module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Data.String.Basic | {
"line": 70,
"column": 17
} | {
"line": 70,
"column": 28
} | [
{
"pp": "c : Char\ns₁ s₂ : Legacy.Iterator\nh₁ : s₂.hasNext = true\nh₂ : s₁.hasNext = true\nh : s₁.curr = s₂.curr\nih :\n ltb { s := ofList (c :: s₁.next.s.toList), i := s₁.next.i + c }\n { s := ofList (c :: s₂.next.s.toList), i := s₂.next.i + c } =\n ltb s₁.next s₂.next\n⊢ { s := ofList s₂.s.toList, i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.String.Basic | {
"line": 70,
"column": 45
} | {
"line": 70,
"column": 56
} | [
{
"pp": "c : Char\ns₁ s₂ : Legacy.Iterator\nh₁ : s₂.hasNext = true\nh₂ : s₁.hasNext = true\nh : s₁.curr = s₂.curr\nih :\n ltb { s := ofList (c :: s₁.next.s.toList), i := s₁.next.i + c }\n { s := ofList (c :: s₂.next.s.toList), i := s₂.next.i + c } =\n ltb s₁.next s₂.next\n⊢ { s := ofList s₁.s.toList, i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.String.Basic | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 78
} | [
{
"pp": "case case1.hc\nc : Char\ns₁ s₂ : Legacy.Iterator\nh₁ : s₂.hasNext = true\nh₂ : s₁.hasNext = true\nh : s₁.curr = s₂.curr\nih :\n ltb { s := ofList (c :: s₁.next.s.toList), i := s₁.next.i + c }\n { s := ofList (c :: s₂.next.s.toList), i := s₂.next.i + c } =\n ltb s₁.next s₂.next\n⊢ { s := ofList... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.String.Basic | {
"line": 76,
"column": 17
} | {
"line": 76,
"column": 28
} | [
{
"pp": "c : Char\ns₁ s₂ : Legacy.Iterator\nh₁ : s₂.hasNext = true\nh₂ : s₁.hasNext = true\nh : ¬s₁.curr = s₂.curr\n⊢ { s := ofList s₂.s.toList, i := s₂.i }.hasNext = true",
"usedConstants": [
"Eq.mpr",
"congrArg",
"String.Legacy.Iterator.mk",
"String",
"String.Legacy.Iterator.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.String.Basic | {
"line": 76,
"column": 45
} | {
"line": 76,
"column": 56
} | [
{
"pp": "c : Char\ns₁ s₂ : Legacy.Iterator\nh₁ : s₂.hasNext = true\nh₂ : s₁.hasNext = true\nh : ¬s₁.curr = s₂.curr\n⊢ { s := ofList s₁.s.toList, i := s₁.i }.hasNext = true",
"usedConstants": [
"Eq.mpr",
"congrArg",
"String.Legacy.Iterator.mk",
"String",
"String.Legacy.Iterator.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.String.Basic | {
"line": 78,
"column": 6
} | {
"line": 78,
"column": 78
} | [
{
"pp": "case case2.hnc\nc : Char\ns₁ s₂ : Legacy.Iterator\nh₁ : s₂.hasNext = true\nh₂ : s₁.hasNext = true\nh : ¬s₁.curr = s₂.curr\n⊢ ¬{ s := ofList (c :: s₁.s.toList), i := s₁.i + c }.curr = { s := ofList (c :: s₂.s.toList), i := s₂.i + c }.curr",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.String.Basic | {
"line": 81,
"column": 17
} | {
"line": 81,
"column": 28
} | [
{
"pp": "c : Char\ns₁ s₂ : Legacy.Iterator\nh₁ : s₂.hasNext = true\nh₂ : ¬s₁.hasNext = true\n⊢ { s := ofList s₂.s.toList, i := s₂.i }.hasNext = true",
"usedConstants": [
"Eq.mpr",
"congrArg",
"String.Legacy.Iterator.mk",
"String",
"String.Legacy.Iterator.hasNext",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.String.Basic | {
"line": 81,
"column": 45
} | {
"line": 81,
"column": 56
} | [
{
"pp": "c : Char\ns₁ s₂ : Legacy.Iterator\nh₁ : s₂.hasNext = true\nh₂ : ¬s₁.hasNext = true\n⊢ ¬{ s := ofList s₁.s.toList, i := s₁.i }.hasNext = true",
"usedConstants": [
"Eq.mpr",
"congrArg",
"String.Legacy.Iterator.mk",
"String",
"String.Legacy.Iterator.hasNext",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.WSeq.Relation | {
"line": 440,
"column": 8
} | {
"line": 440,
"column": 19
} | [
{
"pp": "α : Type u\nβ : Type v\nR : α → β → Prop\nS✝ : WSeq (WSeq α)\nT✝ : WSeq (WSeq β)\nh✝ : LiftRel (LiftRel R) S✝ T✝\ns1 : WSeq α\ns2 : WSeq β\nx✝ :\n (fun s1 s2 ↦ ∃ s t S T, s1 = s.append S.join ∧ s2 = t.append T.join ∧ LiftRel R s t ∧ LiftRel (LiftRel R) S T) s1 s2\ns✝ : WSeq α\nt✝ : WSeq β\nS : WSeq (W... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.String.Basic | {
"line": 83,
"column": 62
} | {
"line": 83,
"column": 73
} | [
{
"pp": "c : Char\ns₁ s₂ : Legacy.Iterator\nh₁ : ¬s₂.hasNext = true\n⊢ ¬{ s := ofList s₂.s.toList, i := s₂.i }.hasNext = true",
"usedConstants": [
"Eq.mpr",
"congrArg",
"String.Legacy.Iterator.mk",
"String",
"String.Legacy.Iterator.hasNext",
"id",
"String.ofList_toL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.WSeq.Relation | {
"line": 468,
"column": 56
} | {
"line": 468,
"column": 73
} | [
{
"pp": "α : Type u\ns : WSeq α\n⊢ (ret s).join ~ʷ s",
"usedConstants": [
"Eq.mpr",
"Stream'.WSeq.join",
"congrArg",
"Stream'.WSeq.cons",
"Stream'.WSeq.ofList_cons",
"id",
"Stream'.WSeq.join_cons",
"Stream'.WSeq.think",
"Stream'.WSeq",
"Stream'.WSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.WSeq.Relation | {
"line": 515,
"column": 18
} | {
"line": 515,
"column": 29
} | [
{
