module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Topology.Connected.Basic | {
"line": 403,
"column": 10
} | {
"line": 403,
"column": 41
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns u v : Set α\nhu : IsOpen u\nhv : IsOpen v\nhuv : Disjoint u v\nhsuv : s ⊆ u ∪ v\nhsu : (s ∩ u).Nonempty\nhs : IsPreconnected s\nhsv : ¬Disjoint s v\n⊢ False",
"usedConstants": [
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | not_disjoint_iff_nonempty_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.OrderClosed | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 60
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : ClosedIicTopology α\ninst✝ : TopologicalSpace β\na b : α\nf : α → β\nh : a < b\n⊢ ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b",
"usedConstants": [
"Set.Ioc",
"ContinuousWithinAt",... | simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsLE h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Order.OrderClosed | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 60
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : ClosedIicTopology α\ninst✝ : TopologicalSpace β\na b : α\nf : α → β\nh : a < b\n⊢ ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b",
"usedConstants": [
"Set.Ioc",
"ContinuousWithinAt",... | simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsLE h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.OrderClosed | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 60
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : ClosedIicTopology α\ninst✝ : TopologicalSpace β\na b : α\nf : α → β\nh : a < b\n⊢ ContinuousWithinAt f (Ioc a b) b ↔ ContinuousWithinAt f (Iic b) b",
"usedConstants": [
"Set.Ioc",
"ContinuousWithinAt",... | simp only [ContinuousWithinAt, nhdsWithin_Ioc_eq_nhdsLE h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.LocallyConnected | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 60
} | [
{
"pp": "case mpr\nα : Type u\ninst✝ : TopologicalSpace α\n⊢ (∀ (x : α), ∀ U ∈ 𝓝 x, ∃ V ∈ 𝓝 x, IsPreconnected V ∧ V ⊆ U) → LocallyConnectedSpace α",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Membership.mem",
"Exists",
"nhds",
"id",
"... | rw [locallyConnectedSpace_iff_connectedComponentIn_open] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Connected.Clopen | {
"line": 58,
"column": 2
} | {
"line": 61,
"column": 42
} | [
{
"pp": "case refine_1\nι : Type u_1\nX : ι → Type u_2\nhι : Nonempty ι\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : Set ((i : ι) × X i)\nhs : IsPreconnected s\n⊢ ∃ i t, IsPreconnected t ∧ s = mk i '' t",
"usedConstants": [
"IsConnected",
"Classical.choice",
"Exists",
"instTopologi... | · obtain rfl | h := s.eq_empty_or_nonempty
· exact ⟨Classical.choice hι, ∅, isPreconnected_empty, (Set.image_empty _).symm⟩
· obtain ⟨a, t, ht, rfl⟩ := Sigma.isConnected_iff.1 ⟨h, hs⟩
exact ⟨a, t, ht.isPreconnected, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Connected.Clopen | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 71
} | [
{
"pp": "α : Type u\nι : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : PreconnectedSpace α\ns : ι → Set α\nh_nonempty : ∀ (i : ι), (s i).Nonempty\nh_disj : Pairwise (Disjoint on s)\ninst✝ : Finite ι\nh_closed : ∀ (i : ι), IsClosed (s i)\nh_Union : ⋃ i, s i = univ\ni : ι\n⊢ IsOpen (s i)",
"usedConstants": ... | rw [← isClosed_compl_iff, compl_eq_univ_diff, ← h_Union, iUnion_diff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 32
} | [
{
"pp": "case h.left.a\nX : Type u\nι : Type u_1\ninst✝ : TopologicalSpace X\ns : Set ι\nf : ι → Set X\nhs : s.Countable\nhf : ∀ i ∈ s, IsLindelof (f i)\ni : Type u\nU : i → Set X\nhU : ∀ (i : i), IsOpen (U i)\nhUcover : ⋃ i ∈ s, f i ⊆ ⋃ i, U i\nhiU : ∀ i_1 ∈ s, f i_1 ⊆ ⋃ i, U i\nr : ι → Set i\nhr : ∀ i_1 ∈ s, ... | exact fun s hs ↦ (hr s hs).1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Compactness.Lindelof | {
"line": 446,
"column": 4
} | {
"line": 446,
"column": 63
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ny : Y\nhf : Tendsto f (coLindelof X) (𝓝 y)\nhfc : Continuous f\nl : Filter Y\nhne : l.NeBot\ninst✝ : CountableInterFilter l\nhle : l ≤ 𝓟 (insert y (range f))\ns : Set Y\nhsy : s ∈ 𝓝 y\nt : Set Y\nhtl : t ∈ l... | filter_upwards [htl, le_principal_iff.1 hle] with y hyt hyf | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Topology.Compactness.Lindelof | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 27
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsSigmaCompact s\n⊢ IsLindelof s",
"usedConstants": [
"congrArg",
"IsSigmaCompact",
"Exists",
"Eq.mp",
"And",
"Nat",
"IsSigmaCompact.eq_1",
"Eq",
"Set.iUnion",
"IsCompact",
"Set... | rw [IsSigmaCompact] at hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 521,
"column": 2
} | {
"line": 525,
"column": 28
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsSigmaCompact s\n⊢ IsLindelof s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"IsSigmaCompact",
"Exists",
"Eq.mp",
"id",
"And.casesOn",
"And",
"Exists.casesOn",
"Nat",
"IsSigmaCom... | rw [IsSigmaCompact] at hs
rcases hs with ⟨K, ⟨hc, huniv⟩⟩
rw [← huniv]
have hl : ∀ n, IsLindelof (K n) := fun n ↦ IsCompact.isLindelof (hc n)
exact isLindelof_iUnion hl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Lindelof | {
"line": 521,
"column": 2
} | {
"line": 525,
"column": 28
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsSigmaCompact s\n⊢ IsLindelof s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"IsSigmaCompact",
"Exists",
"Eq.mp",
"id",
"And.casesOn",
"And",
"Exists.casesOn",
"Nat",
"IsSigmaCom... | rw [IsSigmaCompact] at hs
