ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Order.Ideal.mem_sInf
@[simp]
theorem mem_sInf : x ∈ sInf S ↔ ∀ s ∈ S, x ∈ s 
Arguments: {P : Type u_1} [SemilatticeSup P] [OrderBot P] {x : P} {S : Set (Order.Ideal P)}
Goal: x ∈ sInf S ↔ ∀ s ∈ S, x ∈ s
-- Order.Ideal.eq_sup_of_le_sup
theorem eq_sup_of_le_sup {x i j : P} (hi : i ∈ I) (hj : j ∈ J) (hx : x ≤ i ⊔ j) :
    ∃ i' ∈ I, ∃ j' ∈ J, x = i' ⊔ j' 
Arguments: {P : Type u_1} [DistribLattice P] {I J : Order.Ideal P} {x i j : P} (hi : i ∈ I) (hj : j ∈ J) (hx : x ≤ i ⊔ j)
Goal: ∃ i' ∈ I, ∃ j' ∈ J, x = i' ⊔ j'
-- Order.Ideal.coe_sup_eq
theorem coe_sup_eq : ↑(I ⊔ J) = { x | ∃ i ∈ I, ∃ j ∈ J, x = i ⊔ j } 
Arguments: {P : Type u_1} [DistribLattice P] {I J : Order.Ideal P}
Goal: ↑(I ⊔ J) = {x | ∃ i ∈ I, ∃ j ∈ J, x = i ⊔ j}
-- Order.Ideal.IsProper.not_mem_of_compl_mem
theorem IsProper.not_mem_of_compl_mem (hI : IsProper I) (hxc : xᶜ ∈ I) : x ∉ I 
Arguments: {P : Type u_1} [BooleanAlgebra P] {x : P} {I : Order.Ideal P} (hI : I.IsProper) (hxc : xᶜ ∈ I)
Goal: x ∉ I
-- Order.Ideal.IsProper.not_mem_or_compl_not_mem
theorem IsProper.not_mem_or_compl_not_mem (hI : IsProper I) : x ∉ I ∨ xᶜ ∉ I 
Arguments: {P : Type u_1} [BooleanAlgebra P] {x : P} {I : Order.Ideal P} (hI : I.IsProper)
Goal: x ∉ I ∨ xᶜ ∉ I
-- Order.Cofinal.mem_gt
structure Cofinal (P) [Preorder P] where
    carrier : Set P
    mem_gt : ∀ x : P, ∃ y ∈ carrier, x ≤ y
Arguments: {P : Type u_2} [Preorder P] (self : Order.Cofinal P) (x : P)
Goal: ∃ y ∈ self.carrier, x ≤ y
-- Order.Cofinal.above_mem
theorem above_mem : D.above x ∈ D 
Arguments: {P : Type u_1} [Preorder P] (D : Order.Cofinal P) (x : P)
Goal: D.above x ∈ D
-- Order.Cofinal.le_above
theorem le_above : x ≤ D.above x 
Arguments: {P : Type u_1} [Preorder P] (D : Order.Cofinal P) (x : P)
Goal: x ≤ D.above x
-- Order.sequenceOfCofinals.monotone
theorem sequenceOfCofinals.monotone : Monotone (sequenceOfCofinals p 𝒟) 
Arguments: {P : Type u_1} [Preorder P] (p : P) {ι : Type u_2} [Encodable ι] (𝒟 : ι → Order.Cofinal P)
Goal: Monotone (Order.sequenceOfCofinals p 𝒟)
-- Order.sequenceOfCofinals.encode_mem
theorem sequenceOfCofinals.encode_mem (i : ι) :
    sequenceOfCofinals p 𝒟 (Encodable.encode i + 1) ∈ 𝒟 i 
Arguments: {P : Type u_1} [Preorder P] (p : P) {ι : Type u_2} [Encodable ι] (𝒟 : ι → Order.Cofinal P) (i : ι)
Goal: Order.sequenceOfCofinals p 𝒟 (Encodable.encode i + 1) ∈ 𝒟 i
-- Order.mem_idealOfCofinals
theorem mem_idealOfCofinals : p ∈ idealOfCofinals p 𝒟 
Arguments: {P : Type u_1} [Preorder P] (p : P) {ι : Type u_2} [Encodable ι] (𝒟 : ι → Order.Cofinal P)
Goal: p ∈ Order.idealOfCofinals p 𝒟
-- Order.cofinal_meets_idealOfCofinals
theorem cofinal_meets_idealOfCofinals (i : ι) : ∃ x : P, x ∈ 𝒟 i ∧ x ∈ idealOfCofinals p 𝒟 
Arguments: {P : Type u_1} [Preorder P] (p : P) {ι : Type u_2} [Encodable ι] (𝒟 : ι → Order.Cofinal P) (i : ι)
Goal: ∃ x ∈ 𝒟 i, x ∈ Order.idealOfCofinals p 𝒟
