module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.LiminfLimsup | {
"line": 1024,
"column": 6
} | {
"line": 1024,
"column": 53
} | {
"line": 1024,
"column": 54
} | [
{
"pp": "case pos\nα : Type u_1\nι : Type u_4\nι' : Type u_5\ninst✝² : ConditionallyCompleteLinearOrder α\nv : Filter ι\np : ι' → Prop\ns : ι' → Set ι\ninst✝¹ : Countable (Subtype p)\ninst✝ : Nonempty (Subtype p)\nhv : v.HasBasis p s\nf : ι → α\nhs : ∀ (j : Subtype p), (s ↑j).Nonempty\nj0 : Subtype p\nhj0 : Bdd... | [
"case pos\nα : Type u_1\nι : Type u_4\nι' : Type u_5\ninst✝² : ConditionallyCompleteLinearOrder α\nv : Filter ι\np : ι' → Prop\ns : ι' → Set ι\ninst✝¹ : Countable (Subtype p)\ninst✝ : Nonempty (Subtype p)\nhv : v.HasBasis p s\nf : ι → α\nhs : ∀ (j : Subtype p), (s ↑j).Nonempty\nj0 : Subtype p\nhj0 : BddBelow (range... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 648,
"column": 2
} | {
"line": 648,
"column": 57
} | {
"line": 649,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\ninst✝ : CompleteLattice β\nf : α → β\nhf : Monotone f\na : α\nhb : ∃ b, a < b\n⊢ limsup f (𝓝[>] a) = ⨅ r, ⨅ (_ : r > a), f r",
"ppTerm": "?m.30",
"assigned": tr... | [
"α : Type u_2\nβ : Type u_3\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\ninst✝ : CompleteLattice β\nf : α → β\nhf : Monotone f\na : α\nhb : ∃ b, a < b\n⊢ ⨅ i, ⨅ (_ : a < i), ⨆ a_1 ∈ Set.Ioo a i, f a_1 = ⨅ r, ⨅ (_ : r > a), f r"
] | rw [(nhdsGT_basis_of_exists_gt hb).limsup_eq_iInf_iSup] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.EReal.Operations | {
"line": 837,
"column": 2
} | {
"line": 837,
"column": 35
} | {
"line": 837,
"column": 36
} | [
{
"pp": "x : EReal\nhx_nonneg : 0 ≤ x\nhx_ne_top : x ≠ ⊤\ny z : EReal\n⊢ (y + z) * x = y * x + z * x",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"EReal",
"id",
"instAddCommMonoidEReal",
"instHAdd",
"HAdd... | [
"x : EReal\nhx_nonneg : 0 ≤ x\nhx_ne_top : x ≠ ⊤\ny z : EReal\n⊢ x * (y + z) = x * y + x * z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 55
} | {
"line": 337,
"column": 2
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\ns : Set α\nf : α → β\ng : α → γ\nKf Kg : ℝ≥0\nhf : LipschitzOnWith Kf f s\nhg : LipschitzOnWith Kg g s\nx✝ : α\nhx : x✝ ∈ s\ny✝ : α\nhy : y✝ ∈ s\n⊢ edist ((fun x ↦ (f x, g x))... | [
"α : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\ns : Set α\nf : α → β\ng : α → γ\nKf Kg : ℝ≥0\nhf : LipschitzOnWith Kf f s\nhg : LipschitzOnWith Kg g s\nx✝ : α\nhx : x✝ ∈ s\ny✝ : α\nhy : y✝ ∈ s\n⊢ max (edist ((fun x ↦ (f x, g x)) x✝).1 ... | rw [ENNReal.coe_mono.map_max, Prod.edist_eq, max_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.AtTopBot.Finset | {
"line": 64,
"column": 41
} | {
"line": 64,
"column": 90
} | {
"line": 64,
"column": 91
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ns : Set (Finset α)\nt : Finset α\nH : ∀ (b : Finset α), t ⊆ b → b ∈ s\nb : Finset (α ⊕ β)\nhb : t.disjSum ∅ ⊆ b\n⊢ t ⊆ b.toLeft",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Finset.toLeft",
... | [
"α : Type u_3\nβ : Type u_4\ns : Set (Finset α)\nt : Finset α\nH : ∀ (b : Finset α), t ⊆ b → b ∈ s\nb : Finset (α ⊕ β)\nhb : t.disjSum ∅ ⊆ b\n⊢ ∀ x ∈ t, Sum.inl x ∈ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Finset | {
"line": 71,
"column": 42
} | {
"line": 71,
"column": 91
} | {
"line": 71,
"column": 92
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ns : Set (Finset β)\nt : Finset β\nH : ∀ (b : Finset β), t ⊆ b → b ∈ s\nb : Finset (α ⊕ β)\nhb : ∅.disjSum t ⊆ b\n⊢ t ⊆ b.toRight",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"_private.Mathlib.Ord... | [
"α : Type u_3\nβ : Type u_4\ns : Set (Finset β)\nt : Finset β\nH : ∀ (b : Finset β), t ⊆ b → b ∈ s\nb : Finset (α ⊕ β)\nhb : ∅.disjSum t ⊆ b\n⊢ ∀ x ∈ t, Sum.inr x ∈ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Finset | {
"line": 85,
"column": 33
} | {
"line": 85,
"column": 44
} | {
"line": 85,
"column": 45
} | [
{
"pp": "α : Type u_3\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrderBot α\nh✝ : Nonempty α\nh : IsDirectedOrder α\ns : Finset α\na : α\nha : ∀ i ∈ s, i ≤ a\nb : α\nhb : a ≤ b\nc : α\nhc : c ∈ s\n⊢ c ∈ Finset.Iic b",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fin... | [
"α : Type u_3\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrderBot α\nh✝ : Nonempty α\nh : IsDirectedOrder α\ns : Finset α\na : α\nha : ∀ i ∈ s, i ≤ a\nb : α\nhb : a ≤ b\nc : α\nhc : c ∈ s\n⊢ c ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Finset | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 13
} | {
"line": 101,
"column": 14
} | [
{
"pp": "α : Type u_3\ni : α\n⊢ ∀ᶠ (s : Finset α) in atTop, i ∈ s",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.inhabitedFinset",
"Finset",
"Filter.Eventually",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
... | [
"α : Type u_3\ni : α\n⊢ ∃ a, ∀ (b : Finset α), a ⊆ b → i ∈ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.SummationFilter | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 46
} | {
"line": 67,
"column": 4
} | [
{
"pp": "β : Type u_2\nL : SummationFilter β\nc : Set β\n⊢ L.support ⊆ c ↔ (map SetLike.coe L.filter).limsInf ⊆ c",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Filter.limsInf",
"congrArg",
"Filter.map",
"Finset",
"Filter.Eventually",
"... | [
"β : Type u_2\nL : SummationFilter β\nc : Set β\n⊢ (∀ (x : β), (∀ᶠ (s : Finset β) in L.filter, x ∈ s) → x ∈ c) ↔\n ∀ (t' : Set β), (∀ᶠ (a : Finset β) in L.filter, t' ⊆ ↑a) → t' ⊆ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.SummationFilter | {
"line": 68,
"column": 58
} | {
"line": 68,
"column": 69
} | {
"line": 68,
"column": 70
} | [
{
"pp": "β : Type u_2\nL : SummationFilter β\nc : Set β\nhL : ∀ (t' : Set β), (∀ᶠ (a : Finset β) in L.filter, t' ⊆ ↑a) → t' ⊆ c\nx : β\nhx : ∀ᶠ (s : Finset β) in L.filter, x ∈ s\n⊢ ∀ᶠ (a : Finset β) in L.filter, {x} ⊆ ↑a",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"β : Type u_2\nL : SummationFilter β\nc : Set β\nhL : ∀ (t' : Set β), (∀ᶠ (a : Finset β) in L.filter, t' ⊆ ↑a) → t' ⊆ c\nx : β\nhx : ∀ᶠ (s : Finset β) in L.filter, x ∈ s\n⊢ ∀ᶠ (a : Finset β) in L.filter, x ∈ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.LiminfLimsup | {
"line": 1056,
"column": 4
} | {
"line": 1056,
"column": 44
} | {
"line": 1056,
"column": 45
} | [
{
"pp": "case neg.hs\nα : Type u_1\nι : Type u_4\nι' : Type u_5\ninst✝² : ConditionallyCompleteLinearOrder α\nv : Filter ι\np : ι' → Prop\ns : ι' → Set ι\ninst✝¹ : Countable (Subtype p)\ninst✝ : Nonempty (Subtype p)\nhv : v.HasBasis p s\nf : ι → α\nH' : ¬∀ (j : Subtype p), ¬BddBelow (range fun i ↦ f ↑i)\nH : ∀ ... | [
