ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Ordinal.nfpFamily_lt_ord
theorem nfpFamily_lt_ord {ι} {f : ι → Ordinal → Ordinal} {c} (hc : ℵ₀ < cof c) (hc' : #ι < cof c)
    (hf : ∀ (i), ∀ b < c, f i b < c) {a} : a < c → nfpFamily.{u, u} f a < c 
Arguments: {ι : Type u} {f : ι → Ordinal.{u} → Ordinal.{u}} {c : Ordinal.{u}} (hc : ℵ₀ < c.cof) (hc' : #ι < c.cof) (hf : ∀ (i : ι), ∀ b < c, f i b < c) {a : Ordinal.{u}}
Goal: a < c → Ordinal.nfpFamily f a < c
-- Ordinal.nfpBFamily_lt_ord_lift
theorem nfpBFamily_lt_ord_lift {o : Ordinal} {f : ∀ a < o, Ordinal → Ordinal} {c} (hc : ℵ₀ < cof c)
    (hc' : Cardinal.lift.{v, u} o.card < cof c) (hf : ∀ (i hi), ∀ b < c, f i hi b < c) {a} :
    a < c → nfpBFamily.{u, v} o f a < c 
Arguments: {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v} → Ordinal.{max u v}} {c : Ordinal.{max u v}} (hc : ℵ₀ < c.cof) (hc' : Cardinal.lift.{v, u} o.card < c.cof) (hf : ∀ (i : Ordinal.{u}) (hi : i < o), ∀ b < c, f i hi b < c) {a : Ordinal.{max u v}}
Goal: a < c → o.nfpBFamily f a < c
-- Ordinal.nfpBFamily_lt_ord
theorem nfpBFamily_lt_ord {o : Ordinal} {f : ∀ a < o, Ordinal → Ordinal} {c} (hc : ℵ₀ < cof c)
    (hc' : o.card < cof c) (hf : ∀ (i hi), ∀ b < c, f i hi b < c) {a} :
    a < c → nfpBFamily.{u, u} o f a < c 
Arguments: {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{u} → Ordinal.{u}} {c : Ordinal.{u}} (hc : ℵ₀ < c.cof) (hc' : o.card < c.cof) (hf : ∀ (i : Ordinal.{u}) (hi : i < o), ∀ b < c, f i hi b < c) {a : Ordinal.{u}}
Goal: a < c → o.nfpBFamily f a < c
-- Ordinal.nfp_lt_ord
theorem nfp_lt_ord {f : Ordinal → Ordinal} {c} (hc : ℵ₀ < cof c) (hf : ∀ i < c, f i < c) {a} :
    a < c → nfp f a < c 
Arguments: {f : Ordinal.{u_2} → Ordinal.{u_2}} {c : Ordinal.{u_2}} (hc : ℵ₀ < c.cof) (hf : ∀ i < c, f i < c) {a : Ordinal.{u_2}}
Goal: a < c → Ordinal.nfp f a < c
-- Ordinal.exists_blsub_cof
theorem exists_blsub_cof (o : Ordinal) :
    ∃ f : ∀ a < (cof o).ord, Ordinal, blsub.{u, u} _ f = o 
Arguments: (o : Ordinal.{u})
Goal: ∃ f, o.cof.ord.blsub f = o
-- Ordinal.le_cof_iff_blsub
theorem le_cof_iff_blsub {b : Ordinal} {a : Cardinal} :
    a ≤ cof b ↔ ∀ {o} (f : ∀ a < o, Ordinal), blsub.{u, u} o f = b → a ≤ o.card 
Arguments: {b : Ordinal.{u}} {a : Cardinal.{u}}
Goal: a ≤ b.cof ↔ ∀ {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{u}), o.blsub f = b → a ≤ o.card
-- Ordinal.cof_blsub_le_lift
theorem cof_blsub_le_lift {o} (f : ∀ a < o, Ordinal) :
    cof (blsub.{u, v} o f) ≤ Cardinal.lift.{v, u} o.card 
Arguments: {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v})
Goal: (o.blsub f).cof ≤ Cardinal.lift.{v, u} o.card
-- Ordinal.cof_blsub_le
theorem cof_blsub_le {o} (f : ∀ a < o, Ordinal) : cof (blsub.{u, u} o f) ≤ o.card 
Arguments: {o : Ordinal.{u}} (f : (a : Ordinal.{u}) → a < o → Ordinal.{u})
Goal: (o.blsub f).cof ≤ o.card
-- Ordinal.blsub_lt_ord_lift
theorem blsub_lt_ord_lift {o : Ordinal.{u}} {f : ∀ a < o, Ordinal} {c : Ordinal}
    (ho : Cardinal.lift.{v, u} o.card < c.cof) (hf : ∀ i hi, f i hi < c) : blsub.{u, v} o f < c 
