module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 445,
"column": 49
} | {
"line": 445,
"column": 60
} | {
"line": 445,
"column": 61
} | [
{
"pp": "a : Ordinal.{u_4}\n⊢ 0 - a = 0",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Ordinal.{u_4}\n⊢ 0 - a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 448,
"column": 49
} | {
"line": 448,
"column": 76
} | {
"line": 448,
"column": 77
} | [
{
"pp": "a : Ordinal.{u_4}\n⊢ a - a = 0",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Ordinal.{u_4}\n⊢ a - a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 13
} | {
"line": 454,
"column": 14
} | [
{
"pp": "a b : Ordinal.{u_4}\n⊢ a - b ≠ 0 ↔ b < a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Ordinal.partialOrder",
"PartialOrder.toPreorder",
"HSub.hSub",
"id",
"Ne",
"Iff",
"instHSub",
"Ordinal.sub",
"LT.l... | [
"a b : Ordinal.{u_4}\n⊢ ¬a - b = 0 ↔ b < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 478,
"column": 2
} | {
"line": 478,
"column": 13
} | {
"line": 478,
"column": 14
} | [
{
"pp": "a b c : Ordinal.{u_4}\nhb : b ≠ 0\n⊢ a + b ≤ c ↔ ∀ d < b, a + d < c",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b c : Ordinal.{u_4}\nhb : b ≠ 0\n⊢ a + b ≤ c ↔ ∀ d < b, a + d < c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 490,
"column": 2
} | {
"line": 490,
"column": 13
} | {
"line": 490,
"column": 14
} | [
{
"pp": "a b c : Ordinal.{u_4}\nhb : IsSuccLimit b\n⊢ a + b ≤ c ↔ ∀ d < b, a + d ≤ c",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b c : Ordinal.{u_4}\nhb : IsSuccLimit b\n⊢ a + b ≤ c ↔ ∀ d < b, a + d ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Enum | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 13
} | {
"line": 112,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : WellFoundedLT α\ninst✝¹ : IsRegularCardinalOrder α\ns : Set α\nhs : IsCofinal s\ninst✝ : SuccOrder α\na o : α\nha : o ∈ s\nH : ↑((enum s hs) a) < o\nb : α\nhb : b < succ a\n⊢ ↑((enum s hs) b) ≤ ↑((enum s hs) a)",
"ppTerm": "?m.43",
"assigned": true... | [
"α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : WellFoundedLT α\ninst✝¹ : IsRegularCardinalOrder α\ns : Set α\nhs : IsCofinal s\ninst✝ : SuccOrder α\na o : α\nha : o ∈ s\nH : ↑((enum s hs) a) < o\nb : α\nhb : b < succ a\n⊢ b ≤ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 892,
"column": 24
} | {
"line": 892,
"column": 35
} | {
"line": 892,
"column": 36
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal.{u_1}\n⊢ a ≤ b + a",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nβ : Type v\nγ : Type w\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal.{u_1}\n⊢ a ≤ b + a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 893,
"column": 24
} | {
"line": 893,
"column": 35
} | {
"line": 893,
"column": 36
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal.{u_1}\n⊢ a ≤ a + b",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nβ : Type v\nγ : Type w\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : Ordinal.{u_1}\n⊢ a ≤ a + b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Enum | {
"line": 158,
"column": 22
} | {
"line": 158,
"column": 33
} | {
"line": 158,
"column": 34
} | [
{
"pp": "α : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\ninst✝ : IsRegularCardinalOrder α\ns : Set α\nhs : IsCofinal s\nH : StrictMono (Subtype.val ∘ ⇑(enum s hs)) :=\n StrictMono.comp (Subtype.strictMono_coe fun x ↦ x ∈ s) (OrderIso.strictMono (enum s hs))\nhe : IsNormal (Subtype.val ∘ ⇑(enum ... | [
"α : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\ninst✝ : IsRegularCardinalOrder α\ns : Set α\nhs : IsCofinal s\nH : StrictMono (Subtype.val ∘ ⇑(enum s hs)) :=\n StrictMono.comp (Subtype.strictMono_coe fun x ↦ x ∈ s) (OrderIso.strictMono (enum s hs))\nhe : IsNormal (Subtype.val ∘ ⇑(enum s hs))\n⊢ Di... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 922,
"column": 4
} | {
"line": 922,
"column": 15
} | {
"line": 922,
"column": 16
} | [
{
"pp": "case refine_1\na b : Ordinal.{u_1}\nα : Type u_1\nr : α → α → Prop\nx✝¹ : IsWellOrder α r\nβ : Type u_1\ns : β → β → Prop\nx✝ : IsWellOrder β s\nf : Sum.Lex r emptyRelation ≼i s\nh : Surjective ⇑((InitialSeg.leAdd r emptyRelation).trans f)\n⊢ False",
"ppTerm": "?refine_1",
"assigned": false,
... | [
"case refine_1\na b : Ordinal.{u_1}\nα : Type u_1\nr : α → α → Prop\nx✝¹ : IsWellOrder α r\nβ : Type u_1\ns : β → β → Prop\nx✝ : IsWellOrder β s\nf : Sum.Lex r emptyRelation ≼i s\nh : Surjective ⇑((InitialSeg.leAdd r emptyRelation).trans f)\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 924,
"column": 4
} | {
"line": 924,
"column": 31
} | {
"line": 924,
"column": 32
} | [
{
"pp": "a b : Ordinal.{u_1}\nα : Type u_1\nr : α → α → Prop\nx✝¹ : IsWellOrder α r\nβ : Type u_1\ns : β → β → Prop\nx✝ : IsWellOrder β s\nf : r ≺i s\n⊢ ∀ (a b : α ⊕ PUnit.{u_1 + 1}),\n Sum.Lex r emptyRelation a b →\n s (Sum.recOn a ⇑f.toRelEmbedding fun x ↦ f.top) (Sum.recOn b ⇑f.toRelEmbedding fun x ↦... | [
"a b : Ordinal.{u_1}\nα : Type u_1\nr : α → α → Prop\nx✝¹ : IsWellOrder α r\nβ : Type u_1\ns : β → β → Prop\nx✝ : IsWellOrder β s\nf : r ≺i s\n⊢ ∀ (a : α), s (f.toRelEmbedding a) f.top"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 421,
"column": 2
} | {
"line": 421,
