ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Set.ordConnected_iff_uIcc_subset_right
theorem ordConnected_iff_uIcc_subset_right (hx : x ∈ s) :
    OrdConnected s ↔ ∀ ⦃y⦄, y ∈ s → [[y, x]] ⊆ s 
Arguments: {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} (hx : x ∈ s)
Goal: s.OrdConnected ↔ ∀ ⦃y : α⦄, y ∈ s → [[y, x]] ⊆ s
-- surjOn_Ioo_of_monotone_surjective
theorem surjOn_Ioo_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
    (a b : α) : SurjOn f (Ioo a b) (Ioo (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [PartialOrder β] {f : α → β} (h_mono : Monotone f) (h_surj : Function.Surjective f) (a b : α)
Goal: Set.SurjOn f (Set.Ioo a b) (Set.Ioo (f a) (f b))
-- surjOn_Ico_of_monotone_surjective
theorem surjOn_Ico_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
    (a b : α) : SurjOn f (Ico a b) (Ico (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [PartialOrder β] {f : α → β} (h_mono : Monotone f) (h_surj : Function.Surjective f) (a b : α)
Goal: Set.SurjOn f (Set.Ico a b) (Set.Ico (f a) (f b))
-- surjOn_Ioc_of_monotone_surjective
theorem surjOn_Ioc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
    (a b : α) : SurjOn f (Ioc a b) (Ioc (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [PartialOrder β] {f : α → β} (h_mono : Monotone f) (h_surj : Function.Surjective f) (a b : α)
Goal: Set.SurjOn f (Set.Ioc a b) (Set.Ioc (f a) (f b))
-- surjOn_Icc_of_monotone_surjective
theorem surjOn_Icc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
    {a b : α} (hab : a ≤ b) : SurjOn f (Icc a b) (Icc (f a) (f b)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [PartialOrder β] {f : α → β} (h_mono : Monotone f) (h_surj : Function.Surjective f) {a b : α} (hab : a ≤ b)
Goal: Set.SurjOn f (Set.Icc a b) (Set.Icc (f a) (f b))
-- surjOn_Ioi_of_monotone_surjective
theorem surjOn_Ioi_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
    (a : α) : SurjOn f (Ioi a) (Ioi (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [PartialOrder β] {f : α → β} (h_mono : Monotone f) (h_surj : Function.Surjective f) (a : α)
Goal: Set.SurjOn f (Set.Ioi a) (Set.Ioi (f a))
-- surjOn_Iio_of_monotone_surjective
theorem surjOn_Iio_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
    (a : α) : SurjOn f (Iio a) (Iio (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [PartialOrder β] {f : α → β} (h_mono : Monotone f) (h_surj : Function.Surjective f) (a : α)
Goal: Set.SurjOn f (Set.Iio a) (Set.Iio (f a))
-- surjOn_Ici_of_monotone_surjective
theorem surjOn_Ici_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
    (a : α) : SurjOn f (Ici a) (Ici (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [PartialOrder β] {f : α → β} (h_mono : Monotone f) (h_surj : Function.Surjective f) (a : α)
Goal: Set.SurjOn f (Set.Ici a) (Set.Ici (f a))
-- surjOn_Iic_of_monotone_surjective
theorem surjOn_Iic_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