Arguments: {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}} {c : Ordinal.{max u v}} (ho : Cardinal.lift.{v, u} o.card < c.cof) (hf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c)
Goal: o.blsub f < c
-- Ordinal.blsub_lt_ord
theorem blsub_lt_ord {o : Ordinal} {f : ∀ a < o, Ordinal} {c : Ordinal} (ho : o.card < c.cof)
    (hf : ∀ i hi, f i hi < c) : blsub.{u, u} o f < c 
Arguments: {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{u}} {c : Ordinal.{u}} (ho : o.card < c.cof) (hf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c)
Goal: o.blsub f < c
-- Ordinal.cof_bsup_le_lift
theorem cof_bsup_le_lift {o : Ordinal} {f : ∀ a < o, Ordinal} (H : ∀ i h, f i h < bsup.{u, v} o f) :
    cof (bsup.{u, v} o f) ≤ Cardinal.lift.{v, u} o.card 
Arguments: {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}} (H : ∀ (i : Ordinal.{u}) (h : i < o), f i h < o.bsup f)
Goal: (o.bsup f).cof ≤ Cardinal.lift.{v, u} o.card
-- Ordinal.cof_bsup_le
theorem cof_bsup_le {o : Ordinal} {f : ∀ a < o, Ordinal} :
    (∀ i h, f i h < bsup.{u, u} o f) → cof (bsup.{u, u} o f) ≤ o.card 
Arguments: {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{u}}
Goal: (∀ (i : Ordinal.{u}) (h : i < o), f i h < o.bsup f) → (o.bsup f).cof ≤ o.card
-- Ordinal.bsup_lt_ord_lift
theorem bsup_lt_ord_lift {o : Ordinal} {f : ∀ a < o, Ordinal} {c : Ordinal}
    (ho : Cardinal.lift.{v, u} o.card < c.cof) (hf : ∀ i hi, f i hi < c) : bsup.{u, v} o f < c 
Arguments: {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}} {c : Ordinal.{max u v}} (ho : Cardinal.lift.{v, u} o.card < c.cof) (hf : ∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c)
Goal: o.bsup f < c
-- Ordinal.bsup_lt_ord
theorem bsup_lt_ord {o : Ordinal} {f : ∀ a < o, Ordinal} {c : Ordinal} (ho : o.card < c.cof) :
    (∀ i hi, f i hi < c) → bsup.{u, u} o f < c 
Arguments: {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{u}} {c : Ordinal.{u}} (ho : o.card < c.cof)
Goal: (∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c) → o.bsup f < c
-- Ordinal.cof_zero
@[simp]
theorem cof_zero : cof 0 = 0 
Goal: Ordinal.cof 0 = 0
-- Ordinal.cof_eq_zero
@[simp]
theorem cof_eq_zero {o} : cof o = 0 ↔ o = 0 
Arguments: {o : Ordinal.{u_2}}
Goal: o.cof = 0 ↔ o = 0
-- Ordinal.cof_ne_zero
theorem cof_ne_zero {o} : cof o ≠ 0 ↔ o ≠ 0 
Arguments: {o : Ordinal.{u_2}}
Goal: o.cof ≠ 0 ↔ o ≠ 0
-- Ordinal.cof_succ
@[simp]
theorem cof_succ (o) : cof (succ o) = 1 
Arguments: (o : Ordinal.{u_2})
Goal: (Order.succ o).cof = 1
-- Ordinal.cof_eq_one_iff_is_succ
@[simp]
theorem cof_eq_one_iff_is_succ {o} : cof.{u} o = 1 ↔ ∃ a, o = succ a 
Arguments: {o : Ordinal.{u}}
Goal: o.cof = 1 ↔ ∃ a, o = Order.succ a
-- Ordinal.IsFundamentalSequence.cof_eq
protected theorem cof_eq (hf : IsFundamentalSequence a o f) : a.cof.ord = o 
Arguments: {a o : Ordinal.{u}} {f : (b : Ordinal.{u}) → b < o → Ordinal.{u}} (hf : a.IsFundamentalSequence o f)
Goal: a.cof.ord = o
-- Ordinal.IsFundamentalSequence.strict_mono
protected theorem strict_mono (hf : IsFundamentalSequence a o f) {i j} :