"case neg.hs\nα : Type u_1\nι : Type u_4\nι' : Type u_5\ninst✝² : ConditionallyCompleteLinearOrder α\nv : Filter ι\np : ι' → Prop\ns : ι' → Set ι\ninst✝¹ : Countable (Subtype p)\ninst✝ : Nonempty (Subtype p)\nhv : v.HasBasis p s\nf : ι → α\nH' : ¬∀ (j : Subtype p), ¬BddBelow (range fun i ↦ f ↑i)\nH : ∀ (j : Subtype... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.SummationFilter | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 46
} | {
"line": 188,
"column": 47
} | [
{
"pp": "case mp\nγ : Type u_3\nβ : Type u_4\nf : γ ↪ β\ns : Set (Finset γ)\nt : Finset β\nht : ∀ (b : Finset β), t ⊆ b → b.preimage ⇑f ⋯ ∈ s\nx : Finset γ\nhx : t.preimage ⇑f ⋯ ⊆ x\n⊢ x ∈ s",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nγ : Type u_3\nβ : Type u_4\nf : γ ↪ β\ns : Set (Finset γ)\nt : Finset β\nht : ∀ (b : Finset β), t ⊆ b → b.preimage ⇑f ⋯ ∈ s\nx : Finset γ\nhx : t.preimage ⇑f ⋯ ⊆ x\n⊢ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.SummationFilter | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 48
} | {
"line": 208,
"column": 49
} | [
{
"pp": "case mk\nβ : Type u_4\ninst✝² : Finite β\nthis : Fintype β\nF : Filter (Finset β)\ninst✝¹ : { filter := F }.LeAtTop\ninst✝ : { filter := F }.NeBot\nhAtTop : True\nhL : F ≤ pure Finset.univ\ns : Set (Finset β)\nhs : s ∈ F\nhs' : Finset.univ ∉ s\n⊢ ∅ ∈ F",
"ppTerm": "?mk",
"assigned": false,
... | [
"case mk\nβ : Type u_4\ninst✝² : Finite β\nthis : Fintype β\nF : Filter (Finset β)\ninst✝¹ : { filter := F }.LeAtTop\ninst✝ : { filter := F }.NeBot\nhAtTop : True\nhL : F ≤ pure Finset.univ\ns : Set (Finset β)\nhs : s ∈ F\nhs' : Finset.univ ∉ s\n⊢ ∅ ∈ F"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 239,
"column": 48
} | {
"line": 239,
"column": 59
} | {
"line": 239,
"column": 60
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nL : SummationFilter β\nf : β → α\na : α\ns : Set β\nhf : mulSupport f ⊆ s\n⊢ ∀ x ∉ Set.range fun a ↦ ↑a, f x = 1",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nL : SummationFilter β\nf : β → α\na : α\ns : Set β\nhf : mulSupport f ⊆ s\n⊢ ∀ x ∉ s, f x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 13
} | {
"line": 244,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf : β → α\na : α\ns : Set β\nhf : mulSupport f ⊆ s\n⊢ HasProd (f ∘ Subtype.val) a ↔ HasProd f a",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf : β → α\na : α\ns : Set β\nhf : mulSupport f ⊆ s\n⊢ HasProd (f ∘ Subtype.val) a ↔ HasProd f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 263,
"column": 2
} | {
"line": 263,
"column": 13
} | {
"line": 263,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : Fintype β\nf : β → α\nL : SummationFilter β\ninst✝ : L.LeAtTop\n⊢ HasProd f (∏ b, f b) L",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : Fintype β\nf : β → α\nL : SummationFilter β\ninst✝ : L.LeAtTop\n⊢ HasProd f (∏ b, f b) L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 27
} | {
"line": 270,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ns : Finset β\nf : β → α\nL : SummationFilter ↑↑s\ninst✝¹ : L.HasSupport\ninst✝ : DecidablePred fun x ↦ x ∈ L.support\n⊢ HasProd (f ∘ Subtype.val) (∏ b ∈ map (Embedding.subtype fun x ↦ x ∈ ↑s) L.support.toFinset, f b) L",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ns : Finset β\nf : β → α\nL : SummationFilter ↑↑s\ninst✝¹ : L.HasSupport\ninst✝ : DecidablePred fun x ↦ x ∈ L.support\n⊢ HasProd (f ∘ Subtype.val) (∏ x ∈ L.support.toFinset, f ↑x) L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 46
} | {
"line": 277,
"column": 47
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\ns : Finset β\nf : β → α\nL : SummationFilter ↑↑s\ninst✝ : L.LeAtTop\n⊢ HasProd (f ∘ Subtype.val) (∏ b ∈ s, f b) L",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Finset",
"HasProd",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\ns : Finset β\nf : β → α\nL : SummationFilter ↑↑s\ninst✝ : L.LeAtTop\n⊢ HasProd (f ∘ Subtype.val) (∏ b ∈ s, f b) L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 83
} | {
"line": 306,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nL : SummationFilter β\nf : β → α\ns : Finset β\nhf : ∀ b ∉ s, f b = 1\ninst✝ : L.HasSupport\n⊢ Multipliable f L",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"hasProd_prod_support_of_ne_finset_... | [] | exact (hasProd_prod_support_of_ne_finset_one (fun b _ hb ↦ hf b hb)).multipliable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 83
} | {
"line": 306,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nL : SummationFilter β\nf : β → α\ns : Finset β\nhf : ∀ b ∉ s, f b = 1\ninst✝ : L.HasSupport\n⊢ Multipliable f L",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"hasProd_prod_support_of_ne_finset_... | [] | exact (hasProd_prod_support_of_ne_finset_one (fun b _ hb ↦ hf b hb)).multipliable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 83
} | {
"line": 306,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nL : SummationFilter β\nf : β → α\ns : Finset β\nhf : ∀ b ∉ s, f b = 1\ninst✝ : L.HasSupport\n⊢ Multipliable f L",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"hasProd_prod_support_of_ne_finset_... | [] | exact (hasProd_prod_support_of_ne_finset_one (fun b _ hb ↦ hf b hb)).multipliable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 28
} | {
"line": 46,
"column": 29
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalGroup α\nf : β → α\nhf : Multipliable (fun b ↦ (f b)⁻¹) L\n⊢ Multipliable f L",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalGroup α\nf : β → α\nhf : Multipliable (fun b ↦ (f b)⁻¹) L\n⊢ Multipliable f L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 35
} | {
"line": 66,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalGroup α\nf g : β → α\nhg : Multipliable g L\nhfg : Multipliable (fun b ↦ f b / g b) L\n⊢ Multipliable f L",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalGroup α\nf g : β → α\nhg : Multipliable g L\nhfg : Multipliable (fun b ↦ f b / g b) L\n⊢ Multipliable f L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 71,
"column": 31
} | {
"line": 71,
"column": 57
} | {
"line": 71,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalGroup α\nf g : β → α\nhfg : Multipliable (fun b ↦ f b / g b) L\nhf : Multipliable f L\n⊢ Multipliable (fun b ↦ g b / f b) L",
"ppTerm": "?m.31",
"assigned": false,
"use... | [
"α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalGroup α\nf g : β → α\nhfg : Multipliable (fun b ↦ f b / g b) L\nhf : Multipliable f L\n⊢ Multipliable (fun b ↦ g b / f b) L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 35
} | {
"line": 74,
"column": 36
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\nm : M\ninst✝ : ContinuousMul M\nf : ℕ → M\nh : HasProd (fun n ↦ f (n + 1)) m\n⊢ HasProd f (f 0 * m)",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\nm : M\ninst✝ : ContinuousMul M\nf : ℕ → M\nh : HasProd (fun n ↦ f (n + 1)) m\n⊢ HasProd f (f 0 * m)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 33
