module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 42
} | {
"line": 145,
"column": 2
} | [
{
"pp": "o : Ordinal.{u_1}\nho : IsSuccLimit o\nho' : o < ω_ 1\n⊢ o.cof = ℵ₀",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Cardinal",
"PartialOrder.toPreorder",
"LE.le.antisymm",
"Ordinal.cof_le_card",
"Cardinal.aleph0",
"Ordinal.card",
"Cardinal... | [
"o : Ordinal.{u_1}\nho : IsSuccLimit o\nho' : o < ω_ 1\n⊢ ℵ₀ ≤ o.cof",
"o : Ordinal.{u_1}\nho : IsSuccLimit o\nho' : o < ω_ 1\n⊢ o.card ≤ ℵ₀"
] | apply ((cof_le_card _).trans _).antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 43
} | {
"line": 139,
"column": 44
} | [
{
"pp": "a b : Cardinal.{u_1}\nh : ℵ₀ ≤ b\n⊢ a * b ≤ max a b",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : Cardinal.{u_1}\nh : ℵ₀ ≤ b\n⊢ a * b ≤ max a b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 762,
"column": 2
} | {
"line": 762,
"column": 13
} | {
"line": 762,
"column": 14
} | [
{
"pp": "c : Cardinal.{u_1}\n⊢ c ≤ ℶ_ c.ord",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u_1}\n⊢ c ≤ ℶ_ c.ord"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 791,
"column": 2
} | {
"line": 791,
"column": 13
} | {
"line": 791,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\n⊢ lift.{v, u} c ≤ ℵ_ 1 ↔ c ≤ ℵ_ 1",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\n⊢ lift.{v, u} c ≤ ℵ_ 1 ↔ c ≤ ℵ_ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 797,
"column": 2
} | {
"line": 797,
"column": 13
} | {
"line": 797,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\n⊢ ℵ_ 1 < lift.{v, u} c ↔ ℵ_ 1 < c",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\n⊢ ℵ_ 1 < lift.{v, u} c ↔ ℵ_ 1 < c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 809,
"column": 2
} | {
"line": 809,
"column": 13
} | {
"line": 809,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\n⊢ ℵ_ 1 = lift.{v, u} c ↔ ℵ_ 1 = c",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\n⊢ ℵ_ 1 = lift.{v, u} c ↔ ℵ_ 1 = c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 821,
"column": 2
} | {
"line": 821,
"column": 13
} | {
"line": 821,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ ℵ_ ↑n ≤ lift.{v, u} c ↔ ℵ_ ↑n ≤ c",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\nn : ℕ\n⊢ ℵ_ ↑n ≤ lift.{v, u} c ↔ ℵ_ ↑n ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 825,
"column": 2
} | {
"line": 825,
"column": 13
} | {
"line": 825,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ lift.{v, u} c ≤ ℵ_ ↑n ↔ c ≤ ℵ_ ↑n",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\nn : ℕ\n⊢ lift.{v, u} c ≤ ℵ_ ↑n ↔ c ≤ ℵ_ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 13
} | {
"line": 36,
"column": 14
} | [
{
"pp": "β : Type v\no : Ordinal.{u}\nc : Cardinal.{v}\nho : lift.{v, u} o.card ≤ lift.{u, v} c\nhc : ℵ₀ ≤ c\nA : Ordinal.{u} → Set β\nhA : ∀ j < o, #↑(A j) ≤ c\ni : o.ToType\n⊢ lift.{u, v} #↑(((fun x ↦ A ↑x) ∘ ⇑ToType.mk.symm) i) ≤ lift.{u, v} c",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants... | [
"β : Type v\no : Ordinal.{u}\nc : Cardinal.{v}\nho : lift.{v, u} o.card ≤ lift.{u, v} c\nhc : ℵ₀ ≤ c\nA : Ordinal.{u} → Set β\nhA : ∀ j < o, #↑(A j) ≤ c\ni : o.ToType\n⊢ #↑(A ↑(ToType.mk.symm i)) ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 829,
"column": 2
} | {
"line": 829,
"column": 13
} | {
"line": 829,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ ℵ_ ↑n < lift.{v, u} c ↔ ℵ_ ↑n < c",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\nn : ℕ\n⊢ ℵ_ ↑n < lift.{v, u} c ↔ ℵ_ ↑n < c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 828,
"column": 64
} | {
"line": 829,
"column": 33
} | {
"line": 831,
"column": 0
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ ℵ_ ↑n < lift.{v, u} c ↔ ℵ_ ↑n < c",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Cardinal.aleph",
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Cardinal.lift",
... | [] | by
simpa using lift_lt (a := ℵ_ n) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 833,
"column": 2
} | {
"line": 833,
"column": 13
} | {
"line": 833,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ lift.{v, u} c < ℵ_ ↑n ↔ c < ℵ_ ↑n",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\nn : ℕ\n⊢ lift.{v, u} c < ℵ_ ↑n ↔ c < ℵ_ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 837,
"column": 2
} | {
"line": 837,
"column": 13
} | {
"line": 837,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ ℵ_ ↑n = lift.{v, u} c ↔ ℵ_ ↑n = c",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\nn : ℕ\n⊢ ℵ_ ↑n = lift.{v, u} c ↔ ℵ_ ↑n = c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 869,
"column": 2
} | {
"line": 869,
"column": 13
} | {
"line": 869,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ ℶ_ ↑n ≤ lift.{v, u} c ↔ ℶ_ ↑n ≤ c",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\nn : ℕ\n⊢ ℶ_ ↑n ≤ lift.{v, u} c ↔ ℶ_ ↑n ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 85
} | {
"line": 183,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\ns : Set α\nhs : IsCofinal s\nt : Set ↑s\nht : IsCofinal t\nht' : typeLT ↑t = (Order.cof α).ord\n⊢ typeLT ↑(Subtype.val '' t) = (Order.cof α).ord",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"E... | [] | · rw [← ht']
exact ((Subtype.strictMono_coe _).strictMonoOn _).orderIso.ordinalType_congr.symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 873,
"column": 2
} | {
"line": 873,
"column": 13
} | {
"line": 873,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ lift.{v, u} c ≤ ℶ_ ↑n ↔ c ≤ ℶ_ ↑n",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\nn : ℕ\n⊢ lift.{v, u} c ≤ ℶ_ ↑n ↔ c ≤ ℶ_ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 13
} | {
"line": 191,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ o.cof.ord.cof = o.cof",
