ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ULift.down_iSup
theorem down_iSup [SupSet α] (f : ι → ULift.{v} α) : (⨆ i, f i).down = ⨆ i, (f i).down 
Arguments: {α : Type u_1} {ι : Sort u_5} [SupSet α] (f : ι → ULift.{v, u_1} α)
Goal: (⨆ i, f i).down = ⨆ i, (f i).down
-- ULift.up_iSup
theorem up_iSup [SupSet α] (f : ι → α) : up (⨆ i, f i) = ⨆ i, up (f i) 
Arguments: {α : Type u_1} {ι : Sort u_5} [SupSet α] (f : ι → α)
Goal: { down := ⨆ i, f i } = ⨆ i, { down := f i }
-- ULift.down_iInf
theorem down_iInf [InfSet α] (f : ι → ULift.{v} α) : (⨅ i, f i).down = ⨅ i, (f i).down 
Arguments: {α : Type u_1} {ι : Sort u_5} [InfSet α] (f : ι → ULift.{v, u_1} α)
Goal: (⨅ i, f i).down = ⨅ i, (f i).down
-- ULift.up_iInf
theorem up_iInf [InfSet α] (f : ι → α) : up (⨅ i, f i) = ⨅ i, up (f i) 
Arguments: {α : Type u_1} {ι : Sort u_5} [InfSet α] (f : ι → α)
Goal: { down := ⨅ i, f i } = ⨅ i, { down := f i }
-- Quotient.mk'_eq_mk
theorem Quotient.mk'_eq_mk [s : Setoid α] : Quotient.mk' = Quotient.mk s 
Arguments: {α : Sort u} [s : Setoid α]
Goal: Quotient.mk' = Quotient.mk s
-- Order.radical_le_coatom
lemma Order.radical_le_coatom {a : α} (h : IsCoatom a) : radical α ≤ a 
Arguments: {α : Type u_1} [CompleteLattice α] {a : α} (h : IsCoatom a)
Goal: Order.radical α ≤ a
-- OrderIso.map_radical
theorem OrderIso.map_radical (f : α ≃o β) : f (Order.radical α) = Order.radical β 
Arguments: {α : Type u_1} [CompleteLattice α] {β : Type u_2} [CompleteLattice β] (f : α ≃o β)
Goal: f (Order.radical α) = Order.radical β
-- Order.radical_nongenerating
theorem Order.radical_nongenerating [IsCoatomic α] {a : α} (h : a ⊔ radical α = ⊤) : a = ⊤ 
Arguments: {α : Type u_1} [CompleteLattice α] [IsCoatomic α] {a : α} (h : a ⊔ Order.radical α = ⊤)
Goal: a = ⊤
-- DistribLattice.mem_ideal_sup_principal
lemma mem_ideal_sup_principal (a b : α) (J : Ideal α) : b ∈ J ⊔ principal a ↔ ∃ j ∈ J, b ≤ j ⊔ a 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] (a b : α) (J : Order.Ideal α)
Goal: b ∈ J ⊔ Order.Ideal.principal a ↔ ∃ j ∈ J, b ≤ j ⊔ a
-- DistribLattice.prime_ideal_of_disjoint_filter_ideal
theorem prime_ideal_of_disjoint_filter_ideal (hFI : Disjoint (F : Set α) (I : Set α)) :
    ∃ J : Ideal α, (IsPrime J) ∧ I ≤ J ∧ Disjoint (F : Set α) J 
Arguments: {α : Type u_1} [DistribLattice α] [BoundedOrder α] {F : Order.PFilter α} {I : Order.Ideal α} (hFI : Disjoint ↑F ↑I)
Goal: ∃ J, J.IsPrime ∧ I ≤ J ∧ Disjoint ↑F ↑J
-- RelEmbedding.coe_natLT
@[simp]
theorem coe_natLT {f : ℕ → α} {H : ∀ n : ℕ, r (f n) (f (n + 1))} : ⇑(natLT f H) = f 
Arguments: {α : Type u_1} {r : α → α → Prop} [IsStrictOrder α r] {f : ℕ → α} {H : ∀ (n : ℕ), r (f n) (f (n + 1))}
Goal: ⇑(RelEmbedding.natLT f H) = f
-- RelEmbedding.coe_natGT
@[simp]
theorem coe_natGT {f : ℕ → α} {H : ∀ n : ℕ, r (f (n + 1)) (f n)} : ⇑(natGT f H) = f 
Arguments: {α : Type u_1} {r : α → α → Prop} [IsStrictOrder α r] {f : ℕ → α} {H : ∀ (n : ℕ), r (f (n + 1)) (f n)}
Goal: ⇑(RelEmbedding.natGT f H) = f
