module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.NoetherianSpace | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 13
} | {
"line": 219,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : NoetherianSpace α\nZ : Set α\nH : Z ∈ irreducibleComponents α\n⊢ ∃ o, IsOpen[inst✝¹] o ∧ o.Nonempty ∧ o ⊆ Z",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : NoetherianSpace α\nZ : Set α\nH : Z ∈ irreducibleComponents α\n⊢ ∃ o, IsOpen[inst✝¹] o ∧ o.Nonempty ∧ o ⊆ Z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.QuasiSeparated | {
"line": 122,
"column": 6
} | {
"line": 122,
"column": 17
} | {
"line": 122,
"column": 18
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nι : Type u_3\nb : ι → Set α\nbasis : IsTopologicalBasis (range b)\nisCompact_inter : ∀ (i j : ι), IsCompact (b i ∩ b j)\nU V : Set α\nhUopen : IsOpen[inst✝] U\nhUcomp : IsCompact U\nhVopen : IsOpen[inst✝] V\nhVcomp : IsCompact V\ni : ι\n⊢ IsCompact (b i)",
... | [
"α : Type u_1\ninst✝ : TopologicalSpace α\nι : Type u_3\nb : ι → Set α\nbasis : IsTopologicalBasis (range b)\nisCompact_inter : ∀ (i j : ι), IsCompact (b i ∩ b j)\nU V : Set α\nhUopen : IsOpen[inst✝] U\nhUcomp : IsCompact U\nhVopen : IsOpen[inst✝] V\nhVcomp : IsCompact V\ni : ι\n⊢ IsCompact (b i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Ideal | {
"line": 422,
"column": 40
} | {
"line": 422,
"column": 88
} | {
"line": 422,
"column": 89
} | [
{
"pp": "P : Type u_1\ninst✝¹ : SemilatticeSup P\ninst✝ : IsCodirectedOrder P\nx : P\nI : Ideal P\nhx : x ∉ I\nh : I = I ⊔ principal x\n⊢ x ∈ I",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"P : Type u_1\ninst✝¹ : SemilatticeSup P\ninst✝ : IsCodirectedOrder P\nx : P\nI : Ideal P\nhx : x ∉ I\nh : I = I ⊔ principal x\n⊢ x ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Ideal | {
"line": 515,
"column": 2
} | {
"line": 516,
"column": 9
} | {
"line": 516,
"column": 10
} | [
{
"pp": "P : Type u_1\ninst✝¹ : CompleteLattice P\nI : Ideal P\nα : Sort u_2\ninst✝ : Finite α\nf : α → P\n⊢ ⨆ i, f i ∈ I ↔ ∀ (i : α), f i ∈ I",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"P : Type u_1\ninst✝¹ : CompleteLattice P\nI : Ideal P\nα : Sort u_2\ninst✝ : Finite α\nf : α → P\n⊢ ⨆ i, f i ∈ I ↔ ∀ (i : α), f i ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Ideal | {
"line": 637,
"column": 4
} | {
"line": 637,
"column": 73
} | {
"line": 637,
"column": 74
} | [
{
"pp": "case refine_3\nP : Type u_1\ninst✝¹ : LE P\ninst✝ : OrderTop P\nS : Set (Ideal P)\nhS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S\nhS₂ : S.Nonempty\nhS₃ : ⊤ ∉ S\n⊢ ⋯.toIdeal ≠ ⊤",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"SetLike.m... | [
"case refine_3\nP : Type u_1\ninst✝¹ : LE P\ninst✝ : OrderTop P\nS : Set (Ideal P)\nhS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S\nhS₂ : S.Nonempty\nhS₃ : ⊤ ∉ S\n⊢ ∀ x ∈ S, ⊤ ∉ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Ideal | {
"line": 639,
"column": 4
} | {
"line": 639,
"column": 28
} | {
"line": 639,
"column": 29
} | [
{
"pp": "case refine_4\nP : Type u_1\ninst✝¹ : LE P\ninst✝ : OrderTop P\nS : Set (Ideal P)\nhS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S\nhS₂ : S.Nonempty\nhS₃ : ⊤ ∉ S\nJ : Ideal P\nhJ : J ∈ S\n⊢ J ≤ ⋯.toIdeal",
"ppTerm": "?refine_4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.coe_... | [
"case refine_4\nP : Type u_1\ninst✝¹ : LE P\ninst✝ : OrderTop P\nS : Set (Ideal P)\nhS₁ : IsChain (fun x1 x2 ↦ x1 ≤ x2) S\nhS₂ : S.Nonempty\nhS₃ : ⊤ ∉ S\nJ : Ideal P\nhJ : J ∈ S\n⊢ ↑J ⊆ ⋃ a ∈ S, ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sober | {
"line": 100,
"column": 11
} | {
"line": 100,
"column": 26
} | {
"line": 100,
"column": 27
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nx : α\nS : Set α\nhS : IsClosed S\nhxS : x ∈ S\nthis : closure {x} ⊆ S\n⊢ IsGenericPoint x S ↔ ∀ (Z : Set α), IsClosed Z → x ∈ Z → S ⊆ Z",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"IsGenericPoint",
"Membership.mem",
... | [
"α : Type u_1\ninst✝ : TopologicalSpace α\nx : α\nS : Set α\nhS : IsClosed S\nhxS : x ∈ S\nthis : closure {x} ⊆ S\n⊢ closure {x} = S ↔ ∀ (Z : Set α), IsClosed Z → x ∈ Z → S ⊆ Z"
] | IsGenericPoint, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Constructible | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 15
} | {
"line": 140,
"column": 16
} | [
{
"pp": "case refine_2\nX : Type u_2\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsSpectralMap Subtype.val\nt : Set X\nhtcomp : IsCompact t\nhtopen : IsOpen[inst✝] t\n⊢ IsCompact (s ∩ t)",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_2\nX : Type u_2\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsSpectralMap Subtype.val\nt : Set X\nhtcomp : IsCompact t\nhtopen : IsOpen[inst✝] t\n⊢ IsCompact (s ∩ t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructible | {
"line": 163,
"column": 4
} | {
"line": 164,
"column": 11
} | {
"line": 164,
"column": 12
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set Y\nhf : IsClosedEmbedding f\nhf' : IsCompact (range f)ᶜ\nhs : IsRetrocompact s\nU : Set X\nhUcomp : IsCompact U\nhUopen : IsOpen[inst✝¹] U\n⊢ IsOpen[inst✝] (f '' U ∪ (range f)ᶜ)",
"ppTerm": "?m.3... | [
"X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set Y\nhf : IsClosedEmbedding f\nhf' : IsCompact (range f)ᶜ\nhs : IsRetrocompact s\nU : Set X\nhUcomp : IsCompact U\nhUopen : IsOpen[inst✝¹] U\n⊢ IsOpen[inst✝] (f '' U ∪ (range f)ᶜ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructible | {
"line": 166,
"column": 2
} | {
"line": 167,
"column": 49
} | {
"line": 167,
"column": 50
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set Y\nhf : IsClosedEmbedding f\nhf' : IsCompact (range f)ᶜ\nhs : IsRetrocompact s\nU : Set X\nhUcomp : IsCompact U\nhUopen : IsOpen[inst✝¹] U\nhfUopen : IsOpen[inst✝] (f '' U ∪ (range f)ᶜ)\nhfUcomp : Is... | [
"X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set Y\nhf : IsClosedEmbedding f\nhf' : IsCompact (range f)ᶜ\nhs : IsRetrocompact s\nU : Set X\nhUcomp : IsCompact U\nhUopen : IsOpen[inst✝¹] U\nhfUopen : IsOpen[inst✝] (f '' U ∪ (range f)ᶜ)\nhfUcomp : IsCompact (f '... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sober | {
"line": 204,
"column": 6
} | {
"line": 204,
"column": 17
} | {
"line": 204,
"column": 18
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : IsOpenEmbedding f\ninst✝ : QuasiSober β\nx : β\nT : Set β\nhT : IsClosed[inst✝¹] T\nhS : IsIrreducible (f ⁻¹' T)\nhS' : IsClosed[inst✝²] (f ⁻¹' T)\nhS'' : IsIrreducible (T ∩ range f)\nhx : IsGenericPoi... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\nhf : IsOpenEmbedding f\ninst✝ : QuasiSober β\nx : β\nT : Set β\nhT : IsClosed[inst✝¹] T\nhS : IsIrreducible (f ⁻¹' T)\nhS' : IsClosed[inst✝²] (f ⁻¹' T)\nhS'' : IsIrreducible (T ∩ range f)\nhx : IsGenericPoint x (closur... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructible | {
"line": 281,
"column": 6
} | {
"line": 282,
"column": 13
} | {
"line": 282,
"column": 14
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhf : IsClosedEmbedding f\nhfcomp : IsRetrocompact (range f)ᶜ\nU : Set X\nhUopen : IsOpen[inst✝¹] U\nhUcomp : IsRetrocompact U\n⊢ IsOpen[inst✝] (f '' U ∪ (range f)ᶜ)",
"ppTerm": "?m.44",
"a... | [
"X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhf : IsClosedEmbedding f\nhfcomp : IsRetrocompact (range f)ᶜ\nU : Set X\nhUopen : IsOpen[inst✝¹] U\nhUcomp : IsRetrocompact U\n⊢ IsOpen[inst✝] (f '' U ∪ (range f)ᶜ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructible | {
"line": 284,
"column": 6
} | {
"line": 285,
"column": 13
} | {
"line": 285,
"column": 14
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhf : IsClosedEmbedding f\nhfcomp : IsRetrocompact (range f)ᶜ\nU : Set X\nhUopen : IsOpen[inst✝¹] U\nhUcomp : IsRetrocompact U\nhfU : IsOpen[inst✝] (f '' U ∪ (range f)ᶜ)\nh : IsRetrocompact (f '' U... | [