rcases hs with ⟨K, ⟨hc, huniv⟩⟩
rw [← huniv]
have hl : ∀ n, IsLindelof (K n) := fun n ↦ IsCompact.isLindelof (hc n)
exact isLindelof_iUnion hl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 742,
"column": 4
} | {
"line": 742,
"column": 11
} | [
{
"pp": "case a\nX : Type u\nY : Type v\nι : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t✝ : Set X\ninst✝ : SecondCountableTopology X\nt : Set X\nx✝¹ : t ⊆ univ\nx✝ : Filter X\n⊢ ∀ {ι : Type u} (U : ι → Set X), (∀ (i : ι), IsOpen (U i)) → t ⊆ ⋃ i, U i → ∃ t_1, t_1.Countable ∧ t ⊆ ⋃ i ... | intro ι | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Topology.Separation.Regular | {
"line": 748,
"column": 6
} | {
"line": 749,
"column": 95
} | [
{
"pp": "case neg.h\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\nx : X\nhs : IsClosed (⋂ s, ↑s)\na b : Set X\nha : IsClosed a\nhb : IsClosed b\nhab : ⋂ s, ↑s ⊆ a ∪ b\nab_disj : Disjoint a b\nu v : Set X\nhu : IsOpen u\nhv : IsOpen v\nhau : a ⊆ u\nhbv : b ⊆ v\nhuv : Dis... | have h1 : x ∈ v :=
(hab.trans (union_subset_union hau hbv) (mem_iInter.2 fun i => i.2.2)).resolve_left hxu | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Separation.Regular | {
"line": 760,
"column": 2
} | {
"line": 779,
"column": 77
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\n⊢ T2Space (ConnectedComponents X)",
"usedConstants": [
"Iff.mpr",
"False",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"ConnectedComponents.mk",
... | refine ⟨ConnectedComponents.surjective_coe.forall₂.2 fun a b ne => ?_⟩
rw [ConnectedComponents.coe_ne_coe] at ne
have h := connectedComponent_disjoint ne
-- write ↑b as the intersection of all clopen subsets containing it
rw [connectedComponent_eq_iInter_isClopen b, disjoint_iff_inter_eq_empty] at h
-- Now we... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.Regular | {
"line": 760,
"column": 2
} | {
"line": 779,
"column": 77
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\n⊢ T2Space (ConnectedComponents X)",
"usedConstants": [
"Iff.mpr",
"False",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"ConnectedComponents.mk",
... | refine ⟨ConnectedComponents.surjective_coe.forall₂.2 fun a b ne => ?_⟩
rw [ConnectedComponents.coe_ne_coe] at ne
have h := connectedComponent_disjoint ne
-- write ↑b as the intersection of all clopen subsets containing it
rw [connectedComponent_eq_iInter_isClopen b, disjoint_iff_inter_eq_empty] at h
-- Now we... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.Clopen | {
"line": 685,
"column": 2
} | {
"line": 686,
"column": 31
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nhs : ∀ (f : α → Bool), ContinuousOn f s → ∀ x ∈ s, ∀ y ∈ s, f x = f y\nu v : Set α\nu_op : IsOpen u\nv_op : IsOpen v\nhsuv : s ⊆ u ∪ v\nx : α\nx_in_s : x ∈ s\nx_in_u : x ∈ u\nH : s ∩ (u ∩ v) = ∅\ny : α\ny_in_s : y ∈ s\ny_in_v : y ∈ v\nhy : y ∉ u\nthis ... | simpa [(u.mem_iff_boolIndicator _).mp x_in_u, (u.notMem_iff_boolIndicator _).mp hy] using
hs _ this x x_in_s y y_in_s | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.UniformSpace.Defs | {
"line": 250,
"column": 30
} | {
"line": 250,
"column": 82
} | [
{
"pp": "α : Type ua\nU V : SetRel α α\nhU : IsSymmetricRel U\nhV : IsSymmetricRel V\n⊢ IsSymmetricRel (U ∩ V)",
"usedConstants": [
"Set.preimage_inter",
"Eq.mpr",
"SetRel",
"congrArg",
"IsSymmetricRel.eq_1",
"id",
"Set.instInter",
"Inter.inter",
"Set.pr... | by rw [IsSymmetricRel, preimage_inter, hU.eq, hV.eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 358,
"column": 4
} | {
"line": 362,
"column": 53
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\nι : Sort u_1\ns : ι → Set α\nhs : ∀ (i : ι), IsComplete (s i)\nU : SetRel α α\nhU : U ∈ 𝓤 α\nhd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j\nS : Set α := ⋃ i, s i\nl : Filter α\nhl : Cauchy l\nhls : S ∈ l\nhl_ne : l.NeBot\nhl' : ∀ s ∈ 𝓤 α, ∃ t ∈ ... | rcases Filter.nonempty_of_mem htl with ⟨x, hx⟩
rcases mem_iUnion.1 (htS hx) with ⟨i, hi⟩
refine ⟨i, fun y hy => ?_⟩
rcases mem_iUnion.1 (htS hy) with ⟨j, hj⟩
rwa [hd i j x hi y hj (htU <| mk_mem_prod hx hy)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 358,
"column": 4
} | {
"line": 362,
"column": 53
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\nι : Sort u_1\ns : ι → Set α\nhs : ∀ (i : ι), IsComplete (s i)\nU : SetRel α α\nhU : U ∈ 𝓤 α\nhd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j\nS : Set α := ⋃ i, s i\nl : Filter α\nhl : Cauchy l\nhls : S ∈ l\nhl_ne : l.NeBot\nhl' : ∀ s ∈ 𝓤 α, ∃ t ∈ ... | rcases Filter.nonempty_of_mem htl with ⟨x, hx⟩
rcases mem_iUnion.1 (htS hx) with ⟨i, hi⟩
refine ⟨i, fun y hy => ?_⟩
rcases mem_iUnion.1 (htS hy) with ⟨j, hj⟩
rwa [hd i j x hi y hj (htU <| mk_mem_prod hx hy)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Basic | {
"line": 771,
"column": 2
} | {
"line": 771,
"column": 91
} | [
{
"pp": "α : Type ua\nβ : Type ub\nt₁ : UniformSpace α\nt₂ : UniformSpace β\nu : SetRel α α\nv : SetRel β β\nhu : u ∈ 𝓤 α\nhv : v ∈ 𝓤 β\n⊢ entourageProd u v ∈ 𝓤 (α × β)",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"entourageProd",
"instUniformSpaceProd",
"SetRel",... | rw [uniformity_prod]; exact inter_mem_inf (preimage_mem_comap hu) (preimage_mem_comap hv) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Basic | {
"line": 771,
"column": 2
} | {
"line": 771,
"column": 91
} | [
{
"pp": "α : Type ua\nβ : Type ub\nt₁ : UniformSpace α\nt₂ : UniformSpace β\nu : SetRel α α\nv : SetRel β β\nhu : u ∈ 𝓤 α\nhv : v ∈ 𝓤 β\n⊢ entourageProd u v ∈ 𝓤 (α × β)",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"entourageProd",
"instUniformSpaceProd",