} | {
"line": 82,
"column": 34
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\nm m' : M\ninst✝ : ContinuousMul M\nf : ℕ → M\nhe : HasProd (fun k ↦ f (2 * k)) m\nthis : Injective fun x ↦ 2 * x\nho : HasProd (fun x ↦ f ↑x) m'\n⊢ IsCompl (Set.range fun x ↦ 2 * x) (Set.range ((fun x ↦ x + 1) ∘ fun x ↦ 2 * x))",
"pp... | [
"M : Type u_1\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\nm m' : M\ninst✝ : ContinuousMul M\nf : ℕ → M\nhe : HasProd (fun k ↦ f (2 * k)) m\nthis : Injective fun x ↦ 2 * x\nho : HasProd (fun x ↦ f ↑x) m'\n⊢ IsCompl {a | Even a} {a | Odd a}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 36
} | {
"line": 134,
"column": 0
} | [
{
"pp": "case e'_6\nα : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝⁴ : CommGroup α\ninst✝³ : TopologicalSpace α\ninst✝² : IsTopologicalGroup α\nf : β → α\na : α\ninst✝¹ : L.LeAtTop\ninst✝ : DecidableEq β\nhf : HasProd f a L\nb : β\n⊢ a / f b = 1 / f b * a",
"ppTerm": "?e'_6",
"assigned": true,
... | [] | rw [div_mul_eq_mul_div, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 36
} | {
"line": 134,
"column": 0
} | [
{
"pp": "case e'_6\nα : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝⁴ : CommGroup α\ninst✝³ : TopologicalSpace α\ninst✝² : IsTopologicalGroup α\nf : β → α\na : α\ninst✝¹ : L.LeAtTop\ninst✝ : DecidableEq β\nhf : HasProd f a L\nb : β\n⊢ a / f b = 1 / f b * a",
"ppTerm": "?e'_6",
"assigned": true,
... | [] | rw [div_mul_eq_mul_div, one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 36
} | {
"line": 134,
"column": 0
} | [
{
"pp": "case e'_6\nα : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝⁴ : CommGroup α\ninst✝³ : TopologicalSpace α\ninst✝² : IsTopologicalGroup α\nf : β → α\na : α\ninst✝¹ : L.LeAtTop\ninst✝ : DecidableEq β\nhf : HasProd f a L\nb : β\n⊢ a / f b = 1 / f b * a",
"ppTerm": "?e'_6",
"assigned": true,
... | [] | rw [div_mul_eq_mul_div, one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 214,
"column": 2
} | {
"line": 215,
"column": 96
} | {
"line": 216,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : CommGroup α\ninst✝ : IsUniformGroup α\nf : β → α\n⊢ (CauchySeq fun s ↦ ∏ b ∈ s, f b) ↔ ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Filter.ins... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : CommGroup α\ninst✝ : IsUniformGroup α\nf : β → α\n⊢ Tendsto (fun x ↦ (∏ b ∈ x.2, f b) / ∏ b ∈ x.1, f b) atTop (𝓝 1) ↔\n ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e"
] | simp only [CauchySeq, cauchy_map_iff, prod_atTop_atTop_eq,
uniformity_eq_comap_nhds_one α, tendsto_comap_iff, Function.comp_def, atTop_neBot, true_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 35
} | {
"line": 196,
"column": 36
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : TopologicalSpace M\ninst✝¹ : T2Space M\ninst✝ : ContinuousMul M\nf : ℕ → M\nhf : Multipliable fun n ↦ f (n + 1)\n⊢ ∏' (b : ℕ), f b = f 0 * ∏' (b : ℕ), f (b + 1)",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"M : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : TopologicalSpace M\ninst✝¹ : T2Space M\ninst✝ : ContinuousMul M\nf : ℕ → M\nhf : Multipliable fun n ↦ f (n + 1)\n⊢ ∏' (b : ℕ), f b = f 0 * ∏' (b : ℕ), f (b + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 49
} | {
"line": 218,
"column": 2
} | [
{
"pp": "G : Type u_2\ninst✝² : CommGroup G\ng : G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nf : ℕ → G\nk : ℕ\n⊢ HasProd (fun n ↦ f (n + k)) g ↔ HasProd f (g * ∏ i ∈ range k, f i)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneCla... | [
"G : Type u_2\ninst✝² : CommGroup G\ng : G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nf : ℕ → G\nk : ℕ\n⊢ HasProd (fun n ↦ f (n + k)) g ↔ HasProd (fun x ↦ f ↑x) g"
] | refine Iff.trans ?_ (range k).hasProd_compl_iff | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 69
} | {
"line": 223,
"column": 70
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : CommGroup α\ninst✝ : IsUniformGroup α\nf : β → α\nh✝ : ∀ s ∈ 𝓝 1, ∃ a, ∀ (b : Finset β × Finset β), a ≤ b → (∏ b ∈ b.2, f b) / ∏ b ∈ b.1, f b ∈ s\ne : Set α\nhe : e ∈ 𝓝 1\ns₁ s₂ t : Finset β\nh : (∏ b ∈ (s₁ ∪ s₂, s₁ ∪ s₂ ∪ t).2, f ... | [
"case h\nα : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : CommGroup α\ninst✝ : IsUniformGroup α\nf : β → α\nh✝ : ∀ s ∈ 𝓝 1, ∃ a, ∀ (b : Finset β × Finset β), a ≤ b → (∏ b ∈ b.2, f b) / ∏ b ∈ b.1, f b ∈ s\ne : Set α\nhe : e ∈ 𝓝 1\ns₁ s₂ t : Finset β\nh : (∏ b ∈ (s₁ ∪ s₂, s₁ ∪ s₂ ∪ t).2, f b) / ∏ b ∈ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 36
} | {
"line": 253,
"column": 37
} | [
{
"pp": "case pos\nG : Type u_2\ninst✝³ : CommGroup G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : T2Space G\nf : ℕ → G\nhf : Multipliable f\nh₀ : (fun i ↦ (∏' (i : ℕ), f i) / ∏ j ∈ range i, f j) = fun i ↦ ∏' (k : ℕ), f (k + i)\nh₁ : Tendsto (fun x ↦ ∏' (i : ℕ), f i) atTop (𝓝 (∏' (i : ℕ... | [
"case pos\nG : Type u_2\ninst✝³ : CommGroup G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : T2Space G\nf : ℕ → G\nhf : Multipliable f\nh₀ : (fun i ↦ (∏' (i : ℕ), f i) / ∏ j ∈ range i, f j) = fun i ↦ ∏' (k : ℕ), f (k + i)\nh₁ : Tendsto (fun x ↦ ∏' (i : ℕ), f i) atTop (𝓝 (∏' (i : ℕ), f i))\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 157,
"column": 45
} | {
"line": 157,
"column": 56
} | {
"line": 157,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\nb : β\nhf : ∀ (b' : β), b' ≠ b → f b' = 1\nL : SummationFilter β\ninst✝ : L.LeAtTop\nthis : HasProd f (∏ b' ∈ {b}, f b') L\n⊢ HasProd f (f b) L",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants":... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\nb : β\nhf : ∀ (b' : β), b' ≠ b → f b' = 1\nL : SummationFilter β\ninst✝ : L.LeAtTop\nthis : HasProd f (∏ b' ∈ {b}, f b') L\n⊢ HasProd f (f b) L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 157,
"column": 6
} | {
"line": 157,
"column": 54
} | {
"line": 157,
"column": 55
} | [
{
"pp": "case pos\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝³ : CommMonoid α\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\na₂ : α\nha₂ : 1 ≤ a₂\nh : ∀ (s : Finset ι), ∏ i ∈ s, f i ≤ a₂\nhL : ¬L.NeBot\nhf : (mulSupport f).Finite\n⊢ ∏'[L] (i : ι), f i ... | [
"case pos\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝³ : CommMonoid α\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\na₂ : α\nha₂ : 1 ≤ a₂\nh : ∀ (s : Finset ι), ∏ i ∈ s, f i ≤ a₂\nhL : ¬L.NeBot\nhf : (mulSupport f).Finite\n⊢ ∏ i ∈ hf.toFinset, f i ≤ a₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 273,
"column": 4
} | {
"line": 273,
"column": 30
} | {
"line": 274,
"column": 6
} | [
{
