module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Topology.Connected.Basic | {
"line": 570,
"column": 45
} | {
"line": 570,
"column": 70
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\nx : α\nF : Set α\nhx : x ∈ F\ny : α\nhy : y ∈ F\nh2y : ⟨y, hy⟩ ∈ connectedComponent ⟨x, hx⟩\n⊢ Subtype.val '' connectedComponent ⟨x, hx⟩ = Subtype.val '' connectedComponent ⟨y, hy⟩",
"usedConstants": [
"congrArg",
"Membership.mem",
"Subtype"... | connectedComponent_eq h2y | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Connected.Clopen | {
"line": 80,
"column": 4
} | {
"line": 82,
"column": 50
} | [
{
"pp": "case refine_2\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set (α ⊕ β)\n⊢ ((∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t) → IsConnected s",
"usedConstants": [
"Continuous.continuousOn",
"continuous_inl",
"IsConnect... | rintro (⟨t, ht, rfl⟩ | ⟨t, ht, rfl⟩)
· exact ht.image _ continuous_inl.continuousOn
· exact ht.image _ continuous_inr.continuousOn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Connected.Clopen | {
"line": 80,
"column": 4
} | {
"line": 82,
"column": 50
} | [
{
"pp": "case refine_2\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set (α ⊕ β)\n⊢ ((∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t) → IsConnected s",
"usedConstants": [
"Continuous.continuousOn",
"continuous_inl",
"IsConnect... | rintro (⟨t, ht, rfl⟩ | ⟨t, ht, rfl⟩)
· exact ht.image _ continuous_inl.continuousOn
· exact ht.image _ continuous_inr.continuousOn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.Clopen | {
"line": 93,
"column": 4
} | {
"line": 95,
"column": 50
} | [
{
"pp": "case refine_2\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set (α ⊕ β)\n⊢ ((∃ t, IsPreconnected t ∧ s = inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = inr '' t) → IsPreconnected s",
"usedConstants": [
"Continuous.continuousOn",
"continuous_inl",
"... | rintro (⟨t, ht, rfl⟩ | ⟨t, ht, rfl⟩)
· exact ht.image _ continuous_inl.continuousOn
· exact ht.image _ continuous_inr.continuousOn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Connected.Clopen | {
"line": 93,
"column": 4
} | {
"line": 95,
"column": 50
} | [
{
"pp": "case refine_2\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set (α ⊕ β)\n⊢ ((∃ t, IsPreconnected t ∧ s = inl '' t) ∨ ∃ t, IsPreconnected t ∧ s = inr '' t) → IsPreconnected s",
"usedConstants": [
"Continuous.continuousOn",
"continuous_inl",
"... | rintro (⟨t, ht, rfl⟩ | ⟨t, ht, rfl⟩)
· exact ht.image _ continuous_inl.continuousOn
· exact ht.image _ continuous_inr.continuousOn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 18
} | [
{
"pp": "case h\nX : Type u\nι : Type u_1\ninst✝ : TopologicalSpace X\ns : Set ι\nf : ι → Set X\nhs : s.Countable\nhf : ∀ i ∈ s, IsLindelof (f i)\ni : Type u\nU : i → Set X\nhU : ∀ (i : i), IsOpen[inst✝] (U i)\nhUcover : ⋃ i ∈ s, f i ⊆ ⋃ i, U i\nhiU : ∀ i_1 ∈ s, f i_1 ⊆ ⋃ i, U i\nr : ι → Set i\nhr : ∀ i_1 ∈ s, ... | use ⋃ i ∈ s, r i | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Separation.Regular | {
"line": 260,
"column": 85
} | {
"line": 262,
"column": 43
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : RegularSpace X\ns t : Set X\nhs : IsCompact s\nht : IsClosed[inst✝¹] t\nhst : Disjoint s t\n⊢ SeparatedNhds s t",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Topology.Separation.Regular.0.SeparatedNhds.of_isCompact_isClosed._simp_1_2... | by
simpa only [separatedNhds_iff_disjoint, hs.disjoint_nhdsSet_left, disjoint_nhds_nhdsSet,
ht.closure_eq, disjoint_left] using hst | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactness.Lindelof | {
"line": 440,
"column": 2
} | {
"line": 440,
"column": 29
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ny : Y\nhf : Tendsto f (coLindelof X) (𝓝 y)\nhfc : Continuous[inst✝², inst✝¹] f\nl : Filter Y\nhne : l.NeBot\ninst✝ : CountableInterFilter l\nhle : l ≤ 𝓟 (insert y (range f))\n⊢ ∃ x ∈ insert y (range f), Clust... | by_cases hy : ClusterPt y l | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Topology.Compactness.Lindelof | {
"line": 597,
"column": 50
} | {
"line": 597,
"column": 80
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : LindelofSpace X\nf : X → Y\nhf : Continuous[inst✝², inst✝¹] f\n⊢ IsLindelof (f '' univ)",
"usedConstants": [
"Set.univ",
"IsLindelof.image",
"isLindelof_univ"
]
}
] | exact isLindelof_univ.image hf | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Constructions | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 20
} | [
{
"pp": "case refine_1\nM : Type u_1\nN : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace N\ninst✝ : Monoid N\nf : M →* N\nhf_inj : Function.Injective ⇑f\nhf : IsOpenMap ⇑f\nU : Set (M × Mᵐᵒᵖ)\nhU : IsOpen[inferInstance] U\nhg_openMap : IsOpenMap (Prod.map (⇑f) (⇑opHomeomorph... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.Constructions | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 20
} | [
{
"pp": "case refine_1\nM : Type u_1\nN : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace N\ninst✝ : Monoid N\nf : M →* N\nhf_inj : Function.Injective ⇑f\nhf : IsOpenMap ⇑f\nU : Set (M × Mᵐᵒᵖ)\nhU : IsOpen[inferInstance] U\nhg_openMap : IsOpenMap (Prod.map (⇑f) (⇑opHomeomorph... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Constructions | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 20