"SetRel",... | rw [uniformity_prod]; exact inter_mem_inf (preimage_mem_comap hu) (preimage_mem_comap hv) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 980,
"column": 4
} | {
"line": 981,
"column": 83
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\ninst✝ : (𝓤 α).IsCountablyGenerated\ns : Set α\nh : TotallyBounded s\nU : Set (α × α)\nhU : U ∈ 𝓤 α\n⊢ ∃ t, t.Countable ∧ s ⊆ ⋃ x ∈ t, ball x U",
"usedConstants": [
"SetRel.symmetrize",
"SetRel.inv",
"uniformity",
"symmetrize_mem_u... | obtain ⟨t, ht, hst⟩ := h (SetRel.inv U)
(mem_of_superset (symmetrize_mem_uniformity hU) SetRel.symmetrize_subset_inv) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.GroupWithZero.Indicator | {
"line": 46,
"column": 2
} | {
"line": 49,
"column": 17
} | [
{
"pp": "ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroClass M₀\ni : ι\ns : Set ι\nf g : ι → M₀\n⊢ s.indicator (fun j ↦ f j * g j) i = f i * s.indicator g i",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"Set.indicator",
"Classical.propDecidab... | simp only [indicator]
split_ifs
· rfl
· rw [mul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Indicator | {
"line": 46,
"column": 2
} | {
"line": 49,
"column": 17
} | [
{
"pp": "ι : Type u_1\nM₀ : Type u_4\ninst✝ : MulZeroClass M₀\ni : ι\ns : Set ι\nf g : ι → M₀\n⊢ s.indicator (fun j ↦ f j * g j) i = f i * s.indicator g i",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"Set.indicator",
"Classical.propDecidab... | simp only [indicator]
split_ifs
· rfl
· rw [mul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Pointwise | {
"line": 592,
"column": 7
} | {
"line": 592,
"column": 48
} | [
{
"pp": "α : Type u_2\ninst✝ : Monoid α\nf : Filter α\nhf : 1 ≤ f\ns t : Set α\nht : t ∈ f\nhs : t * univ ⊆ s\n⊢ s = univ",
"usedConstants": [
"Filter.instMembership",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Set.univ",
"Filter.mem_one",
... | mul_univ_of_one_mem (mem_one.1 <| hf ht), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.Pointwise | {
"line": 637,
"column": 2
} | {
"line": 638,
"column": 35
} | [
{
"pp": "case refine_2\nα : Type u_2\ninst✝ : DivisionMonoid α\nf g : Filter α\n⊢ (∃ a b, f = pure a ∧ g = pure b ∧ a * b = 1) → f * g = 1",
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass"... | · rintro ⟨a, b, rfl, rfl, h⟩
rw [pure_mul_pure, h, pure_one] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Filter.Pointwise | {
"line": 724,
"column": 64
} | {
"line": 729,
"column": 88
} | [
{
"pp": "α : Type u_2\ninst✝ : Group α\nf g : Filter α\n⊢ 1 ≤ f / g ↔ ¬Disjoint f g",
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Disjoint.le_bot",
"False",
"instHDiv",
"Filter.instDiv",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"... | by
refine ⟨fun h hfg => ?_, ?_⟩
· obtain ⟨s, hs, t, ht, hst⟩ := hfg.le_bot (mem_bot : ∅ ∈ ⊥)
exact Set.one_mem_div_iff.1 (h <| div_mem_div hs ht) (disjoint_iff.2 hst.symm)
· rintro h s ⟨t₁, h₁, t₂, h₂, hs⟩
exact hs (Set.one_mem_div_iff.2 fun ht => h <| disjoint_of_disjoint_of_mem ht h₁ h₂) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 871,
"column": 72
} | {
"line": 875,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : α\nha : a < 0\n⊢ (Ioo a 0)⁻¹ = Iio a⁻¹",
"usedConstants": [
"Iff.mpr",
"Set.ext",
"GroupWithZero.toMonoidWithZero",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNonAssocCommRin... | by
ext x
refine ⟨fun h ↦ (lt_inv_of_neg (inv_neg''.1 h.2) ha).2 h.1, fun h ↦ ?_⟩
have h' := (h.trans (inv_neg''.2 ha))
exact ⟨(lt_inv_of_neg ha h').2 h, inv_neg''.2 h'⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 26
} | [
{
"pp": "Gᵣ : Type u_3\ninst✝² : UniformSpace Gᵣ\ninst✝¹ : Group Gᵣ\ninst✝ : IsRightUniformGroup Gᵣ\n⊢ 𝓤 Gᵣ = comap (fun x ↦ x.2 / x.1) (𝓝 1)",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClas... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 803,
"column": 2
} | {
"line": 803,
"column": 26
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : IsTopologicalGroup G\nι : Sort u_1\np : ι → Prop\ns : ι → Set G\nhb : (𝓝 1).HasBasis p s\nx : G\n⊢ (Filter.comap (fun x_1 ↦ x_1 * x⁻¹) (𝓝 1)).HasBasis p fun i ↦ {y | y / x ∈ s i}",
"usedConstants": [
"Eq.mpr",
"DivInvM... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 28
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : TopologicalSpace M\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Countable β\ninst✝ : CompleteLattice α\nm : α → M\nm0 : m ⊥ = 1\nR : M → M → Prop\nm_iSup : ∀ (s : ℕ → α), R (m (⨆ i, s i)) (∏' (i : ℕ), m (s i))\ns : β → α\n⊢ R (m (⨆ b, s b)) (∏' (b : β), m (s b))",... | cases nonempty_encodable β | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1123,
"column": 14
} | {
"line": 1123,
"column": 15
} | [
{
"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)\nK : Set G\n⊢ ∀ {L : Set G}, IsCompact K → IsCompact L → {γ | ((fun x ↦ γ • x) '' K ∩ L).Nonempty}.Finite",
"usedConstants": [
"Set"
]
... | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1147,
"column": 14
} | {
"line": 1147,
"column": 15
} | [
{
"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)\nK : Set G\n⊢ ∀ {L : Set G}, IsCompact K → IsCompact L → {γ | ((fun x ↦ γ • x) '' K ∩ L).Nonempty}.Finite",
"usedConstants": [
"Set"
]
... | L | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 1242,
"column": 6
} | {
"line": 1242,
"column": 61
} | [
{
"pp": "G : 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\n⊢ (range (denseSeq G) ∩ (fun y ↦ x * y) ⁻¹' L).Nonempty",
... | rw [← (Homeomorph.mulLeft x).apply_symm_apply 1] at hL1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.AtTopBot.CompleteLattice | {
"line": 28,
"column": 13
} | {
"line": 28,