"ppTerm": "?m.2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u_1}\n⊢ o.cof.ord.cof = o.cof"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 877,
"column": 2
} | {
"line": 877,
"column": 13
} | {
"line": 877,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ ℶ_ ↑n < lift.{v, u} c ↔ ℶ_ ↑n < c",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\nn : ℕ\n⊢ ℶ_ ↑n < lift.{v, u} c ↔ ℶ_ ↑n < c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 881,
"column": 2
} | {
"line": 881,
"column": 13
} | {
"line": 881,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ lift.{v, u} c < ℶ_ ↑n ↔ c < ℶ_ ↑n",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\nn : ℕ\n⊢ lift.{v, u} c < ℶ_ ↑n ↔ c < ℶ_ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 885,
"column": 2
} | {
"line": 885,
"column": 13
} | {
"line": 885,
"column": 14
} | [
{
"pp": "c : Cardinal.{u}\nn : ℕ\n⊢ ℶ_ ↑n = lift.{v, u} c ↔ ℶ_ ↑n = c",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u}\nn : ℕ\n⊢ ℶ_ ↑n = lift.{v, u} c ↔ ℶ_ ↑n = c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 924,
"column": 2
} | {
"line": 924,
"column": 13
} | {
"line": 924,
"column": 14
} | [
{
"pp": "o : Ordinal.{u}\n⊢ ω_ 1 ≤ lift.{v, u} o ↔ ω_ 1 ≤ o",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\n⊢ ω_ 1 ≤ lift.{v, u} o ↔ ω_ 1 ≤ o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 13
} | {
"line": 77,
"column": 14
} | [
{
"pp": "o : Ordinal.{u}\nf : ↑(Iio o) → Ordinal.{max u v}\n⊢ ⨆ i, f i.toOrd = ⨆ a, f a",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\nf : ↑(Iio o) → Ordinal.{max u v}\n⊢ ⨆ i, f i.toOrd = ⨆ a, f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 928,
"column": 2
} | {
"line": 928,
"column": 13
} | {
"line": 928,
"column": 14
} | [
{
"pp": "o : Ordinal.{u}\n⊢ lift.{v, u} o ≤ ω_ 1 ↔ o ≤ ω_ 1",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\n⊢ lift.{v, u} o ≤ ω_ 1 ↔ o ≤ ω_ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 932,
"column": 2
} | {
"line": 932,
"column": 13
} | {
"line": 932,
"column": 14
} | [
{
"pp": "o : Ordinal.{u}\n⊢ ω_ 1 < lift.{v, u} o ↔ ω_ 1 < o",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\n⊢ ω_ 1 < lift.{v, u} o ↔ ω_ 1 < o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 936,
"column": 2
} | {
"line": 936,
"column": 13
} | {
"line": 936,
"column": 14
} | [
{
"pp": "o : Ordinal.{u}\n⊢ lift.{v, u} o < ω_ 1 ↔ o < ω_ 1",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\n⊢ lift.{v, u} o < ω_ 1 ↔ o < ω_ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 940,
"column": 2
} | {
"line": 940,
"column": 13
} | {
"line": 940,
"column": 14
} | [
{
"pp": "o : Ordinal.{u}\n⊢ ω_ 1 = lift.{v, u} o ↔ ω_ 1 = o",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\n⊢ ω_ 1 = lift.{v, u} o ↔ ω_ 1 = o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 13
} | {
"line": 216,
"column": 14
} | [
{
"pp": "γ : Type u\ninst✝ : LinearOrder γ\nf : γ → Ordinal.{u}\nhf : StrictMono f\n⊢ (⨆ i, f i + 1).cof = Order.cof γ",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"γ : Type u\ninst✝ : LinearOrder γ\nf : γ → Ordinal.{u}\nhf : StrictMono f\n⊢ (⨆ i, f i + 1).cof = Order.cof γ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 13
} | {
"line": 224,
"column": 14
} | [
{
"pp": "γ : Type u\ninst✝¹ : LinearOrder γ\ninst✝ : NoMaxOrder γ\nf : γ → Ordinal.{u}\nhf : StrictMono f\n⊢ (⨆ i, f i).cof = Order.cof γ",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"γ : Type u\ninst✝¹ : LinearOrder γ\ninst✝ : NoMaxOrder γ\nf : γ → Ordinal.{u}\nhf : StrictMono f\n⊢ (⨆ i, f i).cof = Order.cof γ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 948,
"column": 2
} | {
"line": 948,
"column": 13
} | {
"line": 948,
"column": 14
} | [
{
"pp": "o : Ordinal.{u}\nn : ℕ\n⊢ ω_ ↑n ≤ lift.{v, u} o ↔ ω_ ↑n ≤ o",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\nn : ℕ\n⊢ ω_ ↑n ≤ lift.{v, u} o ↔ ω_ ↑n ≤ o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 26
} | {
"line": 230,
"column": 27
} | [
{
"pp": "a : Ordinal.{u_1}\nf : ↑(Iio a) → Ordinal.{u_1}\nhf : StrictMono f\n⊢ (⨆ i, f i + 1).cof = a.cof",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : Ordinal.{u_1}\nf : ↑(Iio a) → Ordinal.{u_1}\nhf : StrictMono f\n⊢ (⨆ i, f i + 1).cof = a.cof"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 952,
"column": 2
} | {
"line": 952,
"column": 13
} | {
"line": 952,
"column": 14
} | [
{
"pp": "o : Ordinal.{u}\nn : ℕ\n⊢ lift.{v, u} o ≤ ω_ ↑n ↔ o ≤ ω_ ↑n",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\nn : ℕ\n⊢ lift.{v, u} o ≤ ω_ ↑n ↔ o ≤ ω_ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 956,
"column": 2
} | {
"line": 956,
"column": 13
} | {
"line": 956,
"column": 14
} | [
{
"pp": "o : Ordinal.{u}\nn : ℕ\n⊢ ω_ ↑n < lift.{v, u} o ↔ ω_ ↑n < o",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\nn : ℕ\n⊢ ω_ ↑n < lift.{v, u} o ↔ ω_ ↑n < o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 15
} | {
"line": 93,
"column": 16
} | [
{
"pp": "case pos\nι : Type u\nf : ι → Ordinal.{v}\nn : ℕ\nhι : Cardinal.lift.{v, u} #ι ≤ Cardinal.lift.{u, v} ↑n\nhf : ∀ (i : ι), (f i).card ≤ ↑n\nhc : ↑n < ℵ₀\ni : ι\n⊢ f i ≤ ↑n",
"ppTerm": "?pos✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case pos\nι : Type u\nf : ι → Ordinal.{v}\nn : ℕ\nhι : Cardinal.lift.{v, u} #ι ≤ Cardinal.lift.{u, v} ↑n\nhf : ∀ (i : ι), (f i).card ≤ ↑n\nhc : ↑n < ℵ₀\ni : ι\n⊢ f i ≤ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 960,
"column": 2
} | {
"line": 960,
"column": 13
} | {
"line": 960,
"column": 14
} | [
{
"pp": "o : Ordinal.{u}\nn : ℕ\n⊢ lift.{v, u} o < ω_ ↑n ↔ o < ω_ ↑n",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\nn : ℕ\n⊢ lift.{v, u} o < ω_ ↑n ↔ o < ω_ ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 964,
"column": 2
} | {
"line": 964,
"column": 13
} | {
"line": 964,
"column": 14
} | [
{
