module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 723,
"column": 4
} | {
"line": 723,
"column": 15
} | {
"line": 723,
"column": 16
} | [
{
"pp": "case seq.left\nbasis_hd : ℝ → ℝ\nbasis_tl : Basis\nexp : ℝ\ncoef : MultiseriesExpansion basis_tl\ntl : Multiseries basis_hd basis_tl\nh_coef : ∀ x ∈ (mk (Multiseries.cons exp coef tl) 0).seq, x.2.Sorted\nh_Pairwise : Seq.Pairwise (fun x1 x2 ↦ x1 > x2) (mk (Multiseries.cons exp coef tl) 0).seq\n⊢ coef.S... | [
"case seq.left\nbasis_hd : ℝ → ℝ\nbasis_tl : Basis\nexp : ℝ\ncoef : MultiseriesExpansion basis_tl\ntl : Multiseries basis_hd basis_tl\nh_coef : ∀ x ∈ (mk (Multiseries.cons exp coef tl) 0).seq, x.2.Sorted\nh_Pairwise : Seq.Pairwise (fun x1 x2 ↦ x1 > x2) (mk (Multiseries.cons exp coef tl) 0).seq\n⊢ coef.Sorted"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | {
"line": 296,
"column": 70
} | {
"line": 296,
"column": 81
} | {
"line": 296,
"column": 82
} | [
{
"pp": "exps_hd : ℝ\nexps_tl : List ℝ\nbasis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nh_length : (exps_hd :: exps_tl).length = (basis_hd :: basis_tl).length\nh : exps_hd = 0 ∧ FirstNonzeroIsPos exps_tl\nh_eq : toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) = toF... | [
"exps_hd : ℝ\nexps_tl : List ℝ\nbasis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nh_length : (exps_hd :: exps_tl).length = (basis_hd :: basis_tl).length\nh : exps_hd = 0 ∧ FirstNonzeroIsPos exps_tl\nh_eq : toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) = toFun exps_tl b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | {
"line": 315,
"column": 71
} | {
"line": 315,
"column": 82
} | {
"line": 315,
"column": 83
} | [
{
"pp": "exps_hd : ℝ\nexps_tl : List ℝ\nbasis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nh_length : (exps_hd :: exps_tl).length = (basis_hd :: basis_tl).length\nh : exps_hd = 0 ∧ FirstNonzeroIsNeg exps_tl\nh_eq : toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) = toF... | [
"exps_hd : ℝ\nexps_tl : List ℝ\nbasis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nh_length : (exps_hd :: exps_tl).length = (basis_hd :: basis_tl).length\nh : exps_hd = 0 ∧ FirstNonzeroIsNeg exps_tl\nh_eq : toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl) = toFun exps_tl b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | {
"line": 353,
"column": 43
} | {
"line": 353,
"column": 54
} | {
"line": 353,
"column": 55
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nexp1 : ℝ\nm1 : List ℝ\nh1 : (exp1 :: m1).length = (basis_hd :: basis_tl).length\nm2 : List ℝ\nh2 : (exp1 :: m2).length = (basis_hd :: basis_tl).length\nh : List.Lex (fun x1 x2 ↦ x1 < x2) m1 m2\n⊢ m1.length = ba... | [
"basis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nexp1 : ℝ\nm1 : List ℝ\nh1 : (exp1 :: m1).length = (basis_hd :: basis_tl).length\nm2 : List ℝ\nh2 : (exp1 :: m2).length = (basis_hd :: basis_tl).length\nh : List.Lex (fun x1 x2 ↦ x1 < x2) m1 m2\n⊢ m1.length = basis_tl.lengt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | {
"line": 353,
"column": 63
} | {
"line": 353,
"column": 74
} | {
"line": 353,
"column": 75
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nexp1 : ℝ\nm1 : List ℝ\nh1 : (exp1 :: m1).length = (basis_hd :: basis_tl).length\nm2 : List ℝ\nh2 : (exp1 :: m2).length = (basis_hd :: basis_tl).length\nh : List.Lex (fun x1 x2 ↦ x1 < x2) m1 m2\n⊢ m2.length = ba... | [
"basis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nexp1 : ℝ\nm1 : List ℝ\nh1 : (exp1 :: m1).length = (basis_hd :: basis_tl).length\nm2 : List ℝ\nh2 : (exp1 :: m2).length = (basis_hd :: basis_tl).length\nh : List.Lex (fun x1 x2 ↦ x1 < x2) m1 m2\n⊢ m2.length = basis_tl.lengt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | {
"line": 353,
"column": 63
} | {
"line": 353,
"column": 77
} | {
"line": 353,
"column": 77
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nexp1 : ℝ\nm1 : List ℝ\nh1 : (exp1 :: m1).length = (basis_hd :: basis_tl).length\nm2 : List ℝ\nh2 : (exp1 :: m2).length = (basis_hd :: basis_tl).length\nh : List.Lex (fun x1 x2 ↦ x1 < x2) m1 m2\n⊢ m2.length = ba... | [] | simpa using h2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | {
"line": 353,
"column": 63
} | {
"line": 353,
"column": 77
} | {
"line": 353,
"column": 77
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nexp1 : ℝ\nm1 : List ℝ\nh1 : (exp1 :: m1).length = (basis_hd :: basis_tl).length\nm2 : List ℝ\nh2 : (exp1 :: m2).length = (basis_hd :: basis_tl).length\nh : List.Lex (fun x1 x2 ↦ x1 < x2) m1 m2\n⊢ m2.length = ba... | [] | simpa using h2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | {
"line": 353,
"column": 63
} | {
"line": 353,
"column": 77
} | {
"line": 353,
"column": 77
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nexp1 : ℝ\nm1 : List ℝ\nh1 : (exp1 :: m1).length = (basis_hd :: basis_tl).length\nm2 : List ℝ\nh2 : (exp1 :: m2).length = (basis_hd :: basis_tl).length\nh : List.Lex (fun x1 x2 ↦ x1 < x2) m1 m2\n⊢ m2.length = ba... | [] | simpa using h2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 737,
"column": 24
} | {
"line": 737,
"column": 35
} | {
"line": 737,
"column": 36
} | [
{
"pp": "case cons\nbasis_hd : ℝ → ℝ\nbasis_tl : Basis\nexp : ℝ\ncoef : MultiseriesExpansion basis_tl\ntl : Multiseries basis_hd basis_tl\nh : (Multiseries.cons exp coef tl).Sorted\n⊢ (Multiseries.cons exp coef tl).tail.Sorted",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"case cons\nbasis_hd : ℝ → ℝ\nbasis_tl : Basis\nexp : ℝ\ncoef : MultiseriesExpansion basis_tl\ntl : Multiseries basis_hd basis_tl\nh : (Multiseries.cons exp coef tl).Sorted\n⊢ tl.Sorted"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Corecursion | {
"line": 534,
"column": 4
} | {
"line": 534,
"column": 40
} | {
"line": 535,
"column": 4
} | [
{
"pp": "case h.nil\nα : Type u_1\nmotive : (Stream'.Seq α → Stream'.Seq α) → Prop\nh_step :\n ∀ (op : Stream'.Seq α → Stream'.Seq α),\n motive op →\n ∃ T,\n ∀ (s : Stream'.Seq α),\n (op s).destruct =\n Option.map\n (fun x ↦\n match x with\n ... | [
"case h.nil\nα : Type u_1\nmotive : (Stream'.Seq α → Stream'.Seq α) → Prop\nh_step :\n ∀ (op : Stream'.Seq α → Stream'.Seq α),\n motive op →\n ∃ T,\n ∀ (s : Stream'.Seq α),\n (op s).destruct =\n Option.map\n (fun x ↦\n match x with\n | (... | simp only [tail_nil, head_nil] at hT | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | {
"line": 367,
"column": 8
} | {
"line": 367,
"column": 39
} | {
"line": 368,
"column": 8
} | [
{
"pp": "case cons.cons.cons.rel.h.h_firstIsPos\nbasis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nexp1 : ℝ\nm1 : List ℝ\nexp2 : ℝ\nm2 : List ℝ\nh : exp1 < exp2\nh1 : m1.length = basis_tl.length\nh2 : m2.length = basis_tl.length\n⊢ FirstNonzeroIsPos ((exp2 - exp1) :: m... | [
"case cons.cons.cons.rel.h.h_firstIsPos\nbasis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_basis : WellFormedBasis (basis_hd :: basis_tl)\nexp1 : ℝ\nm1 : List ℝ\nexp2 : ℝ\nm2 : List ℝ\nh : exp1 < exp2\nh1 : m1.length = basis_tl.length\nh2 : m2.length = basis_tl.length\n⊢ 0 < exp2 - exp1"
] | apply FirstNonzeroIsPos.of_head | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 798,
"column": 41
} | {
"line": 798,
"column": 52
} | {
"line": 798,
"column": 53
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : Basis\nexp : ℝ\ncoef : MultiseriesExpansion basis_tl\ntl : Multiseries basis_hd basis_tl\nf : ℝ → ℝ\nh : (mk (Multiseries.cons exp coef tl) f).Sorted\n⊢ (Multiseries.cons exp coef tl).Sorted",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedF... | [
"basis_hd : ℝ → ℝ\nbasis_tl : Basis\nexp : ℝ\ncoef : MultiseriesExpansion basis_tl\ntl : Multiseries basis_hd basis_tl\nf : ℝ → ℝ\nh : (mk (Multiseries.cons exp coef tl) f).Sorted\n⊢ (Multiseries.cons exp coef tl).Sorted"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 803,