"X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhf : IsClosedEmbedding f\nhfcomp : IsRetrocompact (range f)ᶜ\nU : Set X\nhUopen : IsOpen[inst✝¹] U\nhUcomp : IsRetrocompact U\nhfU : IsOpen[inst✝] (f '' U ∪ (range f)ᶜ)\nh : IsRetrocompact (f '' U ∪ (range f)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.Compacts | {
"line": 873,
"column": 8
} | {
"line": 873,
"column": 29
} | {
"line": 873,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : CompactSpace α\ninst✝ : T2Space α\ns t : CompactOpens α\n⊢ IsClosed (↑s ⇨ ↑t)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : CompactSpace α\ninst✝ : T2Space α\ns t : CompactOpens α\n⊢ IsClosed (↑t ∪ (↑s)ᶜ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.Compacts | {
"line": 874,
"column": 7
} | {
"line": 874,
"column": 28
} | {
"line": 874,
"column": 29
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : CompactSpace α\ninst✝ : T2Space α\ns t : CompactOpens α\n⊢ IsOpen { carrier := ↑s ⇨ ↑t, isCompact' := ⋯ }.carrier",
"ppTerm": "?m.44",
"assigned": true,
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : CompactSpace α\ninst✝ : T2Space α\ns t : CompactOpens α\n⊢ IsOpen (↑t ∪ (↑s)ᶜ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructible | {
"line": 298,
"column": 14
} | {
"line": 299,
"column": 9
} | {
"line": 299,
"column": 10
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set Y\nhf : IsOpenEmbedding f\nhfcomp : IsRetrocompact (range f)\nhsf : s ⊆ range f\nhs : IsConstructible (f ⁻¹' s)\n⊢ IsConstructible s",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants... | [
"X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set Y\nhf : IsOpenEmbedding f\nhfcomp : IsRetrocompact (range f)\nhsf : s ⊆ range f\nhs : IsConstructible (f ⁻¹' s)\n⊢ IsConstructible s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sober | {
"line": 239,
"column": 4
} | {
"line": 239,
"column": 86
} | {
"line": 239,
"column": 87
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\ninst✝ : ∀ (s : ↑S), QuasiSober ↑↑s\nhS' : ⋃₀ S = ⊤\n⊢ TopologicalSpace.IsOpenCover fun s ↦ { carrier := ↑s, is_open' := ⋯ }",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\ninst✝¹ : TopologicalSpace α\nS : Set (Set α)\nhS : ∀ (s : ↑S), IsOpen ↑s\ninst✝ : ∀ (s : ↑S), QuasiSober ↑↑s\nhS' : ⋃₀ S = ⊤\n⊢ ⋃ i ∈ S, i = univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructible | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 13
} | {
"line": 323,
"column": 14
} | [
{
"pp": "X : Type u_2\ninst✝¹ : TopologicalSpace X\ns : Set X\ninst✝ : CompactSpace X\nhs : IsRetrocompact s\n⊢ IsCompact s",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_2\ninst✝¹ : TopologicalSpace X\ns : Set X\ninst✝ : CompactSpace X\nhs : IsRetrocompact s\n⊢ IsCompact s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructible | {
"line": 357,
"column": 4
} | {
"line": 357,
"column": 60
} | {
"line": 358,
"column": 4
} | [
{
"pp": "case sdiff\nX : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : CompactSpace X\nP : (s : Set X) → IsConstructible s → Prop\ninst✝¹ : QuasiSeparatedSpace X\nι : Type u_4\ninst✝ : Nonempty ι\nb : ι → Set X\nbasis : IsTopologicalBasis (range b)\nisCompact_basis : ∀ (i : ι), IsCompact (b i)\nsdiff : ∀ (i :... | [
"case sdiff\nX : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : CompactSpace X\nP : (s : Set X) → IsConstructible s → Prop\ninst✝¹ : QuasiSeparatedSpace X\nι : Type u_4\ninst✝ : Nonempty ι\nb : ι → Set X\nbasis : IsTopologicalBasis (range b)\nisCompact_basis : ∀ (i : ι), IsCompact (b i)\nsdiff : ∀ (i : ι) (s : Set... | obtain ⟨t, ht, rfl⟩ := (this _).1 ⟨hV.2.isCompact, hV.1⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Spectral.Prespectral | {
"line": 193,
"column": 23
} | {
"line": 193,
"column": 34
} | {
"line": 193,
"column": 35
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : PrespectralSpace X\nZ U₁ U₂ : Set X\nhU₁ : ∀ a ∈ U₁, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₁\nhU₂ : ∀ a ∈ U₂, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₂\nhU₁Z : (Z ∩ U₁).Nonempty\nhU₂Z : (Z ∩ U₂).Nonempty\nhU₁₂ : Z ∩ ... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : PrespectralSpace X\nZ U₁ U₂ : Set X\nhU₁ : ∀ a ∈ U₁, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₁\nhU₂ : ∀ a ∈ U₂, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₂\nhU₁Z : (Z ∩ U₁).Nonempty\nhU₂Z : (Z ∩ U₂).Nonempty\nhU₁₂ : Z ∩ (U₁ ∩ U₂) ⊆ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructible | {
"line": 360,
"column": 15
} | {
"line": 360,
"column": 26
} | {
"line": 360,
"column": 27
} | [
{
"pp": "case sdiff.empty\nX : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : CompactSpace X\nP : (s : Set X) → IsConstructible s → Prop\ninst✝¹ : QuasiSeparatedSpace X\nι : Type u_4\ninst✝ : Nonempty ι\nb : ι → Set X\nbasis : IsTopologicalBasis (range b)\nisCompact_basis : ∀ (i : ι), IsCompact (b i)\nsdiff : ... | [
"case sdiff.empty\nX : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : CompactSpace X\nP : (s : Set X) → IsConstructible s → Prop\ninst✝¹ : QuasiSeparatedSpace X\nι : Type u_4\ninst✝ : Nonempty ι\nb : ι → Set X\nbasis : IsTopologicalBasis (range b)\nisCompact_basis : ∀ (i : ι), IsCompact (b i)\nsdiff : ∀ (i : ι) (s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Spectral.Prespectral | {
"line": 193,
"column": 47
} | {
"line": 193,
"column": 58
} | {
"line": 193,
"column": 59
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : PrespectralSpace X\nZ U₁ U₂ : Set X\nhU₁ : ∀ a ∈ U₁, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₁\nhU₂ : ∀ a ∈ U₂, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₂\nhU₁Z : (Z ∩ U₁).Nonempty\nhU₂Z : (Z ∩ U₂).Nonempty\nhU₁₂ : Z ∩ ... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : PrespectralSpace X\nZ U₁ U₂ : Set X\nhU₁ : ∀ a ∈ U₁, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₁\nhU₂ : ∀ a ∈ U₂, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₂\nhU₁Z : (Z ∩ U₁).Nonempty\nhU₂Z : (Z ∩ U₂).Nonempty\nhU₁₂ : Z ∩ (U₁ ∩ U₂) ⊆ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructible | {
"line": 477,
"column": 12
} | {
"line": 477,
"column": 23
} | {
"line": 477,
"column": 24
} | [
{
"pp": "X : Type u_2\ninst✝² : TopologicalSpace X\ns t : Set X\ninst✝¹ : PrespectralSpace X\ninst✝ : QuasiSeparatedSpace X\nhs : IsLocallyConstructible s\nhst : s ⊆ t\nht : IsCompact t\nx : X\nU : Set X\nhxU : U ∈ 𝓝 x\nhU : IsOpen[inst✝²] U\nhUs : IsConstructible (U ↓∩ s)\nV : Set X\nhV₁ : IsOpen[inst✝²] V\nh... | [
"X : Type u_2\ninst✝² : TopologicalSpace X\ns t : Set X\ninst✝¹ : PrespectralSpace X\ninst✝ : QuasiSeparatedSpace X\nhs : IsLocallyConstructible s\nhst : s ⊆ t\nht : IsCompact t\nx : X\nU : Set X\nhxU : U ∈ 𝓝 x\nhU : IsOpen[inst✝²] U\nhUs : IsConstructible (U ↓∩ s)\nV : Set X\nhV₁ : IsOpen[inst✝²] V\nhV₂ : IsCompa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.GoingDown | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 15
} | {
"line": 83,
"column": 16
} | [
{
"pp": "case h\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingDown R S\nq : PrimeSpectrum R\nP : Ideal S\ninst✝ : P.IsPrime\nlo : P.LiesOver (RelSeries.singleton {(a, b) | a < b} q).last.asIdeal\n⊢ (PrimeSpectrum.comap (algebraMap R S) { as... | [
"case h\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingDown R S\nq : PrimeSpectrum R\nP : Ideal S\ninst✝ : P.IsPrime\nlo : P.LiesOver (RelSeries.singleton {(a, b) | a < b} q).last.asIdeal\n⊢ comap (algebraMap R S) P = q.asIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.GoingDown | {
"line": 89,
"column": 43
} | {
"line": 89,
"column": 75
} | {
"line": 89,
"column": 76
