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.SetTheory.Cardinal.EventuallyConst | {
"line": 67,
"column": 83
} | {
"line": 67,
"column": 85
} | {
"line": 68,
"column": 2
} | [
{
"pp": "β : Type v\ninst✝¹ : PartialOrder β\nf : Ordinal.{v} → β\ninst✝ : Small.{v, v} β\nhf : Monotone f\n⊢ EventuallyConst f atTop",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal.lift_univ",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"in... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 128,
"column": 48
} | {
"line": 128,
"column": 50
} | {
"line": 129,
"column": 2
} | [
{
"pp": "α : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\ns : Set (Set α)\nhα : cof α ≠ ℵ₀\nhsα : s.Countable\nhs : ∀ x ∈ s, IsClub x\n⊢ IsClub (⋂₀ s)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"IsClub.sInter",
"Cardinal.i... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.NatCount | {
"line": 25,
"column": 80
} | {
"line": 25,
"column": 82
} | {
"line": 26,
"column": 2
} | [
{
"pp": "p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\n⊢ ↑(count p n) ≤ Cardinal.mk ↑{k | p k}",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal.mk_subtype_mono",
"Cardinal",
"Nat.CountSet.fintype",
"congrArg",
"Set.ofPred",
"Nat.c... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.NatCount | {
"line": 29,
"column": 67
} | {
"line": 29,
"column": 69
} | {
"line": 30,
"column": 2
} | [
{
"pp": "p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\n⊢ ↑(count p n) ≤ {k | p k}.encard",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"ENat.instNatCast",
"Cardinal",
"Set.ofPred",
"Nat.count_le_cardinal",
"Cardinal.mk",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.NatCount | {
"line": 34,
"column": 68
} | {
"line": 34,
"column": 70
} | {
"line": 34,
"column": 71
} | [
{
"pp": "p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh : {k | p k}.Finite\n⊢ {k | p k}.encard ≠ ⊤",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard_eq_top_iff._simp_1",
"Set.encard",
"instTopENat",
"congrArg",
"Set.ofPred",
"S... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.NatCount | {
"line": 33,
"column": 90
} | {
"line": 33,
"column": 92
} | {
"line": 34,
"column": 2
} | [
{
"pp": "p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh : {k | p k}.Finite\n⊢ count p n ≤ {k | p k}.ncard",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard_eq_top_iff._simp_1",
"Set.encard",
"Set.ncard_def",
"Nat.count_le_setENCard",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.UnivLE | {
"line": 23,
"column": 86
} | {
"line": 23,
"column": 88
} | {
"line": 24,
"column": 2
} | [
{
"pp": "⊢ UnivLE.{u, v} ↔ univ.{u, v + 1} ≤ univ.{v, u + 1}",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Eq.mpr",
"Cardinal.lift_univ",
"Preorder.toLT",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
"UnivLE",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.UnivLE | {
"line": 34,
"column": 94
} | {
"line": 34,
"column": 96
} | {
"line": 35,
"column": 2
} | [
{
"pp": "⊢ UnivLE.{u, v} ↔ Nonempty (Ordinal.{u} ↪ Ordinal.{v})",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"UnivLE",
"Cardinal",
"congrArg",
"Cardinal.univ",
"univLE_iff_cardinal_le",
"id",
"LE.le",
"Cardinal.instLE",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.UnivLE | {
"line": 43,
"column": 56
} | {
"line": 43,
"column": 58
} | {
"line": 44,
"column": 2
} | [
{
"pp": "⊢ UnivLE.{u, v} ∨ UnivLE.{v, u}",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"_private.Mathlib.SetTheory.Cardinal.UnivLE.0.univLE_total._simp_1_1",
"Eq.mpr",
"UnivLE",
"Cardinal",
"congrArg",
"Cardinal.univ",
"id",
"le_total",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 31,
"column": 5
} | {
"line": 31,
"column": 7
} | {
"line": 31,
"column": 8
} | [
{
"pp": "A : Type u_1\n⊢ ∀ ⦃s : Set (Set (List A))⦄,\n s ⊆ {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T} → sSup s ∈ {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T}",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.ofPred",
"Membership.mem",
"CompleteLattice.... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 32,
"column": 5
} | {
"line": 32,
"column": 7
} | {
"line": 32,
"column": 8
} | [
{