"pp": "case nil.cons\nα : Type u\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)\nT : WSeq (WSeq α)\ns : WSeq α\nS : WSeq (WSeq α)\n⊢ LiftRelAux\n (LiftRelO (fun x1 x2 ↦ x1 = x2) fun s1 s2 ↦\n ∃ s S T, s1 = s.append (S.append ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Seq.Parallel | {
"line": 282,
"column": 39
} | {
"line": 282,
"column": 50
} | [
{
"pp": "case inr.nil\nα : Type u\nβ : Type v\nf : α → β\nS : WSeq (Computation α)\nc1 c2 : Computation β\nh : ∃ l S, c1 = map f (corec parallel.aux1 (l, S)) ∧ c2 = corec parallel.aux1 (List.map (map f) l, WSeq.map (map f) S)\nl : List (Computation α)\nthis : parallel.aux2 (List.map (map f) l) = lmap f (rmap (L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Seq.Parallel | {
"line": 282,
"column": 39
} | {
"line": 282,
"column": 50
} | [
{
"pp": "case inr.cons\nα : Type u\nβ : Type v\nf : α → β\nS : WSeq (Computation α)\nc1 c2 : Computation β\nh : ∃ l S, c1 = map f (corec parallel.aux1 (l, S)) ∧ c2 = corec parallel.aux1 (List.map (map f) l, WSeq.map (map f) S)\nl : List (Computation α)\nthis : parallel.aux2 (List.map (map f) l) = lmap f (rmap (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Seq.Parallel | {
"line": 282,
"column": 39
} | {
"line": 282,
"column": 50
} | [
{
"pp": "case inr.think\nα : Type u\nβ : Type v\nf : α → β\nS : WSeq (Computation α)\nc1 c2 : Computation β\nh : ∃ l S, c1 = map f (corec parallel.aux1 (l, S)) ∧ c2 = corec parallel.aux1 (List.map (map f) l, WSeq.map (map f) S)\nl : List (Computation α)\nthis : parallel.aux2 (List.map (map f) l) = lmap f (rmap ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Sym.Sym2.Finsupp | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 13
} | [
{
"pp": "case mk\nα : Type u_1\nM₀ : Type u_2\ninst✝ : CommMonoidWithZero M₀\nf : α →₀ M₀\np : Sym2 α\na b : α\nhp : ¬f a * f b = 0\n⊢ Quot.mk (Rel α) (a, b) ∈ f.support.sym2",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Sym2.Rel",
"Sym2.mem_iff._simp_1",
"Sym2.mk",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.WSeq.Relation | {
"line": 517,
"column": 16
} | {
"line": 517,
"column": 27
} | [
{
"pp": "case cons\nα : Type u\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)\nS T : WSeq (WSeq α)\na : α\ns : WSeq α\n⊢ LiftRelAux\n (LiftRelO (fun x1 x2 ↦ x1 = x2) fun s1 s2 ↦\n ∃ s S T, s1 = s.append (S.append T).join ∧ s2 =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.WSeq.Relation | {
"line": 518,
"column": 15
} | {
"line": 518,
"column": 26
} | [
{
"pp": "case think\nα : Type u\nS✝ T✝ : WSeq (WSeq α)\ns1 s2 : WSeq α\nh : ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.append (S.join.append T.join)\nS T : WSeq (WSeq α)\ns : WSeq α\n⊢ LiftRelAux\n (LiftRelO (fun x1 x2 ↦ x1 = x2) fun s1 s2 ↦\n ∃ s S T, s1 = s.append (S.append T).join ∧ s2 = s.app... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.WSeq.Relation | {
"line": 561,
"column": 22
} | {
"line": 561,
"column": 33
} | [
{
"pp": "case nil.cons\nα : Type u\nSS✝ : WSeq (WSeq (WSeq α))\ns1 s2 : WSeq α\nh : ∃ s S SS, s1 = s.append (S.append SS.join).join ∧ s2 = s.append (S.join.append (map join SS).join)\nc1 c2 : Computation (Option (α × WSeq α))\nSS : WSeq (WSeq (WSeq α))\ns : WSeq α\nS : WSeq (WSeq α)\n⊢ LiftRelAux\n (LiftRelO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.WSeq.Relation | {
"line": 563,
"column": 20
} | {
"line": 563,
"column": 31
} | [
{
"pp": "case cons\nα : Type u\nSS✝ : WSeq (WSeq (WSeq α))\ns1 s2 : WSeq α\nh : ∃ s S SS, s1 = s.append (S.append SS.join).join ∧ s2 = s.append (S.join.append (map join SS).join)\nc1 c2 : Computation (Option (α × WSeq α))\nS : WSeq (WSeq α)\nSS : WSeq (WSeq (WSeq α))\na : α\ns : WSeq α\n⊢ LiftRelAux\n (LiftR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.WSeq.Relation | {
"line": 564,
"column": 19
} | {
"line": 564,
"column": 30
} | [
{
"pp": "case think\nα : Type u\nSS✝ : WSeq (WSeq (WSeq α))\ns1 s2 : WSeq α\nh : ∃ s S SS, s1 = s.append (S.append SS.join).join ∧ s2 = s.append (S.join.append (map join SS).join)\nc1 c2 : Computation (Option (α × WSeq α))\nS : WSeq (WSeq α)\nSS : WSeq (WSeq (WSeq α))\ns : WSeq α\n⊢ LiftRelAux\n (LiftRelO (f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Vector.Mem | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 71
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nb : β\nv : Vector α 0\nf : α → β\n⊢ ¬b ∈ (map f v).toList",
"usedConstants": [
"Eq.mpr",
"congrArg",
"List.Vector.map",
"List.Vector.eq_nil",
"List.Vector",
"Membership.mem",
"id",
"instOfNatNat",
"List",
"List.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Vector.MapLemmas | {
"line": 64,
"column": 67
} | {
"line": 64,
"column": 78
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nζ : Type u_4\nσ : Type u_5\nσ₁ : Type u_6\nσ₂ : Type u_7\nφ : Type u_8\nn : ℕ\ns : σ\ns₁ : σ₁\ns₂ : σ₂\nxs : Vector α n\nf₁✝ : β → σ₁ → σ₁ × γ\nf₂✝ : α → σ₂ → σ₂ × β\np : β → Prop\nf₁ : (b : β) → p b → γ\nf₂ : α → β\nH : ∀ (x : β), x ∈ (map f₂ xs).toList → p x\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Vector.MapLemmas | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nn : ℕ\nxs : Vector α n\nf : α → β\n⊢ map f xs = (mapAccumr (fun x x_1 ↦ ((), f x)) xs ()).2",
