module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 1415,
"column": 2
} | {
"line": 1415,
"column": 38
} | {
"line": 1416,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : AddCommGroup α\ninst✝² : LinearOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Archimedean α\np : α\nhp : 0 < p\na b : α\nhl : a ≤ b - toIcoDiv hp a b • p\nhr : b - toIcoDiv hp a b • p < a + p\n⊢ b < a + (toIcoDiv hp a b + 1) • p",
"ppTerm": "?m.87",
"assigned": true,
... | [
"α : Type u_1\ninst✝³ : AddCommGroup α\ninst✝² : LinearOrder α\ninst✝¹ : IsOrderedAddMonoid α\ninst✝ : Archimedean α\np : α\nhp : 0 < p\na b : α\nhl : a ≤ b - toIcoDiv hp a b • p\nhr : b - toIcoDiv hp a b • p < a + p\n⊢ b < a + toIcoDiv hp a b • p + p"
] | rw [add_smul, one_smul, ← add_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Order | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 48
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case a\nα : Type u\nn : α → Filter α\nι : α → Sort u_1\np : (a : α) → ι a → Prop\ns : (a : α) → ι a → Set α\nhb : ∀ (a : α), (n a).HasBasis (p a) (s a)\nhopen : ∀ (a : α) (i : ι a), p a i → ∀ᶠ (x : α) in n a, s a i ∈ n x\na : α\nt : TopologicalSpace α := TopologicalSpace.mkOfNhds n\nU : Set α\nhU : U ∈... | [] | exact (hopen x i hpi).mono fun y ↦ by gcongr | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Order | {
"line": 1057,
"column": 2
} | {
"line": 1057,
"column": 96
} | {
"line": 1059,
"column": 0
} | [
{
"pp": "α : Type u\nι : Sort v\nt : ι → TopologicalSpace α\ns : Set α\n⊢ IsClosed[⨆ i, t i] s ↔ ∀ (i : ι), IsClosed[t i] s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"congrArg",
"iSup",
"Compl.compl",
"CompleteLattice.toConditionallyCompleteLattice",
"Top... | [] | simp only [← @isOpen_compl_iff _ _ (⨆ i, t i), ← @isOpen_compl_iff _ _ (t _), isOpen_iSup_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Order | {
"line": 1057,
"column": 2
} | {
"line": 1057,
"column": 96
} | {
"line": 1059,
"column": 0
} | [
{
"pp": "α : Type u\nι : Sort v\nt : ι → TopologicalSpace α\ns : Set α\n⊢ IsClosed[⨆ i, t i] s ↔ ∀ (i : ι), IsClosed[t i] s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"congrArg",
"iSup",
"Compl.compl",
"CompleteLattice.toConditionallyCompleteLattice",
"Top... | [] | simp only [← @isOpen_compl_iff _ _ (⨆ i, t i), ← @isOpen_compl_iff _ _ (t _), isOpen_iSup_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order | {
"line": 1057,
"column": 2
} | {
"line": 1057,
"column": 96
} | {
"line": 1059,
"column": 0
} | [
{
"pp": "α : Type u\nι : Sort v\nt : ι → TopologicalSpace α\ns : Set α\n⊢ IsClosed[⨆ i, t i] s ↔ ∀ (i : ι), IsClosed[t i] s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"congrArg",
"iSup",
"Compl.compl",
"CompleteLattice.toConditionallyCompleteLattice",
"Top... | [] | simp only [← @isOpen_compl_iff _ _ (⨆ i, t i), ← @isOpen_compl_iff _ _ (t _), isOpen_iSup_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Maps.Basic | {
"line": 445,
"column": 43
} | {
"line": 447,
"column": 95
} | {
"line": 449,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nhf : IsOpenMap f\ns : Set Y\n⊢ f ⁻¹' closure[inst✝] s ⊆ closure[inst✝¹] (f ⁻¹' s)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
... | [] | by
rw [← compl_subset_compl]
simp only [← interior_compl, ← preimage_compl, hf.interior_preimage_subset_preimage_interior] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.NhdsWithin | {
"line": 214,
"column": 6
} | {
"line": 214,
"column": 21
} | {
"line": 214,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\na : α\ns t : Set α\n⊢ 𝓝 a ⊓ 𝓟 (s ∪ t) = 𝓝 a ⊓ 𝓟 s ⊔ 𝓝 a ⊓ 𝓟 t",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"nhds",
"Set.instUnion",
"Semilattice... | [
"α : Type u_1\ninst✝ : TopologicalSpace α\na : α\ns t : Set α\n⊢ 𝓝 a ⊓ 𝓟 (s ∪ t) = 𝓝 a ⊓ (𝓟 s ⊔ 𝓟 t)"
] | ← inf_sup_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.NhdsWithin | {
"line": 313,
"column": 2
} | {
"line": 314,
"column": 70
} | {
"line": 316,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nx : α\ns t : Set α\nhst : s =ᶠ[𝓝 x] t\nh : x ∈ interior s\n⊢ x ∈ interior t",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"_private.Mathlib.Topology.NhdsWithin.0.Filter.EventuallyEq.m... | [] | rw [← nhdsWithin_eq_iff_eventuallyEq] at hst
simpa [mem_interior_iff_mem_nhds, ← nhdsWithin_eq_nhds, hst] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.NhdsWithin | {
"line": 313,
"column": 2
} | {
"line": 314,
"column": 70
} | {
"line": 316,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nx : α\ns t : Set α\nhst : s =ᶠ[𝓝 x] t\nh : x ∈ interior s\n⊢ x ∈ interior t",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"_private.Mathlib.Topology.NhdsWithin.0.Filter.EventuallyEq.m... | [] | rw [← nhdsWithin_eq_iff_eventuallyEq] at hst
simpa [mem_interior_iff_mem_nhds, ← nhdsWithin_eq_nhds, hst] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Constructions.SumProd | {
"line": 850,
"column": 2
} | {
"line": 850,
"column": 53
} | {
"line": 852,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\nZ : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Z\ng : Y → Z\nhf : Continuous[inst✝², inst✝] f ∧ Injective f ∧ IsOpenMap f\nhg : Continuous[inst✝¹, inst✝] g ∧ Injective g ∧ IsOpenMap g\nh : Injective (Sum.elim f g)\n⊢ C... | [] | exact ⟨hf.1.sumElim hg.1, h, hf.2.2.sumElim hg.2.2⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Constructions.SumProd | {
"line": 880,
"column": 2
} | {
"line": 880,
"column": 53
} | {
"line": 884,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\nZ : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Z\ng : Y → Z\nhf : Continuous[inst✝², inst✝] f ∧ Injective f ∧ IsClosedMap f\nhg : Continuous[inst✝¹, inst✝] g ∧ Injective g ∧ IsClosedMap g\nh : Injective (Sum.elim f g)\... | [] | exact ⟨hf.1.sumElim hg.1, h, hf.2.2.sumElim hg.2.2⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.ContinuousOn | {
"line": 668,
"column": 72
} | {
"line": 669,
"column": 99
} | {
"line": 671,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : DiscreteTopology β\nf : α × β → γ\ns : Set (α × β)\n⊢ ContinuousOn f s ↔ ∀ (b : β), ContinuousOn (fun x ↦ f (x, b)) {a | (a, b) ∈ s}",
"ppTerm": "?m.21",
"ass... | [] | by
simp_rw [ContinuousOn, Prod.forall, continuousWithinAt_prod_of_discrete_right]; apply forall_comm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Bases | {
"line": 239,
"column": 6
} | {
"line": 239,
"column": 28
} | {
"line": 239,
"column": 29
} | [
{
"pp": "α : Type u\nβ : Type u_1\nt : TopologicalSpace α\ninst✝ : TopologicalSpace β\nB : Set (Set α)\nhB : IsTopologicalBasis B\nf : α → β\nhf : ∀ s ∈ B, IsOpen[inst✝] (f '' s)\no : Set α\nho : IsOpen[t] o\n⊢ IsOpen[inst✝] (f '' o)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"E... | [
"α : Type u\nβ : Type u_1\nt : TopologicalSpace α\ninst✝ : TopologicalSpace β\nB : Set (Set α)\nhB : IsTopologicalBasis B\nf : α → β\nhf : ∀ s ∈ B, IsOpen[inst✝] (f '' s)\no : Set α\nho : IsOpen[t] o\n⊢ IsOpen[inst✝] (f '' ⋃₀ {s | s ∈ B ∧ s ⊆ o})"
] | hB.open_eq_sUnion' ho, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Bases | {
"line": 288,
"column": 14
} | {
"line": 288,
"column": 16
} | {
"line": 288,
"column": 17
} | [
{
"pp": "case refine_2\nα : Type u\nt : TopologicalSpace α\nι : Sort u_2\nU : ι → Set α\nUo : ∀ (i : ι), IsOpen[t] (U i)\nUc : ⋃ i, U i = univ\nb : (i : ι) → Set (Set ↑(U i))\nhb : ∀ (i : ι), IsTopologicalBasis (b i)\na : α\nu : Set α\n⊢ a ∈ u → IsOpen[t] u → ∃ v ∈ ⋃ i, image Subtype.val '' b i, a ∈ v ∧ v ⊆ u",... | [