} | [
{
"pp": "case refine_1\nM : Type u_1\nN : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace N\ninst✝ : Monoid N\nf : M →* N\nhf_inj : Function.Injective ⇑f\nhf : IsOpenMap ⇑f\nU : Set (M × Mᵐᵒᵖ)\nhU : IsOpen[inferInstance] U\nhg_openMap : IsOpenMap (Prod.map (⇑f) (⇑opHomeomorph... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Constructions | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 20
} | [
{
"pp": "case refine_2\nM : Type u_1\nN : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace N\ninst✝ : Monoid N\nf : M →* N\nhf_inj : Function.Injective ⇑f\nhf : IsOpenMap ⇑f\nU : Set (M × Mᵐᵒᵖ)\nhU : IsOpen[inferInstance] U\nhg_openMap : IsOpenMap (Prod.map (⇑f) (⇑opHomeomorph... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.Constructions | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 20
} | [
{
"pp": "case refine_2\nM : Type u_1\nN : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace N\ninst✝ : Monoid N\nf : M →* N\nhf_inj : Function.Injective ⇑f\nhf : IsOpenMap ⇑f\nU : Set (M × Mᵐᵒᵖ)\nhU : IsOpen[inferInstance] U\nhg_openMap : IsOpenMap (Prod.map (⇑f) (⇑opHomeomorph... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Constructions | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 20
} | [
{
"pp": "case refine_2\nM : Type u_1\nN : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace N\ninst✝ : Monoid N\nf : M →* N\nhf_inj : Function.Injective ⇑f\nhf : IsOpenMap ⇑f\nU : Set (M × Mᵐᵒᵖ)\nhU : IsOpen[inferInstance] U\nhg_openMap : IsOpenMap (Prod.map (⇑f) (⇑opHomeomorph... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Constructions | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 20
} | [
{
"pp": "case refine_3\nM : Type u_1\nN : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace N\ninst✝ : Monoid N\nf : M →* N\nhf_inj : Function.Injective ⇑f\nhf : IsOpenMap ⇑f\nU : Set (M × Mᵐᵒᵖ)\nhU : IsOpen[inferInstance] U\nhg_openMap : IsOpenMap (Prod.map (⇑f) (⇑opHomeomorph... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.Constructions | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 20
} | [
{
"pp": "case refine_3\nM : Type u_1\nN : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace N\ninst✝ : Monoid N\nf : M →* N\nhf_inj : Function.Injective ⇑f\nhf : IsOpenMap ⇑f\nU : Set (M × Mᵐᵒᵖ)\nhU : IsOpen[inferInstance] U\nhg_openMap : IsOpenMap (Prod.map (⇑f) (⇑opHomeomorph... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Constructions | {
"line": 179,
"column": 12
} | {
"line": 179,
"column": 20
} | [
{
"pp": "case refine_3\nM : Type u_1\nN : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace N\ninst✝ : Monoid N\nf : M →* N\nhf_inj : Function.Injective ⇑f\nhf : IsOpenMap ⇑f\nU : Set (M × Mᵐᵒᵖ)\nhU : IsOpen[inferInstance] U\nhg_openMap : IsOpenMap (Prod.map (⇑f) (⇑opHomeomorph... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.Clopen | {
"line": 710,
"column": 69
} | {
"line": 710,
"column": 77
} | [
{
"pp": "case inl\nα : Type u\ninst✝ : TopologicalSpace α\nh✝ : ∀ (s : Set α), IsClopen s → s = ∅ ∨ s = univ\nf : α → Bool\nhf : Continuous[inst✝, _] f\nx y : α\nthis : f ⁻¹' {false} = (f ⁻¹' {true})ᶜ\nh : f ⁻¹' {true} = ∅\n⊢ f x = f y",
"usedConstants": [
"congrArg",
"Compl.compl",
"Set.u... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Connected.Clopen | {
"line": 710,
"column": 69
} | {
"line": 710,
"column": 77
} | [
{
"pp": "case inr\nα : Type u\ninst✝ : TopologicalSpace α\nh✝ : ∀ (s : Set α), IsClopen s → s = ∅ ∨ s = univ\nf : α → Bool\nhf : Continuous[inst✝, _] f\nx y : α\nthis : f ⁻¹' {false} = (f ⁻¹' {true})ᶜ\nh : f ⁻¹' {true} = univ\n⊢ f x = f y",
"usedConstants": [
"congrArg",
"Set.univ",
"Membe... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.MulAction | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 12
} | [
{
"pp": "case neg\nM : Type u_1\nX : Type u_2\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : TopologicalSpace X\ninst✝³ : Group M\ninst✝² : IsTopologicalGroup M\ninst✝¹ : MulAction M X\ninst✝ : DiscreteTopology X\nh : ∀ (x : X), IsOpen[inst✝⁵] ↑(MulAction.stabilizer M x)\ny x : X\nU : Set M := {m' | m' • y = x}\nhU' : ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.ConstMulAction | {
"line": 596,
"column": 2
} | {
"line": 596,
"column": 45
} | [
{
"pp": "case mk.mk\nM : Type u_1\nα : Type u_2\nβ : Type u_3\nΓ : Type u_4\ninst✝⁶ : Group Γ\nT : Type u_5\ninst✝⁵ : TopologicalSpace T\ninst✝⁴ : MulAction Γ T\ninst✝³ : T2Space T\ninst✝² : LocallyCompactSpace T\ninst✝¹ : ContinuousConstSMul Γ T\ninst✝ : ProperlyDiscontinuousSMul Γ T\nthis : Setoid T := MulAct... | let U₀₀ := ⋂ γ ∈ bad_Γ_set, (γ • ·) ⁻¹' u γ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : UniformSpace α\ninst✝² : Group α\ninst✝¹ : IsUniformGroup α\ninst✝ : UniformSpace β\nf : β → α\nhf : UniformContinuous f\nthis : UniformContinuous fun x ↦ 1 / f x\n⊢ UniformContinuous fun x ↦ (f x)⁻¹",
"usedConstants": [
"UniformContinuous",
"MulOne.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : UniformSpace α\ninst✝² : Group α\ninst✝¹ : IsUniformGroup α\ninst✝ : UniformSpace β\nf g : β → α\nhf : UniformContinuous f\nhg : UniformContinuous g\nthis : UniformContinuous fun x ↦ f x / (g x)⁻¹\n⊢ UniformContinuous fun x ↦ f x * g x",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 369,