"usedConstants": [
"Unit.unit",
"List.Vector.mapAccumr",
"List.Vector.mapAccumr_snoc",
"congrArg",
"List.Vector.map",
"List.Vector",
"P... | induction xs using Vector.revInductionOn <;> simp_all | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Vector.MapLemmas | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nn : ℕ\nxs : Vector α n\nf : α → β\n⊢ map f xs = (mapAccumr (fun x x_1 ↦ ((), f x)) xs ()).2",
"usedConstants": [
"Unit.unit",
"List.Vector.mapAccumr",
"List.Vector.mapAccumr_snoc",
"congrArg",
"List.Vector.map",
"List.Vector",
"P... | induction xs using Vector.revInductionOn <;> simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Vector.MapLemmas | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nn : ℕ\nxs : Vector α n\nf : α → β\n⊢ map f xs = (mapAccumr (fun x x_1 ↦ ((), f x)) xs ()).2",
"usedConstants": [
"Unit.unit",
"List.Vector.mapAccumr",
"List.Vector.mapAccumr_snoc",
"congrArg",
"List.Vector.map",
"List.Vector",
"P... | induction xs using Vector.revInductionOn <;> simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Vector.MapLemmas | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 26
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nσ : Type u_5\nn : ℕ\nxs : Vector α n\nf : α → σ → σ × β\ns₀ : σ\nS : Set σ\nh₀ : s₀ ∈ S\nclosure : ∀ (a : α) (s : σ), s ∈ S → (f a s).1 ∈ S\nout : ∀ (a : α) (s s' : σ), s ∈ S → s' ∈ S → (f a s).2 = (f a s').2\n⊢ ∃ R, R s₀ () ∧ ∀ {s : σ} {q : Unit} (a : α), R s q → R ... | use fun s _ => s ∈ S, h₀ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Data.Vector.MapLemmas | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 26
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nσ : Type u_5\nn : ℕ\nxs : Vector α n\nys : Vector β n\nf : α → β → σ → σ × γ\ns₀ : σ\nS : Set σ\nh₀ : s₀ ∈ S\nclosure : ∀ (a : α) (b : β) (s : σ), s ∈ S → (f a b s).1 ∈ S\nout : ∀ (a : α) (b : β) (s s' : σ), s ∈ S → s' ∈ S → (f a b s).2 = (f a b s').2\n... | use fun s _ => s ∈ S, h₀ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Data.Vector3 | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 18
} | [
{
"pp": "case refine_2\nα : Type u_1\nm n : ℕ\na : α\nt✝ : Vector3 α m\nv : Vector3 α n\ni : Fin2 (n + 1)\ne : n + 1 + m = n + m + 1\nk : ℕ\nb : α\nt : Vector3 α k\nIH : ∀ (x : n + 1 + k = n + k + 1), insert a (t +-+ v) (Eq.recOn x (i.add k)) = Eq.recOn x (t +-+ insert a v i)\nx✝ : n + 1 + (k + 1) = n + (k + 1)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Sum.Interval | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 28
} | [
{
"pp": "case refine_4.inl.inr\nα₁ : Type u_1\nα₂ : Type u_2\nβ₁ : Type u_3\nβ₂ : Type u_4\nγ₁ : Type u_5\nγ₂ : Type u_6\nf₁ : α₁ → β₁ → Finset γ₁\nf₂ : α₂ → β₂ → Finset γ₂\ng₁ : α₁ → β₂ → Finset γ₁\ng₂ : α₁ → β₂ → Finset γ₂\nval✝¹ : α₁\nval✝ : β₂\nh :\n (∀ (a₁ : α₁) (b₁ : β₁), inl val✝¹ = inl a₁ → inr val✝ = ... | · simp [h.2.1 _ _ rfl rfl] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Vector3 | {
"line": 254,
"column": 6
} | {
"line": 254,
"column": 17
} | [
{
"pp": "case refine_2.refine_1\nα : Type u_1\nn✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (n + 1)), p ((a :: v) i)\n⊢ p a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Vector3 | {
"line": 255,
"column": 6
} | {
"line": 255,
"column": 17
} | [
{
"pp": "case refine_2.refine_2\nα : Type u_1\nn✝ : ℕ\np : α → Prop\nv✝ : Vector3 α n✝\nn : ℕ\na : α\nv : Vector3 α n\nIH : VectorAllP p v ↔ ∀ (i : Fin2 n), p (v i)\nh : ∀ (i : Fin2 (n + 1)), p ((a :: v) i)\ni : Fin2 n\n⊢ p (v i)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ZMod.Coprime | {
"line": 31,
"column": 2
} | {
"line": 31,
"column": 49
} | [
{
"pp": "n : ℤ\nm : ℕ\n⊢ Associated ↑n ↑n.natAbs",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.WSeq.Defs | {
"line": 243,
"column": 16
} | {
"line": 243,
"column": 27
} | [
{
"pp": "case cons\nα : Type u\ns✝ : WSeq α\ns1 s2 : Computation ℕ\nl : List α\na : α\ns : WSeq α\nh :\n s1 =\n Computation.corec\n (fun x ↦\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.WSeq.Defs | {
"line": 244,
"column": 15
} | {
"line": 244,
"column": 26
} | [
{
"pp": "case think\nα : Type u\ns✝ : WSeq α\ns1 s2 : Computation ℕ\nl : List α\ns : WSeq α\nh :\n s1 =\n Computation.corec\n (fun x ↦\n match x with\n | (n, s) =>\n match Seq.destruct s with\n | none => Sum.inl n\n | some (none, s') => Sum.inr (n,... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SemiconjSup | {
"line": 68,
"column": 14
} | {
"line": 68,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf : α → β\ng : β → α\nh : IsOrderRightAdjoint f g\ne : β ≃o γ\ny : γ\n⊢ IsLUB {x | (⇑e ∘ f) x ≤ y} ((g ∘ ⇑e.symm) y)",