    (a : α) : SurjOn f (Iic a) (Iic (f a)) 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [PartialOrder β] {f : α → β} (h_mono : Monotone f) (h_surj : Function.Surjective f) (a : α)
Goal: Set.SurjOn f (Set.Iic a) (Set.Iic (f a))
-- IsWeakUpperModularLattice.covBy_sup_of_inf_covBy_covBy
class IsWeakUpperModularLattice (α : Type*) [Lattice α] : Prop where
  covBy_sup_of_inf_covBy_covBy {a b : α} : a ⊓ b ⋖ a → a ⊓ b ⋖ b → a ⋖ a ⊔ b
Arguments: {α : Type u_2} [Lattice α] [self : IsWeakUpperModularLattice α] {a b : α}
Goal: a ⊓ b ⋖ a → a ⊓ b ⋖ b → a ⋖ a ⊔ b
-- IsWeakLowerModularLattice.inf_covBy_of_covBy_covBy_sup
class IsWeakLowerModularLattice (α : Type*) [Lattice α] : Prop where
  inf_covBy_of_covBy_covBy_sup {a b : α} : a ⋖ a ⊔ b → b ⋖ a ⊔ b → a ⊓ b ⋖ a
Arguments: {α : Type u_2} [Lattice α] [self : IsWeakLowerModularLattice α] {a b : α}
Goal: a ⋖ a ⊔ b → b ⋖ a ⊔ b → a ⊓ b ⋖ a
-- IsUpperModularLattice.covBy_sup_of_inf_covBy
class IsUpperModularLattice (α : Type*) [Lattice α] : Prop where
  covBy_sup_of_inf_covBy {a b : α} : a ⊓ b ⋖ a → b ⋖ a ⊔ b
Arguments: {α : Type u_2} [Lattice α] [self : IsUpperModularLattice α] {a b : α}
Goal: a ⊓ b ⋖ a → b ⋖ a ⊔ b
-- IsLowerModularLattice.inf_covBy_of_covBy_sup
class IsLowerModularLattice (α : Type*) [Lattice α] : Prop where
  inf_covBy_of_covBy_sup {a b : α} : a ⋖ a ⊔ b → a ⊓ b ⋖ b
Arguments: {α : Type u_2} [Lattice α] [self : IsLowerModularLattice α] {a b : α}
Goal: a ⋖ a ⊔ b → a ⊓ b ⋖ b
-- IsModularLattice.sup_inf_le_assoc_of_le
class IsModularLattice (α : Type*) [Lattice α] : Prop where
  sup_inf_le_assoc_of_le : ∀ {x : α} (y : α) {z : α}, x ≤ z → (x ⊔ y) ⊓ z ≤ x ⊔ y ⊓ z
Arguments: {α : Type u_2} [Lattice α] [self : IsModularLattice α] {x : α} (y : α) {z : α}
Goal: x ≤ z → (x ⊔ y) ⊓ z ≤ x ⊔ y ⊓ z
-- covBy_sup_of_inf_covBy_of_inf_covBy_left
theorem covBy_sup_of_inf_covBy_of_inf_covBy_left : a ⊓ b ⋖ a → a ⊓ b ⋖ b → a ⋖ a ⊔ b 
Arguments: {α : Type u_1} [Lattice α] [IsWeakUpperModularLattice α] {a b : α}
Goal: a ⊓ b ⋖ a → a ⊓ b ⋖ b → a ⋖ a ⊔ b
-- covBy_sup_of_inf_covBy_of_inf_covBy_right
theorem covBy_sup_of_inf_covBy_of_inf_covBy_right : a ⊓ b ⋖ a → a ⊓ b ⋖ b → b ⋖ a ⊔ b 
Arguments: {α : Type u_1} [Lattice α] [IsWeakUpperModularLattice α] {a b : α}
Goal: a ⊓ b ⋖ a → a ⊓ b ⋖ b → b ⋖ a ⊔ b
-- inf_covBy_of_covBy_sup_of_covBy_sup_left
theorem inf_covBy_of_covBy_sup_of_covBy_sup_left : a ⋖ a ⊔ b → b ⋖ a ⊔ b → a ⊓ b ⋖ a 
Arguments: {α : Type u_1} [Lattice α] [IsWeakLowerModularLattice α] {a b : α}
Goal: a ⋖ a ⊔ b → b ⋖ a ⊔ b → a ⊓ b ⋖ a
-- inf_covBy_of_covBy_sup_of_covBy_sup_right
theorem inf_covBy_of_covBy_sup_of_covBy_sup_right : a ⋖ a ⊔ b → b ⋖ a ⊔ b → a ⊓ b ⋖ b 
Arguments: {α : Type u_1} [Lattice α] [IsWeakLowerModularLattice α] {a b : α}
Goal: a ⋖ a ⊔ b → b ⋖ a ⊔ b → a ⊓ b ⋖ b
-- covBy_sup_of_inf_covBy_left
theorem covBy_sup_of_inf_covBy_left : a ⊓ b ⋖ a → b ⋖ a ⊔ b 
Arguments: {α : Type u_1} [Lattice α] [IsUpperModularLattice α] {a b : α}