"column": 34
} | [
{
"pp": "α : Type u_6\ninst✝¹ : Subsingleton α\ninst✝ : Preorder α\ns : Set α\nhs : s ∈ ⨅ a, 𝓟 (Ici a)\nx : α\n⊢ x ∈ s",
"usedConstants": [
"Filter.instMembership",
"iInf",
"Set.Ici",
"congrArg",
"Filter.instCompleteLatticeFilter",
"Membership.mem",
"CompleteLattic... | ciInf_subsingleton x, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.AtTopBot.CompleteLattice | {
"line": 97,
"column": 19
} | {
"line": 97,
"column": 43
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝² : Preorder β\ninst✝¹ : ConditionallyCompleteLinearOrder γ\nl : Filter α\ninst✝ : l.NeBot\nf : β → γ\nhf : Monotone f\ng : α → β\nhg : Tendsto g l atTop\nhb : ¬BddAbove (range f)\n⊢ ¬(upperBounds (range fun a ↦ f (g a))).Nonempty",
"usedConstants": [
... | ← Function.comp_def f g, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.OrdContinuous | {
"line": 73,
"column": 18
} | {
"line": 73,
"column": 58
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\ng : β → γ\nf : α → β\nhg : LeftOrdContinuous g\nhf : LeftOrdContinuous f\ns : Set α\nx : α\nh : IsLUB s x\n⊢ IsLUB (g ∘ f '' s) ((g ∘ f) x)",
"usedConstants": [
"Set.image_image",
"congrArg... | simpa only [image_image] using hg (hf h) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.OrdContinuous | {
"line": 73,
"column": 18
} | {
"line": 73,
"column": 58
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\ng : β → γ\nf : α → β\nhg : LeftOrdContinuous g\nhf : LeftOrdContinuous f\ns : Set α\nx : α\nh : IsLUB s x\n⊢ IsLUB (g ∘ f '' s) ((g ∘ f) x)",
"usedConstants": [
"Set.image_image",
"congrArg... | simpa only [image_image] using hg (hf h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.OrdContinuous | {
"line": 73,
"column": 18
} | {
"line": 73,
"column": 58
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\ng : β → γ\nf : α → β\nhg : LeftOrdContinuous g\nhf : LeftOrdContinuous f\ns : Set α\nx : α\nh : IsLUB s x\n⊢ IsLUB (g ∘ f '' s) ((g ∘ f) x)",
"usedConstants": [
"Set.image_image",
"congrArg... | simpa only [image_image] using hg (hf h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.Pi | {
"line": 124,
"column": 4
} | {
"line": 127,
"column": 89
} | [
{
"pp": "case h.refine_3\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → PartialOrder (α i)\nf : (i : ι) → α i\ni : ι\na b : α i\nx : (a : ι) → α a\nh : x ∈ univ.pi fun i_1 ↦ Icc (update f i a i_1) (update f i b i_1)\n⊢ update f i (x i) = x",
"usedConstants": [
"Eq.mpr",
... | ext j
obtain rfl | hij := eq_or_ne i j
· exact Function.update_self ..
· simpa only [Function.update_of_ne hij.symm, le_antisymm_iff] using h j (mem_univ j) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.Pi | {
"line": 124,
"column": 4
} | {
"line": 127,
"column": 89
} | [
{
"pp": "case h.refine_3\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → PartialOrder (α i)\nf : (i : ι) → α i\ni : ι\na b : α i\nx : (a : ι) → α a\nh : x ∈ univ.pi fun i_1 ↦ Icc (update f i a i_1) (update f i b i_1)\n⊢ update f i (x i) = x",
"usedConstants": [
"Eq.mpr",
... | ext j
obtain rfl | hij := eq_or_ne i j
· exact Function.update_self ..
· simpa only [Function.update_of_ne hij.symm, le_antisymm_iff] using h j (mem_univ j) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Field | {
"line": 67,
"column": 8
} | {
"line": 67,
"column": 56
} | [
{
"pp": "case inr\nK✝ : Type u_1\ninst✝⁴ : DivisionRing K✝\ninst✝³ : TopologicalSpace K✝\nα : Type u_2\ninst✝² : Field α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalDivisionRing α\nK : Subfield α\nx : α\nhx : x ∈ closure ↑K\nh : x ≠ 0\n⊢ x⁻¹ ∈ closure ((fun x ↦ x⁻¹) '' ↑K)",
"usedConstants": [
... | exact mem_closure_image (continuousAt_inv₀ h) hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Order.Group | {
"line": 41,
"column": 6
} | {
"line": 41,
"column": 28
} | [
{
"pp": "case h\nG : Type u_1\ninst✝⁴ : TopologicalSpace G\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : OrderTopology G\na b ε : G\nhε : ε > 1\nδ : G\nhδ₁ : 1 < δ\nhδε : δ < ε\n⊢ ∀ (a_1 : G × G), |a_1.1 / a|ₘ < δ ∧ |a_1.2 / b|ₘ < ε / δ → |a_1.1 * a_1.2 / (a * b)|ₘ < ε",
... | rintro ⟨c, d⟩ ⟨hc, hd⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.Algebra.Order.Field | {
"line": 243,
"column": 13
} | {
"line": 243,
"column": 22
} | [
{
"pp": "case h.e'_5.h.e'_3\n𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nC : 𝕜\nhf : ∀ᶠ (x : α) in l, |f x| ≤ C\nhg : Tendsto g l (𝓝 0)\n⊢ 0 = |C * 0|",
"usedCo... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Order.LeftRightNhds | {
"line": 214,
"column": 2
} | {
"line": 215,
"column": 11
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\n⊢ 𝓝[<] a = ⊥ ↔ IsBot a ∨ ∃ b, b ⋖ a",
"usedConstants": [
"OrderDual.toDual",
"Eq.mpr",
"Set.Ioi",
"Preorder.toLT",
"Equiv.instEquivLike",
"HEq.refl",
"CovBy"... | convert (config := { preTransparency := .default }) nhdsGT_eq_bot_iff (a := OrderDual.toDual a)
using 4 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 44
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝⁴ : CommMonoid α\ninst✝³ : Preorder α\ninst✝² : IsOrderedMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\nh : ∀ (i : ι), f i ≤ 1\nhf : ¬Multipliable f L\n⊢ ∏'[L] (i : ι), f i ≤ 1",
"usedConstants": [
... | rw [tprod_eq_one_of_not_multipliable hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 44
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝⁴ : CommMonoid α\ninst✝³ : Preorder α\ninst✝² : IsOrderedMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\nh : ∀ (i : ι), f i ≤ 1\nhf : ¬Multipliable f L\n⊢ ∏'[L] (i : ι), f i ≤ 1",
"usedConstants": [
