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.UniformSpace.Basic | {
"line": 875,
"column": 2
} | {
"line": 875,
"column": 97
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\nf : α → β → γ\nua1 ua2 : UniformSpace α\nub1 ub2 : UniformSpace β\nuc1 : UniformSpace γ\nh : UniformContinuous fun p ↦ f p.1 p.2\n⊢ UniformContinuous fun p ↦ f p.1 p.2",
"usedConstants": [
"UniformContinuous",
"UniformSpace",
"id",
"... | have ha := @UniformContinuous.inf_dom_right _ _ id ua1 ua2 ua2 (@uniformContinuous_id _ (id _)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.Filter.Ultrafilter.Defs | {
"line": 117,
"column": 52
} | {
"line": 117,
"column": 88
} | [
{
"pp": "α : Type u\nf : Ultrafilter α\ns : Set α\n⊢ sᶜ ∈ f ↔ s ∉ f",
"usedConstants": [
"Eq.mpr",
"compl_compl",
"congrArg",
"Compl.compl",
"Iff.rfl",
"Membership.mem",
"BooleanAlgebra.toCompl",
"id",
"Set.instCompl",
"Iff",
"Ultrafilter.com... | rw [← compl_notMem_iff, compl_compl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.Ultrafilter.Defs | {
"line": 117,
"column": 52
} | {
"line": 117,
"column": 88
} | [
{
"pp": "α : Type u\nf : Ultrafilter α\ns : Set α\n⊢ sᶜ ∈ f ↔ s ∉ f",
"usedConstants": [
"Eq.mpr",
"compl_compl",
"congrArg",
"Compl.compl",
"Iff.rfl",
"Membership.mem",
"BooleanAlgebra.toCompl",
"id",
"Set.instCompl",
"Iff",
"Ultrafilter.com... | rw [← compl_notMem_iff, compl_compl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Ultrafilter.Defs | {
"line": 117,
"column": 52
} | {
"line": 117,
"column": 88
} | [
{
"pp": "α : Type u\nf : Ultrafilter α\ns : Set α\n⊢ sᶜ ∈ f ↔ s ∉ f",
"usedConstants": [
"Eq.mpr",
"compl_compl",
"congrArg",
"Compl.compl",
"Iff.rfl",
"Membership.mem",
"BooleanAlgebra.toCompl",
"id",
"Set.instCompl",
"Iff",
"Ultrafilter.com... | rw [← compl_notMem_iff, compl_compl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.AtTopBot.Prod | {
"line": 84,
"column": 42
} | {
"line": 86,
"column": 33
} | [
{
"pp": "α : Type u_3\nγ : Type u_5\ninst✝¹ : Preorder α\ninst✝ : Preorder γ\nf g : α → γ\nhf : Tendsto f atTop atTop\nhg : Tendsto g atTop atTop\n⊢ Tendsto (Prod.map f g) atTop atTop",
"usedConstants": [
"Eq.mpr",
"Filter.Tendsto.prod_map_prod_atTop",
"SProd.sprod",
"congrArg",
... | by
rw [← prod_atTop_atTop_eq]
exact hf.prod_map_prod_atTop hg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 719,
"column": 2
} | {
"line": 719,
"column": 48
} | [
{
"pp": "K : Type u_2\ninst✝⁴ : DivisionSemiring K\ninst✝³ : PartialOrder K\ninst✝² : PosMulReflectLT K\ninst✝¹ : IsOrderedCancelAddMonoid K\ninst✝ : ExistsAddOfLE K\na : K\nh : 0 < a\nb c d : K\n⊢ (fun x ↦ x + b) '' (fun x ↦ a * x) '' Ico c d = Ico (a * c + b) (a * d + b)",
"usedConstants": [
"Eq.mpr... | rw [image_mul_left_Ico h, image_add_const_Ico] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.Pointwise | {
"line": 592,
"column": 49
} | {
"line": 592,
"column": 64
} | [
{
"pp": "α : Type u_2\ninst✝ : Monoid α\nf : Filter α\nhf : 1 ≤ f\ns t : Set α\nht : t ∈ f\nhs : univ ⊆ s\n⊢ s = univ",
"usedConstants": [
"Set.univ_subset_iff",
"congrArg",
"Set.univ",
"Eq.mp",
"HasSubset.Subset",
"propext",
"Eq",
"Set.instHasSubset",
"... | univ_subset_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.Pointwise | {
"line": 599,
"column": 49
} | {
"line": 599,
"column": 64
} | [
{
"pp": "α : Type u_2\ninst✝ : Monoid α\nf : Filter α\nhf : 1 ≤ f\ns t : Set α\nht : t ∈ f\nhs : univ ⊆ s\n⊢ s = univ",
"usedConstants": [
"Set.univ_subset_iff",
"congrArg",
"Set.univ",
"Eq.mp",
"HasSubset.Subset",
"propext",
"Eq",
"Set.instHasSubset",
"... | univ_subset_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Compactness.LocallyCompact | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 44
} | [
{
"pp": "case refine_2\nX✝ : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝⁴ : TopologicalSpace X✝\ninst✝³ : TopologicalSpace Y\ns✝ t✝ : Set X✝\nX : ι → Type u_4\ninst✝² : (i : ι) → TopologicalSpace (X i)\ninst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)\ninst✝ : Finite ι\nt : (i : ι) → X i\nn : Set ((i : ι) → X i)\n... | · exact fun i _ => hsub' i (h i trivial) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Compactness.LocallyCompact | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 42
} | [
{
"pp": "case refine_3\nX✝ : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝⁴ : TopologicalSpace X✝\ninst✝³ : TopologicalSpace Y\ns✝ t✝ : Set X✝\nX : ι → Type u_4\ninst✝² : (i : ι) → TopologicalSpace (X i)\ninst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)\ninst✝ : ∀ (i : ι), CompactSpace (X i)\nt : (i : ι) → X i\nn : ... | refine isCompact_univ_pi fun i => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Bases | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 69
} | [
{
"pp": "case refine_4\nα : Type u\ns : Set (Set α)\nthis : TopologicalSpace α := generateFrom s\nt : Set α\nht : t ∈ s\n⊢ ⋂₀ {t} ∈ (fun f ↦ ⋂₀ f) '' {f | f.Finite ∧ f ⊆ s}",
"usedConstants": [
"Iff.mpr",
"Set.singleton_subset_iff",
"setOf",
"Set.Finite",
"Membership.mem",
... | exact ⟨{t}, ⟨finite_singleton t, singleton_subset_iff.2 ht⟩, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Bases | {
"line": 215,
"column": 30
} | {
"line": 215,
"column": 55
} | [
{
"pp": "α : Type u\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns t : Set α\nhB : IsTopologicalBasis B\nhs : IsOpen[t✝] s\nh : ∀ U ∈ B, U ⊆ s → U ⊆ t\n⊢ ⋃₀ {s_1 | s_1 ∈ B ∧ s_1 ⊆ s} ⊆ t",
"usedConstants": [
"Eq.mpr",
"setOf",
"Set.sUnion",
"Membership.mem",
"id",
"HasSubse... | simpa [sUnion_subset_iff] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 189,
"column": 2
} | {
"line": 192,
"column": 20
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ns t : Set X\ninst✝¹ : LocallyCompactSpace X\ninst✝ : SecondCountableTopology X\n⊢ SigmaCompactSpace X",