"column": 23
} | {
"line": 369,
"column": 36
} | [
{
"pp": "G : Type w\nα : Type u\ninst✝⁵ : TopologicalSpace G\ninst✝⁴ : Group G\ninst✝³ : IsTopologicalGroup G\ninst✝² : TopologicalSpace α\ninst✝¹ : Zero α\ninst✝ : T1Space α\nf : G → α\nk : Set G\nhk : IsCompact k\nhf : support f ⊆ k\nh'f : Continuous[inst✝⁵, inst✝²] f\nh : ¬∀ (x : G), f x = 0 x\n⊢ LocallyComp... | Pi.zero_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.UniformSpace.DiscreteUniformity | {
"line": 40,
"column": 59
} | {
"line": 40,
"column": 74
} | [
{
"pp": "X : Type u_2\ninst✝ : UniformSpace X\n⊢ (∀ (s : Set (X × X)), s ∈ uniformity X ↔ s ∈ uniformity X) ↔ uniformity X = 𝓟 SetRel.id",
"usedConstants": [
"Filter.instMembership",
"UniformSpace",
"Eq.mpr",
"SetRel.id",
"congrArg",
"uniformity",
"Membership.mem",... | Filter.ext_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 862,
"column": 2
} | {
"line": 862,
"column": 35
} | [
{
"pp": "G : Type w\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : IsTopologicalGroup G\nH : Type u_1\ninst✝⁴ : Group H\ninst✝³ : TopologicalSpace H\ninst✝² : IsTopologicalGroup H\nF : Type u_2\ninst✝¹ : FunLike F G H\ninst✝ : MonoidHomClass F G H\nf : F\n⊢ IsInducing ⇑f ↔ 𝓝 1 = comap (⇑f) (𝓝 1)",
... | rw [Topology.isInducing_iff_nhds] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 911,
"column": 8
} | {
"line": 911,
"column": 16
} | [
{
"pp": "case h.mp\nA : Type u_1\ninst✝⁶ : Group A\ninst✝⁵ : TopologicalSpace A\ninst✝⁴ : ContinuousMul A\nB : Type u_2\ninst✝³ : Group B\ninst✝² : TopologicalSpace B\nF : Type u_3\ninst✝¹ : FunLike F A B\ninst✝ : MonoidHomClass F A B\nφ : F\nhφ : IsQuotientMap ⇑φ\nU : Set A\nhU : IsOpen[inst✝⁵] U\nx y : A\nhyU... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.UniformSpace.Separation | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 66
} | [
{
"pp": "α : Type u\ninst✝ : UniformSpace α\nx y : α\nh : ClusterPt (x, y) (𝓤 α)\n⊢ ∀ (i : SetRel α α), (i ∈ 𝓤 α ∧ ∀ (a : α × α), ClusterPt a (𝓟 i) → a ∈ i) → (x, y) ∈ id i",
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"ClusterPt.mono",
"Filter.le_principal_iff",
... | exact fun U ⟨hU, hUc⟩ ↦ hUc _ <| h.mono <| le_principal_iff.2 hU | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.Separation | {
"line": 258,
"column": 4
} | {
"line": 258,
"column": 48
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nt : Set (SeparationQuotient α × SeparationQuotient α)\nht : t ∈ map (Prod.map mk mk) (𝓤 α)\nU : Set (α × α)\nhU : U ∈ 𝓤 α\nhUo : IsOpen[instTopologicalSpaceProd] U\nhUt : SetRel.comp U U ⊆ Pr... | have : y' ⤳ y := (mk_eq_mk.1 hy).specializes | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.UniformSpace.Separation | {
"line": 300,
"column": 42
} | {
"line": 300,
"column": 52
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : T0Space β\nf : α → β\nh : UniformContinuous f\na : α\n⊢ (if hc : UniformContinuous f then lift f ⋯ else fun x ↦ f ⋯.some) (mk a) = f a",
"usedConstants": [
"UniformContinuous",
"Eq.mpr",
"congrArg",
... | dif_pos h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 43,
"column": 67
} | {
"line": 43,
"column": 83
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\n⊢ comap (fun x ↦ (f x.1, f x.2)) (𝓤 β) = 𝓤 α ↔\n 𝓤 α ≤ comap (fun x ↦ (f x.1, f x.2)) (𝓤 β) ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α",
"usedConstants": [
"Eq.mpr",
"congrArg",
"uniformity",
... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 108,
"column": 3
} | {
"line": 108,
"column": 87
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\nα' : Type u_1\nβ' : Type u_2\ninst✝¹ : UniformSpace α'\ninst✝ : UniformSpace β'\ne₁ : α → α'\ne₂ : β → β'\nh₁ : IsUniformInducing e₁\nh₂ : IsUniformInducing e₂\n⊢ comap (fun x ↦ ((e₁ x.1.1, e₂ x.1.2), e₁ x.2.1, e₂ x.2.2)) (𝓤 (α'... | by simp [Function.comp_def, uniformity_prod, ← h₁.1, ← h₂.1, comap_inf, comap_comap] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 380,
"column": 11
} | {
"line": 380,
"column": 73
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nF : Filter α\nhf : IsUniformInducing f\nhs : (map f F).TotallyBounded\n⊢ F.TotallyBounded",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"uniformity",
"Set.Finite",
"Membership... | (hf.basis_uniformity (basis_sets _)).filter_totallyBounded_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1310,
"column": 2
} | {
"line": 1310,
"column": 21
} | [
{
"pp": "G : Type w\nH : Type x\ninst✝⁴ : Group G\ninst✝³ : Monoid H\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalSpace H\ninst✝ : ContinuousInv G\nf : G →* Hˣ\nhf : Continuous[inst✝², inst✝¹] ⇑((Units.coeHom H).comp f)\n⊢ Continuous[inst✝², instTopologicalSpaceMulOpposite] fun x ↦ op ↑(f x)⁻¹",
"usedC... | simp_rw [← map_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 81
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\nf : Filter α\nx : α\nadhs : ∀ s ∈ 𝓤 α, ∃ t ∈ f, t ×ˢ t ⊆ s ∧ ∃ y, (x, y) ∈ s ∧ y ∈ t\ns : Set α\nhs : s ∈ 𝓝 x\nU : Set (α × α)\nU_mem : U ∈ 𝓤 α\nhU : U ○ U ⊆ {p | p.1 = x → p.2 ∈ s}\nt : Set α\nt_mem : t ∈ f\nht : t ×ˢ t ⊆ U\ny : α\nhxy : (x, y) ∈ U\nhy : y... | exact fun z hz => hU (SetRel.prodMk_mem_comp hxy (ht <| mk_mem_prod hy hz)) rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 928,
"column": 4