-- RelEmbedding.exists_not_acc_lt_of_not_acc
theorem exists_not_acc_lt_of_not_acc {a : α} {r} (h : ¬Acc r a) : ∃ b, ¬Acc r b ∧ r b a 
Arguments: {α : Type u_1} {a : α} {r : α → α → Prop} (h : ¬Acc r a)
Goal: ∃ b, ¬Acc r b ∧ r b a
-- RelEmbedding.acc_iff_no_decreasing_seq
theorem acc_iff_no_decreasing_seq {x} :
    Acc r x ↔ IsEmpty { f : ((· > ·) : ℕ → ℕ → Prop) ↪r r // x ∈ Set.range f } 
Arguments: {α : Type u_1} {r : α → α → Prop} [IsStrictOrder α r] {x : α}
Goal: Acc r x ↔ IsEmpty { f // x ∈ Set.range ⇑f }
-- RelEmbedding.not_acc_of_decreasing_seq
theorem not_acc_of_decreasing_seq (f : ((· > ·) : ℕ → ℕ → Prop) ↪r r) (k : ℕ) : ¬Acc r (f k) 
Arguments: {α : Type u_1} {r : α → α → Prop} [IsStrictOrder α r] (f : (fun x x_1 => x > x_1) ↪r r) (k : ℕ)
Goal: ¬Acc r (f k)
-- RelEmbedding.wellFounded_iff_no_descending_seq
theorem wellFounded_iff_no_descending_seq :
    WellFounded r ↔ IsEmpty (((· > ·) : ℕ → ℕ → Prop) ↪r r) 
Arguments: {α : Type u_1} {r : α → α → Prop} [IsStrictOrder α r]
Goal: WellFounded r ↔ IsEmpty ((fun x x_1 => x > x_1) ↪r r)
-- RelEmbedding.not_wellFounded_of_decreasing_seq
theorem not_wellFounded_of_decreasing_seq (f : ((· > ·) : ℕ → ℕ → Prop) ↪r r) : ¬WellFounded r 
Arguments: {α : Type u_1} {r : α → α → Prop} [IsStrictOrder α r] (f : (fun x x_1 => x > x_1) ↪r r)
Goal: ¬WellFounded r
-- Nat.coe_orderEmbeddingOfSet
@[simp]
theorem coe_orderEmbeddingOfSet [DecidablePred (· ∈ s)] :
    ⇑(orderEmbeddingOfSet s) = (↑) ∘ Subtype.ofNat s 
Arguments: {s : Set ℕ} [Infinite ↑s] [DecidablePred fun x => x ∈ s]
Goal: ⇑(Nat.orderEmbeddingOfSet s) = Subtype.val ∘ Nat.Subtype.ofNat s
-- Nat.orderEmbeddingOfSet_apply
theorem orderEmbeddingOfSet_apply [DecidablePred (· ∈ s)] {n : ℕ} :
    orderEmbeddingOfSet s n = Subtype.ofNat s n 
Arguments: {s : Set ℕ} [Infinite ↑s] [DecidablePred fun x => x ∈ s] {n : ℕ}
Goal: (Nat.orderEmbeddingOfSet s) n = ↑(Nat.Subtype.ofNat s n)
-- Nat.Subtype.orderIsoOfNat_apply
@[simp]
theorem Subtype.orderIsoOfNat_apply [dP : DecidablePred (· ∈ s)] {n : ℕ} :
    Subtype.orderIsoOfNat s n = Subtype.ofNat s n 
Arguments: {s : Set ℕ} [Infinite ↑s] [dP : DecidablePred fun x => x ∈ s] {n : ℕ}
Goal: (Nat.Subtype.orderIsoOfNat s) n = Nat.Subtype.ofNat s n
-- Nat.orderEmbeddingOfSet_range
theorem orderEmbeddingOfSet_range [DecidablePred (· ∈ s)] :
    Set.range (Nat.orderEmbeddingOfSet s) = s 
Arguments: (s : Set ℕ) [Infinite ↑s] [DecidablePred fun x => x ∈ s]
Goal: Set.range ⇑(Nat.orderEmbeddingOfSet s) = s
-- Nat.exists_subseq_of_forall_mem_union
theorem exists_subseq_of_forall_mem_union {s t : Set α} (e : ℕ → α) (he : ∀ n, e n ∈ s ∪ t) :
    ∃ g : ℕ ↪o ℕ, (∀ n, e (g n) ∈ s) ∨ ∀ n, e (g n) ∈ t 
Arguments: {α : Type u_1} {s t : Set α} (e : ℕ → α) (he : ∀ (n : ℕ), e n ∈ s ∪ t)
Goal: ∃ g, (∀ (n : ℕ), e (g n) ∈ s) ∨ ∀ (n : ℕ), e (g n) ∈ t
-- exists_increasing_or_nonincreasing_subseq'
theorem exists_increasing_or_nonincreasing_subseq' (r : α → α → Prop) (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ n : ℕ, r (f (g n)) (f (g (n + 1)))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) 