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace",
"Filter.in... | choose K hKc hxK using fun x : X => exists_compact_mem_nhds x
rcases countable_cover_nhds hxK with ⟨s, hsc, hsU⟩
refine SigmaCompactSpace.of_countable _ (hsc.image K) (forall_mem_image.2 fun x _ => hKc x) ?_
rwa [sUnion_image] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 189,
"column": 2
} | {
"line": 192,
"column": 20
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ns t : Set X\ninst✝¹ : LocallyCompactSpace X\ninst✝ : SecondCountableTopology X\n⊢ SigmaCompactSpace X",
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace",
"Filter.in... | choose K hKc hxK using fun x : X => exists_compact_mem_nhds x
rcases countable_cover_nhds hxK with ⟨s, hsc, hsU⟩
refine SigmaCompactSpace.of_countable _ (hsc.image K) (forall_mem_image.2 fun x _ => hKc x) ?_
rwa [sUnion_image] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.CountableInter | {
"line": 239,
"column": 2
} | {
"line": 240,
"column": 35
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝² : CountableInterFilter l\nl₁ l₂ : Filter α\ninst✝¹ : CountableInterFilter l₁\ninst✝ : CountableInterFilter l₂\n⊢ CountableInterFilter (l₁ ⊔ l₂)",
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Membership.mem",... | refine ⟨fun S hSc hS => ⟨?_, ?_⟩⟩ <;> refine (countable_sInter_mem hSc).2 fun s hs => ?_
exacts [(hS s hs).1, (hS s hs).2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.CountableInter | {
"line": 239,
"column": 2
} | {
"line": 240,
"column": 35
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝² : CountableInterFilter l\nl₁ l₂ : Filter α\ninst✝¹ : CountableInterFilter l₁\ninst✝ : CountableInterFilter l₂\n⊢ CountableInterFilter (l₁ ⊔ l₂)",
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Membership.mem",... | refine ⟨fun S hSc hS => ⟨?_, ?_⟩⟩ <;> refine (countable_sInter_mem hSc).2 fun s hs => ?_
exacts [(hS s hs).1, (hS s hs).2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.GDelta.Basic | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 41
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsNowhereDense s\n⊢ ∃ S, (∀ t ∈ S, IsNowhereDense t) ∧ S.Countable ∧ s ⊆ ⋃₀ S",
"usedConstants": [
"subset_refl._simp_1",
"Eq.mpr",
"congrArg",
"Set.countable_singleton._simp_1",
"Set.sUnion",
"Membership.m... | exact ⟨{s}, by simpa, by simp, by simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Compactness.Compact | {
"line": 878,
"column": 2
} | {
"line": 878,
"column": 59
} | [
{
"pp": "X : Type u\nY : Type v\nι : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t : Set X\nf : X → Y\ninst✝ : NoncompactSpace X\n⊢ (cocompact X).NeBot",
"usedConstants": [
"Iff.mpr",
"Compl.compl",
"Filter.HasBasis.neBot_iff",
"Filter.NeBot",
"Set.ins... | refine Filter.hasBasis_cocompact.neBot_iff.2 fun hs => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Baire.Lemmas | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 77
} | [
{
"pp": "X : Type u_1\nα : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : BaireSpace X\nS : Set α\nf : α → Set X\nho : ∀ s ∈ S, IsOpen[inst✝¹] (f s)\nhS : S.Countable\nhd : ∀ s ∈ S, Dense (f s)\n⊢ Dense (⋂₀ (f '' S))",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"Dense... | refine dense_sInter_of_isOpen ?_ (hS.image _) ?_ <;> rwa [forall_mem_image] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Irreducible | {
"line": 215,
"column": 23
} | {
"line": 215,
"column": 30
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set X\nH : IsPreirreducible s\nf : X → Y\nhf : ContinuousOn f s\nu v : Set Y\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nx : X\nhx : x ∈ s\nhxu : x ∈ f ⁻¹' u\ny : X\nhy : y ∈ s\nhyv : y ∈ f ⁻¹' v\nu' : Set X\nhu' ... | ← u'_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Irreducible | {
"line": 291,
"column": 6
} | {
"line": 291,
"column": 43
} | [
{
"pp": "case refine_1.insert\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsIrreducible s\nu : Set X\nU : Finset (Set X)\na✝ : u ∉ U\nIH : (∀ u ∈ U, IsOpen[inst✝] u) → (∀ u ∈ U, (s ∩ u).Nonempty) → (s ∩ ⋂₀ ↑U).Nonempty\nhu : ∀ u_1 ∈ insert u U, IsOpen[inst✝] u_1\nhU : ∀ u_1 ∈ insert u U, (s ∩ u_1)... | rw [Finset.coe_insert, sInter_insert] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Separation.Basic | {
"line": 368,
"column": 2
} | {
"line": 371,
"column": 28
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\nx y : X\n⊢ 𝓝[≠] x ≤ 𝓝[{y}ᶜ] x",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"nhdsWithin",
"PartialOrder.toPreorder",
"Preorder.toLE",
"nhds",
"Set.instSingletonSet",
"id... | rcases eq_or_ne x y with rfl | hy
· exact Eq.le rfl
· rw [Ne.nhdsWithin_compl_singleton hy]
exact nhdsWithin_le_nhds | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.Basic | {
"line": 368,
"column": 2
} | {
"line": 371,
"column": 28
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\nx y : X\n⊢ 𝓝[≠] x ≤ 𝓝[{y}ᶜ] x",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"nhdsWithin",
"PartialOrder.toPreorder",
"Preorder.toLE",
"nhds",
"Set.instSingletonSet",
"id... | rcases eq_or_ne x y with rfl | hy
· exact Eq.le rfl
· rw [Ne.nhdsWithin_compl_singleton hy]
exact nhdsWithin_le_nhds | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.DiscreteSubset | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 42
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ codiscreteWithin s ≤ codiscrete X ⊓ 𝓟 s",
"usedConstants": [
"Filter.instMembership",
"congrArg",
"and_self",
"Filter.instCompleteLatticeFilter",
"Set.univ",
"PartialOrder.toPreorder",