"column": 39
} | {
"line": 421,
"column": 40
} | [
{
"pp": "o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}\na : Ordinal.{max u v}\n⊢ a < o.bsup f ↔ ∃ i, ∃ (hi : i < o), a < f i hi",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}\na : Ordinal.{max u v}\n⊢ a < o.bsup f ↔ ∃ i, ∃ (hi : i < o), a < f i hi"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 420,
"column": 48
} | {
"line": 421,
"column": 77
} | {
"line": 423,
"column": 0
} | [
{
"pp": "o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}\na : Ordinal.{max u v}\n⊢ a < o.bsup f ↔ ∃ i, ∃ (hi : i < o), a < f i hi",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.parti... | [] | by
simpa only [not_forall, not_le] using not_congr (@bsup_le_iff.{_, v} _ f a) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Cofinality.Enum | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 29
} | {
"line": 163,
"column": 30
} | [
{
"pp": "case h\nα : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\ninst✝ : IsRegularCardinalOrder α\ns : Set α\nhs : IsCofinal s\nH : StrictMono (Subtype.val ∘ ⇑(enum s hs)) := ⋯\nhs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.Nonempty → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s\na : α\nha : IsSuccLimit a\nb : α\nhb : ∀ b_1 <... | [
"case h\nα : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\ninst✝ : IsRegularCardinalOrder α\ns : Set α\nhs : IsCofinal s\nH : StrictMono (Subtype.val ∘ ⇑(enum s hs)) :=\n StrictMono.comp (Subtype.strictMono_coe fun x ↦ x ∈ s) (OrderIso.strictMono (enum s hs))\nhs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.Nonempty ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1008,
"column": 6
} | {
"line": 1008,
"column": 29
} | {
"line": 1009,
"column": 2
} | [
{
"pp": "case mp.refine_2\nα : Type u\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\na : Ordinal.{u}\nha : succ a = typeLT α\nb : α\n⊢ (typein fun x1 x2 ↦ x1 < x2).toRelEmbedding b < typeLT α",
"ppTerm": "?mp.refine_2",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"isWellOrde... | [] | exact typein_lt_type .. | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 606,
"column": 4
} | {
"line": 606,
"column": 29
} | {
"line": 606,
"column": 30
} | [
{
"pp": "case refine_1\na : Ordinal.{u_4}\nh : 0 < a\nb : Ordinal.{u_4}\n⊢ a * b < a * succ b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"HMul.hMul",
"Order.succ",
"Order.succ_eq_add_one",
... | [
"case refine_1\na : Ordinal.{u_4}\nh : 0 < a\nb : Ordinal.{u_4}\n⊢ 0 < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 607,
"column": 4
} | {
"line": 607,
"column": 88
} | {
"line": 608,
"column": 6
} | [
{
"pp": "case refine_2\na : Ordinal.{u_4}\nh : 0 < a\na✝ : Ordinal.{u_4}\nhb : IsSuccLimit a✝\n⊢ IsLUB ((fun x ↦ a * x) '' Iio a✝) (a * a✝)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"HMul.hMul",
"... | [
"case refine_2\na : Ordinal.{u_4}\nh : 0 < a\na✝ : Ordinal.{u_4}\nhb : IsSuccLimit a✝\n⊢ ∀ a_1 < a✝, a * a_1 ≤ a * a✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 519,
"column": 2
} | {
"line": 519,
"column": 47
} | {
"line": 519,
"column": 48
} | [
{
"pp": "ι : Type u_3\nf : ι → Ordinal.{max u_4 u_3}\na : Ordinal.{max u_3 u_4}\n⊢ a < lsub f ↔ ∃ i, a ≤ f i",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_3\nf : ι → Ordinal.{max u_4 u_3}\na : Ordinal.{max u_3 u_4}\n⊢ a < lsub f ↔ ∃ i, a ≤ f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Enum | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 17
} | {
"line": 170,
"column": 18
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\ninst✝ : IsRegularCardinalOrder α\ns : Set α\nhs : IsCofinal s\nH : StrictMono (Subtype.val ∘ ⇑(enum s hs)) :=\n StrictMono.comp (Subtype.strictMono_coe fun x ↦ x ∈ s) (OrderIso.strictMono (enum s hs))\nhs' : ∀ ⦃d : Set α⦄, ... | [
"case refine_2\nα : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : WellFoundedLT α\ninst✝ : IsRegularCardinalOrder α\ns : Set α\nhs : IsCofinal s\nH : StrictMono (Subtype.val ∘ ⇑(enum s hs)) :=\n StrictMono.comp (Subtype.strictMono_coe fun x ↦ x ∈ s) (OrderIso.strictMono (enum s hs))\nhs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 612,
"column": 2
} | {
"line": 612,
"column": 13
} | {
"line": 612,
"column": 14
} | [
{
"pp": "a b c : Ordinal.{u_4}\nh : IsSuccLimit c\n⊢ a < b * c ↔ ∃ c' < c, a < b * c'",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b c : Ordinal.{u_4}\nh : IsSuccLimit c\n⊢ a < b * c ↔ ∃ c' < c, a < b * c'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1179,
"column": 34
} | {
"line": 1179,
"column": 45
} | {
"line": 1179,
"column": 46
} | [
{
"pp": "⊢ ord 1 = 1",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ ord 1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1187,
"column": 4
} | {
"line": 1187,
"column": 15
} | {
"line": 1187,
"column": 16
} | [
{
"pp": "a : Cardinal.{u_1}\nha : IsSuccLimit a\nb : Ordinal.{u_1}\nH : ∀ ⦃x : Cardinal.{u_1}⦄, x ∈ Iio a → x ≤ b.card\nc : Cardinal.{u_1}\nhc : c < a\n⊢ c < b.card",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Cardinal.{u_1}\nha : IsSuccLimit a\nb : Ordinal.{u_1}\nH : ∀ ⦃x : Cardinal.{u_1}⦄, x ∈ Iio a → x ≤ b.card\nc : Cardinal.{u_1}\nhc : c < a\n⊢ c < b.card"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 631,