"pp": "o : Ordinal.{u}\nn : ℕ\n⊢ ω_ ↑n = lift.{v, u} o ↔ ω_ ↑n = o",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"o : Ordinal.{u}\nn : ℕ\n⊢ ω_ ↑n = lift.{v, u} o ↔ ω_ ↑n = o"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 13
} | {
"line": 103,
"column": 14
} | [
{
"pp": "ι : Type u_1\nc : Cardinal.{u_1}\nf : ι → Ordinal.{u_1}\nhι : Cardinal.lift.{?u.23, u_1} #ι ≤ Cardinal.lift.{?u.23, u_1} c\nhf : ∀ (i : ι), (f i).card ≤ c\n⊢ (⨆ i, f i).card ≤ c",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nc : Cardinal.{u_1}\nf : ι → Ordinal.{u_1}\nhι : Cardinal.lift.{?u.23, u_1} #ι ≤ Cardinal.lift.{?u.23, u_1} c\nhf : ∀ (i : ι), (f i).card ≤ c\n⊢ (⨆ i, f i).card ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 228,
"column": 8
} | {
"line": 228,
"column": 44
} | {
"line": 228,
"column": 45
} | [
{
"pp": "c : Cardinal.{u_1}\nh : ℵ₀ ≤ c\n⊢ c + c ≤ c",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"c : Cardinal.{u_1}\nh : ℵ₀ ≤ c\n⊢ c + c ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 13
} | {
"line": 116,
"column": 14
} | [
{
"pp": "o : Ordinal.{u_1}\nc : Cardinal.{u_1}\nf : ↑(Iio o) → Ordinal.{u_1}\nhι : Cardinal.lift.{?u.27, u_1} o.card ≤ Cardinal.lift.{?u.27, u_1} c\nhf : ∀ (i : ↑(Iio o)), (f i).card ≤ c\n⊢ (⨆ i, f i).card ≤ c",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"o : Ordinal.{u_1}\nc : Cardinal.{u_1}\nf : ↑(Iio o) → Ordinal.{u_1}\nhι : Cardinal.lift.{?u.27, u_1} o.card ≤ Cardinal.lift.{?u.27, u_1} c\nhf : ∀ (i : ↑(Iio o)), (f i).card ≤ c\n⊢ (⨆ i, f i).card ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 128,
"column": 12
} | {
"line": 128,
"column": 23
} | {
"line": 128,
"column": 24
} | [
{
"pp": "case zero\na : Ordinal.{u_1}\nha : ω ≤ a\n⊢ (a ^ 0).card ≤ max a.card (card 0)",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Lattice.toSemilatticeSup",
"Cardinal.instOne",
"Ordinal.card_one",
... | [
"case zero\na : Ordinal.{u_1}\nha : ω ≤ a\n⊢ 1 ≤ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.FundamentalSequence | {
"line": 151,
"column": 75
} | {
"line": 151,
"column": 84
} | {
"line": 151,
"column": 84
} | [
{
"pp": "a✝ : Ordinal.{?u.2}\nb : Ordinal.{?u.4}\no✝ : Ordinal.{?u.6}\na o : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{u}\nhf : a.IsFundamentalSequence o f\ni : Ordinal.{u}\nhi : i < a.cof.ord\n⊢ a.cof.ord ≤ o",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"a✝ : Ordinal.{?u.2}\nb : Ordinal.{?u.4}\no✝ : Ordinal.{?u.6}\na o : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{u}\nhf : a.IsFundamentalSequence o f\ni : Ordinal.{u}\nhi : i < a.cof.ord\n⊢ o ≤ o"
] | hf.cof_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 26
} | {
"line": 290,
"column": 27
} | [
{
"pp": "β : Type v\nf : β → Ordinal.{u}\na : Ordinal.{u}\nha : Cardinal.lift.{u + 1, max u v} (Cardinal.lift.{u, v} #β) < Cardinal.lift.{u + 1, max u v} (lift.{v, u} a).cof\nhf : ∀ (i : β), f i < a\n⊢ Cardinal.lift.{u + 1, v} #β < Cardinal.lift.{v, u + 1} (lift.{u + 1, u} a).cof",
"ppTerm": "?m.66",
"a... | [
"β : Type v\nf : β → Ordinal.{u}\na : Ordinal.{u}\nha : Cardinal.lift.{u + 1, max u v} (Cardinal.lift.{u, v} #β) < Cardinal.lift.{u + 1, max u v} (lift.{v, u} a).cof\nhf : ∀ (i : β), f i < a\n⊢ Cardinal.lift.{u + 1, v} #β < Cardinal.lift.{max (u + 1) v, u} a.cof"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Ordinal.FundamentalSequence | {
"line": 185,
"column": 8
} | {
"line": 185,
"column": 17
} | {
"line": 185,
"column": 17
} | [
{
"pp": "case refine_1\na o o' : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{u}\nhf : a.IsFundamentalSequence o f\ng : (b : Ordinal.{u}) → b < o' → Ordinal.{u}\nhg : o.IsFundamentalSequence o' g\n⊢ o' ≤ a.cof.ord",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"case refine_1\na o o' : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{u}\nhf : a.IsFundamentalSequence o f\ng : (b : Ordinal.{u}) → b < o' → Ordinal.{u}\nhg : o.IsFundamentalSequence o' g\n⊢ o' ≤ o"
] | hf.cof_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 311,
"column": 16
} | {
"line": 311,
"column": 31
} | {
"line": 311,
"column": 31
} | [
{
"pp": "a b : Cardinal.{u_1}\n⊢ b + a = b ↔ max ℵ₀ a ≤ b ∨ a = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
"SemilatticeSup.toMa... | [
"a b : Cardinal.{u_1}\n⊢ max ℵ₀ a ≤ b ∨ a = 0 ↔ max ℵ₀ a ≤ b ∨ a = 0"
] | add_eq_left_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 13
} | {
"line": 295,
"column": 14
} | [
{
"pp": "α : Type u\nf : α → Ordinal.{u}\na : Ordinal.{u}\nha : Cardinal.lift.{u, u} #α < (lift.{u, u} a).cof\nhf : ∀ (i : α), f i < a\n⊢ ⨆ i, f i + 1 < a",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nf : α → Ordinal.{u}\na : Ordinal.{u}\nha : Cardinal.lift.{u, u} #α < (lift.{u, u} a).cof\nhf : ∀ (i : α), f i < a\n⊢ ⨆ i, f i + 1 < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 13
} | {
"line": 304,
"column": 14
} | [
{
"pp": "α : Type u\nf : α → Ordinal.{u}\na : Ordinal.{u}\nha : Cardinal.lift.{u, u} #α < (lift.{u, u} a).cof\nhf : ∀ (i : α), f i < a\n⊢ ⨆ i, f i < a",
"ppTerm": "?m.29",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nf : α → Ordinal.{u}\na : Ordinal.{u}\nha : Cardinal.lift.{u, u} #α < (lift.{u, u} a).cof\nhf : ∀ (i : α), f i < a\n⊢ ⨆ i, f i < a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 13
} | {
"line": 312,
"column": 14
} | [
{
"pp": "α : Type u\nf : α → Ordinal.{u}\n⊢ (⨆ i, f i + 1).cof ≤ #α",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nf : α → Ordinal.{u}\n⊢ (⨆ i, f i + 1).cof ≤ #α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 53,
"column": 19
} | {
"line": 53,
"column": 28
} | {
"line": 53,
"column": 28
} | [
{
"pp": "o : Ordinal.{u_1}\nH : (ℵ_ o).IsRegular\n⊢ (ℵ_ o).ord.cof = ℵ_ o",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal.aleph",
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"... | [