"pp": "A : Type u_1\n⊢ ∀ ⦃s : Set (Set (List A))⦄,\n s ⊆ {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T} → sInf s ∈ {T | ∀ ⦃x : List A⦄ ⦃a : A⦄, x ++ [a] ∈ T → x ∈ T}",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Set.ofPred",
"Membership.mem",
"CompleteLattice.... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 44,
"column": 38
} | {
"line": 44,
"column": 40
} | {
"line": 44,
"column": 41
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\ny : A\nys : List A\nih : ∀ {x : List A}, x ++ ys ∈ T → x ∈ T\nx : List A\nh : x ++ y :: ys ∈ T\n⊢ x ++ [?m.43] ++ ys ∈ T",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.append_assoc",
"congrArg",
"Membership.mem"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 41,
"column": 63
} | {
"line": 41,
"column": 65
} | {
"line": 42,
"column": 2
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx y : List A\nh : x ++ y ∈ T\n⊢ x ∈ T",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.append_assoc",
"congrArg",
"Membership.mem",
"Eq.mp",
"id",
"Subtype",
"List.rec",
"List.append_... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 46,
"column": 73
} | {
"line": 46,
"column": 75
} | {
"line": 47,
"column": 2
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx y : List A\nh' : x <+: y\nh : y ∈ T\n⊢ x ∈ T",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Subtype",
"Descriptive.Tree.mem_of_append",
"Descriptive.tree",
"instHAppendOfAppend",
"List",
"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 56,
"column": 8
} | {
"line": 56,
"column": 10
} | {
"line": 56,
"column": 11
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\n⊢ T = ⊥ → [] ∉ T",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"Membership.mem",
"Subtype",
"Bot.bot",
"Descriptive.tree",
"List",
"CompleteSublattice.instSetLike",
"True",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 57,
"column": 11
} | {
"line": 57,
"column": 13
} | {
"line": 57,
"column": 14
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nh : [] ∉ T\n⊢ T = ⊥",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Descriptive.Tree.mem_of_prefix",
"False",
"iff_false",
"congrArg",
"Membership.mem",
"Eq.mp",
"id",
"Subtype",
"B... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 57
} | {
"line": 139,
"column": 2
} | [
{
"pp": "α : Type v\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\nι : Sort u_1\nf : ι → Set α\ninst✝ : Countable ι\nhα : cof α ≠ ℵ₀\nhf : ∀ (i : ι), IsClub (f i)\n⊢ IsClub (⋂₀ range f)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.countable_range",
"IsClub.sInter_of... | [
"α : Type v\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\nι : Sort u_1\nf : ι → Set α\ninst✝ : Countable ι\nhα : cof α ≠ ℵ₀\nhf : ∀ (i : ι), IsClub (f i)\n⊢ ∀ x ∈ range f, IsClub x"
] | apply IsClub.sInter_of_countable hα (countable_range f) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 137,
"column": 2
} | {
"line": 139,
"column": 7
} | {
"line": 141,
"column": 0
} | [
{
"pp": "α : Type v\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\nι : Sort u_1\nf : ι → Set α\ninst✝ : Countable ι\nhα : cof α ≠ ℵ₀\nhf : ∀ (i : ι), IsClub (f i)\n⊢ IsClub (⋂ i, f i)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.iInter",
... | [] | rw [← sInter_range]
apply IsClub.sInter_of_countable hα (countable_range f)
simpa | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 137,
"column": 2
} | {
"line": 139,
"column": 7
} | {
"line": 141,
"column": 0
} | [
{
"pp": "α : Type v\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\nι : Sort u_1\nf : ι → Set α\ninst✝ : Countable ι\nhα : cof α ≠ ℵ₀\nhf : ∀ (i : ι), IsClub (f i)\n⊢ IsClub (⋂ i, f i)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.iInter",
... | [] | rw [← sInter_range]
apply IsClub.sInter_of_countable hα (countable_range f)
simpa | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 65,
"column": 82
} | {
"line": 65,
"column": 84
} | {
"line": 66,
"column": 2
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nm n : ℕ\nx : ↥T\n⊢ take m (take n x) = take (min m n) x",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"Descriptive.Tree.take_coe",
"Subtype",
"Descriptive.tree",
"List",