"column": 2
} | {
"line": 803,
"column": 13
} | {
"line": 803,
"column": 14
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : Basis\nms : MultiseriesExpansion (basis_hd :: basis_tl)\nf : ℝ → ℝ\nh_sorted : ms.Sorted\n⊢ (ms.replaceFun f).Sorted",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Tactic.ComputeAsymptotics.MultiseriesExpansion.sorted_iff_seq_sor... | [
"basis_hd : ℝ → ℝ\nbasis_tl : Basis\nms : MultiseriesExpansion (basis_hd :: basis_tl)\nf : ℝ → ℝ\nh_sorted : ms.Sorted\n⊢ ms.seq.Sorted"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Corecursion | {
"line": 559,
"column": 27
} | {
"line": 559,
"column": 38
} | {
"line": 559,
"column": 39
} | [
{
"pp": "α : Type u_1\nγ : Type u_3\ns t : Stream'.Seq α\nop : γ → Stream'.Seq α → Stream'.Seq α\ninst✝ : FriendlyOperationClass op\nmotive : Stream'.Seq α → Stream'.Seq α → Prop\nbase : motive s t\nstep :\n ∀ (u v : Stream'.Seq α),\n motive u v → u = v ∨ ∃ hd u' v' c, u = Seq.cons hd (op c u') ∧ v = Seq.co... | [
"α : Type u_1\nγ : Type u_3\ns t : Stream'.Seq α\nop : γ → Stream'.Seq α → Stream'.Seq α\ninst✝ : FriendlyOperationClass op\nmotive : Stream'.Seq α → Stream'.Seq α → Prop\nbase : motive s t\nstep :\n ∀ (u v : Stream'.Seq α),\n motive u v → u = v ∨ ∃ hd u' v' c, u = Seq.cons hd (op c u') ∧ v = Seq.cons hd (op c ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Trimming | {
"line": 62,
"column": 11
} | {
"line": 62,
"column": 22
} | {
"line": 62,
"column": 23
} | [
{
"pp": "case nil\nbasis_hd✝ : ℝ → ℝ\nbasis_tl✝ : List (ℝ → ℝ)\nf✝ : ℝ → ℝ\nh_approx : (mk Multiseries.nil f✝).Approximates\n⊢ (mk Multiseries.nil f✝).toFun =ᶠ[atTop] 0",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Filter.EventuallyEq",
"i... | [
"case nil\nbasis_hd✝ : ℝ → ℝ\nbasis_tl✝ : List (ℝ → ℝ)\nf✝ : ℝ → ℝ\nh_approx : (mk Multiseries.nil f✝).Approximates\n⊢ f✝ =ᶠ[atTop] 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Corecursion | {
"line": 578,
"column": 15
} | {
"line": 579,
"column": 27
} | {
"line": 581,
"column": 0
} | [
{
"pp": "α : Type u_1\nγ : Type u_3\nop : γ → Stream'.Seq α → Stream'.Seq α\ninst✝ : FriendlyOperationClass op\nmotive : Stream'.Seq α → Stream'.Seq α → Prop\nstep :\n ∀ (u v : Stream'.Seq α),\n motive u v → u = v ∨ ∃ hd u' v' c, u = Seq.cons hd (op c u') ∧ v = Seq.cons hd (op c v') ∧ motive u' v'\nn : ℕ\nh... | [] | by
grw [ih, pow_succ'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Trimming | {
"line": 122,
"column": 2
} | {
"line": 126,
"column": 9
} | {
"line": 128,
"column": 0
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : Basis\nexp : ℝ\ncoef : MultiseriesExpansion basis_tl\ntl ms : Multiseries basis_hd basis_tl\nh_ms : Multiseries.cons exp coef tl = ms\nh : ms.Trimmed\n⊢ coef.Trimmed ∧ ¬coef.IsZero",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Stream'.Seq",
... | [] | cases h with
| nil => simp at h_ms
| cons h_trimmed h_ne_zero =>
simp at h_ms
grind | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Tactic.DeriveCountable | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 41
} | {
"line": 83,
"column": 42
} | [
{
"pp": "p : Prop\na b b' : ℕ\nh : b = b' → p\n⊢ Nat.pair a b = Nat.pair a b' → p",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.pair_eq_pair._simp_1",
"congrArg",
"id",
"Nat.pair",
"And",
"implies_congr",
"Nat",
"True",
... | [
"p : Prop\na b b' : ℕ\nh : b = b' → p\n⊢ b = b' → p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1074,
"column": 8
} | {
"line": 1074,
"column": 56
} | {
"line": 1075,
"column": 8
} | [
{
"pp": "case oadd.inl.none.inl.some.succ.refine_2\na : ONote\nm : ℕ+\nb : ONote\nihb : b = 0\ne✝ : b.fundamentalSequence = Sum.inl none\na' : ONote\niha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)\ne : a.fundamentalSequence = Sum.inl (some a')\nm' : ℕ\ne' : m.natPred = m' + 1\nthis : 0 < ω ^ a'.repr\ni : ℕ\nH : (... | [
"case oadd.inl.none.inl.some.succ.refine_2\na : ONote\nm : ℕ+\nb : ONote\nihb : b = 0\ne✝ : b.fundamentalSequence = Sum.inl none\na' : ONote\niha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)\ne : a.fundamentalSequence = Sum.inl (some a')\nm' : ℕ\ne' : m.natPred = m' + 1\nthis : 0 < ω ^ a'.repr\ni : ℕ\nH : (a.oadd (m' +... | rw [repr, repr_zero, add_zero, iha.1, opow_succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Tactic.NormNum.Irrational | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 24
} | {
"line": 55,
"column": 25
} | [
{
"pp": "case inl\np q : ℕ\nh_coprime : p.Coprime q\nhq : 0 < q\nm : ℕ\nh : ∀ (m : ℕ), 0 ≠ m ^ q\n⊢ 0 ^ p ≠ m ^ q",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Nat.instMonoid",
"id",
"Ne",
"instOfNatNat",
"NPow.toPow",
"HPow.hPow",
"Nat",
"... | [
"case inl\np q : ℕ\nh_coprime : p.Coprime q\nhq : 0 < q\nm : ℕ\nh : ∀ (m : ℕ), 0 ≠ m ^ q\n⊢ ¬0 ^ p = m ^ q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.NormNum.Irrational | {
"line": 59,
"column": 29
} | {
"line": 59,
"column": 49
} | {
"line": 59,
"column": 50
} | [
{
"pp": "n p q : ℕ\nh_coprime : p.Coprime q\nhq : 0 < q\nm : ℕ\nhn : n ≠ 0\nh : n ^ p = m ^ q\nf : ℕ →₀ ℕ := Finsupp.mapRange (fun x ↦ x / q) ⋯ n.factorization\nhf : n.factorization = q • f\n⊢ f 0 = 0",
"ppTerm": "?m.86",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"n p q : ℕ\nh_coprime : p.Coprime q\nhq : 0 < q\nm : ℕ\nhn : n ≠ 0\nh : n ^ p = m ^ q\nf : ℕ →₀ ℕ := Finsupp.mapRange (fun x ↦ x / q) ⋯ n.factorization\nhf : n.factorization = q • f\n⊢ f 0 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.NormNum.Irrational | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 13
} | {
"line": 65,
"column": 14
} | [
{
"pp": "case inr\nn p q : ℕ\nh_coprime : p.Coprime q\nhq : 0 < q\nm : ℕ\nhn : n ≠ 0\nh : n ^ p = m ^ q\nf : ℕ →₀ ℕ := Finsupp.mapRange (fun x ↦ x / q) ⋯ n.factorization\nz : ℕ\n⊢ q ∣ n.factorization z",
"ppTerm": "?inr✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"case inr\nn p q : ℕ\nh_coprime : p.Coprime q\nhq : 0 < q\nm : ℕ\nhn : n ≠ 0\nh : n ^ p = m ^ q\nf : ℕ →₀ ℕ := Finsupp.mapRange (fun x ↦ x / q) ⋯ n.factorization\nz : ℕ\n⊢ q ∣ n.factorization z"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.NormNum.Irrational | {
"line": 65,
"column": 56
} | {
"line": 65,
"column": 67
} | {
"line": 65,
"column": 68
} | [
{
"pp": "n p q : ℕ\nh_coprime : p.Coprime q\nhq : 0 < q\nm : ℕ\nhn : n ≠ 0\nh : n ^ p = m ^ q\nf : ℕ →₀ ℕ := Finsupp.mapRange (fun x ↦ x / q) ⋯ n.factorization\nz : ℕ\n⊢ p * n.factorization z = q * ?m.185",
"ppTerm": "?m.186",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"n p q : ℕ\nh_coprime : p.Coprime q\nhq : 0 < q\nm : ℕ\nhn : n ≠ 0\nh : n ^ p = m ^ q\nf : ℕ →₀ ℕ := Finsupp.mapRange (fun x ↦ x / q) ⋯ n.factorization\nz : ℕ\n⊢ p * n.factorization z = q * ?m.185"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.NormNum.LegendreSymbol | {
"line": 75,
"column": 98
} | {
"line": 81,
"column": 41
} | {
"line": 83,
"column": 0
} | [
{
"pp": "b : ℕ\nhb : (b / 2).beq 0 = false\n⊢ jacobiSymNat 0 b = 0",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Trans.trans",
"instHDiv",
"HMul.hMul",
"Nat.ne_of_beq_eq_false",
"of_decide_eq_true",
"congrArg",
"Nat.succ_le_of_lt... | [] | by
rw [jacobiSymNat, Nat.cast_zero, jacobiSym.zero_left ?_]
calc
1 < 2 * 1 := by decide
_ ≤ 2 * (b / 2) :=
Nat.mul_le_mul_left _ (Nat.succ_le_of_lt (Nat.pos_of_ne_zero (Nat.ne_of_beq_eq_false hb)))
_ ≤ b := Nat.mul_div_le b 2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Tactic.NormNum.LegendreSymbol | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 28
} | {
"line": 121,
"column": 4
} | [
{
"pp": "case inr\na b c : ℕ\nr : ℤ\nha : a % 2 = 1\nhb : b % 2 = 0\nhc : b / 2 = c\nhr : jacobiSymNat a c = r\nha' : legendreSym 2 ↑a = 1\nhc' : c ≠ 0\n⊢ jacobiSymNat a b = r",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"NeZero.mk",