} | {
"line": 928,
"column": 38
} | [
{
"pp": "case refine_1\nα : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝¹ : (𝓤 α).IsCountablyGenerated\ninst✝ : SeparableSpace α\ns : Set α\nhsc : s.Countable\nhsd : Dense s\nt : ℕ → SetRel α α\nh_basis : (𝓤 α).HasAntitoneBasis t\nht_mem : ∀ (i : ℕ), t i ∈ (𝓤 α).sets\nhto : ∀ (i : ℕ), IsOpen[inst... | simp only [mem_iUnion₂, mem_range] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 971,
"column": 4
} | {
"line": 971,
"column": 41
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\ninst✝ : (𝓤 α).IsCountablyGenerated\nhs : ∀ U ∈ 𝓤 α, ∃ t, t.Countable ∧ ⋃ x ∈ t, ball x U = univ\n⊢ ∀ U ∈ 𝓤 α, ∃ t, t.Countable ∧ univ ⊆ ⋃ x ∈ t, ball x U",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Set.un... | simpa only [univ_subset_iff] using hs | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Category.TopCat.Limits.Basic | {
"line": 260,
"column": 58
} | {
"line": 262,
"column": 46
} | [
{
"pp": "J : Type v\ninst✝ : Category.{w, v} J\nF : J ⥤ TopCat\nc : Cocone F\nhc : IsColimit c\nX : Set ↑c.pt\n⊢ IsClosed X ↔ ∀ (j : J), IsClosed (⇑(ConcreteCategory.hom (c.ι.app j)) ⁻¹' X)",
"usedConstants": [
"CategoryTheory.Functor",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
... | by
simp only [← isOpen_compl_iff, isOpen_iff_of_isColimit _ hc,
Functor.const_obj_obj, Set.preimage_compl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 855,
"column": 2
} | {
"line": 858,
"column": 80
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh : S₁.LeftHomologyData\ninst✝² : Epi φ.τ₁\ninst✝¹ : IsIso φ.τ₂\ninst✝ : Mono φ.τ₃\ni : h.K ⟶ S₂.X₂ := h.i ≫ φ.τ₂\nwi : i ≫ S₂.g = 0\nhi : IsLimit (KernelFork.ofι i wi)\nf' : (KernelFork... | have hf' : φ.τ₁ ≫ f' = h.f' := by
have eq := @Fork.IsLimit.lift_ι _ _ _ _ _ _ _ ((KernelFork.ofι S₂.f S₂.zero)) hi
simp only [Fork.ι_ofι] at eq
rw [← cancel_mono h.i, ← cancel_mono φ.τ₂, assoc, assoc, eq, f'_i, φ.comm₁₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 841,
"column": 2
} | {
"line": 841,
"column": 50
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.RightHomologyData\ninst✝ : S.HasRightHomology\n⊢ h.p ≫ h.opcyclesIso.inv = S.pOpcycles",
"usedConstants": [
"CategoryTheory.ShortComplex.opcycles",
"CategoryTheory.CategoryStruct.toQuive... | dsimp [pOpcycles, RightHomologyData.opcyclesIso] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 705,
"column": 2
} | {
"line": 705,
"column": 54
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.HomologyData\ninst✝ : S.HasHomology\n⊢ h.left.homologyIso = h.right.homologyIso ≪≫ h.iso.symm",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ShortComplex.HomologyData.iso",
"CategoryTheo... | rw [right_homologyIso_eq_left_homologyIso_trans_iso] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 924,
"column": 65
} | {
"line": 927,
"column": 78
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\n⊢ S₁.homologyπ ≫ homologyMap φ = cyclesMap φ ≫ S₂.homologyπ",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"C... | by
simp only [← cancel_mono S₂.leftHomologyIso.inv, assoc, ← leftHomologyIso_inv_naturality φ,
homologyπ_comp_leftHomologyIso_inv]
simp only [homologyπ, assoc, Iso.hom_inv_id_assoc, leftHomologyπ_naturality] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 1326,
"column": 68
} | {
"line": 1326,
"column": 80
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasRightHomology\n⊢ S.g = S.g ≫ 𝟙 S.X₃",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryS... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 1326,
"column": 68
} | {
"line": 1326,
"column": 80
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasRightHomology\n⊢ S.g = S.g ≫ 𝟙 S.X₃",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryS... | rw [comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 1326,
"column": 68
} | {
"line": 1326,
"column": 80
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\ninst✝ : S.HasRightHomology\n⊢ S.g = S.g ≫ 𝟙 S.X₃",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryS... | rw [comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Limits | {
"line": 213,
"column": 8
} | {
"line": 213,
"column": 39
} | [
{
"pp": "case h₃\nJ : Type u_1\nC : Type u_2\ninst✝⁵ : Category.{v_1, u_1} J\ninst✝⁴ : Category.{v_2, u_2} C\ninst✝³ : HasZeroMorphisms C\nF : J ⥤ ShortComplex C\ninst✝² : HasColimit (F ⋙ π₁)\ninst✝¹ : HasColimit (F ⋙ π₂)\ninst✝ : HasColimit (F ⋙ π₃)\nx✝¹ x✝ : J\nf : x✝¹ ⟶ x✝\n⊢ (F.map f ≫\n { τ₁ := coli... | · simp [← colimit.w (F ⋙ π₃) f] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 443,
"column": 18
} | {
"line": 443,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ φ₁.τ₃ = h₁₂.h₃ + h₂₃.h₃ + (h₁₂.h₂ + h₂₃.h₂) ≫ S₂.g + φ₃.τ₃",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.t... | h₁₂.comm₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Preadditive | {
"line": 443,
"column": 29
} | {
"line": 443,
"column": 39
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nS₁ S₂ S₃ : ShortComplex C\nφ₁ φ₂ φ₃ φ₄ : S₁ ⟶ S₂\nh₁₂ : Homotopy φ₁ φ₂\nh₂₃ : Homotopy φ₂ φ₃\n⊢ h₁₂.h₃ + h₁₂.h₂ ≫ S₂.g + φ₂.τ₃ = h₁₂.h₃ + h₂₃.h₃ + (h₁₂.h₂ + h₂₃.h₂) ≫ S₂.g + φ₃.τ₃",
"usedConstants": [
"Eq.mpr",
"Catego... | h₂₃.comm₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.QuasiIso | {
"line": 184,
"column": 49
} | {