"Complet... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 136,
"column": 52
} | {
"line": 136,
"column": 54
} | {
"line": 137,
"column": 2
} | [
{
"pp": "α : Type v\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\nι : Sort u_1\nf : ι → Set α\ninst✝ : Countable ι\nhα : cof α ≠ ℵ₀\nhf : ∀ (i : ι), IsClub (f i)\n⊢ IsClub (⋂ i, f i)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.iInter",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 141,
"column": 94
} | {
"line": 141,
"column": 96
} | {
"line": 142,
"column": 2
} | [
{
"pp": "α : Type v\ns t : Set α\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nhα : cof α ≠ ℵ₀\nhs : IsClub s\nht : IsClub t\n⊢ IsClub (s ∩ t)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"congrArg",
"and_self",
"forall_eq_or_imp._simp_1",
"Set.sInter_insert"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 69,
"column": 65
} | {
"line": 69,
"column": 67
} | {
"line": 69,
"column": 68
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx : ↥T\nm n : ℕ\n⊢ take m x = take n x ↔ min m (↑x).length = min n (↑x).length",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"_private.Mathlib.SetTheory.Descriptive.Tree.0.Descriptive.Tree.take_eq_take._simp_1_1",
"congrArg",
"Membe... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 77,
"column": 56
} | {
"line": 77,
"column": 58
} | {
"line": 77,
"column": 59
} | [
{
"pp": "A : Type u_1\nS T : ↥(tree A)\nx y x✝¹ : List A\na : A\nx✝ : x✝¹ ++ [a] ∈ (fun x_1 ↦ x ++ x_1) ⁻¹' ↑T\n⊢ (fun x_1 ↦ x ++ x_1) x✝¹ ++ [a] ∈ T",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.append_assoc",
"congrArg",
"Membership.mem",
"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 83,
"column": 71
} | {
"line": 83,
"column": 73
} | {
"line": 83,
"column": 74
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx y : List A\n⊢ subAt (subAt T x) y = subAt T (x ++ y)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"List.append_assoc",
"congrArg",
"Membership.mem",
"Subtype",
"Descriptive.tree",
"iff_self",
"instHAppendO... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 90,
"column": 17
} | {
"line": 90,
"column": 19
} | {
"line": 90,
"column": 20
} | [
{
"pp": "A : Type u_1\nS T : ↥(tree A)\nx✝ y : List A\nn : ℕ\nx : ↥T\n⊢ List.drop n ↑x ∈ subAt T ↑(take n x)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"congrArg",
"List.take_append_drop",
"Membership.mem",
"SetLike.coe_mem._simp_1",
"Descriptive.Tree.tak... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 105,
"column": 91
} | {
"line": 105,
"column": 93
} | {
"line": 106,
"column": 2
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx y : List A\nhl : y.length ≤ x.length\n⊢ y ∈ pullSub T x ↔ y <+: x ∧ [] ∈ T",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"SetLike.mem_mk_set._simp_1",
"congrArg",
"Set.ofPred",
"Descriptive.Tree.pullSub._pro... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 110,
"column": 8
} | {
"line": 110,
"column": 10
} | {
"line": 111,
"column": 4
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx y : List A\nhl : x.length ≤ y.length\n⊢ y ∈ pullSub T x → ∃ z ∈ T, y = x ++ z",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.SetTheory.Descriptive.Tree.0.Descriptive.Tree.mem_pullSub_long.match_1_1",
"in... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 114,
"column": 9
} | {
"line": 114,
"column": 11
} | {
"line": 114,
"column": 12
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx y : List A\nhl : x.length ≤ y.length\n⊢ (∃ z ∈ T, y = x ++ z) → y ∈ pullSub T x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"SetLike.mem_mk_set._simp_1",
"congrArg",
"and_self",
"Set.ofPred",
"Descriptive.Tree.pullSu... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 116,
"column": 67
} | {
"line": 116,
"column": 69
} | {
"line": 116,
"column": 70
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx y : List A\n⊢ x ++ y ∈ pullSub T x ↔ y ∈ T",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Nat.le_add_right._simp_1",
"congrArg",
"_private.Mathlib.SetTheory.Descriptive.Tree.0.Descriptive.Tree.mem_pullSub_append._simp_1_1",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 118,
"column": 61
} | {
"line": 118,
"column": 63
} | {
"line": 119,
