mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.SetTheory.Ordinal.Arithmetic

Goal state before:
pp: case inl
a b c : Ordinal.{u_4}
h : a < b + c
hc : 0 < c
hab : a < b
⊢ a - b < c

pp: case inr
a b c : Ordinal.{u_4}
h : a < b + c
hc : 0 < c
hba : b ≤ a
⊢ a - b < c

Tactic:
·

Goal state after:
case inl
a b c : Ordinal.{u_4}
h : a < b + c
hc : 0 < c
hab : a < b
⊢ a - b < c
Module: Mathlib.SetTheory.Ordinal.Arithmetic

Goal state before:
pp: case inr
a b c : Ordinal.{u_4}
h : a < b + c
hc : 0 < c
hba : b ≤ a
⊢ a - b < c

Tactic:
·

Goal state after:
case inr
a b c : Ordinal.{u_4}
h : a < b + c
hc : 0 < c
hba : b ≤ a
⊢ a - b < c
Module: Mathlib.SetTheory.Cardinal.Cofinality.Enum

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
f : α → α
hf : StrictMono f
x : α
⊢ StrictMono fun x ↦ ↑((StrictMono.orderIso f hf) x)

pp: α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
f : α → α
hf : StrictMono f
x : α
⊢ (range fun x ↦ ↑((StrictMono.orderIso f hf) x)) = range f

Tactic:
·

Goal state after:
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
f : α → α
hf : StrictMono f
x : α
⊢ StrictMono fun x ↦ ↑((StrictMono.orderIso f hf) x)
Module: Mathlib.SetTheory.Ordinal.Family

Goal state before:
pp: case refine_1
α : Type u_1
ι : Type u
r : ι → ι → Prop
inst✝ : IsWellOrder ι r
o : Ordinal.{u}
ho : type r = o
f : (a : Ordinal.{u}) → a < o → α
a : α
⊢ a ∈ range (familyOfBFamily' r ho f) → a ∈ o.brange f

pp: case refine_2
α : Type u_1
ι : Type u
r : ι → ι → Prop
inst✝ : IsWellOrder ι r
o : Ordinal.{u}
ho : type r = o
f : (a : Ordinal.{u}) → a < o → α
a : α
⊢ a ∈ o.brange f → a ∈ range (familyOfBFamily' r ho f)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
ι : Type u
r : ι → ι → Prop
inst✝ : IsWellOrder ι r
o : Ordinal.{u}
ho : type r = o
f : (a : Ordinal.{u}) → a < o → α
a : α
⊢ a ∈ range (familyOfBFamily' r ho f) → a ∈ o.brange f
Module: Mathlib.SetTheory.Cardinal.Cofinality.Enum

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
f : α → α
hf : StrictMono f
x : α
⊢ (range fun x ↦ ↑((StrictMono.orderIso f hf) x)) = range f

Tactic:
·

Goal state after:
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
f : α → α
hf : StrictMono f
x : α
⊢ (range fun x ↦ ↑((StrictMono.orderIso f hf) x)) = range f
Module: Mathlib.SetTheory.Ordinal.Family

Goal state before:
pp: case refine_2
α : Type u_1
ι : Type u
r : ι → ι → Prop
inst✝ : IsWellOrder ι r
o : Ordinal.{u}
ho : type r = o
f : (a : Ordinal.{u}) → a < o → α
a : α
⊢ a ∈ o.brange f → a ∈ range (familyOfBFamily' r ho f)

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
ι : Type u
r : ι → ι → Prop
inst✝ : IsWellOrder ι r
o : Ordinal.{u}
ho : type r = o
f : (a : Ordinal.{u}) → a < o → α
a : α
⊢ a ∈ o.brange f → a ∈ range (familyOfBFamily' r ho f)
Module: Mathlib.SetTheory.Ordinal.Arithmetic

Goal state before:
pp: case mp
a b c : Ordinal.{u_4}
hc : IsSuccLimit c
d : Ordinal.{u_4}
hd : d < c
ha : a ≤ b + d
⊢ ∃ d < c, a < b + d

pp: case mpr
a b c : Ordinal.{u_4}
hc : IsSuccLimit c
d : Ordinal.{u_4}
hd : d < c
ha : a < b + d
⊢ ∃ d < c, a ≤ b + d