"o : Ordinal.{u_1}\nH : (ℵ_ o).IsRegular\n⊢ ℵ_ o = ℵ_ o"
] | H.cof_ord | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 34
} | {
"line": 81,
"column": 2
} | [
{
"pp": "c : Cardinal.{u_1}\nhc : ℵ₀ ≤ c\n⊢ (succ c).IsRegular",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Order.succ",
"Cardinal",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Cardinal.instSuccOrder",
"Cardinal.aleph0",
"LE.le",
"Cardina... | [
"c : Cardinal.{u_1}\nhc : ℵ₀ ≤ c\nhc₀ : ℵ₀ ≤ succ c\n⊢ (succ c).IsRegular"
] | have hc₀ := hc.trans (le_succ c) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 15
} | {
"line": 87,
"column": 16
} | [
{
"pp": "case le_cof_ord.refine_1\nc : Cardinal.{u_1}\nhc : ℵ₀ ≤ c\nhc₀ : ℵ₀ ≤ succ c\nhc' : (succ c).ord.cof < succ c\nf : ↑(Iio (succ c).ord.cof.ord) → ↑(Iio (succ c).ord)\nhf : IsFundamentalSeq f\n⊢ (succ c).ord.cof.ord.card ≤ c",
"ppTerm": "?le_cof_ord.refine_1",
"assigned": true,
"usedConstants... | [
"case le_cof_ord.refine_1\nc : Cardinal.{u_1}\nhc : ℵ₀ ≤ c\nhc₀ : ℵ₀ ≤ succ c\nhc' : (succ c).ord.cof < succ c\nf : ↑(Iio (succ c).ord.cof.ord) → ↑(Iio (succ c).ord)\nhf : IsFundamentalSeq f\n⊢ (succ c).ord.cof ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 13
} | {
"line": 97,
"column": 14
} | [
{
"pp": "⊢ (ω_ 1).cof = ℵ_ 1",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ (ω_ 1).cof = ℵ_ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 144,
"column": 22
} | {
"line": 144,
"column": 33
} | {
"line": 144,
"column": 34
} | [
{
"pp": "κ : Cardinal.{v}\nx✝ : (lift.{u, v} κ).IsRegular\nh₁ : ℵ₀ ≤ lift.{u, v} κ\nh₂ : lift.{u, v} κ ≤ (lift.{u, v} κ).ord.cof\n⊢ ℵ₀ ≤ κ",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"κ : Cardinal.{v}\nx✝ : (lift.{u, v} κ).IsRegular\nh₁ : ℵ₀ ≤ lift.{u, v} κ\nh₂ : lift.{u, v} κ ≤ (lift.{u, v} κ).ord.cof\n⊢ ℵ₀ ≤ κ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 42
} | {
"line": 399,
"column": 2
} | [
{
"pp": "ι : Type u\nf : ι → Ordinal.{max u v}\nH : ∀ (i : ι), f i < iSup f\nhf : Cardinal.lift.{v, u} #ι < (iSup f).cof\n⊢ False",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Ordinal.partialOrder",
"iSup",
"PartialOrder.toPreorder",
"ConditionallyCompleteLinearO... | [
"ι : Type u\nf : ι → Ordinal.{max u v}\nH : ∀ (i : ι), f i < iSup f\nhf : Cardinal.lift.{v, u} #ι < (iSup f).cof\n⊢ Cardinal.lift.{max u v, u} #ι < (lift.{u, max u v} (iSup f)).cof"
] | apply (lift_iSup_lt_of_lt_cof _ H).false | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 24
} | {
"line": 486,
"column": 25
} | [
{
"pp": "a b c : Cardinal.{u_1}\nhc : c < ℵ₀\n⊢ c + a < c + b ↔ a < b",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
"PartialOrder.toPreorder",
... | [
"a b c : Cardinal.{u_1}\nhc : c < ℵ₀\n⊢ a + c < b + c ↔ a < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Pigeonhole | {
"line": 115,
"column": 43
} | {
"line": 115,
"column": 54
} | {
"line": 115,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : ⋃ a, ↑(f a) = Set.univ\nh : #↑(range f) < #β\nb : β\n⊢ ∃ a, b ∈ f a",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : ⋃ a, ↑(f a) = Set.univ\nh : #↑(range f) < #β\nb : β\n⊢ ∃ a, b ∈ f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 444,
"column": 31
} | {
"line": 444,
"column": 42
} | {
"line": 444,
"column": 43
} | [
{
"pp": "f : Ordinal.{u_1} → Ordinal.{u_1}\nc : Ordinal.{u_1}\nhc : ℵ₀ < c.cof\nhf : ∀ i < c, f i < c\na : Ordinal.{u_1}\n⊢ Cardinal.lift.{u_1, 0} #Unit < c.cof",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder.... | [
"f : Ordinal.{u_1} → Ordinal.{u_1}\nc : Ordinal.{u_1}\nhc : ℵ₀ < c.cof\nhf : ∀ i < c, f i < c\na : Ordinal.{u_1}\n⊢ 1 < c.cof"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Pigeonhole | {
"line": 119,
"column": 40
} | {
"line": 119,
"column": 56
} | {
"line": 119,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : ⋃ a, ↑(f a) = Set.univ\nh : #↑(range f) < #β\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b ↦\n Eq.mp (Eq.trans mem_iUnion._simp_1 (congrArg Exists (funext fun i ↦ SetLike.mem_coe._simp_1)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(ran... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : Infinite β\nf : α → Finset β\nw : ⋃ a, ↑(f a) = Set.univ\nh : #↑(range f) < #β\nu : ∀ (b : β), ∃ a, b ∈ f a :=\n fun b ↦\n Eq.mp (Eq.trans mem_iUnion._simp_1 (congrArg Exists (funext fun i ↦ SetLike.mem_coe._simp_1)))\n (Eq.ge w (mem_univ b))\nu' : β → ↑(range f) := fun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 459,
"column": 24
} | {
"line": 459,
"column": 35
} | {
"line": 459,
"column": 36
} | [
{
"pp": "b : Ordinal.{u}\na : Cardinal.{u}\nH : ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = b → a ≤ #ι\no : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{u}\nhf : o.blsub f = b\n⊢ a ≤ o.card",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"b : Ordinal.{u}\na : Cardinal.{u}\nH : ∀ {ι : Type u} (f : ι → Ordinal.{u}), lsub f = b → a ≤ #ι\no : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{u}\nhf : o.blsub f = b\n⊢ a ≤ o.card"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 495,
"column": 6
} | {
"line": 495,
"column": 28
} | {
"line": 495,
"column": 29
} | [
{
"pp": "case inr.hbc\nκ₁ κ₂ μ₁ μ₂ : Cardinal.{u_1}\nhκ : κ₁ < κ₂\nhμ : μ₁ < μ₂\nhfin : κ₂ + μ₂ < ℵ₀\nhfin_ : κ₂ < ℵ₀ ∧ μ₂ < ℵ₀\n⊢ κ₂ + μ₁ < κ₂ + μ₂",
"ppTerm": "?inr.hbc",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr.hbc\nκ₁ κ₂ μ₁ μ₂ : Cardinal.{u_1}\nhκ : κ₁ < κ₂\nhμ : μ₁ < μ₂\nhfin : κ₂ + μ₂ < ℵ₀\nhfin_ : κ₂ < ℵ₀ ∧ μ₂ < ℵ₀\n⊢ κ₂ + μ₁ < κ₂ + μ₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 462,
"column": 6
} | {
"line": 462,
"column": 17
} | {
"line": 462,
"column": 18
} | [
{