Goal: a ⊓ b ⋖ a → b ⋖ a ⊔ b
-- covBy_sup_of_inf_covBy_right
theorem covBy_sup_of_inf_covBy_right : a ⊓ b ⋖ b → a ⋖ a ⊔ b 
Arguments: {α : Type u_1} [Lattice α] [IsUpperModularLattice α] {a b : α}
Goal: a ⊓ b ⋖ b → a ⋖ a ⊔ b
-- inf_covBy_of_covBy_sup_left
theorem inf_covBy_of_covBy_sup_left : a ⋖ a ⊔ b → a ⊓ b ⋖ b 
Arguments: {α : Type u_1} [Lattice α] [IsLowerModularLattice α] {a b : α}
Goal: a ⋖ a ⊔ b → a ⊓ b ⋖ b
-- inf_covBy_of_covBy_sup_right
theorem inf_covBy_of_covBy_sup_right : b ⋖ a ⊔ b → a ⊓ b ⋖ a 
Arguments: {α : Type u_1} [Lattice α] [IsLowerModularLattice α] {a b : α}
Goal: b ⋖ a ⊔ b → a ⊓ b ⋖ a
-- sup_inf_assoc_of_le
theorem sup_inf_assoc_of_le {x : α} (y : α) {z : α} (h : x ≤ z) : (x ⊔ y) ⊓ z = x ⊔ y ⊓ z 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] {x : α} (y : α) {z : α} (h : x ≤ z)
Goal: (x ⊔ y) ⊓ z = x ⊔ y ⊓ z
-- IsModularLattice.inf_sup_inf_assoc
theorem IsModularLattice.inf_sup_inf_assoc {x y z : α} : x ⊓ z ⊔ y ⊓ z = (x ⊓ z ⊔ y) ⊓ z 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] {x y z : α}
Goal: x ⊓ z ⊔ y ⊓ z = (x ⊓ z ⊔ y) ⊓ z
-- inf_sup_assoc_of_le
theorem inf_sup_assoc_of_le {x : α} (y : α) {z : α} (h : z ≤ x) : x ⊓ y ⊔ z = x ⊓ (y ⊔ z) 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] {x : α} (y : α) {z : α} (h : z ≤ x)
Goal: x ⊓ y ⊔ z = x ⊓ (y ⊔ z)
-- IsModularLattice.sup_inf_sup_assoc
theorem IsModularLattice.sup_inf_sup_assoc : (x ⊔ z) ⊓ (y ⊔ z) = (x ⊔ z) ⊓ y ⊔ z 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] {x y z : α}
Goal: (x ⊔ z) ⊓ (y ⊔ z) = (x ⊔ z) ⊓ y ⊔ z
-- eq_of_le_of_inf_le_of_le_sup
theorem eq_of_le_of_inf_le_of_le_sup (hxy : x ≤ y) (hinf : y ⊓ z ≤ x) (hsup : y ≤ x ⊔ z) :
    x = y 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] {x y z : α} (hxy : x ≤ y) (hinf : y ⊓ z ≤ x) (hsup : y ≤ x ⊔ z)
Goal: x = y
-- eq_of_le_of_inf_le_of_sup_le
theorem eq_of_le_of_inf_le_of_sup_le (hxy : x ≤ y) (hinf : y ⊓ z ≤ x ⊓ z) (hsup : y ⊔ z ≤ x ⊔ z) :
    x = y 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] {x y z : α} (hxy : x ≤ y) (hinf : y ⊓ z ≤ x ⊓ z) (hsup : y ⊔ z ≤ x ⊔ z)
Goal: x = y
-- sup_lt_sup_of_lt_of_inf_le_inf
theorem sup_lt_sup_of_lt_of_inf_le_inf (hxy : x < y) (hinf : y ⊓ z ≤ x ⊓ z) : x ⊔ z < y ⊔ z 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] {x y z : α} (hxy : x < y) (hinf : y ⊓ z ≤ x ⊓ z)
Goal: x ⊔ z < y ⊔ z
-- inf_lt_inf_of_lt_of_sup_le_sup
theorem inf_lt_inf_of_lt_of_sup_le_sup (hxy : x < y) (hinf : y ⊔ z ≤ x ⊔ z) : x ⊓ z < y ⊓ z 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] {x y z : α} (hxy : x < y) (hinf : y ⊔ z ≤ x ⊔ z)
Goal: x ⊓ z < y ⊓ z
-- wellFounded_lt_exact_sequence
theorem wellFounded_lt_exact_sequence {β γ : Type*} [PartialOrder β] [Preorder γ]
    (h₁ : WellFounded ((· < ·) : β → β → Prop)) (h₂ : WellFounded ((· < ·) : γ → γ → Prop)) (K : α)
    (f₁ : β → α) (f₂ : α → β) (g₁ : γ → α) (g₂ : α → γ) (gci : GaloisCoinsertion f₁ f₂)