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingDown R S\nl : RelSeries {(a, b) | a < b}\nq : PrimeSpectrum R\nlt : (q, l.head) ∈ {(a, b) | a < b}\nih :\n ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.last.asIdeal],\n ∃ L,... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingDown R S\nl : RelSeries {(a, b) | a < b}\nq : PrimeSpectrum R\nlt : (q, l.head) ∈ {(a, b) | a < b}\nih :\n ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.last.asIdeal],\n ∃ L,\n L.le... | ← l.toList_getElem_zero_eq_head, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.GoingDown | {
"line": 97,
"column": 25
} | {
"line": 97,
"column": 43
} | {
"line": 97,
"column": 44
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingDown R S\nl : RelSeries {(a, b) | a < b}\nq : PrimeSpectrum R\nlt : (q, l.head) ∈ {(a, b) | a < b}\nih :\n ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.last.asIdeal],\n ∃ L,... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingDown R S\nl : RelSeries {(a, b) | a < b}\nq : PrimeSpectrum R\nlt : (q, l.head) ∈ {(a, b) | a < b}\nih :\n ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.last.asIdeal],\n ∃ L,\n L.le... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.GoingDown | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 39
} | {
"line": 121,
"column": 40
} | [
{
"pp": "case refine_2\nR : Type u_3\nS : Type u_4\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nh : GeneralizingMap (PrimeSpectrum.comap (algebraMap R S))\np : Ideal R\nhp : p.IsPrime\nQ : Ideal S\nhQ : Q.IsPrime\nhlt : p < Ideal.under R Q\nthis : { asIdeal := p, isPrime := hp } ⤳ PrimeSpectr... | [
"case refine_2\nR : Type u_3\nS : Type u_4\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nh : GeneralizingMap (PrimeSpectrum.comap (algebraMap R S))\np : Ideal R\nhp : p.IsPrime\nQ : Ideal S\nhQ : Q.IsPrime\nhlt : p < Ideal.under R Q\nthis : { asIdeal := p, isPrime := hp } ⤳ PrimeSpectrum.comap (al... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.GoingDown | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 12
} | {
"line": 157,
"column": 2
} | [
{
"pp": "R✝ : Type u_1\nS✝ : Type u_2\ninst✝⁶ : CommRing R✝\ninst✝⁵ : CommRing S✝\ninst✝⁴ : Algebra R✝ S✝\nR : Type u_3\nS : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Module.Flat R S\n⊢ ∀ (P : Ideal S) [inst : P.IsPrime],\n Function.Surjective (PrimeSpectrum.comap (Loc... | [
"R✝ : Type u_1\nS✝ : Type u_2\ninst✝⁶ : CommRing R✝\ninst✝⁵ : CommRing S✝\ninst✝⁴ : Algebra R✝ S✝\nR : Type u_3\nS : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Module.Flat R S\nP : Ideal S\nhP : P.IsPrime\n⊢ Function.Surjective (PrimeSpectrum.comap (Localization.localRingHom (... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Derivation.ToSquareZero | {
"line": 36,
"column": 17
} | {
"line": 36,
"column": 48
} | {
"line": 36,
"column": 49
} | [
{
"pp": "R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal B\nf₁ f₂ : A →ₐ[R] B\ne : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂\nx : A\n⊢ (f₁.toLinearMap - f₂.toLinearMap) x ∈ S... | [
"R : Type u\nA : Type v\nB : Type w\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal B\nf₁ f₂ : A →ₐ[R] B\ne : (Ideal.Quotient.mkₐ R I).comp f₁ = (Ideal.Quotient.mkₐ R I).comp f₂\nx : A\n⊢ f₁ x - f₂ x ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Derivation.ToSquareZero | {
"line": 118,
"column": 2
} | {
"line": 122,
"column": 72
} | {
"line": 124,
"column": 0
} | [
{
"pp": "R : Type u\nA : Type v\nB : Type w\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\nI : Ideal B\ninst✝¹ : Algebra A B\nhI : I ^ 2 = ⊥\ninst✝ : IsScalarTower R A B\n⊢ Derivation R A ↥I ≃ { f // (Ideal.Quotient.mkₐ R I).comp f = IsScalarT... | [] | refine ⟨fun d => ⟨liftOfDerivationToSquareZero I hI d, ?_⟩, fun f =>
(derivationToSquareZeroOfLift I hI f.1 f.2 :), ?_, ?_⟩
· ext x; exact liftOfDerivationToSquareZero_mk_apply I hI d x
· intro d; ext x; exact add_sub_cancel_right (d x : B) (algebraMap A B x)
· rintro ⟨f, hf⟩; ext x; exact sub_add_cancel (f x... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Derivation.ToSquareZero | {
"line": 118,
"column": 2
} | {
"line": 122,
"column": 72
} | {
"line": 124,
"column": 0
} | [
{
"pp": "R : Type u\nA : Type v\nB : Type w\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\nI : Ideal B\ninst✝¹ : Algebra A B\nhI : I ^ 2 = ⊥\ninst✝ : IsScalarTower R A B\n⊢ Derivation R A ↥I ≃ { f // (Ideal.Quotient.mkₐ R I).comp f = IsScalarT... | [] | refine ⟨fun d => ⟨liftOfDerivationToSquareZero I hI d, ?_⟩, fun f =>
(derivationToSquareZeroOfLift I hI f.1 f.2 :), ?_, ?_⟩
· ext x; exact liftOfDerivationToSquareZero_mk_apply I hI d x
· intro d; ext x; exact add_sub_cancel_right (d x : B) (algebraMap A B x)
· rintro ⟨f, hf⟩; ext x; exact sub_add_cancel (f x... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RingHom.Flat | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 35
} | {
"line": 192,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\n⊢ (ulift.{u₁, u₂, u_1, u_2} f).Flat ↔ f.Flat",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"RingHom.Flat",
"ULift",
"RingHom.ulift",
"ULift.commRing",
"Iff.intro"
... | [
"case refine_1\nR : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : (ulift.{u₁, u₂, u_1, u_2} f).Flat\n⊢ f.Flat",
"case refine_2\nR : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : f.Flat\n⊢ (ulift.{u₁, u₂, u_1, u_2} f).Flat"
] | refine ⟨fun hf ↦ ?_, fun hf ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.RingHom.Flat | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 94
} | {
"line": 209,
"column": 2
} | [
{
"pp": "R S T : CommRingCat\nf : R ⟶ S\ng : R ⟶ T\nhf : Function.Injective ⇑(ConcreteCategory.hom f)\nhg : (Hom.hom g).Flat\nalgInst✝¹ : Algebra ↑R ↑S := (Hom.hom f).toAlgebra\nalgInst✝ : Algebra ↑R ↑T := (Hom.hom g).toAlgebra\nalgebraizeInst✝ : Module.Flat ↑R ↑T\n⊢ Function.Injective ⇑(ConcreteCategory.hom (p... | [
"R S T : CommRingCat\nf : R ⟶ S\ng : R ⟶ T\nhf : Function.Injective ⇑(ConcreteCategory.hom f)\nhg : (Hom.hom g).Flat\nalgInst✝¹ : Algebra ↑R ↑S := (Hom.hom f).toAlgebra\nalgInst✝ : Algebra ↑R ↑T := (Hom.hom g).toAlgebra\nalgebraizeInst✝ : Module.Flat ↑R ↑T\nthis : ofHom Algebra.TensorProduct.includeRight.toRingHom ... | have : _ = pushout.inr f g := (CommRingCat.isPushout_tensorProduct R S T).inr_isoPushout_hom | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.RingHom.Flat | {
"line": 268,
"column": 30
} | {
"line": 268,
"column": 73
} | {
"line": 268,
"column": 73
} | [
{
"pp": "R S T A : Type u\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing T\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R T\ninst✝⁴ : Algebra S T\ninst✝³ : IsScalarTower R S T\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nh : CommRingCat.flat (CommRingC... | [] | by simp [IsScalarTower.algebraMap_eq R S T] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 21
} | {
"line": 44,
"column": 22
} | [
{
"pp": "case pos\na b p : ℕ\na✝ : p ∈ (a.lcm b).factorization.support\nh : b.factorization p ≤ a.factorization p\nH : p = 0 ∧ ¬(a.lcm b).factorization p = 0\n⊢ False",
"ppTerm": "?pos✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case pos\na b p : ℕ\na✝ : p ∈ (a.lcm b).factorization.support\nh : b.factorization p ≤ a.factorization p\nH : p = 0 ∧ ¬(a.lcm b).factorization p = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 21
} | {
"line": 54,
"column": 22
} | [
{
"pp": "case neg\na b p : ℕ\na✝ : p ∈ (a.lcm b).factorization.support\nh : ¬b.factorization p ≤ a.factorization p\nH : p = 0 ∧ ¬(a.lcm b).factorization p = 0\n⊢ False",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case neg\na b p : ℕ\na✝ : p ∈ (a.lcm b).factorization.support\nh : ¬b.factorization p ≤ a.factorization p\nH : p = 0 ∧ ¬(a.lcm b).factorization p = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 248,