"pp": "b : Ordinal.{u}\na : Cardinal.{u}\nH : ∀ {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), o.blsub f = b → a ≤ o.card\nι : Type u\nf : ι → Ordinal.{u}\nr : ι → ι → Prop\nhr : IsWellOrder ι r\nhf : (type r).blsub (bfamilyOfFamily' r f) = b\nhι' : (#ι).ord = type r\n⊢ a ≤ #ι",
"ppTerm"... | [
"b : Ordinal.{u}\na : Cardinal.{u}\nH : ∀ {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), o.blsub f = b → a ≤ o.card\nι : Type u\nf : ι → Ordinal.{u}\nr : ι → ι → Prop\nhr : IsWellOrder ι r\nhf : (type r).blsub (bfamilyOfFamily' r f) = b\nhι' : (#ι).ord = type r\n⊢ a ≤ #ι"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 273,
"column": 23
} | {
"line": 273,
"column": 34
} | {
"line": 273,
"column": 35
} | [
{
"pp": "case limit.ha\nι : Type u\nf : ι → Ordinal.{max u v} → Ordinal.{max u v}\nc : Cardinal.{max u v}\nhc : c.IsRegular\nhι : lift.{v, u} #ι < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι), ∀ b < c.ord, f i b < c.ord\nhω : ℵ₀ < c.ord.cof\nb : Ordinal.{max u v}\nhb : IsSuccLimit b\nH : ∀ o' < b, o' < c.ord → derivFamily f... | [
"case limit.ha\nι : Type u\nf : ι → Ordinal.{max u v} → Ordinal.{max u v}\nc : Cardinal.{max u v}\nhc : c.IsRegular\nhι : lift.{v, u} #ι < c\nhc' : c ≠ ℵ₀\nhf : ∀ (i : ι), ∀ b < c.ord, f i b < c.ord\nhω : ℵ₀ < c.ord.cof\nb : Ordinal.{max u v}\nhb : IsSuccLimit b\nH : ∀ o' < b, o' < c.ord → derivFamily f o' < c.ord\... | hc.cof_ord, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 515,
"column": 20
} | {
"line": 515,
"column": 42
} | {
"line": 515,
"column": 43
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nx✝¹ x✝ : Cardinal.{u_1}\nhlt : x✝¹ < x✝\n⊢ (fun a ↦ a * ↑n) x✝¹ < (fun a ↦ a * ↑n) x✝",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"HMul.hMul",
"Cardinal",
"PartialOrder.toPreorder",
"Cardinal.instMul",
"id... | [
"n : ℕ\nhn : n ≠ 0\nx✝¹ x✝ : Cardinal.{u_1}\nhlt : x✝¹ < x✝\n⊢ x✝¹ * ↑n < x✝ * ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 479,
"column": 18
} | {
"line": 479,
"column": 49
} | {
"line": 479,
"column": 50
} | [
{
"pp": "o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}\nc : Ordinal.{max u v}\nho : Cardinal.lift.{v, u} o.card < c.cof\nhf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c\nh : o.blsub f = c\n⊢ c.cof ≤ Cardinal.lift.{v, u} o.card",
"ppTerm": "?m.24",
"assigned": true,
"usedConsta... | [
"o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}\nc : Ordinal.{max u v}\nho : Cardinal.lift.{v, u} o.card < c.cof\nhf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c\nh : o.blsub f = c\n⊢ c.cof ≤ (lift.{v, u} o).card"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 284,
"column": 8
} | {
"line": 284,
"column": 19
} | {
"line": 284,
"column": 20
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\nc : Cardinal.{u}\nhc : c.IsRegular\nhc' : c ≠ ℵ₀\nhf : ∀ i < c.ord, f i < c.ord\na : Ordinal.{u}\n⊢ lift.{u, 0} #Unit < c",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder... | [
"f : Ordinal.{u} → Ordinal.{u}\nc : Cardinal.{u}\nhc : c.IsRegular\nhc' : c ≠ ℵ₀\nhf : ∀ i < c.ord, f i < c.ord\na : Ordinal.{u}\n⊢ 1 < c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 479,
"column": 18
} | {
"line": 479,
"column": 76
} | {
"line": 479,
"column": 76
} | [
{
"pp": "o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}\nc : Ordinal.{max u v}\nho : Cardinal.lift.{v, u} o.card < c.cof\nhf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c\nh : o.blsub f = c\n⊢ c.cof ≤ Cardinal.lift.{v, u} o.card",
"ppTerm": "?m.24",
"assigned": true,
"usedConsta... | [] | simpa [← iSup_ord, hf, h] using cof_blsub_le_lift.{u, v} f | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 479,
"column": 18
} | {
"line": 479,
"column": 76
} | {
"line": 479,
"column": 76
} | [
{
"pp": "o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}\nc : Ordinal.{max u v}\nho : Cardinal.lift.{v, u} o.card < c.cof\nhf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c\nh : o.blsub f = c\n⊢ c.cof ≤ Cardinal.lift.{v, u} o.card",
"ppTerm": "?m.24",
"assigned": true,
"usedConsta... | [] | simpa [← iSup_ord, hf, h] using cof_blsub_le_lift.{u, v} f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 479,
"column": 18
} | {
"line": 479,
"column": 76
} | {
"line": 479,
"column": 76
} | [
{
"pp": "o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}\nc : Ordinal.{max u v}\nho : Cardinal.lift.{v, u} o.card < c.cof\nhf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c\nh : o.blsub f = c\n⊢ c.cof ≤ Cardinal.lift.{v, u} o.card",
"ppTerm": "?m.24",
"assigned": true,
"usedConsta... | [] | simpa [← iSup_ord, hf, h] using cof_blsub_le_lift.{u, v} f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 484,
"column": 29
} | {
"line": 484,
"column": 43
} | {
"line": 484,
"column": 43
} | [
{
"pp": "o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{u}\nc : Ordinal.{u}\nho : o.card < c.cof\nhf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c\n⊢ Cardinal.lift.{u, u} o.card < c.cof",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
... | [
"o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{u}\nc : Ordinal.{u}\nho : o.card < c.cof\nhf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c\n⊢ o.card < c.cof"
] | o.card.lift_id | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 442,
"column": 2
} | {
"line": 442,
"column": 13
} | {
"line": 442,
"column": 14
} | [
{
"pp": "⊢ preBeth Ordinal.univ.{u, v} = univ.{u, v}",
"ppTerm": "?m.2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ preBeth Ordinal.univ.{u, v} = univ.{u, v}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 446,
"column": 2
} | {
"line": 446,
"column": 13
} | {
"line": 446,
"column": 14
} | [
{
"pp": "⊢ ℶ_ Ordinal.univ.{u, v} = univ.{u, v}",
"ppTerm": "?m.2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ ℶ_ Ordinal.univ.{u, v} = univ.{u, v}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 507,
"column": 28