"Preorder.toLE",... | simp [codiscrete, codiscreteWithin_mono] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.DiscreteSubset | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 42
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ codiscreteWithin s ≤ codiscrete X ⊓ 𝓟 s",
"usedConstants": [
"Filter.instMembership",
"congrArg",
"and_self",
"Filter.instCompleteLatticeFilter",
"Set.univ",
"PartialOrder.toPreorder",
"Preorder.toLE",... | simp [codiscrete, codiscreteWithin_mono] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.DiscreteSubset | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 42
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ codiscreteWithin s ≤ codiscrete X ⊓ 𝓟 s",
"usedConstants": [
"Filter.instMembership",
"congrArg",
"and_self",
"Filter.instCompleteLatticeFilter",
"Set.univ",
"PartialOrder.toPreorder",
"Preorder.toLE",... | simp [codiscrete, codiscreteWithin_mono] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.Basic | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 22
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\nx : α\nc : Set (Set α)\nH1 : ∀ s ∈ c, x ∈ s\nH2 : ∀ s ∈ c, IsPreconnected s\n⊢ ∀ y ∈ ⋃₀ c, ∃ t ⊆ ⋃₀ c, x ∈ t ∧ y ∈ t ∧ IsPreconnected t",
"usedConstants": []
}
] | rintro y ⟨s, sc, ys⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.Connected.Basic | {
"line": 541,
"column": 2
} | {
"line": 541,
"column": 45
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nx : α\nF : Set α\nhs : IsPreconnected s\nhxs : x ∈ s\nhsF : s ⊆ F\nthis : IsPreconnected (Subtype.val ⁻¹' s)\nh2xs : ⟨x, ⋯⟩ ∈ Subtype.val ⁻¹' s\n⊢ s ⊆ connectedComponentIn F x",
"usedConstants": [
"Membership.mem",
"Subtype",
"con... | have := this.subset_connectedComponent h2xs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.DenseEmbedding | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ninst✝ : TopologicalSpace δ\nf : γ → α\ng : γ → δ\nh : δ → β\nd : δ\na : α\ndi : IsDenseInducing i\nH : Tendsto h (𝓝 d) (𝓝 (i a))\ncomm : h ∘ g = i ∘ f\nlim1 : map f (comap g (�... | exact le_trans lim1 lim2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Connected.Clopen | {
"line": 52,
"column": 2
} | {
"line": 53,
"column": 52
} | [
{
"pp": "case refine_2\nι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : Set ((i : ι) × X i)\n⊢ (∃ i t, IsConnected t ∧ s = mk i '' t) → IsConnected s",
"usedConstants": [
"Continuous.continuousOn",
"IsConnected",
"IsConnected.image",
"continuous_sigmaMk"... | · rintro ⟨i, t, ht, rfl⟩
exact ht.image _ continuous_sigmaMk.continuousOn | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Compactness.Lindelof | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 66
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ns t : Set X\nhs : IsLindelof s\nht : IsClosed[inst✝¹] t\nf : Filter X\nhnf : f.NeBot\ninst✝ : CountableInterFilter f\nhstf : f ≤ 𝓟 s ∧ f ≤ 𝓟 t\nx : X\nhsx : x ∈ s\nhx : ClusterPt x f\n⊢ ∃ x ∈ s ∩ t, ClusterPt x f",
"usedConstants": [
"ClusterPt.mono"... | have hxt : x ∈ t := ht.mem_of_nhdsWithin_neBot <| hx.mono hstf.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Connected.Clopen | {
"line": 69,
"column": 2
} | {
"line": 79,
"column": 50
} | [
{
"pp": "case refine_1\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set (α ⊕ β)\nhs : IsConnected s\n⊢ (∃ t, IsConnected t ∧ s = inl '' t) ∨ ∃ t, IsConnected t ∧ s = inr '' t",
"usedConstants": [
"Sum.inr_injective",
"IsConnected",
"Sum.inl_injectiv... | · obtain ⟨x | x, hx⟩ := hs.nonempty
· have h : s ⊆ range Sum.inl :=
hs.isPreconnected.subset_isClopen isClopen_range_inl ⟨.inl x, hx, x, rfl⟩
refine Or.inl ⟨Sum.inl ⁻¹' s, ?_, ?_⟩
· exact hs.preimage_of_isOpenMap Sum.inl_injective isOpenMap_inl h
· exact (image_preimage_eq_of_subset h).sym... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Separation.Regular | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 33
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (Set X)\nh : inst✝ = generateFrom s\nh' : ∀ t ∈ s, ∀ a ∈ t, Disjoint (𝓝ˢ tᶜ) (𝓝 a)\na : X\nt : Set X\nht : IsOpen[generateFrom s] t\nha : a ∉ tᶜ\n⊢ Disjoint (𝓝ˢ tᶜ) (𝓝 a)",
"usedConstants": [
"congrArg",
"Compl.compl",
"Members... | rw [Set.notMem_compl_iff] at ha | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 762,
"column": 4
} | {
"line": 762,
"column": 47
} | [
{
"pp": "case inr\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : HereditarilyLindelofSpace X\nι : Type u_2\ninst✝ : Nonempty ι\nU : ι → Set X\nh : ∀ (i : ι), IsOpen[inst✝²] (U i)\nt : Set ι\nhtc : t.Countable\nhtu : ⋃ i ∈ t, U i = ⋃ i, U i\nt_ne : t.Nonempty\n⊢ ∃ k, ⋃ n, U (k n) = ⋃ i, U i",
"usedConsta... | obtain ⟨k, rfl⟩ := htc.exists_eq_range t_ne | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Separation.Regular | {
"line": 488,
"column": 2
} | {
"line": 488,
"column": 62
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\nu s : Set X\nh : s ∈ 𝓝ˢ u\nhu : IsClosed[inst✝¹] u\n⊢ ∃ t ∈ 𝓝ˢ u, IsClosed[inst✝¹] t ∧ t ⊆ s",
"usedConstants": [
"Filter.instMembership",
"Membership.mem",
"Exists",
"mem_nhdsSet_iff_exists",
"HasSubs... | obtain ⟨o, ho_open, huo, hos⟩ := mem_nhdsSet_iff_exists.mp h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Connected.Clopen | {
"line": 457,
"column": 4
} | {
"line": 457,
"column": 27
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : IsCoinducing f\nt : Set β\nht : IsClosed[inst✝] t\nht'✝ : IsConnected t\nhf : Surjective f\nhT : IsClosed[inst✝¹] (f ⁻¹' t)\nu v : Set α\nhu : IsClose... | rw [mem_union t' T₁ T₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Support | {
"line": 188,
"column": 60
} | {
"line": 188,
"column": 87
} | [
{
"pp": "α : Type u_2\nβ : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : One β\nf : α → β\nx : α\n⊢ (∃ x_1 ∈ 𝓝 x, EqOn f 1 x_1) ↔ f =ᶠ[𝓝 x] 1",