"column": 8
} | {
"line": 248,
"column": 63
} | {
"line": 248,
"column": 64
} | [
{
"pp": "case mp\nR : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nhI : I.IsRadical\nh : ∀ (x y : Ideal R), x ⊓ y ≤ I → x ≤ I ∨ y ≤ I\nx y : R\nh' : (Ideal.span {x} * Ideal.span {y}).radical ≤ I\n⊢ (Ideal.span {x} ⊓ Ideal.span {y}).radical ≤ I",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
... | [
"case mp\nR : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nhI : I.IsRadical\nh : ∀ (x y : Ideal R), x ⊓ y ≤ I → x ≤ I ∨ y ≤ I\nx y : R\nh' : (Ideal.span {x} * Ideal.span {y}).radical ≤ I\n⊢ (Ideal.span {x}).radical ⊓ (Ideal.span {y}).radical ≤ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 55
} | {
"line": 280,
"column": 56
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : IsDomain R\n⊢ IrreducibleSpace (PrimeSpectrum R)",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"IsDomain.to_noZeroDivisors",
"Semiring.toModule",
... | [
"R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : IsDomain R\n⊢ Ideal.IsPrime ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 285,
"column": 10
} | {
"line": 285,
"column": 25
} | {
"line": 285,
"column": 26
} | [
{
"pp": "R : Type u\nS✝ : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S✝\nS : Set (PrimeSpectrum R)\nh₁ : IsIrreducible S\nh₂ : IsClosed S\n⊢ IsGenericPoint { asIdeal := vanishingIdeal S, isPrime := ⋯ } S",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Prim... | [
"R : Type u\nS✝ : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S✝\nS : Set (PrimeSpectrum R)\nh₁ : IsIrreducible S\nh₂ : IsClosed S\n⊢ closure {{ asIdeal := vanishingIdeal S, isPrime := ⋯ }} = S"
] | IsGenericPoint, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Derivation.Basic | {
"line": 371,
"column": 18
} | {
"line": 374,
"column": 15
} | {
"line": 375,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing M\ninst✝³ : Algebra R A\ninst✝² : Algebra R M\nF : Type u_4\ninst✝¹ : FunLike F A M\ninst✝ : AlgHomClass F R A M\nf : F\nf_inv : M → A\nhf : Function.RightInverse f_inv ⇑f\nd : Derivation R A A\nhd... | [] | by
suffices f (d (f_inv (x + y) - (f_inv x + f_inv y))) = 0 by simpa [sub_eq_zero]
apply hd
simp [hf _] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 472,
"column": 44
} | {
"line": 472,
"column": 68
} | {
"line": 472,
"column": 69
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.fst R S))\np : Ideal R\nhp : p.IsPrime\neq : p✝.asIdeal = p.prod ⊤\n⊢ (comap (RingHom.fst R S) { asIdeal := p, isPrime := hp }).asIdeal = p✝.asIdeal",
"ppTe... | [
"R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.fst R S))\np : Ideal R\nhp : p.IsPrime\neq : p✝.asIdeal = p.prod ⊤\n⊢ Ideal.comap (RingHom.fst R S) p = p✝.asIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 474,
"column": 4
} | {
"line": 474,
"column": 20
} | {
"line": 474,
"column": 21
} | [
{
"pp": "case refine_2.inr\nR : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.fst R S))\np : Ideal S\nhp : p.IsPrime\neq : p✝.asIdeal = ⊤.prod p\n⊢ 1 ∈ p",
"ppTerm": "?refine_2.inr",
"assigned": false,
"used... | [
"case refine_2.inr\nR : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.fst R S))\np : Ideal S\nhp : p.IsPrime\neq : p✝.asIdeal = ⊤.prod p\n⊢ 1 ∈ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 482,
"column": 4
} | {
"line": 482,
"column": 20
} | {
"line": 482,
"column": 21
} | [
{
"pp": "case refine_2.inl\nR : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.snd R S))\np : Ideal R\nhp : p.IsPrime\neq : p✝.asIdeal = p.prod ⊤\n⊢ 1 ∈ p",
"ppTerm": "?refine_2.inl",
"assigned": false,
"used... | [
"case refine_2.inl\nR : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.snd R S))\np : Ideal R\nhp : p.IsPrime\neq : p✝.asIdeal = p.prod ⊤\n⊢ 1 ∈ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 483,
"column": 44
} | {
"line": 483,
"column": 68
} | {
"line": 483,
"column": 69
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.snd R S))\np : Ideal S\nhp : p.IsPrime\neq : p✝.asIdeal = ⊤.prod p\n⊢ (comap (RingHom.snd R S) { asIdeal := p, isPrime := hp }).asIdeal = p✝.asIdeal",
"ppTe... | [
"R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\np✝ : PrimeSpectrum (R × S)\nh : p✝ ∈ zeroLocus ↑(RingHom.ker (RingHom.snd R S))\np : Ideal S\nhp : p.IsPrime\neq : p✝.asIdeal = ⊤.prod p\n⊢ Ideal.comap (RingHom.snd R S) p = p✝.asIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Derivation.Basic | {
"line": 426,
"column": 4
} | {
"line": 426,
"column": 40
} | {
"line": 426,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra R A\nM : Type u_3\ninst✝² : AddCancelCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module A M\nD : A →ₗ[R] M\nh : ∀ (a b : A), D (a * b) = a • D b + b • D a\n⊢ D 1 + D 1 = D 1",
"ppTerm": "?m.63",
"assig... | [
"R : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra R A\nM : Type u_3\ninst✝² : AddCancelCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module A M\nD : A →ₗ[R] M\nh : ∀ (a b : A), D (a * b) = a • D b + b • D a\n⊢ D 1 + D 1 = D 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 565,
"column": 35
} | {
"line": 565,
"column": 46
} | {
"line": 565,
"column": 47
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nf : R\nn : ℕ\nhn : 0 < n\n⊢ ↑(basicOpen (f ^ n)) = ↑(basicOpen f)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.zeroLocus",
"PrimeSpectrum.basicOpen",
"congrArg",
"CommSemiring.toSemiring"... | [
"R : Type u\ninst✝ : CommSemiring R\nf : R\nn : ℕ\nhn : 0 < n\n⊢ zeroLocus {f ^ n} = zeroLocus {f}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Derivation.Basic | {
"line": 506,
"column": 4
} | {
"line": 506,
"column": 23
} | {
"line": 507,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Module K M\ninst✝ : Algebra R K\nD : Derivation R K M\na : K\nn : ℤ\nhn : ¬n = 0\nha : ¬a = 0\nh : n = -↑n.natAbs\n⊢ (-↑n.natAbs * (a ^ ↑(n.natAbs - 1) / a ^ ↑(n.natAb... | [
"R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Module K M\ninst✝ : Algebra R K\nD : Derivation R K M\na : K\nn : ℤ\nhn : ¬n = 0\nha : ¬a = 0\nh : n = -↑n.natAbs\n⊢ (-↑n.natAbs * a ^ (↑(n.natAbs - 1) - ↑(n.natAbs * 2))) • D a =... | rw [← zpow_sub₀ ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 79
} | {
"line": 220,
"column": 2
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\nG_cyclic : IsCyclic G\nk : ℕ\nk_pos : k ≠ 0\nk_lt_card_G : k < Nat.card G\n⊢ ∃ a, a ^ k ≠ 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Finite",
"Nat.card",
"Nat.ne_zero_of_lt",
"Nat.finite_of_card_ne_zero"
],
"us... | [
"G : Type u_2\ninst✝ : Group G\nG_cyclic : IsCyclic G\nk : ℕ\nk_pos : k ≠ 0\nk_lt_card_G : k < Nat.card G\nthis : Finite G\n⊢ ∃ a, a ^ k ≠ 1"
] | have : Finite G := Nat.finite_of_card_ne_zero (Nat.ne_zero_of_lt k_lt_card_G) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 673,
"column": 4
} | {
"line": 674,
"column": 11
} | {
"line": 674,
"column": 12
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\ni j : R\n⊢ IsCompact (↑(basicOpen i) ∩ ↑(basicOpen j))",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"PrimeSpectrum.basicOpen",
"TopologicalSpace.Opens.instCom... | [
"R : Type u\nS : Type v\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\ni j : R\n⊢ IsCompact ↑(basicOpen (i * j))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 702,
"column": 17
} | {
"line": 702,
"column": 28
} | {
"line": 702,
"column": 29
} | [
{
"pp": "ι : Type u_1\nR : ι → Type u_2\ninst✝¹ : (i : ι) → CommRing (R i)\ninst✝ : DecidableEq ι\ni : ι\nf : R i\nq : PrimeSpectrum (R i)\nhp : comap (Pi.evalRingHom R i) q ∈ ↑(basicOpen (Pi.single i f))\n⊢ q ∈ ↑(basicOpen f)",