"usedConstants": [
"setOf",
"Preorder.toLE",
"Function.com... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SemiconjSup | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 24
} | [
{
"pp": "α : Type u_1\nG : Type u_4\ninst✝¹ : PartialOrder α\ninst✝ : Group G\nf₁ f₂ : G →* α ≃o α\nh : α → α\nH : ∀ (x : α), IsLUB (range fun g' ↦ (f₁ g')⁻¹ ((f₂ g') x)) (h x)\ng : G\ny : α\nthis : IsLUB (range ((⇑(f₁ g) ∘ fun g' ↦ (f₁ g')⁻¹ ((f₂ g') y)) ∘ ⇑(Equiv.mulRight g))) ((f₁ g) (h y))\n⊢ IsLUB (range f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Ergodic | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 41
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : Set α\nf : α → α\nμ : Measure α\nhf : PreErgodic f μ\nhs : MeasurableSet s\nhfs : f ⁻¹' s = s\n⊢ s =ᶠ[ae μ] ∅ ∨ s =ᶠ[ae μ] univ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Ergodic | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 13
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : Set α\nf : α → α\nμ : Measure α\nhf : PreErgodic f μ\nhs : MeasurableSet s\nhs' : f ⁻¹' s = s\n⊢ μ s = 0 ∨ μ sᶜ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Ergodic | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 18
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : Set α\nf : α → α\nμ : Measure α\ninst✝ : IsProbabilityMeasure μ\nhf : PreErgodic f μ\nhs : MeasurableSet s\nhs' : f ⁻¹' s = s\n⊢ μ s = 0 ∨ μ s = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.Basic | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 36
} | [
{
"pp": "G : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : ErgodicSMul G α μ\ns : Set α\nhm : NullMeasurableSet s μ\nh : ∀ (g : G), g • s =ᶠ[ae μ] s\ng : G\n⊢ (fun x ↦ g • x) ⁻¹' s =ᶠ[ae μ] s",
"usedConstants": [
"MeasureTheory.ae",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.Basic | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 40
} | [
{
"pp": "G : Type u_1\nα : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : SMulInvariantMeasure G α μ\nh : ∀ (s : Set α), MeasurableSet s → aestabilizer G μ s = ⊤ → EventuallyConst s (ae μ)\ns✝ : Set α\nhm : MeasurableSet s✝\nhs : ∀ (g : G), (fun x ↦ g • x) ⁻¹' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.AEStabilizer | {
"line": 48,
"column": 31
} | {
"line": 48,
"column": 59
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝² : Group G\ninst✝¹ : MulAction G α\nx✝ : MeasurableSpace α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\ns : Set α\ng₁ g₂ : G\nh₁ : g₁ ∈ {g | g • s =ᶠ[ae μ] s}\nh₂ : g₂ ∈ {g | g • s =ᶠ[ae μ] s}\n⊢ g₁ * g₂ ∈ {g | g • s =ᶠ[ae μ] s}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.AEStabilizer | {
"line": 49,
"column": 23
} | {
"line": 49,
"column": 34
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝² : Group G\ninst✝¹ : MulAction G α\nx✝ : MeasurableSpace α\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\ns : Set α\ng : G\nh : g ∈ {g | g • s =ᶠ[ae μ] s}\n⊢ g⁻¹ ∈ {g | g • s =ᶠ[ae μ] s}",
"usedConstants": [
"MeasureTheory.ae",
"instHSMul",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.Regular | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 19
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : MeasurableMul₂ G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : SFinite μ\ninst✝ : μ.IsMulLeftInvariant\ns : Set G\nhsm : MeasurableSet s\nhs : ∀ (g : G), (fun x ↦ g • x) ⁻¹' s =ᶠ[ae μ] s\nhμs : ∃ᵐ (x : G) ∂μ, x ∈ s\na : G\nhas : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.Regular | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 19
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : MeasurableMul₂ G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : SFinite μ\ninst✝ : μ.IsMulRightInvariant\ns : Set G\nhsm : MeasurableSet s\nhs : ∀ (g : Gᵐᵒᵖ), (fun x ↦ g • x) ⁻¹' s =ᶠ[ae μ] s\nhμs : ∃ᵐ (x : G) ∂μ, x ∈ s\na : G\nha... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.FixedPoints.Prufer | {
"line": 38,
"column": 4
} | {
"line": 38,
"column": 65
} | [
{
"pp": "G : Type u_1\ninst✝ : CommGroup G\nn : ℤ\ns : Set G\nhs : (fun x ↦ x ^ n) ⁻¹' s = s\ng : G\nj : ℕ\nthis : ∀ {g' : G}, g' ^ n ^ j = 1 → g' • s ⊆ s\nhg : g⁻¹ ^ n ^ j = 1\n⊢ g • g⁻¹ • s ≤ g • s",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHSMul",
"instSMulOfMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 206,
"column": 35
} | {
"line": 206,
"column": 46
} | [
{
"pp": "f✝ g : CircleDeg1Lift\nf : CircleDeg1Liftˣ\na✝ b✝ : ℝ\nh :\n { toFun := ⇑↑f, invFun := ⇑↑f⁻¹, left_inv := ⋯, right_inv := ⋯ } a✝ ≤\n { toFun := ⇑↑f, invFun := ⇑↑f⁻¹, left_inv := ⋯, right_inv := ⋯ } b✝\n⊢ a✝ ≤ b✝",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 48
} | [
{
"pp": "f : CircleDeg1Lift\nn : ℕ\n⊢ Function.Commute ⇑f fun x ↦ ↑n + x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 35