    ∀ hi hj, i < j → f i hi < f j hj 
Arguments: {a o : Ordinal.{u}} {f : (b : Ordinal.{u}) → b < o → Ordinal.{u}} (hf : a.IsFundamentalSequence o f) {i j : Ordinal.{u}} (hi : i < o) (hj : j < o)
Goal: i < j → f i hi < f j hj
-- Ordinal.IsFundamentalSequence.blsub_eq
theorem blsub_eq (hf : IsFundamentalSequence a o f) : blsub.{u, u} o f = a 
Arguments: {a o : Ordinal.{u}} {f : (b : Ordinal.{u}) → b < o → Ordinal.{u}} (hf : a.IsFundamentalSequence o f)
Goal: o.blsub f = a
-- Ordinal.IsFundamentalSequence.ord_cof
theorem ord_cof (hf : IsFundamentalSequence a o f) :
    IsFundamentalSequence a a.cof.ord fun i hi => f i (hi.trans_le (by rw [hf.cof_eq])) 
Arguments: {a o : Ordinal.{u}} {f : (b : Ordinal.{u}) → b < o → Ordinal.{u}} (hf : a.IsFundamentalSequence o f)
Goal: a.IsFundamentalSequence a.cof.ord fun i hi => f i ⋯
-- Ordinal.IsFundamentalSequence.id_of_le_cof
theorem id_of_le_cof (h : o ≤ o.cof.ord) : IsFundamentalSequence o o fun a _ => a 
Arguments: {o : Ordinal.{u}} (h : o ≤ o.cof.ord)
Goal: o.IsFundamentalSequence o fun a x => a
-- Ordinal.IsFundamentalSequence.zero
protected theorem zero {f : ∀ b < (0 : Ordinal), Ordinal} : IsFundamentalSequence 0 0 f 
Arguments: {f : (b : Ordinal.{u_2}) → b < 0 → Ordinal.{u_2}}
Goal: Ordinal.IsFundamentalSequence 0 0 f
-- Ordinal.IsFundamentalSequence.succ
protected theorem succ : IsFundamentalSequence (succ o) 1 fun _ _ => o 
Arguments: {o : Ordinal.{u}}
Goal: (Order.succ o).IsFundamentalSequence 1 fun x x => o
-- Ordinal.IsFundamentalSequence.monotone
protected theorem monotone (hf : IsFundamentalSequence a o f) {i j : Ordinal} (hi : i < o)
    (hj : j < o) (hij : i ≤ j) : f i hi ≤ f j hj 
Arguments: {a o : Ordinal.{u}} {f : (b : Ordinal.{u}) → b < o → Ordinal.{u}} (hf : a.IsFundamentalSequence o f) {i j : Ordinal.{u}} (hi : i < o) (hj : j < o) (hij : i ≤ j)
Goal: f i hi ≤ f j hj
-- Ordinal.IsFundamentalSequence.trans
theorem trans {a o o' : Ordinal.{u}} {f : ∀ b < o, Ordinal.{u}} (hf : IsFundamentalSequence a o f)
    {g : ∀ b < o', Ordinal.{u}} (hg : IsFundamentalSequence o o' g) :
    IsFundamentalSequence a o' fun i hi =>
      f (g i hi) (by rw [← hg.2.2]; apply lt_blsub) 
Arguments: {a o o' : Ordinal.{u}} {f : (b : Ordinal.{u}) → b < o → Ordinal.{u}} (hf : a.IsFundamentalSequence o f) {g : (b : Ordinal.{u}) → b < o' → Ordinal.{u}} (hg : o.IsFundamentalSequence o' g)
Goal: a.IsFundamentalSequence o' fun i hi => f (g i hi) ⋯
-- Ordinal.exists_fundamental_sequence
theorem exists_fundamental_sequence (a : Ordinal.{u}) :
    ∃ f, IsFundamentalSequence a a.cof.ord f 
Arguments: (a : Ordinal.{u})
Goal: ∃ f, a.IsFundamentalSequence a.cof.ord f
-- Ordinal.cof_cof
@[simp]
theorem cof_cof (a : Ordinal.{u}) : cof (cof a).ord = cof a 
Arguments: (a : Ordinal.{u})
Goal: a.cof.ord.cof = a.cof
-- Ordinal.IsNormal.isFundamentalSequence
protected theorem IsNormal.isFundamentalSequence {f : Ordinal.{u} → Ordinal.{u}} (hf : IsNormal f)
    {a o} (ha : IsLimit a) {g} (hg : IsFundamentalSequence a o g) :