Tactic:
·

Goal state after:
case mp
a b c : Ordinal.{u_4}
hc : IsSuccLimit c
d : Ordinal.{u_4}
hd : d < c
ha : a ≤ b + d
⊢ ∃ d < c, a < b + d
Module: Mathlib.SetTheory.Ordinal.Family

Goal state before:
pp: case refine_1
α : Type u_1
ι : Type u
r : ι → ι → Prop
inst✝ : IsWellOrder ι r
f : ι → α
a : α
⊢ a ∈ (type r).brange (bfamilyOfFamily' r f) → a ∈ range f

pp: case refine_2
α : Type u_1
ι : Type u
r : ι → ι → Prop
inst✝ : IsWellOrder ι r
f : ι → α
a : α
⊢ a ∈ range f → a ∈ (type r).brange (bfamilyOfFamily' r f)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
ι : Type u
r : ι → ι → Prop
inst✝ : IsWellOrder ι r
f : ι → α
a : α
⊢ a ∈ (type r).brange (bfamilyOfFamily' r f) → a ∈ range f
Module: Mathlib.SetTheory.Ordinal.Family

Goal state before:
pp: case refine_2
α : Type u_1
ι : Type u
r : ι → ι → Prop
inst✝ : IsWellOrder ι r
f : ι → α
a : α
⊢ a ∈ range f → a ∈ (type r).brange (bfamilyOfFamily' r f)

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
ι : Type u
r : ι → ι → Prop
inst✝ : IsWellOrder ι r
f : ι → α
a : α
⊢ a ∈ range f → a ∈ (type r).brange (bfamilyOfFamily' r f)
Module: Mathlib.SetTheory.Ordinal.Arithmetic

Goal state before:
pp: case mpr
a b c : Ordinal.{u_4}
hc : IsSuccLimit c
d : Ordinal.{u_4}
hd : d < c
ha : a < b + d
⊢ ∃ d < c, a ≤ b + d

Tactic:
·

Goal state after:
case mpr
a b c : Ordinal.{u_4}
hc : IsSuccLimit c
d : Ordinal.{u_4}
hd : d < c
ha : a < b + d
⊢ ∃ d < c, a ≤ b + d
Module: Mathlib.SetTheory.Ordinal.Basic

Goal state before:
pp: case mp
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : lift.{max v w, u} (type r) ≤ lift.{max u w, v} (type s)
f : ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≼i ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r
⊢ r ≼i s

pp: case mpr
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : Nonempty (r ≼i s)
f : r ≼i s
⊢ ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≼i ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r

Tactic:
·

Goal state after:
case mp
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : lift.{max v w, u} (type r) ≤ lift.{max u w, v} (type s)
f : ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≼i ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r
⊢ r ≼i s
Module: Mathlib.SetTheory.Ordinal.Basic

Goal state before:
pp: case mpr
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : Nonempty (r ≼i s)
f : r ≼i s
⊢ ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≼i ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r

Tactic:
·

Goal state after:
case mpr
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : Nonempty (r ≼i s)
f : r ≼i s
⊢ ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≼i ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r
Module: Mathlib.SetTheory.Ordinal.Basic

Goal state before:
pp: case refine_1
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ :
  isEquivalent
    { α := ULift.{max v w, u} { α := α, r := r, wo := inst✝¹ }.α,
      r := ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r, wo := ⋯ }
    { α := ULift.{max u w, v} { α := β, r := s, wo := inst✝ }.α, r := ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r,
      wo := ⋯ }
f : ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≃r ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r
⊢ r ≃r s

pp: case refine_2
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : Nonempty (r ≃r s)
f : r ≃r s
⊢ ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≃r ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r

Tactic:
·

Goal state after:
case refine_1
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ :
  isEquivalent
    { α := ULift.{max v w, u} { α := α, r := r, wo := inst✝¹ }.α,
      r := ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r, wo := ⋯ }
    { α := ULift.{max u w, v} { α := β, r := s, wo := inst✝ }.α, r := ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r,
      wo := ⋯ }
f : ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≃r ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r
⊢ r ≃r s
Module: Mathlib.SetTheory.Ordinal.Basic