"Nat",
... | [
"case inr\na b c : ℕ\nr : ℤ\nha : a % 2 = 1\nhb : b % 2 = 0\nhc : b / 2 = c\nhr : jacobiSymNat a c = r\nha' : legendreSym 2 ↑a = 1\nhc' : c ≠ 0\nthis : NeZero c\n⊢ jacobiSymNat a b = r"
] | have : NeZero c := ⟨hc'⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Tactic.NormNum.Irrational | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 19
} | {
"line": 125,
"column": 20
} | [
{
"pp": "a b d : ℕ\nh_coprime : a.Coprime b\nq : ℚ\nhq : 0 ≤ q\nhb_zero : ¬b = 0\nx' : ℤ := ⋯\ny : ℕ := ⋯\nx : ℕ\nha : a ≠ x ^ d\nhx' : x' = ↑x\nh : a * y ^ d = b * x ^ d\n⊢ x.Coprime y",
"ppTerm": "?m.203",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b d : ℕ\nh_coprime : a.Coprime b\nq : ℚ\nhq : 0 ≤ q\nhb_zero : ¬b = 0\nx' : ℤ := ⋯\ny : ℕ := ⋯\nx : ℕ\nha : a ≠ x ^ d\nhx' : x' = ↑x\nh : a * y ^ d = b * x ^ d\n⊢ x.Coprime y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.NormNum.Irrational | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 71
} | {
"line": 180,
"column": 2
} | [
{
"pp": "x y : ℝ\nx_num x_den y_num y_den k_den : ℕ\nhy_isNNRat : IsNNRat y y_num y_den\nhx_coprime : x_num.Coprime x_den\nhy_coprime : y_num.Coprime y_den\nhd1 : k_den ^ y_den < x_den\nhd2 : x_den < (k_den + 1) ^ y_den\nhx_inv : Invertible ↑x_den\nhx_eq : x = ↑x_num * ⅟↑x_den\n⊢ Irrational (x ^ y)⁻¹",
"ppT... | [
"x y : ℝ\nx_num x_den y_num y_den k_den : ℕ\nhy_isNNRat : IsNNRat y y_num y_den\nhx_coprime : x_num.Coprime x_den\nhy_coprime : y_num.Coprime y_den\nhd1 : k_den ^ y_den < x_den\nhd2 : x_den < (k_den + 1) ^ y_den\nhx_inv : Invertible ↑x_den\nhx_eq : x = ↑x_num * ⅟↑x_den\n⊢ Irrational (x⁻¹ ^ y)"
] | rw [← Real.inv_rpow (by simp only [hx_eq, invOf_eq_inv]; positivity)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Tactic.NormNum.RealSqrt | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 89
} | {
"line": 46,
"column": 90
} | [
{
"pp": "num denom : ℕ\ninv : Invertible ↑denom\nh₁ : 0 ≤ ↑↑num * ⅟↑denom\n⊢ √(↑(Int.negOfNat num) * ⅟↑denom) = ↑0",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
... | [
"num denom : ℕ\ninv : Invertible ↑denom\nh₁ : 0 ≤ ↑↑num * ⅟↑denom\n⊢ 0 ≤ ↑num * (↑denom)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Simproc.FinsetInterval | {
"line": 27,
"column": 69
} | {
"line": 27,
"column": 80
} | {
"line": 27,
"column": 81
} | [
{
"pp": "m n : ℕ\nhnm : n.blt m = true\n⊢ Icc m n = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Nat.instLocallyFiniteOrder",
"id",
"Finset.Icc_eq_empty_iff... | [
"m n : ℕ\nhnm : n.blt m = true\n⊢ n < m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Simproc.FinsetInterval | {
"line": 30,
"column": 76
} | {
"line": 30,
"column": 87
} | {
"line": 30,
"column": 88
} | [
{
"pp": "m n : ℕ\ns : Finset ℕ\nhmn : m.ble n = true\nhs : Icc (m + 1) n = s\n⊢ m ≤ n",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m n : ℕ\ns : Finset ℕ\nhmn : m.ble n = true\nhs : Icc (m + 1) n = s\n⊢ m ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Simproc.FinsetInterval | {
"line": 55,
"column": 67
} | {
"line": 55,
"column": 78
} | {
"line": 55,
"column": 79
} | [
{
"pp": "m n : ℤ\nhnm : n < m\n⊢ Icc m n = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Int.instLinearOrder",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"id",
"... | [
"m n : ℤ\nhnm : n < m\n⊢ n < m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Tactic.Simproc.FinsetInterval | {
"line": 58,
"column": 76
} | {
"line": 58,
"column": 87
} | {
"line": 58,
"column": 88
} | [
{
"pp": "m n : ℤ\ns : Finset ℤ\nhmn : m ≤ n\nhs : Icc (m + 1) n = s\n⊢ m ≤ n",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m n : ℤ\ns : Finset ℤ\nhmn : m ≤ n\nhs : Icc (m + 1) n = s\n⊢ m ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Testing.Plausible.Functions | {
"line": 114,
"column": 8
} | {
"line": 114,
"column": 19
} | {
"line": 114,
"column": 20
} | [
{
"pp": "case mp\nα : Type u\nβ : Type v\ninst✝² : DecidableEq α\ninst✝¹ : Zero β\ninst✝ : DecidableEq β\na : α\nA : List ((_ : α) × β)\ny od : β\nhval : ⟨a, od⟩ ∈ A.dedupKeys\nhod : (decide ¬od = 0) = true\nthis : od ∈ List.dlookup a A.dedupKeys\n⊢ ¬(some od).getD 0 = 0",
"ppTerm": "?mp",
"assigned": t... | [
"case mp\nα : Type u\nβ : Type v\ninst✝² : DecidableEq α\ninst✝¹ : Zero β\ninst✝ : DecidableEq β\na : α\nA : List ((_ : α) × β)\ny od : β\nhval : ⟨a, od⟩ ∈ A.dedupKeys\nhod : (decide ¬od = 0) = true\nthis : od ∈ List.dlookup a A.dedupKeys\n⊢ ¬od = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.OpenMapping | {
"line": 63,
"column": 8
} | {
"line": 63,
"column": 44
} | {
"line": 63,
"column": 45
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : MulAction G X\ninst✝⁴ : SigmaCompactSpace G\ninst✝³ : BaireSpace X\ninst✝² : T2Space X\ninst✝¹ : ContinuousSMul G X\ninst✝ : IsPretransitive G X\nU : Set G\nhU... | [
"G : Type u_1\nX : Type u_2\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : MulAction G X\ninst✝⁴ : SigmaCompactSpace G\ninst✝³ : BaireSpace X\ninst✝² : T2Space X\ninst✝¹ : ContinuousSMul G X\ninst✝ : IsPretransitive G X\nU : Set G\nhU : U ∈ 𝓝 1\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.OpenMapping | {
"line": 80,
"column": 53
} | {
"line": 80,
"column": 64
} | {
"line": 80,
"column": 65
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : MulAction G X\ninst✝⁴ : SigmaCompactSpace G\ninst✝³ : BaireSpace X\ninst✝² : T2Space X\ninst✝¹ : ContinuousSMul G X\ninst✝ : IsPretransitive G X\nU : Set G\nhU... | [
"G : Type u_1\nX : Type u_2\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : MulAction G X\ninst✝⁴ : SigmaCompactSpace G\ninst✝³ : BaireSpace X\ninst✝² : T2Space X\ninst✝¹ : ContinuousSMul G X\ninst✝ : IsPretransitive G X\nU : Set G\nhU : U ∈ 𝓝 1\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Ascoli | {
"line": 431,
"column": 2
} | {
"line": 431,
"column": 59
} | {
"line": 435,
"column": 2
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nα : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : UniformSpace α\nF : ι → X → α\ninst✝ : TopologicalSpace ι\n𝔖 : Set (Set X)\n𝔖_compact : ∀ K ∈ 𝔖, IsCompact K\nF_ind : IsInducing (⇑(UniformOnFun.ofFun 𝔖) ∘ F)\nF_cl : IsClosed[UniformOnFun.topologicalSpace X α 𝔖] (rang... | [
"ι : Type u_1\nX : Type u_2\nα : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : UniformSpace α\nF : ι → X → α\ninst✝ : TopologicalSpace ι\n𝔖 : Set (Set X)\n𝔖_compact : ∀ K ∈ 𝔖, IsCompact K\nF_ind : IsInducing (⇑(UniformOnFun.ofFun 𝔖) ∘ F)\nF_cl : IsClosed[UniformOnFun.topologicalSpace X α 𝔖] (range (⇑(Uniform... | rw [← isCompact_univ_iff, this.isCompact_iff, image_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Group.SubmonoidClosure | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 13
} | {
"line": 54,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : CompactSpace G\ninst✝ : IsTopologicalGroup G\nx : G\n⊢ MapClusterPt 1 atTop fun x_1 ↦ x ^ x_1",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : CompactSpace G\ninst✝ : IsTopologicalGroup G\nx : G\n⊢ MapClusterPt 1 atTop fun x_1 ↦ x ^ x_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Group.SubmonoidClosure | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 13
} | {
"line": 58,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : CompactSpace G\ninst✝ : IsTopologicalGroup G\nx : G\n⊢ MapClusterPt x atTop fun x_1 ↦ x ^ x_1",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : CompactSpace G\ninst✝ : IsTopologicalGroup G\nx : G\n⊢ MapClusterPt x atTop fun x_1 ↦ x ^ x_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Ascoli | {
"line": 501,
"column": 2
} | {
"line": 503,
"column": 64
} | {
"line": 504,
"column": 2
} | [
{