"case refine_2\nα : Type u\nt : TopologicalSpace α\nι : Sort u_2\nU : ι → Set α\nUo : ∀ (i : ι), IsOpen[t] (U i)\nUc : ⋃ i, U i = univ\nb : (i : ι) → Set (Set ↑(U i))\nhb : ∀ (i : ι), IsTopologicalBasis (b i)\na : α\nu : Set α\nha : a ∈ u\n⊢ IsOpen[t] u → ∃ v ∈ ⋃ i, image Subtype.val '' b i, a ∈ v ∧ v ⊆ u"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Constructions | {
"line": 1078,
"column": 2
} | {
"line": 1078,
"column": 42
} | {
"line": 1079,
"column": 2
} | [
{
"pp": "ι : Type u_5\nA : ι → Type u_6\nT : (i : ι) → TopologicalSpace (A i)\ns : Set ((a : ι) → A a)\n⊢ (∀ x ∈ s, ∃ I t, (∀ (i : ι), ∃ t_1 ⊆ t i, IsOpen[T i] t_1 ∧ x i ∈ t_1) ∧ (↑I).pi t ⊆ s) ↔\n ∀ f ∈ s, ∃ I u, (∀ a ∈ I, IsOpen[T a] (u a) ∧ f a ∈ u a) ∧ (↑I).pi u ⊆ s",
"ppTerm": "?m.34",
"assigned... | [
"case refine_1\nι : Type u_5\nA : ι → Type u_6\nT : (i : ι) → TopologicalSpace (A i)\ns : Set ((a : ι) → A a)\na : (a : ι) → A a\nx✝ : a ∈ s\n⊢ (∃ I t, (∀ (i : ι), ∃ t_1 ⊆ t i, IsOpen[T i] t_1 ∧ a i ∈ t_1) ∧ (↑I).pi t ⊆ s) →\n ∃ I u, (∀ a_2 ∈ I, IsOpen[T a_2] (u a_2) ∧ a a_2 ∈ u a_2) ∧ (↑I).pi u ⊆ s",
"case re... | refine forall₂_congr fun a _ => ⟨?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.SetAccumulate | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 62
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : ℕ → Set α\n⊢ ⇑(partialSups f) = accumulate f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Finset.mem_range._simp_1",
"Lattice.toSemilatticeSup",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"CompleteLattice.toLat... | [] | ext n
simp [partialSups_eq_sup_range, accumulate, Nat.lt_succ_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SetAccumulate | {
"line": 92,
"column": 2
} | {
"line": 93,
"column": 62
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : ℕ → Set α\n⊢ ⇑(partialSups f) = accumulate f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"Finset.mem_range._simp_1",
"Lattice.toSemilatticeSup",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"CompleteLattice.toLat... | [] | ext n
simp [partialSups_eq_sup_range, accumulate, Nat.lt_succ_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Bases | {
"line": 422,
"column": 4
} | {
"line": 423,
"column": 62
} | {
"line": 424,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns : Set α\nι : Type u_2\nX : ι → Type u_3\ninst✝² : (i : ι) → TopologicalSpace (X i)\ninst✝¹ : ∀ (i : ι), SeparableSpace (X i)\ninst✝ : Countable ι\nt : (x : ι) → Set (X x)\nhtc : ∀ (x : ι), (t x).Countable\nhtd : ∀ (x : ι), Dense (t x... | [
"α : Type u\nβ : Type u_1\nt✝ : TopologicalSpace α\nB : Set (Set α)\ns : Set α\nι : Type u_2\nX : ι → Type u_3\ninst✝² : (i : ι) → TopologicalSpace (X i)\ninst✝¹ : ∀ (i : ι), SeparableSpace (X i)\ninst✝ : Countable ι\nt : (x : ι) → Set (X x)\nhtc : ∀ (x : ι), (t x).Countable\nhtd : ∀ (x : ι), Dense (t x)\nthis : ∀ ... | set f : (Σ I : Finset ι, ∀ i : I, t i) → ∀ i, X i := fun ⟨I, g⟩ i ↦
if hi : i ∈ I then g ⟨i, hi⟩ else (default : ∀ i, X i) i | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Topology.Constructions | {
"line": 1089,
"column": 6
} | {
"line": 1095,
"column": 51
} | {
"line": 1096,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nι : Type u_5\nA : ι → Type u_6\nT : (i : ι) → TopologicalSpace (A i)\ns : Set ((a : ι) → A a)\na : (a : ι) → A a\nx✝ : a ∈ s\nI : Finset ι\nt : (a : ι) → Set (A a)\nh1 : ∀ a_1 ∈ I, IsOpen[T a_1] (t a_1) ∧ a a_1 ∈ t a_1\nh2 : (↑I).pi t ⊆ s\ni : ι\n⊢ ∃ t_1 ⊆ (fun a ↦ if a ∈ I then... | [] | by_cases hi : i ∈ I
· use t i
simp_rw [if_pos hi]
exact ⟨Subset.rfl, (h1 i) hi⟩
· use univ
simp_rw [if_neg hi]
exact ⟨Subset.rfl, isOpen_univ, mem_univ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Constructions | {
"line": 1089,
"column": 6
} | {
"line": 1095,
"column": 51
} | {
"line": 1096,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nι : Type u_5\nA : ι → Type u_6\nT : (i : ι) → TopologicalSpace (A i)\ns : Set ((a : ι) → A a)\na : (a : ι) → A a\nx✝ : a ∈ s\nI : Finset ι\nt : (a : ι) → Set (A a)\nh1 : ∀ a_1 ∈ I, IsOpen[T a_1] (t a_1) ∧ a a_1 ∈ t a_1\nh2 : (↑I).pi t ⊆ s\ni : ι\n⊢ ∃ t_1 ⊆ (fun a ↦ if a ∈ I then... | [] | by_cases hi : i ∈ I
· use t i
simp_rw [if_pos hi]
exact ⟨Subset.rfl, (h1 i) hi⟩
· use univ
simp_rw [if_neg hi]
exact ⟨Subset.rfl, isOpen_univ, mem_univ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Bases | {
"line": 461,
"column": 2
} | {
"line": 466,
"column": 41
} | {
"line": 468,
"column": 0
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\ninst✝ : SeparableSpace α\nι : Type u_2\ns : ι → Set α\nhd : Pairwise (Disjoint on s)\nho : ∀ (i : ι), IsOpen[t] (s i)\nhne : ∀ (i : ι), (s i).Nonempty\n⊢ Countable ι",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Functio... | [] | rcases exists_countable_dense α with ⟨u, u_countable, u_dense⟩
choose f hfu hfs using fun i ↦ u_dense.exists_mem_open (ho i) (hne i)
have f_inj : Injective f := fun i j hij ↦
hd.eq <| not_disjoint_iff.2 ⟨f i, hfs i, hij.symm ▸ hfs j⟩
have := u_countable.to_subtype
exact (f_inj.codRestrict hfu).countable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bases | {
"line": 461,
"column": 2
} | {
"line": 466,
"column": 41
} | {
"line": 468,
"column": 0
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\ninst✝ : SeparableSpace α\nι : Type u_2\ns : ι → Set α\nhd : Pairwise (Disjoint on s)\nho : ∀ (i : ι), IsOpen[t] (s i)\nhne : ∀ (i : ι), (s i).Nonempty\n⊢ Countable ι",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Functio... | [] | rcases exists_countable_dense α with ⟨u, u_countable, u_dense⟩
choose f hfu hfs using fun i ↦ u_dense.exists_mem_open (ho i) (hne i)
have f_inj : Injective f := fun i j hij ↦
hd.eq <| not_disjoint_iff.2 ⟨f i, hfs i, hij.symm ▸ hfs j⟩
have := u_countable.to_subtype
exact (f_inj.codRestrict hfu).countable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Constructions | {
"line": 1105,
"column": 2
} | {
"line": 1105,
"column": 42
} | {
"line": 1106,
"column": 2
} | [
{
"pp": "case intro\nι : Type u_5\nA : ι → Type u_6\nT : (i : ι) → TopologicalSpace (A i)\ninst✝ : Finite ι\ns : Set ((a : ι) → A a)\nval✝ : Fintype ι\n⊢ (∀ x ∈ s, ∃ I t, (∀ (i : ι), ∃ t_1 ⊆ t i, IsOpen[T i] t_1 ∧ x i ∈ t_1) ∧ (↑I).pi t ⊆ s) ↔\n ∀ f ∈ s, ∃ u, (∀ (a : ι), IsOpen[T a] (u a) ∧ f a ∈ u a) ∧ univ... | [
"case intro.refine_1\nι : Type u_5\nA : ι → Type u_6\nT : (i : ι) → TopologicalSpace (A i)\ninst✝ : Finite ι\ns : Set ((a : ι) → A a)\nval✝ : Fintype ι\na : (a : ι) → A a\nx✝ : a ∈ s\n⊢ (∃ I t, (∀ (i : ι), ∃ t_1 ⊆ t i, IsOpen[T i] t_1 ∧ a i ∈ t_1) ∧ (↑I).pi t ⊆ s) →\n ∃ u, (∀ (a_2 : ι), IsOpen[T a_2] (u a_2) ∧ a... | refine forall₂_congr fun a _ => ⟨?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Bases | {
"line": 597,
"column": 14
} | {
"line": 597,
"column": 16
} | {
"line": 597,
"column": 17
} | [
{
"pp": "case refine_2\nβ : Type u_1\nι : Type u_2\nt : ι → TopologicalSpace β\nT : ι → Set (Set β)\nh_basis : ∀ (i : ι), IsTopologicalBasis (T i)\nx✝ : TopologicalSpace β := ⨅ i, t i\na : β\nu : Set β\n⊢ a ∈ u → IsOpen[⨅ i, t i] u → ∃ v ∈ {S | ∃ U F, (∀ i ∈ F, U i ∈ T i) ∧ S = ⋂ i ∈ F, U i}, a ∈ v ∧ v ⊆ u",
... | [
"case refine_2\nβ : Type u_1\nι : Type u_2\nt : ι → TopologicalSpace β\nT : ι → Set (Set β)\nh_basis : ∀ (i : ι), IsTopologicalBasis (T i)\nx✝ : TopologicalSpace β := ⨅ i, t i\na : β\nu : Set β\nha : a ∈ u\n⊢ IsOpen[⨅ i, t i] u → ∃ v ∈ {S | ∃ U F, (∀ i ∈ F, U i ∈ T i) ∧ S = ⋂ i ∈ F, U i}, a ∈ v ∧ v ⊆ u"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Ultrafilter | {
"line": 31,
"column": 57
} | {
"line": 31,
"column": 71
} | {
"line": 31,
"column": 72
} | [
{
