummationFilter β\ninst✝ : ContinuousMul α\nhf : Tendsto (fun s ↦ ∏ b ∈ s, f b) L.filter (𝓝 a)\nhg : Tendsto (fun s ↦ ∏ b ∈ s, g b) L.filter (𝓝 b)\n⊢ Tendsto (fun s ↦ ∏ b ∈ s, f b * g b) L.filter (𝓝 (a * b)... | dsimp only [HasProd] at hf hg ⊢ | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Order.LiminfLimsup | {
"line": 1176,
"column": 6
} | {
"line": 1176,
"column": 29
} | {
"line": 1177,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : ConditionallyCompleteLinearOrder β\nf : Filter α\nF : ι → α → β\ns : Finset ι\nhs : s.Nonempty\nh₁ : ∀ i ∈ s, IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (F i)\nh₂ : ∀ i ∈ s, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (F i)\nbddsup : IsBoundedUnder (fun x1 x2 ... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : ConditionallyCompleteLinearOrder β\nf : Filter α\nF : ι → α → β\ns : Finset ι\nh₁ : ∀ i ∈ s, IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (F i)\nh₂ : ∀ i ∈ s, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (F i)\ni : ι\ni_s : i ∈ s\nbddsup : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2... | rcases hs with ⟨i, i_s⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.UniformSpace.Equiv | {
"line": 167,
"column": 29
} | {
"line": 167,
"column": 33
} | {
"line": 167,
"column": 33
} | [
{
"pp": "α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ninst✝ : UniformSpace δ\nf : α ≃ᵤ β\ng : β → α\nhg : Function.RightInverse g ⇑f\nx : β\n⊢ f.symm (f (g x)) = f.symm x",
"ppTerm": "?m.57",
"assigned": true,
"usedC... | [
"α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ninst✝ : UniformSpace δ\nf : α ≃ᵤ β\ng : β → α\nhg : Function.RightInverse g ⇑f\nx : β\n⊢ f.symm x = f.symm x"
] | hg x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.CompactOpen | {
"line": 497,
"column": 2
} | {
"line": 498,
"column": 53
} | {
"line": 499,
"column": 2
} | [
{
"pp": "X : Type u_2\nY : Type u_3\nT : Type u_5\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace T\ns : Set T\nf : T → X → Y\ng : C(X, Y)\nf_cont : ContinuousOn (Function.uncurry f) (s ×ˢ univ)\n⊢ ContinuousOn (fun x ↦ mkD (f x) g) s",
"ppTerm": "?m.28",
"assigned": ... | [
"X : Type u_2\nY : Type u_3\nT : Type u_5\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace T\ns : Set T\nf : T → X → Y\ng : C(X, Y)\nf_cont : ContinuousOn (Function.uncurry f) (s ×ˢ univ)\nthis : ∀ x ∈ s, Continuous[inst✝², inst✝¹] (f x)\n⊢ ContinuousOn (fun x ↦ mkD (f x) g) s"
] | have (x) (hx : x ∈ s) : Continuous (f x) := f_cont.comp_continuous
(Continuous.prodMk_right x) fun _ ↦ ⟨hx, trivial⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 144,
"column": 65
} | {
"line": 148,
"column": 30
} | {
"line": 150,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\nx : α\np : Filter ι\np' : Filter α\nh : TendstoUniformlyOnFilter F f p p'\nhx : 𝓟 {x} ≤ p'\n⊢ Tendsto (fun n ↦ F n x) p (𝓝 (f x))",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Pu... | [] | by
refine Uniform.tendsto_nhds_right.mpr fun u hu => mem_map.mpr ?_
filter_upwards [(h u hu).curry]
intro i h
simpa using h.filter_mono hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 91
} | {
"line": 133,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\np : Filter ι\ninst✝ : CompactSpace α\nh : TendstoLocallyUniformly F f p\nV : Set (β × β)\nhV : V ∈ 𝓤 β\nU : α → Set α\nhU : ∀ (x : α), U x ∈ 𝓝 x ∧ ∀ᶠ (n : ι) in p, ∀ y ∈ U x, (f y... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\np : Filter ι\ninst✝ : CompactSpace α\nh : TendstoLocallyUniformly F f p\nV : Set (β × β)\nhV : V ∈ 𝓤 β\nU : α → Set α\nhU : ∀ (x : α), U x ∈ 𝓝 x ∧ ∀ᶠ (n : ι) in p, ∀ y ∈ U x, (f y, F n y) ∈ V... | obtain ⟨t, ht⟩ := isCompact_univ.elim_nhds_subcover' (fun k _ => U k) fun k _ => (hU k).1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 47
} | {
"line": 285,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nh :\n ∀ x ∈ s,\n ∀ u ∈ 𝓤 β,\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x →... | [
"case mpr\nα : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nh✝ :\n ∀ x ∈ s,\n ∀ u ∈ 𝓤 β,\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x → ∀ {y : α},... | obtain ⟨pa, hpa, pb, hpb, h⟩ := h x hx u hu | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1134,
"column": 4
} | {
"line": 1134,
"column": 41
} | {
"line": 1135,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ₁ : Type u_5\nδ₂ : Type u_6\nφ₁ : δ₁ → α\nφ₂ : δ₂ → α\n𝔗₁ : Set (Set δ₁)\n𝔗₂ : Set (Set δ₂)\nh_image₁ : MapsTo (fun x ↦ φ₁ '' x) 𝔗₁ 𝔖\nh_image₂ : MapsTo (fun x ↦ φ₂ '' x) 𝔗₂ 𝔖\nh_preimage₁ : MapsTo (fun x ↦ φ₁ ⁻... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ₁ : Type u_5\nδ₂ : Type u_6\nφ₁ : δ₁ → α\nφ₂ : δ₂ → α\n𝔗₁ : Set (Set δ₁)\n𝔗₂ : Set (Set δ₂)\nh_image₁ : MapsTo (fun x ↦ φ₁ '' x) 𝔗₁ 𝔖\nh_image₂ : MapsTo (fun x ↦ φ₂ '' x) 𝔗₂ 𝔖\nh_preimage₁ : MapsTo (fun x ↦ φ₁ ⁻¹' x) 𝔖 𝔗₁... | rw [← uniformContinuous_iff_le_comap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1136,
"column": 4
} | {
"line": 1136,
"column": 41
} | {
"line": 1137,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ₁ : Type u_5\nδ₂ : Type u_6\nφ₁ : δ₁ → α\nφ₂ : δ₂ → α\n𝔗₁ : Set (Set δ₁)\n𝔗₂ : Set (Set δ₂)\nh_image₁ : MapsTo (fun x ↦ φ₁ '' x) 𝔗₁ 𝔖\nh_image₂ : MapsTo (fun x ↦ φ₂ '' x) 𝔗₂ 𝔖\nh_preimage₁ : MapsTo (fun x ↦ φ₁ ⁻... | [
"case refine_2\nα : Type u_1\nβ : Type u_2\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ₁ : Type u_5\nδ₂ : Type u_6\nφ₁ : δ₁ → α\nφ₂ : δ₂ → α\n𝔗₁ : Set (Set δ₁)\n𝔗₂ : Set (Set δ₂)\nh_image₁ : MapsTo (fun x ↦ φ₁ '' x) 𝔗₁ 𝔖\nh_image₂ : MapsTo (fun x ↦ φ₂ '' x) 𝔗₂ 𝔖\nh_preimage₁ : MapsTo (fun x ↦ φ₁ ⁻¹' x) 𝔖 𝔗₁... | rw [← uniformContinuous_iff_le_comap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1159,
"column": 4
} | {
"line": 1159,
"column": 41
} | {