"column": 2
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx : List A\n⊢ x ∈ pullSub T x ↔ [] ∈ T",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"Eq.mp",
"Subtype",
"List.append_nil",
"Descriptive.tree",
"instHAppendOfAppend",
"List... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 126,
"column": 61
} | {
"line": 126,
"column": 63
} | {
"line": 126,
"column": 64
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx y : List A\nh : y <+: x ∧ [] ∈ subAt T x\nh' : y.length ≤ x.length\n⊢ x ∈ T",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"Eq.mp",
"Subtype",
"List.append_nil",
"Descriptive.tree",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
"line": 148,
"column": 28
} | {
"line": 148,
"column": 30
} | {
"line": 149,
"column": 2
} | [
{
"pp": "α : Type v\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nf : α → α\nhα : cof α ≠ ℵ₀\nhf : IsNormal f\n⊢ IsClub (Function.fixedPoints f)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PSum.casesOn",
"Preorder.toLT",
"IsClub.mk",
"topOrd... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 124,
"column": 51
} | {
"line": 124,
"column": 53
} | {
"line": 125,
"column": 2
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx : List A\n⊢ pullSub (subAt T x) x ≤ T",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Descriptive.Tree.mem_of_prefix",
"Descriptive.Tree.mem_pullSub_long",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"M... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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Mathlib.SetTheory.Descriptive.Tree | {
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Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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Mathlib.SetTheory.Descriptive.Tree | {
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Mathlib.SetTheory.Descriptive.Tree | {
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Mathlib.SetTheory.Descriptive.Tree | {
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Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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Mathlib.SetTheory.Descriptive.Tree | {
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Mathlib.SetTheory.Descriptive.Tree | {
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Mathlib.SetTheory.Descriptive.Tree | {
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Mathlib.SetTheory.Descriptive.Tree | {
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Mathlib.SetTheory.Descriptive.Tree | {
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Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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Mathlib.SetTheory.Descriptive.Tree | {
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Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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Mathlib.SetTheory.Cardinal.Cofinality.Club | {
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{
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Mathlib.RingTheory.Smooth.Quotient | {
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{
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Mathlib.SetTheory.Ordinal.Commute | {
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{
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Mathlib.SetTheory.Ordinal.Commute | {
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{
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Mathlib.SetTheory.Ordinal.Commute | {
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{
"pp": "o₁ o₂ : Ordinal.{u_1}\nhcomm : AddCommute o₁ o₂\nih : ∀ y < o₁ + o₂, ∀ {o₁ o₂ : Ordinal.{u_1}}, AddCommute o₁ o₂ → o₁ + o₂ = y → ∃ o n₁ n₂, o * ↑n₁ = o₁ ∧ o * ↑n₂ = o₂\nhle : o₁ ≤ o₂\nh₁ : o₁ ≠ 0\no₃ : Ordinal.{u_1} := o₂ - o₁\nhsub : o₁ + o₃ = o₂\n⊢ o₁ + (o₁ + o₃) = o₁ + (o₃ + o₁)",
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Mathlib.SetTheory.Ordinal.Commute | {
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{
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"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
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{