"pp": "X : Type u_2\nα : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : UniformSpace α\nS : Set C(X, α)\nhS1 : IsCompact (ContinuousMap.toFun '' S)\nhS2 : Equicontinuous fun x ↦ ⇑↑x\n⊢ IsCompact S",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike"... | [
"X : Type u_2\nα : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : UniformSpace α\nS : Set C(X, α)\nhS1 : IsCompact (ContinuousMap.toFun '' S)\nhS2 : Equicontinuous fun x ↦ ⇑↑x\n⊢ IsInducing ⇑(Equiv.Set.image DFunLike.coe S ⋯)"
] | suffices h : IsInducing (Equiv.Set.image _ S DFunLike.coe_injective) by
rw [isCompact_iff_compactSpace] at hS1 ⊢
exact (Equiv.toHomeomorphOfIsInducing _ h).symm.compactSpace | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Topology.Algebra.Group.OpenMapping | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 13
} | {
"line": 108,
"column": 14
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : MulAction G X\ninst✝⁴ : SigmaCompactSpace G\ninst✝³ : BaireSpace X\ninst✝² : T2Space X\ninst✝¹ : ContinuousSMul G X\ninst✝ : IsPretransitive G X\nx : X\ng : G\... | [
"G : Type u_1\nX : Type u_2\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : MulAction G X\ninst✝⁴ : SigmaCompactSpace G\ninst✝³ : BaireSpace X\ninst✝² : T2Space X\ninst✝¹ : ContinuousSMul G X\ninst✝ : IsPretransitive G X\nx : X\ng : G\nU : Set G\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 27
} | {
"line": 175,
"column": 28
} | [
{
"pp": "M : Type u_1\nS : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_6\ninst✝¹⁰ : Monoid M\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : AddCommMonoid E'\ninst✝⁶ : AddCommMonoid F\ninst✝⁵ : Module S E\ninst✝⁴ : Module S E'\ninst✝³ : Module S F\ninst✝² : TopologicalSpace F\ninst✝¹ : T2Spac... | [
"M : Type u_1\nS : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_6\ninst✝¹⁰ : Monoid M\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : AddCommMonoid E'\ninst✝⁶ : AddCommMonoid F\ninst✝⁵ : Module S E\ninst✝⁴ : Module S E'\ninst✝³ : Module S F\ninst✝² : TopologicalSpace F\ninst✝¹ : T2Space F\ninst✝ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 27
} | {
"line": 188,
"column": 28
} | [
{
"pp": "M : Type u_1\nS : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_6\ninst✝¹⁰ : Monoid M\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : AddCommMonoid E'\ninst✝⁶ : AddCommMonoid F\ninst✝⁵ : Module S E\ninst✝⁴ : Module S E'\ninst✝³ : Module S F\ninst✝² : TopologicalSpace F\ninst✝¹ : T2Spac... | [
"M : Type u_1\nS : Type u_2\nE : Type u_3\nE' : Type u_4\nF : Type u_6\ninst✝¹⁰ : Monoid M\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : AddCommMonoid E'\ninst✝⁶ : AddCommMonoid F\ninst✝⁵ : Module S E\ninst✝⁴ : Module S E'\ninst✝³ : Module S F\ninst✝² : TopologicalSpace F\ninst✝¹ : T2Space F\ninst✝ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 27
} | {
"line": 203,
"column": 28
} | [
{
"pp": "M : Type u_1\nS : Type u_2\nE : Type u_3\nE' : Type u_4\ninst✝¹⁰ : Monoid M\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : AddCommMonoid E'\ninst✝⁶ : Module S E\ninst✝⁵ : Module S E'\nF : Type u_10\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module S F\ninst✝² : TopologicalSpace F\ninst✝¹ : Conti... | [
"M : Type u_1\nS : Type u_2\nE : Type u_3\nE' : Type u_4\ninst✝¹⁰ : Monoid M\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : AddCommMonoid E'\ninst✝⁶ : Module S E\ninst✝⁵ : Module S E'\nF : Type u_10\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module S F\ninst✝² : TopologicalSpace F\ninst✝¹ : ContinuousConstSM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.DiscreteConvolution | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 27
} | {
"line": 209,
"column": 28
} | [
{
"pp": "M : Type u_1\nS : Type u_2\nE : Type u_3\nE' : Type u_4\ninst✝¹⁰ : Monoid M\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : AddCommMonoid E'\ninst✝⁶ : Module S E\ninst✝⁵ : Module S E'\nF : Type u_10\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module S F\ninst✝² : TopologicalSpace F\ninst✝¹ : Conti... | [
"M : Type u_1\nS : Type u_2\nE : Type u_3\nE' : Type u_4\ninst✝¹⁰ : Monoid M\ninst✝⁹ : CommSemiring S\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : AddCommMonoid E'\ninst✝⁶ : Module S E\ninst✝⁵ : Module S E'\nF : Type u_10\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module S F\ninst✝² : TopologicalSpace F\ninst✝¹ : ContinuousConstSM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.IsOpenUnits | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 81
} | {
"line": 79,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalRing R\nI : Ideal R\nhR : IsAdic I\nhI : I ≤ ⊥.jacobson\ns : Set R\nhs : s ∈ Filter.map (⇑(Units.coeHom R)) (Filter.comap (⇑(Units.embedProduct R)) (𝓝 1 ×ˢ 𝓝 1))\nH : (𝓝 1).HasBasis (fun _n ↦ True) fun n ↦ (fun y ↦ ... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalRing R\nI : Ideal R\nhR : IsAdic I\nhI : I ≤ ⊥.jacobson\ns : Set R\nhs : s ∈ Filter.map (⇑(Units.coeHom R)) (Filter.comap (⇑(Units.embedProduct R)) (𝓝 1 ×ˢ 𝓝 1))\nH : (𝓝 1).HasBasis (fun _n ↦ True) fun n ↦ (fun y ↦ 1 + y) '' ↑(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.IsOpenUnits | {
"line": 91,
"column": 27
} | {
"line": 91,
"column": 38
} | {
"line": 91,
"column": 39
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalRing R\nI : Ideal R\nhR : IsAdic I\nhI : I ≤ ⊥.jacobson\ns : Set R\nhs : s ∈ Filter.map (⇑(Units.coeHom R)) (Filter.comap (⇑(Units.embedProduct R)) (𝓝 1 ×ˢ 𝓝 1))\nH : (𝓝 1).HasBasis (fun _n ↦ True) fun n ↦ (fun y ↦ ... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalRing R\nI : Ideal R\nhR : IsAdic I\nhI : I ≤ ⊥.jacobson\ns : Set R\nhs : s ∈ Filter.map (⇑(Units.coeHom R)) (Filter.comap (⇑(Units.embedProduct R)) (𝓝 1 ×ˢ 𝓝 1))\nH : (𝓝 1).HasBasis (fun _n ↦ True) fun n ↦ (fun y ↦ 1 + y) '' ↑(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Nonarchimedean.TotallyDisconnected | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 25
} | {
"line": 53,
"column": 26
} | [
{
"pp": "case h\nG : Type u_1\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : NonarchimedeanGroup G\ninst✝ : T2Space G\na b : G\nh : a ≠ b\nu v : Set G\nleft✝ : IsOpen u\nopen_v : IsOpen v\nmem_u : a⁻¹ * b ∈ u\nmem_v : 1 ∈ v\ndis : Disjoint u v\nV : OpenSubgroup G\nhV : ↑V ⊆ v\nx : Set G\nmem_aV : x ⊆ ... | [
"case h\nG : Type u_1\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : NonarchimedeanGroup G\ninst✝ : T2Space G\na b : G\nh : a ≠ b\nu v : Set G\nleft✝ : IsOpen u\nopen_v : IsOpen v\nmem_u : a⁻¹ * b ∈ u\nmem_v : 1 ∈ v\ndis : Disjoint u v\nV : OpenSubgroup G\nhV : ↑V ⊆ v\nx : Set G\nmem_aV : x ⊆ a • ↑V\nmem_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.PontryaginDual | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 20
} | {
"line": 56,
"column": 21
} | [
{
"pp": "case refine_2\nA : Type u_1\nB : Type u_2\nC : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : CommGroup G\ninst✝⁸ : Group H\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace G\nin... | [
"case refine_2\nA : Type u_1\nB : Type u_2\nC : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : CommGroup G\ninst✝⁸ : Group H\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace G\ninst✝³ : Topol... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.PontryaginDual | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 17
} | {
"line": 96,
"column": 18
} | [
{
"pp": "case refine_1\nA : Type u_1\nB : Type u_2\nC : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : CommGroup G\ninst✝⁸ : Group H\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace G\nin... | [
"case refine_1\nA : Type u_1\nB : Type u_2\nC : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : CommGroup G\ninst✝⁸ : Group H\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace G\ninst✝³ : Topol... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.PontryaginDual | {