"pp": "X : Type u\nα : Type u_1\nx : X\ninst✝ : TopologicalSpace X\nF : Filter α\nu : α → X\n⊢ (map u (comap u (𝓝 x) ⊓ F)).NeBot ↔ ∃ U, ↑U ≤ F ∧ Tendsto u (↑U) (𝓝 x)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Filter.map",
"Filter.NeBo... | [
"X : Type u\nα : Type u_1\nx : X\ninst✝ : TopologicalSpace X\nF : Filter α\nu : α → X\n⊢ (comap u (𝓝 x) ⊓ F).NeBot ↔ ∃ U, ↑U ≤ F ∧ Tendsto u (↑U) (𝓝 x)"
] | map_neBot_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.LocallyFinite | {
"line": 136,
"column": 50
} | {
"line": 137,
"column": 66
} | {
"line": 138,
"column": 2
} | [
{
"pp": "ι : Type u_1\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\nhf : LocallyFinite f\nhc : ∀ (i : ι), IsClosed[inst✝] (f i)\nx : X\nthis : IsClosed[inst✝] (⋃ i, f ↑i)\n⊢ IsOpen[inst✝] (⋂ i, ⋂ (_ : x ∉ f i), (f i)ᶜ)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"cong... | [] | by
rwa [← isOpen_compl_iff, compl_iUnion, iInter_subtype] at this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.LocallyFinite | {
"line": 136,
"column": 2
} | {
"line": 137,
"column": 66
} | {
"line": 138,
"column": 2
} | [
{
"pp": "ι : Type u_1\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\nhf : LocallyFinite f\nhc : ∀ (i : ι), IsClosed[inst✝] (f i)\nx : X\n⊢ IsOpen[inst✝] (⋂ i, ⋂ (_ : x ∉ f i), (f i)ᶜ)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.iInter",
"Co... | [
"ι : Type u_1\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\nhf : LocallyFinite f\nhc : ∀ (i : ι), IsClosed[inst✝] (f i)\nx : X\n⊢ IsClosed[inst✝] (⋃ i, f ↑i)"
] | suffices IsClosed (⋃ i : { i // x ∉ f i }, f i) by
rwa [← isOpen_compl_iff, compl_iUnion, iInter_subtype] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Topology.Irreducible | {
"line": 234,
"column": 4
} | {
"line": 236,
"column": 50
} | {
"line": 238,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsPreirreducible s\n⊢ IsPreirreducible univ",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.univ",
"Subtype.casesOn",
"Membership.mem",
"Set.Elem",
"Subtype",
"Set.instInter",
"S... | [] | rintro _ _ ⟨u, hu, rfl⟩ ⟨v, hv, rfl⟩ ⟨⟨x, hxs⟩, -, hxu⟩ ⟨⟨y, hys⟩, -, hyv⟩
rcases h u v hu hv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩ with ⟨x, hxs, ⟨hxu, hxv⟩⟩
exact ⟨⟨x, hxs⟩, ⟨Set.mem_univ _, ⟨hxu, hxv⟩⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Irreducible | {
"line": 234,
"column": 4
} | {
"line": 236,
"column": 50
} | {
"line": 238,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsPreirreducible s\n⊢ IsPreirreducible univ",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.univ",
"Subtype.casesOn",
"Membership.mem",
"Set.Elem",
"Subtype",
"Set.instInter",
"S... | [] | rintro _ _ ⟨u, hu, rfl⟩ ⟨v, hv, rfl⟩ ⟨⟨x, hxs⟩, -, hxu⟩ ⟨⟨y, hys⟩, -, hyv⟩
rcases h u v hu hv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩ with ⟨x, hxs, ⟨hxu, hxv⟩⟩
exact ⟨⟨x, hxs⟩, ⟨Set.mem_univ _, ⟨hxu, hxv⟩⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Irreducible | {
"line": 339,
"column": 14
} | {
"line": 339,
"column": 16
} | {
"line": 339,
"column": 17
} | [
{
"pp": "case refine_2\nX : Type u_1\ninst✝ : TopologicalSpace X\nS : Set X\nh : ∀ (U : Set X), IsOpen[inst✝] U → (S ∩ U).Nonempty → S ⊆ closure[inst✝] (S ∩ U)\na b : Set X\n⊢ IsOpen[inst✝] a → IsOpen[inst✝] b → (S ∩ a).Nonempty → (S ∩ b).Nonempty → (S ∩ (a ∩ b)).Nonempty",
"ppTerm": "?refine_2",
"assig... | [
"case refine_2\nX : Type u_1\ninst✝ : TopologicalSpace X\nS : Set X\nh : ∀ (U : Set X), IsOpen[inst✝] U → (S ∩ U).Nonempty → S ⊆ closure[inst✝] (S ∩ U)\na b : Set X\nha : IsOpen[inst✝] a\n⊢ IsOpen[inst✝] b → (S ∩ a).Nonempty → (S ∩ b).Nonempty → (S ∩ (a ∩ b)).Nonempty"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Irreducible | {
"line": 360,
"column": 2
} | {
"line": 362,
"column": 32
} | {
"line": 363,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nZ : Set X\nhZ : Z ∈ irreducibleComponents X\nS : Set (Set X)\nhS : S.Finite\nhSα : S ⊆ irreducibleComponents X\nhZS : Z ⊆ ⋃₀ S\n⊢ Z ∈ S",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"isClosed_of_mem_irreducibleComponents",
"i... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nZ : Set X\nhZ : Z ∈ irreducibleComponents X\nS : Set (Set X)\nhS : S.Finite\nhSα : S ⊆ irreducibleComponents X\nhZS : Z ⊆ ⋃₀ S\nW : Set X\nhWS : W ∈ hS.toFinset\nhZW : Z ⊆ W\n⊢ Z ∈ S"
] | obtain ⟨W, hWS, hZW⟩ := isIrreducible_iff_sUnion_isClosed.mp hZ.1 hS.toFinset
(fun W hW ↦ isClosed_of_mem_irreducibleComponents W (hSα (hS.mem_toFinset.mp hW)))
(hS.coe_toFinset.symm ▸ hZS) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.DiscreteSubset | {
"line": 292,
"column": 12
} | {
"line": 292,
"column": 14
} | {
"line": 293,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \\ s))\nhU : IsClosed[inst✝] U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∈ U\na : X\n⊢ (a ∈ {x}ᶜ → a ∉ U \\ s) → a ∈ (U \\ s)ᶜ",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Compl.comp... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \\ s))\nhU : IsClosed[inst✝] U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∈ U\na : X\nha : a ∈ {x}ᶜ → a ∉ U \\ s\n⊢ a ∈ (U \\ s)ᶜ"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Separation.Basic | {
"line": 854,
"column": 2
} | {
"line": 854,
"column": 97
} | {
"line": 856,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nι : Type u_3\np : ι → Prop\nt : ι → Set X\ns : Set X\nhs : IsDiscrete s\nx : X\nhb : (𝓝 x).HasBasis p t\nhx : x ∈ s\ni : ι\nhi : p i\nhix : t i ∩ s ⊆ {x}\n⊢ ∃ i, p i ∧ t i ∩ s = {x}",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"I... | [] | exact ⟨i, hi, hix.antisymm <| singleton_subset_iff.2 ⟨mem_of_mem_nhds <| hb.mem_of_mem hi, hx⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Connected.Basic | {
"line": 247,
"column": 37
} | {
"line": 256,
"column": 58
} | {
"line": 258,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder β\ninst✝¹ : SuccOrder β\ninst✝ : IsSuccArchimedean β\ns : β → Set α\nt : Set β\nht : t.OrdConnected\nH : ∀ n ∈ t, IsPreconnected (s n)\nK : ∀ n ∈ t, succ n ∈ t → (s n ∩ s (succ n)).Nonempty\n⊢ IsPreconnected (⋃ n ∈ t, s n)",
... | [] | by
have h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t := fun hi hj hk =>
ht.out hi hj (Ico_subset_Icc_self hk)
have h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t := fun hi hj hk =>
ht.out hi hj ⟨hk.1.trans <| le_succ _, succ_le_of_lt hk.2⟩
have h3 : ∀ {i j k : β}, i ∈ t → j ∈ t → ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Connected.Clopen | {
"line": 216,
"column": 30
} | {
"line": 216,
"column": 32
} | {
"line": 217,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : PreconnectedSpace α\nP : α → α → Prop\ninst✝ : Std.Symm P\nh : ∀ (x : α), ∀ᶠ (y : α) in 𝓝 x, P x y\nh' : IsTrans α P\nx y z a : α\n⊢ P z a → P z a ∧ P a z",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [],
"usedFVars": [
... | [
"α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : PreconnectedSpace α\nP : α → α → Prop\ninst✝ : Std.Symm P\nh : ∀ (x : α), ∀ᶠ (y : α) in 𝓝 x, P x y\nh' : IsTrans α P\nx y z a : α\nha : P z a\n⊢ P z a ∧ P a z"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Connected.Clopen | {
"line": 244,
"column": 54
} | {
"line": 244,
"column": 56
} | {
"line": 244,
"column": 57
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nhs : IsPreconnected s\nP : α → α → Prop\nh : ∀ x ∈ s, ∀ᶠ (y : α) in 𝓝[s] x, P x y\nh' : ∀ (x y z : α), x ∈ s → y ∈ s → z ∈ s → P x y → P y z → P x z\nh'' : ∀ (x y : α), x ∈ s → y ∈ s → P x y → P y x\nx y : α\nhx : x ∈ s\nhy : y ∈ s\nz : α\nhz : z ∈ s\... | [