"ppTerm": "?m.200",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [
"ι : Type u_1\nR : ι → Type u_2\ninst✝¹ : (i : ι) → CommRing (R i)\ninst✝ : DecidableEq ι\ni : ι\nf : R i\nq : PrimeSpectrum (R i)\nhp : comap (Pi.evalRingHom R i) q ∈ ↑(basicOpen (Pi.single i f))\n⊢ f ∉ q.asIdeal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 710,
"column": 76
} | {
"line": 722,
"column": 26
} | {
"line": 724,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : ι → Type u_2\ninst✝ : (i : ι) → CommRing (R i)\n⊢ IsOpenEmbedding (sigmaToPi R)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"TopologicalSpace.IsTopologicalBasis.isOpenMap_iff",
"Eq.mpr",
"Continuous",
"PrimeSpectrum.basicOpen",
"c... | [] | by
classical
refine .of_continuous_injective_isOpenMap ?_ ?_ ?_
· rw [continuous_sigma_iff]
intro i
exact continuous_comap (Pi.evalRingHom R i)
· exact sigmaToPi_injective R
· rw [isOpenMap_sigma]
intro i
simp only [sigmaToPi_apply, PrimeSpectrum.isTopologicalBasis_basic_opens.isOpenMap_iff]
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Exponent | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 24
} | {
"line": 313,
"column": 25
} | [
{
"pp": "case refine_3\nG : Type u_1\ninst✝¹ : Monoid G\ninst✝ : Nontrivial G\np : ℕ\nhp : Nat.Prime p\nh : ∀ (g : G), g ≠ 1 → orderOf g = p\ng : G\nhg : g ≠ 1\n⊢ p ∣ exponent G",
"ppTerm": "?refine_3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_3\nG : Type u_1\ninst✝¹ : Monoid G\ninst✝ : Nontrivial G\np : ℕ\nhp : Nat.Prime p\nh : ∀ (g : G), g ≠ 1 → orderOf g = p\ng : G\nhg : g ≠ 1\n⊢ p ∣ exponent G"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1012,
"column": 43
} | {
"line": 1012,
"column": 54
} | {
"line": 1012,
"column": 55
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ns✝ : Set (PrimeSpectrum R)\nx✝ : ∃ s, s.Finite ∧ s✝ = (zeroLocus s)ᶜ\ns : Set R\nhs : s.Finite\ne : s✝ = (zeroLocus s)ᶜ\n⊢ (zeroLocus ↑hs.toFinset)ᶜ = s✝",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.zeroLo... | [
"R : Type u\ninst✝ : CommSemiring R\ns✝ : Set (PrimeSpectrum R)\nx✝ : ∃ s, s.Finite ∧ s✝ = (zeroLocus s)ᶜ\ns : Set R\nhs : s.Finite\ne : s✝ = (zeroLocus s)ᶜ\n⊢ (zeroLocus s)ᶜ = s✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1013,
"column": 40
} | {
"line": 1013,
"column": 51
} | {
"line": 1013,
"column": 52
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ns✝ : Set (PrimeSpectrum R)\nx✝ : ∃ t, (zeroLocus ↑t)ᶜ = s✝\ns : Finset R\ne : (zeroLocus ↑s)ᶜ = s✝\n⊢ s✝ = (zeroLocus ↑s)ᶜ",
"ppTerm": "?m.68",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : CommSemiring R\ns✝ : Set (PrimeSpectrum R)\nx✝ : ∃ t, (zeroLocus ↑t)ᶜ = s✝\ns : Finset R\ne : (zeroLocus ↑s)ᶜ = s✝\n⊢ s✝ = (zeroLocus ↑s)ᶜ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 329,
"column": 44
} | {
"line": 329,
"column": 69
} | {
"line": 329,
"column": 70
} | [
{
"pp": "α : Type u_1\ninst✝³ : Group α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : IsCyclic α\nn : ℕ\nhn0 : 0 < n\ng : α\nhg : ∀ (x : α), x ∈ zpowers g\nx : α\nhx : x ∈ {a | a ^ n = 1}\nm : ℕ\nhm : g ^ m = x\n⊢ g ^ (m * n.gcd (Fintype.card α)) = 1",
"ppTerm": "?m.175",
"assigned": true,
"u... | [
"α : Type u_1\ninst✝³ : Group α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : IsCyclic α\nn : ℕ\nhn0 : 0 < n\ng : α\nhg : ∀ (x : α), x ∈ zpowers g\nx : α\nhx : x ∈ {a | a ^ n = 1}\nm : ℕ\nhm : g ^ m = x\n⊢ x ^ n = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1018,
"column": 45
} | {
"line": 1018,
"column": 56
} | {
"line": 1018,
"column": 57
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ns✝ : Set (PrimeSpectrum R)\nx✝ : ∃ t, (zeroLocus ↑t)ᶜ = s✝\ns : Finset R\ne : (zeroLocus ↑s)ᶜ = s✝\n⊢ (zeroLocus ↑(Ideal.span ↑s))ᶜ = s✝",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"PrimeSpect... | [
"R : Type u\ninst✝ : CommSemiring R\ns✝ : Set (PrimeSpectrum R)\nx✝ : ∃ t, (zeroLocus ↑t)ᶜ = s✝\ns : Finset R\ne : (zeroLocus ↑s)ᶜ = s✝\n⊢ (zeroLocus ↑s)ᶜ = s✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1019,
"column": 33
} | {
"line": 1019,
"column": 54
} | {
"line": 1019,
"column": 55
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ns✝ : Set (PrimeSpectrum R)\nx✝ : ∃ I, I.FG ∧ (zeroLocus ↑I)ᶜ = s✝\nI : Ideal R\ns : Finset R\nhs : Ideal.span ↑s = I\ne : (zeroLocus ↑I)ᶜ = s✝\n⊢ (zeroLocus ↑s)ᶜ = s✝",
"ppTerm": "?m.63",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"R : Type u\ninst✝ : CommSemiring R\ns✝ : Set (PrimeSpectrum R)\nx✝ : ∃ I, I.FG ∧ (zeroLocus ↑I)ᶜ = s✝\nI : Ideal R\ns : Finset R\nhs : Ideal.span ↑s = I\ne : (zeroLocus ↑I)ᶜ = s✝\n⊢ (zeroLocus ↑s)ᶜ = s✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Exponent | {
"line": 413,
"column": 4
} | {
"line": 413,
"column": 57
} | {
"line": 413,
"column": 58
} | [
{
"pp": "case refine_1\nG : Type u\ninst✝¹ : LeftCancelMonoid G\ninst✝ : Finite G\n_inst : Fintype G := Fintype.ofFinite G\n⊢ 0 < Finset.univ.lcm orderOf",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"Nat.instMulZeroClass",
"Pre... | [
"case refine_1\nG : Type u\ninst✝¹ : LeftCancelMonoid G\ninst✝ : Finite G\n_inst : Fintype G := Fintype.ofFinite G\n⊢ ∀ (x : G), IsOfFinOrder x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Exponent | {
"line": 422,
"column": 64
} | {
"line": 424,
"column": 79
} | {
"line": 426,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝² : LeftCancelMonoid G\ninst✝¹ : Finite G\ninst✝ : Nontrivial G\n⊢ 1 < exponent G",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"mt",
"id",
"LeftCancelMonoid.toMonoid",
"Ne",
"instOfNatNat",
"A... | [] | by
rw [Nat.one_lt_iff_ne_zero_and_ne_one]
exact ⟨exponent_ne_zero_of_finite, mt exp_eq_one_iff.mp (not_subsingleton G)⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Cyclic.Basic | {
"line": 422,
"column": 2
} | {
"line": 422,
"column": 47
} | {
"line": 422,
"column": 48
} | [
{
"pp": "G : Type u_2\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : IsCyclic G\ng : G\nhg : ∀ (x : G), x ∈ zpowers g\na b : ZMod (Nat.card G)\nh : ↑a.val = ↑b.val\n⊢ a = b",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_2\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : IsCyclic G\ng : G\nhg : ∀ (x : G), x ∈ zpowers g\na b : ZMod (Nat.card G)\nh : ↑a.val = ↑b.val\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.PGroup | {
"line": 180,
"column": 18
} | {
"line": 180,
"column": 43
} | {
"line": 180,
"column": 43
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\nthis✝ : Fintype α\nthis : Fintype ↑(fixedPoints G α)\nx : α\n⊢ card { y // Quotient.mk'' y = Quotient.mk'' x } = card ↑(orbit G x)",
"ppTerm": "?m.122",
"as... | [
"p : ℕ\nG : Type u_1\ninst✝² : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\nthis✝ : Fintype α\nthis : Fintype ↑(fixedPoints G α)\nx : α\n⊢ card { y // (orbitRel G α) y x } = card ↑(orbit G x)"
] | simp only [Quotient.eq''] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Exponent | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 13
} | {
"line": 569,
"column": 14
} | [
{
"pp": "case h.h\nι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nj : ι\nhj : ∀ (n : ℕ), 0 < n → ∃ g, g ^ n ≠ 1\nn : ℕ\nhn : 0 < n\nm : M j\nhm : m ^ n ≠ 1\nh : Pi.mulSingle j m ^ n = 1\n⊢ m ^ n = 1",
"ppTerm": "?h.h",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"case h.h\nι : Type u_1\nM : ι → Type u_2\ninst✝ : (i : ι) → Monoid (M i)\nj : ι\nhj : ∀ (n : ℕ), 0 < n → ∃ g, g ^ n ≠ 1\nn : ℕ\nhn : 0 < n\nm : M j\nhm : m ^ n ≠ 1\nh : Pi.mulSingle j m ^ n = 1\n⊢ m ^ n = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Exponent | {
"line": 608,
"column": 2
} | {
"line": 608,
"column": 79
} | {
"line": 609,
"column": 2
} | [
{