"line": 100,
"column": 53
} | {
"line": 100,
"column": 71
} | {
"line": 100,
"column": 72
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : CommGroup G\ninst✝⁸ : Group H\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace G\ninst✝³ : Topologi... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\nG : Type u_4\nH : Type u_5\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : CommGroup G\ninst✝⁸ : Group H\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace G\ninst✝³ : TopologicalSpace H\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.TopCat.Sphere | {
"line": 76,
"column": 27
} | {
"line": 76,
"column": 68
} | {
"line": 76,
"column": 69
} | [
{
"pp": "n : ℕ\nx : EuclideanSpace ℝ (Fin n)\nhx : x ∈ Metric.sphere 0 1\ny : EuclideanSpace ℝ (Fin n)\nhy : y ∈ Metric.sphere 0 1\nh :\n (ConcreteCategory.hom (diskBoundaryInclusion n)) { down := ⟨x, hx⟩ } =\n (ConcreteCategory.hom (diskBoundaryInclusion n)) { down := ⟨y, hy⟩ }\n⊢ x = y",
"ppTerm": "?m... | [
"n : ℕ\nx : EuclideanSpace ℝ (Fin n)\nhx : x ∈ Metric.sphere 0 1\ny : EuclideanSpace ℝ (Fin n)\nhy : y ∈ Metric.sphere 0 1\nh :\n (ConcreteCategory.hom (diskBoundaryInclusion n)) { down := ⟨x, hx⟩ } =\n (ConcreteCategory.hom (diskBoundaryInclusion n)) { down := ⟨y, hy⟩ }\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.TopCat.Sphere | {
"line": 82,
"column": 27
} | {
"line": 82,
"column": 60
} | {
"line": 82,
"column": 61
} | [
{
"pp": "n : ℕ\nx : EuclideanSpace ℝ (Fin n)\nhx : x ∈ Metric.ball 0 1\ny : EuclideanSpace ℝ (Fin n)\nhy : y ∈ Metric.ball 0 1\nh :\n (ConcreteCategory.hom (ballInclusion n)) { down := ⟨x, hx⟩ } =\n (ConcreteCategory.hom (ballInclusion n)) { down := ⟨y, hy⟩ }\n⊢ x = y",
"ppTerm": "?m.67",
"assigned"... | [
"n : ℕ\nx : EuclideanSpace ℝ (Fin n)\nhx : x ∈ Metric.ball 0 1\ny : EuclideanSpace ℝ (Fin n)\nhy : y ∈ Metric.ball 0 1\nh :\n (ConcreteCategory.hom (ballInclusion n)) { down := ⟨x, hx⟩ } =\n (ConcreteCategory.hom (ballInclusion n)) { down := ⟨y, hy⟩ }\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DerivedSet | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 13
} | {
"line": 83,
"column": 14
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nA : Set X\nhA : IsClosed[inst✝] A\n⊢ relDerivedSet A = derivedSet A",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"CompleteLattice.toConditionallyCompleteLattice",
"id",
"... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nA : Set X\nhA : IsClosed[inst✝] A\n⊢ derivedSet A ⊆ A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DerivedSet | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 18
} | {
"line": 116,
"column": 19
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nU : Set X\n⊢ Perfect U ↔ U = derivedSet U",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Perfect",
"Preperfect",
"id",
"derivedSet",
"IsClosed",
"perfect_def",
"A... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nU : Set X\n⊢ IsClosed[inst✝] U ∧ Preperfect U ↔ U = derivedSet U"
] | perfect_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.CantorBendixson | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 34
} | {
"line": 87,
"column": 4
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\na : Ordinal.{u}\n⊢ sᵈ[a + 1] = relDerivedSet sᵈ[a]",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"CantorBendixson.iteratedDerivedSet",
"CompleteLattice.toConditionallyCompleteLattice"... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\na : Ordinal.{u}\n⊢ gfpApprox relDerivedSet s (a + 1) = derivedSet (gfpApprox relDerivedSet s a) ∩ gfpApprox relDerivedSet s a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CantorBendixson | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 34
} | {
"line": 91,
"column": 35
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\na : Ordinal.{u}\nha : Order.IsSuccLimit a\n⊢ sᵈ[a] = ⋂ b, sᵈ[↑b]",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partialOrder",
"Iff.of_eq",
"congrArg",
"Set.... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\na : Ordinal.{u}\nha : Order.IsSuccLimit a\n⊢ gfpApprox relDerivedSet s a = ⋂ i, ⋂ (_ : i < a), gfpApprox relDerivedSet s i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CantorBendixson | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 45
} | {
"line": 110,
"column": 6
} | [
{
"pp": "case limit\nX : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsClosed s\na : Ordinal.{u}\nha : Order.IsSuccLimit a\nih : ∀ o' < a, IsClosed sᵈ[o']\n⊢ IsClosed sᵈ[a]",
"ppTerm": "?limit",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partia... | [
"case limit\nX : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsClosed s\na : Ordinal.{u}\nha : Order.IsSuccLimit a\nih : ∀ o' < a, IsClosed sᵈ[o']\n⊢ IsClosed (⋂ i, ⋂ (_ : i < a), sᵈ[i])"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CantorBendixson | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 39
} | {
"line": 124,
"column": 40
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\na : Ordinal.{u}\nha : sᵈ[a + 1] = sᵈ[a]\n⊢ relDerivedSet sᵈ[a] = sᵈ[a]",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.inter_eq_right._simp_1",
"Eq.mpr",
"PartialOrder.toPreorder",
"CantorBendixson.ite... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\na : Ordinal.{u}\nha : sᵈ[a + 1] = sᵈ[a]\n⊢ sᵈ[a] ⊆ derivedSet sᵈ[a]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CantorBendixson | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 39
} | {
"line": 142,
"column": 40
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ perfectKernel s ⊆ s",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ perfectKernel s ⊆ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CantorBendixson | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 29
} | {
"line": 146,
"column": 30
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns t : Set X\nhst : s ⊆ t\n⊢ perfectKernel s ⊆ perfectKernel t",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.iInter",
"CantorBendixson.iteratedDerivedSet",
"id",
"LE.le",
"Set.subset_iInter_... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns t : Set X\nhst : s ⊆ t\n⊢ ∀ (i : Ordinal.{u}), ⋂ a, sᵈ[a] ⊆ tᵈ[i]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CantorBendixson | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 13
} | {
"line": 155,
"column": 14
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\n⊢ perfectKernel ∅ = ∅",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\ninst✝ : TopologicalSpace X\n⊢ perfectKernel ∅ = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CantorBendixson | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 70
} | {
"line": 173,
"column": 4
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns P : Set X\nhP : Perfect P\nhPs : P ⊆ s\ni : Ordinal.{u}\n⊢ P ⊆ sᵈ[i]",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\ninst✝ : TopologicalSpace X\ns P : Set X\nhP : Perfect P\nhPs : P ⊆ s\ni : Ordinal.{u}\n⊢ P ⊆ sᵈ[i]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CantorBendixson | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 67
} | {
"line": 182,
"column": 4
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsClosed s\na : Ordinal.{u}\nha : sᵈ[a] ∈ fixedPoints ⇑relDerivedSet\n⊢ sᵈ[a] = derivedSet sᵈ[a]",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsClosed s\na : Ordinal.{u}\nha : sᵈ[a] ∈ fixedPoints ⇑relDerivedSet\n⊢ sᵈ[a] = derivedSet sᵈ[a]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Finite | {
"line": 332,
"column": 4
} | {
"line": 332,
"column": 30
} | {
"line": 333,
"column": 2
} | [
{