} | [
{
"pp": "f : CircleDeg1Lift\nn : ℕ\n⊢ Function.Commute ⇑f fun x ↦ x - ↑n",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instSub",
"Function.Commute",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
"CircleDeg1Lift.instFunLikeReal",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 306,
"column": 17
} | {
"line": 306,
"column": 45
} | [
{
"pp": "f : CircleDeg1Lift\nn : ℕ\n⊢ Function.Commute ⇑f fun x ↦ x + ↑-[n+1]",
"usedConstants": [
"neg_add_rev",
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"AddMonoid.toAddSemigroup",
"AddGroupWithOne.toAddGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 39
} | [
{
"pp": "f : CircleDeg1Lift\nn : ℤ\n⊢ Function.Commute ⇑f fun x ↦ ↑n + x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 35
} | [
{
"pp": "f : CircleDeg1Lift\nn : ℤ\n⊢ Function.Commute ⇑f fun x ↦ x - ↑n",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real",
"congrArg",
"Real.instSub",
"Function.Commute",
"AddMonoid.toAddZeroClass",
"sub_eq_add_neg",
"HSub.hSub",
"CircleDeg1Lift.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 479,
"column": 6
} | {
"line": 479,
"column": 34
} | [
{
"pp": "f : CircleDeg1Lift\n⊢ Tendsto (fun x ↦ x - 1) atTop atTop",
"usedConstants": [
"Eq.mpr",
"Real",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"AddGroupWithOne.toAddMonoidWithOne",
"sub_eq_add_neg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 497,
"column": 2
} | {
"line": 497,
"column": 52
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nh : f x ≤ x + ↑m\nn : ℕ\n⊢ (⇑f)^[n] x ≤ x + ↑n * ↑m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 52
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nh : x + ↑m ≤ f x\nn : ℕ\n⊢ x + ↑n * ↑m ≤ (⇑f)^[n] x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 507,
"column": 2
} | {
"line": 507,
"column": 52
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nh : f x = x + ↑m\nn : ℕ\n⊢ (⇑f)^[n] x = x + ↑n * ↑m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 52
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ (⇑f)^[n] x ≤ x + ↑n * ↑m ↔ f x ≤ x + ↑m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 516,
"column": 2
} | {
"line": 516,
"column": 52
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ (⇑f)^[n] x < x + ↑n * ↑m ↔ f x < x + ↑m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 52
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ (⇑f)^[n] x = x + ↑n * ↑m ↔ f x = x + ↑m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 526,
"column": 2
} | {
"line": 526,
"column": 27
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ x + ↑n * ↑m ≤ (⇑f)^[n] x ↔ x + ↑m ≤ f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 530,
"column": 2
} | {
"line": 530,
"column": 27
} | [
{
"pp": "f : CircleDeg1Lift\nx : ℝ\nm : ℤ\nn : ℕ\nhn : 0 < n\n⊢ x + ↑n * ↑m < (⇑f)^[n] x ↔ x + ↑m < f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 569,
"column": 4
} | {
"line": 569,
"column": 64
} | [
{
"pp": "f : CircleDeg1Lift\nτ' : ℝ\nh : Tendsto (fun n ↦ (⇑f)^[n] 0 / ↑n) atTop (𝓝 τ')\n⊢ Tendsto f.transnumAuxSeq atTop (𝓝 τ')",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.instZero",
"congrArg",
"Nat.instMonoid",
"Real.instDivInvMonoid",
"Nat.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.AddCircle | {
"line": 67,
"column": 30
} | {
"line": 67,
"column": 75
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nI : Set (AddCircle T)\nu x : AddCircle T\nhu : IsOfFinAddOrder u\nG : AddSubgroup (AddCircle T) := AddSubgroup.zmultiples u\nn : ℕ := addOrderOf u\nB : Set (AddCircle T) := ball x (T / (2 * ↑n))\nhI : I =ᶠ[ae volume] B\nhn : 1 ≤ ↑n\ng : AddCircle T\nhg : g ∈ G\nhg' : ⟨g, hg⟩ ≠... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.AddCircle.DenseSubgroup | {
"line": 34,
"column": 4
} | {
"line": 34,
"column": 53
} | [
{
"pp": "case inl\na : ℝ\n⊢ Dense ↑(AddSubgroup.closure {0, a}) ↔ Irrational (a / 0)",
"usedConstants": [
"_private.Mathlib.Topology.Instances.AddCircle.DenseSubgroup.0.dense_addSubgroupClosure_pair_iff._simp_1_1",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWithZero.toMonoidWith... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Circle.RotationNumber.TranslationNumber | {
"line": 874,
"column": 4
} | {
"line": 874,
"column": 15
} | [
{
"pp": "f₁ f₂ : CircleDeg1Liftˣ\nh : τ ↑f₁ = τ ↑f₂\nthis :\n ∀ (n : Multiplicative ℤ),\n τ (((Units.coeHom CircleDeg1Lift).comp ((zpowersHom CircleDeg1Liftˣ) f₁)) n) =\n τ (((Units.coeHom CircleDeg1Lift).comp ((zpowersHom CircleDeg1Liftˣ) f₂)) n)\nF : CircleDeg1Lift\nhF :\n ∀ (g : Multiplicative ℤ),\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Conservative | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 29
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nx✝¹ : Set α\nx✝ : MeasurableSet x✝¹\nh0 : μ x✝¹ ≠ 0\n⊢ ∃ x ∈ x✝¹, ∃ m, m ≠ 0 ∧ id^[m] x ∈ x✝¹",