Goal state before:
pp: case refine_2
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : Nonempty (r ≃r s)
f : r ≃r s
⊢ ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≃r ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r

Tactic:
·

Goal state after:
case refine_2
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : Nonempty (r ≃r s)
f : r ≃r s
⊢ ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≃r ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r
Module: Mathlib.SetTheory.Ordinal.Family

Goal state before:
pp: α : Type u_3
β : Type u_4
f : α ⊕ β → Ordinal.{u}
inst✝¹ : Small.{u, u_3} α
inst✝ : Small.{u, u_4} β
⊢ ∀ (i : α ⊕ β), f i ≤ max (⨆ a, f (Sum.inl a)) (⨆ b, f (Sum.inr b))

pp: α : Type u_3
β : Type u_4
f : α ⊕ β → Ordinal.{u}
inst✝¹ : Small.{u, u_3} α
inst✝ : Small.{u, u_4} β
⊢ ⨆ a, f (Sum.inl a) ≤ ⨆ i, f i

pp: α : Type u_3
β : Type u_4
f : α ⊕ β → Ordinal.{u}
inst✝¹ : Small.{u, u_3} α
inst✝ : Small.{u, u_4} β
⊢ ⨆ b, f (Sum.inr b) ≤ ⨆ i, f i

Tactic:
·

Goal state after:
α : Type u_3
β : Type u_4
f : α ⊕ β → Ordinal.{u}
inst✝¹ : Small.{u, u_3} α
inst✝ : Small.{u, u_4} β
⊢ ∀ (i : α ⊕ β), f i ≤ max (⨆ a, f (Sum.inl a)) (⨆ b, f (Sum.inr b))
Module: Mathlib.SetTheory.Ordinal.Family

Goal state before:
pp: case inl
α : Type u_3
β : Type u_4
f : α ⊕ β → Ordinal.{u}
inst✝¹ : Small.{u, u_3} α
inst✝ : Small.{u, u_4} β
i : α
⊢ f (Sum.inl i) ≤ max (⨆ a, f (Sum.inl a)) (⨆ b, f (Sum.inr b))

pp: case inr
α : Type u_3
β : Type u_4
f : α ⊕ β → Ordinal.{u}
inst✝¹ : Small.{u, u_3} α
inst✝ : Small.{u, u_4} β
i : β
⊢ f (Sum.inr i) ≤ max (⨆ a, f (Sum.inl a)) (⨆ b, f (Sum.inr b))

Tactic:
·

Goal state after:
case inl
α : Type u_3
β : Type u_4
f : α ⊕ β → Ordinal.{u}
inst✝¹ : Small.{u, u_3} α
inst✝ : Small.{u, u_4} β
i : α
⊢ f (Sum.inl i) ≤ max (⨆ a, f (Sum.inl a)) (⨆ b, f (Sum.inr b))
Module: Mathlib.SetTheory.Ordinal.Family

Goal state before:
pp: case inr
α : Type u_3
β : Type u_4
f : α ⊕ β → Ordinal.{u}
inst✝¹ : Small.{u, u_3} α
inst✝ : Small.{u, u_4} β
i : β
⊢ f (Sum.inr i) ≤ max (⨆ a, f (Sum.inl a)) (⨆ b, f (Sum.inr b))

Tactic:
·

Goal state after:
case inr
α : Type u_3
β : Type u_4
f : α ⊕ β → Ordinal.{u}
inst✝¹ : Small.{u, u_3} α
inst✝ : Small.{u, u_4} β
i : β
⊢ f (Sum.inr i) ≤ max (⨆ a, f (Sum.inl a)) (⨆ b, f (Sum.inr b))
Module: Mathlib.SetTheory.Cardinal.Cofinality.Enum