... | rw [tprod_eq_one_of_not_multipliable hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 44
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝⁴ : CommMonoid α\ninst✝³ : Preorder α\ninst✝² : IsOrderedMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\nh : ∀ (i : ι), f i ≤ 1\nhf : ¬Multipliable f L\n⊢ ∏'[L] (i : ι), f i ≤ 1",
"usedConstants": [
... | rw [tprod_eq_one_of_not_multipliable hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 72,
"column": 2
} | {
"line": 73,
"column": 24
} | [
{
"pp": "case h₂\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\nhab : a ≠ b\n⊢ Icc a b ⊆ closure (Ioc a b)",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"congrArg",
"Set.Subset.trans",
"PartialOrder.t... | · apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)
rw [closure_Ioo hab] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 91
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\np : Filter ι\ninst✝ : CompactSpace α\nh : TendstoLocallyUniformly F f p\nV : Set (β × β)\nhV : V ∈ 𝓤 β\nU : α → Set α\nhU : ∀ (x : α), U x ∈ 𝓝 x ∧ ∀ᶠ (n : ι) in p, ∀ y ∈ U x, (f y... | obtain ⟨t, ht⟩ := isCompact_univ.elim_nhds_subcover' (fun k _ => U k) fun k _ => (hU k).1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 144,
"column": 65
} | {
"line": 148,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\nx : α\np : Filter ι\np' : Filter α\nh : TendstoUniformlyOnFilter F f p p'\nhx : 𝓟 {x} ≤ p'\n⊢ Tendsto (fun n ↦ F n x) p (𝓝 (f x))",
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
... | by
refine Uniform.tendsto_nhds_right.mpr fun u hu => mem_map.mpr ?_
filter_upwards [(h u hu).curry]
intro i h
simpa using h.filter_mono hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.IntermediateValue | {
"line": 305,
"column": 2
} | {
"line": 315,
"column": 89
} | [
{
"pp": "α : Type u\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty\n⊢ b ∈ s",
"usedConstants": [
"Iff.mpr",
"Set.Ioc",
... | let S := s ∩ Icc a b
replace ha : a ∈ S := ⟨ha, left_mem_Icc.2 hab⟩
have Sbd : BddAbove S := ⟨b, fun z hz => hz.2.2⟩
let c := sSup (s ∩ Icc a b)
have c_mem : c ∈ S := hs.csSup_mem ⟨_, ha⟩ Sbd
have c_le : c ≤ b := csSup_le ⟨_, ha⟩ fun x hx => hx.2.2
rcases eq_or_lt_of_le c_le with hc | hc
· exact hc ▸ c_me... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.IntermediateValue | {
"line": 305,
"column": 2
} | {
"line": 315,
"column": 89
} | [
{
"pp": "α : Type u\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ico a b, (s ∩ Ioc x b).Nonempty\n⊢ b ∈ s",
"usedConstants": [
"Iff.mpr",
"Set.Ioc",
... | let S := s ∩ Icc a b
replace ha : a ∈ S := ⟨ha, left_mem_Icc.2 hab⟩
have Sbd : BddAbove S := ⟨b, fun z hz => hz.2.2⟩
let c := sSup (s ∩ Icc a b)
have c_mem : c ∈ S := hs.csSup_mem ⟨_, ha⟩ Sbd
have c_le : c ≤ b := csSup_le ⟨_, ha⟩ fun x hx => hx.2.2
rcases eq_or_lt_of_le c_le with hc | hc
· exact hc ▸ c_me... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 47
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nh :\n ∀ x ∈ s,\n ∀ u ∈ 𝓤 β,\n ∃ pa,\n (∀ᶠ (x : ι) in p, pa x) ∧\n ∃ pb, (∀ᶠ (y : α) in 𝓝[s] x, pb y) ∧ ∀ {x : ι}, pa x →... | obtain ⟨pa, hpa, pb, hpb, h⟩ := h x hx u hu | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Order.IntermediateValue | {
"line": 416,
"column": 2
} | {
"line": 416,
"column": 88
} | [
{
"pp": "α : Type u\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b x y : α\ns t : Set α\nhxy : x ≤ y\nhs : IsClosed s\nht : IsClosed t\nhab : Icc a b ⊆ s ∪ t\nhx : x ∈ Icc a b ∩ s\nhy : y ∈ Icc a b ∩ t\nxyab : Icc x y ⊆ Icc a b\... | have : Ioc z y ⊆ s ∪ t := fun w hw => hab (xyab ⟨le_trans hz.1 (le_of_lt hw.1), hw.2⟩) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 49
} | [
{
"pp": "α : Type u_3\nm' : PseudoMetricSpace α\ntoDist✝ : Dist α\ndist_self✝ : ∀ (x : α), dist x x = 0\ndist_comm✝ : ∀ (x y : α), dist x y = dist y x\ndist_triangle✝ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z\nedist✝ : α → α → ℝ≥0∞\nhed : ∀ (x y : α), edist✝ x y = ENNReal.ofReal (dist x y)\ntoUniformSpace... | obtain ⟨_, _, _, _, hed', _, hU', _, hB'⟩ := m' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 26
} | [
{
"pp": "α : Type u_3\ntoDist✝¹ : Dist α\ndist_self✝¹ : ∀ (x : α), dist x x = 0\ndist_comm✝¹ : ∀ (x y : α), dist x y = dist y x\ndist_triangle✝¹ : ∀ (x y z : α), dist x z ≤ dist x y + dist y z\nedist✝¹ : α → α → ℝ≥0∞\nhed : ∀ (x y : α), edist✝¹ x y = ENNReal.ofReal (dist x y)\ntoUniformSpace✝¹ : UniformSpace α\... | obtain rfl : d = d' := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.MetricSpace.Pseudo.Constructions | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 57
} | [
{
"pp": "a b : ℝ≥0\nthis : ↑a ≤ ↑b + dist a b\n⊢ a ≤ b + nndist a b",
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"Real.instLE",
"Real",
"NNReal.coe_add",
"coe_nndist",
"congrArg",
"PartialOrder.toPreorder",
"PseudoMetricSpace.toNNDist",
"Preorde... | rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Topology.MetricSpace.Pseudo.Constructions | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 57
} | [
{
"pp": "a b : ℝ≥0\nthis : ↑a ≤ ↑b + dist a b\n⊢ a ≤ b + nndist a b",
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"Real.instLE",
"Real",
"NNReal.coe_add",
"coe_nndist",
"congrArg",
"PartialOrder.toPreorder",
"PseudoMetricSpace.toNNDist",
"Preorde... | rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Pseudo.Constructions | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 57
} | [
{
"pp": "a b : ℝ≥0\nthis : ↑a ≤ ↑b + dist a b\n⊢ a ≤ b + nndist a b",
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"Real.instLE",
"Real",
"NNReal.coe_add",
"coe_nndist",
"congrArg",