"usedConstants": [
"Function.iterate_id",
"Eq.mpr",
"Nat.instNontrivial",
"congrArg",
"and_self",
"Membership.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Conservative | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 34
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nf : α → α\ns : Set α\nμ : Measure α\nhf : Conservative f μ\nhs : NullMeasurableSet s μ\nh0 : μ s ≠ 0\nt : ℕ → Set α := fun n ↦ s ∩ f^[n] ⁻¹' s\nH : ¬∃ᶠ (m : ℕ) in atTop, μ (s ∩ f^[m] ⁻¹' s) ≠ 0\nN : ℕ\nhN : μ (t N) ≠ 0\nhmax : ∀ n > N, μ (t n) = 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 36
} | [
{
"pp": "M : Type u_1\nX : Type u_2\ninst✝¹⁵ : Monoid M\ninst✝¹⁴ : SMul M X\ninst✝¹³ : TopologicalSpace X\ninst✝¹² : R1Space X\ninst✝¹¹ : MeasurableSpace X\ninst✝¹⁰ : BorelSpace X\nμ : Measure X\ninst✝⁹ : IsFiniteMeasure μ\ninst✝⁸ : μ.InnerRegular\nN : Type u_3\ninst✝⁷ : MulAction M N\ninst✝⁶ : Monoid N\ninst✝⁵... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 15
} | [
{
"pp": "M : Type u_1\nX : Type u_2\ninst✝¹⁵ : Monoid M\ninst✝¹⁴ : SMul M X\ninst✝¹³ : TopologicalSpace X\ninst✝¹² : R1Space X\ninst✝¹¹ : MeasurableSpace X\ninst✝¹⁰ : BorelSpace X\nμ : Measure X\ninst✝⁹ : IsFiniteMeasure μ\ninst✝⁸ : μ.InnerRegular\nN : Type u_3\ninst✝⁷ : MulAction M N\ninst✝⁶ : Monoid N\ninst✝⁵... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 36
} | [
{
"pp": "G : Type u_1\ninst✝¹¹ : Group G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : ContinuousInv G\nX : Type u_2\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : R1Space X\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : BorelSpace X\ninst✝⁴ : MulAction G X\ninst✝³ : ContinuousSMul G X\nμ : Measure X\ninst✝² : IsFiniteMeasure μ\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 138,
"column": 2
} | {
"line": 142,
"column": 52
} | [
{
"pp": "G : Type u_1\ninst✝¹¹ : Group G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : ContinuousInv G\nX : Type u_2\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : R1Space X\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : BorelSpace X\ninst✝⁴ : MulAction G X\ninst✝³ : ContinuousSMul G X\ng : G\nhg : DenseRange fun x ↦ g ^ x\nμ : Measu... | borelize G
refine ⟨measurePreserving_smul _ _, ⟨fun s hsm hs ↦ ?_⟩⟩
refine aeconst_of_dense_aestabilizer_smul hsm.nullMeasurableSet (hg.mono ?_)
rw [← Subgroup.coe_zpowers, SetLike.coe_subset_coe, ← Subgroup.zpowers_inv, Subgroup.zpowers_le,
MulAction.mem_aestabilizer, ← preimage_smul, hs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 138,
"column": 2
} | {
"line": 142,
"column": 52
} | [
{
"pp": "G : Type u_1\ninst✝¹¹ : Group G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : ContinuousInv G\nX : Type u_2\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : R1Space X\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : BorelSpace X\ninst✝⁴ : MulAction G X\ninst✝³ : ContinuousSMul G X\ng : G\nhg : DenseRange fun x ↦ g ^ x\nμ : Measu... | borelize G
refine ⟨measurePreserving_smul _ _, ⟨fun s hsm hs ↦ ?_⟩⟩
refine aeconst_of_dense_aestabilizer_smul hsm.nullMeasurableSet (hg.mono ?_)
rw [← Subgroup.coe_zpowers, SetLike.coe_subset_coe, ← Subgroup.zpowers_inv, Subgroup.zpowers_le,
MulAction.mem_aestabilizer, ← preimage_smul, hs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Dynamics.Ergodic.Action.OfMinimal | {
"line": 167,
"column": 4
} | {
"line": 167,
"column": 90
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : IsTopologicalGroup G\ninst✝² : MeasurableSpace G\ninst✝¹ : OpensMeasurableSpace G\nμ : Measure G\ninst✝ : μ.IsOpenPosMeasure\ng : G\nhg : Ergodic (fun x ↦ g * x) μ\na : G\nh : (range fun x ↦ g ^ x)ᶜ ∈ 𝓝 a\nthis :\n Tendsto\n (fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.CountableSeparating | {
"line": 42,
"column": 6
} | {
"line": 42,
"column": 33
} | [
{
"pp": "case exists_countable_separating.refine_3.inr\nX : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder X\ninst✝¹ : OrderTopology X\ninst✝ : SecondCountableTopology X\ns✝ s : Set X\nhsc : s.Countable\nhsd : Dense s\nt : Set X := s ∪ {x | ∃ y, y ⋖ x}\nx y : X\nh : ∀ s ∈ Iio '' t, x ∈ s ↔ y ∈ s\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.CountableSeparating | {
"line": 45,
"column": 65
} | {
"line": 45,
"column": 76
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder X\ninst✝¹ : OrderTopology X\ninst✝ : SecondCountableTopology X\ns✝ s : Set X\nhsc : s.Countable\nhsd : Dense s\nt : Set X := s ∪ {x | ∃ y, y ⋖ x}\nx y : X\nh : ∀ s ∈ Iio '' t, x ∈ s ↔ y ∈ s\nhne : x ≠ y\nhlt : x < y\nhe : Ioo x y = ∅\n⊢ ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.CountableSeparating | {
"line": 49,
"column": 6
} | {
"line": 49,
"column": 35
} | [
{