"line": 1160,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ : ι → Type u_5\nφ : (i : ι) → δ i → α\n𝔗 : (i : ι) → Set (Set (δ i))\nh_image : ∀ (i : ι), MapsTo (fun x ↦ φ i '' x) (𝔗 i) 𝔖\nh_preimage : ∀ (i : ι), MapsTo (fun x ↦ φ i ⁻¹' x) 𝔖 (𝔗 i)\nh_cover : ∀... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nδ : ι → Type u_5\nφ : (i : ι) → δ i → α\n𝔗 : (i : ι) → Set (Set (δ i))\nh_image : ∀ (i : ι), MapsTo (fun x ↦ φ i '' x) (𝔗 i) 𝔖\nh_preimage : ∀ (i : ι), MapsTo (fun x ↦ φ i ⁻¹' x) 𝔖 (𝔗 i)\nh_cover : ∀ S ∈ 𝔖, ∃ I... | rw [← uniformContinuous_iff_le_comap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1215,
"column": 9
} | {
"line": 1215,
"column": 23
} | {
"line": 1215,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\np : Filter ι\ns : Set β\nF : ι → α → β\nf : α → β\nt : Set α\nhF : ∀ᶠ (i : ι) in p, ∀ x ∈ t, F i x ∈ s\nhf : ∀ x ∈ t, f x ∈ s\ng : β → γ\nhg : UniformContinuousOn g s\nh : TendstoUniformlyOn F f p t... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1215,
"column": 9
} | {
"line": 1215,
"column": 23
} | {
"line": 1215,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\np : Filter ι\ns : Set β\nF : ι → α → β\nf : α → β\nt : Set α\nhF : ∀ᶠ (i : ι) in p, ∀ x ∈ t, F i x ∈ s\nhf : ∀ x ∈ t, f x ∈ s\ng : β → γ\nhg : UniformContinuousOn g s\nh : TendstoUniformlyOn F f p t... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1215,
"column": 9
} | {
"line": 1215,
"column": 23
} | {
"line": 1215,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\np : Filter ι\ns : Set β\nF : ι → α → β\nf : α → β\nt : Set α\nhF : ∀ᶠ (i : ι) in p, ∀ x ∈ t, F i x ∈ s\nhf : ∀ x ∈ t, f x ∈ s\ng : β → γ\nhg : UniformContinuousOn g s\nh : TendstoUniformlyOn F f p t... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.ErdosKaplansky | {
"line": 134,
"column": 2
} | {
"line": 135,
"column": 33
} | {
"line": 137,
"column": 0
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nh : ℵ₀ ≤ Module.rank K V\n⊢ lift.{u, v} (Module.rank K V) < Module.rank K (V →ₗ[K] K)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.to_smulCommClass",
"Pre... | [] | rw [rank_dual_eq_card_dual_of_aleph0_le_rank h, ← rank_dual_eq_card_dual_of_aleph0_le_rank' h]
exact lift_rank_lt_rank_dual' h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.ErdosKaplansky | {
"line": 134,
"column": 2
} | {
"line": 135,
"column": 33
} | {
"line": 137,
"column": 0
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nh : ℵ₀ ≤ Module.rank K V\n⊢ lift.{u, v} (Module.rank K V) < Module.rank K (V →ₗ[K] K)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.to_smulCommClass",
"Pre... | [] | rw [rank_dual_eq_card_dual_of_aleph0_le_rank h, ← rank_dual_eq_card_dual_of_aleph0_le_rank' h]
exact lift_rank_lt_rank_dual' h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.DivisionRing | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 47
} | {
"line": 69,
"column": 47
} | [
{
"pp": "K : Type u\nV✝ V₁ V₂ V₃ : Type v\nι : Type w\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V✝\ninst✝² : Module K V✝\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\nV : Type u₀\nx✝¹ : AddCommGroup V\nx✝ : Module K V\nb : Module.Basis (Module.Free.ChooseBasisIndex K V) K V := Module.Free.chooseBasis K ... | [
"K : Type u\nV✝ V₁ V₂ V₃ : Type v\nι : Type w\ninst✝⁴ : DivisionRing K\ninst✝³ : AddCommGroup V✝\ninst✝² : Module K V✝\ninst✝¹ : AddCommGroup V₁\ninst✝ : Module K V₁\nV : Type u₀\nx✝¹ : AddCommGroup V\nx✝ : Module K V\nb : Module.Basis (Module.Free.ChooseBasisIndex K V) K V := Module.Free.chooseBasis K V\n⊢ lift.{u... | Module.Free.rank_eq_card_chooseBasisIndex | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.RankNullity | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 34
} | {
"line": 265,
"column": 35
} | [
{
"pp": "R : Type u_2\nM : Type u\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : HasRankNullity.{u, u_2} R\ninst✝⁵ : StrongRankCondition R\ninst✝⁴ : IsDomain R\ninst✝³ : IsTorsionFree R M\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nL : Submodule R M\ninst✝ : Module.Fin... | [
"R : Type u_2\nM : Type u\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : HasRankNullity.{u, u_2} R\ninst✝⁵ : StrongRankCondition R\ninst✝⁴ : IsDomain R\ninst✝³ : IsTorsionFree R M\nN : Type u_1\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nL : Submodule R M\ninst✝ : Module.Finite R ↥L\nf ... | ← Submodule.finrank_eq_rank, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 310,
"column": 2
} | {
"line": 314,
"column": 49
} | {
"line": 316,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝¹ : DivisionRing K\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → ι → K\nhb : LinearIndependent K b\n⊢ ⇑(basisOfPiSpaceOfLinearIndependent hb) = b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"Pi.Function.module",
... | [] | by_cases hι : Nonempty ι
· simp [hι, basisOfPiSpaceOfLinearIndependent]
· rw [basisOfPiSpaceOfLinearIndependent, dif_neg hι]
ext i
exact ((not_nonempty_iff.mp hι).false i).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.FiniteDimensional.Lemmas | {
"line": 310,
"column": 2
} | {
"line": 314,
"column": 49
} | {
"line": 316,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝¹ : DivisionRing K\nι : Type u_1\ninst✝ : Fintype ι\nb : ι → ι → K\nhb : LinearIndependent K b\n⊢ ⇑(basisOfPiSpaceOfLinearIndependent hb) = b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Eq.mpr",
"Pi.Function.module",
... | [] | by_cases hι : Nonempty ι
· simp [hι, basisOfPiSpaceOfLinearIndependent]
· rw [basisOfPiSpaceOfLinearIndependent, dif_neg hι]
ext i
exact ((not_nonempty_iff.mp hι).false i).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Products | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 47
} | {
"line": 90,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nι : Type v\nZ : ι → ModuleCat R\ninst✝ : DecidableEq ι\ns : Cocone (Discrete.functor Z)\nf : (coproductCocone Z).pt ⟶ s.pt\nh : ∀ (j : Discrete ι), (coproductCocone Z).ι.app j ≫ f = s.ι.app j\n⊢ Hom.hom f = Hom.hom (↟(toModule R ι ↑s.1 fun i ↦ Hom.hom (s.ι.app { as := i }))... | [
"R : Type u\ninst✝¹ : Ring R\nι : Type v\nZ : ι → ModuleCat R\ninst✝ : DecidableEq ι\ns : Cocone (Discrete.functor Z)\nf : (coproductCocone Z).pt ⟶ s.pt\nh : ∀ (j : Discrete ι), (coproductCocone Z).ι.app j ≫ f = s.ι.app j\ni : ι\n⊢ Hom.hom f ∘ₗ lof R ι (fun i ↦ ↑(Z i)) i =\n Hom.hom (↟(toModule R ι ↑s.1 fun i ↦ ... | refine DirectSum.linearMap_ext _ fun i ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Homology.ShortComplex.ShortExact | {