"α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nhs : IsPreconnected s\nP : α → α → Prop\nh : ∀ x ∈ s, ∀ᶠ (y : α) in 𝓝[s] x, P x y\nh' : ∀ (x y z : α), x ∈ s → y ∈ s → z ∈ s → P x y → P y z → P x z\nh'' : ∀ (x y : α), x ∈ s → y ∈ s → P x y → P y x\nx y : α\nhx : x ∈ s\nhy : y ∈ s\nz : α\nhz : z ∈ s\na : α\nha :... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Connected.Basic | {
"line": 453,
"column": 2
} | {
"line": 454,
"column": 73
} | {
"line": 456,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set α\nt : Set β\nhs : IsPreconnected s\nht : IsPreconnected t\na₁ : α\nb₁ : β\nha₁ : (a₁, b₁).1 ∈ s\nhb₁ : (a₁, b₁).2 ∈ t\na₂ : α\nb₂ : β\nha₂ : (a₂, b₂).1 ∈ s\nhb₂ : (a₂, b₂).2 ∈ t\n⊢ IsPreconnected (P... | [] | · exact (ht.image _ (by fun_prop)).union (a₁, b₂) ⟨b₂, hb₂, rfl⟩
⟨a₁, ha₁, rfl⟩ (hs.image _ (Continuous.prodMk_left _).continuousOn) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Compactness.Lindelof | {
"line": 405,
"column": 6
} | {
"line": 405,
"column": 17
} | {
"line": 406,
"column": 6
} | [
{
"pp": "case mp.refine_2\nX : Type u\nι : Type u_1\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsLindelof (b i)\nY : Type u\nf' : Y → ι\nh₁ : IsLindelof (⋃ i, (b ∘ f') i)\nh₂ : IsOpen[inst✝] (⋃ i, (b ∘ f') i)\nhf' : ∀ (i : Y), b (f' i) = (b ∘ f') i\nt : Set Y... | [
"case mp.refine_2\nX : Type u\nι : Type u_1\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsLindelof (b i)\nY : Type u\nf' : Y → ι\nh₁ : IsLindelof (⋃ i, (b ∘ f') i)\nh₂ : IsOpen[inst✝] (⋃ i, (b ∘ f') i)\nhf' : ∀ (i : Y), b (f' i) = (b ∘ f') i\nt : Set Y\nht : t.Cou... | rintro i hi | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.Separation.Regular | {
"line": 643,
"column": 4
} | {
"line": 643,
"column": 19
} | {
"line": 644,
"column": 4
} | [
{
"pp": "case refine_3\nX : Type u_1\ninst✝ : TopologicalSpace X\ns t : Set X\nhSt : Disjoint (closure[inst✝] s) t\nhsT : Disjoint s (closure[inst✝] t)\ne : Set X := (closure[inst✝] s ∩ closure[inst✝] t)ᶜ\nh : NormalSpace ↑e\nhe : IsOpen[inst✝] e\nhst : Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻... | [
"case refine_3\nX : Type u_1\ninst✝ : TopologicalSpace X\ns t : Set X\nhSt : Disjoint (closure[inst✝] s) t\nhsT : Disjoint s (closure[inst✝] t)\ne : Set X := (closure[inst✝] s ∩ closure[inst✝] t)ᶜ\nh : NormalSpace ↑e\nhe : IsOpen[inst✝] e\nhst : Disjoint (Subtype.val ⁻¹' closure[inst✝] s) (Subtype.val ⁻¹' closure[i... | intro x hxU hxV | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.DenseEmbedding | {
"line": 221,
"column": 42
} | {
"line": 221,
"column": 77
} | {
"line": 221,
"column": 77
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ni : α → β\nc : Continuous[inst✝¹, inst✝] i\ndense : ∀ (x : β), x ∈ closure[inst✝] (range i)\nH : ∀ (a : α), ∀ s ∈ 𝓝 a, ∃ t ∈ 𝓝 (i a), ∀ (b : α), i b ∈ t → b ∈ s\na : α\n⊢ comap i (𝓝 (i a)) ≤ 𝓝 a",
"ppTerm": "?m... | [] | by simpa [Filter.le_def] using! H a | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactness.Lindelof | {
"line": 752,
"column": 2
} | {
"line": 752,
"column": 78
} | {
"line": 753,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : HereditarilyLindelofSpace X\nι : Type u_2\nU : ι → Set X\nh : ∀ (i : ι), IsOpen[inst✝¹] (U i)\nthis : IsLindelof (⋃ i, U i)\n⊢ ∃ t, t.Countable ∧ ⋃ i ∈ t, U i = ⋃ i, U i",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Partial... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : HereditarilyLindelofSpace X\nι : Type u_2\nU : ι → Set X\nh : ∀ (i : ι), IsOpen[inst✝¹] (U i)\nthis : IsLindelof (⋃ i, U i)\nt : Set ι\nhtc : t.Countable\nhtu : ⋃ i, U i ⊆ ⋃ i ∈ t, U i\n⊢ ∃ t, t.Countable ∧ ⋃ i ∈ t, U i = ⋃ i, U i"
] | rcases this.elim_countable_subcover U h (Eq.subset rfl) with ⟨t, ⟨htc, htu⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Separation.Regular | {
"line": 736,
"column": 12
} | {
"line": 736,
"column": 14
} | {
"line": 736,
"column": 15
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\nx : X\nhs : IsClosed[inst✝²] (⋂ s, ↑s)\na b : Set X\n⊢ IsClosed[inst✝²] a → IsClosed[inst✝²] b → ⋂ s, ↑s ⊆ a ∪ b → Disjoint a b → ⋂ s, ↑s ⊆ a ∨ ⋂ s, ↑s ⊆ b",
"ppTerm": "?m.69",
"assigned": true,
"usedCons... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\nx : X\nhs : IsClosed[inst✝²] (⋂ s, ↑s)\na b : Set X\nha : IsClosed[inst✝²] a\n⊢ IsClosed[inst✝²] b → ⋂ s, ↑s ⊆ a ∪ b → Disjoint a b → ⋂ s, ↑s ⊆ a ∨ ⋂ s, ↑s ⊆ b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 66,
"column": 14
} | {
"line": 66,
"column": 28
} | {
"line": 66,
"column": 29
} | [
{
"pp": "α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\nl : Filter β\nf : β → α\n⊢ (map f l).NeBot ∧ map f l ×ˢ map f l ≤ 𝓤 α ↔ l.NeBot ∧ Tendsto (fun p ↦ (f p.1, f p.2)) (l ×ˢ l) (𝓤 α)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SProd.sprod",
"co... | [
"α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\nl : Filter β\nf : β → α\n⊢ l.NeBot ∧ map f l ×ˢ map f l ≤ 𝓤 α ↔ l.NeBot ∧ Tendsto (fun p ↦ (f p.1, f p.2)) (l ×ˢ l) (𝓤 α)"
] | map_neBot_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Defs | {
"line": 472,
"column": 63
} | {
"line": 479,
"column": 35
} | {
"line": 481,
"column": 0
} | [
{
"pp": "β : Type ub\nV W M : SetRel β β\ninst✝ : W.IsSymm\np : β × β\n⊢ p ∈ V ○ M ○ W ↔ (ball p.1 V ×ˢ ball p.2 W ∩ M).Nonempty",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SetRel",
"SProd.sprod",
"congrArg",
"Membership.m... | [] | by
obtain ⟨x, y⟩ := p
constructor
· rintro ⟨z, ⟨w, hpw, hwz⟩, hzy⟩
exact ⟨(w, z), ⟨hpw, by rwa [mem_ball_symmetry]⟩, hwz⟩
· rintro ⟨⟨w, z⟩, ⟨w_in, z_in⟩, hwz⟩
rw [mem_ball_symmetry] at z_in
exact ⟨z, ⟨w, w_in, hwz⟩, z_in⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 290,
"column": 12
} | {
"line": 290,
"column": 14
} | {
"line": 290,
"column": 15
} | [
{
"pp": "case h\nα : Type u\nuniformSpace : UniformSpace α\nu : ℕ → α\nk : ℕ\nh✝ : ∀ V ∈ 𝓤 α, ∃ N, ∀ k_1 ≥ N, ∀ l ≥ N, (u (k_1 + k), u (l + k)) ∈ V\nV : Set (α × α)\nmV : V ∈ 𝓤 α\nN : ℕ\nh : ∀ k_1 ≥ N, ∀ l ≥ N, (u (k_1 + k), u (l + k)) ∈ V\na : ℕ\n⊢ a ≥ N + k → ∀ l ≥ N + k, (u a, u l) ∈ V",
"ppTerm": "?h"... | [
"case h\nα : Type u\nuniformSpace : UniformSpace α\nu : ℕ → α\nk : ℕ\nh✝ : ∀ V ∈ 𝓤 α, ∃ N, ∀ k_1 ≥ N, ∀ l ≥ N, (u (k_1 + k), u (l + k)) ∈ V\nV : Set (α × α)\nmV : V ∈ 𝓤 α\nN : ℕ\nh : ∀ k_1 ≥ N, ∀ l ≥ N, (u (k_1 + k), u (l + k)) ∈ V\na : ℕ\nha : a ≥ N + k\n⊢ ∀ l ≥ N + k, (u a, u l) ∈ V"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.UniformSpace.Basic | {
"line": 718,
"column": 2
} | {
"line": 718,
"column": 70
} | {
"line": 720,
"column": 0
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\n⊢ comap (fun p ↦ (MulOpposite.op p.1, MulOpposite.op p.2)) (𝓤 αᵐᵒᵖ) = 𝓤 α",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"MulOpposite",
"uniformity",
"Function.comp",
"id",
"ins... | [] | simpa [uniformity_mulOpposite, comap_comap, (· ∘ ·)] using! comap_id | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Topology.UniformSpace.Basic | {
"line": 718,
"column": 2
} | {
"line": 718,
"column": 70
} | {
"line": 720,
"column": 0
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\n⊢ comap (fun p ↦ (MulOpposite.op p.1, MulOpposite.op p.2)) (𝓤 αᵐᵒᵖ) = 𝓤 α",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"MulOpposite",
"uniformity",
"Function.comp",
"id",
"ins... | [] | simpa [uniformity_mulOpposite, comap_comap, (· ∘ ·)] using! comap_id | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Basic | {
"line": 718,
"column": 2
} | {
"line": 718,
"column": 70
} | {
"line": 720,
"column": 0
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\n⊢ comap (fun p ↦ (MulOpposite.op p.1, MulOpposite.op p.2)) (𝓤 αᵐᵒᵖ) = 𝓤 α",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"MulOpposite",
"uniformity",
"Function.comp",
"id",