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... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
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{
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Mathlib.SetTheory.Ordinal.Commute | {
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{
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Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
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{
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Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
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{
"pp": "α : Type u\ng : Ordinal.{u} → α\n⊢ ¬InjOn g (Iio (succ #α).ord)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Order.succ",
"SuccOrder.succ",
"Ordinal.partialOrder",
"Cardinal",
"Cardinal.lift_lift",
"congrArg",
"PartialOrder.toPreorder",... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 79,
"column": 59
} | {
"line": 79,
"column": 61
} | {
"line": 80,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\n⊢ Monotone (lfpApprox f x)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"sup_le_sup_left",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"ChainCompletePartialOrder.instOfCompleteLatt... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 87,
"column": 48
} | {
"line": 87,
"column": 50
} | {
"line": 88,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\n⊢ lfpApprox f x 0 = x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"not_lt_zero._simp_1",
"Eq.mpr",
"False",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"ChainCompletePartialOrder.inst... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 91,
"column": 60
} | {
"line": 91,
"column": 62
} | {
"line": 92,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\na : Ordinal.{u}\n⊢ x ≤ lfpApprox f x a",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"ChainCompletePartialOrder.instOfCompleteLattice",
"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 96,
"column": 45
} | {
"line": 96,
"column": 47
} | {
"line": 97,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\na b : Ordinal.{u}\nh : a < b\n⊢ f (lfpApprox f x a) ≤ lfpApprox f x b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"le_rfl",
"ChainComp... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 85,
"column": 49
} | {
"line": 85,
"column": 51
} | {
"line": 86,
"column": 2
} | [
{
"pp": "b e x y : Ordinal.{u_1}\nhb : 1 < b\nhx : x ≠ 0\nhxb : x < b\nhy : y < b ^ e\n⊢ CNF b (b ^ e * x + y) = (e, x) :: CNF b y",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Ordinal.noZeroDivisors",
"Ordinal.instLinearOrder",
"instHDiv... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 93,
"column": 79
} | {
"line": 93,
"column": 81
} | {
"line": 94,
"column": 2
} | [
{
"pp": "o : Ordinal.{u_1}\nho : o ≠ 0\n⊢ CNF 0 o = [(0, o)]",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Ordinal.log_of_left_le_one",
"instHDiv",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"zero_le._simp_1",
"AddMonoid.toAddZero... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 96,
"column": 78
} | {
"line": 96,
"column": 80
} | {
"line": 97,
"column": 2
} | [
{
"pp": "o : Ordinal.{u_1}\nho : o ≠ 0\n⊢ CNF 1 o = [(0, o)]",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Ordinal.log_of_left_le_one",
"instHDiv",
"Ordinal.partialOrder",
"instReflLe",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 99,
"column": 94
} | {
"line": 99,
"column": 96
} | {
"line": 100,
"column": 2
} | [
{
"pp": "b o : Ordinal.{u_1}\nhb : b ≤ 1\nho : o ≠ 0\n⊢ CNF b o = [(0, o)]",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Ordinal.partialOrder",
"instIsBotZeroClass",
"Ordinal.CNF.zero_left",
"AddMonoid.toAddZeroClass",
"Part... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 103,
"column": 90
} | {
"line": 103,
"column": 92
} | {
"line": 104,
"column": 2
} | [
{
"pp": "b o : Ordinal.{u_1}\nho : o ≠ 0\nhb : o < b\n⊢ CNF b o = [(0, o)]",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"Ordinal.log_eq_zero",
"congrArg",
"Ordinal.mod_one",
"Ordinal.mod",
"id",
"HDiv.hDiv",
"in... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 101,