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Membership.mem",
"Exists",
"nhds",
"Filter.EventuallyEq",
"Set... | eventuallyEq_iff_exists_mem | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.Support | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 55
} | [
{
"pp": "α : Type u_2\nα' : Type u_3\nβ : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace α'\ninst✝ : One β\nf : α → β\nhf : HasCompactMulSupport f\ng : α' → α\nhg : IsClosedEmbedding g\n⊢ closure[inst✝¹] (g ⁻¹' mulSupport f) ⊆ g ⁻¹' mulTSupport f",
"usedConstants": [
"Eq.mpr",
... | rw [hg.isEmbedding.closure_eq_preimage_closure_image] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Support | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 27
} | [
{
"pp": "α' : Type u_3\nβ : Type u_4\ninst✝³ : TopologicalSpace α'\ninst✝² : One β\ninst✝¹ : T2Space α'\ninst✝ : TopologicalSpace β\nU : Set α'\nhU : IsOpen[inst✝³] U\nf : ↑U → β\ncont : Continuous[instTopologicalSpaceSubtype, inst✝] f\nsupp : HasCompactMulSupport f\nx : α'\nh : x ∈ mulTSupport (extend Subtype.... | exact cont.continuousAt | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.ConstMulAction | {
"line": 535,
"column": 2
} | {
"line": 536,
"column": 86
} | [
{
"pp": "Γ : Type u_4\nT : Type u_5\ninst✝⁵ : TopologicalSpace T\ninst✝⁴ : SMul Γ T\ninst✝³ : ProperlyDiscontinuousSMul Γ T\ninst✝² : T2Space T\ninst✝¹ : LocallyCompactSpace T\ninst✝ : ContinuousConstSMul Γ T\nx : T\nV : Set T\nV_cpt : IsCompact V\nV_nhd : V ∈ 𝓝 x\nΓ₀ : Set Γ := {γ | ((fun x ↦ γ • x) '' V ∩ V)... | refine ⟨V ∩ ⋂ γ : Γ₀, (γ.1 • ·) ⁻¹' u γ ∩ v γ, inter_mem V_nhd (iInter_mem.mpr fun γ ↦
inter_mem ((continuous_const_smul _).continuousAt <| hu γ) (hv γ)), fun γ hγ ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Maps.Proper.Basic | {
"line": 154,
"column": 51
} | {
"line": 159,
"column": 15
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\nf : X → Y\ng : Y → Z\ninst✝ : T2Space Y\nhf : Continuous[inst✝³, inst✝²] f\nhg : Continuous[inst✝², inst✝¹] g\nhgf : IsProperMap (g ∘ f)\n⊢ IsProperMap f",
"usedConstants... | by
rw [isProperMap_iff_ultrafilter_of_t2]
refine ⟨hf, fun 𝒰 y h ↦ ?_⟩
rw [isProperMap_iff_ultrafilter] at hgf
rcases hgf.2 ((hg.tendsto y).comp h) with ⟨x, -, hx⟩
exact ⟨x, hx⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 279,
"column": 6
} | {
"line": 279,
"column": 25
} | [
{
"pp": "G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\na : G\ns : Set G\nhs : s ∈ 𝓝 a\nthis : Tendsto (fun p ↦ p.1 * p.2) (𝓝 (a, 1)) (𝓝 a)\nU : Set G\nhU : U ∈ 𝓝 a\nV : Set G\nhV : V ∈ 𝓝 1\nhUV : U ×ˢ V ⊆ (fun p ↦ p.1 * p.2) ⁻¹'... | ← image_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 449,
"column": 8
} | {
"line": 449,
"column": 27
} | [
{
"pp": "case refine_4\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : MulZeroClass M\ninst✝ : ContinuousMul M\nK U : Set M\nhK : IsCompact K\nhU : U ∈ 𝓝 0\nx : M\nhx : x ∈ K\nt : Set M\nht : t ∈ 𝓝 x\ns : Set M\nhs : s ∈ 𝓝 0\nh : t ×ˢ s ⊆ (fun p ↦ p.1 * p.2) ⁻¹' U\n⊢ ∃ t ∈ 𝓝[K] x, ∃ V ∈ 𝓝 0, t * V ⊆ U... | ← image_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Group.Quotient | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 52
} | [
{
"pp": "G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\nN : Subgroup G\ns : Set G\n⊢ Dense (mk '' s) ↔ Dense (s * ↑N)",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"QuotientGroup.mk",
"Quot... | rw [← dense_preimage_mk, preimage_image_mk_eq_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Group.Quotient | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 52
} | [
{
"pp": "G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\nN : Subgroup G\ns : Set G\n⊢ Dense (mk '' s) ↔ Dense (s * ↑N)",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"QuotientGroup.mk",
"Quot... | rw [← dense_preimage_mk, preimage_image_mk_eq_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Group.Quotient | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 52
} | [
{
"pp": "G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\nN : Subgroup G\ns : Set G\n⊢ Dense (mk '' s) ↔ Dense (s * ↑N)",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"QuotientGroup.mk",
"Quot... | rw [← dense_preimage_mk, preimage_image_mk_eq_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Group.Quotient | {
"line": 76,
"column": 29
} | {
"line": 78,
"column": 55
} | [
{
"pp": "G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\nN : Subgroup G\nγ : G\n⊢ Continuous[instTopologicalSpace N, instTopologicalSpace N] fun x ↦ γ • x",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Continuous",
"ContinuousConstSMul.co... | by
rw [← isOpenQuotientMap_mk.continuous_comp_iff]
exact continuous_mk.comp <| continuous_const_smul γ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Monoid | {
"line": 971,
"column": 69
} | {
"line": 975,
"column": 19
} | [
{
"pp": "ι : Type u_1\nX : Type u_5\ninst✝¹ : TopologicalSpace X\nM : Type u_6\ninst✝ : One M\nf : ι → X → M\nhf : LocallyFinite fun i ↦ mulSupport (f i)\nx₀ : X\n⊢ ∃ I, ∀ᶠ (x : X) in 𝓝 x₀, (mulSupport fun i ↦ f i x) ⊆ ↑I",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",... | by
rcases hf x₀ with ⟨U, hxU, hUf⟩
refine ⟨hUf.toFinset, mem_of_superset hxU fun y hy i hi => ?_⟩
rw [hUf.coe_toFinset]
exact ⟨y, hi, hy⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 167,
"column": 2
} | {
"line": 168,
"column": 69
} | [
{
"pp": "G : Type w\nα : Type u\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\nb c : G\nf : α → G\nl : Filter α\n⊢ Tendsto (fun x ↦ b * f x) l (𝓝 (b * c)) ↔ Tendsto f l (𝓝 c)",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivI... | refine ⟨?_, Tendsto.const_mul b⟩