    IsFundamentalSequence (f a) o fun b hb => f (g b hb) 
Arguments: {f : Ordinal.{u} → Ordinal.{u}} (hf : Ordinal.IsNormal f) {a o : Ordinal.{u}} (ha : a.IsLimit) {g : (b : Ordinal.{u}) → b < o → Ordinal.{u}} (hg : a.IsFundamentalSequence o g)
Goal: (f a).IsFundamentalSequence o fun b hb => f (g b hb)
-- Ordinal.IsNormal.cof_eq
theorem IsNormal.cof_eq {f} (hf : IsNormal f) {a} (ha : IsLimit a) : cof (f a) = cof a 
Arguments: {f : Ordinal.{u_2} → Ordinal.{u_2}} (hf : Ordinal.IsNormal f) {a : Ordinal.{u_2}} (ha : a.IsLimit)
Goal: (f a).cof = a.cof
-- Ordinal.IsNormal.cof_le
theorem IsNormal.cof_le {f} (hf : IsNormal f) (a) : cof a ≤ cof (f a) 
Arguments: {f : Ordinal.{u_2} → Ordinal.{u_2}} (hf : Ordinal.IsNormal f) (a : Ordinal.{u_2})
Goal: a.cof ≤ (f a).cof
-- Ordinal.cof_add
@[simp]
theorem cof_add (a b : Ordinal) : b ≠ 0 → cof (a + b) = cof b 
Arguments: (a b : Ordinal.{u_2})
Goal: b ≠ 0 → (a + b).cof = b.cof
-- Ordinal.aleph0_le_cof
theorem aleph0_le_cof {o} : ℵ₀ ≤ cof o ↔ IsLimit o 
Arguments: {o : Ordinal.{u_2}}
Goal: ℵ₀ ≤ o.cof ↔ o.IsLimit
-- Ordinal.aleph'_cof
@[simp]
theorem aleph'_cof {o : Ordinal} (ho : o.IsLimit) : (aleph' o).ord.cof = o.cof 
Arguments: {o : Ordinal.{u_2}} (ho : o.IsLimit)
Goal: (Cardinal.aleph' o).ord.cof = o.cof
-- Ordinal.aleph_cof
@[simp]
theorem aleph_cof {o : Ordinal} (ho : o.IsLimit) : (aleph o).ord.cof = o.cof 
Arguments: {o : Ordinal.{u_2}} (ho : o.IsLimit)
Goal: (Cardinal.aleph o).ord.cof = o.cof
-- Ordinal.cof_omega
@[simp]
theorem cof_omega : cof ω = ℵ₀ 
Goal: ω.cof = ℵ₀
-- Ordinal.cof_eq'
theorem cof_eq' (r : α → α → Prop) [IsWellOrder α r] (h : IsLimit (type r)) :
    ∃ S : Set α, (∀ a, ∃ b ∈ S, r a b) ∧ #S = cof (type r) 
Arguments: {α : Type u_1} (r : α → α → Prop) [IsWellOrder α r] (h : (Ordinal.type r).IsLimit)
Goal: ∃ S, (∀ (a : α), ∃ b ∈ S, r a b) ∧ #↑S = (Ordinal.type r).cof
-- Ordinal.cof_univ
@[simp]
theorem cof_univ : cof univ.{u, v} = Cardinal.univ.{u, v} 
Goal: Ordinal.univ.{u, v}.cof = Cardinal.univ.{u, v}
-- Ordinal.unbounded_of_unbounded_sUnion
theorem unbounded_of_unbounded_sUnion (r : α → α → Prop) [wo : IsWellOrder α r] {s : Set (Set α)}
    (h₁ : Unbounded r <| ⋃₀ s) (h₂ : #s < StrictOrder.cof r) : ∃ x ∈ s, Unbounded r x 
Arguments: {α : Type u_1} (r : α → α → Prop) [wo : IsWellOrder α r] {s : Set (Set α)} (h₁ : Set.Unbounded r (⋃₀ s)) (h₂ : #↑s < StrictOrder.cof r)
Goal: ∃ x ∈ s, Set.Unbounded r x
-- Ordinal.unbounded_of_unbounded_iUnion
theorem unbounded_of_unbounded_iUnion {α β : Type u} (r : α → α → Prop) [wo : IsWellOrder α r]
    (s : β → Set α) (h₁ : Unbounded r <| ⋃ x, s x) (h₂ : #β < StrictOrder.cof r) :
    ∃ x : β, Unbounded r (s x) 
Arguments: {α β : Type u} (r : α → α → Prop) [wo : IsWellOrder α r] (s : β → Set α) (h₁ : Set.Unbounded r (⋃ x, s x)) (h₂ : #β < StrictOrder.cof r)
Goal: ∃ x, Set.Unbounded r (s x)
-- Ordinal.infinite_pigeonhole
theorem infinite_pigeonhole {β α : Type u} (f : β → α) (h₁ : ℵ₀ ≤ #β) (h₂ : #α < (#β).ord.cof) :
    ∃ a : α, #(f ⁻¹' {a}) = #β 
Arguments: {β α : Type u} (f : β → α) (h₁ : ℵ₀ ≤ #β) (h₂ : #α < (#β).ord.cof)