"pp": "case h\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nC D : Set X\ninst✝ : RelCWComplex C D\nfinite : _root_.Finite ((n : ℕ) × cell C n)\nh✝ : Nonempty ((n : ℕ) × cell C n)\nx✝ : Fintype ((n : ℕ) × cell C n)\nA : Finset ℕ := Finset.image Sigma.fst Finset.univ\nm : ℕ\na✝ : A.max' ⋯ + 1 ≤ m\nh' : Nonempty (... | [] | linarith [A.le_max' m hmA] | Mathlib.Tactic._aux_Mathlib_Tactic_Linarith_Frontend___elabRules_Mathlib_Tactic_linarith_1 | Mathlib.Tactic.linarith |
Mathlib.Topology.Category.Compactum | {
"line": 151,
"column": 6
} | {
"line": 151,
"column": 11
} | {
"line": 151,
"column": 11
} | [
{
"pp": "X Y : Compactum\nf : X ⟶ Y\nxs : Ultrafilter X.A\n⊢ (ConcreteCategory.hom (X.a ≫ f.f)) xs = Y.str (Ultrafilter.map (⇑(ConcreteCategory.hom f)) xs)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Monad.Algebra.Hom.h",
"CategoryTheory.Cate... | [
"X Y : Compactum\nf : X ⟶ Y\nxs : Ultrafilter X.A\n⊢ (ConcreteCategory.hom (β.map f.f ≫ Y.a)) xs = Y.str (Ultrafilter.map (⇑(ConcreteCategory.hom f)) xs)"
] | ← f.h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Convenient.ContinuousMapGeneratedBy | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 41
} | {
"line": 72,
"column": 42
} | [
{
"pp": "ι : Type t\nX : ι → Type u\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ninst✝ : TopologicalSpace Y\n⊢ ContinuousGeneratedBy X _root_.id",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"Equiv.instEquivLike",
"Equiv.symm_comp... | [
"ι : Type t\nX : ι → Type u\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ninst✝ : TopologicalSpace Y\n⊢ Continuous _root_.id"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.Compactum | {
"line": 257,
"column": 6
} | {
"line": 257,
"column": 16
} | {
"line": 258,
"column": 6
} | [
{
"pp": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F ∈ basic (cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x ↦ ↑x\nC0 : ssu := {Z | ∃ B ∈ F, X.str ⁻¹' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A ∈ G}\nC1 : ssu := insert... | [
"X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F ∈ basic (cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x ↦ ↑x\nC0 : ssu := {Z | ∃ B ∈ F, X.str ⁻¹' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A ∈ G}\nC1 : ssu := insert AA C0\nC2 :... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Category.Compactum | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 14
} | {
"line": 263,
"column": 4
} | [
{
"pp": "X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F ∈ basic (cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x ↦ ↑x\nC0 : ssu := {Z | ∃ B ∈ F, X.str ⁻¹' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A ∈ G}\nC1 : ssu := insert... | [
"X : Compactum\nA : Set X.A\nF : Ultrafilter X.A\nhF : F ∈ basic (cl A)\nfsu : Type u_1 := Finset (Set (Ultrafilter X.A))\nssu : Type u_1 := Set (Set (Ultrafilter X.A))\nι : fsu → ssu := fun x ↦ ↑x\nC0 : ssu := {Z | ∃ B ∈ F, X.str ⁻¹' B = Z}\nAA : Set (Ultrafilter X.A) := {G | A ∈ G}\nC1 : ssu := insert AA C0\nC2 :... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 175,
"column": 15
} | {
"line": 175,
"column": 45
} | {
"line": 175,
"column": 46
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\n⊢ ∀ (n : ℕ) (i : CWComplex.cell C n),\n ∃ I,\n MapsTo (↑(CWComplex.map n i)) (sphere 0 1)\n (∅ ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(CWComplex.map m j) '' closedBall 0 1)",
"ppTerm": "?m.41",
"assigned": true... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\n⊢ ∀ (n : ℕ) (i : CWComplex.cell C n),\n ∃ I,\n MapsTo (↑(CWComplex.map n i)) (sphere 0 1) (⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(CWComplex.map m j) '' closedBall 0 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 176,
"column": 16
} | {
"line": 176,
"column": 72
} | {
"line": 176,
"column": 73
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\n⊢ ∀ A ⊆ C,\n (∀ (n : ℕ) (j : CWComplex.cell C n), IsClosed[inst✝¹] (A ∩ ↑(CWComplex.map n j) '' closedBall 0 1)) ∧\n IsClosed[inst✝¹] (A ∩ ∅) →\n IsClosed[inst✝¹] A",
"ppTerm": "?m.52",
"assigned": true,
... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\n⊢ ∀ A ⊆ C,\n (∀ (n : ℕ) (j : CWComplex.cell C n), IsClosed[inst✝¹] (A ∩ ↑(CWComplex.map n j) '' closedBall 0 1)) →\n IsClosed[inst✝¹] A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 178,
"column": 15
} | {
"line": 178,
"column": 45
} | {
"line": 178,
"column": 46
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\n⊢ ∅ ∪ ⋃ n, ⋃ j, ↑(CWComplex.map n j) '' closedBall 0 1 = C",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"pseudoMetricSpacePi",
"outParam",
"Real.instZero",... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\n⊢ ⋃ n, ⋃ j, ↑(CWComplex.map n j) '' closedBall 0 1 = C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 190,
"column": 16
} | {
"line": 190,
"column": 27
} | {
"line": 190,
"column": 28
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : RelCWComplex C ∅\n⊢ ∀ (n : ℕ) (i : cell C n),\n ∃ I, MapsTo (↑(map n i)) (sphere 0 1) (⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(map m j) '' closedBall 0 1)",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [],
"usedFVars": ... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : RelCWComplex C ∅\n⊢ ∀ (n : ℕ) (i : cell C n),\n ∃ I, MapsTo (↑(map n i)) (sphere 0 1) (⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, ↑(map m j) '' closedBall 0 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 191,
"column": 16
} | {
"line": 191,
"column": 27
} | {
"line": 191,
"column": 28
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : RelCWComplex C ∅\n⊢ ∀ A ⊆ C, (∀ (n : ℕ) (j : cell C n), IsClosed[inst✝¹] (A ∩ ↑(map n j) '' closedBall 0 1)) → IsClosed[inst✝¹] A",
"ppTerm": "?m.56",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : RelCWComplex C ∅\n⊢ ∀ A ⊆ C, (∀ (n : ℕ) (j : cell C n), IsClosed[inst✝¹] (A ∩ ↑(map n j) '' closedBall 0 1)) → IsClosed[inst✝¹] A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 192,
"column": 15
} | {
"line": 192,
"column": 26
} | {
"line": 192,
"column": 27
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : RelCWComplex C ∅\n⊢ ⋃ n, ⋃ j, ↑(map n j) '' closedBall 0 1 = C",
"ppTerm": "?m.70",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : RelCWComplex C ∅\n⊢ ⋃ n, ⋃ j, ↑(map n j) '' closedBall 0 1 = C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.Profinite.Nobeling.Span | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 64
} | {
"line": 159,
"column": 2
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝ : LinearOrder I\ns : Finset I\na : I\nas : List I\nha : List.IsChain (fun x1 x2 ↦ x1 > x2) (a :: as)\nc : Products I →₀ ℤ\nhc : ↑c.support ⊆ {m | ↑m ≤ as}\nm : Products I\nhm : c m ≠ 0\n⊢ e (π C fun x ↦ x ∈ s) a * c m • Products.eval (π C fun x ↦ x ∈ s) m ∈\n Su... | [
"I : Type u\nC : Set (I → Bool)\ninst✝ : LinearOrder I\ns : Finset I\na : I\nas : List I\nha : List.IsChain (fun x1 x2 ↦ x1 > x2) (a :: as)\nc : Products I →₀ ℤ\nhc : ↑c.support ⊆ {m | ↑m ≤ as}\nm : Products I\nhm : c m ≠ 0\nhsm :\n ∀ (c : ℤ) (x : LocallyConstant ↑(π C fun x ↦ x ∈ s) ℤ),\n (LinearMap.mulLeft ℤ ... | have hsm := (LinearMap.mulLeft ℤ (e (π C (· ∈ s)) a)).map_smul | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Category.Profinite.Nobeling.Span | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 62
} | {
"line": 165,
"column": 63
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝ : LinearOrder I\ns : Finset I\na : I\nas : List I\nha : List.IsChain (fun x1 x2 ↦ x1 > x2) (a :: as)\nc : Products I →₀ ℤ\nhc : ↑c.support ⊆ {m | ↑m ≤ as}\nm : Products I\nhm : c m ≠ 0\nhsm :\n ∀ (c : ℤ) (x : LocallyConstant ↑(π C fun x ↦ x ∈ s) ℤ),\n e (π C fu... | [