"PartialOrder.toPreorder",
"PseudoMetricSpace.toNNDist",
"Preorde... | rwa [← NNReal.coe_le_coe, NNReal.coe_add, coe_nndist] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 60,
"column": 84
} | {
"line": 65,
"column": 35
} | [
{
"pp": "β : Type u_2\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\nf g : (b : β) → X b\nr : ℝ≥0\nhr : 0 < r\n⊢ nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r",
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"nndist_pi_... | by
rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm]
simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]
intro h
refine exists_congr fun b => ?_
apply (and_iff_right <| h _).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 66
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx : α\nε : ℝ\n⊢ closedBall x ε = ⋂ δ, ⋂ (_ : δ > ε), ball x δ",
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"congrArg",
"Set.iInter",
"Iff.rfl",
"PartialOrder.toPreorde... | ext y; rw [mem_closedBall, ← forall_gt_iff_le, mem_iInter₂]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 66
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx : α\nε : ℝ\n⊢ closedBall x ε = ⋂ δ, ⋂ (_ : δ > ε), ball x δ",
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"congrArg",
"Set.iInter",
"Iff.rfl",
"PartialOrder.toPreorde... | ext y; rw [mem_closedBall, ← forall_gt_iff_le, mem_iInter₂]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1210,
"column": 4
} | {
"line": 1210,
"column": 67
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\ne : β → α\na : α\n⊢ (∀ (i : ℕ), True → ∃ y ∈ range e, y ∈ ball a (1 / (↑i + 1))) ↔ ∀ (n : ℕ), ∃ k, dist a (e k) < 1 / (↑n + 1)",
"usedConstants": [
"Real",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"_private.M... | simp only [mem_ball, dist_comm, exists_range_iff, forall_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1210,
"column": 4
} | {
"line": 1210,
"column": 67
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\ne : β → α\na : α\n⊢ (∀ (i : ℕ), True → ∃ y ∈ range e, y ∈ ball a (1 / (↑i + 1))) ↔ ∀ (n : ℕ), ∃ k, dist a (e k) < 1 / (↑n + 1)",
"usedConstants": [
"Real",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"_private.M... | simp only [mem_ball, dist_comm, exists_range_iff, forall_const] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1210,
"column": 4
} | {
"line": 1210,
"column": 67
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\ne : β → α\na : α\n⊢ (∀ (i : ℕ), True → ∃ y ∈ range e, y ∈ ball a (1 / (↑i + 1))) ↔ ∀ (n : ℕ), ∃ k, dist a (e k) < 1 / (↑n + 1)",
"usedConstants": [
"Real",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"_private.M... | simp only [mem_ball, dist_comm, exists_range_iff, forall_const] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.Diam | {
"line": 80,
"column": 34
} | {
"line": 80,
"column": 77
} | [
{
"pp": "X : Type u_2\ns : Set X\nx : X\ninst✝ : PseudoEMetricSpace X\nd : ℝ≥0∞\n⊢ ediam (insert x s) ≤ d ↔ max (⨆ y ∈ s, edist x y) (ediam s) ≤ d",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"PseudoEMetricSpace.edist_comm",
"congrArg",
"iSup",
"zero_le._simp_1",
... | simp +contextual [ediam_le_iff, edist_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.EMetricSpace.Diam | {
"line": 80,
"column": 34
} | {
"line": 80,
"column": 77
} | [
{
"pp": "X : Type u_2\ns : Set X\nx : X\ninst✝ : PseudoEMetricSpace X\nd : ℝ≥0∞\n⊢ ediam (insert x s) ≤ d ↔ max (⨆ y ∈ s, edist x y) (ediam s) ≤ d",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"PseudoEMetricSpace.edist_comm",
"congrArg",
"iSup",
"zero_le._simp_1",
... | simp +contextual [ediam_le_iff, edist_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.Diam | {
"line": 80,
"column": 34
} | {
"line": 80,
"column": 77
} | [
{
"pp": "X : Type u_2\ns : Set X\nx : X\ninst✝ : PseudoEMetricSpace X\nd : ℝ≥0∞\n⊢ ediam (insert x s) ≤ d ↔ max (⨆ y ∈ s, edist x y) (ediam s) ≤ d",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"PseudoEMetricSpace.edist_comm",
"congrArg",
"iSup",
"zero_le._simp_1",
... | simp +contextual [ediam_le_iff, edist_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Defs | {
"line": 126,
"column": 2
} | {
"line": 130,
"column": 32
} | [
{
"pp": "γ : Type w\ninst✝ : MetricSpace γ\nx : γ\nr : ℝ\nhr : r ≤ 0\n⊢ (closedBall x r).Subsingleton",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"PartialOrder.toPreo... | rcases hr.lt_or_eq with (hr | rfl)
· rw [closedBall_eq_empty.2 hr]
exact subsingleton_empty
· rw [closedBall_zero]
exact subsingleton_singleton | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Defs | {
"line": 126,
"column": 2
} | {
"line": 130,
"column": 32
} | [
{
"pp": "γ : Type w\ninst✝ : MetricSpace γ\nx : γ\nr : ℝ\nhr : r ≤ 0\n⊢ (closedBall x r).Subsingleton",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"PartialOrder.toPreo... | rcases hr.lt_or_eq with (hr | rfl)
· rw [closedBall_eq_empty.2 hr]
exact subsingleton_empty
· rw [closedBall_zero]
exact subsingleton_singleton | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.DirectedInverseSystem | {
"line": 304,
"column": 54
} | {
"line": 306,
"column": 5
} | [
{
"pp": "ι : Type u_6\ninst✝ : LinearOrder ι\nX : ι → Type u_7\ni : ι\nhi : IsSuccPrelimit i\nf : ↑(limit piLTProj i)\nk l : ↑(Iio i)\nhl : ↑l < ↑k\n⊢ (piLTLim hi).symm f l = ↑f k ⟨↑l, hl⟩",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Equiv.right_inv",
"congrArg",
"Par... | by
conv_rhs => rw [← (piLTLim hi).right_inv f]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Finiteness.Small | {
"line": 30,
"column": 2
} | {
"line": 30,
"column": 29