"line": 170,
"column": 49
} | {
"line": 172,
"column": 28
} | {
"line": 174,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Category.{v_2, u_2} D\ninst✝¹ : Preadditive C\ninst✝ : Balanced C\nS : ShortComplex C\nhS : S.ShortExact\n⊢ IsColimit (CokernelCofork.ofπ S.g ⋯)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheo... | [] | by
have := hS.epi_g
exact hS.exact.gIsCokernel | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.Grp.EpiMono | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 19
} | {
"line": 193,
"column": 0
} | [
{
"pp": "A B : GrpCat\nf : A ⟶ B\nb1 b2 : ↑B\nx✝ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ { toFun := fun x ↦ (b1 * b2) • x, invFun := fun x ↦ (b1 * b2)⁻¹ • x, left_inv := ⋯, right_inv := ⋯ } x✝ =\n ({ toFun := fun x ↦ b1 • x, invFun := fun x ↦ b1⁻¹ •... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Preadditive.LeftExact | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 84
} | {
"line": 177,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : Preadditive C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\ninst✝³ : Preadditive D\nF : C ⥤ D\ninst✝² : F.PreservesZeroMorphisms\ninst✝¹ : HasBinaryBiproducts C\ninst✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesColimit (parallelPair f 0) F\nX Y : C\nf g : X ⟶ ... | [
"C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\ninst✝⁵ : Preadditive C\nD : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D\ninst✝³ : Preadditive D\nF : C ⥤ D\ninst✝² : F.PreservesZeroMorphisms\ninst✝¹ : HasBinaryBiproducts C\ninst✝ : ∀ {X Y : C} (f : X ⟶ Y), PreservesColimit (parallelPair f 0) F\nX Y : C\nf g : X ⟶ Y\nthis✝ : P... | let c' := isColimitCokernelCoforkOfCofork (i.ofIsoColimit (Cofork.isoCoforkOfπ c)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.Dual.Lemmas | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 43
} | {
"line": 328,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np : Submodule R M\nx : M\nhx : ¬p.mkQ x = 0\nhp' : Projective R (M ⧸ p)\n⊢ ∃ f, f (p.mkQ x) ≠ 0",
"ppTerm": "?m.126",
"assigned": true,
"usedConstants": [
"Submodule",
"Submodule.Quotient.a... | [] | exact Projective.exists_dual_ne_zero R hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Abelian.Exact | {
"line": 105,
"column": 2
} | {
"line": 114,
"column": 38
} | {
"line": 116,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Abelian C\nS : ShortComplex C\ncg : KernelFork S.g\nhg : IsLimit cg\ncf : CokernelCofork S.f\nhf : IsColimit cf\n⊢ S.Exact ↔ Fork.ι cg ≫ Cofork.π cf = 0",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"CategoryTheory.Preaddi... | [] | rw [exact_iff_kernel_ι_comp_cokernel_π_zero]
let e₁ := IsLimit.conePointUniqueUpToIso (kernelIsKernel S.g) hg
let e₂ := IsColimit.coconePointUniqueUpToIso (cokernelIsCokernel S.f) hf
have : cg.ι ≫ cf.π = e₁.inv ≫ kernel.ι S.g ≫ cokernel.π S.f ≫ e₂.hom := by
have eq₁ := IsLimit.conePointUniqueUpToIso_inv_comp ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Exact | {
"line": 105,
"column": 2
} | {
"line": 114,
"column": 38
} | {
"line": 116,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : Abelian C\nS : ShortComplex C\ncg : KernelFork S.g\nhg : IsLimit cg\ncf : CokernelCofork S.f\nhf : IsColimit cf\n⊢ S.Exact ↔ Fork.ι cg ≫ Cofork.π cf = 0",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"CategoryTheory.Preaddi... | [] | rw [exact_iff_kernel_ι_comp_cokernel_π_zero]
let e₁ := IsLimit.conePointUniqueUpToIso (kernelIsKernel S.g) hg
let e₂ := IsColimit.coconePointUniqueUpToIso (cokernelIsCokernel S.f) hf
have : cg.ι ≫ cf.π = e₁.inv ≫ kernel.ι S.g ≫ cokernel.π S.f ≫ e₂.hom := by
have eq₁ := IsLimit.conePointUniqueUpToIso_inv_comp ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.KanExtension.Adjunction | {
"line": 402,
"column": 6
} | {
"line": 403,
"column": 60
} | {
"line": 404,
"column": 6
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\nL✝ : C ⥤ D\nH : Type u_3\ninst✝⁴ : Category.{v_3, u_3} H\ninst✝³ : ∀ (F : C ⥤ H), L✝.HasRightKanExtension F\nL : C ⥤ D\ninst✝² : ∀ (G : C ⥤ H), L.HasRightKanExtension G\ninst✝¹ : HasLimitsOfShape C H\ninst✝ : Ha... | [
"C : Type u_1\nD : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C\ninst✝⁵ : Category.{v_2, u_2} D\nL✝ : C ⥤ D\nH : Type u_3\ninst✝⁴ : Category.{v_3, u_3} H\ninst✝³ : ∀ (F : C ⥤ H), L✝.HasRightKanExtension F\nL : C ⥤ D\ninst✝² : ∀ (G : C ⥤ H), L.HasRightKanExtension G\ninst✝¹ : HasLimitsOfShape C H\ninst✝ : HasLimitsOfSha... | rw [assoc, assoc, limMap_π, limitIsoOfIsRightKanExtension_hom_π_assoc,
limitIsoOfIsRightKanExtension_hom_π, limMap_π_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Abelian.Subobject | {
"line": 51,
"column": 33
} | {
"line": 51,
"column": 91
} | {
"line": 51,
"column": 91
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nA : Cᵒᵖ\nf : A ⟶ op X\nhf : Mono f\n⊢ ({ hom := (epiDesc f.unop (cokernel.π (kernel.ι f.unop)) ⋯).op, inv := (cokernel.desc (kernel.ι f.unop) f.unop ⋯).op,\n hom_inv_id := ⋯, inv_hom_id := ⋯ }.hom ≫\n f).unop =\n (co... | [] | by simp only [unop_comp, Quiver.Hom.unop_op, comp_epiDesc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 207,
"column": 2
} | {
"line": 214,
"column": 44
} | {
"line": 216,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\ninst✝ : Balanced C\nhP : P.IsSeparating\n⊢ P.IsDetecting",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.IsIso",
"Cat... | [] | intro X Y f hf
refine
(isIso_iff_mono_and_epi _).2 ⟨⟨fun g h hgh => hP _ _ fun G hG i => ?_⟩, ⟨fun g h hgh => ?_⟩⟩
· obtain ⟨t, -, ht⟩ := hf G hG (i ≫ g ≫ f)
rw [ht (i ≫ g) (Category.assoc _ _ _), ht (i ≫ h) (hgh.symm ▸ Category.assoc _ _ _)]
· refine hP _ _ fun G hG i => ?_
obtain ⟨t, rfl, -⟩ := hf G... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 207,
"column": 2
} | {
"line": 214,
"column": 44
} | {