"column": 6
} | {
"line": 631,
"column": 31
} | {
"line": 631,
"column": 32
} | [
{
"pp": "o : Ordinal.{u}\nh✝ : lsub ⇑(typein fun x1 x2 ↦ x1 < x2).toRelEmbedding < o\nh : lsub ⇑(typein fun x1 x2 ↦ x1 < x2).toRelEmbedding < typeLT o.ToType\n⊢ False",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\nh✝ : lsub ⇑(typein fun x1 x2 ↦ x1 < x2).toRelEmbedding < o\nh : lsub ⇑(typein fun x1 x2 ↦ x1 < x2).toRelEmbedding < typeLT o.ToType\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 710,
"column": 2
} | {
"line": 710,
"column": 29
} | {
"line": 710,
"column": 30
} | [
{
"pp": "a b : Ordinal.{u_4}\nh : b ≠ 0\n⊢ a < b * (a / b) + b",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : Ordinal.{u_4}\nh : b ≠ 0\n⊢ a < b * (a / b) + b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1220,
"column": 2
} | {
"line": 1220,
"column": 13
} | {
"line": 1220,
"column": 14
} | [
{
"pp": "c : Cardinal.{u_1}\ni : c.ord.ToType\n⊢ #↑(Iio i) < c",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u_1}\ni : c.ord.ToType\n⊢ #↑(Iio i) < c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 638,
"column": 2
} | {
"line": 638,
"column": 23
} | {
"line": 638,
"column": 24
} | [
{
"pp": "o : Ordinal.{u}\nho : IsSuccPrelimit o\n⊢ sSup {b | b < (typein fun x1 x2 ↦ x1 < x2).top} = o",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"isWellOrder_lt",
"Ordinal.partialOrder",
"congrArg",
"PartialOrder.toPreord... | [
"o : Ordinal.{u}\nho : IsSuccPrelimit o\n⊢ sSup (Iio o) = o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1286,
"column": 2
} | {
"line": 1286,
"column": 13
} | {
"line": 1286,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ 1 ≤ o.card ↔ 1 ≤ o",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u_1}\n⊢ 1 ≤ o.card ↔ 1 ≤ o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1308,
"column": 2
} | {
"line": 1308,
"column": 13
} | {
"line": 1308,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ 0 < o.card ↔ 0 < o",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u_1}\n⊢ 0 < o.card ↔ 0 < o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 718,
"column": 2
} | {
"line": 718,
"column": 47
} | {
"line": 718,
"column": 48
} | [
{
"pp": "o : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{max u v}\na : Ordinal.{max u v}\n⊢ a < o.blsub f ↔ ∃ i, ∃ (hi : i < o), a ≤ f i hi",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{max u v}\na : Ordinal.{max u v}\n⊢ a < o.blsub f ↔ ∃ i, ∃ (hi : i < o), a ≤ f i hi"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1312,
"column": 2
} | {
"line": 1312,
"column": 13
} | {
"line": 1312,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ 1 < o.card ↔ 1 < o",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u_1}\n⊢ 1 < o.card ↔ 1 < o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1334,
"column": 2
} | {
"line": 1334,
"column": 13
} | {
"line": 1334,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ o.card ≤ 1 ↔ o ≤ 1",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u_1}\n⊢ o.card ≤ 1 ↔ o ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1347,
"column": 2
} | {
"line": 1347,
"column": 13
} | {
"line": 1347,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ o.card = 0 ↔ o = 0",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u_1}\n⊢ o.card = 0 ↔ o = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1351,
"column": 2
} | {
"line": 1351,
"column": 13
} | {
"line": 1351,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ o.card = 1 ↔ o = 1",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u_1}\n⊢ o.card = 1 ↔ o = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1371,
"column": 38
} | {
"line": 1371,
"column": 52
} | {
"line": 1371,
"column": 53
} | [
{
"pp": "α : Type u\nr : α → α → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : Fintype α\n⊢ type r = ↑(Fintype.card α)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Ordinal.type",
"Fintype.card",
"id",
"AddMonoidWithOn... | [
"α : Type u\nr : α → α → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : Fintype α\n⊢ (type r).card = ↑(Fintype.card α)"
] | ← card_eq_nat, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 762,
"column": 2
} | {
"line": 762,
"column": 13
} | {
"line": 762,
"column": 14
} | [
{
"pp": "a b : Ordinal.{u_4}\nb0 : b ≠ 0\n⊢ b * a / b = a",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : Ordinal.{u_4}\nb0 : b ≠ 0\n⊢ b * a / b = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 796,
"column": 45
} | {
"line": 796,
"column": 75
} | {
"line": 796,
"column": 76
} | [
{
"pp": "α : Type u\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nf : (a : Ordinal.{u}) → a < type r → Ordinal.{max u v}\no : Ordinal.{max u v}\nH : ∀ (i : α), f ((typein r).toRelEmbedding i) ⋯ < o\ni : Ordinal.{u}\nh : i < type r\n⊢ f i h < o",
"ppTerm": "?m.42",
"assigned": false,
"usedConstants": [... | [
"α : Type u\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nf : (a : Ordinal.{u}) → a < type r → Ordinal.{max u v}\no : Ordinal.{max u v}\nH : ∀ (i : α), f ((typein r).toRelEmbedding i) ⋯ < o\ni : Ordinal.{u}\nh : i < type r\n⊢ f i h < o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 803,