"pp": "case refine_2\nM₁ : Type u_1\nM₂ : Type u_2\ninst✝¹ : Monoid M₁\ninst✝ : Monoid M₂\n⊢ exponent M₁ ∣ exponent (M₁ × M₂)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Prod.fst_surjective",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"MonoidHom.inst... | [
"case refine_3\nM₁ : Type u_1\nM₂ : Type u_2\ninst✝¹ : Monoid M₁\ninst✝ : Monoid M₂\n⊢ exponent M₂ ∣ exponent (M₁ × M₂)"
] | · exact MonoidHom.exponent_dvd (f := MonoidHom.fst M₁ M₂) Prod.fst_surjective | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1151,
"column": 2
} | {
"line": 1151,
"column": 13
} | {
"line": 1151,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nf : R\n⊢ IsRetrocompact ↑(basicOpen f)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.zeroLocus",
"PrimeSpectrum.basicOpen",
"congrArg",
"Compl.compl",
"IsRetrocompact",
"Topolog... | [
"R : Type u\ninst✝ : CommSemiring R\nf : R\n⊢ IsRetrocompact (zeroLocus {f})ᶜ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1176,
"column": 2
} | {
"line": 1176,
"column": 13
} | {
"line": 1176,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : f.IsIntegral\nx : PrimeSpectrum S\nhx : IsClosed {x}\n⊢ IsClosed {comap f x}",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nS : Type u_2\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : f.IsIntegral\nx : PrimeSpectrum S\nhx : IsClosed {x}\n⊢ IsClosed {comap f x}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 52
} | {
"line": 105,
"column": 53
} | [
{
"pp": "case h.inr\nα : Type u_1\ninst✝² : Group α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhn : ∀ (n : ℕ), 0 < n → #{a | a ^ n = 1} ≤ n\nd : ℕ\nIH : ∀ m < d, m ∣ Fintype.card α → 0 < #{a | orderOf a = m} → #{a | orderOf a = m} = φ m\nhd : d ∣ Fintype.card α\nhpos : 0 < #{a | orderOf a = d}\nhd0 : d ≠ 0\na ... | [
"case h.inr\nα : Type u_1\ninst✝² : Group α\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhn : ∀ (n : ℕ), 0 < n → #{a | a ^ n = 1} ≤ n\nd : ℕ\nIH : ∀ m < d, m ∣ Fintype.card α → 0 < #{a | orderOf a = m} → #{a | orderOf a = m} = φ m\nhd : d ∣ Fintype.card α\nhpos : 0 < #{a | orderOf a = d}\nhd0 : d ≠ 0\na : α\nha' : a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.PGroup | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 47
} | {
"line": 400,
"column": 4
} | [
{
"pp": "n : ℕ\nG : Type u_2\ninst✝¹ : AddCommGroup G\ninst✝ : Module (ZMod n) G\n⊢ IsPGroup n (Multiplicative G)",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Multiplicative.group",
"Eq.mpr",
"InvOneClass.toOne",
"Equiv.instEquivLike",
"DivInvOneMonoid.toIn... | [
"n : ℕ\nG : Type u_2\ninst✝¹ : AddCommGroup G\ninst✝ : Module (ZMod n) G\n⊢ ∀ (a : G), ∃ k, Multiplicative.ofAdd a ^ n ^ k = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Rank | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 47
} | {
"line": 55,
"column": 48
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : FG G\nh : rank G = 0\ns : Finset G\nhs : s = ∅\nhs' : Subgroup.closure ↑s = ⊤\n⊢ Subsingleton G",
"ppTerm": "?m.50",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : FG G\nh : rank G = 0\ns : Finset G\nhs : s = ∅\nhs' : Subgroup.closure ↑s = ⊤\n⊢ Subsingleton G"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Sylow | {
"line": 289,
"column": 33
} | {
"line": 289,
"column": 56
} | {
"line": 289,
"column": 57
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝ : Group G\ng : G\nP : Sylow p G\nh : G\nx✝ : h ∈ g • P\na : G\nb : a ∈ ↑↑P\nc : ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) a = h\n⊢ g⁻¹ * ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) a * g ∈ ↑P",
"ppTerm": "?m.61",
"assigned":... | [
"p : ℕ\nG : Type u_1\ninst✝ : Group G\ng : G\nP : Sylow p G\nh : G\nx✝ : h ∈ g • P\na : G\nb : a ∈ ↑↑P\nc : ((MulDistribMulAction.toMonoidEnd (MulAut G) G) (MulAut.conj g)) a = h\n⊢ a ∈ P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 187,
"column": 30
} | {
"line": 187,
"column": 59
} | {
"line": 187,
"column": 60
} | [
{
"pp": "G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b, ... | [
"G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b, ⋯⟩\n⊢ x ^ m ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 188,
"column": 30
} | {
"line": 188,
"column": 59
} | {
"line": 188,
"column": 60
} | [
{
"pp": "G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b,... | [
"G : Type u_2\nG' : Type u_3\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : IsCyclic G'\nf : G →* G'\nhf : f.ker ≤ center G\na b : G\nx : G'\ny : G\nhxy : f y = x\nhx : ∀ (a : ↥f.range), a ∈ zpowers ⟨x, ⋯⟩\nm : ℤ\nhm✝ : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) m = ⟨f a, ⋯⟩\nn : ℤ\nhn : (fun x_1 ↦ ⟨x, ⋯⟩ ^ x_1) n = ⟨f b, ⋯⟩\nhm : x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 286,
"column": 46
} | {
"line": 286,
"column": 57
} | {
"line": 286,
"column": 58
} | [
{
"pp": "α : Type u_1\nG : Type u_2\nG' : Type u_3\na : α\np : ℕ\nhp : Fact (Nat.Prime p)\n⊢ Nat.Prime (Nat.card (ZMod p))",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Nat.Prime",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass... | [
"α : Type u_1\nG : Type u_2\nG' : Type u_3\na : α\np : ℕ\nhp : Fact (Nat.Prime p)\n⊢ Nat.Prime p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 374,
"column": 4
} | {
"line": 374,
"column": 45
} | {
"line": 374,
"column": 46
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝¹ : Finite α\nthis✝ : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\nthis : orderOf g ∈ (p ^ 2).divisors\n⊢ ∃ a < 3, p ^ a = orderOf g",
"ppTerm": "?m.151",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝¹ : Finite α\nthis✝ : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\nthis : orderOf g ∈ (p ^ 2).divisors\n⊢ ∃ a < 3, p ^ a = orderOf g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 375,
"column": 28
} | {
"line": 375,
"column": 39
} | {
"line": 375,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝² : Finite α\nthis✝¹ : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\nthis✝ : orderOf g ∈ (p ^ 2).divisors\nthis : ∃ a < 3, p ^ a = orderOf g\n⊢ ?m.163",
"ppTerm": "?m.168",
"assigned": false,
"used... | [
"α : Type u_1\ninst✝ : Group α\np : ℕ\nhp : Nat.Prime p\nhα : Nat.card α = p ^ 2\nthis✝² : Finite α\nthis✝¹ : Nontrivial α\nh_cyc : ¬IsCyclic α\ng : α\nhg : g ≠ 1\nthis✝ : orderOf g ∈ (p ^ 2).divisors\nthis : ∃ a < 3, p ^ a = orderOf g\n⊢ ?m.163"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 1281,
"column": 2
} | {
"line": 1282,
"column": 9
} | {
"line": 1282,
"column": 10
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\nhs : s ∈ irreducibleComponents (PrimeSpectrum R)\n⊢ ((zeroLocus ∘ SetLike.coe) ∘ vanishingIdeal) s = id s",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"PrimeSpectrum.... | [
"R : Type u\ninst✝ : CommSemiring R\ns : Set (PrimeSpectrum R)\nhs : s ∈ irreducibleComponents (PrimeSpectrum R)\n⊢ IsClosed s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 13
} | {
"line": 428,
"column": 14
} | [
{
"pp": "G : Type u_2\ninst✝ : AddGroup G\ng : G\nhg : ∀ (x : G), x ∈ zmultiples g\nn : ℕ\nhn : Nat.card G = n\n⊢ (zmodAddEquivOfGenerator hg hn) 1 = g",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_2\ninst✝ : AddGroup G\ng : G\nhg : ∀ (x : G), x ∈ zmultiples g\nn : ℕ\nhn : Nat.card G = n\n⊢ (zmodAddEquivOfGenerator hg hn) 1 = g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 13
} | {
"line": 433,
"column": 14
} | [
{
"pp": "G : Type u_2\ninst✝ : AddGroup G\ng : G\nhg : ∀ (x : G), x ∈ zmultiples g\nn : ℕ\nhn : Nat.card G = n\n⊢ (zmodAddEquivOfGenerator hg hn).symm g = 1",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_2\ninst✝ : AddGroup G\ng : G\nhg : ∀ (x : G), x ∈ zmultiples g\nn : ℕ\nhn : Nat.card G = n\n⊢ (zmodAddEquivOfGenerator hg hn).symm g = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 448,