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
s : Set α
hs : IsCofinal s
H : StrictMono (Subtype.val ∘ ⇑(enum s hs)) := ⋯
hs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.Nonempty → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s
a : α
ha : IsSuccLimit a
b : α
hb : ∀ b_1 < a, (Subtype.val ∘ ⇑(enum s hs)) b_1 ≤ b
bdd : BddAbove (Subtype.val ∘ ⇑(enum s hs) '' Iio a)
this✝¹ : Nonempty α
this✝ : OrderBot α := ⋯
this : ConditionallyCompleteLinearOrderBot α := ⋯
⊢ (Subtype.val ∘ ⇑(enum s hs)) a ≤ sSup (Subtype.val ∘ ⇑(enum s hs) '' Iio a)

pp: α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
s : Set α
hs : IsCofinal s
H : StrictMono (Subtype.val ∘ ⇑(enum s hs)) := ⋯
hs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.Nonempty → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s
a : α
ha : IsSuccLimit a
b : α
hb : ∀ b_1 < a, (Subtype.val ∘ ⇑(enum s hs)) b_1 ≤ b
bdd : BddAbove (Subtype.val ∘ ⇑(enum s hs) '' Iio a)
this✝¹ : Nonempty α
this✝ : OrderBot α := ⋯
this : ConditionallyCompleteLinearOrderBot α := ⋯
⊢ sSup (Subtype.val ∘ ⇑(enum s hs) '' Iio a) ≤ b

Tactic:
·

Goal state after:
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
s : Set α
hs : IsCofinal s
H : StrictMono (Subtype.val ∘ ⇑(enum s hs)) := ⋯
hs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.Nonempty → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s
a : α
ha : IsSuccLimit a
b : α
hb : ∀ b_1 < a, (Subtype.val ∘ ⇑(enum s hs)) b_1 ≤ b
bdd : BddAbove (Subtype.val ∘ ⇑(enum s hs) '' Iio a)
this✝¹ : Nonempty α
this✝ : OrderBot α := ⋯
this : ConditionallyCompleteLinearOrderBot α := ⋯
⊢ (Subtype.val ∘ ⇑(enum s hs)) a ≤ sSup (Subtype.val ∘ ⇑(enum s hs) '' Iio a)
Module: Mathlib.SetTheory.Ordinal.Basic

Goal state before:
pp: case mp
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : lift.{max v w, u} (type r) < lift.{max u w, v} (type s)
f : ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≺i ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r
⊢ r ≺i s

pp: case mpr
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : Nonempty (r ≺i s)
f : r ≺i s
⊢ ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≺i ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r

Tactic:
·

Goal state after:
case mp
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : lift.{max v w, u} (type r) < lift.{max u w, v} (type s)
f : ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≺i ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r
⊢ r ≺i s
Module: Mathlib.SetTheory.Ordinal.Basic

Goal state before:
pp: case mpr
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : Nonempty (r ≺i s)
f : r ≺i s
⊢ ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≺i ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r

Tactic:
·

Goal state after:
case mpr
α : Type u
β : Type v
r : α → α → Prop
s : β → β → Prop
inst✝¹ : IsWellOrder α r
inst✝ : IsWellOrder β s
x✝ : Nonempty (r ≺i s)
f : r ≺i s
⊢ ULift.down ⁻¹'o { α := α, r := r, wo := inst✝¹ }.r ≺i ULift.down ⁻¹'o { α := β, r := s, wo := inst✝ }.r
Module: Mathlib.SetTheory.Ordinal.Arithmetic

Goal state before:
pp: case left
α✝ : Type u_1
β✝ : Type u_2
γ✝ : Type u_3
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t✝ : γ✝ → γ✝ → Prop
c a b : Ordinal.{u}
x✝³ x✝² x✝¹ : WellOrder
α : Type u
r : α → α → Prop
wo✝² : IsWellOrder α r
β : Type u
s : β → β → Prop
wo✝¹ : IsWellOrder β s
x✝ : ⟦{ α := α, r := r, wo := wo✝² }⟧ ≤ ⟦{ α := β, r := s, wo := wo✝¹ }⟧
γ : Type u
t : γ → γ → Prop
wo✝ : IsWellOrder γ t
f : InitialSeg r s
a₁✝ : α
b₁✝ : γ
a₂✝ : α
b₂✝ : γ
h' : r a₁✝ a₂✝
⊢ Prod.Lex s t (f (a₁✝, b₁✝).1, (a₁✝, b₁✝).2) (f (a₂✝, b₂✝).1, (a₂✝, b₂✝).2)