    (gi : GaloisInsertion g₂ g₁) (hf : ∀ a, f₁ (f₂ a) = a ⊓ K) (hg : ∀ a, g₁ (g₂ a) = a ⊔ K) :
    WellFounded ((· < ·) : α → α → Prop) 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] {β : Type u_2} {γ : Type u_3} [PartialOrder β] [Preorder γ] (h₁ : WellFounded fun x x_1 => x < x_1) (h₂ : WellFounded fun x x_1 => x < x_1) (K : α) (f₁ : β → α) (f₂ : α → β) (g₁ : γ → α) (g₂ : α → γ) (gci : GaloisCoinsertion f₁ f₂) (gi : GaloisInsertion g₂ g₁) (hf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K) (hg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K)
Goal: WellFounded fun x x_1 => x < x_1
-- wellFounded_gt_exact_sequence
theorem wellFounded_gt_exact_sequence {β γ : Type*} [Preorder β] [PartialOrder γ]
    (h₁ : WellFounded ((· > ·) : β → β → Prop)) (h₂ : WellFounded ((· > ·) : γ → γ → Prop)) (K : α)
    (f₁ : β → α) (f₂ : α → β) (g₁ : γ → α) (g₂ : α → γ) (gci : GaloisCoinsertion f₁ f₂)
    (gi : GaloisInsertion g₂ g₁) (hf : ∀ a, f₁ (f₂ a) = a ⊓ K) (hg : ∀ a, g₁ (g₂ a) = a ⊔ K) :
    WellFounded ((· > ·) : α → α → Prop) 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] {β : Type u_2} {γ : Type u_3} [Preorder β] [PartialOrder γ] (h₁ : WellFounded fun x x_1 => x > x_1) (h₂ : WellFounded fun x x_1 => x > x_1) (K : α) (f₁ : β → α) (f₂ : α → β) (g₁ : γ → α) (g₂ : α → γ) (gci : GaloisCoinsertion f₁ f₂) (gi : GaloisInsertion g₂ g₁) (hf : ∀ (a : α), f₁ (f₂ a) = a ⊓ K) (hg : ∀ (a : α), g₁ (g₂ a) = a ⊔ K)
Goal: WellFounded fun x x_1 => x > x_1
-- infIccOrderIsoIccSup_apply_coe
@[simps]
def infIccOrderIsoIccSup (a b : α) : Set.Icc (a ⊓ b) a ≃o Set.Icc b (a ⊔ b) 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] (a b : α) (x : ↑(Set.Icc (a ⊓ b) a))
Goal: ↑((infIccOrderIsoIccSup a b) x) = ↑x ⊔ b
-- infIccOrderIsoIccSup_symm_apply_coe
@[simps]
def infIccOrderIsoIccSup (a b : α) : Set.Icc (a ⊓ b) a ≃o Set.Icc b (a ⊔ b) 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] (a b : α) (x : ↑(Set.Icc b (a ⊔ b)))
Goal: ↑((RelIso.symm (infIccOrderIsoIccSup a b)) x) = a ⊓ ↑x
-- inf_strictMonoOn_Icc_sup
theorem inf_strictMonoOn_Icc_sup {a b : α} : StrictMonoOn (fun c => a ⊓ c) (Icc b (a ⊔ b)) 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] {a b : α}
Goal: StrictMonoOn (fun c => a ⊓ c) (Set.Icc b (a ⊔ b))
-- sup_strictMonoOn_Icc_inf
theorem sup_strictMonoOn_Icc_inf {a b : α} : StrictMonoOn (fun c => c ⊔ b) (Icc (a ⊓ b) a) 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] {a b : α}
Goal: StrictMonoOn (fun c => c ⊔ b) (Set.Icc (a ⊓ b) a)
-- infIooOrderIsoIooSup_apply_coe
@[simps]
def infIooOrderIsoIooSup (a b : α) : Ioo (a ⊓ b) a ≃o Ioo b (a ⊔ b) 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] (a b : α) (c : ↑(Set.Ioo (a ⊓ b) a))
Goal: ↑((infIooOrderIsoIooSup a b) c) = ↑c ⊔ b
-- infIooOrderIsoIooSup_symm_apply_coe
@[simps]
def infIooOrderIsoIooSup (a b : α) : Ioo (a ⊓ b) a ≃o Ioo b (a ⊔ b) 