"pp": "case neg\nι : Type u_4\nα : Type u_5\nα' : Type u_6\nG : Type u_7\ninst✝⁶ : CommMonoid α\ninst✝⁵ : CommMonoid α'\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace α'\ninst✝² : T2Space α'\nf : ι → α\nL : SummationFilter ι\ng : G\ninst✝¹ : FunLike G α α'\ninst✝ : MonoidHomClass G α α'\nhge : IsClos... | [
"case neg\nι : Type u_4\nα : Type u_5\nα' : Type u_6\nG : Type u_7\ninst✝⁶ : CommMonoid α\ninst✝⁵ : CommMonoid α'\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace α'\ninst✝² : T2Space α'\nf : ι → α\nL : SummationFilter ι\ng : G\ninst✝¹ : FunLike G α α'\ninst✝ : MonoidHomClass G α α'\nhge : IsClosedEmbedding ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 173,
"column": 6
} | {
"line": 173,
"column": 32
} | {
"line": 173,
"column": 33
} | [
{
"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\ng : ι → α\nh : ∀ (i : ι), 1 ≤ g i\nhg : Multipliable g L\nhL : ¬L.NeBot\n⊢ 1 ≤ ∏'[L] (i : ι), g i",
"ppTer... | [
"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\ng : ι → α\nh : ∀ (i : ι), 1 ≤ g i\nhg : Multipliable g L\nhL : ¬L.NeBot\n⊢ 1 ≤ ∏ᶠ (i : ι), g i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 49
} | {
"line": 210,
"column": 50
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝⁷ : CommGroup α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedMonoid α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : IsTopologicalGroup α\ninst✝² : OrderClosedTopology α\nf g : ι → α\na₁ a₂ : α\ni : ι\ninst✝¹ : L.NeBot\ninst✝ : L.LeAtTop\nh : f ≤ g\nhi : f... | [
"ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝⁷ : CommGroup α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedMonoid α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : IsTopologicalGroup α\ninst✝² : OrderClosedTopology α\nf g : ι → α\na₁ a₂ : α\ni : ι\ninst✝¹ : L.NeBot\ninst✝ : L.LeAtTop\nh : f ≤ g\nhi : f i < g i\nhf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 337,
"column": 17
} | {
"line": 337,
"column": 39
} | {
"line": 337,
"column": 40
} | [
{
"pp": "case succ\nα : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\na : α\nL : SummationFilter β\ninst✝ : ContinuousMul α\nhf : HasProd f a L\nn : ℕ\nhn : HasProd (fun x ↦ f x ^ n) (a ^ n) L\n⊢ HasProd (fun x ↦ f x ^ (n + 1)) (a ^ (n + 1)) L",
"ppTerm": "?succ",
... | [
"case succ\nα : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\na : α\nL : SummationFilter β\ninst✝ : ContinuousMul α\nhf : HasProd f a L\nn : ℕ\nhn : HasProd (fun x ↦ f x ^ n) (a ^ n) L\n⊢ HasProd (fun x ↦ f x ^ n * f x) (a ^ n * a) L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 326,
"column": 4
} | {
"line": 326,
"column": 15
} | {
"line": 326,
"column": 16
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\ninst✝⁵ : AddCommGroup α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsOrderedAddMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x ↦ b\ns : Finset ι\n⊢ #s • b ≤ ∑' (x : ι), b",
"ppTerm": "?m.29",
... | [
"ι : Type u_1\nα : Type u_3\ninst✝⁵ : AddCommGroup α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsOrderedAddMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x ↦ b\ns : Finset ι\n⊢ #s • b ≤ ∑' (x : ι), b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 329,
"column": 4
} | {
"line": 329,
"column": 44
} | {
"line": 329,
"column": 45
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\ninst✝⁵ : AddCommGroup α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsOrderedAddMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x ↦ b\nH : ∀ (s : Finset ι), #s • b ≤ ∑' (x : ι), b\nn : ℕ\nhn : ∑' (x : ι... | [
"ι : Type u_1\nα : Type u_3\ninst✝⁵ : AddCommGroup α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsOrderedAddMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x ↦ b\nH : ∀ (s : Finset ι), #s • b ≤ ∑' (x : ι), b\nn : ℕ\nhn : ∑' (x : ι), b ≤ n • b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 347,
"column": 2
} | {
"line": 350,
"column": 43
} | {
"line": 352,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nL : SummationFilter β\ninst✝ : ContinuousMul α\nf : γ → β → α\na : γ → α\ns : Finset γ\n⊢ (∀ i ∈ s, HasProd (f i) (a i) L) → HasProd (fun b ↦ ∏ i ∈ s, f i b) (∏ i ∈ s, a i) L",
"ppTerm": "?m.27",
"assi... | [] | exact Finset.induction_on s (by simp) <| by
simp +contextual only [mem_insert, forall_eq_or_imp, not_false_iff,
prod_insert, and_imp]
exact fun x s _ IH hx h ↦ hx.mul (IH h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 347,
"column": 2
} | {
"line": 350,
"column": 43
} | {
"line": 352,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nL : SummationFilter β\ninst✝ : ContinuousMul α\nf : γ → β → α\na : γ → α\ns : Finset γ\n⊢ (∀ i ∈ s, HasProd (f i) (a i) L) → HasProd (fun b ↦ ∏ i ∈ s, f i b) (∏ i ∈ s, a i) L",
"ppTerm": "?m.27",
"assi... | [] | exact Finset.induction_on s (by simp) <| by
simp +contextual only [mem_insert, forall_eq_or_imp, not_false_iff,
prod_insert, and_imp]
exact fun x s _ IH hx h ↦ hx.mul (IH h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 347,
"column": 2
} | {
"line": 350,
"column": 43
} | {
"line": 352,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nL : SummationFilter β\ninst✝ : ContinuousMul α\nf : γ → β → α\na : γ → α\ns : Finset γ\n⊢ (∀ i ∈ s, HasProd (f i) (a i) L) → HasProd (fun b ↦ ∏ i ∈ s, f i b) (∏ i ∈ s, a i) L",
"ppTerm": "?m.27",
"assi... | [] | exact Finset.induction_on s (by simp) <| by
simp +contextual only [mem_insert, forall_eq_or_imp, not_false_iff,
prod_insert, and_imp]
exact fun x s _ IH hx h ↦ hx.mul (IH h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 328,
"column": 31
} | {
"line": 328,
"column": 42
} | {
"line": 328,
"column": 43
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nm : M\nf : ℤ → M\nhf : HasProd f m\nthis : Injective Int.negSucc\nu : Finset ℤ\nv' : Finset ℕ\nhv' : u.preimage Nat.cast ⋯ ∪ u.preimage Int.negSucc ⋯ ⊆ v'\na✝ : ℕ\nhx : Int.ofNat a✝ ∈ u\n⊢ a✝ ∈ u.preimage Nat.cast ⋯ ∪ u.preimage Int.negSu... | [
"M : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nm : M\nf : ℤ → M\nhf : HasProd f m\nthis : Injective Int.negSucc\nu : Finset ℤ\nv' : Finset ℕ\nhv' : u.preimage Nat.cast ⋯ ∪ u.preimage Int.negSucc ⋯ ⊆ v'\na✝ : ℕ\nhx : Int.ofNat a✝ ∈ u\n⊢ ↑a✝ ∈ u ∨ Int.negSucc a✝ ∈ u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 40
} | {
"line": 347,
"column": 41
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\ninst✝⁴ : CommRing α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : ι → α\nx : α\nhfx : HasProd f x\n⊢ HasProd (fun x ↦ |f x|) |x|",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"... | [
"ι : Type u_1\nα : Type u_3\ninst✝⁴ : CommRing α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : ι → α\nx : α\nhfx : HasProd f x\n⊢ Tendsto (fun s ↦ |∏ x ∈ s, f x|) (SummationFilter.unconditional ι).filter (nhds |x|)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 329,