"line": 216,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nP : ObjectProperty C\ninst✝ : Balanced C\nhP : P.IsSeparating\n⊢ P.IsDetecting",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.IsIso",
"Cat... | [] | intro X Y f hf
refine
(isIso_iff_mono_and_epi _).2 ⟨⟨fun g h hgh => hP _ _ fun G hG i => ?_⟩, ⟨fun g h hgh => ?_⟩⟩
· obtain ⟨t, -, ht⟩ := hf G hG (i ≫ g ≫ f)
rw [ht (i ≫ g) (Category.assoc _ _ _), ht (i ≫ h) (hgh.symm ▸ Category.assoc _ _ _)]
· refine hP _ _ fun G hG i => ?_
obtain ⟨t, rfl, -⟩ := hf G... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 95,
"column": 49
} | {
"line": 95,
"column": 88
} | {
"line": 95,
"column": 88
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : D\nT : C ⥤ D\ninst✝¹ : HasFiniteLimits C\ninst✝ : PreservesFiniteLimits T\nA P : StructuredArrow S T\nf : P ⟶ A\nhf : Mono f\nq : S ⟶ T.obj (Subobject.underlying.obj (projectSubobject (Subobject.mk f)))\nhq : q ≫ ... | [] | by dsimp; simpa [← T.map_comp] using hq | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.WideEqualizers | {
"line": 310,
"column": 16
} | {
"line": 310,
"column": 53
} | {
"line": 310,
"column": 54
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\nt : Trident f\nlift : (s : Trident f) → s.pt ⟶ t.pt\nfac : ∀ (s : Trident f), lift s ≫ t.ι = s.ι\nuniq : ∀ (s : Trident f) (m : s.pt ⟶ t.pt), (∀ (j : WalkingParallelFamily J), m ≫ t.π.app j = s.π.app j) → ... | [
"J : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : J → (X ⟶ Y)\ninst✝ : Nonempty J\nt : Trident f\nlift : (s : Trident f) → s.pt ⟶ t.pt\nfac : ∀ (s : Trident f), lift s ≫ t.ι = s.ι\nuniq : ∀ (s : Trident f) (m : s.pt ⟶ t.pt), (∀ (j : WalkingParallelFamily J), m ≫ t.π.app j = s.π.app j) → m = lift s\n... | ← t.w (line (Classical.arbitrary J)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 177,
"column": 19
} | {
"line": 177,
"column": 57
} | {
"line": 177,
"column": 57
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ninst✝ : Mono f.unop.left.op\n⊢ ((Subobject.underlyingIso f.unop.left.op).hom ≫ f.unop.left.op).unop = (Subobject.mk f.unop.left... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\nA : CostructuredArrow S T\nP : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ninst✝ : Mono f.unop.left.op\n⊢ (Subobject.mk f.unop.left.op).arrow.unop = (Subobject.mk f.unop.left.op).arrow.unop"
] | Subobject.underlyingIso_hom_comp_eq_mk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.Comma | {
"line": 229,
"column": 6
} | {
"line": 229,
"column": 42
} | {
"line": 230,
"column": 4
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nS : C ⥤ D\nT : D\ninst✝¹ : HasFiniteColimits C\ninst✝ : PreservesFiniteColimits S\nA : CostructuredArrow S T\nP Q : (CostructuredArrow S T)ᵒᵖ\nf : P ⟶ op A\ng : Q ⟶ op A\nhf : Mono f\nhg : Mono g\nh :\n... | [] | exact unop_left_comp_ofMkLEMk_unop _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 664,
"column": 2
} | {
"line": 664,
"column": 80
} | {
"line": 665,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasCoproduct fun x ↦ G\n⊢ (∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (h : G ⟶ X), h ≫ f = h ≫ g) → f = g) ↔ ∀ (A : C), Epi (Sigma.desc fun f ↦ f)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"CategoryTheory.Epi",
... | [
"case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasCoproduct fun x ↦ G\nh : ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (h : G ⟶ X), h ≫ f = h ≫ g) → f = g\nA Z✝ : C\nu v : A ⟶ Z✝\nhuv : (Sigma.desc fun f ↦ f) ≫ u = (Sigma.desc fun f ↦ f) ≫ v\ni : G ⟶ A\n⊢ i ≫ u = i ≫ v",
"case refine_2\nC ... | refine ⟨fun h A => ⟨fun u v huv => h _ _ fun i => ?_⟩, fun h X Y f g hh => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Generator.Basic | {
"line": 674,
"column": 2
} | {
"line": 674,
"column": 80
} | {
"line": 675,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasProduct fun x ↦ G\n⊢ (∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (h : Y ⟶ G), f ≫ h = g ≫ h) → f = g) ↔ ∀ (A : C), Mono (Pi.lift fun f ↦ f)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"CategoryTheory.Mono",
... | [
"case refine_1\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nG : C\ninst✝ : ∀ (A : C), HasProduct fun x ↦ G\nh : ∀ ⦃X Y : C⦄ (f g : X ⟶ Y), (∀ (h : Y ⟶ G), f ≫ h = g ≫ h) → f = g\nA Z✝ : C\nu v : Z✝ ⟶ A\nhuv : (u ≫ Pi.lift fun f ↦ f) = v ≫ Pi.lift fun f ↦ f\ni : A ⟶ G\n⊢ u ≫ i = v ≫ i",
"case refine_2\nC : Type u₁\... | refine ⟨fun h A => ⟨fun u v huv => h _ _ fun i => ?_⟩, fun h X Y f g hh => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Category.Grp.CartesianMonoidal | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 93
} | {
"line": 51,
"column": 0
} | [
{
"pp": "case snd\nG H : GrpCat\np : ↑G\nq : ↑H\n⊢ ((hom (Functor.LaxMonoidal.μ (forget GrpCat) G H)) (p, q)).2 = (p, q).2",
"ppTerm": "?snd",
"assigned": true,
"usedConstants": [
"GrpCat.instConcreteCategoryMonoidHomCarrier",
"GrpCat",
"MonoidHom.instFunLike",
"CategoryTheor... | [] | exact congr_hom (CC := fun X ↦ X) (Functor.Monoidal.μ_snd (forget GrpCat.{u}) G H) (p, q) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.Grp.CartesianMonoidal | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 93
} | {
"line": 51,
"column": 0
} | [
{
"pp": "case snd\nG H : GrpCat\np : ↑G\nq : ↑H\n⊢ ((hom (Functor.LaxMonoidal.μ (forget GrpCat) G H)) (p, q)).2 = (p, q).2",
"ppTerm": "?snd",
"assigned": true,
"usedConstants": [
"GrpCat.instConcreteCategoryMonoidHomCarrier",
"GrpCat",
"MonoidHom.instFunLike",
"CategoryTheor... | [] | exact congr_hom (CC := fun X ↦ X) (Functor.Monoidal.μ_snd (forget GrpCat.{u}) G H) (p, q) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.Grp.CartesianMonoidal | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 93
} | {
"line": 51,
"column": 0
} | [
{
"pp": "case snd\nG H : GrpCat\np : ↑G\nq : ↑H\n⊢ ((hom (Functor.LaxMonoidal.μ (forget GrpCat) G H)) (p, q)).2 = (p, q).2",
"ppTerm": "?snd",
"assigned": true,
"usedConstants": [
"GrpCat.instConcreteCategoryMonoidHomCarrier",
"GrpCat",
"MonoidHom.instFunLike",