Goal: ∃ a, #↑(f ⁻¹' {a}) = #β
-- Ordinal.infinite_pigeonhole_card
theorem infinite_pigeonhole_card {β α : Type u} (f : β → α) (θ : Cardinal) (hθ : θ ≤ #β)
    (h₁ : ℵ₀ ≤ θ) (h₂ : #α < θ.ord.cof) : ∃ a : α, θ ≤ #(f ⁻¹' {a}) 
Arguments: {β α : Type u} (f : β → α) (θ : Cardinal.{u}) (hθ : θ ≤ #β) (h₁ : ℵ₀ ≤ θ) (h₂ : #α < θ.ord.cof)
Goal: ∃ a, θ ≤ #↑(f ⁻¹' {a})
-- Ordinal.infinite_pigeonhole_set
theorem infinite_pigeonhole_set {β α : Type u} {s : Set β} (f : s → α) (θ : Cardinal)
    (hθ : θ ≤ #s) (h₁ : ℵ₀ ≤ θ) (h₂ : #α < θ.ord.cof) :
    ∃ (a : α) (t : Set β) (h : t ⊆ s), θ ≤ #t ∧ ∀ ⦃x⦄ (hx : x ∈ t), f ⟨x, h hx⟩ = a 
Arguments: {β α : Type u} {s : Set β} (f : ↑s → α) (θ : Cardinal.{u}) (hθ : θ ≤ #↑s) (h₁ : ℵ₀ ≤ θ) (h₂ : #α < θ.ord.cof)
Goal: ∃ a t, ∃ (h : t ⊆ s), θ ≤ #↑t ∧ ∀ ⦃x : β⦄ (hx : x ∈ t), f ⟨x, ⋯⟩ = a
-- Cardinal.IsStrongLimit.ne_zero
theorem IsStrongLimit.ne_zero {c} (h : IsStrongLimit c) : c ≠ 0 
Arguments: {c : Cardinal.{u_2}} (h : c.IsStrongLimit)
Goal: c ≠ 0
-- Cardinal.IsStrongLimit.two_power_lt
theorem IsStrongLimit.two_power_lt {x c} (h : IsStrongLimit c) : x < c → (2^x) < c 
Arguments: {x c : Cardinal.{u_2}} (h : c.IsStrongLimit)
Goal: x < c → 2 ^ x < c
-- Cardinal.isStrongLimit_aleph0
theorem isStrongLimit_aleph0 : IsStrongLimit ℵ₀ 
Goal: ℵ₀.IsStrongLimit
-- Cardinal.IsStrongLimit.isSuccLimit
protected theorem IsStrongLimit.isSuccLimit {c} (H : IsStrongLimit c) : IsSuccLimit c 
Arguments: {c : Cardinal.{u_2}} (H : c.IsStrongLimit)
Goal: Order.IsSuccLimit c
-- Cardinal.IsStrongLimit.isLimit
theorem IsStrongLimit.isLimit {c} (H : IsStrongLimit c) : IsLimit c 
Arguments: {c : Cardinal.{u_2}} (H : c.IsStrongLimit)
Goal: c.IsLimit
-- Cardinal.isStrongLimit_beth
theorem isStrongLimit_beth {o : Ordinal} (H : IsSuccLimit o) : IsStrongLimit (beth o) 
Arguments: {o : Ordinal.{u_2}} (H : Order.IsSuccLimit o)
Goal: (Cardinal.beth o).IsStrongLimit
-- Cardinal.mk_bounded_subset
theorem mk_bounded_subset {α : Type*} (h : ∀ x < #α, (2^x) < #α) {r : α → α → Prop}
    [IsWellOrder α r] (hr : (#α).ord = type r) : #{ s : Set α // Bounded r s } = #α 
Arguments: {α : Type u_2} (h : ∀ x < #α, 2 ^ x < #α) {r : α → α → Prop} [IsWellOrder α r] (hr : (#α).ord = Ordinal.type r)
Goal: #{ s // Set.Bounded r s } = #α
-- Cardinal.mk_subset_mk_lt_cof
theorem mk_subset_mk_lt_cof {α : Type*} (h : ∀ x < #α, (2^x) < #α) :
    #{ s : Set α // #s < cof (#α).ord } = #α 
Arguments: {α : Type u_2} (h : ∀ x < #α, 2 ^ x < #α)
Goal: #{ s // #↑s < (#α).ord.cof } = #α
-- Cardinal.IsRegular.aleph0_le
theorem IsRegular.aleph0_le {c : Cardinal} (H : c.IsRegular) : ℵ₀ ≤ c 
Arguments: {c : Cardinal.{u_2}} (H : c.IsRegular)
Goal: ℵ₀ ≤ c
-- Cardinal.IsRegular.cof_eq
theorem IsRegular.cof_eq {c : Cardinal} (H : c.IsRegular) : c.ord.cof = c 
Arguments: {c : Cardinal.{u_2}} (H : c.IsRegular)
Goal: c.ord.cof = c
-- Cardinal.IsRegular.pos
theorem IsRegular.pos {c : Cardinal} (H : c.IsRegular) : 0 < c 
Arguments: {c : Cardinal.{u_2}} (H : c.IsRegular)
Goal: 0 < c
-- Cardinal.IsRegular.nat_lt