"column": 51
} | {
"line": 101,
"column": 53
} | {
"line": 102,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\nhx : x ≤ f x\na : Ordinal.{u}\n⊢ lfpApprox f x (a + 1) = f (lfpApprox f x a)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Lattice.toSemilatti... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 107,
"column": 97
} | {
"line": 107,
"column": 99
} | {
"line": 108,
"column": 2
} | [
{
"pp": "b o : Ordinal.{u_1}\n⊢ foldr (fun p r ↦ b ^ p.1 * p.2 + r) 0 (CNF b o) = o",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"List.foldr_cons",
"MulZeroClass.toMul",
"congrArg",
"Ordinal.div_add_mod",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 112,
"column": 41
} | {
"line": 112,
"column": 43
} | {
"line": 112,
"column": 44
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\na : Ordinal.{u}\nha : IsSuccLimit a\n⊢ x ≤ lfpApprox f x ↑⟨0, ⋯⟩",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.instOfCompleteLattice",
"Ordinal.partialOrder",
"instReflLe",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 97,
"column": 66
} | {
"line": 97,
"column": 68
} | {
"line": 97,
"column": 69
} | [
{
"pp": "α : Type u_1\nl : List (Lists α)\n⊢ (ofList l).toList = l",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Lists'.toList",
"congrArg",
"Lists'.toList.eq_2",
"Lists",
"List.rec",
"List.cons",
"Bool.true",
"List",
"True",
"e... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 113,
"column": 68
} | {
"line": 113,
"column": 70
} | {
"line": 114,
"column": 4
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\na : Ordinal.{u}\nha : IsSuccLimit a\nb : Ordinal.{u}\nhab : b < a\n⊢ f (lfpApprox f x b) ≤ lfpApprox f x ↑⟨b + 1, ⋯⟩",
"ppTerm": "?m.106",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.instOfCompleteLattice",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 105,
"column": 15
} | {
"line": 105,
"column": 17
} | {
"line": 106,
"column": 4
} | [
{
"pp": "α : Type u_1\nb : Bool\nh : true = b\nl : Lists' α b\n⊢ let l' := ⋯ ▸ l;\n ofList l'.toList = l'",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lists'.toList",
"congrArg",
"Lists'.cons'.injEq",
"HEq.refl",
"False.elim",
"noCon... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 114,
"column": 16
} | {
"line": 114,
"column": 18
} | {
"line": 114,
"column": 19
} | [
{
"pp": "α : Type u_1\nmotive : Lists' α true → Sort u_2\nofList : (l : List (Lists α)) → motive (Lists'.ofList l)\nl : Lists' α true\n⊢ motive (Lists'.ofList l.toList) = motive l",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Lists'.toList",
"congrArg",
"Bool.true",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 149,
"column": 62
} | {
"line": 149,
"column": 64
} | {
"line": 150,
"column": 2
} | [
{
"pp": "α : Type u_1\na y : Lists α\nl : Lists' α true\n⊢ a ∈ cons y l ↔ a ~ y ∨ a ∈ l",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"_private.Mathlib.SetTheory.Lists.0.Lists'.mem_cons._simp_1_3",
"Lists'.toList",
"congrArg",
"_private.Mathlib.SetTheory.Lists.0.L... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 108,
"column": 58
} | {
"line": 108,
"column": 60
} | {
"line": 109,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\na : Ordinal.{u}\nha : IsSuccLimit a\n⊢ lfpApprox f x a = ⨆ b, lfpApprox f x ↑b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"ChainCompletePar... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 152,
"column": 94
} | {
"line": 152,
"column": 96
} | {
"line": 153,
"column": 2
} | [
{
"pp": "α : Type u_1\na : Lists α\nl₁ l₂ : Lists' α true\n⊢ cons a l₁ ⊆ l₂ ↔ a ∈ l₂ ∧ l₁ ⊆ l₂",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Lists'.Subset.cons",
"Lists'.toList",
"HEq.refl",
"False.elim",
"Lists",
"noConfusion_of_Nat",
"Membersh... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 161,
"column": 43
} | {
"line": 161,
"column": 45
} | {
"line": 162,
"column": 2
} | [
{
"pp": "α : Type u_1\nl₁ l₂ : List (Lists α)\nh : l₁ ⊆ l₂\n⊢ ofList l₁ ⊆ ofList l₂",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Lists'.Subset.cons",
"_private.Mathlib.SetTheory.Lists.0.Lists'.ofList_subset._simp_1_1",
"Lists'.toList",
"congrArg",
"List.ins... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 114,