"ins... | [] | simpa [uniformity_mulOpposite, comap_comap, (· ∘ ·)] using! comap_id | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 779,
"column": 10
} | {
"line": 779,
"column": 12
} | {
"line": 780,
"column": 2
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\n⊢ a ∈ 𝓤 α → ∃ t, t.Finite ∧ range s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ a}",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"uniformity",
"Membership.mem",
... | [
"α : Type u\nuniformSpace : UniformSpace α\ns : ℕ → α\nhs : CauchySeq s\na : Set (α × α)\nha : a ∈ 𝓤 α\n⊢ ∃ t, t.Finite ∧ range s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ a}"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 547,
"column": 17
} | {
"line": 547,
"column": 19
} | {
"line": 547,
"column": 20
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : UniformSpace β\nF : ι → α → β\ns : Set α\np : Filter ι\nι' : Type u_5\nα' : Type u_6\nβ' : Type u_7\ninst✝ : UniformSpace β'\nF' : ι' → α' → β'\np' : Filter ι'\ns' : Set α'\nh : UniformCauchySeqOn F p s\nh' : UniformCauchySeqOn F' p' s'\nu : Set ((β × ... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : UniformSpace β\nF : ι → α → β\ns : Set α\np : Filter ι\nι' : Type u_5\nα' : Type u_6\nβ' : Type u_7\ninst✝ : UniformSpace β'\nF' : ι' → α' → β'\np' : Filter ι'\ns' : Set α'\nh : UniformCauchySeqOn F p s\nh' : UniformCauchySeqOn F' p' s'\nu : Set ((β × β') × β × β'... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 466,
"column": 97
} | {
"line": 468,
"column": 65
} | {
"line": 470,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : IsUniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\na : α\n⊢ Tendsto f (comap e (𝓝 a)) (𝓝 (⋯.extend f a))",
... | [] | by
simpa only [IsDenseInducing.extend] using
tendsto_nhds_limUnder (uniformly_extend_exists h_e ‹_› h_f _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 56
} | {
"line": 99,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\np : Filter ι\nh : TendstoLocallyUniformlyOn F f p s\nh' : s' ⊆ s\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nx : α\nhx : x ∈ s'\nt : Set α\nht : t ∈ 𝓝[s] x\nH : ∀ᶠ (n : ι) in p, ... | [] | exact ⟨t, nhdsWithin_mono x h' ht, H.mono fun n => id⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 556,
"column": 85
} | {
"line": 557,
"column": 79
} | {
"line": 559,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nX : Type u_3\nα' : Type u_7\ntX : TopologicalSpace X\nu : κ → UniformSpace α'\nF : ι → X → α'\nS : Set X\n⊢ EquicontinuousOn F S ↔ ∀ (k : κ), EquicontinuousOn F S",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"UniformSpace",
"Eq.mpr",
... | [] | by
simp_rw [EquicontinuousOn, equicontinuousWithinAt_iInf_rng, @forall_comm _ κ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.NAry | {
"line": 132,
"column": 63
} | {
"line": 132,
"column": 96
} | {
"line": 134,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\nm : α → β → γ\na : α\nb : β\n⊢ map₂ m (pure a) (pure b) = pure (m a b)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"congrArg",
"Filter.map",
"id",
"Filter.instPure",
"Eq.... | [] | by rw [map₂_pure_right, map_pure] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Pointwise | {
"line": 589,
"column": 2
} | {
"line": 592,
"column": 71
} | {
"line": 594,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Monoid α\nf : Filter α\nhf : 1 ≤ f\n⊢ f * ⊤ = ⊤",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Set.univ_subset_iff",
"Eq.mpr",
"_private.Mathlib.Order.Filter.Pointwise.0.Filter.mul_top_of_one_le._simp_1_2",
... | [] | refine top_le_iff.1 fun s => ?_
simp only [mem_mul, mem_top, exists_eq_left]
rintro ⟨t, ht, hs⟩
rwa [mul_univ_of_one_mem (mem_one.1 <| hf ht), univ_subset_iff] at hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Pointwise | {
"line": 589,
"column": 2
} | {
"line": 592,
"column": 71
} | {
"line": 594,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Monoid α\nf : Filter α\nhf : 1 ≤ f\n⊢ f * ⊤ = ⊤",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Set.univ_subset_iff",
"Eq.mpr",
"_private.Mathlib.Order.Filter.Pointwise.0.Filter.mul_top_of_one_le._simp_1_2",
... | [] | refine top_le_iff.1 fun s => ?_
simp only [mem_mul, mem_top, exists_eq_left]
rintro ⟨t, ht, hs⟩
rwa [mul_univ_of_one_mem (mem_one.1 <| hf ht), univ_subset_iff] at hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 640,
"column": 63
} | {
"line": 643,
"column": 79
} | {
"line": 645,
"column": 0
} | [
{
"pp": "G₀ : Type u_2\ninst✝³ : GroupWithZero G₀\ninst✝² : PartialOrder G₀\ninst✝¹ : PosMulReflectLT G₀\ninst✝ : MulPosReflectLT G₀\na : G₀\nha : 0 < a\n⊢ (Ioo 0 a)⁻¹ = Ioi a⁻¹",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.ext",
"GroupWithZero.toMonoidW... | [] | by
ext x
exact ⟨fun h ↦ inv_lt_of_inv_lt₀ (inv_pos.1 h.1) h.2,
fun h ↦ ⟨inv_pos.2 <| (inv_pos.2 ha).trans h, inv_lt_of_inv_lt₀ ha h⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Pointwise | {
"line": 1135,
"column": 53
} | {
"line": 1139,
"column": 31
} | {
"line": 1141,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : Zero α\ninst✝¹ : Zero β\ninst✝ : SMulWithZero α β\ng : Filter β\n⊢ 0 • g ≤ 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Eq.mpr",
"instHSMul",
"outParam",
"congrArg",
... | [] | by
refine fun s hs => mem_smul_filter.2 ?_
convert! @univ_mem _ g
refine eq_univ_iff_forall.2 fun a => ?_
rwa [mem_preimage, zero_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Maps.Proper.Basic | {
"line": 106,
"column": 59
} | {
"line": 106,
"column": 73
} | {
"line": 106,
"column": 74
} | [
{
"pp": "case mpr\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nx✝ : Continuous[inst✝¹, inst✝] f\nH : ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x\n⊢ ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, (map f (comap f (𝓝 y) ⊓ ℱ)).NeBot → ∃ x, f x =... | [
"case mpr\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nx✝ : Continuous[inst✝¹, inst✝] f\nH : ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (𝓝 y) → ∃ x, f x = y ∧ ↑𝒰 ≤ 𝓝 x\n⊢ ∀ ⦃ℱ : Filter X⦄ ⦃y : Y⦄, (comap f (𝓝 y) ⊓ ℱ).NeBot → ∃ x, f x = y ∧ (𝓝 x ⊓ ℱ).NeBo... | map_neBot_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.Monoid | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 94
} | {
"line": 510,
"column": 0
} | [
{
"pp": "M : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : MulZeroClass M\ninst✝ : ContinuousMul M\nl : Filter (M × M)\nhl : Disjoint l (cocompact (M × M))\nh'l : l ≤ (𝓝 0).coprod (𝓝 0)\n⊢ Tendsto (fun x ↦ x.1 * x.2) l (𝓝 0)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Iff.m... | [] | simpa [inf_eq_right.mpr h'l] using tendsto_mul_coprod_nhds_zero_inf_of_disjoint_cocompact hl | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Algebra.Monoid | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 94
} | {
"line": 510,
"column": 0
} | [
{
"pp": "M : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : MulZeroClass M\ninst✝ : ContinuousMul M\nl : Filter (M × M)\nhl : Disjoint l (cocompact (M × M))\nh'l : l ≤ (𝓝 0).coprod (𝓝 0)\n⊢ Tendsto (fun x ↦ x.1 * x.2) l (𝓝 0)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Iff.m... | [] | simpa [inf_eq_right.mpr h'l] using tendsto_mul_coprod_nhds_zero_inf_of_disjoint_cocompact hl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Monoid | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 94
} | {
"line": 510,
"column": 0
} | [
{
"pp": "M : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : MulZeroClass M\ninst✝ : ContinuousMul M\nl : Filter (M × M)\nhl : Disjoint l (cocompact (M × M))\nh'l : l ≤ (𝓝 0).coprod (𝓝 0)\n⊢ Tendsto (fun x ↦ x.1 * x.2) l (𝓝 0)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Iff.m... | [] | simpa [inf_eq_right.mpr h'l] using tendsto_mul_coprod_nhds_zero_inf_of_disjoint_cocompact hl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Monoid | {