"column": 17
} | {
"line": 448,
"column": 58
} | {
"line": 449,
"column": 2
} | [
{
"pp": "case h\nG : Type u_2\ninst✝¹ : CommGroup G\ninst✝ : IsSimpleGroup G\ng : G\nhg : zpowers g = ⊤\n⊢ Nat.Prime (orderOf g) ∧ Nonempty (Additive G ≃+ ZMod (orderOf g))",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Prime",
"ZMod.commRing",
"Monoid.... | [
"case h\nG : Type u_2\ninst✝¹ : CommGroup G\ninst✝ : IsSimpleGroup G\ng : G\nhg : zpowers g = ⊤\n⊢ Nat.Prime (Nat.card G) ∧ Nonempty (Additive G ≃+ ZMod (Nat.card G))"
] | rw [orderOf_eq_card_of_zpowers_eq_top hg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 559,
"column": 2
} | {
"line": 560,
"column": 75
} | {
"line": 562,
"column": 0
} | [
{
"pp": "G : Type u_2\ninst✝¹ : Infinite G\ninst✝ : AddGroup G\ng : G\nhg : zmultiples g = ⊤\n⊢ Function.Bijective ⇑((zmultiplesHom G) g)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"AddSubgroup.instBot",
"Equiv.instEquivLike",
"A... | [] | refine ⟨(AddMonoidHom.ker_eq_bot_iff _).mp ?_, AddMonoidHom.range_eq_top.mp hg⟩
simp [zmultiplesHom_ker_eq, ← infinite_zmultiples, hg, Set.infinite_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 559,
"column": 2
} | {
"line": 560,
"column": 75
} | {
"line": 562,
"column": 0
} | [
{
"pp": "G : Type u_2\ninst✝¹ : Infinite G\ninst✝ : AddGroup G\ng : G\nhg : zmultiples g = ⊤\n⊢ Function.Bijective ⇑((zmultiplesHom G) g)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"AddSubgroup.instBot",
"Equiv.instEquivLike",
"A... | [] | refine ⟨(AddMonoidHom.ker_eq_bot_iff _).mp ?_, AddMonoidHom.range_eq_top.mp hg⟩
simp [zmultiplesHom_ker_eq, ← infinite_zmultiples, hg, Set.infinite_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Sylow | {
"line": 556,
"column": 32
} | {
"line": 556,
"column": 68
} | {
"line": 556,
"column": 69
} | [
{
"pp": "G : Type u\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Finite ↑↑H\nx : G\nhx : ∀ (n : G), n ∈ H ↔ x * n * x⁻¹ ∈ H\ny✝ : G ⧸ H\ny : G\nhy : Quotient.mk'' y ∈ orbit ↥H ↑x\nb : G\nhb₁ : b ∈ H\nhb₂✝ : (fun m ↦ m • ↑x) ⟨b, hb₁⟩ = Quotient.mk'' y\nhb₂ : x * ((b * x)⁻¹ * y) * x⁻¹ ∈ H\n⊢ b⁻¹ * (x * (y⁻¹ * x)⁻¹ ... | [
"G : Type u\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : Finite ↑↑H\nx : G\nhx : ∀ (n : G), n ∈ H ↔ x * n * x⁻¹ ∈ H\ny✝ : G ⧸ H\ny : G\nhy : Quotient.mk'' y ∈ orbit ↥H ↑x\nb : G\nhb₁ : b ∈ H\nhb₂✝ : (fun m ↦ m • ↑x) ⟨b, hb₁⟩ = Quotient.mk'' y\nhb₂ : x * ((b * x)⁻¹ * y) * x⁻¹ ∈ H\n⊢ b⁻¹ * (y * x⁻¹) ∈ H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Torsion | {
"line": 398,
"column": 46
} | {
"line": 398,
"column": 57
} | {
"line": 398,
"column": 58
} | [
{
"pp": "G : Type u_1\ninst✝ : CommGroup G\np : ℕ\ng : ↥(primaryComponent G p)\nx✝ : ℕ\nhk : ↑g ^ p ^ x✝ = 1\n⊢ ↑(g ^ p ^ x✝) = ↑1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Nat.instMonoid",
"Group.toDivisi... | [
"G : Type u_1\ninst✝ : CommGroup G\np : ℕ\ng : ↥(primaryComponent G p)\nx✝ : ℕ\nhk : ↑g ^ p ^ x✝ = 1\n⊢ ↑g ^ p ^ x✝ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Atoms.Finite | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 13
} | {
"line": 58,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝³ : LE α\ninst✝² : BoundedOrder α\ninst✝¹ : IsSimpleOrder α\ninst✝ : DecidableEq α\na✝ : α\n⊢ a✝ ∈ Finset.univ ↔ a✝ ∈ {⊤, ⊥}",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congrArg",
"Finset",
"OrderBot.to... | [
"α : Type u_1\ninst✝³ : LE α\ninst✝² : BoundedOrder α\ninst✝¹ : IsSimpleOrder α\ninst✝ : DecidableEq α\na✝ : α\n⊢ a✝ = ⊤ ∨ a✝ = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Atoms.Finite | {
"line": 99,
"column": 4
} | {
"line": 100,
"column": 55
} | {
"line": 101,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na b : α\nhab : a < b\n⊢ ∃ x, a ⋖ x ∧ x ≤ b",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Order.Atoms.Finite.0.instIsStronglyAtomic._simp_1",
"Preorder.toLT... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\na b : α\nhab : a < b\nx : α\nhx : x ∈ LocallyFiniteOrder.finsetIoc a b\nhxmin : ∀ ⦃y : α⦄, (fun x ↦ x ∈ LocallyFiniteOrder.finsetIoc a b) y → y ≤ x → x ≤ y\n⊢ ∃ x, a ⋖ x ∧ x ≤ b"
] | obtain ⟨x, hx, hxmin⟩ := (LocallyFiniteOrder.finsetIoc a b).exists_minimal
⟨b, by simpa [LocallyFiniteOrder.finset_mem_Ioc]⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.Atoms.Finite | {
"line": 117,
"column": 43
} | {
"line": 117,
"column": 54
} | {
"line": 117,
"column": 55
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\na : α\ninst✝ : IsStronglyAtomic α\nha : (Set.Ici a).Infinite\nhfin : {x | a ⋖ x}.Finite\nh : ∀ (b : α), a ⋖ b → (Set.Ici b).Finite\n⊢ ∀ i ∈ {x | a ⋖ x}, (Set.Ici i).Finite",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Preorder.toLT",... | [
"α : Type u_1\ninst✝¹ : PartialOrder α\na : α\ninst✝ : IsStronglyAtomic α\nha : (Set.Ici a).Infinite\nhfin : {x | a ⋖ x}.Finite\nh : ∀ (b : α), a ⋖ b → (Set.Ici b).Finite\n⊢ ∀ (i : α), a ⋖ i → (Set.Ici i).Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CompactlyGenerated.Intervals | {
"line": 28,
"column": 16
} | {
"line": 28,
"column": 27
} | {
"line": 28,
"column": 28
} | [
{
"pp": "α : Type u_2\ninst✝ : CompleteLattice α\na : α\nb : ↑(Iic a)\nh : ∀ (ι : Type u_2) (s : ι → α), ↑b ≤ iSup s → ∃ t, ↑b ≤ ⨆ a ∈ t, s a\nι : Type u_2\ns : ι → ↑(Iic a)\nhb : ↑b ≤ iSup (Subtype.val ∘ s)\nt : Finset ι\nht : ↑b ≤ ⨆ a_1 ∈ t, (Subtype.val ∘ s) a_1\n⊢ ↑b ≤ ↑(⨆ a_1 ∈ t, s a_1)",
"ppTerm": "?... | [
"α : Type u_2\ninst✝ : CompleteLattice α\na : α\nb : ↑(Iic a)\nh : ∀ (ι : Type u_2) (s : ι → α), ↑b ≤ iSup s → ∃ t, ↑b ≤ ⨆ a ∈ t, s a\nι : Type u_2\ns : ι → ↑(Iic a)\nhb : ↑b ≤ iSup (Subtype.val ∘ s)\nt : Finset ι\nht : ↑b ≤ ⨆ a_1 ∈ t, (Subtype.val ∘ s) a_1\n⊢ ↑b ≤ ⨆ i ∈ t, ↑(s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CompactlyGenerated.Intervals | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 19
} | {
"line": 43,
"column": 20
} | [
{
"pp": "case refine_2\nι : Type u_1\nα : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na : α\nx✝ : ↑(Iic a)\ns : Set α\nhs : ∀ x ∈ s, IsCompactElement x\nhx✝ : sSup s ≤ a\nhx : ∀ b ∈ s, b ≤ a\nf : ↑s → ↑(Iic a) := fun y ↦ ⟨↑y, ⋯⟩\nb : α\n⊢ b ∈ Subtype.val '' range f ↔ b ∈ s",
"ppTer... | [
"case refine_2\nι : Type u_1\nα : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na : α\nx✝ : ↑(Iic a)\ns : Set α\nhs : ∀ x ∈ s, IsCompactElement x\nhx✝ : sSup s ≤ a\nhx : ∀ b ∈ s, b ≤ a\nf : ↑s → ↑(Iic a) := fun y ↦ ⟨↑y, ⋯⟩\nb : α\n⊢ b ∈ s → b ≤ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 778,
"column": 2
} | {
"line": 778,
"column": 75
} | {
"line": 778,
"column": 76
} | [
{
"pp": "M : Type u_4\nN : Type u_5\ninst✝³ : Group M\ninst✝² : Group N\ncyc✝ : IsCyclic (M × N)\ninst✝¹ : Finite M\ninst✝ : Finite N\nhM✝ : IsCyclic M\nhN✝ : IsCyclic N\nx✝² : CommGroup (M × N) := IsCyclic.commGroup\ncyc : (Monoid.exponent M).lcm (Monoid.exponent N) = Nat.card M * Nat.card N\nx✝¹ : CommGroup M... | [
"M : Type u_4\nN : Type u_5\ninst✝³ : Group M\ninst✝² : Group N\ncyc✝ : IsCyclic (M × N)\ninst✝¹ : Finite M\ninst✝ : Finite N\nhM✝ : IsCyclic M\nhN✝ : IsCyclic N\nx✝² : CommGroup (M × N) := IsCyclic.commGroup\ncyc : (Monoid.exponent M).lcm (Monoid.exponent N) = Nat.card M * Nat.card N\nx✝¹ : CommGroup M := IsCyclic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 791,
"column": 4
} | {
"line": 791,
"column": 15
} | {
"line": 791,
"column": 16
} | [
{
"pp": "case inl\nM : Type u_4\nN : Type u_5\ninst✝³ : AddGroup M\ninst✝² : AddGroup N\ninst✝¹ : Infinite M\ninst✝ : Nontrivial N\nhMN : IsAddCyclic (ZMod 0 × ZMod 0)\nh✝ : Infinite N\nf : ZMod 0 →+ ZMod 2 := (ZMod.castHom ⋯ (ZMod 2)).toAddMonoidHom\nhf : Function.Surjective ⇑(ZMod.castHom ⋯ (ZMod 2))\nthis : ... | [