"line": 184,
"column": 80
} | [
{
"pp": "C : Type ?u.18729\ninst✝⁵ : Category.{v_1, ?u.18729} C\ninst✝⁴ : HasZeroMorphisms C\nS₁ S₂ S₃ S₄ : ShortComplex C\ninst✝³ : S₁.HasHomology\ninst✝² : S₂.HasHomology\ninst✝¹ : S₃.HasHomology\ninst✝ : S₄.HasHomology\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0",
"use... | rw [← φ.comm₁₂, hf₂, comp_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.QuasiIso | {
"line": 184,
"column": 49
} | {
"line": 184,
"column": 80
} | [
{
"pp": "C : Type ?u.18729\ninst✝⁵ : Category.{v_1, ?u.18729} C\ninst✝⁴ : HasZeroMorphisms C\nS₁ S₂ S₃ S₄ : ShortComplex C\ninst✝³ : S₁.HasHomology\ninst✝² : S₂.HasHomology\ninst✝¹ : S₃.HasHomology\ninst✝ : S₄.HasHomology\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0",
"use... | rw [← φ.comm₁₂, hf₂, comp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.QuasiIso | {
"line": 184,
"column": 49
} | {
"line": 184,
"column": 80
} | [
{
"pp": "C : Type ?u.18729\ninst✝⁵ : Category.{v_1, ?u.18729} C\ninst✝⁴ : HasZeroMorphisms C\nS₁ S₂ S₃ S₄ : ShortComplex C\ninst✝³ : S₁.HasHomology\ninst✝² : S₂.HasHomology\ninst✝¹ : S₃.HasHomology\ninst✝ : S₄.HasHomology\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0",
"use... | rw [← φ.comm₁₂, hf₂, comp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.QuasiIso | {
"line": 188,
"column": 37
} | {
"line": 188,
"column": 68
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruc... | rw [← φ.comm₁₂, hf₂, comp_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.QuasiIso | {
"line": 188,
"column": 37
} | {
"line": 188,
"column": 68
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruc... | rw [← φ.comm₁₂, hf₂, comp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.QuasiIso | {
"line": 188,
"column": 37
} | {
"line": 188,
"column": 68
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruc... | rw [← φ.comm₁₂, hf₂, comp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.QuasiIso | {
"line": 189,
"column": 37
} | {
"line": 189,
"column": 68
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruc... | rw [← φ.comm₁₂, hf₂, comp_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.QuasiIso | {
"line": 189,
"column": 37
} | {
"line": 189,
"column": 68
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruc... | rw [← φ.comm₁₂, hf₂, comp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.QuasiIso | {
"line": 189,
"column": 37
} | {
"line": 189,
"column": 68
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\ninst✝¹ : S₁.HasHomology\ninst✝ : S₂.HasHomology\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruc... | rw [← φ.comm₁₂, hf₂, comp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Abelian | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 41
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\nS : ShortComplex C\nkf : KernelFork S.g\ncc : CokernelCofork S.f\nhkf : IsLimit kf\nhcc : IsColimit cc\nH : C\nπ : kf.pt ⟶ H\nι : H ⟶ cc.pt\nfac : Fork.ι kf ≫ Cofork.π cc = π ≫ ι\ninst✝¹ : Epi π\ninst✝ : Mono ι\n⊢ (isoImage S hkf hcc fac).hom ... | apply image.isoStrongEpiMono_hom_comp_ι | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | {
"line": 655,
"column": 2
} | {
"line": 657,
"column": 54
} | [
{
"pp": "case w\nC : Type u_1\nD : Type u_2\ninst✝⁷ : Category.{v_1, u_1} C\ninst✝⁶ : Category.{v_2, u_2} D\ninst✝⁵ : HasZeroMorphisms C\ninst✝⁴ : HasZeroMorphisms D\nS : ShortComplex C\nF : C ⥤ D\ninst✝³ : F.PreservesZeroMorphisms\ninst✝² : S.HasHomology\ninst✝¹ : F.PreservesLeftHomologyOf S\ninst✝ : F.Preserv... | dsimp only [Iso.trans, Iso.symm, Iso.refl, Functor.mapIso, RightHomologyData.homologyIso,
rightHomologyIso, RightHomologyData.rightHomologyIso, LeftHomologyData.homologyIso,
leftHomologyIso, LeftHomologyData.leftHomologyIso] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 467,
"column": 7
} | {
"line": 467,
"column": 15
} | [
{
"pp": "case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasPullbacks C\nX Y : C\ng : X ⟶ Y\nf₂ : Subobject Y\nf₁ : MonoOver Y\n⊢ (pullback g).obj (Quotient.mk'' f₁ ⊓ f₂) = (pullback g).obj (Quotient.mk'' f₁) ⊓ (pullback g).obj f₂",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Over",
... | inf_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 467,
"column": 16
} | {
"line": 467,
"column": 24
} | [
{
"pp": "case h\nC : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\ninst✝ : HasPullbacks C\nX Y : C\ng : X ⟶ Y\nf₂ : Subobject Y\nf₁ : MonoOver Y\n⊢ (pullback g).obj ((inf.obj (Quotient.mk'' f₁)).obj f₂) = (pullback g).obj (Quotient.mk'' f₁) ⊓ (pullback g).obj f₂",
"usedConstants": [
"Eq.mpr",
"Category... | inf_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 478,
"column": 7
} | {
"line": 478,
"column": 15
} | [
{
"pp": "case h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : HasPullbacks C\nX Y : C\ng : Y ⟶ X\ninst✝ : Mono g\nf₂ : Subobject Y\nf₁ : MonoOver Y\n⊢ (map g).obj (Quotient.mk'' f₁ ⊓ f₂) = (map g).obj (Quotient.mk'' f₁) ⊓ (map g).obj f₂",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Over"... | inf_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Subobject.Lattice | {
"line": 478,
"column": 16
} | {
"line": 478,
"column": 24
} | [
{