"column": 2
} | {
"line": 803,
"column": 28
} | {
"line": 803,
"column": 29
} | [
{
"pp": "a : Ordinal.{u_4}\n⊢ a / 1 = a",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Ordinal.{u_4}\n⊢ a / 1 = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 807,
"column": 2
} | {
"line": 807,
"column": 28
} | {
"line": 807,
"column": 29
} | [
{
"pp": "a : Ordinal.{u_4}\nh : a ≠ 0\n⊢ a / a = 1",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Ordinal.{u_4}\nh : a ≠ 0\n⊢ a / a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 841,
"column": 2
} | {
"line": 841,
"column": 13
} | {
"line": 841,
"column": 14
} | [
{
"pp": "a b : Ordinal.{u_4}\nb0 : a * b ≠ 0\n⊢ a ≤ a * b",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : Ordinal.{u_4}\nb0 : a * b ≠ 0\n⊢ a ≤ a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 890,
"column": 13
} | {
"line": 890,
"column": 28
} | {
"line": 890,
"column": 29
} | [
{
"pp": "a b : Ordinal.{u_4}\nH : a % b = 0\n⊢ a = b * (a / b)",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : Ordinal.{u_4}\nH : a % b = 0\n⊢ a = b * (a / b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 13
} | {
"line": 80,
"column": 14
} | [
{
"pp": "a b c : Ordinal.{u_1}\nb0 : b ≠ 0\nh : IsSuccLimit c\n⊢ a < b ^ c ↔ ∃ c' < c, a < b ^ c'",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b c : Ordinal.{u_1}\nb0 : b ≠ 0\nh : IsSuccLimit c\n⊢ a < b ^ c ↔ ∃ c' < c, a < b ^ c'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 13
} | {
"line": 84,
"column": 14
} | [
{
"pp": "a : Ordinal.{u_1}\n⊢ a ^ 1 = a",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Ordinal.{u_1}\n⊢ a ^ 1 = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 909,
"column": 2
} | {
"line": 909,
"column": 13
} | {
"line": 909,
"column": 14
} | [
{
"pp": "x y : Ordinal.{u_4}\n⊢ x * y % x = 0",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : Ordinal.{u_4}\n⊢ x * y % x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 94,
"column": 23
} | {
"line": 94,
"column": 51
} | {
"line": 94,
"column": 52
} | [
{
"pp": "b : Ordinal.{u_1}\nl : IsSuccLimit b\nIH : ∀ o' < b, 1 ^ o' = 1\nc : Ordinal.{u_1}\nH : ∀ b' < b, 1 ^ b' ≤ c\n⊢ 1 ≤ c",
"ppTerm": "?m.44",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"b : Ordinal.{u_1}\nl : IsSuccLimit b\nIH : ∀ o' < b, 1 ^ o' = 1\nc : Ordinal.{u_1}\nH : ∀ b' < b, 1 ^ b' ≤ c\n⊢ 1 ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 100,
"column": 20
} | {
"line": 100,
"column": 31
} | {
"line": 100,
"column": 32
} | [
{
"pp": "case add_one\na : Ordinal.{u_1}\na0 : 0 < a\nh0 : 0 < a ^ 0\nb : Ordinal.{u_1}\nIH : 0 < a ^ b\n⊢ 0 < a ^ (b + 1)",
"ppTerm": "?add_one",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
... | [
"case add_one\na : Ordinal.{u_1}\na0 : 0 < a\nh0 : 0 < a ^ 0\nb : Ordinal.{u_1}\nIH : 0 < a ^ b\n⊢ 0 < a ^ b * a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 41
} | {
"line": 119,
"column": 42
} | [
{
"pp": "case refine_1\na : Ordinal.{u_1}\nh : 1 < a\nha : 0 < a\n⊢ ∀ (a_1 : Ordinal.{u_1}), a ^ a_1 < a ^ succ a_1",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"HMul.hMul",
"Order.succ",
"MulZ... | [
"case refine_1\na : Ordinal.{u_1}\nh : 1 < a\nha : 0 < a\n⊢ ∀ (a_1 : Ordinal.{u_1}), a ^ a_1 < a ^ a_1 * a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 15
} | {
"line": 155,
"column": 16
} | [
{
"pp": "case inr\na b : Ordinal.{u_1}\nh : a ≠ 1 ∧ b ≠ 0\nha : 1 < a\n⊢ a ^ b ≠ 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"Ordinal.one",
"HPow.hPow",
"One.toOfNat1",
"instHPow",
"OfNat.ofNat",
"Ordinal.instPow",
"Ord... | [
"case inr\na b : Ordinal.{u_1}\nh : a ≠ 1 ∧ b ≠ 0\nha : 1 < a\n⊢ ¬a ^ b = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 183,
"column": 22
} | {
"line": 183,
"column": 33
} | {
"line": 183,
"column": 34
} | [
{
"pp": "case neg.add_one\na b : Ordinal.{u_1}\nab : a ≤ b\nha : ¬a = 0\nc : Ordinal.{u_1}\nIH : a ^ c ≤ b ^ c\n⊢ a ^ (c + 1) ≤ b ^ (c + 1)",
"ppTerm": "?neg.add_one✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
... | [
"case neg.add_one\na b : Ordinal.{u_1}\nab : a ≤ b\nha : ¬a = 0\nc : Ordinal.{u_1}\nIH : a ^ c ≤ b ^ c\n⊢ a ^ c * a ≤ b ^ c * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 874,
"column": 6
} | {
"line": 874,
"column": 73
} | {
"line": 874,
"column": 74
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{max u v}\nx✝ : (∀ (a : Ordinal.{u}), f a < f (succ a)) ∧ ∀ (o : Ordinal.{u}), IsSuccLimit o → (o.bsup fun x x_1 ↦ f x) = f o\nh₁ : ∀ (a : Ordinal.{u}), f a < f (succ a)\nh₂ : ∀ (o : Ordinal.{u}), IsSuccLimit o → (o.bsup fun x x_1 ↦ f x) = f o\na✝ : Ordinal.{u}\nho : IsSuccLim... | [
"f : Ordinal.{u} → Ordinal.{max u v}\nx✝ : (∀ (a : Ordinal.{u}), f a < f (succ a)) ∧ ∀ (o : Ordinal.{u}), IsSuccLimit o → (o.bsup fun x x_1 ↦ f x) = f o\nh₁ : ∀ (a : Ordinal.{u}), f a < f (succ a)\nh₂ : ∀ (o : Ordinal.{u}), IsSuccLimit o → (o.bsup fun x x_1 ↦ f x) = f o\na✝ : Ordinal.{u}\nho : IsSuccLimit a✝\n⊢ ∀ a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Log | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 68
} | {
"line": 225,
"column": 0
} | [
{
"pp": "b n : ℕ\n⊢ log b n = 1 ↔ b ≤ n ∧ n < b * b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"Nat.pow_add",
"HMul.hMul",