"case inl\nM : Type u_4\nN : Type u_5\ninst✝³ : AddGroup M\ninst✝² : AddGroup N\ninst✝¹ : Infinite M\ninst✝ : Nontrivial N\nhMN : IsAddCyclic (ZMod 0 × ZMod 0)\nh✝ : Infinite N\nf : ZMod 0 →+ ZMod 2 := (ZMod.castHom ⋯ (ZMod 2)).toAddMonoidHom\nhf : Function.Surjective ⇑(ZMod.castHom ⋯ (ZMod 2))\nthis : IsAddCyclic ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 795,
"column": 44
} | {
"line": 795,
"column": 60
} | {
"line": 795,
"column": 61
} | [
{
"pp": "M : Type u_4\nN : Type u_5\ninst✝³ : AddGroup M\ninst✝² : AddGroup N\ninst✝¹ : Infinite M\ninst✝ : Nontrivial N\nhMN : IsAddCyclic (ZMod 0 × ZMod (Nat.card N))\nh✝ : Finite N\nZN : Type := ZMod (Nat.card N)\nthis : IsAddCyclic (ZMod (Nat.card N) × ZN)\n⊢ Nat.card N = 1",
"ppTerm": "?m.209",
"as... | [
"M : Type u_4\nN : Type u_5\ninst✝³ : AddGroup M\ninst✝² : AddGroup N\ninst✝¹ : Infinite M\ninst✝ : Nontrivial N\nhMN : IsAddCyclic (ZMod 0 × ZMod (Nat.card N))\nh✝ : Finite N\nZN : Type := ZMod (Nat.card N)\nthis : IsAddCyclic (ZMod (Nat.card N) × ZN)\n⊢ Nat.card N = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Sylow | {
"line": 706,
"column": 2
} | {
"line": 706,
"column": 38
} | {
"line": 706,
"column": 39
} | [
{
"pp": "G : Type u\ninst✝ : Group G\nk p : ℕ\nhp : Nat.Prime p\nh : IsPGroup p G\nH : Subgroup G\nhk : k ≤ Nat.card ↥H\nhk₀ : k ≠ 0\nm : ℕ\nhmk : p ^ m ≤ k\nhkm : k < p ^ (m + 1)\nH' : Subgroup G\nH'H : H' ≤ H\nH'card : Nat.card ↥H' = p ^ m\n⊢ Nat.card ↥H' ≤ k ∧ k < p * Nat.card ↥H'",
"ppTerm": "?m.111",
... | [
"G : Type u\ninst✝ : Group G\nk p : ℕ\nhp : Nat.Prime p\nh : IsPGroup p G\nH : Subgroup G\nhk : k ≤ Nat.card ↥H\nhk₀ : k ≠ 0\nm : ℕ\nhmk : p ^ m ≤ k\nhkm : k < p ^ (m + 1)\nH' : Subgroup G\nH'H : H' ≤ H\nH'card : Nat.card ↥H' = p ^ m\n⊢ p ^ m ≤ k ∧ k < p * p ^ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Sylow | {
"line": 747,
"column": 41
} | {
"line": 747,
"column": 52
} | {
"line": 747,
"column": 53
} | [
{
"pp": "G : Type u\ninst✝¹ : Group G\ninst✝ : Fintype G\np k : ℕ\nhp : Nat.Prime p\nh : p ^ k ∣ Nat.card G\nthis : Fact (Nat.Prime p)\nH : Subgroup G\nhH : Nat.card ↥H = p ^ k\ng : G\nhg : g ∈ (↑H \\ {1}).toFinset\n⊢ g ∈ H ∧ g ≠ 1",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Inv... | [
"G : Type u\ninst✝¹ : Group G\ninst✝ : Fintype G\np k : ℕ\nhp : Nat.Prime p\nh : p ^ k ∣ Nat.card G\nthis : Fact (Nat.Prime p)\nH : Subgroup G\nhH : Nat.card ↥H = p ^ k\ng : G\nhg : g ∈ (↑H \\ {1}).toFinset\n⊢ g ∈ H ∧ ¬g = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Sylow | {
"line": 748,
"column": 4
} | {
"line": 748,
"column": 15
} | {
"line": 748,
"column": 16
} | [
{
"pp": "G : Type u\ninst✝¹ : Group G\ninst✝ : Fintype G\np k : ℕ\nhp : Nat.Prime p\nh : p ^ k ∣ Nat.card G\nthis : Fact (Nat.Prime p)\nH : Subgroup G\nhH : Nat.card ↥H = p ^ k\ng : G\nhg✝ : g ∈ (↑H \\ {1}).toFinset\nhg : g ∈ H\nhg1 : g ≠ 1\n⊢ p ∣ orderOf g",
"ppTerm": "?m.81",
"assigned": false,
"u... | [
"G : Type u\ninst✝¹ : Group G\ninst✝ : Fintype G\np k : ℕ\nhp : Nat.Prime p\nh : p ^ k ∣ Nat.card G\nthis : Fact (Nat.Prime p)\nH : Subgroup G\nhH : Nat.card ↥H = p ^ k\ng : G\nhg✝ : g ∈ (↑H \\ {1}).toFinset\nhg : g ∈ H\nhg1 : g ≠ 1\n⊢ p ∣ orderOf g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Sylow | {
"line": 824,
"column": 30
} | {
"line": 824,
"column": 41
} | {
"line": 824,
"column": 42
} | [
{
"pp": "G : Type u\ninst✝¹ : Group G\ninst✝ : Finite G\nhn : ∀ {p : ℕ} [Fact (Nat.Prime p)] (P : Sylow p G), (↑P).Normal\nthis✝ : Fintype G\nps : Finset ℕ := (Nat.card G).primeFactors\nP : (p : ℕ) → Sylow p G := default\nthis : (p : ℕ) → Fintype ↥↑(P p)\np₁ : ℕ\nhp₁ : p₁ ∈ ps\np₂ : ℕ\nhp₂ : p₂ ∈ ps\nhne : ⟨p₁,... | [
"G : Type u\ninst✝¹ : Group G\ninst✝ : Finite G\nhn : ∀ {p : ℕ} [Fact (Nat.Prime p)] (P : Sylow p G), (↑P).Normal\nthis✝ : Fintype G\nps : Finset ℕ := (Nat.card G).primeFactors\nP : (p : ℕ) → Sylow p G := default\nthis : (p : ℕ) → Fintype ↥↑(P p)\np₁ : ℕ\nhp₁ : p₁ ∈ ps\np₂ : ℕ\nhp₂ : p₂ ∈ ps\nhne : ⟨p₁, hp₁⟩ ≠ ⟨p₂,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Sylow | {
"line": 843,
"column": 30
} | {
"line": 843,
"column": 41
} | {
"line": 843,
"column": 42
} | [
{
"pp": "G : Type u\ninst✝¹ : Group G\ninst✝ : Finite G\nhn : ∀ {p : ℕ} [Fact (Nat.Prime p)] (P : Sylow p G), (↑P).Normal\nthis✝ : Fintype G\nps : Finset ℕ := (Nat.card G).primeFactors\nP : (p : ℕ) → Sylow p G := default\nthis : (p : ℕ) → Fintype ↥↑(P p)\nhcomm : _root_.Pairwise fun p₁ p₂ ↦ ∀ (x y : G), x ∈ P ↑... | [
"G : Type u\ninst✝¹ : Group G\ninst✝ : Finite G\nhn : ∀ {p : ℕ} [Fact (Nat.Prime p)] (P : Sylow p G), (↑P).Normal\nthis✝ : Fintype G\nps : Finset ℕ := (Nat.card G).primeFactors\nP : (p : ℕ) → Sylow p G := default\nthis : (p : ℕ) → Fintype ↥↑(P p)\nhcomm : _root_.Pairwise fun p₁ p₂ ↦ ∀ (x y : G), x ∈ P ↑p₁ → y ∈ P ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.JordanHolder | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 41
} | {
"line": 160,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\nx y : X\nhx : x ∈ s\nhy : y ∈ s\n⊢ x ≤ y ∨ y ≤ x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.mem_range",
"PartialOrder.toPreorder",
"CompositionSeries",
"setOf"... | [
"X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ny : X\nhy : y ∈ s\ni : Fin (s.length + 1)\nhx : s.toFun i ∈ s\n⊢ s.toFun i ≤ y ∨ y ≤ s.toFun i"
] | rcases Set.mem_range.1 hx with ⟨i, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Order.JordanHolder | {
"line": 206,
"column": 53
} | {
"line": 206,
"column": 91
} | {
"line": 206,
"column": 92
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ni : Fin (s.length + 1)\nhx : s.toFun i ≠ last s\n⊢ ↑i ≠ s.length",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"setOf",
"id",
"RelSeries.length",
"Ne",
"instOfNa... | [
"X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns : CompositionSeries X\ni : Fin (s.length + 1)\nhx : s.toFun i ≠ last s\n⊢ ¬↑i = s.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.JordanHolder | {
"line": 260,
"column": 40
} | {
"line": 260,
"column": 51
} | {
"line": 260,
"column": 52
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nh : s₁.Equivalent s₂\ni : Fin s₂.length\n⊢ Iso (s₁.toFun ((Exists.choose h).symm i).castSucc, s₁.toFun ((Exists.choose h).symm i).succ)\n (s₂.toFun i.castSucc, s₂.toFun i.succ)",
"ppTerm": "?m.24",
"a... | [
"X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ : CompositionSeries X\nh : s₁.Equivalent s₂\ni : Fin s₂.length\n⊢ Iso (s₁.toFun ((Exists.choose h).symm i).castSucc, s₁.toFun ((Exists.choose h).symm i).succ)\n (s₂.toFun i.castSucc, s₂.toFun i.succ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.JordanHolder | {
"line": 283,
"column": 6
} | {
"line": 283,
"column": 56
} | {
"line": 283,
"column": 57
} | [
{
"pp": "case refine_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : last s₁ = head s₂\nht : last t₁ = head t₂\nh₁ : s₁.Equivalent t₁\nh₂ : s₂.Equivalent t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans fin... | [
"case refine_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : last s₁ = head s₂\nht : last t₁ = head t₂\nh₁ : s₁.Equivalent t₁\nh₂ : s₂.Equivalent t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.JordanHolder | {
"line": 285,
"column": 6
} | {
"line": 285,
"column": 76
} | {
"line": 285,
"column": 77
} | [
{
"pp": "case refine_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : last s₁ = head s₂\nht : last t₁ = head t₂\nh₁ : s₁.Equivalent t₁\nh₂ : s₂.Equivalent t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans fin... | [
"case refine_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : last s₁ = head s₂\nht : last t₁ = head t₂\nh₁ : s₁.Equivalent t₁\nh₂ : s₂.Equivalent t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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