Arguments: {α : Type u_1} (r : α → α → Prop) (f : ℕ → α)
Goal: ∃ g, (∀ (n : ℕ), r (f (g n)) (f (g (n + 1)))) ∨ ∀ (m n : ℕ), m < n → ¬r (f (g m)) (f (g n))
-- exists_increasing_or_nonincreasing_subseq
theorem exists_increasing_or_nonincreasing_subseq (r : α → α → Prop) [IsTrans α r] (f : ℕ → α) :
    ∃ g : ℕ ↪o ℕ,
      (∀ m n : ℕ, m < n → r (f (g m)) (f (g n))) ∨ ∀ m n : ℕ, m < n → ¬r (f (g m)) (f (g n)) 
Arguments: {α : Type u_1} (r : α → α → Prop) [IsTrans α r] (f : ℕ → α)
Goal: ∃ g, (∀ (m n : ℕ), m < n → r (f (g m)) (f (g n))) ∨ ∀ (m n : ℕ), m < n → ¬r (f (g m)) (f (g n))
-- WellFounded.monotone_chain_condition'
theorem WellFounded.monotone_chain_condition' [Preorder α] :
    WellFounded ((· > ·) : α → α → Prop) ↔ ∀ a : ℕ →o α, ∃ n, ∀ m, n ≤ m → ¬a n < a m 
Arguments: {α : Type u_1} [Preorder α]
Goal: (WellFounded fun x x_1 => x > x_1) ↔ ∀ (a : ℕ →o α), ∃ n, ∀ (m : ℕ), n ≤ m → ¬a n < a m
-- WellFounded.monotone_chain_condition
theorem WellFounded.monotone_chain_condition [PartialOrder α] :
    WellFounded ((· > ·) : α → α → Prop) ↔ ∀ a : ℕ →o α, ∃ n, ∀ m, n ≤ m → a n = a m 
Arguments: {α : Type u_1} [PartialOrder α]
Goal: (WellFounded fun x x_1 => x > x_1) ↔ ∀ (a : ℕ →o α), ∃ n, ∀ (m : ℕ), n ≤ m → a n = a m
-- WellFounded.iSup_eq_monotonicSequenceLimit
theorem WellFounded.iSup_eq_monotonicSequenceLimit [CompleteLattice α]
    (h : WellFounded ((· > ·) : α → α → Prop)) (a : ℕ →o α) :
    iSup a = monotonicSequenceLimit a 
Arguments: {α : Type u_1} [CompleteLattice α] (h : WellFounded fun x x_1 => x > x_1) (a : ℕ →o α)
Goal: iSup ⇑a = monotonicSequenceLimit a
-- Estimator.bound_le
class Estimator [Preorder α] (a : Thunk α) (ε : Type*) extends EstimatorData a ε where
    bound_le e : bound e ≤ a.get
    improve_spec e : match improve e with
    | none => bound e = a.get
    | some e' => bound e < bound e'
Arguments: {α : Type u_1} [Preorder α] {a : Thunk α} {ε : Type u_3} [self : Estimator a ε] (e : ε)
Goal: EstimatorData.bound a e ≤ a.get
-- Estimator.improve_spec
class Estimator [Preorder α] (a : Thunk α) (ε : Type*) extends EstimatorData a ε where
    bound_le e : bound e ≤ a.get
    improve_spec e : match improve e with
    | none => bound e = a.get
    | some e' => bound e < bound e'
Arguments: {α : Type u_1} [Preorder α] {a : Thunk α} {ε : Type u_3} [self : Estimator a ε] (e : ε)
Goal: match EstimatorData.improve a e with
| none => EstimatorData.bound a e = a.get
| some e' => EstimatorData.bound a e < EstimatorData.bound a e'
-- Estimator.improveUntilAux_spec
theorem Estimator.improveUntilAux_spec (a : Thunk α) (p : α → Bool)
    [Estimator a ε] [WellFoundedGT (range (bound a : ε → α))] (e : ε) (r : Bool) :
    match Estimator.improveUntilAux a p e r with
    | .error _ => ¬ p a.get
    | .ok e' => p (bound a e') 
Arguments: {α : Type u_1} {ε : Type u_2} [Preorder α] (a : Thunk α) (p : α → Bool) [Estimator a ε] [WellFoundedGT ↑(Set.range (EstimatorData.bound a))] (e : ε) (r : Bool)