"CategoryTheor... | [] | exact congr_hom (CC := fun X ↦ X) (Functor.Monoidal.μ_snd (forget GrpCat.{u}) G H) (p, q) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 770,
"column": 48
} | {
"line": 770,
"column": 58
} | {
"line": 770,
"column": 58
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : LocallySmall.{w, v₁, u₁} C\nF : C ⥤ Type w\nX : C\nG : F.Elementsᵒᵖ ⥤ Type w := (CategoryOfElements.π F).op ⋙ shrinkYoneda.{w, v₁, u₁}.obj X\nc : G.CoconeTypes := G.coconeTypesEquiv.symm (coconeπOpCompShrinkYonedaObj F X)\nthis :\n ∀ (u : G.ColimitTyp... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : LocallySmall.{w, v₁, u₁} C\nF : C ⥤ Type w\nX : C\nG : F.Elementsᵒᵖ ⥤ Type w := (CategoryOfElements.π F).op ⋙ shrinkYoneda.{w, v₁, u₁}.obj X\nc : G.CoconeTypes := G.coconeTypesEquiv.symm (coconeπOpCompShrinkYonedaObj F X)\nthis :\n ∀ (u : G.ColimitType) (x : F.ob... | rw [← hx₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Presheaf | {
"line": 847,
"column": 4
} | {
"line": 847,
"column": 20
} | {
"line": 848,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : LocallySmall.{w, v₁, u₁} C\nF : C ⥤ Type w\nX : C\n⊢ ∀ (Y : (shrinkYoneda.{w, v₁, u₁}.flip.obj (op X)).Elements)\n (m : (shrinkYoneda.{w, v₁, u₁}.flip.obj (op X)).elementsMk X (shrinkYonedaObjObjEquiv.symm (𝟙 X)) ⟶ Y),\n m = ⟨shrinkYonedaObjObjE... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : LocallySmall.{w, v₁, u₁} C\nF : C ⥤ Type w\nX : C\nu : (shrinkYoneda.{w, v₁, u₁}.flip.obj (op X)).Elements\nm : ((shrinkYoneda.{w, v₁, u₁}.flip.obj (op X)).elementsMk X (shrinkYonedaObjObjEquiv.symm (𝟙 X))).fst ⟶ u.fst\nhm :\n (ConcreteCategory.hom ((shrinkYoned... | rintro u ⟨m, hm⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.GroupTheory.Divisible | {
"line": 236,
"column": 26
} | {
"line": 238,
"column": 9
} | {
"line": 240,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Group A\ninst✝ : RootableBy A ℤ\nn : ℕ\na : A\nhn : n ≠ 0\n⊢ RootableBy.root a ↑n ^ n = a",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"RootableBy.root_cancel",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddGrou... | [] | by
have := RootableBy.root_cancel a (show (n : ℤ) ≠ 0 from mod_cast hn)
simpa | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Injective | {
"line": 282,
"column": 76
} | {
"line": 289,
"column": 86
} | {
"line": 291,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝⁷ : Ring R\nQ : Type v\ninst✝⁶ : AddCommGroup Q\ninst✝⁵ : Module R Q\nM : Type u_1\nN : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\ni : M →ₗ[R] N\nf : M →ₗ[R] Q\ninst✝ : Fact (Function.Injective ⇑i)\nh : Baer R Q\ny : N\nr : R\n... | [] | by
have : r ∈ ideal i f y := by
change (r • y) ∈ (extensionOfMax i f).toLinearPMap.domain
rw [eq1]
apply Submodule.zero_mem _
rw [ExtensionOfMaxAdjoin.extendIdealTo_is_extension i f h y r this]
dsimp [ExtensionOfMaxAdjoin.idealTo]
simp only [eq1, ← ZeroMemClass.zero_def, (extensionOfMax i f).toLinea... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Periodic | {
"line": 318,
"column": 52
} | {
"line": 318,
"column": 62
} | {
"line": 318,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝¹ : NonAssocRing α\ninst✝ : SubtractionMonoid β\nh : Antiperiodic f c\nhi : f 0 = 0\nn : ℕ\n⊢ f (-(↑(n + 1) * c)) = 0",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"AddG... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝¹ : NonAssocRing α\ninst✝ : SubtractionMonoid β\nh : Antiperiodic f c\nhi : f 0 = 0\nn : ℕ\n⊢ f (↑(n + 1) * -c) = 0"
] | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Totient | {
"line": 205,
"column": 6
} | {
"line": 206,
"column": 66
} | {
"line": 208,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Prime p\nn : ℕ\nh1 : Function.Injective fun x ↦ x * p\nh2 : image (fun x ↦ x * p) (range (p ^ n)) ⊆ range (p ^ (n + 1))\n⊢ #(range (p ^ (n + 1)) \\ image (fun x ↦ x * p) (range (p ^ n))) = p ^ n * (p - 1)",
"ppTerm": "?m.236",
"assigned": true,
"usedConstants": [
"Nat.pow_... | [] | rw [card_sdiff_of_subset h2, Finset.card_image_of_injective _ h1, card_range, card_range, ←
one_mul (p ^ n), pow_succ', ← tsub_mul, one_mul, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 823,
"column": 41
} | {
"line": 823,
"column": 82
} | {
"line": 825,
"column": 0
} | [
{
"pp": "X : Type u_3\nY : Type u_5\nα : Type u_6\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\nuα : UniformSpace α\nA : Set Y\nu : Y → X → α\nS : Set X\nhA : EquicontinuousOn (u ∘ Subtype.val) S\nhu : Continuous[tY, Pi.topologicalSpace] (S.restrict ∘ u)\nx : X\nhx : x ∈ S\n⊢ Continuous[tY, uα.toTopologica... | [] | exact continuous_apply ⟨x, hx⟩ |>.comp hu | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 823,
"column": 41
} | {
"line": 823,
"column": 82
} | {
"line": 825,
"column": 0
} | [
{
"pp": "X : Type u_3\nY : Type u_5\nα : Type u_6\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\nuα : UniformSpace α\nA : Set Y\nu : Y → X → α\nS : Set X\nhA : EquicontinuousOn (u ∘ Subtype.val) S\nhu : Continuous[tY, Pi.topologicalSpace] (S.restrict ∘ u)\nx : X\nhx : x ∈ S\n⊢ Continuous[tY, uα.toTopologica... | [] | exact continuous_apply ⟨x, hx⟩ |>.comp hu | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 823,
"column": 41
} | {
"line": 823,
"column": 82
} | {
"line": 825,
"column": 0
} | [
{
"pp": "X : Type u_3\nY : Type u_5\nα : Type u_6\ntX : TopologicalSpace X\ntY : TopologicalSpace Y\nuα : UniformSpace α\nA : Set Y\nu : Y → X → α\nS : Set X\nhA : EquicontinuousOn (u ∘ Subtype.val) S\nhu : Continuous[tY, Pi.topologicalSpace] (S.restrict ∘ u)\nx : X\nhx : x ∈ S\n⊢ Continuous[tY, uα.toTopologica... | [] | exact continuous_apply ⟨x, hx⟩ |>.comp hu | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Field | {
"line": 67,
"column": 8
} | {
"line": 67,
"column": 56
} | {
"line": 67,
"column": 57
} | [
{
"pp": "case inr\nK✝ : Type u_1\ninst✝⁴ : DivisionRing K✝\ninst✝³ : TopologicalSpace K✝\nα : Type u_2\ninst✝² : Field α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalDivisionRing α\nK : Subfield α\nx : α\nhx : x ∈ closure ↑K\nh : x ≠ 0\n⊢ x⁻¹ ∈ closure ((fun x ↦ x⁻¹) '' ↑K)",