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nP Q : Submodule R M\nsmallP : Small.{u, u_2} ↥P\nsmallQ : Small.{u, u_2} ↥Q\n⊢ Small.{u, u_2} ↥(P ⊔ Q)",
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.ids",
"LinearM... | rw [Submodule.sup_eq_range] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.Basic | {
"line": 178,
"column": 57
} | {
"line": 180,
"column": 5
} | [
{
"pp": "σ : Type u\nR : Type v\ninst✝ : CommSemiring R\nm : ℕ\np : MvPolynomial σ R\n⊢ p ∈ restrictTotalDegree σ R m ↔ p.totalDegree ≤ m",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Nat.instMulZeroClass",
"Nat.instLattice",
"Lattice.toSemilatticeSup",
"Semiring.toModule"... | by
rw [totalDegree, Finset.sup_le_iff]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 126,
"column": 4
} | {
"line": 128,
"column": 32
} | [
{
"pp": "case add\nR : Type u_1\ninst✝⁶ : CommSemiring R\nN : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid P\ninst✝³ : AddCommMonoid Q\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ng : N →ₗ[R] P\nhg : Function.Surjective ⇑g\nx y : Q ⊗[R] P\nhx : ∃ a, (l... | obtain ⟨x, rfl⟩ := hx
obtain ⟨y, rfl⟩ := hy
exact ⟨x + y, map_add _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 126,
"column": 4
} | {
"line": 128,
"column": 32
} | [
{
"pp": "case add\nR : Type u_1\ninst✝⁶ : CommSemiring R\nN : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid P\ninst✝³ : AddCommMonoid Q\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ng : N →ₗ[R] P\nhg : Function.Surjective ⇑g\nx y : Q ⊗[R] P\nhx : ∃ a, (l... | obtain ⟨x, rfl⟩ := hx
obtain ⟨y, rfl⟩ := hy
exact ⟨x + y, map_add _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 151,
"column": 4
} | {
"line": 153,
"column": 32
} | [
{
"pp": "case add\nR : Type u_1\ninst✝⁶ : CommSemiring R\nN : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid P\ninst✝³ : AddCommMonoid Q\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ng : N →ₗ[R] P\nhg : Function.Surjective ⇑g\nx y : P ⊗[R] Q\nhx : ∃ a, (r... | obtain ⟨x, rfl⟩ := hx
obtain ⟨y, rfl⟩ := hy
exact ⟨x + y, map_add _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 151,
"column": 4
} | {
"line": 153,
"column": 32
} | [
{
"pp": "case add\nR : Type u_1\ninst✝⁶ : CommSemiring R\nN : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : AddCommMonoid P\ninst✝³ : AddCommMonoid Q\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\ng : N →ₗ[R] P\nhg : Function.Surjective ⇑g\nx y : P ⊗[R] Q\nhx : ∃ a, (r... | obtain ⟨x, rfl⟩ := hx
obtain ⟨y, rfl⟩ := hy
exact ⟨x + y, map_add _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 469,
"column": 8
} | {
"line": 470,
"column": 51
} | [
{
"pp": "case a.refine_4.zero\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal A\nx✝ : A ⊗[R] B\nhx : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))\nx : ↥(Subm... | use 0
simp only [map_zero, smul_eq_mul, zero_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 469,
"column": 8
} | {
"line": 470,
"column": 51
} | [
{
"pp": "case a.refine_4.zero\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal A\nx✝ : A ⊗[R] B\nhx : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))\nx : ↥(Subm... | use 0
simp only [map_zero, smul_eq_mul, zero_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Colimit.DirectLimit | {
"line": 250,
"column": 53
} | {
"line": 250,
"column": 67
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝⁶ : Preorder ι\nG : ι → Type u_3\nT : ⦃i j : ι⦄ → i ≤ j → Type u_4\nf : (x x_1 : ι) → (h : x ≤ x_1) → T h\ninst✝⁵ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)\ninst✝⁴ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)\ninst✝³ : IsDirectedOrder ι\ninst✝² : Nonempty... | Nat.cast_succ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 533,
"column": 8
} | {
"line": 534,
"column": 51
} | [
{
"pp": "case a.refine_4.zero\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal B\nx✝ : A ⊗[R] B\nhx : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)))\nx : A ⊗[R... | use 0
simp only [map_zero, smul_eq_mul, zero_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 533,
"column": 8
} | {
"line": 534,
"column": 51
} | [
{
"pp": "case a.refine_4.zero\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal B\nx✝ : A ⊗[R] B\nhx : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)))\nx : A ⊗[R... | use 0
simp only [map_zero, smul_eq_mul, zero_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 630,
"column": 6
} | {
"line": 630,
"column": 25
} | [
{
"pp": "R : Type u_4\nS : Type u_5\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\nA : Type u_6\nB : Type u_7\nC : Type u_8\nD : Type u_9\ninst✝¹¹ : Ring A\ninst✝¹⁰ : Ring B\ninst✝⁹ : Ring C\ninst✝⁸ : Ring D\ninst✝⁷ : Algebra R A\ninst✝⁶ : Algebra R B\ninst✝⁵ : Algebra R C\ninst✝⁴ : Algebra... | ← RingHom.comap_ker | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 160,
"column": 11
} | {
"line": 160,
"column": 34
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ Flat R M ↔\n ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P),\n Function.Injective ⇑(lTensor M N.subtype)",
"usedConstants": [
"Eq.mpr",
"Submodule"... | iff_rTensor_injectiveₛ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 176,
"column": 18
} | {
"line": 176,
"column": 39
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nf :\n ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P),\n Function.Injective ⇑(rTensor M N.subtype)\ni : N →ₗ[... | lTensor_comp_rTensor, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Flat.Basic | {
"line": 209,
"column": 11
} | {