"pp": "case h\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : HasPullbacks C\nX Y : C\ng : Y ⟶ X\ninst✝ : Mono g\nf₂ : Subobject Y\nf₁ : MonoOver Y\n⊢ (map g).obj ((inf.obj (Quotient.mk'' f₁)).obj f₂) = (map g).obj (Quotient.mk'' f₁) ⊓ (map g).obj f₂",
"usedConstants": [
"Eq.mpr",
"Catego... | inf_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.Single | {
"line": 136,
"column": 70
} | {
"line": 138,
"column": 30
} | [
{
"pp": "V : Type u\ninst✝³ : Category.{v, u} V\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasZeroObject V\nι : Type u_1\ninst✝ : DecidableEq ι\nc : ComplexShape ι\nj : ι\nA B : V\nf : (single V c j).obj A ⟶ (single V c j).obj B\n⊢ (single V c j).map ((singleObjXSelf c j A).inv ≫ f.f j ≫ (singleObjXSelf c j B).hom)... | by
ext
simp [single_map_f_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 35
} | [
{
"pp": "case neg\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j k : ι\nhij : ¬c.Rel i j\n⊢ C.d i j ≫ C.d j k = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | rw [C.shape i j hij, zero_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 35
} | [
{
"pp": "case neg\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j k : ι\nhij : ¬c.Rel i j\n⊢ C.d i j ≫ C.d j k = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | rw [C.shape i j hij, zero_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 35
} | [
{
"pp": "case neg\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j k : ι\nhij : ¬c.Rel i j\n⊢ C.d i j ≫ C.d j k = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | rw [C.shape i j hij, zero_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 790,
"column": 8
} | {
"line": 790,
"column": 34
} | [
{
"pp": "case e_a\nC : Type u\nA : Type u_1\ninst✝³ : Category.{v, u} C\nD : Type u_2\ninst✝² : Category.{v_1, u_2} D\ninst✝¹ : AddMonoid A\ninst✝ : HasShift D A\nF : C ⥤ D\nhF : F.FullyFaithful\ns : A → C ⥤ C\ni : (i : A) → s i ⋙ F ≅ F ⋙ shiftFunctor D i\nn : A\nX : C\nthis : (i (0 + n)).hom.app X = eqToHom ⋯ ... | erw [(i n).hom.naturality] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 297,
"column": 8
} | {
"line": 299,
"column": 49
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nS : ShortComplex C\ninst✝ : HasZeroObject C\nhg : S.g = 0\na✝ : Epi S.f\n⊢ S.Exact",
"usedConstants": [
"CategoryTheory.ShortComplex.HomologyData.exact_iff",
"Eq.mpr",
"CategoryTheory.Epi",
"Cate... | (HomologyData.ofIsColimitCokernelCofork S hg _
(CokernelCofork.IsColimit.ofEpiOfIsZero (CokernelCofork.ofπ (0 : S.X₂ ⟶ 0) comp_zero)
inferInstance (isZero_zero C))).exact_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 900,
"column": 37
} | {
"line": 900,
"column": 68
} | [
{
"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\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
... | rw [← φ.comm₁₂, hf₂, comp_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 900,
"column": 37
} | {
"line": 900,
"column": 68
} | [
{
"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\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
... | rw [← φ.comm₁₂, hf₂, comp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.Exact | {
"line": 900,
"column": 37
} | {
"line": 900,
"column": 68
} | [
{
"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\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nhg₁ : S₁.g = 0\nhf₂ : S₂.f = 0\nhg₂ : S₂.g = 0\n⊢ S₁.f ≫ φ.τ₂ = 0",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
... | rw [← φ.comm₁₂, hf₂, comp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.QuasiIso | {
"line": 312,
"column": 4
} | {
"line": 313,
"column": 39
} | [
{
"pp": "ι : Type u_1\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nc : ComplexShape ι\nK L M K' L' : HomologicalComplex C c\ninst✝ : CategoryWithHomology C\nX✝ Y✝ Z✝ : HomologicalComplex C c\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhg : quasiIso C c g\nhfg : quasiIso C c (f ≫ g)\n⊢ quasiIso C c f",
... | rw [mem_quasiIso_iff] at hg hfg ⊢
rwa [← quasiIso_iff_comp_right f g] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.QuasiIso | {
"line": 312,
"column": 4
} | {
"line": 313,
"column": 39
} | [
{
"pp": "ι : Type u_1\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasZeroMorphisms C\nc : ComplexShape ι\nK L M K' L' : HomologicalComplex C c\ninst✝ : CategoryWithHomology C\nX✝ Y✝ Z✝ : HomologicalComplex C c\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\nhg : quasiIso C c g\nhfg : quasiIso C c (f ≫ g)\n⊢ quasiIso C c f",
... | rw [mem_quasiIso_iff] at hg hfg ⊢
rwa [← quasiIso_iff_comp_right f g] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex | {
"line": 673,
"column": 29
} | {
"line": 674,
"column": 37
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nι : Type u_2\nc : ComplexShape ι\nK : HomologicalComplex C c\ni : ι\ninst✝ : K.HasHomology i\nh : K.ExactAt i\n⊢ IsZero (K.homology i)",
"usedConstants": [
"HomologicalComplex.exactAt_iff_isZero_homology",
"Eq.mp... | by
rwa [← exactAt_iff_isZero_homology] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.Homotopy | {
"line": 305,
"column": 2
} | {
"line": 310,
"column": 19
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nW : Type u_2\ninst✝² : Category.{v_1, u_2} W\ninst✝¹ : Preadditive W\nG : V ⥤ W\ninst✝ : G.Additive\nhom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)\n⊢ (G.mapHomologicalComplex ... | rw [nullHomotopicMap', map_nullHomotopicMap]
congr