"column": 31
} | {
"line": 329,
"column": 42
} | {
"line": 329,
"column": 43
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nm : M\nf : ℤ → M\nhf : HasProd f m\nthis : Injective Int.negSucc\nu : Finset ℤ\nv' : Finset ℕ\nhv' : u.preimage Nat.cast ⋯ ∪ u.preimage Int.negSucc ⋯ ⊆ v'\na✝ : ℕ\nhx : Int.negSucc a✝ ∈ u\n⊢ a✝ ∈ u.preimage Nat.cast ⋯ ∪ u.preimage Int.neg... | [
"M : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nm : M\nf : ℤ → M\nhf : HasProd f m\nthis : Injective Int.negSucc\nu : Finset ℤ\nv' : Finset ℕ\nhv' : u.preimage Nat.cast ⋯ ∪ u.preimage Int.negSucc ⋯ ⊆ v'\na✝ : ℕ\nhx : Int.negSucc a✝ ∈ u\n⊢ ↑a✝ ∈ u ∨ Int.negSucc a✝ ∈ u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 368,
"column": 2
} | {
"line": 370,
"column": 23
} | {
"line": 372,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\ninst✝ : ContinuousMul α\nι : Type u_4\ns : Finset ι\nt : ι → Set β\na : ι → α\nhs : (↑s).Pairwise (Disjoint on t)\nhf : ∀ i ∈ s, HasProd (f ∘ Subtype.val) (a i)\n⊢ HasProd (f ∘ Subtype.val) (∏ i ∈ s, a i)",
"... | [] | simp_rw [hasProd_subtype_iff_mulIndicator] at *
rw [Finset.mulIndicator_biUnion _ _ hs]
exact hasProd_prod hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 368,
"column": 2
} | {
"line": 370,
"column": 23
} | {
"line": 372,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\ninst✝ : ContinuousMul α\nι : Type u_4\ns : Finset ι\nt : ι → Set β\na : ι → α\nhs : (↑s).Pairwise (Disjoint on t)\nhf : ∀ i ∈ s, HasProd (f ∘ Subtype.val) (a i)\n⊢ HasProd (f ∘ Subtype.val) (∏ i ∈ s, a i)",
"... | [] | simp_rw [hasProd_subtype_iff_mulIndicator] at *
rw [Finset.mulIndicator_biUnion _ _ hs]
exact hasProd_prod hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 29
} | {
"line": 376,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\na b : α\ninst✝ : ContinuousMul α\ns t : Set β\nhs : IsCompl s t\nha : HasProd (f ∘ Subtype.val) a\nhb : HasProd (f ∘ Subtype.val) b\n⊢ HasProd f (a * b)",
"ppTerm": "?m.30",
"assigned": false,
"usedCo... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\na b : α\ninst✝ : ContinuousMul α\ns t : Set β\nhs : IsCompl s t\nha : HasProd (f ∘ Subtype.val) a\nhb : HasProd (f ∘ Subtype.val) b\n⊢ HasProd f (a * b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 487,
"column": 2
} | {
"line": 488,
"column": 58
} | {
"line": 490,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\ns : Finset β\nL : SummationFilter β\ninst✝ : L.LeAtTop\n⊢ ∏ x ∈ s, f x = ∏'[L] (x : β), (↑s).mulIndicator f x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.pr... | [] | rw [tprod_eq_prod' (Set.mulSupport_mulIndicator_subset),
Finset.prod_mulIndicator_subset _ Finset.Subset.rfl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 487,
"column": 2
} | {
"line": 488,
"column": 58
} | {
"line": 490,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\ns : Finset β\nL : SummationFilter β\ninst✝ : L.LeAtTop\n⊢ ∏ x ∈ s, f x = ∏'[L] (x : β), (↑s).mulIndicator f x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.pr... | [] | rw [tprod_eq_prod' (Set.mulSupport_mulIndicator_subset),
Finset.prod_mulIndicator_subset _ Finset.Subset.rfl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 487,
"column": 2
} | {
"line": 488,
"column": 58
} | {
"line": 490,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\ns : Finset β\nL : SummationFilter β\ninst✝ : L.LeAtTop\n⊢ ∏ x ∈ s, f x = ∏'[L] (x : β), (↑s).mulIndicator f x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.pr... | [] | rw [tprod_eq_prod' (Set.mulSupport_mulIndicator_subset),
Finset.prod_mulIndicator_subset _ Finset.Subset.rfl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 498,
"column": 31
} | {
"line": 498,
"column": 42
} | {
"line": 498,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nL : SummationFilter β\ninst✝ : L.LeAtTop\nf : β → α\nb : β\nhf : ∀ (b' : β), b' ≠ b → f b' = 1\nb' : β\nhb' : b' ∉ {b}\n⊢ b' ≠ b",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"id",
"Ne"
... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nL : SummationFilter β\ninst✝ : L.LeAtTop\nf : β → α\nb : β\nhf : ∀ (b' : β), b' ≠ b → f b' = 1\nb' : β\nhb' : b' ∉ {b}\n⊢ ¬b' = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 48
} | {
"line": 296,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : CommGroup α\ninst✝¹ : IsUniformGroup α\nf : β → α\ninst✝ : CompleteSpace α\ni : γ → β\nhf : Multipliable f\nhi : Injective i\n⊢ Multipliable (f ∘ i)",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : CommGroup α\ninst✝¹ : IsUniformGroup α\nf : β → α\ninst✝ : CompleteSpace α\ni : γ → β\nhf : Multipliable f\nhi : Injective i\n⊢ Multipliable (f ∘ i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 370,
"column": 4
} | {
"line": 370,
"column": 15
} | {
"line": 370,
"column": 16
} | [
{
"pp": "α : Type u_1\nG : Type u_4\ninst✝² : TopologicalSpace G\ninst✝¹ : CommGroup G\ninst✝ : IsTopologicalGroup G\nf : α → G\nhf : Multipliable f\ne : Set G\nhe : e ∈ 𝓝 1\ns : Finset α\nhs : ∀ (t : Finset α), Disjoint t s → ∏ k ∈ t, f k ∈ e\nx : α\nhx : x ∉ s\n⊢ f x ∈ e",
"ppTerm": "?m.49",
"assigne... | [
"α : Type u_1\nG : Type u_4\ninst✝² : TopologicalSpace G\ninst✝¹ : CommGroup G\ninst✝ : IsTopologicalGroup G\nf : α → G\nhf : Multipliable f\ne : Set G\nhe : e ∈ 𝓝 1\ns : Finset α\nhs : ∀ (t : Finset α), Disjoint t s → ∏ k ∈ t, f k ∈ e\nx : α\nhx : x ∉ s\n⊢ f x ∈ e"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 399,
"column": 6
} | {
"line": 399,
"column": 41
} | {
"line": 399,
"column": 42
} | [
{
"pp": "β : Type u_2\nG : Type u_4\ninst✝⁴ : TopologicalSpace G\ninst✝³ : CommGroup G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : Infinite β\ninst✝ : T2Space G\na : G\nh : Multipliable fun x ↦ a\nha : ¬a = 1\nthis : {a}ᶜ ∈ 𝓝 1\n⊢ Finite β",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
... | [
"β : Type u_2\nG : Type u_4\ninst✝⁴ : TopologicalSpace G\ninst✝³ : CommGroup G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : Infinite β\ninst✝ : T2Space G\na : G\nh : Multipliable fun x ↦ a\nha : ¬a = 1\nthis : {a}ᶜ ∈ 𝓝 1\n⊢ Set.univ.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 414,
"column": 6
} | {
"line": 414,
"column": 42
} | {
"line": 414,
"column": 43
} | [
{
"pp": "case inr.inr\nβ : Type u_2\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : CommGroup G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : T2Space G\na : G\nhβ : Infinite β\nha : a ≠ 1\n⊢ ¬Multipliable fun b ↦ a",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"In... | [
"case inr.inr\nβ : Type u_2\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : CommGroup G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : T2Space G\na : G\nhβ : Infinite β\nha : a ≠ 1\n⊢ ¬a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 570,