convert! Tendsto.const_mul b⁻¹ using 3 <;> rw [inv_mul_cancel_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 167,
"column": 2
} | {
"line": 168,
"column": 69
} | [
{
"pp": "G : Type w\nα : Type u\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\nb c : G\nf : α → G\nl : Filter α\n⊢ Tendsto (fun x ↦ b * f x) l (𝓝 (b * c)) ↔ Tendsto f l (𝓝 c)",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivI... | refine ⟨?_, Tendsto.const_mul b⟩
convert! Tendsto.const_mul b⁻¹ using 3 <;> rw [inv_mul_cancel_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 581,
"column": 2
} | {
"line": 581,
"column": 58
} | [
{
"pp": "H : Type x\ninst✝⁴ : TopologicalSpace H\ninst✝³ : CommGroup H\ninst✝² : PartialOrder H\ninst✝¹ : IsOrderedMonoid H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[≥] a⁻¹) (𝓝[≤] a)",
"usedConstants": [
"Set.Ici",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInv... | simpa only [inv_inv] using tendsto_inv_nhdsGE (a := a⁻¹) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 581,
"column": 2
} | {
"line": 581,
"column": 58
} | [
{
"pp": "H : Type x\ninst✝⁴ : TopologicalSpace H\ninst✝³ : CommGroup H\ninst✝² : PartialOrder H\ninst✝¹ : IsOrderedMonoid H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[≥] a⁻¹) (𝓝[≤] a)",
"usedConstants": [
"Set.Ici",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInv... | simpa only [inv_inv] using tendsto_inv_nhdsGE (a := a⁻¹) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 581,
"column": 2
} | {
"line": 581,
"column": 58
} | [
{
"pp": "H : Type x\ninst✝⁴ : TopologicalSpace H\ninst✝³ : CommGroup H\ninst✝² : PartialOrder H\ninst✝¹ : IsOrderedMonoid H\ninst✝ : ContinuousInv H\na : H\n⊢ Tendsto Inv.inv (𝓝[≥] a⁻¹) (𝓝[≤] a)",
"usedConstants": [
"Set.Ici",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInv... | simpa only [inv_inv] using tendsto_inv_nhdsGE (a := a⁻¹) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 862,
"column": 6
} | {
"line": 862,
"column": 34
} | [
{
"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)",
... | Topology.isInducing_iff_nhds | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1014,
"column": 59
} | {
"line": 1016,
"column": 98
} | [
{
"pp": "α : Type u\nG : Type u_1\ninst✝² : GroupWithZero G\ninst✝¹ : TopologicalSpace G\ninst✝ : ContinuousDiv G\nb : G\nhb : b ≠ 0\nc : G\nf : α → G\nl : Filter α\n⊢ Tendsto (fun x ↦ f x / b) l (𝓝 (c / b)) ↔ Tendsto f l (𝓝 c)",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
... | by
refine ⟨fun h ↦ ?_, fun h ↦ Filter.Tendsto.div_const' h b⟩
convert! h.div_const' b⁻¹ with k <;> rw [← div_mul_eq_div_div_swap, inv_mul_cancel₀ hb, div_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1184,
"column": 8
} | {
"line": 1184,
"column": 27
} | [
{
"pp": "case refine_4\nG : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\nK U : Set G\nhK : IsCompact K\nhU : IsOpen[inst✝²] U\nhKU : K ⊆ U\nx : G\nhx : x ∈ K\nt : Set G\nht : t ∈ 𝓝 x\ns : Set G\nhs : s ∈ 𝓝 1\nh : t ×ˢ s ⊆ (fun p ↦ p.1 * p.2) ⁻¹' U\n⊢ ∃ t ∈ 𝓝[K] x, ∃ V... | ← image_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 407,
"column": 57
} | {
"line": 407,
"column": 79
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : DiscreteUniformity β\nf : α → β\nhf : IsUniformEmbedding f\n⊢ 𝓤 α = 𝓟 SetRel.id",
"usedConstants": [
"Eq.mpr",
"IsUniformInducing.comap_uniformity",
"SetRel.id",
"congrArg",
"uniformity... | ← hf.comap_uniformity, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1359,
"column": 2
} | {
"line": 1360,
"column": 62
} | [
{
"pp": "α : Type u\ninst✝³ : Monoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : ContinuousMul α\nS : Submonoid α\nhS : IsCompact ↑S\n⊢ IsCompact ↑S.units",
"usedConstants": [
"Set.instSProd",
"MonoidHom.instFunLike",
"MulOpposite.opHomeomorph",
"Units.instTopological... | have : IsCompact (S ×ˢ S.op) := hS.prod (opHomeomorph.isCompact_preimage.mp hS)
exact isClosedEmbedding_embedProduct.isCompact_preimage this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1359,
"column": 2
} | {
"line": 1360,
"column": 62
} | [
{
"pp": "α : Type u\ninst✝³ : Monoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : T1Space α\ninst✝ : ContinuousMul α\nS : Submonoid α\nhS : IsCompact ↑S\n⊢ IsCompact ↑S.units",
"usedConstants": [
"Set.instSProd",
"MonoidHom.instFunLike",
"MulOpposite.opHomeomorph",
"Units.instTopological... | have : IsCompact (S ×ˢ S.op) := hS.prod (opHomeomorph.isCompact_preimage.mp hS)
exact isClosedEmbedding_embedProduct.isCompact_preimage this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 503,
"column": 2
} | {
"line": 503,
"column": 50
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\nf : Filter α\nhf : f.TotallyBounded\ns : Set α\nhs : s ∈ f\nU : SetRel α α\nhU : U ∈ 𝓤 α\nr : Set (α × α)\nhr : r ∈ 𝓤 α\nrs : ∀ {a b : α}, (a, b) ∈ r → (b, a) ∈ r\nrU : r ○ r ⊆ U\nk : Set α\nfk : k.Finite\nks : SetRel.preimage r k ∈ f\nu : Set α := k ∩ {y | ... | choose g hgs hgr using fun x : u => x.coe_prop.2 | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.Topology.Category.TopCat.Limits.Basic | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 28
} | [
{
"pp": "J : Type v\ninst✝ : Category.{w, v} J\nF : J ⥤ TopCat\nc : Cocone F\nhc : IsColimit c\nc' : Cocone F := coconeOfCoconeForget (forget.mapCocone c)\nhc' : IsColimit c' := isColimitCoconeOfForget (forget.mapCocone c) (isColimitOfPreserves forget hc)\ne : c'.pt ≅ c.pt := hc'.coconePointUniqueUpToIso hc\nhe... | simp only [coinduced_iSup] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic | {
"line": 875,
"column": 24
} | {
"line": 875,
"column": 54
} | [
{