"line": 1005,
"column": 51
} | {
"line": 1007,
"column": 88
} | {
"line": 1009,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nX : Type u_5\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace M\ninst✝¹ : CommMonoid M\ninst✝ : ContinuousMul M\nf : ι → X → M\np : ι → Prop\nhc : ∀ (i : ι), p i → Continuous[inst✝³, inst✝²] (f i)\nhf : LocallyFinite fun i ↦ mulSupport (f i)\n⊢ Continuous[inst✝³, inst... | [] | by
simp only [← finprod_subtype_eq_finprod_cond]
exact continuous_finprod (fun i => hc i i.2) (hf.comp_injective Subtype.coe_injective) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Group.Pointwise | {
"line": 312,
"column": 79
} | {
"line": 323,
"column": 16
} | {
"line": 325,
"column": 0
} | [
{
"pp": "G : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\n⊢ RegularSpace G",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Set.instSProd",
"MulOne.toOne",... | [] | by
refine .of_exists_mem_nhds_isClosed_subset fun a s hs ↦ ?_
have : Tendsto (fun p : G × G => p.1 * p.2) (𝓝 (a, 1)) (𝓝 a) :=
continuous_mul.tendsto' _ _ (mul_one a)
rcases mem_nhds_prod_iff.mp (this hs) with ⟨U, hU, V, hV, hUV⟩
rw [← image_subset_iff, image_prod] at hUV
refine ⟨closure U, mem_of_supers... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 746,
"column": 8
} | {
"line": 746,
"column": 34
} | {
"line": 746,
"column": 34
} | [
{
"pp": "G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\ng h : G\nhg : g ∈ connectedComponent 1\nhh : h ∈ connectedComponent 1\n⊢ g ∈ (fun x ↦ g * x) '' connectedComponent h",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"M... | [
"G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\ng h : G\nhg : g ∈ connectedComponent 1\nhh : h ∈ connectedComponent 1\n⊢ g ∈ (fun x ↦ g * x) '' connectedComponent 1"
] | ← connectedComponent_eq hh | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1234,
"column": 35
} | {
"line": 1244,
"column": 42
} | {
"line": 1244,
"column": 43
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nS : Subgroup G\nhS : Tendsto (⇑S.subtype) cofinite (cocompact G)\n⊢ ∀ {K L : Set G}, IsCompact K → IsCompact L → {γ | ((fun x ↦ γ • x) '' K ∩ L).Nonempty}.Finite",
"ppTerm": "?m.24",
"assigned": true,
"... | [] | by
intro K L hK hL
have : Continuous fun p : G × G => (p.1⁻¹, p.2) := continuous_inv.prodMap continuous_id
have H : Set.Finite _ :=
hS ((hK.prod hL).image (continuous_mul.comp this)).compl_mem_cocompact
simp only [preimage_compl, compl_compl, coe_subtype, comp_apply] at H
apply Fin... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1333,
"column": 4
} | {
"line": 1333,
"column": 17
} | {
"line": 1335,
"column": 0
} | [
{
"pp": "case refine_2\nG : Type w\nH : Type x\nα : Type u\nβ : Type v\ninst✝⁴ : TopologicalSpace G\ninst✝³ : Group G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : SeparableSpace G\ninst✝ : WeaklyLocallyCompactSpace G\nL : Set G\nhLc : IsCompact L\nhL1 : L ∈ 𝓝 1\nx : G\nn : ℕ\nhn : denseSeq G n ∈ (fun y ↦ x * y) ⁻¹... | [] | exact ⟨n, hn⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.ContinuousMonoidHom | {
"line": 250,
"column": 15
} | {
"line": 250,
"column": 49
} | {
"line": 251,
"column": 2
} | [
{
"pp": "A : Type u_2\nE : Type u_6\ninst✝⁴ : Monoid A\ninst✝³ : TopologicalSpace A\ninst✝² : CommMonoid E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousMul E\nf : A →ₜ* E\na : A\n⊢ (f ^ 0) a = f a ^ 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [] | rw [pow_zero, pow_zero, one_toFun] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.ContinuousMonoidHom | {
"line": 250,
"column": 15
} | {
"line": 250,
"column": 49
} | {
"line": 251,
"column": 2
} | [
{
"pp": "A : Type u_2\nE : Type u_6\ninst✝⁴ : Monoid A\ninst✝³ : TopologicalSpace A\ninst✝² : CommMonoid E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousMul E\nf : A →ₜ* E\na : A\n⊢ (f ^ 0) a = f a ^ 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [] | rw [pow_zero, pow_zero, one_toFun] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.ContinuousMonoidHom | {
"line": 250,
"column": 15
} | {
"line": 250,
"column": 49
} | {
"line": 251,
"column": 2
} | [
{
"pp": "A : Type u_2\nE : Type u_6\ninst✝⁴ : Monoid A\ninst✝³ : TopologicalSpace A\ninst✝² : CommMonoid E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousMul E\nf : A →ₜ* E\na : A\n⊢ (f ^ 0) a = f a ^ 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [] | rw [pow_zero, pow_zero, one_toFun] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 432,
"column": 2
} | {
"line": 433,
"column": 82
} | {
"line": 435,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝¹ : Group G\nu : UniformSpace G\ninst✝ : IsTopologicalGroup G\nF : ι → α → G\nf : α → G\np : Filter ι\ns : Set α\nhu : rightUniformSpace G = u\n⊢ TendstoUniformlyOn F f p s ↔ ∀ u_1 ∈ 𝓝 1, ∀ᶠ (i : ι) in p, ∀ a ∈ s, F i a * (f a)⁻¹ ∈ u_1",
"ppTerm": "?m... | [] | exact hu ▸ ⟨fun h u hu => h _ ⟨u, hu, fun _ => id⟩,
fun h _ ⟨u, hu, hv⟩ => mem_of_superset (h u hu) fun _ hi a ha => hv (hi a ha)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 545,
"column": 4
} | {
"line": 545,
"column": 37
} | {
"line": 546,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nG : Type u_5\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : AddCommGroup α\ninst✝¹⁰ : IsTopologicalAddGroup α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : AddCommGroup β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : AddCommGroup γ\ninst✝⁵ : IsTopologicalAddGroup γ\n... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nG : Type u_5\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : AddCommGroup α\ninst✝¹⁰ : IsTopologicalAddGroup α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : AddCommGroup β\ninst✝⁷ : TopologicalSpace γ\ninst✝⁶ : AddCommGroup γ\ninst✝⁵ : IsTopologicalAddGroup γ\ninst✝⁴ : Top... | rcases U_in with ⟨U₁, U₁_in, HU₁⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.IsUniformGroup.Basic | {
"line": 660,
"column": 8
} | {
"line": 660,
"column": 29
} | {
"line": 660,
"column": 30
} | [
{
"pp": "G : Type u\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : IsTopologicalGroup G\ninst✝² : FirstCountableTopology G\nN : Subgroup G\ninst✝¹ : N.Normal\ninst✝ : CompleteSpace G\nthis✝¹ : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)\nthis✝ : UniformSpace G := IsTopological... | [
"G : Type u\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : IsTopologicalGroup G\ninst✝² : FirstCountableTopology G\nN : Subgroup G\ninst✝¹ : N.Normal\ninst✝ : CompleteSpace G\nthis✝¹ : UniformSpace (G ⧸ N) := IsTopologicalGroup.rightUniformSpace (G ⧸ N)\nthis✝ : UniformSpace G := IsTopologicalGroup.rightU... | QuotientGroup.mk_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.LeftRightNhds | {
"line": 155,
"column": 10
} | {
"line": 155,
"column": 12
} | {
"line": 155,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nt : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}\ny : α → α\nhy : ∀ x ∈ t, y x > x\nh'y : ∀ x ∈ t, s ∩ Ioo x (y x) = ∅\na : α\n⊢ a ∈ t → ∀ ⦃y_1 : α⦄, y_1 ∈... | [
"α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nt : Set α := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥ ∧ ¬IsTop x}\ny : α → α\nhy : ∀ x ∈ t, y x > x\nh'y : ∀ x ∈ t, s ∩ Ioo x (y x) = ∅\na : α\nha : a ∈ t\n⊢ ∀ ⦃y_1 : α⦄, y_1 ∈ t → a ≠... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Order.LeftRightNhds | {
"line": 271,
"column": 2
} | {