} | {
"line": 507,
"column": 42
} | {
"line": 507,
"column": 42
} | [
{
"pp": "o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{u}\nc : Ordinal.{u}\nho : o.card < c.cof\n⊢ Cardinal.lift.{u, u} o.card < c.cof",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"congrArg",
"PartialOrder.... | [
"o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{u}\nc : Ordinal.{u}\nho : o.card < c.cof\n⊢ o.card < c.cof"
] | o.card.lift_id | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 13
} | {
"line": 450,
"column": 14
} | [
{
"pp": "⊢ preAleph Ordinal.univ.{u, v} = univ.{u, v}",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ preAleph Ordinal.univ.{u, v} = univ.{u, v}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 458,
"column": 2
} | {
"line": 458,
"column": 13
} | {
"line": 458,
"column": 14
} | [
{
"pp": "⊢ ℵ_ Ordinal.univ.{u, v} = univ.{u, v}",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ ℵ_ Ordinal.univ.{u, v} = univ.{u, v}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 462,
"column": 2
} | {
"line": 462,
"column": 13
} | {
"line": 462,
"column": 14
} | [
{
"pp": "⊢ preOmega Ordinal.univ.{u, v} = Ordinal.univ.{u, v}",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ preOmega Ordinal.univ.{u, v} = Ordinal.univ.{u, v}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 466,
"column": 2
} | {
"line": 466,
"column": 13
} | {
"line": 466,
"column": 14
} | [
{
"pp": "⊢ ω_ Ordinal.univ.{u, v} = Ordinal.univ.{u, v}",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ ω_ Ordinal.univ.{u, v} = Ordinal.univ.{u, v}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.Cardinality | {
"line": 86,
"column": 45
} | {
"line": 86,
"column": 60
} | {
"line": 86,
"column": 61
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Nontrivial R\ninst✝ : Module R M\nι : Type w\nb : Basis ι R M\nκ : Type w'\nv : κ → M\nind : LinearIndependent R v\nm : ind.Maximal\ni : ι\nw : ∀ (x : κ), (b.repr (v x)) i = 0\nrepr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linea... | [
"R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Nontrivial R\ninst✝ : Module R M\nι : Type w\nb : Basis ι R M\nκ : Type w'\nv : κ → M\nind : LinearIndependent R v\nm : ind.Maximal\ni : ι\nw : ∀ (x : κ), (b.repr (v x)) i = 0\nrepr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearCombination... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 595,
"column": 24
} | {
"line": 595,
"column": 40
} | {
"line": 595,
"column": 41
} | [
{
"pp": "a : Cardinal.{u}\nha : a ≤ ℵ₀\nb : ℕ\n⊢ a ^ ↑b ≤ ℵ₀",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Cardinal.instPowCardinal",
"Cardinal",
"id",
"Cardinal.aleph0",
"LE.le",
"Nat.cast",
"Cardinal.instLE",
"HPow.hPow",
"instHPow... | [
"a : Cardinal.{u}\nha : a ≤ ℵ₀\nb : ℕ\n⊢ a ^ b ≤ ℵ₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Basis.Cardinality | {
"line": 100,
"column": 6
} | {
"line": 100,
"column": 32
} | {
"line": 100,
"column": 33
} | [
{
"pp": "case none\nR : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Nontrivial R\ninst✝ : Module R M\nι : Type w\nb : Basis ι R M\nκ : Type w'\nv : κ → M\nind : LinearIndependent R v\nm : ind.Maximal\ni : ι\nw : ∀ (x : κ), (b.repr (v x)) i = 0\nrepr_eq_zero : ∀ (l : κ →₀ R), (b.r... | [
"case none\nR : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Nontrivial R\ninst✝ : Module R M\nι : Type w\nb : Basis ι R M\nκ : Type w'\nv : κ → M\nind : LinearIndependent R v\nm : ind.Maximal\ni : ι\nw : ∀ (x : κ), (b.repr (v x)) i = 0\nrepr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linear... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 620,
"column": 6
} | {
"line": 620,
"column": 21
} | {
"line": 620,
"column": 22
} | [
{
"pp": "α : Type u\nβ' : Type v\n⊢ #(α ≃ β') = 0 ↔ lift.{v, u} #α ≠ lift.{u, v} #β'",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Cardinal.lift",
"Cardinal.mk",
"id",
"Equiv",
"Ne",
"IsEmpty",
... | [
"α : Type u\nβ' : Type v\n⊢ IsEmpty (α ≃ β') ↔ lift.{v, u} #α ≠ lift.{u, v} #β'"
] | mk_eq_zero_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 631,
"column": 6
} | {
"line": 631,
"column": 21
} | {
"line": 631,
"column": 22
} | [
{
"pp": "α : Type u\nβ' : Type v\n⊢ #(α ↪ β') = 0 ↔ lift.{u, v} #β' < lift.{v, u} #α",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Cardinal.lift",
"Cardinal.mk",
... | [
"α : Type u\nβ' : Type v\n⊢ IsEmpty (α ↪ β') ↔ lift.{u, v} #β' < lift.{v, u} #α"
] | mk_eq_zero_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 637,
"column": 11
} | {
"line": 637,
"column": 26
} | {
"line": 637,
"column": 27
} | [
{
"pp": "α : Type u\nβ' : Type v\n⊢ #(α → β') = 0 ↔ #α ≠ 0 ∧ #β' = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.SetTheory.Cardinal.Arithmetic.0.Cardinal.mk_arrow_eq_zero_iff._simp_1_1",
"Cardinal",
"congrArg",
"Cardinal.mk",
... | [
"α : Type u\nβ' : Type v\n⊢ IsEmpty (α → β') ↔ #α ≠ 0 ∧ IsEmpty β'"
] | mk_eq_zero_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 643,
"column": 27
} | {
"line": 643,
"column": 42
} | {
"line": 643,
"column": 43
} | [
{
"pp": "α : Type u\nβ' : Type v\n⊢ Nonempty ↑{f | Surjective f} ↔ (#α = 0 ∨ Nonempty β') ∧ Nonempty (β' ↪ α)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"setOf",
"Cardinal.mk",
"Set.Elem",
"id",
"IsEmpty... | [
"α : Type u\nβ' : Type v\n⊢ Nonempty ↑{f | Surjective f} ↔ (IsEmpty α ∨ Nonempty β') ∧ Nonempty (β' ↪ α)"
] | mk_eq_zero_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Basis.Cardinality | {
"line": 106,
"column": 16
} | {
"line": 106,
"column": 46
} | {
"line": 106,
"column": 47
} | [
{
"pp": "case inr\nR : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Nontrivial R\ninst✝ : Module R M\nι : Type w\nb : Basis ι R M\nκ : Type w'\nv : κ → M\nind : LinearIndependent R v\nm : ind.Maximal\ni : ι\nw : ∀ (x : κ), (b.repr (v x)) i = 0\nrepr_eq_zero : ∀ (l : κ →₀ R), (b.re... | [