theorem IsRegular.nat_lt {c : Cardinal} (H : c.IsRegular) (n : ℕ) : n < c 
Arguments: {c : Cardinal.{u_2}} (H : c.IsRegular) (n : ℕ)
Goal: ↑n < c
-- Cardinal.IsRegular.ord_pos
theorem IsRegular.ord_pos {c : Cardinal} (H : c.IsRegular) : 0 < c.ord 
Arguments: {c : Cardinal.{u_2}} (H : c.IsRegular)
Goal: 0 < c.ord
-- Cardinal.isRegular_cof
theorem isRegular_cof {o : Ordinal} (h : o.IsLimit) : IsRegular o.cof 
Arguments: {o : Ordinal.{u_2}} (h : o.IsLimit)
Goal: o.cof.IsRegular
-- Cardinal.isRegular_aleph0
theorem isRegular_aleph0 : IsRegular ℵ₀ 
Goal: ℵ₀.IsRegular
-- Cardinal.isRegular_succ
theorem isRegular_succ {c : Cardinal.{u}} (h : ℵ₀ ≤ c) : IsRegular (succ c) 
Arguments: {c : Cardinal.{u}} (h : ℵ₀ ≤ c)
Goal: (Order.succ c).IsRegular
-- Cardinal.isRegular_aleph_one
theorem isRegular_aleph_one : IsRegular (aleph 1) 
Goal: (Cardinal.aleph 1).IsRegular
-- Cardinal.isRegular_aleph'_succ
theorem isRegular_aleph'_succ {o : Ordinal} (h : ω ≤ o) : IsRegular (aleph' (succ o)) 
Arguments: {o : Ordinal.{u_2}} (h : ω ≤ o)
Goal: (Cardinal.aleph' (Order.succ o)).IsRegular
-- Cardinal.isRegular_aleph_succ
theorem isRegular_aleph_succ (o : Ordinal) : IsRegular (aleph (succ o)) 
Arguments: (o : Ordinal.{u_2})
Goal: (Cardinal.aleph (Order.succ o)).IsRegular
-- Cardinal.infinite_pigeonhole_card_lt
theorem infinite_pigeonhole_card_lt {β α : Type u} (f : β → α) (w : #α < #β) (w' : ℵ₀ ≤ #α) :
    ∃ a : α, #α < #(f ⁻¹' {a}) 
Arguments: {β α : Type u} (f : β → α) (w : #α < #β) (w' : ℵ₀ ≤ #α)
Goal: ∃ a, #α < #↑(f ⁻¹' {a})
-- Cardinal.exists_infinite_fiber
theorem exists_infinite_fiber {β α : Type u} (f : β → α) (w : #α < #β) (w' : Infinite α) :
    ∃ a : α, Infinite (f ⁻¹' {a}) 
Arguments: {β α : Type u} (f : β → α) (w : #α < #β) (w' : Infinite α)
Goal: ∃ a, Infinite ↑(f ⁻¹' {a})
-- Cardinal.le_range_of_union_finset_eq_top
theorem le_range_of_union_finset_eq_top {α β : Type*} [Infinite β] (f : α → Finset β)
    (w : ⋃ a, (f a : Set β) = ⊤) : #β ≤ #(range f) 
Arguments: {α : Type u_2} {β : Type u_3} [Infinite β] (f : α → Finset β) (w : ⋃ a, ↑(f a) = ⊤)
Goal: #β ≤ #↑(Set.range f)
-- Cardinal.lsub_lt_ord_lift_of_isRegular
theorem lsub_lt_ord_lift_of_isRegular {ι} {f : ι → Ordinal} {c} (hc : IsRegular c)
    (hι : Cardinal.lift.{v, u} #ι < c) : (∀ i, f i < c.ord) → Ordinal.lsub.{u, v} f < c.ord 
Arguments: {ι : Type u} {f : ι → Ordinal.{max u v}} {c : Cardinal.{max u v}} (hc : c.IsRegular) (hι : Cardinal.lift.{v, u} #ι < c)
Goal: (∀ (i : ι), f i < c.ord) → Ordinal.lsub f < c.ord
-- Cardinal.lsub_lt_ord_of_isRegular
theorem lsub_lt_ord_of_isRegular {ι} {f : ι → Ordinal} {c} (hc : IsRegular c) (hι : #ι < c) :
    (∀ i, f i < c.ord) → Ordinal.lsub f < c.ord 
Arguments: {ι : Type (max u_2 u_3)} {f : ι → Ordinal.{max u_2 u_3}} {c : Cardinal.{max u_2 u_3}} (hc : c.IsRegular) (hι : #ι < c)
Goal: (∀ (i : ι), f i < c.ord) → Ordinal.lsub f < c.ord
-- Cardinal.sup_lt_ord_lift_of_isRegular
theorem sup_lt_ord_lift_of_isRegular {ι} {f : ι → Ordinal} {c} (hc : IsRegular c)