Goal: match Estimator.improveUntilAux a p e r with
| Except.error a_1 => ¬p a.get = true
| Except.ok e' => p (EstimatorData.bound a e') = true
-- Estimator.improveUntil_spec
theorem Estimator.improveUntil_spec
    (a : Thunk α) (p : α → Bool) [Estimator a ε] [WellFoundedGT (range (bound a : ε → α))] (e : ε) :
    match Estimator.improveUntil a p e with
    | .error _ => ¬ p a.get
    | .ok e' => p (bound a e') 
Arguments: {α : Type u_1} {ε : Type u_2} [Preorder α] (a : Thunk α) (p : α → Bool) [Estimator a ε] [WellFoundedGT ↑(Set.range (EstimatorData.bound a))] (e : ε)
Goal: match Estimator.improveUntil a p e with
| Except.error a_1 => ¬p a.get = true
| Except.ok e' => p (EstimatorData.bound a e') = true
-- instEstimatorDataHAddThunkProd_improve
@[simps]
instance [Add α] {a b : Thunk α} (εa εb : Type*) [EstimatorData a εa] [EstimatorData b εb] :
    EstimatorData (a + b) (εa × εb) 
Arguments: {α : Type u_1} [Add α] {a b : Thunk α} (εa : Type u_3) (εb : Type u_4) [EstimatorData a εa] [EstimatorData b εb] (e : εa × εb)
Goal: EstimatorData.improve (a + b) e =
  match EstimatorData.improve a e.1 with
  | some e' => some (e', e.2)
  | none =>
    match EstimatorData.improve b e.2 with
    | some e' => some (e.1, e')
    | none => none
-- instEstimatorDataHAddThunkProd_bound
@[simps]
instance [Add α] {a b : Thunk α} (εa εb : Type*) [EstimatorData a εa] [EstimatorData b εb] :
    EstimatorData (a + b) (εa × εb) 
Arguments: {α : Type u_1} [Add α] {a b : Thunk α} (εa : Type u_3) (εb : Type u_4) [EstimatorData a εa] [EstimatorData b εb] (e : εa × εb)
Goal: EstimatorData.bound (a + b) e = EstimatorData.bound a e.1 + EstimatorData.bound b e.2
-- EqvGen.is_equivalence
theorem EqvGen.is_equivalence : Equivalence (@EqvGen α r) 
Arguments: {α : Type u} (r : α → α → Prop)
Goal: Equivalence (EqvGen r)
-- Quot.exact
theorem Quot.exact {a b : α} (H : Quot.mk r a = Quot.mk r b) : EqvGen r a b 
Arguments: {α : Type u} (r : α → α → Prop) {a b : α} (H : Quot.mk r a = Quot.mk r b)
Goal: EqvGen r a b
-- Quot.EqvGen_sound
theorem Quot.EqvGen_sound {r : α → α → Prop} {a b : α} (H : EqvGen r a b) :
    Quot.mk r a = Quot.mk r b 
Arguments: {α : Type u} {r : α → α → Prop} {a b : α} (H : EqvGen r a b)
Goal: Quot.mk r a = Quot.mk r b
-- Classical.cases_true_false
@[elab_as_elim] theorem cases_true_false (p : Prop → Prop)
    (h1 : p True) (h2 : p False) (a : Prop) : p a 
Arguments: (p : Prop → Prop) (h1 : p True) (h2 : p False) (a : Prop)
Goal: p a
-- Classical.cases_on
theorem cases_on (a : Prop) {p : Prop → Prop} (h1 : p True) (h2 : p False) : p a 
Arguments: (a : Prop) {p : Prop → Prop} (h1 : p True) (h2 : p False)
Goal: p a
-- Classical.cases
theorem cases {p : Prop → Prop} (h1 : p True) (h2 : p False) (a) : p a 
Arguments: {p : Prop → Prop} (h1 : p True) (h2 : p False) (a : Prop)
Goal: p a
-- Classical.eq_false_or_eq_true
theorem eq_false_or_eq_true (a : Prop) : a = False ∨ a = True 