"I : Type u\nC : Set (I → Bool)\ninst✝ : LinearOrder I\ns : Finset I\na : I\nas : List I\nha : List.IsChain (fun x1 x2 ↦ x1 > x2) (a :: as)\nc : Products I →₀ ℤ\nhc : ↑c.support ⊆ {m | ↑m ≤ as}\nm : Products I\nhm : c m ≠ 0\nhsm :\n ∀ (c : ℤ) (x : LocallyConstant ↑(π C fun x ↦ x ∈ s) ℤ),\n e (π C fun x ↦ x ∈ s)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.Profinite.Nobeling.Basic | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 28
} | {
"line": 114,
"column": 29
} | [
{
"pp": "I : Type u\nJ : I → Prop\ninst✝ : (i : I) → Decidable (J i)\nx : I → Bool\nh : ∀ (i : I), x i ≠ false → J i\ni : I\n⊢ false ≠ x i → J i",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"I : Type u\nJ : I → Prop\ninst✝ : (i : I) → Decidable (J i)\nx : I → Bool\nh : ∀ (i : I), x i ≠ false → J i\ni : I\n⊢ false ≠ x i → J i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 517,
"column": 4
} | {
"line": 517,
"column": 23
} | {
"line": 517,
"column": 24
} | [
{
"pp": "X : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝ : RelCWComplex C D\nthis : D ∪ ⋃ n, ⋃ j, openCell n j = D ∪ ⋃ m, ⋃ (_ : ↑m < ⊤), ⋃ j, closedCell m j\n⊢ D ∪ ⋃ n, ⋃ j, openCell n j = C",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"X : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝ : RelCWComplex C D\nthis : D ∪ ⋃ n, ⋃ j, openCell n j = D ∪ ⋃ m, ⋃ (_ : ↑m < ⊤), ⋃ j, closedCell m j\n⊢ D ∪ ⋃ n, ⋃ j, openCell n j = C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 522,
"column": 2
} | {
"line": 522,
"column": 13
} | {
"line": 522,
"column": 14
} | [
{
"pp": "X : Type u_1\nt : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\n⊢ ⋃ n, ⋃ j, openCell n j = C",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\nt : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\n⊢ ⋃ n, ⋃ j, openCell n j = C"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.Compactum | {
"line": 379,
"column": 6
} | {
"line": 379,
"column": 32
} | {
"line": 379,
"column": 32
} | [
{
"pp": "X Y : Compactum\nf : X ⟶ Y\n⊢ Continuous ⇑(ConcreteCategory.hom f)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
... | [
"X Y : Compactum\nf : X ⟶ Y\n⊢ ∀ (x : X.A) (g : Ultrafilter X.A),\n ↑g ≤ 𝓝 x → Tendsto (⇑(ConcreteCategory.hom f)) (↑g) (𝓝 ((ConcreteCategory.hom f) x))"
] | continuous_iff_ultrafilter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.Compactum | {
"line": 410,
"column": 8
} | {
"line": 410,
"column": 18
} | {
"line": 411,
"column": 8
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := ⋯\nc1 : x = (Ultrafilter.map Ultrafilter.lim FF).lim\nc2 : ∀ (U : Set X) (F : Ultrafilter X), F.lim ∈ U → IsOpen[inst✝²] U → U ∈ F\nc3 : ↑(Ultrafilter.map Ultrafilter.lim FF... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nFF : Ultrafilter (Ultrafilter X)\nx : X := (Ultrafilter.map Ultrafilter.lim FF).lim\nc1 : x = (Ultrafilter.map Ultrafilter.lim FF).lim\nc2 : ∀ (U : Set X) (F : Ultrafilter X), F.lim ∈ U → IsOpen[inst✝²] U → U ∈ F\nc3 : ↑(Ultrafi... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Category.Compactum | {
"line": 421,
"column": 10
} | {
"line": 421,
"column": 36
} | {
"line": 421,
"column": 36
} | [
{
"pp": "X Y : Compactum\nf : X.A → Y.A\ncont : Continuous f\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Continuous",
"congrArg",
"PartialOrder.toPreorder",
"Categ... | [
"X Y : Compactum\nf : X.A → Y.A\ncont : ∀ (x : X.A) (g : Ultrafilter X.A), ↑g ≤ 𝓝 x → Tendsto f (↑g) (𝓝 (f x))\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)"
] | continuous_iff_ultrafilter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.Profinite.Nobeling.Span | {
"line": 215,
"column": 17
} | {
"line": 215,
"column": 75
} | {
"line": 215,
"column": 76
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ns : Finset I\ninst✝ : WellFoundedLT I\nx : ↑(π C fun x ↦ x ∈ s)\nl : List I := s.sort fun x1 x2 ↦ x1 ≥ x2\na : I\nas : List I\nha : List.IsChain (fun x1 x2 ↦ x1 > x2) (a :: as)\nc : Products I →₀ ℤ\nhc : ↑c.support ⊆ {m | ↑m ≤ as}\nhmap :\n ∀ (g ... | [
"I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ns : Finset I\ninst✝ : WellFoundedLT I\nx : ↑(π C fun x ↦ x ∈ s)\nl : List I := s.sort fun x1 x2 ↦ x1 ≥ x2\na : I\nas : List I\nha : List.IsChain (fun x1 x2 ↦ x1 > x2) (a :: as)\nc : Products I →₀ ℤ\nhc : ↑c.support ⊆ {m | ↑m ≤ as}\nhmap :\n ∀ (g : Products I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 26
} | {
"line": 160,
"column": 27
} | [
{
"pp": "case pos\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\ng : I → Bool\nhg : g ∈ C1 C ho\ni : I\nh : term I ho = i\n⊢ true = g i",
"ppTerm": "?pos✝",
"assigned": true,
... | [
"case pos\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\ng : I → Bool\nhg : g ∈ C1 C ho\ni : I\nh : term I ho = i\n⊢ true = g (term I ho)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 590,
"column": 2
} | {
"line": 590,
"column": 80
} | {
"line": 591,
"column": 2
} | [
{
"pp": "X : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝¹ : RelCWComplex C D\ninst✝ : T2Space X\nA : Set X\nhAC : A ⊆ C\nhDA : IsClosed[t] (A ∩ D)\nh : ∀ (n : ℕ), 0 < n → ∀ (j : cell C n), Disjoint A (openCell n j) ∨ IsClosed[t] (A ∩ closedCell n j)\n⊢ IsClosed[t] A",
"ppTerm": "?m.42",
"assign... | [
"X : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝¹ : RelCWComplex C D\ninst✝ : T2Space X\nA : Set X\nhAC : A ⊆ C\nhDA : IsClosed[t] (A ∩ D)\nh : ∀ (n : ℕ), 0 < n → ∀ (j : cell C n), Disjoint A (openCell n j) ∨ IsClosed[t] (A ∩ closedCell n j)\n⊢ ∀ (n : ℕ), 0 < n → ∀ (j : cell C n), IsClosed[t] (A ∩ openCell... | apply isClosed_of_isClosed_inter_openCell_or_isClosed_inter_closedCell hAC hDA | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Category.Profinite.Nobeling.Basic | {
"line": 430,
"column": 2
} | {
"line": 430,
"column": 24
} | {
"line": 430,
"column": 25
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\nl : Products I\nJ : I → Prop\ninst✝ : (j : I) → Decidable (J j)\nh✝ : isGood (π C J) l\ni : I\nhi : i ∈ ↑l\nh' : ¬J i\nw✝ : I → Bool\nleft✝ : w✝ ∈ C\nh : ∀ i ∈ ↑l, ↑⟨Proj J w✝, ⋯⟩ i = true\n⊢ False",
"ppTerm": "?m.85",
"assigned": false,
... | [
"I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\nl : Products I\nJ : I → Prop\ninst✝ : (j : I) → Decidable (J j)\nh✝ : isGood (π C J) l\ni : I\nhi : i ∈ ↑l\nh' : ¬J i\nw✝ : I → Bool\nleft✝ : w✝ ∈ C\nh : ∀ i ∈ ↑l, ↑⟨Proj J w✝, ⋯⟩ i = true\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.Profinite.Nobeling.ZeroLimit | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 39
} | {
"line": 166,
"column": 40
} | [
{
"pp": "case h\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nx✝ : ↑(smaller C o)\na : LocallyConstant ↑C ℤ\nb : LocallyConstant ↑(π C fun x ↦ ord I x < o) ℤ\nhb : b ∈ range (π C fun x ↦ ord I x < o) ∧ (πs C o) b = a\n⊢ (fun x ↦ ⟨(πs C o) ↑x, ⋯⟩) ⟨b, ⋯⟩ = ⟨a,... | [
"case h\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nx✝ : ↑(smaller C o)\na : LocallyConstant ↑C ℤ\nb : LocallyConstant ↑(π C fun x ↦ ord I x < o) ℤ\nhb : b ∈ range (π C fun x ↦ ord I x < o) ∧ (πs C o) b = a\n⊢ (πs C o) b = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 652,
"column": 2
} | {
"line": 652,
"column": 13
} | {
"line": 652,
"column": 14
} | [
{
"pp": "X : Type u_1\nt : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\nn : ℕ\ni : cell C n\n⊢ ∃ I, cellFrontier n i ⊆ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, openCell m j",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\nt : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\nn : ℕ\ni : cell C n\n⊢ ∃ I, cellFrontier n i ⊆ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, openCell m j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 682,
"column": 4
} | {
"line": 682,
"column": 15
} | {
"line": 682,
"column": 16
} | [
{
"pp": "case e_I\nX : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝ : RelCWComplex C D\nE : Set X\nI✝¹ : (n : ℕ) → Set (cell C n)\nclosed'✝¹ : IsClosed[t] E\nhE : D ∪ ⋃ n, ⋃ j, openCell n ↑j = E\nF : Set X\nI✝ : (n : ℕ) → Set (cell C n)\nclosed'✝ : IsClosed[t] F\nhF : D ∪ ⋃ n, ⋃ j, openCell n ↑j = F\nh ... | [