Arguments: {α : Type u_1} [Lattice α] [IsModularLattice α] (a b : α) (c : ↑(Set.Ioo b (a ⊔ b)))
Goal: ↑((RelIso.symm (infIooOrderIsoIooSup a b)) c) = a ⊓ ↑c
-- isModularLattice_iff_inf_sup_inf_assoc
theorem isModularLattice_iff_inf_sup_inf_assoc [Lattice α] :
    IsModularLattice α ↔ ∀ x y z : α, x ⊓ z ⊔ y ⊓ z = (x ⊓ z ⊔ y) ⊓ z 
Arguments: {α : Type u_1} [Lattice α]
Goal: IsModularLattice α ↔ ∀ (x y z : α), x ⊓ z ⊔ y ⊓ z = (x ⊓ z ⊔ y) ⊓ z
-- Disjoint.disjoint_sup_right_of_disjoint_sup_left
theorem disjoint_sup_right_of_disjoint_sup_left [Lattice α] [OrderBot α]
    [IsModularLattice α] (h : Disjoint a b) (hsup : Disjoint (a ⊔ b) c) :
    Disjoint a (b ⊔ c) 
Arguments: {α : Type u_1} {a b c : α} [Lattice α] [OrderBot α] [IsModularLattice α] (h : Disjoint a b) (hsup : Disjoint (a ⊔ b) c)
Goal: Disjoint a (b ⊔ c)
-- Disjoint.disjoint_sup_left_of_disjoint_sup_right
theorem disjoint_sup_left_of_disjoint_sup_right [Lattice α] [OrderBot α]
    [IsModularLattice α] (h : Disjoint b c) (hsup : Disjoint a (b ⊔ c)) :
    Disjoint (a ⊔ b) c 
Arguments: {α : Type u_1} {a b c : α} [Lattice α] [OrderBot α] [IsModularLattice α] (h : Disjoint b c) (hsup : Disjoint a (b ⊔ c))
Goal: Disjoint (a ⊔ b) c
-- Disjoint.isCompl_sup_right_of_isCompl_sup_left
theorem isCompl_sup_right_of_isCompl_sup_left [Lattice α] [BoundedOrder α] [IsModularLattice α]
    (h : Disjoint a b) (hcomp : IsCompl (a ⊔ b) c) :
    IsCompl a (b ⊔ c) 
Arguments: {α : Type u_1} {a b c : α} [Lattice α] [BoundedOrder α] [IsModularLattice α] (h : Disjoint a b) (hcomp : IsCompl (a ⊔ b) c)
Goal: IsCompl a (b ⊔ c)
-- Disjoint.isCompl_sup_left_of_isCompl_sup_right
theorem isCompl_sup_left_of_isCompl_sup_right [Lattice α] [BoundedOrder α] [IsModularLattice α]
    (h : Disjoint b c) (hcomp : IsCompl a (b ⊔ c)) :
    IsCompl (a ⊔ b) c 
Arguments: {α : Type u_1} {a b c : α} [Lattice α] [BoundedOrder α] [IsModularLattice α] (h : Disjoint b c) (hcomp : IsCompl a (b ⊔ c))
Goal: IsCompl (a ⊔ b) c
-- Set.Iic.isCompl_inf_inf_of_isCompl_of_le
lemma Set.Iic.isCompl_inf_inf_of_isCompl_of_le [Lattice α] [BoundedOrder α] [IsModularLattice α]
    {a b c : α} (h₁ : IsCompl b c) (h₂ : b ≤ a) :
    IsCompl (⟨a ⊓ b, inf_le_left⟩ : Iic a) (⟨a ⊓ c, inf_le_left⟩ : Iic a) 
Arguments: {α : Type u_1} [Lattice α] [BoundedOrder α] [IsModularLattice α] {a b c : α} (h₁ : IsCompl b c) (h₂ : b ≤ a)
Goal: IsCompl ⟨a ⊓ b, ⋯⟩ ⟨a ⊓ c, ⋯⟩
-- LinearLocallyFiniteOrder.succFn_spec
theorem succFn_spec (i : ι) : IsGLB (Set.Ioi i) (succFn i) 
Arguments: {ι : Type u_1} [LinearOrder ι] (i : ι)
Goal: IsGLB (Set.Ioi i) (LinearLocallyFiniteOrder.succFn i)
-- LinearLocallyFiniteOrder.le_succFn
theorem le_succFn (i : ι) : i ≤ succFn i 
Arguments: {ι : Type u_1} [LinearOrder ι] (i : ι)
Goal: i ≤ LinearLocallyFiniteOrder.succFn i
-- LinearLocallyFiniteOrder.isGLB_Ioc_of_isGLB_Ioi
theorem isGLB_Ioc_of_isGLB_Ioi {i j k : ι} (hij_lt : i < j) (h : IsGLB (Set.Ioi i) k) :
    IsGLB (Set.Ioc i j) k 