ext i j
split_ifs
· rfl
· rw [G.map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Homotopy | {
"line": 305,
"column": 2
} | {
"line": 310,
"column": 19
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝⁴ : Category.{v, u} V\ninst✝³ : Preadditive V\nc : ComplexShape ι\nC D : HomologicalComplex V c\nW : Type u_2\ninst✝² : Category.{v_1, u_2} W\ninst✝¹ : Preadditive W\nG : V ⥤ W\ninst✝ : G.Additive\nhom : (i j : ι) → c.Rel j i → (C.X i ⟶ D.X j)\n⊢ (G.mapHomologicalComplex ... | rw [nullHomotopicMap', map_nullHomotopicMap]
congr
ext i j
split_ifs
· rfl
· rw [G.map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.IsBounded | {
"line": 168,
"column": 2
} | {
"line": 174,
"column": 52
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Preorder β\ninst✝¹ : NoMaxOrder β\nf : α → β\nl : Filter α\ninst✝ : l.NeBot\nhf : Tendsto f l atTop\n⊢ ¬IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l f",
"usedConstants": [
"False",
"Preorder.toLT",
"congrArg",
"Filter.map",
"Set.Nonempty.... | rintro ⟨b, hb⟩
rw [eventually_map] at hb
obtain ⟨b', h⟩ := exists_gt b
have hb' := (tendsto_atTop.mp hf) b'
have : { x : α | f x ≤ b } ∩ { x : α | b' ≤ f x } = ∅ :=
eq_empty_of_subset_empty fun x hx => (not_le_of_gt h) (le_trans hx.2 hx.1)
exact (nonempty_of_mem (hb.and hb')).ne_empty this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.IsBounded | {
"line": 168,
"column": 2
} | {
"line": 174,
"column": 52
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : Preorder β\ninst✝¹ : NoMaxOrder β\nf : α → β\nl : Filter α\ninst✝ : l.NeBot\nhf : Tendsto f l atTop\n⊢ ¬IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l f",
"usedConstants": [
"False",
"Preorder.toLT",
"congrArg",
"Filter.map",
"Set.Nonempty.... | rintro ⟨b, hb⟩
rw [eventually_map] at hb
obtain ⟨b', h⟩ := exists_gt b
have hb' := (tendsto_atTop.mp hf) b'
have : { x : α | f x ≤ b } ∩ { x : α | b' ≤ f x } = ∅ :=
eq_empty_of_subset_empty fun x hx => (not_le_of_gt h) (le_trans hx.2 hx.1)
exact (nonempty_of_mem (hb.and hb')).ne_empty this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.IsBounded | {
"line": 599,
"column": 2
} | {
"line": 599,
"column": 27
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : LinearOrder β\nf : Filter α\nF : ι → α → β\ns : Finset ι\nhs : s.Nonempty\ni : ι\ni_s : i ∈ s\nb : β\nhb : ∀ (a : β), (∀ᶠ (x : β) in Filter.map (F i) f, (fun x1 x2 ↦ x1 ≤ x2) x a) → (fun x1 x2 ↦ x1 ≤ x2) b a\n⊢ ∀ (a : β),\n (∀ᶠ (x : β) in Fil... | refine fun c hc ↦ hb c ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Filter.IsBounded | {
"line": 602,
"column": 2
} | {
"line": 602,
"column": 32
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : LinearOrder β\nf : Filter α\nF : ι → α → β\ns : Finset ι\nhs : s.Nonempty\ni : ι\ni_s : i ∈ s\nb : β\nhb : ∀ (a : β), (∀ᶠ (x : β) in Filter.map (F i) f, (fun x1 x2 ↦ x1 ≤ x2) x a) → (fun x1 x2 ↦ x1 ≤ x2) b a\nc : β\nhc : ∀ᶠ (a : α) in f, (fun x1... | simp only [sup'_le_iff] at h ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.InfiniteSum.SummationFilter | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 69
} | [
{
"pp": "β : Type u_2\nL : SummationFilter β\nhL : ¬L.NeBot\n⊢ L.support = univ",
"usedConstants": [
"SummationFilter.neBot_or_eq_bot",
"congrArg",
"Finset",
"Set.univ",
"Filter.Eventually",
"setOf",
"Membership.mem",
"Or.resolve_left",
"Bot.bot",
... | simp [SummationFilter.support, L.neBot_or_eq_bot.resolve_left hL] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 212,
"column": 59
} | {
"line": 218,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nL : SummationFilter β\nf : β → α\na : α\ninst✝ : L.HasSupport\ng : γ → β\nhg : Injective g\nhf : ∀ x ∈ L.support, x ∉ Set.range g → f x = 1\n⊢ HasProd (f ∘ g) a (L.comap { toFun := g, inj' := hg }) ↔ HasProd f... | by
simp only [HasProd, SummationFilter.comap_filter, tendsto_map'_iff, comp_apply,
Embedding.coeFn_mk, Function.comp_def]
refine tendsto_congr' ?_
filter_upwards [L.eventually_le_support] with s hs
rw [s.prod_preimage]
exact fun x h h' ↦ hf x (hs h) h' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.LiminfLimsup | {
"line": 510,
"column": 2
} | {
"line": 510,
"column": 16
} | [
{
"pp": "α : Type u_6\nβ : Type u_7\ninst✝ : CompleteLattice β\nf : Filter α\nu : α → β\nx : β\nh : ∃ᶠ (a : α) in f, u a ≤ x\n⊢ liminf u f ≤ x",
"usedConstants": [
"Filter.liminf_eq",
"Eq.mpr",
"Filter.liminf",
"congrArg",
"Filter.Eventually",
"PartialOrder.toPreorder",
... | rw [liminf_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 316,
"column": 40
} | {
"line": 316,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nL : SummationFilter β\nf : β → α\nha : Multipliable f L\nh : L.HasSupport ∧ ((mulSupport fun b ↦ f b) ∩ L.support).Finite\n⊢ ∀ x ∈ ⋯.toFinset, L.support.mulIndicator (fun b ↦ f b) x = f x",
"usedConstants": [
"MulO... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 316,
"column": 40
} | {
"line": 316,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nL : SummationFilter β\nf : β → α\nha : Multipliable f L\nh : L.HasSupport ∧ ((mulSupport fun b ↦ f b) ∩ L.support).Finite\n⊢ ∀ x ∈ ⋯.toFinset, L.support.mulIndicator (fun b ↦ f b) x = f x",
"usedConstants": [
"MulO... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 316,
"column": 40
} | {