Arguments: (a : Prop)
Goal: a = False ∨ a = True
-- LocallyFiniteOrder.finset_mem_Icc
class LocallyFiniteOrder (α : Type*) [Preorder α] where
    finsetIcc : α → α → Finset α
    finsetIco : α → α → Finset α
    finsetIoc : α → α → Finset α
    finsetIoo : α → α → Finset α
    finset_mem_Icc : ∀ a b x : α, x ∈ finsetIcc a b ↔ a ≤ x ∧ x ≤ b
    finset_mem_Ico : ∀ a b x : α, x ∈ finsetIco a b ↔ a ≤ x ∧ x < b
    finset_mem_Ioc : ∀ a b x : α, x ∈ finsetIoc a b ↔ a < x ∧ x ≤ b
    finset_mem_Ioo : ∀ a b x : α, x ∈ finsetIoo a b ↔ a < x ∧ x < b
Arguments: {α : Type u_1} [Preorder α] [self : LocallyFiniteOrder α] (a b x : α)
Goal: x ∈ LocallyFiniteOrder.finsetIcc a b ↔ a ≤ x ∧ x ≤ b
-- LocallyFiniteOrder.finset_mem_Ioc
class LocallyFiniteOrder (α : Type*) [Preorder α] where
    finsetIcc : α → α → Finset α
    finsetIco : α → α → Finset α
    finsetIoc : α → α → Finset α
    finsetIoo : α → α → Finset α
    finset_mem_Icc : ∀ a b x : α, x ∈ finsetIcc a b ↔ a ≤ x ∧ x ≤ b
    finset_mem_Ico : ∀ a b x : α, x ∈ finsetIco a b ↔ a ≤ x ∧ x < b
    finset_mem_Ioc : ∀ a b x : α, x ∈ finsetIoc a b ↔ a < x ∧ x ≤ b
    finset_mem_Ioo : ∀ a b x : α, x ∈ finsetIoo a b ↔ a < x ∧ x < b
Arguments: {α : Type u_1} [Preorder α] [self : LocallyFiniteOrder α] (a b x : α)
Goal: x ∈ LocallyFiniteOrder.finsetIoc a b ↔ a < x ∧ x ≤ b
-- LocallyFiniteOrder.finset_mem_Ioo
class LocallyFiniteOrder (α : Type*) [Preorder α] where
    finsetIcc : α → α → Finset α
    finsetIco : α → α → Finset α
    finsetIoc : α → α → Finset α
    finsetIoo : α → α → Finset α
    finset_mem_Icc : ∀ a b x : α, x ∈ finsetIcc a b ↔ a ≤ x ∧ x ≤ b
    finset_mem_Ico : ∀ a b x : α, x ∈ finsetIco a b ↔ a ≤ x ∧ x < b
    finset_mem_Ioc : ∀ a b x : α, x ∈ finsetIoc a b ↔ a < x ∧ x ≤ b
    finset_mem_Ioo : ∀ a b x : α, x ∈ finsetIoo a b ↔ a < x ∧ x < b
Arguments: {α : Type u_1} [Preorder α] [self : LocallyFiniteOrder α] (a b x : α)
Goal: x ∈ LocallyFiniteOrder.finsetIoo a b ↔ a < x ∧ x < b
-- LocallyFiniteOrder.finset_mem_Ico
class LocallyFiniteOrder (α : Type*) [Preorder α] where
    finsetIcc : α → α → Finset α
    finsetIco : α → α → Finset α
    finsetIoc : α → α → Finset α
    finsetIoo : α → α → Finset α
    finset_mem_Icc : ∀ a b x : α, x ∈ finsetIcc a b ↔ a ≤ x ∧ x ≤ b
    finset_mem_Ico : ∀ a b x : α, x ∈ finsetIco a b ↔ a ≤ x ∧ x < b
    finset_mem_Ioc : ∀ a b x : α, x ∈ finsetIoc a b ↔ a < x ∧ x ≤ b
    finset_mem_Ioo : ∀ a b x : α, x ∈ finsetIoo a b ↔ a < x ∧ x < b
Arguments: {α : Type u_1} [Preorder α] [self : LocallyFiniteOrder α] (a b x : α)
Goal: x ∈ LocallyFiniteOrder.finsetIco a b ↔ a ≤ x ∧ x < b
-- LocallyFiniteOrderTop.finset_mem_Ici
class LocallyFiniteOrderTop (α : Type*) [Preorder α] where
    finsetIoi : α → Finset α
    finsetIci : α → Finset α
    finset_mem_Ici : ∀ a x : α, x ∈ finsetIci a ↔ a ≤ x
    finset_mem_Ioi : ∀ a x : α, x ∈ finsetIoi a ↔ a < x
Arguments: {α : Type u_1} [Preorder α] [self : LocallyFiniteOrderTop α] (a x : α)