"ppTerm": "?inr",
"as... | [] | exact mem_closure_image (continuousAt_inv₀ h) hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.AtTopBot.CompleteLattice | {
"line": 28,
"column": 13
} | {
"line": 28,
"column": 34
} | {
"line": 28,
"column": 35
} | [
{
"pp": "α : Type u_6\ninst✝¹ : Subsingleton α\ninst✝ : Preorder α\ns : Set α\nhs : s ∈ ⨅ a, 𝓟 (Ici a)\nx : α\n⊢ x ∈ s",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"iInf",
"Set.Ici",
"congrArg",
"Filter.instCompleteLatticeFilter",
... | [
"α : Type u_6\ninst✝¹ : Subsingleton α\ninst✝ : Preorder α\ns : Set α\nx : α\nhs : s ∈ 𝓟 (Ici x)\n⊢ x ∈ s"
] | ciInf_subsingleton x, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 689,
"column": 15
} | {
"line": 689,
"column": 52
} | {
"line": 689,
"column": 53
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\n⊢ toIcoDiv hp a b = toIocDiv hp a b ∨ toIocDiv hp a b + 1 = toIcoDiv hp a b",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\n⊢ ¬a ≡ b [PMOD p] ∨ toIocDiv hp a b + 1 = toIcoDiv hp a b"
] | ← not_modEq_iff_toIcoDiv_eq_toIocDiv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.AtTopBot.CompleteLattice | {
"line": 97,
"column": 19
} | {
"line": 97,
"column": 43
} | {
"line": 97,
"column": 44
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝² : Preorder β\ninst✝¹ : ConditionallyCompleteLinearOrder γ\nl : Filter α\ninst✝ : l.NeBot\nf : β → γ\nhf : Monotone f\ng : α → β\nhg : Tendsto g l atTop\nhb : ¬BddAbove (range f)\n⊢ ¬(upperBounds (range fun a ↦ f (g a))).Nonempty",
"ppTerm": "?m.64",
... | [
"α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝² : Preorder β\ninst✝¹ : ConditionallyCompleteLinearOrder γ\nl : Filter α\ninst✝ : l.NeBot\nf : β → γ\nhf : Monotone f\ng : α → β\nhg : Tendsto g l atTop\nhb : ¬BddAbove (range f)\n⊢ ¬(upperBounds (range (f ∘ g))).Nonempty"
] | ← Function.comp_def f g, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Order.Group | {
"line": 41,
"column": 6
} | {
"line": 41,
"column": 28
} | {
"line": 42,
"column": 6
} | [
{
"pp": "G : Type u_1\ninst✝⁴ : TopologicalSpace G\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : OrderTopology G\na b ε : G\nhε : ε > 1\nδ : G\nhδ₁ : 1 < δ\nhδε : δ < ε\n⊢ ∀ (a_1 : G × G), |a_1.1 / a|ₘ < δ ∧ |a_1.2 / b|ₘ < ε / δ → |a_1.1 * a_1.2 / (a * b)|ₘ < ε",
"ppTerm... | [
"G : Type u_1\ninst✝⁴ : TopologicalSpace G\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : OrderTopology G\na b ε : G\nhε : ε > 1\nδ : G\nhδ₁ : 1 < δ\nhδε : δ < ε\nc d : G\nhc : |(c, d).1 / a|ₘ < δ\nhd : |(c, d).2 / b|ₘ < ε / δ\n⊢ |(c, d).1 * (c, d).2 / (a * b)|ₘ < ε"
] | rintro ⟨c, d⟩ ⟨hc, hd⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.Algebra.Order.Field | {
"line": 243,
"column": 13
} | {
"line": 243,
"column": 22
} | {
"line": 243,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nC : 𝕜\nhf : ∀ᶠ (x : α) in l, |f x| ≤ C\nhg : Tendsto g l (𝓝 0)\n⊢ 0 = |C * 0|",
"ppTerm": "?m.185",
"as... | [
"𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nC : 𝕜\nhf : ∀ᶠ (x : α) in l, |f x| ≤ C\nhg : Tendsto g l (𝓝 0)\n⊢ 0 = |0|"
] | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 141,
"column": 7
} | {
"line": 141,
"column": 44
} | {
"line": 141,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : AddCommGroup 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsOrderedAddMonoid 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ¬x ≡ a [PMOD p]\nx✝ : 𝕜\n⊢ toIcoDiv hp a x = toIocDiv hp a x",
"ppTerm": "?m.185",
... | [
"𝕜 : Type u_1\ninst✝⁵ : AddCommGroup 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsOrderedAddMonoid 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ¬x ≡ a [PMOD p]\nx✝ : 𝕜\n⊢ ¬a ≡ x [PMOD p]"
] | ← not_modEq_iff_toIcoDiv_eq_toIocDiv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 160,
"column": 16
} | {
"line": 160,
"column": 53
} | {
"line": 160,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : AddCommGroup 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsOrderedAddMonoid 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ¬x ≡ a [PMOD p]\nx✝ : 𝕜\n⊢ toIcoDiv hp a x = toIocDiv hp a x",
"ppTerm": "?m.189",
... | [
"𝕜 : Type u_1\ninst✝⁵ : AddCommGroup 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsOrderedAddMonoid 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np : 𝕜\nhp : 0 < p\na x : 𝕜\nhx : ¬x ≡ a [PMOD p]\nx✝ : 𝕜\n⊢ ¬a ≡ x [PMOD p]"
] | ← not_modEq_iff_toIcoDiv_eq_toIocDiv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 84
} | {
"line": 248,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : AddCommGroup 𝕜\np : 𝕜\nn : ℕ\nh0 : n ≠ 0\nhn : IsSMulRegular 𝕜 n\ny : 𝕜\nhx✝ : ↑y ∈ {x | n • x = 0}\nhx : n • y ∈ zmultiples p\nm' : ℤ\nhm : (fun x ↦ x • p) m' = n • y\nthis : NeZero n\n⊢ ∃ k y_1, ↑y_1 = ↑⟨↑y, hx✝⟩ ∧ n • y_1 = ↑k • p",
"ppTerm": "?m.124",
"assigned": ... | [
"𝕜 : Type u_1\ninst✝ : AddCommGroup 𝕜\np : 𝕜\nn : ℕ\nh0 : n ≠ 0\nhn : IsSMulRegular 𝕜 n\ny : 𝕜\nhx✝ : ↑y ∈ {x | n • x = 0}\nhx : n • y ∈ zmultiples p\nm' : ℤ\nthis : NeZero n\nhm : (fun x ↦ x • p) (((Int.divModEquiv n) m').1 * ↑n + ↑↑((Int.divModEquiv n) m').2) = n • y\n⊢ ∃ k y_1, ↑y_1 = ↑⟨↑y, hx✝⟩ ∧ n • y_1 =... | rw [← (Int.divModEquiv n).symm_apply_apply m', Int.divModEquiv_symm_apply] at hm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ExactSequence | {
"line": 257,
"column": 2
} | {
"line": 273,
"column": 24
} | {
"line": 275,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS : ComposableArrows C (n + 2)\n⊢ S.Exact ↔ (mk₂ (S.map' 0 1 isComplex₂_iff._proof_2 ⋯) (S.map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact ∧ S.δ₀.Exact",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
... | [] | constructor
· intro h
constructor
· rw [exact₂_iff]; swap
· rw [isComplex₂_iff]
exact h.toIsComplex.zero 0
exact h.exact 0 (by lia)
· exact Exact.mk (IsComplex.mk (fun i hi => h.toIsComplex.zero (i + 1)))
(fun i hi => h.exact (i + 1))
· rintro ⟨h, h₀⟩
refine Exact.mk (IsC... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ExactSequence | {
"line": 257,
"column": 2
} | {
"line": 273,