"pp": "R : Type u_1\ninst✝¹⁴ : Ring R\nR₂ : Type u_2\ninst✝¹³ : Ring R₂\nR₃ : Type u_3\ninst✝¹² : Ring R₃\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : AddCommGroup M\nM₂ : Type u_5\ninst✝⁹ : TopologicalSpace M₂\ninst✝⁸ : AddCommGroup M₂\nM₃ : Type u_6\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : AddCommG... | ext; simp [add_smul, add_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Basic | {
"line": 875,
"column": 24
} | {
"line": 875,
"column": 54
} | [
{
"pp": "R : Type u_1\ninst✝¹⁴ : Ring R\nR₂ : Type u_2\ninst✝¹³ : Ring R₂\nR₃ : Type u_3\ninst✝¹² : Ring R₃\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : AddCommGroup M\nM₂ : Type u_5\ninst✝⁹ : TopologicalSpace M₂\ninst✝⁸ : AddCommGroup M₂\nM₃ : Type u_6\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : AddCommG... | ext; simp [add_smul, add_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 319,
"column": 16
} | {
"line": 324,
"column": 9
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.LeftHomologyData\nh₂ : S₂.LeftHomologyData\nψ₁ ψ₂ : LeftHomologyMapData φ h₁ h₂\n⊢ ψ₁ = ψ₂",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.HasZeroMorphisms",... | by
have hK : ψ₁.φK = ψ₂.φK := by rw [← cancel_mono h₂.i, commi, commi]
have hH : ψ₁.φH = ψ₂.φH := by rw [← cancel_epi h₁.π, commπ, commπ, hK]
cases ψ₁
cases ψ₂
congr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 452,
"column": 34
} | {
"line": 454,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\ninst✝ : S.HasLeftHomology\n⊢ Epi S.leftHomologyπ",
"usedConstants": [
"CategoryTheory.ShortComplex.leftHomologyData",
"CategoryTheory.ShortComplex.leftHomologyπ",
"CategoryTheor... | by
dsimp only [leftHomologyπ]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Kernels | {
"line": 54,
"column": 68
} | {
"line": 54,
"column": 80
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\ninst✝ : HasZeroMorphisms C\nX Y Q : Cᵒᵖ\np : Y ⟶ Q\nf : X ⟶ Y\nw : f ≫ p = 0\nh : IsColimit (CokernelCofork.ofπ p w)\n⊢ p.unop ≫ f.unop = 0",
"usedConstants": [
"Eq.mpr",
"Opposite",
"CategoryThe... | ← unop_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Opposites.Kernels | {
"line": 79,
"column": 39
} | {
"line": 79,
"column": 51
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nJ : Type u₂\ninst✝¹ : Category.{v₂, u₂} J\ninst✝ : HasZeroMorphisms C\nK X Y : Cᵒᵖ\ni : K ⟶ X\nf : X ⟶ Y\nw : i ≫ f = 0\nh : IsLimit (KernelFork.ofι i w)\n⊢ f.unop ≫ i.unop = 0",
"usedConstants": [
"Eq.mpr",
"Opposite",
"CategoryTheory.Ca... | ← unop_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.LeftHomology | {
"line": 752,
"column": 71
} | {
"line": 755,
"column": 34
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : HasZeroMorphisms C\nS : ShortComplex C\nh : S.LeftHomologyData\ninst✝ : S.HasLeftHomology\n⊢ S.leftHomologyπ ≫ h.leftHomologyIso.hom = h.cyclesIso.hom ≫ h.π",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ShortComplex.leftHomologyDat... | by
dsimp only [leftHomologyπ, leftHomologyIso, cyclesIso, leftHomologyMapIso',
cyclesMapIso', Iso.refl]
rw [← leftHomologyπ_naturality'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 422,
"column": 53
} | {
"line": 422,
"column": 92
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\n⊢ S₁.f ≫ φ.τ₂ ≫ h₂.p = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ShortComplex.RightHomologyData.wp",
"Ca... | rw [← φ.comm₁₂_assoc, h₂.wp, comp_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 422,
"column": 53
} | {
"line": 422,
"column": 92
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\n⊢ S₁.f ≫ φ.τ₂ ≫ h₂.p = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ShortComplex.RightHomologyData.wp",
"Ca... | rw [← φ.comm₁₂_assoc, h₂.wp, comp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 422,
"column": 53
} | {
"line": 422,
"column": 92
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\nφ : S₁ ⟶ S₂\nh₁ : S₁.RightHomologyData\nh₂ : S₂.RightHomologyData\n⊢ S₁.f ≫ φ.τ₂ ≫ h₂.p = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.ShortComplex.RightHomologyData.wp",
"Ca... | rw [← φ.comm₁₂_assoc, h₂.wp, comp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 816,
"column": 19
} | {
"line": 816,
"column": 72
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\ne : S₁ ≅ S₂\ninst✝¹ : S₁.HasRightHomology\ninst✝ : S₂.HasRightHomology\n⊢ opcyclesMap e.inv ≫ opcyclesMap e.hom = 𝟙 S₂.opcycles",
"usedConstants": [
"CategoryTheory.ShortComplex.opcycles",... | rw [← opcyclesMap_comp, e.inv_hom_id, opcyclesMap_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 816,
"column": 19
} | {
"line": 816,
"column": 72
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\ne : S₁ ≅ S₂\ninst✝¹ : S₁.HasRightHomology\ninst✝ : S₂.HasRightHomology\n⊢ opcyclesMap e.inv ≫ opcyclesMap e.hom = 𝟙 S₂.opcycles",
"usedConstants": [
"CategoryTheory.ShortComplex.opcycles",... | rw [← opcyclesMap_comp, e.inv_hom_id, opcyclesMap_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.RightHomology | {
"line": 816,
"column": 19
} | {
"line": 816,
"column": 72
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : HasZeroMorphisms C\nS S₁ S₂ S₃ : ShortComplex C\ne : S₁ ≅ S₂\ninst✝¹ : S₁.HasRightHomology\ninst✝ : S₂.HasRightHomology\n⊢ opcyclesMap e.inv ≫ opcyclesMap e.hom = 𝟙 S₂.opcycles",
"usedConstants": [
"CategoryTheory.ShortComplex.opcycles",... | rw [← opcyclesMap_comp, e.inv_hom_id, opcyclesMap_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 613,
"column": 6
} | {
"line": 613,
"column": 72
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nh₁ h₁' : S.LeftHomologyData\nh₂ h₂' : S.RightHomologyData\n⊢ leftRightHomologyComparison' h₁ h₂ =\n leftHomologyMap' (𝟙 S) h₁ h₁' ≫ leftRightHomologyComparison' h₁' h₂' ≫ rightHomologyMap' (𝟙 S) h₂' h₂",
"u... | leftRightHomologyComparison'_naturality_assoc (𝟙 S) h₁ h₂ h₁' h₂', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.ShortComplex.Homology | {