Goal: x ∈ LocallyFiniteOrderTop.finsetIci a ↔ a ≤ x
-- LocallyFiniteOrderTop.finset_mem_Ioi
class LocallyFiniteOrderTop (α : Type*) [Preorder α] where
    finsetIoi : α → Finset α
    finsetIci : α → Finset α
    finset_mem_Ici : ∀ a x : α, x ∈ finsetIci a ↔ a ≤ x
    finset_mem_Ioi : ∀ a x : α, x ∈ finsetIoi a ↔ a < x
Arguments: {α : Type u_1} [Preorder α] [self : LocallyFiniteOrderTop α] (a x : α)
Goal: x ∈ LocallyFiniteOrderTop.finsetIoi a ↔ a < x
-- LocallyFiniteOrderBot.finset_mem_Iio
class LocallyFiniteOrderBot (α : Type*) [Preorder α] where
    finsetIio : α → Finset α
    finsetIic : α → Finset α
    finset_mem_Iic : ∀ a x : α, x ∈ finsetIic a ↔ x ≤ a
    finset_mem_Iio : ∀ a x : α, x ∈ finsetIio a ↔ x < a
Arguments: {α : Type u_1} [Preorder α] [self : LocallyFiniteOrderBot α] (a x : α)
Goal: x ∈ LocallyFiniteOrderBot.finsetIio a ↔ x < a
-- LocallyFiniteOrderBot.finset_mem_Iic
class LocallyFiniteOrderBot (α : Type*) [Preorder α] where
    finsetIio : α → Finset α
    finsetIic : α → Finset α
    finset_mem_Iic : ∀ a x : α, x ∈ finsetIic a ↔ x ≤ a
    finset_mem_Iio : ∀ a x : α, x ∈ finsetIio a ↔ x < a
Arguments: {α : Type u_1} [Preorder α] [self : LocallyFiniteOrderBot α] (a x : α)
Goal: x ∈ LocallyFiniteOrderBot.finsetIic a ↔ x ≤ a
-- Finset.mem_Icc
@[simp]
theorem mem_Icc : x ∈ Icc a b ↔ a ≤ x ∧ x ≤ b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b x : α}
Goal: x ∈ Finset.Icc a b ↔ a ≤ x ∧ x ≤ b
-- Finset.mem_Ico
@[simp]
theorem mem_Ico : x ∈ Ico a b ↔ a ≤ x ∧ x < b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b x : α}
Goal: x ∈ Finset.Ico a b ↔ a ≤ x ∧ x < b
-- Finset.mem_Ioc
@[simp]
theorem mem_Ioc : x ∈ Ioc a b ↔ a < x ∧ x ≤ b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b x : α}
Goal: x ∈ Finset.Ioc a b ↔ a < x ∧ x ≤ b
-- Finset.mem_Ioo
@[simp]
theorem mem_Ioo : x ∈ Ioo a b ↔ a < x ∧ x < b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] {a b x : α}
Goal: x ∈ Finset.Ioo a b ↔ a < x ∧ x < b
-- Finset.coe_Icc
@[simp, norm_cast]
theorem coe_Icc (a b : α) : (Icc a b : Set α) = Set.Icc a b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α)
Goal: ↑(Finset.Icc a b) = Set.Icc a b
-- Finset.coe_Ico
@[simp, norm_cast]
theorem coe_Ico (a b : α) : (Ico a b : Set α) = Set.Ico a b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α)
Goal: ↑(Finset.Ico a b) = Set.Ico a b
-- Finset.coe_Ioc
@[simp, norm_cast]
theorem coe_Ioc (a b : α) : (Ioc a b : Set α) = Set.Ioc a b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α)
Goal: ↑(Finset.Ioc a b) = Set.Ioc a b
-- Finset.coe_Ioo
@[simp, norm_cast]
theorem coe_Ioo (a b : α) : (Ioo a b : Set α) = Set.Ioo a b 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α)
Goal: ↑(Finset.Ioo a b) = Set.Ioo a b
-- Finset.mem_Ici
@[simp]
theorem mem_Ici : x ∈ Ici a ↔ a ≤ x 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] {a x : α}
Goal: x ∈ Finset.Ici a ↔ a ≤ x
-- Finset.mem_Ioi
@[simp]
theorem mem_Ioi : x ∈ Ioi a ↔ a < x 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] {a x : α}