pp: case right
α✝ : Type u_1
β✝ : Type u_2
γ✝ : Type u_3
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t✝ : γ✝ → γ✝ → Prop
c a b : Ordinal.{u}
x✝³ x✝² x✝¹ : WellOrder
α : Type u
r : α → α → Prop
wo✝² : IsWellOrder α r
β : Type u
s : β → β → Prop
wo✝¹ : IsWellOrder β s
x✝ : ⟦{ α := α, r := r, wo := wo✝² }⟧ ≤ ⟦{ α := β, r := s, wo := wo✝¹ }⟧
γ : Type u
t : γ → γ → Prop
wo✝ : IsWellOrder γ t
f : InitialSeg r s
a✝ : α
b₁✝ b₂✝ : γ
h' : t b₁✝ b₂✝
⊢ Prod.Lex s t (f (a✝, b₁✝).1, (a✝, b₁✝).2) (f (a✝, b₂✝).1, (a✝, b₂✝).2)

Tactic:
·

Goal state after:
case left
α✝ : Type u_1
β✝ : Type u_2
γ✝ : Type u_3
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t✝ : γ✝ → γ✝ → Prop
c a b : Ordinal.{u}
x✝³ x✝² x✝¹ : WellOrder
α : Type u
r : α → α → Prop
wo✝² : IsWellOrder α r
β : Type u
s : β → β → Prop
wo✝¹ : IsWellOrder β s
x✝ : ⟦{ α := α, r := r, wo := wo✝² }⟧ ≤ ⟦{ α := β, r := s, wo := wo✝¹ }⟧
γ : Type u
t : γ → γ → Prop
wo✝ : IsWellOrder γ t
f : InitialSeg r s
a₁✝ : α
b₁✝ : γ
a₂✝ : α
b₂✝ : γ
h' : r a₁✝ a₂✝
⊢ Prod.Lex s t (f (a₁✝, b₁✝).1, (a₁✝, b₁✝).2) (f (a₂✝, b₂✝).1, (a₂✝, b₂✝).2)
Module: Mathlib.SetTheory.Ordinal.Family

Goal state before:
pp: case inl
⊢ (sSup ∅).ord = sSup (ord '' ∅)

pp: case inr
s : Set Cardinal.{u_3}
hn : s.Nonempty
⊢ (sSup s).ord = sSup (ord '' s)

Tactic:
·

Goal state after:
case inl
⊢ (sSup ∅).ord = sSup (ord '' ∅)
Module: Mathlib.SetTheory.Ordinal.Arithmetic

Goal state before:
pp: case right
α✝ : Type u_1
β✝ : Type u_2
γ✝ : Type u_3
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t✝ : γ✝ → γ✝ → Prop
c a b : Ordinal.{u}
x✝³ x✝² x✝¹ : WellOrder
α : Type u
r : α → α → Prop
wo✝² : IsWellOrder α r
β : Type u
s : β → β → Prop
wo✝¹ : IsWellOrder β s
x✝ : ⟦{ α := α, r := r, wo := wo✝² }⟧ ≤ ⟦{ α := β, r := s, wo := wo✝¹ }⟧
γ : Type u
t : γ → γ → Prop
wo✝ : IsWellOrder γ t
f : InitialSeg r s
a✝ : α
b₁✝ b₂✝ : γ
h' : t b₁✝ b₂✝
⊢ Prod.Lex s t (f (a✝, b₁✝).1, (a✝, b₁✝).2) (f (a✝, b₂✝).1, (a✝, b₂✝).2)