"line": 839,
"column": 29
} | {
"line": 841,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\nS S₁ S₂ S₃ S₄ : ShortComplex C\nφ✝ : S₁ ⟶ S₂\nh₁ : S₁.HomologyData\nh₂ : S₂.HomologyData\nφ : S₁ ⟶ S₂\ninst✝² : S₁.HasHomology\ninst✝¹ : S₂.HasHomology\ninst✝ : IsIso φ\n⊢ IsIso (homologyMap φ)",
"usedConstants": [
"Category... | by
dsimp only [homologyMap, homologyMap']
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1141,
"column": 4
} | {
"line": 1143,
"column": 7
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : TopologicalSpace M₂\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommMonoid M₃\nin... | rcases hf with ⟨A, rfl⟩
simp only [ContinuousLinearEquiv.comp_coe, inverse_equiv, ContinuousLinearEquiv.coe_inj]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1141,
"column": 4
} | {
"line": 1143,
"column": 7
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : TopologicalSpace M₂\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommMonoid M₃\nin... | rcases hf with ⟨A, rfl⟩
simp only [ContinuousLinearEquiv.comp_coe, inverse_equiv, ContinuousLinearEquiv.coe_inj]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1149,
"column": 4
} | {
"line": 1151,
"column": 7
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : TopologicalSpace M₂\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommMonoid M₃\nin... | rcases hf with ⟨A, rfl⟩
simp only [ContinuousLinearEquiv.comp_coe, inverse_equiv, ContinuousLinearEquiv.coe_inj]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1149,
"column": 4
} | {
"line": 1151,
"column": 7
} | [
{
"pp": "case pos\nR : Type u_1\nM : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : TopologicalSpace M₂\ninst✝⁷ : TopologicalSpace M₃\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₂\ninst✝¹ : AddCommMonoid M₃\nin... | rcases hf with ⟨A, rfl⟩
simp only [ContinuousLinearEquiv.comp_coe, inverse_equiv, ContinuousLinearEquiv.coe_inj]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1286,
"column": 4
} | {
"line": 1287,
"column": 54
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M₂\ninst✝ : TopologicalSpace M₂\nmodule_M : Module R M\nmodule_M₂ : Module R₂ M₂\nσ₁₂ : R →+* R₂\nσ₂₁ : R₂ →+* R\nre₁₂ : Rin... | have h' : (fun x ↦ x ∈ p) = (fun x ↦ x ∈ q) := by simp [h]
exact (Homeomorph.ofEqSubtypes h').symm.continuous | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Equiv | {
"line": 1286,
"column": 4
} | {
"line": 1287,
"column": 54
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring R₂\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M₂\ninst✝ : TopologicalSpace M₂\nmodule_M : Module R M\nmodule_M₂ : Module R₂ M₂\nσ₁₂ : R →+* R₂\nσ₂₁ : R₂ →+* R\nre₁₂ : Rin... | have h' : (fun x ↦ x ∈ p) = (fun x ↦ x ∈ q) := by simp [h]
exact (Homeomorph.ofEqSubtypes h').symm.continuous | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.Limits | {
"line": 209,
"column": 33
} | {
"line": 213,
"column": 39
} | [
{
"pp": "J : 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 := colimit.ι (F ⋙ π... | by
ext
· simp [← colimit.w (F ⋙ π₁) f]
· simp [← colimit.w (F ⋙ π₂) f]
· simp [← colimit.w (F ⋙ π₃) f] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.ShortComplex.QuasiIso | {
"line": 160,
"column": 26
} | {
"line": 162,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : HasZeroMorphisms C\nS₁ S₂ : ShortComplex C\ninst✝² : S₁.HasHomology\ninst✝¹ : S₂.HasHomology\nφ : S₁ ⟶ S₂\ninst✝ : QuasiIso φ\n⊢ QuasiIso (opMap φ)",
"usedConstants": [
"CategoryTheory.ShortComplex.QuasiIso",
"Eq.mpr",
"Opposi... | by
rw [quasiIso_opMap_iff]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Category.ModuleCat.Topology.Basic | {
"line": 304,
"column": 4
} | {
"line": 305,
"column": 69
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\ninst✝¹ : TopologicalSpace R\nM : ModuleCat R\nI : Type u_1\nX : I → TopModuleCat R\nf : (i : I) → M ⟶ (X i).toModuleCat\nJ : Type u_2\ninst✝ : Category.{v_1, u_2} J\nF : J ⥤ TopModuleCat R\nc : Cone (F ⋙ forget₂ (TopModuleCat R) (ModuleCat R))\nhc : IsLimit c\ns : Cone F\ni... | change Continuous (X := s.pt) (Y := F.obj i)
(hc.lift ((forget₂ _ (ModuleCat R)).mapCone s) ≫ c.π.app i).hom | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | {
"line": 111,
"column": 4
} | {
"line": 112,
"column": 71
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\nh : S.LeftHomologyData\nF : C ⥤ D\ninst✝¹ : F.PreservesZeroMorphisms\ninst✝ : h.IsPreservedBy F\nthis✝ : PreservesLimit (parall... | rw [Fork.IsLimit.lift_ι hi]
simp only [KernelFork.map_ι, Fork.ι_ofι, map_f, ← F.map_comp, f'_i] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | {
"line": 111,
"column": 4
} | {
"line": 112,
"column": 71
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasZeroMorphisms D\nS S₁ S₂ : ShortComplex C\nh : S.LeftHomologyData\nF : C ⥤ D\ninst✝¹ : F.PreservesZeroMorphisms\ninst✝ : h.IsPreservedBy F\nthis✝ : PreservesLimit (parall... | rw [Fork.IsLimit.lift_ι hi]
simp only [KernelFork.map_ι, Fork.ι_ofι, map_f, ← F.map_comp, f'_i] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | {
"line": 530,
"column": 2
} | {
"line": 531,
"column": 21
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁸ : Category.{v_1, u_1} C\ninst✝⁷ : Category.{v_2, u_2} D\ninst✝⁶ : HasZeroMorphisms C\ninst✝⁵ : HasZeroMorphisms D\nS₁ S₂ : ShortComplex C\nφ : S₁ ⟶ S₂\nF : C ⥤ D\ninst✝⁴ : F.PreservesZeroMorphisms\ninst✝³ : S₁.HasLeftHomology\ninst✝² : S₂.HasLeftHomology\ninst✝¹ : F.P... | simp only [LeftHomologyData.map_cyclesMap', Functor.mapShortComplex_obj, ← cyclesMap'_comp,
comp_id, id_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.ShortComplex.PreservesHomology | {
"line": 879,
"column": 6
} | {