Arguments: {ι : Type u_1} [LinearOrder ι] {i j k : ι} (hij_lt : i < j) (h : IsGLB (Set.Ioi i) k)
Goal: IsGLB (Set.Ioc i j) k
-- LinearLocallyFiniteOrder.isMax_of_succFn_le
theorem isMax_of_succFn_le [LocallyFiniteOrder ι] (i : ι) (hi : succFn i ≤ i) : IsMax i 
Arguments: {ι : Type u_1} [LinearOrder ι] [LocallyFiniteOrder ι] (i : ι) (hi : LinearLocallyFiniteOrder.succFn i ≤ i)
Goal: IsMax i
-- LinearLocallyFiniteOrder.succFn_le_of_lt
theorem succFn_le_of_lt (i j : ι) (hij : i < j) : succFn i ≤ j 
Arguments: {ι : Type u_1} [LinearOrder ι] (i j : ι) (hij : i < j)
Goal: LinearLocallyFiniteOrder.succFn i ≤ j
-- LinearLocallyFiniteOrder.le_of_lt_succFn
theorem le_of_lt_succFn (j i : ι) (hij : j < succFn i) : j ≤ i 
Arguments: {ι : Type u_1} [LinearOrder ι] (j i : ι) (hij : j < LinearLocallyFiniteOrder.succFn i)
Goal: j ≤ i
-- toZ_of_ge
theorem toZ_of_ge (hi : i0 ≤ i) : toZ i0 i = Nat.find (exists_succ_iterate_of_le hi) 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 i : ι} (hi : i0 ≤ i)
Goal: toZ i0 i = ↑(Nat.find ⋯)
-- toZ_of_lt
theorem toZ_of_lt (hi : i < i0) : toZ i0 i = -Nat.find (exists_pred_iterate_of_le hi.le) 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 i : ι} (hi : i < i0)
Goal: toZ i0 i = -↑(Nat.find ⋯)
-- toZ_of_eq
@[simp]
theorem toZ_of_eq : toZ i0 i0 = 0 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 : ι}
Goal: toZ i0 i0 = 0
-- iterate_succ_toZ
theorem iterate_succ_toZ (i : ι) (hi : i0 ≤ i) : succ^[(toZ i0 i).toNat] i0 = i 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 : ι} (i : ι) (hi : i0 ≤ i)
Goal: Order.succ^[(toZ i0 i).toNat] i0 = i
-- iterate_pred_toZ
theorem iterate_pred_toZ (i : ι) (hi : i < i0) : pred^[(-toZ i0 i).toNat] i0 = i 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 : ι} (i : ι) (hi : i < i0)
Goal: Order.pred^[(-toZ i0 i).toNat] i0 = i
-- toZ_nonneg
lemma toZ_nonneg (hi : i0 ≤ i) : 0 ≤ toZ i0 i 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 i : ι} (hi : i0 ≤ i)
Goal: 0 ≤ toZ i0 i
-- toZ_neg
theorem toZ_neg (hi : i < i0) : toZ i0 i < 0 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 i : ι} (hi : i < i0)
Goal: toZ i0 i < 0
-- toZ_iterate_succ_le
theorem toZ_iterate_succ_le (n : ℕ) : toZ i0 (succ^[n] i0) ≤ n 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 : ι} (n : ℕ)
Goal: toZ i0 (Order.succ^[n] i0) ≤ ↑n
-- toZ_iterate_pred_ge
theorem toZ_iterate_pred_ge (n : ℕ) : -(n : ℤ) ≤ toZ i0 (pred^[n] i0) 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 : ι} (n : ℕ)
Goal: -↑n ≤ toZ i0 (Order.pred^[n] i0)
-- toZ_iterate_succ_of_not_isMax
theorem toZ_iterate_succ_of_not_isMax (n : ℕ) (hn : ¬IsMax (succ^[n] i0)) :
    toZ i0 (succ^[n] i0) = n 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 : ι} (n : ℕ) (hn : ¬IsMax (Order.succ^[n] i0))
Goal: toZ i0 (Order.succ^[n] i0) = ↑n
-- toZ_iterate_pred_of_not_isMin
theorem toZ_iterate_pred_of_not_isMin (n : ℕ) (hn : ¬IsMin (pred^[n] i0)) :