Tactic:
·

Goal state after:
case right
α✝ : Type u_1
β✝ : Type u_2
γ✝ : Type u_3
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t✝ : γ✝ → γ✝ → Prop
c a b : Ordinal.{u}
x✝³ x✝² x✝¹ : WellOrder
α : Type u
r : α → α → Prop
wo✝² : IsWellOrder α r
β : Type u
s : β → β → Prop
wo✝¹ : IsWellOrder β s
x✝ : ⟦{ α := α, r := r, wo := wo✝² }⟧ ≤ ⟦{ α := β, r := s, wo := wo✝¹ }⟧
γ : Type u
t : γ → γ → Prop
wo✝ : IsWellOrder γ t
f : InitialSeg r s
a✝ : α
b₁✝ b₂✝ : γ
h' : t b₁✝ b₂✝
⊢ Prod.Lex s t (f (a✝, b₁✝).1, (a✝, b₁✝).2) (f (a✝, b₂✝).1, (a✝, b₂✝).2)
Module: Mathlib.SetTheory.Cardinal.Cofinality.Enum

Goal state before:
pp: case refine_1
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
s : Set α
hs : IsCofinal s
H : StrictMono (Subtype.val ∘ ⇑(enum s hs)) :=
  StrictMono.comp (Subtype.strictMono_coe fun x ↦ x ∈ s) (OrderIso.strictMono (enum s hs))
hs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.Nonempty → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s
a : α
ha : IsSuccLimit a
b : α
hb : ∀ b_1 < a, (Subtype.val ∘ ⇑(enum s hs)) b_1 ≤ b
bdd : BddAbove (Subtype.val ∘ ⇑(enum s hs) '' Iio a)
this✝¹ : Nonempty α
this✝ : OrderBot α := WellFoundedLT.toOrderBot α
this : ConditionallyCompleteLinearOrderBot α := WellFoundedLT.conditionallyCompleteLinearOrderBot α
⊢ Subtype.val ∘ ⇑(enum s hs) '' Iio a ⊆ s

pp: case refine_2
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
s : Set α
hs : IsCofinal s
H : StrictMono (Subtype.val ∘ ⇑(enum s hs)) :=
  StrictMono.comp (Subtype.strictMono_coe fun x ↦ x ∈ s) (OrderIso.strictMono (enum s hs))
hs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.Nonempty → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s
a : α
ha : IsSuccLimit a
b : α
hb : ∀ b_1 < a, (Subtype.val ∘ ⇑(enum s hs)) b_1 ≤ b
bdd : BddAbove (Subtype.val ∘ ⇑(enum s hs) '' Iio a)
this✝¹ : Nonempty α
this✝ : OrderBot α := WellFoundedLT.toOrderBot α
this : ConditionallyCompleteLinearOrderBot α := WellFoundedLT.conditionallyCompleteLinearOrderBot α
⊢ (Subtype.val ∘ ⇑(enum s hs) '' Iio a).Nonempty

pp: case refine_3
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
s : Set α
hs : IsCofinal s
H : StrictMono (Subtype.val ∘ ⇑(enum s hs)) :=
  StrictMono.comp (Subtype.strictMono_coe fun x ↦ x ∈ s) (OrderIso.strictMono (enum s hs))
hs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.Nonempty → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s
a : α
ha : IsSuccLimit a
b✝ : α
hb✝ : ∀ b < a, (Subtype.val ∘ ⇑(enum s hs)) b ≤ b✝
bdd : BddAbove (Subtype.val ∘ ⇑(enum s hs) '' Iio a)
this✝¹ : Nonempty α
this✝ : OrderBot α := WellFoundedLT.toOrderBot α
this : ConditionallyCompleteLinearOrderBot α := WellFoundedLT.conditionallyCompleteLinearOrderBot α
b : α
hb : b < a
⊢ ↑((enum s hs) b) < sSup (Subtype.val ∘ ⇑(enum s hs) '' Iio a)

Tactic:
·

Goal state after:
case refine_1
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
s : Set α
hs : IsCofinal s
H : StrictMono (Subtype.val ∘ ⇑(enum s hs)) :=
  StrictMono.comp (Subtype.strictMono_coe fun x ↦ x ∈ s) (OrderIso.strictMono (enum s hs))
hs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.Nonempty → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s
a : α
ha : IsSuccLimit a
b : α
hb : ∀ b_1 < a, (Subtype.val ∘ ⇑(enum s hs)) b_1 ≤ b
bdd : BddAbove (Subtype.val ∘ ⇑(enum s hs) '' Iio a)
this✝¹ : Nonempty α
this✝ : OrderBot α := WellFoundedLT.toOrderBot α
this : ConditionallyCompleteLinearOrderBot α := WellFoundedLT.conditionallyCompleteLinearOrderBot α
⊢ Subtype.val ∘ ⇑(enum s hs) '' Iio a ⊆ s
Module: Mathlib.SetTheory.Ordinal.Family