"case inr\nR : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Nontrivial R\ninst✝ : Module R M\nι : Type w\nb : Basis ι R M\nκ : Type w'\nv : κ → M\nind : LinearIndependent R v\nm : ind.Maximal\ni : ι\nw : ∀ (x : κ), (b.repr (v x)) i = 0\nrepr_eq_zero : ∀ (l : κ →₀ R), (b.repr ((linearC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 583,
"column": 6
} | {
"line": 583,
"column": 46
} | {
"line": 583,
"column": 47
} | [
{
"pp": "case a.refine_2\nα : Type u_1\nh : (#α).IsStrongPrelimit\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nhr : (#α).ord = type r\nha✝ : #α ≠ 0\nh' : (#α).IsStrongLimit\nha : ℵ₀ ≤ #α\na b : α\nhab : (fun x ↦ ⟨{x}, ⋯⟩) a = (fun x ↦ ⟨{x}, ⋯⟩) b\n⊢ a = b",
"ppTerm": "?a.refine_2✝",
"assigned": false,
... | [
"case a.refine_2\nα : Type u_1\nh : (#α).IsStrongPrelimit\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nhr : (#α).ord = type r\nha✝ : #α ≠ 0\nh' : (#α).IsStrongLimit\nha : ℵ₀ ≤ #α\na b : α\nhab : (fun x ↦ ⟨{x}, ⋯⟩) a = (fun x ↦ ⟨{x}, ⋯⟩) b\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 607,
"column": 6
} | {
"line": 607,
"column": 46
} | {
"line": 607,
"column": 47
} | [
{
"pp": "case a.refine_2\nα : Type u_1\nh : (#α).IsStrongPrelimit\nha : #α ≠ 0\nh' : (#α).IsStrongLimit\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : (#α).ord = type r\nthis : LinearOrder α := linearOrderOfSTO r\na b : α\nhab : (fun x ↦ ⟨{x}, ⋯⟩) a = (fun x ↦ ⟨{x}, ⋯⟩) b\n⊢ a = b",
"ppTerm": "?a.refine_2✝",... | [
"case a.refine_2\nα : Type u_1\nh : (#α).IsStrongPrelimit\nha : #α ≠ 0\nh' : (#α).IsStrongLimit\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : (#α).ord = type r\nthis : LinearOrder α := linearOrderOfSTO r\na b : α\nhab : (fun x ↦ ⟨{x}, ⋯⟩) a = (fun x ↦ ⟨{x}, ⋯⟩) b\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 591,
"column": 2
} | {
"line": 607,
"column": 50
} | {
"line": 609,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nh : (#α).IsStrongPrelimit\nha : #α ≠ 0\nh' : (#α).IsStrongLimit\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : (#α).ord = type r\n⊢ #{ s // #↑s < (#α).ord.cof } = #α",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal.isSuccLimit_o... | [] | classical
letI := linearOrderOfSTO r
apply le_antisymm
· conv_rhs => rw [← mk_bounded_subset h hr]
apply mk_subtype_le_of_subset
intro s hs
rw [hr] at hs
contrapose! hs
rw [not_bounded_iff] at hs
apply cof_le
simp_rw [IsCofinal, ← not_lt]
exact hs
· refine @mk_le_of_injective α _... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Data.DFinsupp.Defs | {
"line": 539,
"column": 20
} | {
"line": 539,
"column": 32
} | {
"line": 539,
"column": 32
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : DecidableEq ι\nf : Π₀ (i : ι), β i\ni i✝ : ι\n⊢ (if i = i✝ then f i✝ else 0) = (single i (f i)) i✝",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Eq.recOn",
"congrArg",
"DFinsupp... | [
"ι : Type u\nβ : ι → Type v\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : DecidableEq ι\nf : Π₀ (i : ι), β i\ni i✝ : ι\n⊢ (if i = i✝ then f i✝ else 0) = if h : i = i✝ then Eq.recOn h (f i) else 0"
] | single_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.DFinsupp.Sigma | {
"line": 94,
"column": 6
} | {
"line": 94,
"column": 17
} | {
"line": 94,
"column": 18
} | [
{
"pp": "case inl.inr\nι : Type u\nα : ι → Type u_2\nδ : (i : ι) → α i → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → DecidableEq (α i)\ninst✝ : (i : ι) → (j : α i) → Zero (δ i j)\ni : ι\nj : α i\nx : δ ⟨i, j⟩.fst ⟨i, j⟩.snd\nj' : α i\nhj : j' ≠ j\n⊢ ⟨i, j'⟩ ≠ ⟨i, j⟩",
"ppTerm": "?inl.inr✝",
"assi... | [
"case inl.inr\nι : Type u\nα : ι → Type u_2\nδ : (i : ι) → α i → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → DecidableEq (α i)\ninst✝ : (i : ι) → (j : α i) → Zero (δ i j)\ni : ι\nj : α i\nx : δ ⟨i, j⟩.fst ⟨i, j⟩.snd\nj' : α i\nhj : j' ≠ j\n⊢ ¬j' = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.Sigma | {
"line": 152,
"column": 6
} | {
"line": 152,
"column": 17
} | {
"line": 152,
"column": 18
} | [
{
"pp": "case inl.inr\nι : Type u\nα : ι → Type u_2\nδ : (i : ι) → α i → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → (j : α i) → Zero (δ i j)\ninst✝ : (i : ι) → DecidableEq (α i)\ni : ι\nj : α i\nx : δ i j\nj' : α i\nhj : j' ≠ j\n⊢ ⟨i, j'⟩ ≠ ⟨i, j⟩",
"ppTerm": "?inl.inr✝",
"assigned": true,
"... | [
"case inl.inr\nι : Type u\nα : ι → Type u_2\nδ : (i : ι) → α i → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → (j : α i) → Zero (δ i j)\ninst✝ : (i : ι) → DecidableEq (α i)\ni : ι\nj : α i\nx : δ i j\nj' : α i\nhj : j' ≠ j\n⊢ ¬j' = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.Sigma | {
"line": 163,
"column": 8
} | {
"line": 163,
"column": 27
} | {
"line": 163,
"column": 28
} | [
{
"pp": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\nκ : Type u_1\nα : ι → Type u_2\nδ : (i : ι) → α i → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → (j : α i) → Zero (δ i j)\nf : Π₀ (i : (x : ι) × α x), δ i.fst i.snd\ni : ι\nj : α i\n⊢ f.sigmaCurry.sigmaUncurry ⟨i, j⟩ = f ⟨... | [
"ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\nκ : Type u_1\nα : ι → Type u_2\nδ : (i : ι) → α i → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → (j : α i) → Zero (δ i j)\nf : Π₀ (i : (x : ι) × α x), δ i.fst i.snd\ni : ι\nj : α i\n⊢ (f.sigmaCurry i) j = f ⟨i, j⟩"
] | sigmaUncurry_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.DFinsupp.BigOperators | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 81
} | {
"line": 185,
"column": 82
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝⁵ : DecidableEq ι\nι₁ : Type u₁\ninst✝⁴ : DecidableEq ι₁\nβ₁ : ι₁ → Type v₁\ninst✝³ : (i₁ : ι₁) → Zero (β₁ i₁)\ninst✝² : (i : ι₁) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → AddCommMonoid (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i₁ : ι₁),... | [