"congrArg",
"Iff.rfl",
"Nat.log_eq_iff",
"id",
"Ne",
"instMulNat",
"inst... | [] | rw [log_eq_iff (Or.inl Nat.one_ne_zero), Nat.pow_add, Nat.pow_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Nat.Log | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 68
} | {
"line": 225,
"column": 0
} | [
{
"pp": "b n : ℕ\n⊢ log b n = 1 ↔ b ≤ n ∧ n < b * b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"Nat.pow_add",
"HMul.hMul",
"congrArg",
"Iff.rfl",
"Nat.log_eq_iff",
"id",
"Ne",
"instMulNat",
"inst... | [] | rw [log_eq_iff (Or.inl Nat.one_ne_zero), Nat.pow_add, Nat.pow_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Log | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 68
} | {
"line": 225,
"column": 0
} | [
{
"pp": "b n : ℕ\n⊢ log b n = 1 ↔ b ≤ n ∧ n < b * b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"Nat.pow_add",
"HMul.hMul",
"congrArg",
"Iff.rfl",
"Nat.log_eq_iff",
"id",
"Ne",
"instMulNat",
"inst... | [] | rw [log_eq_iff (Or.inl Nat.one_ne_zero), Nat.pow_add, Nat.pow_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 32
} | {
"line": 227,
"column": 6
} | [
{
"pp": "case inr.inr.limit\na b : Ordinal.{u_1}\nha : 0 < a\nha' : 1 < a\nc : Ordinal.{u_1}\nl : IsSuccLimit c\nIH : ∀ o' < c, a ^ (b + o') = a ^ b * a ^ o'\nd : Ordinal.{u_1}\n⊢ (∀ a' < c, ((fun x ↦ a ^ x) ∘ fun x ↦ b + x) a' ≤ d) ↔ a ^ b * a ^ c ≤ d",
"ppTerm": "?inr.inr.limit",
"assigned": true,
... | [
"case inr.inr.limit\na b : Ordinal.{u_1}\nha : 0 < a\nha' : 1 < a\nc : Ordinal.{u_1}\nl : IsSuccLimit c\nIH : ∀ o' < c, a ^ (b + o') = a ^ b * a ^ o'\nd : Ordinal.{u_1}\n⊢ (∀ a' < c, a ^ b * a ^ a' ≤ d) ↔ a ^ b * a ^ c ≤ d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 32
} | {
"line": 254,
"column": 33
} | [
{
"pp": "case inr.inr.inr.limit\na b : Ordinal.{u_1}\nhb : 0 < b\nha : a ≠ 0\nha' : 1 < a\nc : Ordinal.{u_1}\nl : IsSuccLimit c\nIH : ∀ o' < c, a ^ (b * o') = (a ^ b) ^ o'\nd : Ordinal.{u_1}\n⊢ (∀ a' < c, ((fun x ↦ a ^ x) ∘ fun x ↦ b * x) a' ≤ d) ↔ (a ^ b) ^ c ≤ d",
"ppTerm": "?inr.inr.inr.limit",
"assi... | [
"case inr.inr.inr.limit\na b : Ordinal.{u_1}\nhb : 0 < b\nha : a ≠ 0\nha' : 1 < a\nc : Ordinal.{u_1}\nl : IsSuccLimit c\nIH : ∀ o' < c, a ^ (b * o') = (a ^ b) ^ o'\nd : Ordinal.{u_1}\n⊢ (∀ a' < c, (a ^ b) ^ a' ≤ d) ↔ (a ^ b) ^ c ≤ d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 243,
"column": 2
} | {
"line": 254,
"column": 84
} | {
"line": 256,
"column": 0
} | [
{
"pp": "a b c : Ordinal.{u_1}\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Ordinal.isNormal_mul_right",
"MulOne.toOne",
"LE.le.eq_or_lt",
"False",
"Ordinal.noZeroDivisors",
"Ordinal.instL... | [] | obtain rfl | hb := eq_zero_or_pos b; · simp
obtain rfl | ha := eq_or_ne a 0
· have := hb.ne'
by_cases c = 0 <;> simp_all
obtain rfl | ha' := (one_le_iff_ne_zero.2 ha).eq_or_lt; · simp
induction c using limitRecOn with
| zero => simp
| add_one c IH => rw [mul_add_one, opow_add, IH, opow_add_one]
| limi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 243,
"column": 2
} | {
"line": 254,
"column": 84
} | {
"line": 256,
"column": 0
} | [
{
"pp": "a b c : Ordinal.{u_1}\n⊢ a ^ (b * c) = (a ^ b) ^ c",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Ordinal.isNormal_mul_right",
"MulOne.toOne",
"LE.le.eq_or_lt",
"False",
"Ordinal.noZeroDivisors",
"Ordinal.instL... | [] | obtain rfl | hb := eq_zero_or_pos b; · simp
obtain rfl | ha := eq_or_ne a 0
· have := hb.ne'
by_cases c = 0 <;> simp_all
obtain rfl | ha' := (one_le_iff_ne_zero.2 ha).eq_or_lt; · simp
induction c using limitRecOn with
| zero => simp
| add_one c IH => rw [mul_add_one, opow_add, IH, opow_add_one]
| limi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 13
} | {
"line": 274,
"column": 14
} | [
{
"pp": "b u v x : Ordinal.{u_1}\nhv : v < b\nhu : u < x\n⊢ b ^ u * v < b ^ x",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"b u v x : Ordinal.{u_1}\nhv : v < b\nhu : u < x\n⊢ b ^ u * v < b ^ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 324,
"column": 4
} | {
"line": 324,
"column": 20
} | {
"line": 324,
"column": 21
} | [
{
"pp": "case inl\nb c : Ordinal.{u_1}\nhb : 1 < b\nhc : c ≠ 0\n⊢ b ^ c ≤ 0 ↔ c ≤ log b 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
"and_true",
"iff_false",
"Ordinal.partialOrder",
"Ordinal.log_zero_right",
... | [
"case inl\nb c : Ordinal.{u_1}\nhb : 1 < b\nhc : c ≠ 0\n⊢ ¬b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 30
} | {
"line": 325,
"column": 2
} | [
{
"pp": "case inl\nb c : Ordinal.{u_1}\nhb : 1 < b\nhc : c ≠ 0\n⊢ b ^ c ≤ 0 ↔ c ≤ log b 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Ordinal.instLinearOrder",
"eq_false",
"and_true",
"iff_false",
"Ordinal.partialOrder",
... | [
"case inr\nb x c : Ordinal.{u_1}\nhb : 1 < b\nhc : c ≠ 0\nhx : x ≠ 0\n⊢ b ^ c ≤ x ↔ c ≤ log b x"
] | · simpa [hc] using hb.ne_bot | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 1056,
"column": 36
} | {
"line": 1056,
"column": 67
} | {
"line": 1056,
"column": 68
} | [
{
"pp": "⊢ 1 < ω",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ 1 < ω"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 950,
"column": 85
} | {
"line": 959,
"column": 87
} | {