"column": 2
} | {
"line": 570,
"column": 36
} | {
"line": 570,
"column": 37
} | [
{
"pp": "G : Type u_2\ninst✝³ : CommGroup G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : T2Space G\nf : ℕ → G\nhf : Multipliable f\n⊢ f 0 * ∏' (n : ℕ+), f ↑n = ∏' (n : ℕ), f n",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"PNat.val",
"Eq.mpr",
"H... | [
"G : Type u_2\ninst✝³ : CommGroup G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : T2Space G\nf : ℕ → G\nhf : Multipliable f\n⊢ ∏' (n : ℕ+), f ↑n = ∏' (b : ℕ), f (b + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 616,
"column": 52
} | {
"line": 616,
"column": 63
} | {
"line": 616,
"column": 64
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf : β → α\ns t : Set β\nhf₀ : ∀ b ∈ t, f b = 1\nb : ↑s\nhb : b ∉ Set.range (Set.inclusion ⋯)\n⊢ ↑b ∈ t",
"ppTerm": "?m.49",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf : β → α\ns t : Set β\nhf₀ : ∀ b ∈ t, f b = 1\nb : ↑s\nhb : b ∉ Set.range (Set.inclusion ⋯)\n⊢ ↑b ∈ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 587,
"column": 48
} | {
"line": 587,
"column": 59
} | {
"line": 587,
"column": 60
} | [
{
"pp": "G : Type u_2\ninst✝⁴ : CommGroup G\ninst✝³ : UniformSpace G\ninst✝² : IsUniformGroup G\ninst✝¹ : CompleteSpace G\ninst✝ : T2Space G\nf : ℤ → G\nhf2 : Multipliable f\nh1 : Multipliable fun n ↦ f ↑n\nh2 : Multipliable fun n ↦ f (-↑n)\nh3 : Multipliable fun n ↦ f ↑↑n\nh4 : Multipliable fun n ↦ f (-↑↑n)\nt... | [
"G : Type u_2\ninst✝⁴ : CommGroup G\ninst✝³ : UniformSpace G\ninst✝² : IsUniformGroup G\ninst✝¹ : CompleteSpace G\ninst✝ : T2Space G\nf : ℤ → G\nhf2 : Multipliable f\nh1 : Multipliable fun n ↦ f ↑n\nh2 : Multipliable fun n ↦ f (-↑n)\nh3 : Multipliable fun n ↦ f ↑↑n\nh4 : Multipliable fun n ↦ f (-↑↑n)\nthis : ∏' (n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 595,
"column": 2
} | {
"line": 595,
"column": 42
} | {
"line": 595,
"column": 43
} | [
{
"pp": "G : Type u_2\ninst✝⁴ : CommGroup G\ninst✝³ : UniformSpace G\ninst✝² : IsUniformGroup G\ninst✝¹ : CompleteSpace G\ninst✝ : T2Space G\nf : ℤ → G\nhf : Function.Even f\nhf2 : Multipliable f\n⊢ ∏' (n : ℤ), f n = f 0 * (∏' (n : ℕ+), f ↑↑n) ^ 2",
"ppTerm": "?m.41",
"assigned": true,
"usedConstant... | [
"G : Type u_2\ninst✝⁴ : CommGroup G\ninst✝³ : UniformSpace G\ninst✝² : IsUniformGroup G\ninst✝¹ : CompleteSpace G\ninst✝ : T2Space G\nf : ℤ → G\nhf : Function.Even f\nhf2 : Multipliable f\n⊢ ∏' (n : ℤ), f n = (f 0 * ∏' (n : ℕ+), f ↑↑n) * ∏' (n : ℕ+), f ↑↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 97,
"column": 14
} | {
"line": 97,
"column": 55
} | {
"line": 97,
"column": 56
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na₁ a₂ : α\nh : a₂ ≠ 0\nH : HasSum (fun i ↦ a₂ * f i) (a₂ * a₁) L\n⊢ HasSum f a₁ L",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants":... | [
"ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na₁ a₂ : α\nh : a₂ ≠ 0\nH : HasSum (fun i ↦ a₂ * f i) (a₂ * a₁) L\n⊢ HasSum f a₁ L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 100,
"column": 14
} | {
"line": 100,
"column": 56
} | {
"line": 100,
"column": 57
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na₁ a₂ : α\nh : a₂ ≠ 0\nH : HasSum (fun i ↦ f i * a₂) (a₁ * a₂) L\n⊢ HasSum f a₁ L",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants":... | [
"ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na₁ a₂ : α\nh : a₂ ≠ 0\nH : HasSum (fun i ↦ f i * a₂) (a₁ * a₂) L\n⊢ HasSum f a₁ L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 35
} | {
"line": 104,
"column": 36
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na₁ a₂ : α\nh : a₂ ≠ 0\n⊢ HasSum (fun i ↦ f i / a₂) (a₁ / a₂) L ↔ HasSum f a₁ L",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
... | [
"ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na₁ a₂ : α\nh : a₂ ≠ 0\n⊢ HasSum (fun i ↦ f i * a₂⁻¹) (a₁ * a₂⁻¹) L ↔ HasSum f a₁ L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 107,
"column": 14
} | {
"line": 107,
"column": 55
} | {
"line": 107,
"column": 56
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : a ≠ 0\nH : Summable (fun i ↦ a * f i) L\n⊢ Summable f L",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedF... | [
"ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : a ≠ 0\nH : Summable (fun i ↦ a * f i) L\n⊢ Summable f L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 110,
"column": 14
} | {
"line": 110,
"column": 56
} | {
"line": 110,
"column": 57
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : a ≠ 0\nH : Summable (fun i ↦ f i * a) L\n⊢ Summable f L",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedF... | [
"ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : a ≠ 0\nH : Summable (fun i ↦ f i * a) L\n⊢ Summable f L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 35
} | {
"line": 113,
"column": 36
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : a ≠ 0\n⊢ Summable (fun i ↦ f i / a) L ↔ Summable f L",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : a ≠ 0\n⊢ Summable (fun i ↦ f i * a⁻¹) L ↔ Summable f L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.NNReal.Lemmas | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 28
} | {
"line": 141,
"column": 29
} | [
{
"pp": "case inl\nα : Type u_2\nL : SummationFilter α\nh✝ : L.NeBot\ny : ℝ≥0\nf : α → ℝ≥0\nhy : HasSum (fun i ↦ ↑(f i)) (↑y) L\n⊢ HasSum (fun x ↦ ((fun i ↦ ↑(f i)) x).toNNReal) (↑y).toNNReal L",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mp... | [
"case inl\nα : Type u_2\nL : SummationFilter α\nh✝ : L.NeBot\ny : ℝ≥0\nf : α → ℝ≥0\nhy : HasSum (fun i ↦ ↑(f i)) (↑y) L\n⊢ HasSum (fun x ↦ f x) y L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 35
} | {
"line": 129,
"column": 36
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝³ : DivisionSemiring α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsTopologicalSemiring α\nf : ι → α\na : α\ninst✝ : T2Space α\n⊢ ∑'[L] (x : ι), f x / a = (∑'[L] (x : ι), f x) / a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
... | [
"ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝³ : DivisionSemiring α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsTopologicalSemiring α\nf : ι → α\na : α\ninst✝ : T2Space α\n⊢ ∑'[L] (x : ι), f x * a⁻¹ = (∑'[L] (x : ι), f x) * a⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 44
} | {
"line": 134,
"column": 45
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : HasSum (fun x ↦ 1 / f x) a L\nb : α\nthis : HasSum (fun i ↦ b * (1 / f i)) (b * a) L\n⊢ HasSum (fun i ↦ b / f i) (b * a) L",
"ppTerm": ... | [
"ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : HasSum (fun x ↦ 1 / f x) a L\nb : α\nthis : HasSum (fun i ↦ b * (1 / f i)) (b * a) L\n⊢ HasSum (fun i ↦ b * (f i)⁻¹) (b * a) L"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Real | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 91