    toZ i0 (pred^[n] i0) = -n 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 : ι} (n : ℕ) (hn : ¬IsMin (Order.pred^[n] i0))
Goal: toZ i0 (Order.pred^[n] i0) = -↑n
-- le_of_toZ_le
theorem le_of_toZ_le {j : ι} (h_le : toZ i0 i ≤ toZ i0 j) : i ≤ j 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 i j : ι} (h_le : toZ i0 i ≤ toZ i0 j)
Goal: i ≤ j
-- toZ_mono
theorem toZ_mono {i j : ι} (h_le : i ≤ j) : toZ i0 i ≤ toZ i0 j 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 i j : ι} (h_le : i ≤ j)
Goal: toZ i0 i ≤ toZ i0 j
-- toZ_le_iff
theorem toZ_le_iff (i j : ι) : toZ i0 i ≤ toZ i0 j ↔ i ≤ j 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 : ι} (i j : ι)
Goal: toZ i0 i ≤ toZ i0 j ↔ i ≤ j
-- toZ_iterate_succ
theorem toZ_iterate_succ [NoMaxOrder ι] (n : ℕ) : toZ i0 (succ^[n] i0) = n 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 : ι} [NoMaxOrder ι] (n : ℕ)
Goal: toZ i0 (Order.succ^[n] i0) = ↑n
-- toZ_iterate_pred
theorem toZ_iterate_pred [NoMinOrder ι] (n : ℕ) : toZ i0 (pred^[n] i0) = -n 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 : ι} [NoMinOrder ι] (n : ℕ)
Goal: toZ i0 (Order.pred^[n] i0) = -↑n
-- injective_toZ
theorem injective_toZ : Function.Injective (toZ i0) 
Arguments: {ι : Type u_1} [LinearOrder ι] [SuccOrder ι] [IsSuccArchimedean ι] [PredOrder ι] {i0 : ι}
Goal: Function.Injective (toZ i0)
-- OrderEmbedding.preimage_Ici
@[simp] theorem preimage_Ici : e ⁻¹' Ici (e x) = Ici x 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (x : α)
Goal: ⇑e ⁻¹' Set.Ici (e x) = Set.Ici x
-- OrderEmbedding.preimage_Iic
@[simp] theorem preimage_Iic : e ⁻¹' Iic (e x) = Iic x 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (x : α)
Goal: ⇑e ⁻¹' Set.Iic (e x) = Set.Iic x
-- OrderEmbedding.preimage_Ioi
@[simp] theorem preimage_Ioi : e ⁻¹' Ioi (e x) = Ioi x 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (x : α)
Goal: ⇑e ⁻¹' Set.Ioi (e x) = Set.Ioi x
-- OrderEmbedding.preimage_Iio
@[simp] theorem preimage_Iio : e ⁻¹' Iio (e x) = Iio x 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (x : α)
Goal: ⇑e ⁻¹' Set.Iio (e x) = Set.Iio x
-- OrderEmbedding.preimage_Icc
@[simp] theorem preimage_Icc : e ⁻¹' Icc (e x) (e y) = Icc x y 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (x y : α)
Goal: ⇑e ⁻¹' Set.Icc (e x) (e y) = Set.Icc x y
-- OrderEmbedding.preimage_Ico
@[simp] theorem preimage_Ico : e ⁻¹' Ico (e x) (e y) = Ico x y 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (x y : α)
Goal: ⇑e ⁻¹' Set.Ico (e x) (e y) = Set.Ico x y
-- OrderEmbedding.preimage_Ioc
@[simp] theorem preimage_Ioc : e ⁻¹' Ioc (e x) (e y) = Ioc x y 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (x y : α)
Goal: ⇑e ⁻¹' Set.Ioc (e x) (e y) = Set.Ioc x y
-- OrderEmbedding.preimage_Ioo
@[simp] theorem preimage_Ioo : e ⁻¹' Ioo (e x) (e y) = Ioo x y 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (e : α ↪o β) (x y : α)
Goal: ⇑e ⁻¹' Set.Ioo (e x) (e y) = Set.Ioo x y