"line": 961,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_1}\ns : Set Ordinal.{u_1}\n⊢ o * sSup s = sSup ((fun x ↦ o * x) '' s)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.isNormal_mul_right",
"False",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Lattice.toSemilatt... | [] | by
rcases s.eq_empty_or_nonempty with (rfl | hs)
· simp
rcases eq_zero_or_pos o with (rfl | ho)
· simp [hs.image_const]
by_cases bdd : BddAbove s
· exact (isNormal_mul_right ho).map_sSup hs bdd
· rw [csSup_of_not_bddAbove bdd, csSup_empty, csSup_of_not_bddAbove]
· simp
exact fun ⟨u, hu⟩ ↦ bdd ⟨u, ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 364,
"column": 4
} | {
"line": 364,
"column": 15
} | {
"line": 364,
"column": 16
} | [
{
"pp": "case inl\nb : Ordinal.{u_1}\nhb : 1 < b\n⊢ 0 < b ^ succ (log b 0)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Ordinal.opow_one",
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"Order.succ_eq_add_one",
"Ordinal.partialOrder",
"Ordinal.log_zer... | [
"case inl\nb : Ordinal.{u_1}\nhb : 1 < b\n⊢ 0 < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Log | {
"line": 378,
"column": 6
} | {
"line": 381,
"column": 72
} | {
"line": 382,
"column": 6
} | [
{
"pp": "case succ.inl\nn : ℕ\nhn : 1 < n\nfuel : ℕ\nih :\n ∀ {b : ℕ},\n 1 < b →\n n < b ^ fuel →\n (go n b fuel).fst = b ^ ((go n b fuel).snd + 1) / n ∧\n b ^ (go n b fuel).snd < n ∧ n ≤ b ^ ((go n b fuel).snd + 1)\nb : ℕ\nhb : 1 < b\nhfuel : n < b ^ (fuel + 1)\nhbn : b < n\nih₁ : (go ... | [
"case succ.inl\nn : ℕ\nhn : 1 < n\nfuel : ℕ\nih :\n ∀ {b : ℕ},\n 1 < b →\n n < b ^ fuel →\n (go n b fuel).fst = b ^ ((go n b fuel).snd + 1) / n ∧\n b ^ (go n b fuel).snd < n ∧ n ≤ b ^ ((go n b fuel).snd + 1)\nb : ℕ\nhb : 1 < b\nhfuel : n < b ^ (fuel + 1)\nhbn : b < n\nih₁ : (go n (b ^ 2) fu... | simp_all only [go, if_neg (Nat.not_le_of_gt hbn), ← Nat.pow_two, ← Nat.pow_mul,
Nat.div_lt_iff_lt_mul (Nat.zero_lt_of_lt hbn), Nat.div_div_eq_div_mul,
Nat.mul_comm n b, Nat.mul_add_one, @Nat.pow_add_one' _ (2 * _ + 1),
Nat.mul_lt_mul_left, Nat.mul_div_mul_left, Nat.zero_lt_of_lt hb] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 389,
"column": 4
} | {
"line": 392,
"column": 55
} | {
"line": 394,
"column": 0
} | [
{
"pp": "case inr\nb x y : Ordinal.{u_1}\nxy : x ≤ y\nhx : x ≠ 0\n⊢ log b x ≤ log b y",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Ordinal.log_of_left_le_one",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.partialOrder",
... | [] | obtain hb | hb := lt_or_ge 1 b
· exact (opow_le_iff_le_log hb (hx.bot_lt.trans_le xy).ne').1 <|
(opow_log_le_self _ hx).trans xy
· rw [log_of_left_le_one hb, log_of_left_le_one hb] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 389,
"column": 4
} | {
"line": 392,
"column": 55
} | {
"line": 394,
"column": 0
} | [
{
"pp": "case inr\nb x y : Ordinal.{u_1}\nxy : x ≤ y\nhx : x ≠ 0\n⊢ log b x ≤ log b y",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Ordinal.log_of_left_le_one",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.partialOrder",
... | [] | obtain hb | hb := lt_or_ge 1 b
· exact (opow_le_iff_le_log hb (hx.bot_lt.trans_le xy).ne').1 <|
(opow_log_le_self _ hx).trans xy
· rw [log_of_left_le_one hb, log_of_left_le_one hb] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 409,
"column": 4
} | {
"line": 409,
"column": 15
} | {
"line": 409,
"column": 16
} | [
{
"pp": "case inl\no : Ordinal.{u_1}\nho : o ≠ 0\n⊢ o % 0 ^ log 0 o < o",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.log_of_left_le_one",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"zero_le._simp... | [
"case inl\no : Ordinal.{u_1}\nho : o ≠ 0\n⊢ 0 < o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 1131,
"column": 4
} | {
"line": 1131,
"column": 15
} | {
"line": 1131,
"column": 16
} | [
{
"pp": "case not_isMin\nc : Cardinal.{u_1}\nhc : ℵ₀ ≤ c\n⊢ ¬IsMin c.ord",
"ppTerm": "?not_isMin",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"isMin_iff_eq_bot._simp_1",
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
"Cardinal... | [
"case not_isMin\nc : Cardinal.{u_1}\nhc : ℵ₀ ≤ c\n⊢ ¬c = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 444,
"column": 2
} | {
"line": 444,
"column": 13
} | {
"line": 444,
"column": 14
} | [
{
"pp": "b v : Ordinal.{u_1}\nhb : 1 < b\nu : Ordinal.{u_1}\nhv : v ≠ 0\n⊢ log b (b ^ u * v) = u + log b v",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"b v : Ordinal.{u_1}\nhb : 1 < b\nu : Ordinal.{u_1}\nhv : v ≠ 0\n⊢ log b (b ^ u * v) = u + log b v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 453,
"column": 4
} | {
"line": 453,
"column": 15
} | {
"line": 453,
"column": 16
} | [
{
"pp": "case inl\no : Ordinal.{u_1}\nho : o ≠ 0\n⊢ 0 < o / 0 ^ log 0 o",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.log_of_left_le_one",
"Preorder.toLT",
"instHDiv",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
... | [
"case inl\no : Ordinal.{u_1}\nho : o ≠ 0\n⊢ 0 < o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 463,
"column": 4
} | {
"line": 463,
"column": 38
} | {
"line": 463,
"column": 39
} | [
{
"pp": "case a\no : Ordinal.{u_1}\nho : o ≠ 0\n⊢ o / 2 ^ log 2 o ≤ 1",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"AddMonoid.toAddSemigroup",
"Ordinal.partialOrder",
"congrArg",
"PartialOrder.toPreorder",