"pp": "case inr\nX : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder X\ninst✝¹ : OrderTopology X\ninst✝ : SecondCountableTopology X\ns✝ s : Set X\nhsc : s.Countable\nhsd : Dense s\nt : Set X := s ∪ {x | ∃ y, y ⋖ x}\nx y : X\nh : ∀ s ∈ Iio '' t, x ∈ s ↔ y ∈ s\nhne✝ : x ≠ y\nhlt : x < y\nhne : (Ioo ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.CountableSeparating | {
"line": 56,
"column": 31
} | {
"line": 56,
"column": 73
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\ninst✝² : LinearOrder X\ninst✝¹ : OrderTopology X\ninst✝ : SecondCountableTopology X\ns : Set X\nt : Set (Set X)\nhtc : t.Countable\nht_sub : ∀ s ∈ t, s ∈ range Ioi\nht : ∀ x ∈ s, ∀ y ∈ s, (∀ s ∈ t, x ∈ s ↔ y ∈ s) → x = y\n⊢ ∀ s ∈ compl '' t, s ∈ range Iic",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Extreme | {
"line": 52,
"column": 6
} | {
"line": 52,
"column": 57
} | [
{
"pp": "X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\nf : X → X\nc : ℝ≥0∞\nhc : c ≠ ∞\nhf : MeasurePreserving f μ μ\nhc₀ : c ≠ 0\nthis✝ : IsFiniteMeasure μ\nS : Set (Measure X) := {ν | MeasurePreserving f ν ν ∧ ν univ = c}\nh : μ ∈ extremePoints ℝ≥0∞ S\nthis : ∀ {s : Set X}, MeasurableSet s → f ⁻¹' s = s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Extreme | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 47
} | [
{
"pp": "X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\nf : X → X\nh : μ ∈ extremePoints ℝ≥0∞ {ν | MeasurePreserving f ν ν ∧ IsProbabilityMeasure ν}\n⊢ μ ∈ extremePoints ℝ≥0∞ {ν | MeasurePreserving f ν ν ∧ ν univ = 1}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.Ergodic.Extreme | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 59
} | [
{
"pp": "X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\nf : X → X\ninst✝ : IsProbabilityMeasure μ\nhμ : Ergodic f μ\n⊢ μ ∈ extremePoints ℝ≥0∞ {ν | MeasurePreserving f ν ν ∧ IsProbabilityMeasure ν}",
"usedConstants": [
"MeasureTheory.MeasurePreserving",
"Eq.mpr",
"MeasureTheory.Measure... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 23
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nν : Measure α\ninst✝¹ : μ.HaveLebesgueDecomposition ν\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : AEMeasurable f ν\nhf_ne_zero : ∀ᵐ (x : α) ∂ν, f x ≠ 0\nhf_ne_top : ∀ᵐ (x : α) ∂ν, f x ≠ ∞\nthis : SigmaFinite (ν.withDensity f)\ns : Set α\nh... | exact hf.restrict | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 23
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nν : Measure α\ninst✝¹ : μ.HaveLebesgueDecomposition ν\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : AEMeasurable f ν\nhf_ne_zero : ∀ᵐ (x : α) ∂ν, f x ≠ 0\nhf_ne_top : ∀ᵐ (x : α) ∂ν, f x ≠ ∞\nthis : SigmaFinite (ν.withDensity f)\ns : Set α\nh... | exact hf.restrict | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 23
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nν : Measure α\ninst✝¹ : μ.HaveLebesgueDecomposition ν\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : AEMeasurable f ν\nhf_ne_zero : ∀ᵐ (x : α) ∂ν, f x ≠ 0\nhf_ne_top : ∀ᵐ (x : α) ∂ν, f x ≠ ∞\nthis : SigmaFinite (ν.withDensity f)\ns : Set α\nh... | exact hf.restrict | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 38
} | [
{
"pp": "A : Type u_1\nG : Type u_2\ninst✝¹ : TopologicalSpace A\ninst✝ : DiscreteTopology A\nU : Finset G\nx : G → A\n⊢ IsOpen[Pi.topologicalSpace] (cylinder U x)",
"usedConstants": [
"Eq.mpr",
"Pi.topologicalSpace",
"congrArg",
"Finset",
"Set.instSingletonSet",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 38
} | [
{
"pp": "A : Type u_1\nG : Type u_2\ninst✝¹ : TopologicalSpace A\ninst✝ : T1Space A\nU : Finset G\nx : G → A\n⊢ IsClosed[Pi.topologicalSpace] (cylinder U x)",
"usedConstants": [
"Eq.mpr",
"Pi.topologicalSpace",
"congrArg",
"Finset",
"Set.instSingletonSet",
"id",
"Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 379,
"column": 6
} | {
"line": 379,
"column": 36
} | [
{
"pp": "A : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\nv h : G\nhmem : h ∈ Finset.image (fun x ↦ v * x) p.support\n⊢ ∃ w ∈ p.support, v * w = h",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 375,
"column": 2
} | {
"line": 382,
"column": 17
} | [
{
"pp": "A : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\nv : G\n⊢ G → A",
"usedConstants": [
"Inhabited.default",
"HMul.hMul",
"Monoid.toMulOneClass",
"Finset",
"Classical.propDecidable",
"Membership.mem",
... | intro h
if hmem : h ∈ p.support.image (v * ·) then
-- package existence of a preimage under (v * ·)
let ex : ∃ w, w ∈ p.support ∧ v * w = h := by
simpa [Finset.mem_image] using hmem
exact p.config (Classical.choose ex)
else
exact default | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 375,
"column": 2
} | {
"line": 382,
"column": 17
} | [
{
"pp": "A : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\nv : G\n⊢ G → A",
"usedConstants": [
"Inhabited.default",