-- OrderEmbedding.preimage_uIcc
@[simp] theorem preimage_uIcc [Lattice β] (e : α ↪o β) (x y : α) :
    e ⁻¹' (uIcc (e x) (e y)) = uIcc x y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [Lattice β] (e : α ↪o β) (x y : α)
Goal: ⇑e ⁻¹' Set.uIcc (e x) (e y) = Set.uIcc x y
-- OrderEmbedding.preimage_uIoc
@[simp] theorem preimage_uIoc [LinearOrder β] (e : α ↪o β) (x y : α) :
    e ⁻¹' (uIoc (e x) (e y)) = uIoc x y 
Arguments: {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] (e : α ↪o β) (x y : α)
Goal: ⇑e ⁻¹' Ι (e x) (e y) = Ι x y
-- Set.Iic_disjoint_Ioi
@[simp]
theorem Iic_disjoint_Ioi (h : a ≤ b) : Disjoint (Iic a) (Ioi b) 
Arguments: {α : Type v} [Preorder α] {a b : α} (h : a ≤ b)
Goal: Disjoint (Set.Iic a) (Set.Ioi b)
-- Set.Iio_disjoint_Ici
@[simp]
theorem Iio_disjoint_Ici (h : a ≤ b) : Disjoint (Iio a) (Ici b) 
Arguments: {α : Type v} [Preorder α] {a b : α} (h : a ≤ b)
Goal: Disjoint (Set.Iio a) (Set.Ici b)
-- Set.Iic_disjoint_Ioc
@[simp]
theorem Iic_disjoint_Ioc (h : a ≤ b) : Disjoint (Iic a) (Ioc b c) 
Arguments: {α : Type v} [Preorder α] {a b c : α} (h : a ≤ b)
Goal: Disjoint (Set.Iic a) (Set.Ioc b c)
-- Set.Ioc_disjoint_Ioc_same
@[simp]
theorem Ioc_disjoint_Ioc_same : Disjoint (Ioc a b) (Ioc b c) 
Arguments: {α : Type v} [Preorder α] {a b c : α}
Goal: Disjoint (Set.Ioc a b) (Set.Ioc b c)
-- Set.Ico_disjoint_Ico_same
@[simp]
theorem Ico_disjoint_Ico_same : Disjoint (Ico a b) (Ico b c) 
Arguments: {α : Type v} [Preorder α] {a b c : α}
Goal: Disjoint (Set.Ico a b) (Set.Ico b c)
-- Set.Ici_disjoint_Iic
@[simp]
theorem Ici_disjoint_Iic : Disjoint (Ici a) (Iic b) ↔ ¬a ≤ b 
Arguments: {α : Type v} [Preorder α] {a b : α}
Goal: Disjoint (Set.Ici a) (Set.Iic b) ↔ ¬a ≤ b
-- Set.Iic_disjoint_Ici
@[simp]
theorem Iic_disjoint_Ici : Disjoint (Iic a) (Ici b) ↔ ¬b ≤ a 
Arguments: {α : Type v} [Preorder α] {a b : α}
Goal: Disjoint (Set.Iic a) (Set.Ici b) ↔ ¬b ≤ a
-- Set.Ioc_disjoint_Ioi
@[simp]
theorem Ioc_disjoint_Ioi (h : b ≤ c) : Disjoint (Ioc a b) (Ioi c) 
Arguments: {α : Type v} [Preorder α] {a b c : α} (h : b ≤ c)
Goal: Disjoint (Set.Ioc a b) (Set.Ioi c)
-- Set.Ioc_disjoint_Ioi_same
theorem Ioc_disjoint_Ioi_same : Disjoint (Ioc a b) (Ioi b) 
Arguments: {α : Type v} [Preorder α] {a b : α}
Goal: Disjoint (Set.Ioc a b) (Set.Ioi b)
-- Set.iUnion_Iic
@[simp]
theorem iUnion_Iic : ⋃ a : α, Iic a = univ 
Arguments: {α : Type v} [Preorder α]
Goal: ⋃ a, Set.Iic a = Set.univ
-- Set.iUnion_Ici
@[simp]
theorem iUnion_Ici : ⋃ a : α, Ici a = univ 
Arguments: {α : Type v} [Preorder α]
Goal: ⋃ a, Set.Ici a = Set.univ
-- Set.iUnion_Icc_right
@[simp]
theorem iUnion_Icc_right (a : α) : ⋃ b, Icc a b = Ici a 
Arguments: {α : Type v} [Preorder α] (a : α)
Goal: ⋃ b, Set.Icc a b = Set.Ici a
-- Set.iUnion_Ioc_right
@[simp]
theorem iUnion_Ioc_right (a : α) : ⋃ b, Ioc a b = Ioi a 
Arguments: {α : Type v} [Preorder α] (a : α)
Goal: ⋃ b, Set.Ioc a b = Set.Ioi a