"Nat.instAtLeastTwo... | [
"case a\no : Ordinal.{u_1}\nho : o ≠ 0\n⊢ o / (1 + 1) ^ log (1 + 1) o ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 464,
"column": 4
} | {
"line": 464,
"column": 53
} | {
"line": 464,
"column": 54
} | [
{
"pp": "case a\no : Ordinal.{u_1}\nho : o ≠ 0\n⊢ 1 ≤ o / 2 ^ log 2 o",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"Ordinal.partialOrder",
"instIsBotZeroClass",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"Nat.inst... | [
"case a\no : Ordinal.{u_1}\nho : o ≠ 0\n⊢ ¬o / 2 ^ log 2 o = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 475,
"column": 4
} | {
"line": 475,
"column": 54
} | {
"line": 475,
"column": 55
} | [
{
"pp": "case inr\nx y b : Ordinal.{u_1}\nhx : x ≠ 0\nhy : y ≠ 0\nhb : b ≤ 1\n⊢ log b x + log b y ≤ log b (x * y)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.log_of_left_le_one",
"HMul.hMul",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
... | [
"case inr\nx y b : Ordinal.{u_1}\nhx : x ≠ 0\nhy : y ≠ 0\nhb : b ≤ 1\n⊢ 0 ≤ 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 483,
"column": 24
} | {
"line": 483,
"column": 54
} | {
"line": 483,
"column": 55
} | [
{
"pp": "a : Ordinal.{u_1}\nha : a ≠ 0\nn : ℕ\nhn : a / ω ^ log ω a = ↑n\n| a",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"Ordinal.omega0",
"MulZeroClass.toMul",
"congrArg",
"Ordinal.div_add_mod",
"Ordinal.mod",
"... | [
"a : Ordinal.{u_1}\nha : a ≠ 0\nn : ℕ\nhn : a / ω ^ log ω a = ↑n\n| ω ^ log ω a * (a / ω ^ log ω a) + a % ω ^ log ω a"
] | ← div_add_mod a (ω ^ log ω a), | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 486,
"column": 4
} | {
"line": 486,
"column": 15
} | {
"line": 486,
"column": 16
} | [
{
"pp": "case zero\na : Ordinal.{u_1}\nha : a ≠ 0\nhn : a / ω ^ log ω a = ↑0\n⊢ ω ^ log ω a * ↑0 + a % ω ^ log ω a - ω ^ log ω a < a",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"Ordinal.omega0",
"Ordinal.partialOrder... | [
"case zero\na : Ordinal.{u_1}\nha : a ≠ 0\nhn : a / ω ^ log ω a = ↑0\n⊢ a % ω ^ log ω a - ω ^ log ω a < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 534,
"column": 4
} | {
"line": 534,
"column": 15
} | {
"line": 534,
"column": 16
} | [
{
"pp": "case inl\no : Ordinal.{u_1}\nho : 0 < o\nho₁ : 1 < o\n⊢ ⨆ n, o ^ n = o ^ ω",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inl\no : Ordinal.{u_1}\nho : 0 < o\nho₁ : 1 < o\n⊢ ⨆ n, o ^ n = o ^ ω"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 28
} | {
"line": 256,
"column": 4
} | [
{
"pp": "case a\nf : Ordinal.{u} → Ordinal.{u}\na : Ordinal.{u}\n⊢ ⨆ n, f^[n] a ≤ nfp f a",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.partialOrder",
"congrArg",
"iSup",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"U... | [
"case a\nf : Ordinal.{u} → Ordinal.{u}\na : Ordinal.{u}\n⊢ ∀ (i : ℕ), f^[i] a ≤ nfp f a"
] | rw [Ordinal.iSup_le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 320,
"column": 44
} | {
"line": 320,
"column": 71
} | {
"line": 320,
"column": 72
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\nH : IsNormal f\na b : Ordinal.{u}\nh : b ≤ nfp f a\n⊢ f b ≤ nfp f a",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : Ordinal.{u} → Ordinal.{u}\nH : IsNormal f\na b : Ordinal.{u}\nh : b ≤ nfp f a\n⊢ f b ≤ nfp f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 197,
"column": 12
} | {
"line": 197,
"column": 23
} | {
"line": 197,
"column": 24
} | [
{
"pp": "case zero\na o : Ordinal.{u}\nho : IsPrincipal (fun x1 x2 ↦ x1 + x2) o\nha : a < o\n⊢ a * ↑0 < o",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"congrArg",
... | [
"case zero\na o : Ordinal.{u}\nho : IsPrincipal (fun x1 x2 ↦ x1 + x2) o\nha : a < o\n⊢ 0 < o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 18
} | {
"line": 230,
"column": 4
} | [
{
"pp": "case refine_2\no : Ordinal.{u}\nh : ∀ a < o, a + o = o\na b : Ordinal.{u}\nhao : a < o\nhbo : b < o\n⊢ (fun x1 x2 ↦ x1 + x2) a b < o",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"Pa... | [
"case refine_2\no : Ordinal.{u}\nh : ∀ a < o, a + o = o\na b : Ordinal.{u}\nhao : a < o\nhbo : b < o\n⊢ (fun x1 x2 ↦ x1 + x2) a b < a + o"
] | rw [← h a hao] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 13
} | {
"line": 362,
"column": 14
} | [
{
"pp": "a b : Ordinal.{u_1}\nha : a ≤ 1\nhb : b ≤ 1\n⊢ (fun x1 x2 ↦ x1 * x2) a b ≤ 1",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"HMul.hMul",
"MulZeroClass.toMul",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeIn... | [
"a b : Ordinal.{u_1}\nha : a ≤ 1\nhb : b ≤ 1\n⊢ a * b ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 47,
"column": 28
} | {
"line": 47,
"column": 39
} | {
"line": 47,
"column": 40
} | [
{
"pp": "c : Cardinal.{u_1}\nIH : ∀ y < c, ℵ₀ ≤ y → y * y = y\nhc : ℵ₀ ≤ c\n⊢ c ≤ c * c",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u_1}\nIH : ∀ y < c, ℵ₀ ≤ y → y * y = y\nhc : ℵ₀ ≤ c\n⊢ c ≤ c * c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 306,
"column": 2
} | {
"line": 306,
"column": 13
} | {
"line": 306,
"column": 14
} | [
{
"pp": "⊢ typeLT Cardinal.{u} = Ordinal.univ.{u, u + 1}",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ typeLT Cardinal.{u} = Ordinal.univ.{u, u + 1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 41