Goal: x ∈ Finset.Ioi a ↔ a < x
-- Finset.coe_Ici
@[simp, norm_cast]
theorem coe_Ici (a : α) : (Ici a : Set α) = Set.Ici a 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] (a : α)
Goal: ↑(Finset.Ici a) = Set.Ici a
-- Finset.coe_Ioi
@[simp, norm_cast]
theorem coe_Ioi (a : α) : (Ioi a : Set α) = Set.Ioi a 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] (a : α)
Goal: ↑(Finset.Ioi a) = Set.Ioi a
-- Finset.mem_Iic
@[simp]
theorem mem_Iic : x ∈ Iic a ↔ x ≤ a 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] {a x : α}
Goal: x ∈ Finset.Iic a ↔ x ≤ a
-- Finset.mem_Iio
@[simp]
theorem mem_Iio : x ∈ Iio a ↔ x < a 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] {a x : α}
Goal: x ∈ Finset.Iio a ↔ x < a
-- Finset.coe_Iic
@[simp, norm_cast]
theorem coe_Iic (a : α) : (Iic a : Set α) = Set.Iic a 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] (a : α)
Goal: ↑(Finset.Iic a) = Set.Iic a
-- Finset.coe_Iio
@[simp, norm_cast]
theorem coe_Iio (a : α) : (Iio a : Set α) = Set.Iio a 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] (a : α)
Goal: ↑(Finset.Iio a) = Set.Iio a
-- Finset.Ici_eq_Icc
theorem Ici_eq_Icc (a : α) : Ici a = Icc a ⊤ 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] [OrderTop α] (a : α)
Goal: Finset.Ici a = Finset.Icc a ⊤
-- Finset.Ioi_eq_Ioc
theorem Ioi_eq_Ioc (a : α) : Ioi a = Ioc a ⊤ 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] [OrderTop α] (a : α)
Goal: Finset.Ioi a = Finset.Ioc a ⊤
-- Finset.Iic_eq_Icc
theorem Iic_eq_Icc : Iic = Icc (⊥ : α) 
Arguments: {α : Type u_1} [Preorder α] [OrderBot α] [LocallyFiniteOrder α]
Goal: Finset.Iic = Finset.Icc ⊥
-- Finset.Iio_eq_Ico
theorem Iio_eq_Ico : Iio = Ico (⊥ : α) 
Arguments: {α : Type u_1} [Preorder α] [OrderBot α] [LocallyFiniteOrder α]
Goal: Finset.Iio = Finset.Ico ⊥
-- Finset.mem_uIcc
@[simp]
theorem mem_uIcc : x ∈ uIcc a b ↔ a ⊓ b ≤ x ∧ x ≤ a ⊔ b 
Arguments: {α : Type u_1} [Lattice α] [LocallyFiniteOrder α] {a b x : α}
Goal: x ∈ Finset.uIcc a b ↔ a ⊓ b ≤ x ∧ x ≤ a ⊔ b
-- Finset.coe_uIcc
@[simp, norm_cast]
theorem coe_uIcc (a b : α) : (Finset.uIcc a b : Set α) = Set.uIcc a b 
Arguments: {α : Type u_1} [Lattice α] [LocallyFiniteOrder α] (a b : α)
Goal: ↑(Finset.uIcc a b) = Set.uIcc a b
-- Set.finite_Icc
theorem finite_Icc : (Icc a b).Finite 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α)
Goal: (Set.Icc a b).Finite
-- Set.finite_Ico
theorem finite_Ico : (Ico a b).Finite 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α)
Goal: (Set.Ico a b).Finite
-- Set.finite_Ioc
theorem finite_Ioc : (Ioc a b).Finite 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α)
Goal: (Set.Ioc a b).Finite
-- Set.finite_Ioo
theorem finite_Ioo : (Ioo a b).Finite 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α)
Goal: (Set.Ioo a b).Finite
-- Set.finite_Ici
theorem finite_Ici : (Ici a).Finite 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] (a : α)
Goal: (Set.Ici a).Finite
-- Set.finite_Ioi