    (hι : Cardinal.lift.{v, u} #ι < c) : (∀ i, f i < c.ord) → Ordinal.sup.{u, v} f < c.ord 
Arguments: {ι : Type u} {f : ι → Ordinal.{max u v}} {c : Cardinal.{max u v}} (hc : c.IsRegular) (hι : Cardinal.lift.{v, u} #ι < c)
Goal: (∀ (i : ι), f i < c.ord) → Ordinal.sup f < c.ord
-- Cardinal.sup_lt_ord_of_isRegular
theorem sup_lt_ord_of_isRegular {ι} {f : ι → Ordinal} {c} (hc : IsRegular c) (hι : #ι < c) :
    (∀ i, f i < c.ord) → Ordinal.sup f < c.ord 
Arguments: {ι : Type (max u_2 u_3)} {f : ι → Ordinal.{max u_2 u_3}} {c : Cardinal.{max u_2 u_3}} (hc : c.IsRegular) (hι : #ι < c)
Goal: (∀ (i : ι), f i < c.ord) → Ordinal.sup f < c.ord
-- Cardinal.blsub_lt_ord_lift_of_isRegular
theorem blsub_lt_ord_lift_of_isRegular {o : Ordinal} {f : ∀ a < o, Ordinal} {c} (hc : IsRegular c)
    (ho : Cardinal.lift.{v, u} o.card < c) :
    (∀ i hi, f i hi < c.ord) → Ordinal.blsub.{u, v} o f < c.ord 
Arguments: {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}} {c : Cardinal.{max u v}} (hc : c.IsRegular) (ho : Cardinal.lift.{v, u} o.card < c)
Goal: (∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c.ord) → o.blsub f < c.ord
-- Cardinal.blsub_lt_ord_of_isRegular
theorem blsub_lt_ord_of_isRegular {o : Ordinal} {f : ∀ a < o, Ordinal} {c} (hc : IsRegular c)
    (ho : o.card < c) : (∀ i hi, f i hi < c.ord) → Ordinal.blsub o f < c.ord 
Arguments: {o : Ordinal.{max u_2 u_3}} {f : (a : Ordinal.{max u_2 u_3}) → a < o → Ordinal.{max u_2 u_3}} {c : Cardinal.{max u_2 u_3}} (hc : c.IsRegular) (ho : o.card < c)
Goal: (∀ (i : Ordinal.{max u_2 u_3}) (hi : i < o), f i hi < c.ord) → o.blsub f < c.ord
-- Cardinal.bsup_lt_ord_lift_of_isRegular
theorem bsup_lt_ord_lift_of_isRegular {o : Ordinal} {f : ∀ a < o, Ordinal} {c} (hc : IsRegular c)
    (hι : Cardinal.lift.{v, u} o.card < c) :
    (∀ i hi, f i hi < c.ord) → Ordinal.bsup.{u, v} o f < c.ord 
Arguments: {o : Ordinal.{u}} {f : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}} {c : Cardinal.{max u v}} (hc : c.IsRegular) (hι : Cardinal.lift.{v, u} o.card < c)
Goal: (∀ (i : Ordinal.{u}) (hi : i < o), f i hi < c.ord) → o.bsup f < c.ord
-- Cardinal.bsup_lt_ord_of_isRegular
theorem bsup_lt_ord_of_isRegular {o : Ordinal} {f : ∀ a < o, Ordinal} {c} (hc : IsRegular c)
    (hι : o.card < c) : (∀ i hi, f i hi < c.ord) → Ordinal.bsup o f < c.ord 
Arguments: {o : Ordinal.{max u_2 u_3}} {f : (a : Ordinal.{max u_2 u_3}) → a < o → Ordinal.{max u_2 u_3}} {c : Cardinal.{max u_2 u_3}} (hc : c.IsRegular) (hι : o.card < c)
Goal: (∀ (i : Ordinal.{max u_2 u_3}) (hi : i < o), f i hi < c.ord) → o.bsup f < c.ord
-- Cardinal.iSup_lt_lift_of_isRegular
theorem iSup_lt_lift_of_isRegular {ι} {f : ι → Cardinal} {c} (hc : IsRegular c)
    (hι : Cardinal.lift.{v, u} #ι < c) : (∀ i, f i < c) → iSup.{max u v + 1, u + 1} f < c 
Arguments: {ι : Type u} {f : ι → Cardinal.{max u v}} {c : Cardinal.{max u v}} (hc : c.IsRegular) (hι : Cardinal.lift.{v, u} #ι < c)
Goal: (∀ (i : ι), f i < c) → iSup f < c
-- Cardinal.iSup_lt_of_isRegular
theorem iSup_lt_of_isRegular {ι} {f : ι → Cardinal} {c} (hc : IsRegular c) (hι : #ι < c) :