} | {
"line": 37,
"column": 4
} | [
{
"pp": "x y x' y' : ℝ\nhx : x ∈ Icc x' y'\nhy : y ∈ Icc x' y'\n⊢ dist x y ≤ y' - x'",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Real.instSub",
"HSub.hSub",
"id",
"LE.le",
"instHSub",
"Real.pseudoMetricSpace",
... | [
"x y x' y' : ℝ\nhx : x ∈ Icc x' y'\nhy : y ∈ Icc x' y'\n⊢ |x - y| ≤ y' - x'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Real | {
"line": 40,
"column": 23
} | {
"line": 40,
"column": 50
} | {
"line": 40,
"column": 51
} | [
{
"pp": "x y : ℝ\nhx : x ∈ Icc 0 1\nhy : y ∈ Icc 0 1\n⊢ dist x y ≤ 1",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℝ\nhx : x ∈ Icc 0 1\nhy : y ∈ Icc 0 1\n⊢ dist x y ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 50
} | {
"line": 282,
"column": 51
} | [
{
"pp": "α : Type u_3\ninst✝³ : Ring α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsTopologicalRing α\ninst✝ : T2Space α\nx : α\nh : Summable fun x_1 ↦ x ^ x_1\n⊢ Tendsto (fun n ↦ ∑ i ∈ range n, x ^ i * (1 - x)) atTop (nhds 1)",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_3\ninst✝³ : Ring α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsTopologicalRing α\ninst✝ : T2Space α\nx : α\nh : Summable fun x_1 ↦ x ^ x_1\n⊢ Tendsto (fun n ↦ 1 - x ^ n) atTop (nhds 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 50
} | {
"line": 287,
"column": 51
} | [
{
"pp": "α : Type u_3\ninst✝³ : Ring α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsTopologicalRing α\ninst✝ : T2Space α\nx : α\nh : Summable fun x_1 ↦ x ^ x_1\n⊢ Tendsto (fun n ↦ ∑ i ∈ range n, (1 - x) * x ^ i) atTop (nhds 1)",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_3\ninst✝³ : Ring α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsTopologicalRing α\ninst✝ : T2Space α\nx : α\nh : Summable fun x_1 ↦ x ^ x_1\n⊢ Tendsto (fun n ↦ 1 - x ^ n) atTop (nhds 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 35
} | {
"line": 72,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\nhf : AntilipschitzWith K f\nx y : α\n⊢ K⁻¹ * nndist x y ≤ nndist (f x) (f y)",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\nhf : AntilipschitzWith K f\nx y : α\n⊢ K⁻¹ * nndist x y ≤ nndist (f x) (f y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 59
} | {
"line": 96,
"column": 60
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\ninst✝¹ : EMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\nf : α → β\nhf : AntilipschitzWith K f\nx y : α\nh : f x = f y\n⊢ x = y",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_4\nβ : Type u_5\ninst✝¹ : EMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\nf : α → β\nhf : AntilipschitzWith K f\nx y : α\nh : f x = f y\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 254,
"column": 43
} | {
"line": 254,
"column": 66
} | {
"line": 254,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\nf : α → β\nhf : LipschitzWith K f\ng : β → α\nhg : Function.RightInverse g f\nx y : β\n⊢ edist x y ≤ ↑K * edist (g x) (g y)",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"use... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\nf : α → β\nhf : LipschitzWith K f\ng : β → α\nhg : Function.RightInverse g f\nx y : β\n⊢ edist x y ≤ ↑K * edist (g x) (g y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Metrizable.Uniformity | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 84
} | {
"line": 120,
"column": 85
} | [
{
"pp": "X : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nhd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\nx y : X\nl : List X\na b c : X\nhab : d a b = 0\nhbc : d b c = 0\n⊢ d a c ≤ 0",
"ppTerm": "?m.85",
"assigned"... | [
"X : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nhd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\nx y : X\nl : List X\na b c : X\nhab : d a b = 0\nhbc : d b c = 0\n⊢ d a c = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sequences | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 30
} | {
"line": 139,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\nh :\n ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a\ns : Set X\nx : X\nhcx : x ∈ closure[inst✝] s\nhx : x ∈ s\n⊢ x ∈ seqClosure s",
"ppTerm": "?pos✝",
"assigned": true,... | [] | exact subset_seqClosure hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Sequences | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 30
} | {
"line": 139,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\nh :\n ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a\ns : Set X\nx : X\nhcx : x ∈ closure[inst✝] s\nhx : x ∈ s\n⊢ x ∈ seqClosure s",
"ppTerm": "?pos✝",
"assigned": true,... | [] | exact subset_seqClosure hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sequences | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 30
} | {
"line": 139,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\nh :\n ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a\ns : Set X\nx : X\nhcx : x ∈ closure[inst✝] s\nhx : x ∈ s\n⊢ x ∈ seqClosure s",
"ppTerm": "?pos✝",
"assigned": true,... | [] | exact subset_seqClosure hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sequences | {
"line": 140,
"column": 6
} | {
"line": 142,
"column": 32
} | {
"line": 142,
"column": 33
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nh :\n ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a\ns : Set X\nx : X\nhcx : x ∈ closure[inst✝] s\nhx : x ∉ s\n⊢ ∃ u, Tendsto u atTop (𝓝 x) ∧ ∃ᶠ (x : ℕ) in atTop, u x ∈ s",
"ppTerm"... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nh :\n ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a\ns : Set X\nx : X\nhcx : x ∈ closure[inst✝] s\nhx : x ∉ s\n⊢ ∃ u, Tendsto u atTop (𝓝 x) ∧ ∃ᶠ (x : ℕ) in atTop, u x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 139,
"column": 19
} | {
"line": 139,
"column": 57
} | {
"line": 139,
"column": 58
} | [
{
"pp": "α : Type u_1\nm : α → ℝ≥0∞\nf : Filter α\nh : ∀ (x : ℝ≥0), ∀ᶠ (a : α) in f, ↑x < m a\nn : ℕ\n⊢ ∀ᶠ (a : α) in f, ↑n < m a",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nm : α → ℝ≥0∞\nf : Filter α\nh : ∀ (x : ℝ≥0), ∀ᶠ (a : α) in f, ↑x < m a\nn : ℕ\n⊢ ∀ᶠ (a : α) in f, ↑n < m a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sequences | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 13
} | {
"line": 204,
"column": 14
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nx : ℕ → X\ninst✝¹ : SequentialSpace X\ninst✝ : T1Space X\nhx : ∀ (l : X) (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 l)\n⊢ IsClosed[inst✝²] (range x)",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\nx : ℕ → X\ninst✝¹ : SequentialSpace X\ninst✝ : T1Space X\nhx : ∀ (l : X) (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 l)\n⊢ IsClosed[inst✝²] (range x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sequences | {