"column": 96
} | {
"line": 114,
"column": 98
} | {
"line": 115,
"column": 2
} | [
{
"pp": "b o : Ordinal.{u}\nx : Ordinal.{u} × Ordinal.{u}\n⊢ x ∈ CNF b o → x.1 ≤ log b o",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"False",
"instHDiv",
"Ordinal.partialOrder",
"Ordinal.log_zero_right",
"congrArg",
"i... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 169,
"column": 51
} | {
"line": 169,
"column": 53
} | {
"line": 170,
"column": 2
} | [
{
"pp": "α : Type u_1\nl : Lists' α true\n⊢ l ⊆ l",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lists'.toList",
"congrArg",
"Lists",
"id",
"HasSubset.Subset",
"List.Subset.refl",
"Bool.true",
"Lists'.instHasSubsetTrue",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 124,
"column": 20
} | {
"line": 124,
"column": 22
} | {
"line": 124,
"column": 23
} | [
{
"pp": "b o : Ordinal.{u}\nx : Ordinal.{u} × Ordinal.{u}\n⊢ x ∈ CNF b 0 → 0 < x.2",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"PartialOrder.toPreorder",
"Membership.mem",
"List.not_... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 116,
"column": 69
} | {
"line": 116,
"column": 71
} | {
"line": 117,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\n⊢ Monotone lfpApprox",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"sup_le_sup_left",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"ChainCompletePartialOrder.instOfCompleteLattice",
"iSup₂_mono"... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 172,
"column": 76
} | {
"line": 172,
"column": 78
} | {
"line": 173,
"column": 2
} | [
{
"pp": "α : Type u_1\nl : Lists' α true\n⊢ l ⊆ nil → l = nil",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lists'.toList",
"congrArg",
"HEq.refl",
"List.Mem.tail",
"False.elim",
"Lists",
"noConfusion_of_Nat",
"Membership.me... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 180,
"column": 36
} | {
"line": 180,
"column": 38
} | {
"line": 180,
"column": 39
} | [
{
"pp": "α : Type u_1\na : Lists α\nx✝ : Lists' α true\nh : a ∈ nil.toList\n⊢ a ∈ x✝",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Lists'.toList",
"HEq.refl",
"List.Mem.tail",
"False.elim",
"Lists",
"noConfusion_of_Nat",
"Membership.mem",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 181,
"column": 29
} | {
"line": 181,
"column": 31
} | {
"line": 182,
"column": 4
} | [
{
"pp": "α : Type u_1\na : Lists α\nb✝ : Bool\na0 : Lists' α b✝\nl0 l₂ : Lists' α true\ns : a0.cons' l0 ⊆ l₂\nh : a ∈ (a0.cons' l0).toList\n⊢ a ∈ l₂",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Lists'.Subset.cons",
"Lists'.toList",
"congrArg",
"HEq.refl",
... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 189,
"column": 41
} | {
"line": 189,
"column": 43
} | {
"line": 190,
"column": 4
} | [
{
"pp": "α : Type u_1\nl₁ l₂ : Lists' α true\nH : ∀ (a : Lists α), a ∈ l₁.toList → a ∈ l₂\n⊢ l₁ ⊆ l₂",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Lists'.toList",
"_private.Mathlib.SetTheory.Lists.0.Lists'.subset_def._simp_1_2",
"congrArg",
"List... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 227,
"column": 66
} | {
"line": 227,
"column": 68
} | {
"line": 227,
"column": 69
} | [
{
"pp": "α : Type u_1\nl : List (Lists α)\n⊢ (ofList l).toList = l",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"congrArg",
"Lists",
"List",
"True",
"eq_self",
"Lists'.to_ofList",
"of_eq_true",
"congrFun'",
"Lists.toList",
"Eq",... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 122,
"column": 55
} | {
"line": 122,
"column": 57
} | {
"line": 123,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\nf : α →o α\n⊢ Monotone (lfpApprox f)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"sup_le_sup",
"ChainCompletePartialOrder.instOfCompleteLattice",
"i... | [] | by | [anonymous] | by |
Mathlib.SetTheory.Lists | {
"line": 231,
"column": 20
} | {
"line": 231,
"column": 22
} | {
"line": 231,
"column": 23
} | [
{
"pp": "α : Type u_1\nl : Lists' α true\nx✝ : IsList ⟨true, l⟩\n⊢ ofList (toList ⟨true, l⟩) = ⟨true, l⟩",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"congrArg",
"Lists",
"Bool.true",
"True",
"eq_self",
"Bool",
"of_eq_true",
"congrFun'",
... | [] | by | [anonymous] | by |
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