"line": 272,
"column": 37
} | {
"line": 273,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na b : α\nhab : a < b\ns : Set α\ntfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a\n⊢ [s ∈ 𝓝[≥] a, s ∈ 𝓝[Icc a b] a, s ∈ 𝓝[Ico a b] a, ∃ u ∈ Ioc a b, Ico a u ⊆ s, ∃ u ∈ Ioi a, Ico a u ⊆ s].TFAE",
"ppTerm": "... | [
"α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na b : α\nhab : a < b\ns : Set α\ntfae_1_iff_2 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Icc a b] a\ntfae_1_iff_3 : s ∈ 𝓝[≥] a ↔ s ∈ 𝓝[Ico a b] a\n⊢ [s ∈ 𝓝[≥] a, s ∈ 𝓝[Icc a b] a, s ∈ 𝓝[Ico a b] a, ∃ u ∈ Ioc a b, Ico a u ⊆ s, ∃ u ∈ Ioi a,... | tfae_have 1 ↔ 3 := by
rw [nhdsWithin_Ico_eq_nhdsGE hab] | Mathlib.Tactic.TFAE._aux_Mathlib_Tactic_TFAE___macroRules_Mathlib_Tactic_TFAE_tfaeHave_1 | Mathlib.Tactic.TFAE.tfaeHave |
Mathlib.Topology.Algebra.Order.Field | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 61
} | {
"line": 285,
"column": 0
} | [
{
"pp": "case inr.inr.inr\n𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nn : ℕ\nc d : 𝕜\nhn : n ≠ 0\nhc : 0 < c\nthis : Tendsto (fun x ↦ c * x ^ n) atTop atTop\n⊢ Tendsto (fun x ↦ c * x ^ n) atTop (𝓝 d) ↔ (c ... | [] | simp [not_tendsto_nhds_of_tendsto_atTop this, hc.ne', hn] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 806,
"column": 84
} | {
"line": 807,
"column": 21
} | {
"line": 809,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n⊢ (PartialEquiv.refl α).prod (PartialEquiv.refl β) = PartialEquiv.refl (α × β)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.ext",
"congrArg",
"PartialEquiv.prod",
"PartialEquiv",
"Set.mem_univ._simp_1",
"Set.un... | [] | by
ext ⟨x, y⟩ <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.Basic | {
"line": 202,
"column": 10
} | {
"line": 202,
"column": 15
} | {
"line": 202,
"column": 16
} | [
{
"pp": "case refine_2\nα : Type u\nts : TopologicalSpace α\ninst✝ : Preorder α\nc : Set α\nh : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nk : Set (Set α)\n⊢ k.Finite →\n k ⊆ {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a} →\n ∃ f ⊆ c, ∃ x ⊆ c, f.Finite ∧ x.Finite ∧ ⋂₀ k = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ x, ... | [
"case refine_2\nα : Type u\nts : TopologicalSpace α\ninst✝ : Preorder α\nc : Set α\nh : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nk : Set (Set α)\nk_fin : k.Finite\n⊢ k ⊆ {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a} →\n ∃ f ⊆ c, ∃ x ⊆ c, f.Finite ∧ x.Finite ∧ ⋂₀ k = (⋂ a ∈ f, Ioi a) ∩ ⋂ a ∈ x, Iio a"
] | k_fin | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Module.CharacterModule | {
"line": 86,
"column": 46
} | {
"line": 88,
"column": 18
} | {
"line": 90,
"column": 0
} | [
{
"pp": "R : Type uR\ninst✝⁴ : CommRing R\nA : Type uA\ninst✝³ : AddCommGroup A\nB : Type uB\ninst✝² : AddCommGroup B\ninst✝¹ : Module R A\ninst✝ : Module R B\n⊢ dual 0 = 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"SemilinearMapClass.distribMulActionSemi... | [] | by
ext f
exact map_zero f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.Basic | {
"line": 588,
"column": 10
} | {
"line": 588,
"column": 12
} | {
"line": 589,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\na✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ x ∈ s, x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\n⊢ IsOpen[inst✝³] a → {x | x ∈ s ∧ x ∈ a... | [
"α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\na✝ : Nontrivial α\ns : Set α := {x | ∃ y, x ⋖ y}\ny : α → α\nhy : ∀ x ∈ s, x ⋖ y x\nHy : ∀ (x z : α), x ∈ s → z < y x → z ≤ x\na : Set α\nha : IsOpen[inst✝³] a\n⊢ {x | x ∈ s ∧ x ∈ a ∧ y x ∉... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 627,
"column": 72
} | {
"line": 627,
"column": 85
} | {
"line": 628,
"column": 4
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝² : Field 𝕜\np : 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhp : Fact (0 < p)\nu : AddCircle p\nn : ℕ\nh : 0 < n\nk : 𝕜\na : ℤ\nha : ↑(a / ↑n) * (↑n * p / ↑n) + (a % ↑n) • p / ↑n = k\nh0 : ↑n ≠ 0\n⊢ ↑(↑(a.natMod ↑n) / ↑n * p) = ↑k",
"ppTerm": "... | [
"case refine_2\n𝕜 : Type u_1\ninst✝² : Field 𝕜\np : 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhp : Fact (0 < p)\nu : AddCircle p\nn : ℕ\nh : 0 < n\nk : 𝕜\na : ℤ\nha : ↑(a / ↑n) * (p * ↑n / ↑n) + (a % ↑n) • p / ↑n = k\nh0 : ↑n ≠ 0\n⊢ ↑(↑(a.natMod ↑n) / ↑n * p) = ↑k"
] | mul_comm _ p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 673,
"column": 8
} | {
"line": 673,
"column": 59
} | {
"line": 674,
"column": 8
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\ninst✝² : Field 𝕜\np q : 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhp : Fact (0 < p)\nn : ℕ\nhn : 0 < n\n⊢ Bijective fun m ↦ ⟨↑(↑↑m / ↑n * p), ⋯⟩",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"instHDiv",
... | [
"case refine_1\n𝕜 : Type u_1\nB : Type u_2\ninst✝² : Field 𝕜\np q : 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhp : Fact (0 < p)\nn : ℕ\nhn : 0 < n\nm₁ m₂ : ↑{m | m < n ∧ m.gcd n = 1}\nh : (fun m ↦ ⟨↑(↑↑m / ↑n * p), ⋯⟩) m₁ = (fun m ↦ ⟨↑(↑↑m / ↑n * p), ⋯⟩) m₂\n⊢ ↑m₁ = ↑m₂",
"case refine_2\n𝕜 :... | refine ⟨fun m₁ m₂ h => Subtype.ext ?_, fun u => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Category.ModuleCat.Subobject | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 66
} | {
"line": 83,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nM N : ModuleCat R\nf : M ⟶ N\nx : ↥(Hom.hom f).ker\n⊢ (ConcreteCategory.hom (kernelSubobject f).arrow) (toKernelSubobject x) = ↑x",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"Submodule",
"CategoryTh... | [] | simp [toKernelSubobject, -hom_comp, ← CategoryTheory.comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.ModuleCat.Subobject | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 66
} | {
"line": 83,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nM N : ModuleCat R\nf : M ⟶ N\nx : ↥(Hom.hom f).ker\n⊢ (ConcreteCategory.hom (kernelSubobject f).arrow) (toKernelSubobject x) = ↑x",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"Submodule",
"CategoryTh... | [] | simp [toKernelSubobject, -hom_comp, ← CategoryTheory.comp_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Subobject | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 66
} | {
"line": 83,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nM N : ModuleCat R\nf : M ⟶ N\nx : ↥(Hom.hom f).ker\n⊢ (ConcreteCategory.hom (kernelSubobject f).arrow) (toKernelSubobject x) = ↑x",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"CategoryTheory.Category.assoc",
"Submodule",
"CategoryTh... | [] | simp [toKernelSubobject, -hom_comp, ← CategoryTheory.comp_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Refinements | {
"line": 102,
"column": 17
} | {
"line": 102,
"column": 19
} | {
"line": 103,
"column": 4
} | [
{
"pp": "case mp\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : ShortComplex C\nhS : ∀ ⦃A : C⦄ (y : A ⟶ S.cycles), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ S.toCycles\nA : C\na : A ⟶ S.X₂\n⊢ a ≫ S.g = 0 → ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ a = x₁ ≫ S.f",
"ppTerm": "?mp",
"assigned": ... | [
"case mp\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : ShortComplex C\nhS : ∀ ⦃A : C⦄ (y : A ⟶ S.cycles), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ S.toCycles\nA : C\na : A ⟶ S.X₂\nha : a ≫ S.g = 0\n⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x₁, π ≫ a = x₁ ≫ S.f"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 365,
"column": 36
} | {
"line": 365,
"column": 69
} | {