"case e_I\nX : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝ : RelCWComplex C D\nE : Set X\nI✝¹ : (n : ℕ) → Set (cell C n)\nclosed'✝¹ : IsClosed[t] E\nhE : D ∪ ⋃ n, ⋃ j, openCell n ↑j = E\nF : Set X\nI✝ : (n : ℕ) → Set (cell C n)\nclosed'✝ : IsClosed[t] F\nhF : D ∪ ⋃ n, ⋃ j, openCell n ↑j = F\nh :\n (fun E ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CompactSystem | {
"line": 143,
"column": 19
} | {
"line": 143,
"column": 35
} | {
"line": 143,
"column": 36
} | [
{
"pp": "α : Type u_1\nS : Set (Set α)\nhpi : IsPiSystem S\nh : ∀ (C : ℕ → Set α), Directed (fun x1 x2 ↦ x1 ⊇ x2) C → (∀ (i : ℕ), C i ∈ S) → ⋂ i, C i = ∅ → ∃ n, C n = ∅\nC : ℕ → Set α\nh1 : ∀ (i : ℕ), C i ∈ insert ∅ S\nh2 : ⋂ n, ⋂ m, ⋂ (_ : m ≤ n), C m = ∅\nthis : (∀ (n : ℕ), dissipate C n ∈ S ∨ dissipate C n =... | [
"α : Type u_1\nS : Set (Set α)\nhpi : IsPiSystem S\nh : ∀ (C : ℕ → Set α), Directed (fun x1 x2 ↦ x1 ⊇ x2) C → (∀ (i : ℕ), C i ∈ S) → ⋂ i, C i = ∅ → ∃ n, C n = ∅\nC : ℕ → Set α\nh1 : ∀ (i : ℕ), C i ∈ insert ∅ S\nh2 : ⋂ n, ⋂ m, ⋂ (_ : m ≤ n), C m = ∅\nthis : (∀ (n : ℕ), dissipate C n ∈ S ∨ dissipate C n = ∅) ∧ ⋂ n, d... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CompactSystem | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 25
} | {
"line": 146,
"column": 26
} | [
{
"pp": "case pos\nα : Type u_1\nS : Set (Set α)\nhpi : IsPiSystem S\nh : ∀ (C : ℕ → Set α), Directed (fun x1 x2 ↦ x1 ⊇ x2) C → (∀ (i : ℕ), C i ∈ S) → ⋂ i, C i = ∅ → ∃ n, C n = ∅\nC : ℕ → Set α\nh1 : ∀ (i : ℕ), C i ∈ insert ∅ S\nh2 : ⋂ n, ⋂ m, ⋂ (_ : m ≤ n), C m = ∅\nn : ℕ\ng : (dissipate C n).Nonempty\n⊢ dissi... | [
"case pos\nα : Type u_1\nS : Set (Set α)\nhpi : IsPiSystem S\nh : ∀ (C : ℕ → Set α), Directed (fun x1 x2 ↦ x1 ⊇ x2) C → (∀ (i : ℕ), C i ∈ S) → ⋂ i, C i = ∅ → ∃ n, C n = ∅\nC : ℕ → Set α\nh1 : ∀ (i : ℕ), C i ∈ insert ∅ S\nh2 : ⋂ n, ⋂ m, ⋂ (_ : m ≤ n), C m = ∅\nn : ℕ\ng : (dissipate C n).Nonempty\n⊢ dissipate C n = ∅... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CompactSystem | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 15
} | {
"line": 170,
"column": 16
} | [
{
"pp": "case refine_2\nα : Type u_2\ninst✝ : TopologicalSpace α\nC : ℕ → Set α\nhC_cc : ∀ (i : ℕ), C i ∈ {s | IsCompact s ∧ IsClosed s}\nh_nonempty : ∀ (n : ℕ), (dissipate C n).Nonempty\n⊢ IsCompact (dissipate C 0)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Set.dissipate",... | [
"case refine_2\nα : Type u_2\ninst✝ : TopologicalSpace α\nC : ℕ → Set α\nhC_cc : ∀ (i : ℕ), C i ∈ {s | IsCompact s ∧ IsClosed s}\nh_nonempty : ∀ (n : ℕ), (dissipate C n).Nonempty\n⊢ IsCompact (C 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CompactSystem | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 13
} | {
"line": 176,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\ns : Set α\n⊢ IsCompact s ↔ IsCompact s ∧ IsClosed s",
"ppTerm": "?m.123",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"id",
"IsClosed",
"And",
"Iff",
"iff_self_and._simp_1",
"Eq",
... | [
"α : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\ns : Set α\n⊢ IsCompact s → IsClosed s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 528,
"column": 4
} | {
"line": 528,
"column": 84
} | {
"line": 530,
"column": 0
} | [
{
"pp": "case neg\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nq : ↑(GoodProducts (π C fun x ↦ ord I x < o))\nl : ↑(MaxProducts C ho)\nthis : Inhabited I\nh : ¬↑↑q = []\n⊢ (Ordinal.... | [] | exact Products.prop_of_isGood C _ q.prop q.val.val.head! (List.head!_mem_self h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli | {
"line": 62,
"column": 61
} | {
"line": 62,
"column": 72
} | {
"line": 62,
"column": 73
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : CompactSpace β\nA : Set (α →ᵇ β)\nclosed : IsClosed A\nH : ∀ (x₀ : α), ∀ ε > 0, ∃ U ∈ nhds x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ↑A), dist (↑i x) (↑i x') < ε\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 <... | [
"α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : CompactSpace β\nA : Set (α →ᵇ β)\nclosed : IsClosed A\nH : ∀ (x₀ : α), ∀ ε > 0, ∃ U ∈ nhds x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ↑A), dist (↑i x) (↑i x') < ε\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nεε₁ : ε... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.BoundedCompactlySupported | {
"line": 78,
"column": 2
} | {
"line": 79,
"column": 9
} | {
"line": 79,
"column": 10
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : NonUnitalNormedRing γ\ninst✝ : Nontrivial γ\nh : C_cb(α, γ) = ⊤\nx : γ\nhx : x ≠ 0\n⊢ IsCompact univ",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nγ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : NonUnitalNormedRing γ\ninst✝ : Nontrivial γ\nh : C_cb(α, γ) = ⊤\nx : γ\nhx : x ≠ 0\n⊢ IsCompact univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CountablyCompact | {
"line": 98,
"column": 36
} | {
"line": 98,
"column": 47
} | {
"line": 98,
"column": 48
} | [
{
"pp": "E : Type u_2\ninst✝ : TopologicalSpace E\nA : Set E\nhA : ∀ (x : ℕ → E), (∀ᶠ (n : ℕ) in atTop, x n ∈ A) → ∃ a ∈ A, MapClusterPt a atTop x\nf : Filter E\nx✝¹ : f.NeBot\nx✝ : f.IsCountablyGenerated\nhle : f ≤ 𝓟 A\nx : ℕ → E\nhx : Tendsto x atTop f\n⊢ ∀ᶠ (n : ℕ) in atTop, x n ∈ A",
"ppTerm": "?m.57",... | [
"E : Type u_2\ninst✝ : TopologicalSpace E\nA : Set E\nhA : ∀ (x : ℕ → E), (∀ᶠ (n : ℕ) in atTop, x n ∈ A) → ∃ a ∈ A, MapClusterPt a atTop x\nf : Filter E\nx✝¹ : f.NeBot\nx✝ : f.IsCountablyGenerated\nhle : f ≤ 𝓟 A\nx : ℕ → E\nhx : Tendsto x atTop f\n⊢ ∃ a, ∀ (b : ℕ), a ≤ b → x b ∈ A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CountablyCompact | {
"line": 156,
"column": 77
} | {
"line": 156,
"column": 88
} | {
"line": 156,
"column": 89
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\ninst✝ : TopologicalSpace E\nA : Set E\nhA : IsCountablyCompact A\nb : Set ι\nhb : b.Countable\nU : ι → Set E\nhUo : ∀ i ∈ b, IsOpen[inst✝] (U i)\nhAU : A ⊆ ⋃ i ∈ b, U i\nthis : Countable ↑b\n⊢ A ⊆ ⋃ i, U ↑i",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": ... | [
"ι : Type u_1\nE : Type u_2\ninst✝ : TopologicalSpace E\nA : Set E\nhA : IsCountablyCompact A\nb : Set ι\nhb : b.Countable\nU : ι → Set E\nhUo : ∀ i ∈ b, IsOpen[inst✝] (U i)\nhAU : A ⊆ ⋃ i ∈ b, U i\nthis : Countable ↑b\n⊢ A ⊆ ⋃ i ∈ b, U i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CountablyCompact | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 15
} | {
"line": 250,
"column": 16
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nA✝ B : Set E\ninst✝¹ : SequentialSpace E\ninst✝ : CountablyCompactSpace E\nx : ℕ → E\nhx : ∀ (x_1 : E) (x_2 : ℕ → ℕ), StrictMono x_2 → ¬Tendsto (x ∘ x_2) atTop (𝓝 x_1)\nA : Set E := ⋃ i, closure[inst✝³]... | [
"ι : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nA✝ B : Set E\ninst✝¹ : SequentialSpace E\ninst✝ : CountablyCompactSpace E\nx : ℕ → E\nhx : ∀ (x_1 : E) (x_2 : ℕ → ℕ), StrictMono x_2 → ¬Tendsto (x ∘ x_2) atTop (𝓝 x_1)\nA : Set E := ⋃ i, closure[inst✝³] {x i}\nthis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 598,
"column": 4
} | {
"line": 598,
"column": 38
} | {
"line": 599,
"column": 4
} | [
{
"pp": "case h₂\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nl : ↑(MaxProducts C ho)\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x ↦ ord I x < o)))\nthis ... | [
"case h₂\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nl : ↑(MaxProducts C ho)\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x ↦ ord I x < o)))\nthis : Inhabited ... | rw [max_eq_o_cons_tail C hsC ho l] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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