"HMul.hMul",
"Monoid.toMulOneClass",
"Finset",
"Classical.propDecidable",
"Membership.mem",
... | intro h
if hmem : h ∈ p.support.image (v * ·) then
-- package existence of a preimage under (v * ·)
let ex : ∃ w, w ∈ p.support ∧ v * w = h := by
simpa [Finset.mem_image] using hmem
exact p.config (Classical.choose ex)
else
exact default | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 421,
"column": 4
} | {
"line": 421,
"column": 34
} | [
{
"pp": "A : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\nv w : G\nhw : w ∈ p.support\nhmem : v * w ∈ Finset.image (fun x ↦ v * x) p.support\n⊢ ∃ w' ∈ p.support, v * w' = v * w",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Mono... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 462,
"column": 2
} | {
"line": 462,
"column": 38
} | [
{
"pp": "A : Type u_3\nG : Type u_4\ninst✝¹ : Inhabited A\ninst✝ : Monoid G\nF : Set (Pattern A G)\nh : G\nx : G → A\np : Pattern A G\nhp : p ∈ F\ng : G\nhx : p.mulOccursInAt (mulShift h x) g\n⊢ p.mulOccursInAt x (h * g)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 498,
"column": 4
} | {
"line": 498,
"column": 86
} | [
{
"pp": "case h.mp\nA : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\ng : G\nx : G → A\nH : x ∈ {x | p.mulOccursInAt x g}\nw : G\nhw : w ∈ p.support\nhu : g * w ∈ Finset.image (fun x ↦ g * x) p.support\nhx : x (g * w) = p.config w\n⊢ x (g * w) = p.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 505,
"column": 4
} | {
"line": 505,
"column": 86
} | [
{
"pp": "case h.mpr\nA : Type u_1\ninst✝² : Inhabited A\nG : Type u_2\ninst✝¹ : Monoid G\ninst✝ : IsLeftCancelMul G\np : Pattern A G\ng : G\nx : G → A\nH : x ∈ cylinder (Finset.image (fun x ↦ g * x) p.support) (p.mulShift g)\nu : G\nhu : u ∈ p.support\nhx : x (g * u) = p.mulShift g (g * u)\n⊢ x (g * u) = p.conf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 519,
"column": 2
} | {
"line": 519,
"column": 41
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Inhabited A\nG : Type u_2\ninst✝² : Monoid G\ninst✝¹ : IsLeftCancelMul G\ninst✝ : DiscreteTopology A\np : Pattern A G\ng : G\n⊢ IsOpen[Pi.topologicalSpace] {x | p.mulOccursInAt x g}",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Pi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 545,
"column": 2
} | {
"line": 545,
"column": 40
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Inhabited A\nG : Type u_2\ninst✝² : Monoid G\ninst✝¹ : IsLeftCancelMul G\ninst✝ : DiscreteTopology A\nF : Set (Pattern A G)\nh_eq : {x | ∀ p ∈ F, ∀ (v : G), ¬p.mulOccursInAt x v} = ⋂ p ∈ F, ⋂ v, {x | ¬p.mulOccursInAt x v}\np : Pattern A G\nhp : p ∈ F\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.SymbolicDynamics.Basic | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 41
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Inhabited A\nG : Type u_2\ninst✝² : Monoid G\ninst✝¹ : IsLeftCancelMul G\ninst✝ : T1Space A\np : Pattern A G\ng : G\n⊢ IsClosed[Pi.topologicalSpace] {x | p.mulOccursInAt x g}",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Pi.topolo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 382,
"column": 53
} | {
"line": 382,
"column": 92
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\ns : Set α\nhs : MeasurableSet s\n⊢ (ν.withDensity (μ.rnDeriv ν)).real s = μ.real s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
... | Measure.withDensity_rnDeriv_eq _ _ hμν, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Dynamics.TopologicalEntropy.DynamicalEntourage | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 47
} | [
{
"pp": "X : Type u_1\nT : X → X\nU : SetRel X X\ninst✝ : UniformSpace X\nh : Continuous[inst✝.toTopologicalSpace, inst✝.toTopologicalSpace] T\nU_uni : U ∈ 𝓤 X\nn : ℕ\nx : X\nk : ℕ\nb✝ : k ∈ Ico 0 n\n⊢ ball x ((map T T)^[↑⟨k, b✝⟩] ⁻¹' U) ∈ 𝓝 x",
"usedConstants": [
"Filter.instMembership",
"Eq.... | simp only [map_iterate, _root_.ball_preimage] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym | {
"line": 464,
"column": 2
} | {
"line": 465,
"column": 30
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\n⊢ μ.rnDeriv ν =ᶠ[ae ν] fun x ↦ μ.rnDeriv (μ + ν) x / ν.rnDeriv (μ + ν) x",
"usedConstants": [
"MeasureTheory.ae",
"ENNReal.instAdd",
"MeasureTheory.Measure",
"Preorder.toLT",... | filter_upwards [rnDeriv_add_self ν μ, rnDeriv_self_add μ ν, μ.rnDeriv_lt_top ν]
with a ha1 ha2 ha_lt_top | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Dynamics.TopologicalEntropy.NetEntropy | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 15
} | [
{
"pp": "X : Type u_1\nT : X → X\nU : SetRel X X\nn : ℕ\nF : Set X\ns t : Finset X\nhs : IsDynNetIn T F U n ↑s\nht : IsDynCoverOf T F U n ↑t\nx : X\nx_s : x ∈ s\n⊢ ∃ z ∈ t, z ∈ ball x (dynEntourage T U n)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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