    (∀ i, f i < c) → iSup f < c 
Arguments: {ι : Type u_2} {f : ι → Cardinal.{u_2}} {c : Cardinal.{u_2}} (hc : c.IsRegular) (hι : #ι < c)
Goal: (∀ (i : ι), f i < c) → iSup f < c
-- Cardinal.sum_lt_lift_of_isRegular
theorem sum_lt_lift_of_isRegular {ι : Type u} {f : ι → Cardinal} {c : Cardinal} (hc : IsRegular c)
    (hι : Cardinal.lift.{v, u} #ι < c) (hf : ∀ i, f i < c) : sum f < c 
Arguments: {ι : Type u} {f : ι → Cardinal.{max u v}} {c : Cardinal.{max u v}} (hc : c.IsRegular) (hι : Cardinal.lift.{v, u} #ι < c) (hf : ∀ (i : ι), f i < c)
Goal: Cardinal.sum f < c
-- Cardinal.sum_lt_of_isRegular
theorem sum_lt_of_isRegular {ι : Type u} {f : ι → Cardinal} {c : Cardinal} (hc : IsRegular c)
    (hι : #ι < c) : (∀ i, f i < c) → sum f < c 
Arguments: {ι : Type u} {f : ι → Cardinal.{u}} {c : Cardinal.{u}} (hc : c.IsRegular) (hι : #ι < c)
Goal: (∀ (i : ι), f i < c) → Cardinal.sum f < c
-- Cardinal.card_lt_of_card_iUnion_lt
@[simp]
theorem card_lt_of_card_iUnion_lt {ι : Type u} {α : Type u} {t : ι → Set α} {c : Cardinal}
    (h : #(⋃ i, t i) < c) (i : ι) : #(t i) < c 
Arguments: {ι α : Type u} {t : ι → Set α} {c : Cardinal.{u}} (h : #↑(⋃ i, t i) < c) (i : ι)
Goal: #↑(t i) < c
-- Cardinal.card_iUnion_lt_iff_forall_of_isRegular
@[simp]
theorem card_iUnion_lt_iff_forall_of_isRegular {ι : Type u} {α : Type u} {t : ι → Set α}
    {c : Cardinal} (hc : c.IsRegular) (hι : #ι < c) : #(⋃ i, t i) < c ↔ ∀ i, #(t i) < c 
Arguments: {ι α : Type u} {t : ι → Set α} {c : Cardinal.{u}} (hc : c.IsRegular) (hι : #ι < c)
Goal: #↑(⋃ i, t i) < c ↔ ∀ (i : ι), #↑(t i) < c
-- Cardinal.card_lt_of_card_biUnion_lt
theorem card_lt_of_card_biUnion_lt {α β : Type u} {s : Set α} {t : ∀ a ∈ s, Set β} {c : Cardinal}
    (h : #(⋃ a ∈ s, t a ‹_›) < c) (a : α) (ha : a ∈ s) : # (t a ha) < c 
Arguments: {α β : Type u} {s : Set α} {t : (a : α) → a ∈ s → Set β} {c : Cardinal.{u}} (h : #↑(⋃ a, ⋃ (h : a ∈ s), t a h) < c) (a : α) (ha : a ∈ s)
Goal: #↑(t a ha) < c
-- Cardinal.card_biUnion_lt_iff_forall_of_isRegular
theorem card_biUnion_lt_iff_forall_of_isRegular {α β : Type u} {s : Set α} {t : ∀ a ∈ s, Set β}
    {c : Cardinal} (hc : c.IsRegular) (hs : #s < c) :
    #(⋃ a ∈ s, t a ‹_›) < c ↔ ∀ a (ha : a ∈ s), # (t a ha) < c 
Arguments: {α β : Type u} {s : Set α} {t : (a : α) → a ∈ s → Set β} {c : Cardinal.{u}} (hc : c.IsRegular) (hs : #↑s < c)
Goal: #↑(⋃ a, ⋃ (h : a ∈ s), t a h) < c ↔ ∀ (a : α) (ha : a ∈ s), #↑(t a ha) < c
-- Cardinal.nfpFamily_lt_ord_lift_of_isRegular
theorem nfpFamily_lt_ord_lift_of_isRegular {ι} {f : ι → Ordinal → Ordinal} {c} (hc : IsRegular c)
    (hι : Cardinal.lift.{v, u} #ι < c) (hc' : c ≠ ℵ₀) (hf : ∀ (i), ∀ b < c.ord, f i b < c.ord) {a}
    (ha : a < c.ord) : nfpFamily.{u, v} f a < c.ord 
Arguments: {ι : Type u} {f : ι → Ordinal.{max u v} → Ordinal.{max u v}} {c : Cardinal.{max u v}} (hc : c.IsRegular) (hι : Cardinal.lift.{v, u} #ι < c) (hc' : c ≠ ℵ₀) (hf : ∀ (i : ι), ∀ b < c.ord, f i b < c.ord) {a : Ordinal.{max u v}} (ha : a < c.ord)
Goal: Ordinal.nfpFamily f a < c.ord