"line": 330,
"column": 2
} | {
"line": 330,
"column": 13
} | {
"line": 330,
"column": 14
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : SeqCompactSpace X\nf_cont : SeqContinuous f\n⊢ IsSeqCompact (Set.range f)",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : SeqCompactSpace X\nf_cont : SeqContinuous f\n⊢ IsSeqCompact (Set.range f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 36
} | {
"line": 214,
"column": 37
} | [
{
"pp": "x : ℝ≥0∞\nxt : x ≠ ∞\n⊢ (𝓝 x).HasBasis (fun x ↦ 0 < x) fun ε ↦ Icc (x - ε) (x + ε)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"ENNReal.instAdd",
"Preorder.toLT",
"congrArg",
"instIsBotZeroClas... | [
"x : ℝ≥0∞\nxt : x ≠ ∞\n⊢ (𝓝 x).HasBasis (fun x ↦ x ≠ 0) fun ε ↦ Icc (x - ε) (x + ε)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 85
} | {
"line": 224,
"column": 86
} | [
{
"pp": "⊢ 𝓟 (Icc (∞ - 1) (∞ + 1)) ≤ 𝓝 ∞",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"ENNReal.instAdd",
"Set.Icc_self",
"ENNReal.ofNNReal",
"congrArg",
"Filter.instCompleteLatticeFilter",
"PartialOrder.toPreorder"... | [
"⊢ pure ∞ ≤ 𝓝 ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 35
} | {
"line": 84,
"column": 36
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedGroup E\n⊢ comap norm (𝓝 0) = 𝓝 1",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_4\ninst✝ : SeminormedGroup E\n⊢ comap norm (𝓝 0) = 𝓝 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 36
} | {
"line": 112,
"column": 4
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedGroup E\nx : E\n⊢ Tendsto (fun a ↦ ‖a⁻¹ * x‖) (𝓝 x) (𝓝 0)",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_4\ninst✝ : SeminormedGroup E\nx : E\n⊢ Tendsto (fun a ↦ ‖a⁻¹ * x‖) (𝓝 x) (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 13
} | {
"line": 120,
"column": 14
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedGroup E\nx : E\n⊢ Tendsto (fun a ↦ ‖a‖) (𝓝 x) (𝓝 ‖x‖)",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_4\ninst✝ : SeminormedGroup E\nx : E\n⊢ Tendsto (fun a ↦ ‖a‖) (𝓝 x) (𝓝 ‖x‖)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 13
} | {
"line": 125,
"column": 14
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedGroup E\n⊢ Tendsto (fun a ↦ ‖a‖) (𝓝 1) (𝓝 0)",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_4\ninst✝ : SeminormedGroup E\n⊢ Tendsto (fun a ↦ ‖a‖) (𝓝 1) (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 13
} | {
"line": 129,
"column": 14
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedGroup E\n⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun a ↦ ‖a‖",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_4\ninst✝ : SeminormedGroup E\n⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun a ↦ ‖a‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 289,
"column": 4
} | {
"line": 289,
"column": 52
} | {
"line": 289,
"column": 53
} | [
{
"pp": "case refine_1\nι : Type u_2\nκ : Type u_3\nG : Type u_6\ninst✝ : SeminormedGroup G\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : UniformCauchySeqOnFilter f l l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.1 p.2 < ε} → (a, b) ∈ u\nx : (ι × ι) × κ\nhx : (... | [
"case refine_1\nι : Type u_2\nκ : Type u_3\nG : Type u_6\ninst✝ : SeminormedGroup G\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : UniformCauchySeqOnFilter f l l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.1 p.2 < ε} → (a, b) ∈ u\nx : (ι × ι) × κ\nhx : (f x.1.1 x.2,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 294,
"column": 4
} | {
"line": 294,
"column": 52
} | {
"line": 294,
"column": 53
} | [
{
"pp": "case refine_2\nι : Type u_2\nκ : Type u_3\nG : Type u_6\ninst✝ : SeminormedGroup G\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : TendstoUniformlyOnFilter (fun n z ↦ (f n.1 z)⁻¹ * f n.2 z) 1 (l ×ˢ l) l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.1 p.2 <... | [
"case refine_2\nι : Type u_2\nκ : Type u_3\nG : Type u_6\ninst✝ : SeminormedGroup G\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : TendstoUniformlyOnFilter (fun n z ↦ (f n.1 z)⁻¹ * f n.2 z) 1 (l ×ˢ l) l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.1 p.2 < ε} → (a, b)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 10
} | {
"line": 120,
"column": 11
} | [
{
"pp": "case right\nα : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : ContinuousMul α\ninst✝ : RegularSpace α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nha : HasProd f a\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\ns : Set α\nhs : s ∈ 𝓝 a\nhsc : Is... | [
"case right\nα : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : ContinuousMul α\ninst✝ : RegularSpace α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nha : HasProd f a\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\ns : Set α\nhs : s ∈ 𝓝 a\nhsc : IsClosed[inst✝... | intro bs | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 32
} | {
"line": 329,
"column": 4
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedCommGroup E\nx : E\n⊢ Tendsto (fun a ↦ ‖a / x‖) (𝓝 x) (𝓝 0)",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_4\ninst✝ : SeminormedCommGroup E\nx : E\n⊢ Tendsto (fun a ↦ ‖a / x‖) (𝓝 x) (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 40
} | {
"line": 353,
"column": 41
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedCommGroup E\na : E\ns : Subgroup E\nhg : a ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑s\nb : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < b n\nu : ℕ → E\nu_in : ∀ (n : ℕ), u n ∈ s\nlim_u : Tendsto u atTop (𝓝 a)\nn₀ : ℕ\nhn₀ : ∀ n ≥ n₀, ‖(u n)⁻¹ * a‖ < b 0\nz : ℕ →... | [
"E : Type u_4\ninst✝ : SeminormedCommGroup E\na : E\ns : Subgroup E\nhg : a ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑s\nb : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < b n\nu : ℕ → E\nu_in : ∀ (n : ℕ), u n ∈ s\nlim_u : Tendsto u atTop (𝓝 a)\nn₀ : ℕ\nhn₀ : ∀ n ≥ n₀, ‖(u n)⁻¹ * a‖ < b 0\nz : ℕ → E := fun n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 154,
"column": 22
} | {
"line": 154,
"column": 68
} | {
"line": 154,
"column": 69
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : ContinuousMul α\ninst✝ : T3Space α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nha : HasProd g a\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\nhf' : Multipliable f\n⊢ HasProd f a",
"ppTerm":... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : ContinuousMul α\ninst✝ : T3Space α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nha : HasProd g a\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\nhf' : Multipliable f\n⊢ HasProd f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.