"line": 209,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nι : Type v\nM : ι → Type w\ninst✝¹ : (i : ι) → AddCommMonoid (M i)\ninst✝ : (i : ι) → Module R (M i)\n⊢ Flat R (⨁ (i : ι), M i) ↔ ∀ (i : ι), Flat R (M i)",
"usedConstants": [
"Eq.mpr",
"Submodule",
"_private.Mathlib.RingTheory.Flat.Basic.0.Modu... | iff_rTensor_injectiveₛ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.BigOperators.Expect | {
"line": 285,
"column": 43
} | {
"line": 285,
"column": 83
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module ℚ≥0 M\ninst✝³ : AddCommMonoid N\ninst✝² : Module ℚ≥0 N\nF : Type u_5\ninst✝¹ : FunLike F M N\ninst✝ : LinearMapClass F ℚ≥0 M N\ng : F\nf : ι → M\ns : Finset ι\n⊢ g (𝔼 i ∈ s, f i) = 𝔼 i ∈ s, g (f i)",
"usedConstant... | by simp only [expect, map_smul, map_sum] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Group.Finset.Powerset | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Finset α\na : α\ninst✝¹ : CommMonoid β\ninst✝ : DecidableEq α\nha : a ∉ s\nf : Finset α → β\n⊢ Disjoint s.powerset (image (insert a) s.powerset)",
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"False",
"eq_false",
"and_true",
... | · aesop (add simp [disjoint_left, insert_subset_iff]) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Int.Interval | {
"line": 162,
"column": 28
} | {
"line": 162,
"column": 36
} | [
{
"pp": "n a : ℤ\nh : 0 ≤ a\nha : 0 < a\ni : ℤ\nhi₀ : 0 ≤ i\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ n % a + a * (n / a + 1) = n + a",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Int.instDiv",
"instHDiv",
"HMul.hMul",
"congrArg",
"... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 57,
"column": 95
} | {
"line": 58,
"column": 36
} | [
{
"pp": "n : ℕ\ns : Multiset ℕ\nh : ∀ x ∈ s, x ≡ 1 [MOD n]\n⊢ s.prod ≡ 1 [MOD n]",
"usedConstants": [
"Multiset.map",
"congrArg",
"Multiset.prod",
"Multiset",
"Eq.mp",
"instOfNatNat",
"Nat.ModEq.multisetProd_map_one",
"Nat.ModEq",
"Nat",
"Nat.instC... | by
simpa using multisetProd_map_one h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 150,
"column": 97
} | {
"line": 151,
"column": 36
} | [
{
"pp": "n : ℤ\ns : Multiset ℤ\nh : ∀ x ∈ s, x ≡ 1 [ZMOD n]\n⊢ s.prod ≡ 1 [ZMOD n]",
"usedConstants": [
"Int.instCommMonoid",
"Multiset.map",
"congrArg",
"Multiset.prod",
"Multiset",
"Eq.mp",
"Int",
"instOfNat",
"Int.ModEq",
"Int.ModEq.multisetProd... | by
simpa using multisetProd_map_one h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Sym | {
"line": 208,
"column": 4
} | {
"line": 210,
"column": 35
} | [
{
"pp": "case h.refine_1.refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nn : ℕ\ng : α ↪ β\ns : Finset α\nd : Sym β n\nhd : ∀ a ∈ d, ∃ a_2 ∈ s, g a_2 = a\ng' : { x // x ∈ d } → α :=\n fun x ↦\n match x with\n | ⟨x, hx⟩ => ⋯.choose\n⊢ Sym.map (⇑g) ((fun p ↦ Sym.map g' p... | · simp only [Sym.map_map, Function.comp_apply, g']
convert Sym.attach_map_coe d with ⟨x, hx⟩ hx'
exact (hd x hx).choose_spec.2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Opposite | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 47
} | [
{
"pp": "α : Sort u\nX : Type v\ninst✝ : Small.{u, v} X\n⊢ Small.{u, v} Xᵒᵖ",
"usedConstants": [
"Small.equiv_small"
]
}
] | obtain ⟨S, ⟨e⟩⟩ := Small.equiv_small (α := X) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.Sym.Sym2 | {
"line": 282,
"column": 18
} | {
"line": 284,
"column": 40
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\na b₁✝ b₁ : α\nh : (fun b ↦ s(a, b)) b₁✝ = (fun b ↦ s(a, b)) b₁\n⊢ b₁✝ = b₁",
"usedConstants": [
"Sym2.Rel",
"Sym2.eq._simp_1",
"Sym2.mk",
"congrArg",
"Eq.mp",
"Prod.mk",
"Sym2.rel_iff'._simp_1",
"Or.casesOn",
... | by
simp only [Sym2.eq, Sym2.rel_iff', Prod.mk.injEq, true_and, Prod.swap_prod_mk] at h
obtain rfl | ⟨rfl, rfl⟩ := h <;> rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Sym.Sym2 | {
"line": 412,
"column": 2
} | {
"line": 414,
"column": 36
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nf g : α → β\ns : Sym2 α\nh : ∀ x ∈ s, f x = g x\ny : β\n⊢ (∃ a ∈ s, f a = y) ↔ ∃ a ∈ s, g a = y",
"usedConstants": [
"congrArg",
"Membership.mem",
"Exists",
"And.casesOn",
"And",
"Exists.casesOn",
"And.intro",
"True... | constructor <;>
· rintro ⟨w, hw, rfl⟩
exact ⟨w, hw, by simp [hw, h]⟩ | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Sym.Sym2 | {
"line": 537,
"column": 67
} | {
"line": 538,
"column": 34
} | [
{
"pp": "α : Type u_1\n⊢ Set.range diag = diagSet",
"usedConstants": [
"Set.ext",
"Sym2.Rel",
"Sym2.eq._simp_1",
"Sym2.mk",
"congrArg",
"_private.Mathlib.Data.Sym.Sym2.0.Sym2.range_diag._simp_1_2",
"Quot.ind",
"Membership.mem",
"Exists",
"exists_eq... | by
ext ⟨a, b⟩; simp [diag, eq_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.ObjectProperty.Basic | {
"line": 78,
"column": 39
} | {
"line": 78,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝ : CategoryStruct.{v, u} C\nι : Type u'\nX : ι → C\nP : ObjectProperty C\nh : ∀ (i : ι), P (X i)\n⊢ ofObj X ≤ P",
"usedConstants": [
"CategoryTheory.ObjectProperty.ofObj.casesOn",
"CategoryTheory.ObjectProperty.ofObj",
"HEq.refl",
"CategoryTheory.ObjectPrope... | rintro _ ⟨i⟩; exact h i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.ObjectProperty.Basic | {
"line": 78,
"column": 39
} | {
"line": 78,
"column": 62
} | [
{
"pp": "C : Type u\ninst✝ : CategoryStruct.{v, u} C\nι : Type u'\nX : ι → C\nP : ObjectProperty C\nh : ∀ (i : ι), P (X i)\n⊢ ofObj X ≤ P",
"usedConstants": [
"CategoryTheory.ObjectProperty.ofObj.casesOn",
"CategoryTheory.ObjectProperty.ofObj",
"HEq.refl",
"CategoryTheory.ObjectPrope... | rintro _ ⟨i⟩; exact h i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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