} | {
"line": 312,
"column": 42
} | [
{
"pp": "⊢ #Cardinal.{u} = univ.{u, u + 1}",
"ppTerm": "?m.2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ #Cardinal.{u} = univ.{u, u + 1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 13
} | {
"line": 315,
"column": 14
} | [
{
"pp": "⊢ cof Cardinal.{u} = univ.{u, u + 1}",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ cof Cardinal.{u} = univ.{u, u + 1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 13
} | {
"line": 382,
"column": 14
} | [
{
"pp": "c : Cardinal.{u_1}\n⊢ c ≤ preAleph c.ord",
"ppTerm": "?m.4",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u_1}\n⊢ c ≤ preAleph c.ord"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 13
} | {
"line": 411,
"column": 14
} | [
{
"pp": "f : Ordinal.{u_1} → Cardinal.{u_1}\nhf : StrictMono f\no : Ordinal.{u_1}\n⊢ preAleph o ≤ f o",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"f : Ordinal.{u_1} → Cardinal.{u_1}\nhf : StrictMono f\no : Ordinal.{u_1}\n⊢ preAleph o ≤ f o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 501,
"column": 2
} | {
"line": 501,
"column": 13
} | {
"line": 501,
"column": 14
} | [
{
"pp": "c : Cardinal.{u_1}\n⊢ c ≤ ℵ_ c.ord",
"ppTerm": "?m.4",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u_1}\n⊢ c ≤ ℵ_ c.ord"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 559,
"column": 2
} | {
"line": 559,
"column": 25
} | {
"line": 559,
"column": 26
} | [
{
"pp": "o : Ordinal.{u_1}\nho : ω ^ 2 ≤ o\n⊢ preAleph o = ℵ_ o",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.SetTheory.Cardinal.Aleph.0.Cardinal.preAleph_of_omega0_sq_le._simp_1_1",
"Cardinal.aleph",
"Ordinal.partialOrder",
"Card... | [
"o : Ordinal.{u_1}\nho : ω ^ 2 ≤ o\n⊢ preOmega o = ω_ o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 635,
"column": 2
} | {
"line": 635,
"column": 13
} | {
"line": 635,
"column": 14
} | [
{
"pp": "⊢ preBeth 1 = 1",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ preBeth 1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 645,
"column": 4
} | {
"line": 645,
"column": 15
} | {
"line": 645,
"column": 16
} | [
{
"pp": "case a\n⊢ ℵ₀ ≤ preBeth ω",
"ppTerm": "?a✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case a\n⊢ ℵ₀ ≤ preBeth ω"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 649,
"column": 2
} | {
"line": 649,
"column": 13
} | {
"line": 649,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ 0 < preBeth o ↔ 0 < o",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u_1}\n⊢ 0 < preBeth o ↔ 0 < o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 655,
"column": 2
} | {
"line": 655,
"column": 13
} | {
"line": 655,
"column": 14
} | [
{
"pp": "c : Cardinal.{u_1}\n⊢ c ≤ preBeth c.ord",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u_1}\n⊢ c ≤ preBeth c.ord"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 659,
"column": 2
} | {
"line": 659,
"column": 13
} | {
"line": 659,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ preBeth o = 0 ↔ o = 0",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u_1}\n⊢ preBeth o = 0 ↔ o = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 674,
"column": 4
} | {
"line": 674,
"column": 15
} | {
"line": 674,
"column": 16
} | [
{
"pp": "case refine_2\no : Ordinal.{u_1}\nho : IsSuccPrelimit o\nx : Cardinal.{u_1}\nhx : x < ⨆ a, preBeth ↑a\na : ↑(Iio o)\nha : x < preBeth ↑a\n⊢ 2 ^ x ≤ preBeth (↑a + 1)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal.instPowCardinal",
"Ordinal... | [
"case refine_2\no : Ordinal.{u_1}\nho : IsSuccPrelimit o\nx : Cardinal.{u_1}\nhx : x < ⨆ a, preBeth ↑a\na : ↑(Iio o)\nha : x < preBeth ↑a\n⊢ 2 ^ x ≤ 2 ^ preBeth ↑a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 81,
"column": 15
} | {
"line": 81,
"column": 41
} | {
"line": 81,
"column": 42
} | [
{
"pp": "a b : Cardinal.{u_1}\nha : ℵ₀ ≤ a\nhb : ℵ₀ ≤ b\n⊢ a ≤ a * b",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : Cardinal.{u_1}\nha : ℵ₀ ≤ a\nhb : ℵ₀ ≤ b\n⊢ a ≤ a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 82,
"column": 10
} | {
"line": 82,
"column": 36
} | {
"line": 82,
"column": 37
} | [
{
"pp": "a b : Cardinal.{u_1}\nha : ℵ₀ ≤ a\nhb : ℵ₀ ≤ b\n⊢ b ≤ a * b",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : Cardinal.{u_1}\nha : ℵ₀ ≤ a\nhb : ℵ₀ ≤ b\n⊢ b ≤ a * b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 13
} | {
"line": 85,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ o.cof ≤ o.card",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u_1}\n⊢ o.cof ≤ o.card"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 13
} | {
"line": 88,
"column": 14
} | [
{
"pp": "c : Cardinal.{u_1}\n⊢ c.ord.cof ≤ c",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u_1}\n⊢ c.ord.cof ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 13
} | {
"line": 119,
"column": 14
} | [
{
"pp": "⊢ cof 1 = 1",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ cof 1 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 13
} | {
"line": 131,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ o.cof < ℵ₀ ↔ o.cof ≤ 1",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u_1}\n⊢ o.cof < ℵ₀ ↔ o.cof ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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