"line": 316,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nL : SummationFilter β\nf : β → α\nha : Multipliable f L\nh : L.HasSupport ∧ ((mulSupport fun b ↦ f b) ∩ L.support).Finite\n⊢ ∀ x ∈ ⋯.toFinset, L.support.mulIndicator (fun b ↦ f b) x = f x",
"usedConstants": [
"MulO... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.Homotopy | {
"line": 652,
"column": 8
} | {
"line": 652,
"column": 20
} | [
{
"pp": "case e_a.e_a.succ\nι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : CochainComplex V ℕ\ne : P ⟶ Q\nzero : P.X 1 ⟶ Q.X 0\ncomm_zero : e.f 0 = P.d 0 1 ≫ zero\none : P.X 2 ⟶ Q.X 1\ncomm_o... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Action.Basic | {
"line": 426,
"column": 4
} | {
"line": 426,
"column": 18
} | [
{
"pp": "V : Type u_1\ninst✝² : Category.{v_1, u_1} V\nW : Type u_2\ninst✝¹ : Category.{v_2, u_2} W\nF : V ⥤ W\nh : F.FullyFaithful\nG : Type u_3\ninst✝ : Monoid G\nX✝ Y✝ : Action V G\nf : (F.mapAction G).obj X✝ ⟶ (F.mapAction G).obj Y✝\nx✝ : G\n⊢ F.map (X✝.ρ x✝) ≫ f.hom = f.hom ≫ F.map (Y✝.ρ x✝)",
"usedCon... | exact f.comm _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.LiminfLimsup | {
"line": 1050,
"column": 44
} | {
"line": 1050,
"column": 67
} | [
{
"pp": "α : Type u_1\nι : Type u_4\nι' : Type u_5\ninst✝² : ConditionallyCompleteLinearOrder α\nv : Filter ι\np : ι' → Prop\ns : ι' → Set ι\ninst✝¹ : Countable (Subtype p)\ninst✝ : Nonempty (Subtype p)\nhv : v.HasBasis p s\nf : ι → α\nH : ¬∃ j, s ↑j = ∅\nH' : ∀ (j : Subtype p), ¬BddBelow (range fun i ↦ f ↑i)\n... | nonempty_iInter_Iic_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Homology.Homotopy | {
"line": 819,
"column": 19
} | {
"line": 819,
"column": 93
} | [
{
"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.245299\nc : ComplexShape ι\ninst✝² : DecidableR... | rw [← homologyMap_comp, h.homotopyHomInvId.homologyMap_eq, homologyMap_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.Homotopy | {
"line": 819,
"column": 19
} | {
"line": 819,
"column": 93
} | [
{
"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.245299\nc : ComplexShape ι\ninst✝² : DecidableR... | rw [← homologyMap_comp, h.homotopyHomInvId.homologyMap_eq, homologyMap_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.Homotopy | {
"line": 819,
"column": 19
} | {
"line": 819,
"column": 93
} | [
{
"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.245299\nc : ComplexShape ι\ninst✝² : DecidableR... | rw [← homologyMap_comp, h.homotopyHomInvId.homologyMap_eq, homologyMap_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Coherent | {
"line": 85,
"column": 38
} | {
"line": 85,
"column": 46
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nS : Set (Set X)\ninst✝ : SequentialSpace X\nh : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S\nt : Set X\nht : ∀ s ∈ S, IsClosed[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' t)\nu : ℕ → X\nx : X\nhut : ∀ (n : ℕ), u n ∈ t\nhux : Tendst... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Coherent | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 10
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nS : Set (Set X)\ninst✝ : SequentialSpace X\nh : ∀ ⦃u : ℕ → X⦄ ⦃x : X⦄, Tendsto u atTop (𝓝 x) → insert x (range u) ∈ S\nt : Set X\nht : ∀ s ∈ S, IsClosed[instTopologicalSpaceSubtype] (Subtype.val ⁻¹' t)\nu : ℕ → X\nx : X\nhut : ∀ (n : ℕ), u n ∈ t\nhux : Tendst... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 25
} | [
{
"pp": "case neg\nι : Type u_4\nα : Type u_5\nα' : Type u_6\nG : Type u_7\ninst✝⁶ : CommMonoid α\ninst✝⁵ : CommMonoid α'\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace α'\ninst✝² : T2Space α'\nf : ι → α\nL : SummationFilter ι\ng : G\ninst✝¹ : FunLike G α α'\ninst✝ : MonoidHomClass G α α'\nhge : IsClos... | obtain ⟨b, hb⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 303,
"column": 81
} | {
"line": 305,
"column": 5
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\np : Filter ι\np' : Filter α\nc : β\n⊢ Tendsto (↿F) (p ×ˢ p') (𝓝 c) ↔ TendstoUniformlyOnFilter F (fun x ↦ c) p p'",
"usedConstants": [
"Eq.mpr",
"SProd.sprod",
"congrArg",
"Iff.rfl",
"unif... | by
simp_rw [nhds_eq_comap_uniformity, tendsto_comap_iff]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Adjunction.Parametrized | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 63
} | [
{
"pp": "case w.h\nC₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\ninst✝² : Category.{v₁, u₁} C₁\ninst✝¹ : Category.{v₂, u₂} C₂\ninst✝ : Category.{v₃, u₃} C₃\nF : C₁ ⥤ C₂ ⥤ C₃\nG : C₁ᵒᵖ ⥤ C₃ ⥤ C₂\nadj : (X₁ : C₁) → F.obj X₁ ⊣ G.obj (op X₁)\nh :\n ∀ {X₁ Y₁ : C₁} (f : X₁ ⟶ Y₁) {X₂ : C₂} {X₃ : C₃} (g : (F.obj Y₁).obj X... | simpa [Adjunction.homEquiv_unit] using h f (X₂ := X₂) (𝟙 _) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Category.ModuleCat.Monoidal.Closed | {
"line": 34,
"column": 16
} | {
"line": 37,
"column": 17
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nM N P : ModuleCat R\nf : (MonoidalCategory.tensorLeft M).obj N ⟶ P\n⊢ (fun f ↦ (β_ M N).hom ≫ ↟(TensorProduct.lift (Hom.hom₂ f)))\n ((fun f ↦ ofHom₂ ((TensorProduct.mk R ↑N ↑M).compr₂ (Hom.hom ((β_ N M).hom ≫ f)))) f) =\n f",
"usedConstants": [
"NonAsso... | by
ext : 1
apply TensorProduct.ext'
solve_by_elim | [anonymous] | Lean.Parser.Term.byTactic |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.