Goal state before:
pp: case inr
s : Set Cardinal.{u_3}
hn : s.Nonempty
⊢ (sSup s).ord = sSup (ord '' s)

Tactic:
·

Goal state after:
case inr
s : Set Cardinal.{u_3}
hn : s.Nonempty
⊢ (sSup s).ord = sSup (ord '' s)
Module: Mathlib.SetTheory.Ordinal.Family

Goal state before:
pp: case pos
s : Set Cardinal.{u_3}
hn : s.Nonempty
hs : BddAbove s
⊢ (sSup s).ord = sSup (ord '' s)

pp: case neg
s : Set Cardinal.{u_3}
hn : s.Nonempty
hs : ¬BddAbove s
⊢ (sSup s).ord = sSup (ord '' s)

Tactic:
·

Goal state after:
case pos
s : Set Cardinal.{u_3}
hn : s.Nonempty
hs : BddAbove s
⊢ (sSup s).ord = sSup (ord '' s)
Module: Mathlib.SetTheory.Ordinal.Family

Goal state before:
pp: case neg
s : Set Cardinal.{u_3}
hn : s.Nonempty
hs : ¬BddAbove s
⊢ (sSup s).ord = sSup (ord '' s)

Tactic:
·

Goal state after:
case neg
s : Set Cardinal.{u_3}
hn : s.Nonempty
hs : ¬BddAbove s
⊢ (sSup s).ord = sSup (ord '' s)
Module: Mathlib.SetTheory.Ordinal.Arithmetic

Goal state before:
pp: case left
α✝ : Type u_1
β✝ : Type u_2
γ✝ : Type u_3
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t✝ : γ✝ → γ✝ → Prop
c a b : Ordinal.{u}
x✝³ x✝² x✝¹ : WellOrder
α : Type u
r : α → α → Prop
wo✝² : IsWellOrder α r
β : Type u
s : β → β → Prop
wo✝¹ : IsWellOrder β s
x✝ : ⟦{ α := α, r := r, wo := wo✝² }⟧ ≤ ⟦{ α := β, r := s, wo := wo✝¹ }⟧
γ : Type u
t : γ → γ → Prop
wo✝ : IsWellOrder γ t
f : InitialSeg r s
a₁✝ : γ
b₁✝ : α
a₂✝ : γ
b₂✝ : α
h' : t a₁✝ a₂✝
⊢ Prod.Lex t s ((a₁✝, b₁✝).1, f (a₁✝, b₁✝).2) ((a₂✝, b₂✝).1, f (a₂✝, b₂✝).2)

pp: case right
α✝ : Type u_1
β✝ : Type u_2
γ✝ : Type u_3
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t✝ : γ✝ → γ✝ → Prop
c a b : Ordinal.{u}
x✝³ x✝² x✝¹ : WellOrder
α : Type u
r : α → α → Prop
wo✝² : IsWellOrder α r
β : Type u
s : β → β → Prop
wo✝¹ : IsWellOrder β s
x✝ : ⟦{ α := α, r := r, wo := wo✝² }⟧ ≤ ⟦{ α := β, r := s, wo := wo✝¹ }⟧
γ : Type u
t : γ → γ → Prop
wo✝ : IsWellOrder γ t
f : InitialSeg r s
a✝ : γ
b₁✝ b₂✝ : α
h' : r b₁✝ b₂✝
⊢ Prod.Lex t s ((a✝, b₁✝).1, f (a✝, b₁✝).2) ((a✝, b₂✝).1, f (a✝, b₂✝).2)