"column": 24
} | {
"line": 275,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS : ComposableArrows C (n + 2)\n⊢ S.Exact ↔ (mk₂ (S.map' 0 1 isComplex₂_iff._proof_2 ⋯) (S.map' 1 2 isComplex₂_iff._proof_4 ⋯)).Exact ∧ S.δ₀.Exact",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
... | [] | constructor
· intro h
constructor
· rw [exact₂_iff]; swap
· rw [isComplex₂_iff]
exact h.toIsComplex.zero 0
exact h.exact 0 (by lia)
· exact Exact.mk (IsComplex.mk (fun i hi => h.toIsComplex.zero (i + 1)))
(fun i hi => h.exact (i + 1))
· rintro ⟨h, h₀⟩
refine Exact.mk (IsC... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Refinements | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 88
} | {
"line": 226,
"column": 4
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : ∀ ⦃A : C⦄ (y : A ⟶ S₂.homology), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ homologyMap φ\nA : C\ny₂ : A ⟶ S₂.X₂\nhy₂ : y₂ ≫ S₂.g = 0\nA₁ : C\nπ₁ : A₁ ⟶ A\nhπ₁ : Epi π₁\nγ : A₁ ⟶ S₁.homology\... | [
"case mp\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : ∀ ⦃A : C⦄ (y : A ⟶ S₂.homology), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ homologyMap φ\nA : C\ny₂ : A ⟶ S₂.X₂\nhy₂ : y₂ ≫ S₂.g = 0\nA₁ : C\nπ₁ : A₁ ⟶ A\nhπ₁ : Epi π₁\nγ : A₁ ⟶ S₁.homology\nhγ : π₁ ≫ S... | obtain ⟨A₂, π₂, hπ₂, x₂, hx₂, fac⟩ := S₁.eq_liftCycles_homologyπ_up_to_refinements γ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 314,
"column": 2
} | {
"line": 315,
"column": 16
} | {
"line": 317,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\ninst✝ : Reflective R\n⊢ IsIso (reflectorAdjunction R).toMonad.μ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.IsIso",... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 314,
"column": 2
} | {
"line": 315,
"column": 16
} | {
"line": 317,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\ninst✝ : Reflective R\n⊢ IsIso (reflectorAdjunction R).toMonad.μ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.IsIso",... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 320,
"column": 2
} | {
"line": 321,
"column": 16
} | {
"line": 323,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\ninst✝ : Coreflective R\n⊢ IsIso (coreflectorAdjunction R).toComonad.δ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.coreflector",
"CategoryTheo... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monad.Adjunction | {
"line": 320,
"column": 2
} | {
"line": 321,
"column": 16
} | {
"line": 323,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nL : C ⥤ D\nR : D ⥤ C\ninst✝ : Coreflective R\n⊢ IsIso (coreflectorAdjunction R).toComonad.δ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.coreflector",
"CategoryTheo... | [] | dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monad.Comonadicity | {
"line": 235,
"column": 65
} | {
"line": 240,
"column": 18
} | {
"line": 242,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₁, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝¹ : ComonadicLeftAdjoint F\nA B : C\nf g : A ⟶ B\ninst✝ : F.IsCosplitPair f g\n⊢ CreatesLimit (parallelPair f g) F",
"ppTerm": "?m.36",
"assigned": true,
"usedConsta... | [] | by
apply +allowSynthFailures comonadicCreatesLimitOfPreservesLimit
all_goals
apply @preservesLimit_of_iso_diagram _ _ _ _ _ _ _ _ _ (diagramIsoParallelPair.{v₁} _).symm ?_
dsimp
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 131,
"column": 94
} | {
"line": 132,
"column": 51
} | {
"line": 134,
"column": 0
} | [
{
"pp": "M : Type u_1\nR : Type u_2\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\n⊢ x ∈ a⁻¹ • S ↔ a • x ∈ S",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule.... | [] | by
rw [mem_pointwise_smul_iff_inv_smul_mem, inv_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Over | {
"line": 186,
"column": 10
} | {
"line": 186,
"column": 57
} | {
"line": 188,
"column": 0
} | [
{
"pp": "A : Type u_2\ninst✝⁶ : CommSemiring A\nB : Type u_3\ninst✝⁵ : CommSemiring B\nC : Type u_4\ninst✝⁴ : Semiring C\ninst✝³ : Algebra A B\ninst✝² : Algebra B C\ninst✝¹ : Algebra A C\ninst✝ : IsScalarTower A B C\n𝔓 : Ideal C\nP : Ideal B\np : Ideal A\nhp : 𝔓.LiesOver p\nhP : 𝔓.LiesOver P\n⊢ p = under A P... | [] | by rw [𝔓.over_def p, 𝔓.over_def P, under_under] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.IsPrimary | {
"line": 59,
"column": 10
} | {
"line": 59,
"column": 38
} | {
"line": 59,
"column": 39
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nhi : I.radical.IsMaximal\nx y : R\nhxy : x * y ∈ I\nh : I + span {y} = ⊤\n⊢ x ∈ I ∨ y ∈ I.radical",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSe... | [
"case pos\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nhi : I.radical.IsMaximal\nx y : R\nhxy : x * y ∈ I\nh : I + span {y} = ⊤\n⊢ span {x} ≤ I ∨ y ∈ I.radical"
] | ← span_singleton_le_iff_mem, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.IsPrimary | {
"line": 59,
"column": 66
} | {
"line": 59,
"column": 74
} | {
"line": 60,
"column": 8
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nhi : I.radical.IsMaximal\nx y : R\nhxy : x * y ∈ I\nh : I + span {y} = ⊤\n⊢ span {x} * (I + span {y}) ≤ I ∨ y ∈ I.radical",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
... | [
"case pos\nR : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nhi : I.radical.IsMaximal\nx y : R\nhxy : x * y ∈ I\nh : I + span {y} = ⊤\n⊢ span {x} * I + span {x} * span {y} ≤ I ∨ y ∈ I.radical"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 566,
"column": 36
} | {
"line": 566,
"column": 51
} | {
"line": 566,
"column": 52
} | [
{
"pp": "R✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y ⟶ X\ns : S\ny : ↑Y\nt : S\n⊢ ((CoextendScalars.equiv f X)\n ((CoextendSc... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 566,
"column": 36
} | {
"line": 566,
"column": 51
} | {
"line": 566,
"column": 52
} | [
{
"pp": "R✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y ⟶ X\ns : S\ny : ↑Y\nt : S\n⊢ ((CoextendScalars.equiv f X)\n ((CoextendSc... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 566,
"column": 36
} | {
"line": 566,
"column": 51