"ι : Type u\nβ : ι → Type v\ninst✝⁵ : DecidableEq ι\nι₁ : Type u₁\ninst✝⁴ : DecidableEq ι₁\nβ₁ : ι₁ → Type v₁\ninst✝³ : (i₁ : ι₁) → Zero (β₁ i₁)\ninst✝² : (i : ι₁) → (x : β₁ i) → Decidable (x ≠ 0)\ninst✝¹ : (i : ι) → AddCommMonoid (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i₁ : ι₁), β₁ i₁\ng : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.Defs | {
"line": 824,
"column": 32
} | {
"line": 824,
"column": 43
} | {
"line": 824,
"column": 44
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : (i : ι) → β i\ns : Multiset ι\nh : ∀ (i : ι), i ∈ s ∨ f i = 0\ni : ι\nH : i ∈ { toFun := f, support' := Trunc.mk ⟨s, h⟩ }.support\n⊢ i ∈ s.toFinset.val.toFinset",
... | [
"ι : Type u\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : (i : ι) → β i\ns : Multiset ι\nh : ∀ (i : ι), i ∈ s ∨ f i = 0\ni : ι\nH : i ∈ { toFun := f, support' := Trunc.mk ⟨s, h⟩ }.support\n⊢ i ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.BigOperators | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 48
} | {
"line": 238,
"column": 2
} | [
{
"pp": "ι : Type u\nγ : Type w\nβ : ι → Type v\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nv : Π₀ (i : ι), β i\nf : (i : ι) → β i → γ\nhf : ∀ (i : ι), f i 0 = 1\n⊢ ∏ i ∈ v.support, f i (v i) = ∏ i, f i (v i)... | [
"ι : Type u\nγ : Type w\nβ : ι → Type v\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nv : Π₀ (i : ι), β i\nf : (i : ι) → β i → γ\nhf : ∀ (i : ι), f i 0 = 1\n⊢ ∀ x ∈ Finset.univ, x ∉ v.support → f x (v x) = 1"
] | apply Finset.prod_subset v.support.subset_univ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.DFinsupp.Defs | {
"line": 830,
"column": 2
} | {
"line": 830,
"column": 37
} | {
"line": 831,
"column": 2
} | [
{
"pp": "case h\nι : Type u\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ni : ι\nf : (i : ι) → β i\ns : { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\n⊢ i ∈ { toFun := f, support' := Trunc.mk s }.support ↔ { toFun := f, support' := Trunc.mk s } i ... | [
"case h\nι : Type u\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ni : ι\nf : (i : ι) → β i\ns : { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\n⊢ i ∈ {i ∈ (↑s).toFinset | { toFun := f, support' := Trunc.mk s } i ≠ 0} ↔ { toFun := f, support' := Trunc.m... | dsimp only [support, Trunc.lift_mk] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Data.DFinsupp.Defs | {
"line": 851,
"column": 4
} | {
"line": 851,
"column": 15
} | {
"line": 851,
"column": 16
} | [
{
"pp": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\ni : ι\n⊢ ↑((fun f_1 ↦ ⟨mk f.support fun i ↦ ↑(f_1 i), ⋯⟩)\n ((fun x ↦\n match (m... | [
"ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\ni : ι\n⊢ f i = 0 → 0 = f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.Defs | {
"line": 877,
"column": 22
} | {
"line": 877,
"column": 50
} | {
"line": 877,
"column": 51
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\nH : f.support = ∅\n⊢ ∀ (i : ι), f i = 0 i",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"DFinsupp.instDFunLike",
"... | [
"ι : Type u\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\nH : f.support = ∅\n⊢ ∀ (i : ι), f i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.Defs | {
"line": 887,
"column": 8
} | {
"line": 887,
"column": 39
} | {
"line": 888,
"column": 8
} | [
{
"pp": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : (i : ι) → (x : β i) → Decidable (x = 0)\nf : Π₀ (i : ι), β i\ns : { s // ∀ (i : ι), i ∈ s ∨ f.toFun i = 0 }\nhs₁ :... | [
"ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : (i : ι) → (x : β i) → Decidable (x = 0)\nf : Π₀ (i : ι), β i\ns : { s // ∀ (i : ι), i ∈ s ∨ f.toFun i = 0 }\nhs₁ : ∀ i ∈ ↑s, f... | letI := Classical.propDecidable | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.Data.DFinsupp.Defs | {
"line": 893,
"column": 2
} | {
"line": 893,
"column": 30
} | {
"line": 893,
"column": 31
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ns : Set ι\nf : Π₀ (i : ι), β i\n⊢ ↑f.support ⊆ s ↔ ∀ i ∉ s, f i = 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.me... | [
"ι : Type u\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ns : Set ι\nf : Π₀ (i : ι), β i\n⊢ (∀ (x : ι), ¬f x = 0 → x ∈ s) ↔ ∀ i ∉ s, f i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.Defs | {
"line": 918,
"column": 4
} | {
"line": 921,
"column": 18
} | {
"line": 923,
"column": 0
} | [
{
"pp": "ι : Type u\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ni : ι\nb : β₁ i\ni' : ι\n⊢ (mapRange f hf (single i b)) i' = (single i (f i b)) i'",
"ppTerm": "?m.35",
"a... | [] | by_cases h : i = i'
· subst i'
simp
· simp [h, hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.DFinsupp.Defs | {
"line": 918,
"column": 4
} | {
"line": 921,
"column": 18
} | {
"line": 923,
"column": 0
} | [
{
"pp": "ι : Type u\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\ni : ι\nb : β₁ i\ni' : ι\n⊢ (mapRange f hf (single i b)) i' = (single i (f i b)) i'",
"ppTerm": "?m.35",
"a... | [] | by_cases h : i = i'
· subst i'
simp
· simp [h, hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.DFinsupp.Defs | {
"line": 927,
"column": 4
} | {
"line": 927,
"column": 15
} | {
"line": 927,
"column": 16
} | [
{
"pp": "ι : Type u\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\nh : Function.Injective (mapRange f hf)\ni : ι\nx y : β₁ i\neq : f i x = f i y\n⊢ mapRange f hf (single i x) = mapRange f hf (single i y)",... | [
"ι : Type u\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\nh : Function.Injective (mapRange f hf)\ni : ι\nx y : β₁ i\neq : f i x = f i y\n⊢ single i (f i x) = single i (f i y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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