theorem finite_Ioi : (Ioi a).Finite 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] (a : α)
Goal: (Set.Ioi a).Finite
-- Set.finite_Iic
theorem finite_Iic : (Iic b).Finite 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] (b : α)
Goal: (Set.Iic b).Finite
-- Set.finite_Iio
theorem finite_Iio : (Iio b).Finite 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderBot α] (b : α)
Goal: (Set.Iio b).Finite
-- Set.finite_interval
@[simp]
theorem finite_interval : (uIcc a b).Finite 
Arguments: {α : Type u_1} [Lattice α] [LocallyFiniteOrder α] (a b : α)
Goal: (Set.uIcc a b).Finite
-- Icc_toDual
theorem Icc_toDual : Icc (toDual a) (toDual b) = (Icc b a).map toDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α)
Goal: Finset.Icc (OrderDual.toDual a) (OrderDual.toDual b) = Finset.map OrderDual.toDual.toEmbedding (Finset.Icc b a)
-- Ico_toDual
theorem Ico_toDual : Ico (toDual a) (toDual b) = (Ioc b a).map toDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α)
Goal: Finset.Ico (OrderDual.toDual a) (OrderDual.toDual b) = Finset.map OrderDual.toDual.toEmbedding (Finset.Ioc b a)
-- Ioc_toDual
theorem Ioc_toDual : Ioc (toDual a) (toDual b) = (Ico b a).map toDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α)
Goal: Finset.Ioc (OrderDual.toDual a) (OrderDual.toDual b) = Finset.map OrderDual.toDual.toEmbedding (Finset.Ico b a)
-- Ioo_toDual
theorem Ioo_toDual : Ioo (toDual a) (toDual b) = (Ioo b a).map toDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : α)
Goal: Finset.Ioo (OrderDual.toDual a) (OrderDual.toDual b) = Finset.map OrderDual.toDual.toEmbedding (Finset.Ioo b a)
-- Icc_ofDual
theorem Icc_ofDual (a b : αᵒᵈ) : Icc (ofDual a) (ofDual b) = (Icc b a).map ofDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : αᵒᵈ)
Goal: Finset.Icc (OrderDual.ofDual a) (OrderDual.ofDual b) = Finset.map OrderDual.ofDual.toEmbedding (Finset.Icc b a)
-- Ico_ofDual
theorem Ico_ofDual (a b : αᵒᵈ) : Ico (ofDual a) (ofDual b) = (Ioc b a).map ofDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : αᵒᵈ)
Goal: Finset.Ico (OrderDual.ofDual a) (OrderDual.ofDual b) = Finset.map OrderDual.ofDual.toEmbedding (Finset.Ioc b a)
-- Ioc_ofDual
theorem Ioc_ofDual (a b : αᵒᵈ) : Ioc (ofDual a) (ofDual b) = (Ico b a).map ofDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : αᵒᵈ)
Goal: Finset.Ioc (OrderDual.ofDual a) (OrderDual.ofDual b) = Finset.map OrderDual.ofDual.toEmbedding (Finset.Ico b a)
-- Ioo_ofDual
theorem Ioo_ofDual (a b : αᵒᵈ) : Ioo (ofDual a) (ofDual b) = (Ioo b a).map ofDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrder α] (a b : αᵒᵈ)
Goal: Finset.Ioo (OrderDual.ofDual a) (OrderDual.ofDual b) = Finset.map OrderDual.ofDual.toEmbedding (Finset.Ioo b a)
-- Iic_toDual
theorem Iic_toDual (a : α) : Iic (toDual a) = (Ici a).map toDual.toEmbedding 
Arguments: {α : Type u_1} [Preorder α] [LocallyFiniteOrderTop α] (a : α)
Goal: Finset.Iic (OrderDual.toDual a) = Finset.map OrderDual.toDual.toEmbedding (Finset.Ici a)