} | {
"line": 566,
"column": 52
} | [
{
"pp": "R✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nX : ModuleCat R\nY : ModuleCat S\ng : (restrictScalars f).obj Y ⟶ X\ns : S\ny : ↑Y\nt : S\n⊢ ((CoextendScalars.equiv f X)\n ((CoextendSc... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 606,
"column": 8
} | {
"line": 606,
"column": 23
} | {
"line": 606,
"column": 24
} | [
{
"pp": "R✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nY : ModuleCat S\ns : S\ny : ↑Y\nt : S\n⊢ ((CoextendScalars.equiv f ((restrictScalars f).obj Y))\n ((CoextendScalars.equiv f ((restrictSc... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 606,
"column": 8
} | {
"line": 606,
"column": 23
} | {
"line": 606,
"column": 24
} | [
{
"pp": "R✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nY : ModuleCat S\ns : S\ny : ↑Y\nt : S\n⊢ ((CoextendScalars.equiv f ((restrictScalars f).obj Y))\n ((CoextendScalars.equiv f ((restrictSc... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 606,
"column": 8
} | {
"line": 606,
"column": 23
} | {
"line": 606,
"column": 24
} | [
{
"pp": "R✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : Ring R\ninst✝ : Ring S\nf : R →+* S\nY : ModuleCat S\ns : S\ny : ↑Y\nt : S\n⊢ ((CoextendScalars.equiv f ((restrictScalars f).obj Y))\n ((CoextendScalars.equiv f ((restrictSc... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 743,
"column": 19
} | {
"line": 743,
"column": 34
} | {
"line": 744,
"column": 4
} | [
{
"pp": "case refine_3.tmul\nR✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X ⟶ (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Mo... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 743,
"column": 19
} | {
"line": 743,
"column": 34
} | {
"line": 744,
"column": 4
} | [
{
"pp": "case refine_3.tmul\nR✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X ⟶ (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Mo... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 743,
"column": 19
} | {
"line": 743,
"column": 34
} | {
"line": 744,
"column": 4
} | [
{
"pp": "case refine_3.tmul\nR✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X ⟶ (restrictScalars f).obj Y\nm1 : Module R S := Module.compHom S f\nm2 : Mo... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Nilpotent.Lemmas | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 36
} | {
"line": 128,
"column": 0
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : End R M\np q : Submodule R M\nhp : MapsTo ⇑f ↑p ↑p\nhq : MapsTo ⇑f ↑q ↑q\nh : p ≤ q\nn : ℕ\nx : M\nhx : x ∈ p\nhn : ⟨(f ^ n) ↑⟨x, ⋯⟩, ⋯⟩ = 0\n⊢ ↑⟨(f ^ n) ↑⟨x, hx⟩, ⋯⟩ = ↑0",
"ppTerm": "?h",
... | [] | exact (congr_arg Subtype.val hn :) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 834,
"column": 21
} | {
"line": 834,
"column": 36
} | {
"line": 835,
"column": 6
} | [
{
"pp": "case tmul\nR✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nY : ModuleCat S\ns : S\nm1 : Module R S := Module.compHom S f\nm2 : Module R ↑Y := Module.compHom (↑Y) f\ns' : ↑((restrictSc... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 834,
"column": 21
} | {
"line": 834,
"column": 36
} | {
"line": 835,
"column": 6
} | [
{
"pp": "case tmul\nR✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nY : ModuleCat S\ns : S\nm1 : Module R S := Module.compHom S f\nm2 : Module R ↑Y := Module.compHom (↑Y) f\ns' : ↑((restrictSc... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 834,
"column": 21
} | {
"line": 834,
"column": 36
} | {
"line": 835,
"column": 6
} | [
{
"pp": "case tmul\nR✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nY : ModuleCat S\ns : S\nm1 : Module R S := Module.compHom S f\nm2 : Module R ↑Y := Module.compHom (↑Y) f\ns' : ↑((restrictSc... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PolynomialAlgebra | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 15
} | {
"line": 89,
"column": 16
} | [
{
"pp": "R : Type u_1\nA : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\na₁ a₂ : A\np₁ p₂ : R[X]\nk : ℕ\n⊢ (if ¬(p₁ * p₂).coeff k = 0 then a₁ * a₂ * (algebraMap R A) ((p₁ * p₂).coeff k) else 0) =\n ∑ x ∈ Finset.antidiagonal k,\n if ¬p₂.coeff x.2 = 0 then\n (if ¬p₁.c... | [
"R : Type u_1\nA : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\na₁ a₂ : A\np₁ p₂ : R[X]\nk : ℕ\n⊢ (if ¬(p₁ * p₂).coeff k = 0 then a₁ * a₂ * (algebraMap R A) ((p₁ * p₂).coeff k) else 0) =\n ∑ x ∈ Finset.antidiagonal k,\n if ¬p₂.coeff x.2 = 0 then\n (if ¬p₁.coeff x.1 = 0... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.LocalProperties.Exactness | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 40
} | {
"line": 82,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\nL : Type u_4\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : Module R N\ninst✝¹⁰ : AddCommMonoid L\ninst✝⁹ : Module R L\nMₚ : (P : Ideal R) → [P.IsMaximal] → Type u_6\ninst✝⁸ : (P : Ideal R) → [in... | [
"R : Type u_1\nM : Type u_2\nN : Type u_3\nL : Type u_4\ninst✝¹⁵ : CommSemiring R\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Module R M\ninst✝¹² : AddCommMonoid N\ninst✝¹¹ : Module R N\ninst✝¹⁰ : AddCommMonoid L\ninst✝⁹ : Module R L\nMₚ : (P : Ideal R) → [P.IsMaximal] → Type u_6\ninst✝⁸ : (P : Ideal R) → [inst : P.IsMax... | apply eq_of_localization₀_maximal Nₚ g | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.EssentialFiniteness | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 26
} | {
"line": 77,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Finset S\n⊢ IsLocalization (Submonoid.comap (algebraMap (↥(adjoin R ↑σ)) S) (IsUnit.submonoid S)) S ↔\n ∀ (s : S), ∃ t ∈ adjoin R ↑σ, IsUnit t ∧ s * t ∈ adjoin R ↑σ",
"ppTerm": "?m.46",
"assigned":... | [
"case mp\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Finset S\nhσ : IsLocalization (Submonoid.comap (algebraMap (↥(adjoin R ↑σ)) S) (IsUnit.submonoid S)) S\n⊢ ∀ (s : S), ∃ t ∈ adjoin R ↑σ, IsUnit t ∧ s * t ∈ adjoin R ↑σ",
"case mpr\nR : Type u_1\nS : Type u_2\ni... | constructor <;> intro hσ | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
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