"line": 367,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u\ninst✝² : Category.{v, u} D\ninst✝¹ : Preadditive D\ninst✝ : Linear R D\nF : C ⥤ D\nX Y : Free R C\nf : X ⟶ Y\nr : R\n⊢ (sum f fun i a ↦ (r * a) • F.map i) = r • sum f fun f' r ↦ r • F.map f'",
"ppTerm": "?m.50",
... | [] | simp [Finsupp.smul_sum, mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 365,
"column": 36
} | {
"line": 365,
"column": 69
} | {
"line": 367,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u\ninst✝² : Category.{v, u} D\ninst✝¹ : Preadditive D\ninst✝ : Linear R D\nF : C ⥤ D\nX Y : Free R C\nf : X ⟶ Y\nr : R\n⊢ ∀ (i : X ⟶ Y), 0 • F.map i = 0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants"... | [] | simp [Finsupp.smul_sum, mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer | {
"line": 256,
"column": 62
} | {
"line": 256,
"column": 74
} | {
"line": 256,
"column": 74
} | [
{
"pp": "J : MultispanShape\na : J.R\nb : J.L\n⊢ ∀ (a : right a ⟶ left b), False",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"False",
"CategoryTheory.Limits.WalkingMultispan",
"CategoryTheory.Limits.MultispanShape.L",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | by rintro ⟨⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 71,
"column": 72
} | {
"line": 71,
"column": 87
} | {
"line": 72,
"column": 8
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝ : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\nj : J\n⊢ (𝟭 C).map (c.π.app j) ≫ T.η.app (D.obj j).A ≫ (D.obj j).a = 𝟙 c.pt ≫ c.π.app j",
"ppTerm": "?m.110",
"assigned": true,
"usedCo... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝ : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\nj : J\n⊢ c.π.app j ≫ T.η.app (D.obj j).A ≫ (D.obj j).a = 𝟙 c.pt ≫ c.π.app j"
] | Functor.id_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 71,
"column": 6
} | {
"line": 73,
"column": 10
} | {
"line": 74,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝ : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\nj : J\n⊢ (T.η.app c.pt ≫ t.lift (newCone D c)) ≫ c.π.app j = 𝟙 c.pt ≫ c.π.app j",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": ... | [] | rw [Category.assoc, t.fac, newCone_π_app, ← T.η.naturality_assoc, Functor.id_map,
(D.obj j).unit]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 71,
"column": 6
} | {
"line": 73,
"column": 10
} | {
"line": 74,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝ : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\nj : J\n⊢ (T.η.app c.pt ≫ t.lift (newCone D c)) ≫ c.π.app j = 𝟙 c.pt ≫ c.π.app j",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": ... | [] | rw [Category.assoc, t.fac, newCone_π_app, ← T.η.naturality_assoc, Functor.id_map,
(D.obj j).unit]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monad.Comonadicity | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 59
} | {
"line": 359,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : Category.{v₁, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\ninst✝² : F.ReflectsIsomorphisms\ninst✝¹ : HasEqualizerOfIsCosplitPair F\ninst✝ : PreservesLimitOfIsCosplitPair F\nthis : ReflectsLimitOfIsCosplitPair F\n⊢ ComonadicLeftAdjoint F",
... | [] | apply comonadicOfHasPreservesReflectsFSplitEqualizers adj | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 89
} | {
"line": 523,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nJ : Type u\ninst✝³ : Category.{v, u} J\nT : Comonad C\nD : J ⥤ T.Coalgebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\ninst✝² : PreservesLimit (D ⋙ T.forget) T.toFunctor\ninst✝¹ : PreservesLimit ((D ⋙ T.forget) ⋙ T.toF... | [
"C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nJ : Type u\ninst✝³ : Category.{v, u} J\nT : Comonad C\nD : J ⥤ T.Coalgebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\ninst✝² : PreservesLimit (D ⋙ T.forget) T.toFunctor\ninst✝¹ : PreservesLimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.to... | rw [assoc, ← show _ = _ ≫ c.π.app j from T.ε.naturality _, ← assoc, commuting, assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 131,
"column": 12
} | {
"line": 131,
"column": 60
} | {
"line": 131,
"column": 60
} | [
{
"pp": "M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : a • x ∈ a • S\n⊢ x ∈ S",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule.instAddCommMono... | [] | by simpa using smul_mem_pointwise_smul a⁻¹ _ _ h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Pointwise | {
"line": 135,
"column": 12
} | {
"line": 135,
"column": 60
} | {
"line": 135,
"column": 60
} | [
{
"pp": "M : Type u_1\nR : Type u_3\ninst✝² : Group M\ninst✝¹ : Semiring R\ninst✝ : MulSemiringAction M R\na : M\nS : Ideal R\nx : R\nh : x ∈ a • S\n⊢ a⁻¹ • x ∈ S",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule.instAddCommMo... | [] | by simpa using smul_mem_pointwise_smul a⁻¹ _ _ h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.ComposableArrows.Basic | {
"line": 973,
"column": 4
} | {
"line": 973,
"column": 26
} | {
"line": 975,
"column": 0
} | [
{
"pp": "case succ.succ\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn : ℕ\nhn :\n ∀ (obj : Fin (n + 1) → C) (mapSucc : (i : Fin n) → obj i.castSucc ⟶ obj i.succ),\n ∃ F e, ∀ (i : ℕ) (hi : i < n), mapSucc ⟨i, hi⟩ = (e ⟨i, ⋯⟩).inv ≫ F.map' i (i + 1) ⋯ hi ≫ (e ⟨i + 1, ⋯⟩).hom\nobj : Fin (n + 1 + 1) → C\nmapS... | [] | · exact h i (by valid) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.IsPrimary | {
"line": 92,
"column": 96
} | {
"line": 97,
"column": 71
} | {
"line": 99,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nS : Submodule R M\nhI : S.IsPrimary\n⊢ (S.colon Set.univ).radical.IsPrime",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"And.imp",
"Eq.mpr",
"Sub... | [] | by
refine isPrime_iff.mpr <| hI.imp (by simp) fun h x y ⟨n, hn⟩ ↦ ?_
simp_rw [← mem_colon_iff_le, ← mem_radical_iff] at h
refine or_iff_not_imp_left.mpr fun hx ↦ ⟨n, ?_⟩
simp only [mul_pow, mem_colon, Set.mem_univ, true_imp_iff, mul_smul] at hn ⊢
exact fun p ↦ (h (hn p)).resolve_right (mt mem_radical_of_pow_m... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.MinimalPrime.Basic | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 64
} | {
"line": 80,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\nI J : Ideal R\ninst✝ : J.IsPrime\ne : I ≤ J\nS : Set (Ideal R)ᵒᵈ := {p | IsPrime p ∧ I ≤ OrderDual.ofDual p}\np : (Ideal R)ᵒᵈ\nhJp : OrderDual.toDual J ≤ p\nhp : Maximal (fun x ↦ x ∈ S) p\n⊢ ∃ p ∈ I.minimalPrimes, p ≤ J",
"ppTerm": "?m.57",
"assigned": tru... | [] | exact ⟨p, ⟨hp.prop, fun q hq hle ↦ hp.le_of_ge hq hle⟩, hJp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.MinimalPrime.Basic | {
"line": 205,
"column": 4
} | {
"line": 209,
"column": 31
} | {
"line": 210,
"column": 4
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝⁴ : CommSemiring R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nI p : Ideal R\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\nhI : p ∈ I.minimalPrimes\nJ : Ideal S\nhJP : J ≤ P\nhJ : map (Quotient.mk (map (algebraMap R S) p)) P ∈ (map (Quotient.mk ... | [
"case refine_2\nR : Type u_1\ninst✝⁴ : CommSemiring R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nI p : Ideal R\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\nhI : p ∈ I.minimalPrimes\nJ : Ideal S\nhJP : J ≤ P\nhJ : map (Quotient.mk (map (algebraMap R S) p)) P ∈ (map (Quotient.mk (map (algebr... | have h1 : p.map (algebraMap R S) ≤ q := by
rw [Ideal.map_le_iff_le_comap]
refine hI.2 ⟨inferInstance, le_trans Ideal.le_comap_map (Ideal.comap_mono hleq.1)⟩ ?_
convert! Ideal.comap_mono hqle
exact Ideal.LiesOver.over | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
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