Tactic:
·

Goal state after:
case left
α✝ : Type u_1
β✝ : Type u_2
γ✝ : Type u_3
r✝ : α✝ → α✝ → Prop
s✝ : β✝ → β✝ → Prop
t✝ : γ✝ → γ✝ → Prop
c a b : Ordinal.{u}
x✝³ x✝² x✝¹ : WellOrder
α : Type u
r : α → α → Prop
wo✝² : IsWellOrder α r
β : Type u
s : β → β → Prop
wo✝¹ : IsWellOrder β s
x✝ : ⟦{ α := α, r := r, wo := wo✝² }⟧ ≤ ⟦{ α := β, r := s, wo := wo✝¹ }⟧
γ : Type u
t : γ → γ → Prop
wo✝ : IsWellOrder γ t
f : InitialSeg r s
a₁✝ : γ
b₁✝ : α
a₂✝ : γ
b₂✝ : α
h' : t a₁✝ a₂✝
⊢ Prod.Lex t s ((a₁✝, b₁✝).1, f (a₁✝, b₁✝).2) ((a₂✝, b₂✝).1, f (a₂✝, b₂✝).2)
Module: Mathlib.SetTheory.Cardinal.Cofinality.Enum

Goal state before:
pp: case refine_2
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
s : Set α
hs : IsCofinal s
H : StrictMono (Subtype.val ∘ ⇑(enum s hs)) :=
  StrictMono.comp (Subtype.strictMono_coe fun x ↦ x ∈ s) (OrderIso.strictMono (enum s hs))
hs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.Nonempty → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s
a : α
ha : IsSuccLimit a
b : α
hb : ∀ b_1 < a, (Subtype.val ∘ ⇑(enum s hs)) b_1 ≤ b
bdd : BddAbove (Subtype.val ∘ ⇑(enum s hs) '' Iio a)
this✝¹ : Nonempty α
this✝ : OrderBot α := WellFoundedLT.toOrderBot α
this : ConditionallyCompleteLinearOrderBot α := WellFoundedLT.conditionallyCompleteLinearOrderBot α
⊢ (Subtype.val ∘ ⇑(enum s hs) '' Iio a).Nonempty

pp: case refine_3
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
s : Set α
hs : IsCofinal s
H : StrictMono (Subtype.val ∘ ⇑(enum s hs)) :=
  StrictMono.comp (Subtype.strictMono_coe fun x ↦ x ∈ s) (OrderIso.strictMono (enum s hs))
hs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.Nonempty → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s
a : α
ha : IsSuccLimit a
b✝ : α
hb✝ : ∀ b < a, (Subtype.val ∘ ⇑(enum s hs)) b ≤ b✝
bdd : BddAbove (Subtype.val ∘ ⇑(enum s hs) '' Iio a)
this✝¹ : Nonempty α
this✝ : OrderBot α := WellFoundedLT.toOrderBot α
this : ConditionallyCompleteLinearOrderBot α := WellFoundedLT.conditionallyCompleteLinearOrderBot α
b : α
hb : b < a
⊢ ↑((enum s hs) b) < sSup (Subtype.val ∘ ⇑(enum s hs) '' Iio a)

Tactic:
·

Goal state after:
case refine_2
α : Type u_1
inst✝² : LinearOrder α
inst✝¹ : WellFoundedLT α
inst✝ : IsRegularCardinalOrder α
s : Set α
hs : IsCofinal s
H : StrictMono (Subtype.val ∘ ⇑(enum s hs)) :=
  StrictMono.comp (Subtype.strictMono_coe fun x ↦ x ∈ s) (OrderIso.strictMono (enum s hs))
hs' : ∀ ⦃d : Set α⦄, d ⊆ s → d.Nonempty → ∀ ⦃a : α⦄, IsLUB d a → a ∈ s
a : α
ha : IsSuccLimit a
b : α
hb : ∀ b_1 < a, (Subtype.val ∘ ⇑(enum s hs)) b_1 ≤ b
bdd : BddAbove (Subtype.val ∘ ⇑(enum s hs) '' Iio a)
this✝¹ : Nonempty α
this✝ : OrderBot α := WellFoundedLT.toOrderBot α
this : ConditionallyCompleteLinearOrderBot α := WellFoundedLT.conditionallyCompleteLinearOrderBot α
⊢ (Subtype.val ∘ ⇑(enum s hs) '' Iio a).Nonempty