"line": 879,
"column": 26
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasZeroMorphisms D\nF : C ⥤ D\ninst✝¹ : F.PreservesZeroMorphisms\nS : ShortComplex C\nhg : S.g = 0\ninst✝ : PreservesColimit (parallelPair S.f 0) F\nh : S.LeftHomologyData\n... | have := h.isIso_i hg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 131,
"column": 75
} | {
"line": 132,
"column": 27
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\ninst✝¹ : HasZeroMorphisms C\nf : X ⟶ Y\ninst✝ : HasKernel f\n⊢ (kernelSubobjectIso f).hom ≫ kernel.ι f = (kernelSubobject f).arrow",
"usedConstants": [
"CategoryTheory.Subobject.arrow",
"CategoryTheory.Subobject.underlying",
"Catego... | by
simp [kernelSubobjectIso] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Subobject.Limits | {
"line": 136,
"column": 75
} | {
"line": 137,
"column": 27
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\ninst✝¹ : HasZeroMorphisms C\nf : X ⟶ Y\ninst✝ : HasKernel f\n⊢ (kernelSubobjectIso f).inv ≫ (kernelSubobject f).arrow = kernel.ι f",
"usedConstants": [
"CategoryTheory.Subobject.arrow",
"CategoryTheory.Subobject.underlying",
"Catego... | by
simp [kernelSubobjectIso] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.End | {
"line": 240,
"column": 85
} | {
"line": 243,
"column": 45
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nM : Type u_1\ninst✝² : Category.{v_1, u_1} M\ninst✝¹ : MonoidalCategory M\nF : M ⥤ C ⥤ C\nn : M\nX : C\ninst✝ : F.Monoidal\n⊢ (η F).app ((F.obj n).obj X) = (μ F n (𝟙_ M)).app X ≫ (F.map (ρ_ n).hom).app X",
"usedConstants": [
"Eq.mpr",
"CategoryTh... | by
rw [map_rightUnitor]
dsimp
simp only [Category.comp_id, μ_δ_app_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 34
} | [
{
"pp": "ι : Type u_1\nV : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ni j j' : ι\nrij : c.Rel i j\nrij' : c.Rel i j'\n⊢ C.d i j' ≫ eqToHom ⋯ = C.d i j",
"usedConstants": [
"ComplexShape.next_eq"
]
}
] | obtain rfl := c.next_eq rij rij' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 538,
"column": 2
} | {
"line": 552,
"column": 57
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddGroup A\ninst✝ : HasShift C A\nX : C\nm n p m' n' p' : A\nhm : m' + m = 0\nhn : n' + n = 0\nhp : p' + p = 0\nh : m + n = p\n⊢ (shiftFunctorCompIsoId C p' p hp).inv.app X =\n (shiftFunctorCompIsoId C n' n hn).inv.app X ≫\n (shiftF... | dsimp [shiftFunctorCompIsoId]
simp only [Functor.map_comp, Category.assoc]
congr 1
rw [← NatTrans.naturality]
dsimp
rw [← cancel_mono ((shiftFunctorAdd' C p' p 0 hp).inv.app X), Iso.hom_inv_id_app,
Category.assoc, Category.assoc, Category.assoc, Category.assoc,
← shiftFunctorAdd'_assoc_inv_app p' m n ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 538,
"column": 2
} | {
"line": 552,
"column": 57
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddGroup A\ninst✝ : HasShift C A\nX : C\nm n p m' n' p' : A\nhm : m' + m = 0\nhn : n' + n = 0\nhp : p' + p = 0\nh : m + n = p\n⊢ (shiftFunctorCompIsoId C p' p hp).inv.app X =\n (shiftFunctorCompIsoId C n' n hn).inv.app X ≫\n (shiftF... | dsimp [shiftFunctorCompIsoId]
simp only [Functor.map_comp, Category.assoc]
congr 1
rw [← NatTrans.naturality]
dsimp
rw [← cancel_mono ((shiftFunctorAdd' C p' p 0 hp).inv.app X), Iso.hom_inv_id_app,
Category.assoc, Category.assoc, Category.assoc, Category.assoc,
← shiftFunctorAdd'_assoc_inv_app p' m n ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 597,
"column": 2
} | {
"line": 597,
"column": 42
} | [
{
"pp": "C : Type u\nA : Type u_1\ninst✝² : Category.{v, u} C\ninst✝¹ : AddCommMonoid A\ninst✝ : HasShift C A\ni j : A\n⊢ shiftFunctorComm C i j = (shiftFunctorAdd' C i j (i + j) ⋯).symm ≪≫ shiftFunctorAdd' C j i (i + j) ⋯",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"AddMonoid... | rw [shiftFunctorAdd'_eq_shiftFunctorAdd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 724,
"column": 2
} | {
"line": 725,
"column": 35
} | [
{
"pp": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nX₀ X₁ X₂ : V\nd₀ : X₁ ⟶ X₀\nd₁ : X₂ ⟶ X₁\ns : d₁ ≫ d₀ = 0\nsucc : (S : ShortComplex V) → (X₃ : V) ×' (d₂ : X₃ ⟶ S.X₁) ×' d₂ ≫ S.f = 0\n⊢ (mk X₀ X₁ X₂ d₀ d₁ s succ).d 1 0 = d₀",
"usedConstants": [
"Eq.mpr",
"Nat.instOne"... | change ite (1 = 0 + 1) (𝟙 X₁ ≫ d₀) 0 = d₀
rw [if_pos rfl, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 724,
"column": 2
} | {
"line": 725,
"column": 35
} | [
{
"pp": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nX₀ X₁ X₂ : V\nd₀ : X₁ ⟶ X₀\nd₁ : X₂ ⟶ X₁\ns : d₁ ≫ d₀ = 0\nsucc : (S : ShortComplex V) → (X₃ : V) ×' (d₂ : X₃ ⟶ S.X₁) ×' d₂ ≫ S.f = 0\n⊢ (mk X₀ X₁ X₂ d₀ d₁ s succ).d 1 0 = d₀",
"usedConstants": [
"Eq.mpr",
"Nat.instOne"... | change ite (1 = 0 + 1) (𝟙 X₁ ≫ d₀) 0 = d₀
rw [if_pos rfl, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Shift.Basic | {
"line": 776,
"column": 8
} | {
"line": 776,
"column": 35
} | [
{
"pp": "C : 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\nm₁ m₂ m₃ : A\nX : C\nh :\n (shiftFunctorAdd D (m₁ + m₂) m₃).hom.... | erw [(i m₃).hom.naturality] | Lean.Parser.Tactic._aux_Init_Meta___macroRules_Lean_Parser_Tactic_tacticErw____1 | Lean.Parser.Tactic.tacticErw___ |
Mathlib.Algebra.Homology.HomologicalComplex | {
"line": 793,
"column": 2
} | {
"line": 794,
"column": 35
} | [
{
"pp": "V : Type u\ninst✝¹ : Category.{v, u} V\ninst✝ : HasZeroMorphisms V\nX₀ X₁ : V\nd₀ : X₁ ⟶ X₀\nsucc' : {X₀ X₁ : V} → (f : X₁ ⟶ X₀) → (X₂ : V) ×' (d : X₂ ⟶ X₁) ×' d ≫ f = 0\n⊢ (mk' X₀ X₁ d₀ fun {X₀ X₁} ↦ succ').d 1 0 = d₀",
"usedConstants": [
"Eq.mpr",
"Nat.instOne",
"CategoryTheory.... | change ite (1 = 0 + 1) (𝟙 X₁ ≫ d₀) 0 = d₀
rw [if_pos rfl, Category.id_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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