-- Order.isIdeal_sUnion_of_directedOn
lemma isIdeal_sUnion_of_directedOn {C : Set (Set P)} (hidl : ∀ I ∈ C, IsIdeal I)
    (hD : DirectedOn (· ⊆ ·) C) (hNe : C.Nonempty) : IsIdeal C.sUnion 
Arguments: {P : Type u_1} [Preorder P] {C : Set (Set P)} (hidl : ∀ I ∈ C, Order.IsIdeal I) (hD : DirectedOn (fun x x_1 => x ⊆ x_1) C) (hNe : C.Nonempty)
Goal: Order.IsIdeal (⋃₀ C)
-- Order.isIdeal_sUnion_of_isChain
lemma isIdeal_sUnion_of_isChain {C : Set (Set P)} (hidl : ∀ I ∈ C, IsIdeal I)
    (hC : IsChain (· ⊆ ·) C) (hNe : C.Nonempty) : IsIdeal C.sUnion 
Arguments: {P : Type u_1} [Preorder P] {C : Set (Set P)} (hidl : ∀ I ∈ C, Order.IsIdeal I) (hC : IsChain (fun x x_1 => x ⊆ x_1) C) (hNe : C.Nonempty)
Goal: Order.IsIdeal (⋃₀ C)
-- exists_maximal_of_chains_bounded
theorem exists_maximal_of_chains_bounded (h : ∀ c, IsChain r c → ∃ ub, ∀ a ∈ c, a ≺ ub)
    (trans : ∀ {a b c}, a ≺ b → b ≺ c → a ≺ c) : ∃ m, ∀ a, m ≺ a → a ≺ m 
Arguments: {α : Type u_1} {r : α → α → Prop} (h : ∀ (c : Set α), IsChain r c → ∃ ub, ∀ a ∈ c, r a ub) (trans : ∀ {a b c : α}, r a b → r b c → r a c)
Goal: ∃ m, ∀ (a : α), r m a → r a m
-- exists_maximal_of_nonempty_chains_bounded
theorem exists_maximal_of_nonempty_chains_bounded [Nonempty α]
    (h : ∀ c, IsChain r c → c.Nonempty → ∃ ub, ∀ a ∈ c, a ≺ ub)
    (trans : ∀ {a b c}, a ≺ b → b ≺ c → a ≺ c) : ∃ m, ∀ a, m ≺ a → a ≺ m 
Arguments: {α : Type u_1} {r : α → α → Prop} [Nonempty α] (h : ∀ (c : Set α), IsChain r c → c.Nonempty → ∃ ub, ∀ a ∈ c, r a ub) (trans : ∀ {a b c : α}, r a b → r b c → r a c)
Goal: ∃ m, ∀ (a : α), r m a → r a m
-- zorn_preorder
theorem zorn_preorder (h : ∀ c : Set α, IsChain (· ≤ ·) c → BddAbove c) :
    ∃ m : α, ∀ a, m ≤ a → a ≤ m 
Arguments: {α : Type u_1} [Preorder α] (h : ∀ (c : Set α), IsChain (fun x x_1 => x ≤ x_1) c → BddAbove c)
Goal: ∃ m, ∀ (a : α), m ≤ a → a ≤ m
-- zorn_nonempty_preorder
theorem zorn_nonempty_preorder [Nonempty α]
    (h : ∀ c : Set α, IsChain (· ≤ ·) c → c.Nonempty → BddAbove c) : ∃ m : α, ∀ a, m ≤ a → a ≤ m 
Arguments: {α : Type u_1} [Preorder α] [Nonempty α] (h : ∀ (c : Set α), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → BddAbove c)
Goal: ∃ m, ∀ (a : α), m ≤ a → a ≤ m
-- zorn_preorder₀
theorem zorn_preorder₀ (s : Set α)
    (ih : ∀ c ⊆ s, IsChain (· ≤ ·) c → ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub) :
    ∃ m ∈ s, ∀ z ∈ s, m ≤ z → z ≤ m 
Arguments: {α : Type u_1} [Preorder α] (s : Set α) (ih : ∀ c ⊆ s, IsChain (fun x x_1 => x ≤ x_1) c → ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub)
Goal: ∃ m ∈ s, ∀ z ∈ s, m ≤ z → z ≤ m
-- zorn_nonempty_preorder₀
theorem zorn_nonempty_preorder₀ (s : Set α)
    (ih : ∀ c ⊆ s, IsChain (· ≤ ·) c → ∀ y ∈ c, ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub) (x : α)
    (hxs : x ∈ s) : ∃ m ∈ s, x ≤ m ∧ ∀ z ∈ s, m ≤ z → z ≤ m 
Arguments: {α : Type u_1} [Preorder α] (s : Set α) (ih : ∀ c ⊆ s, IsChain (fun x x_1 => x ≤ x_1) c → ∀ y ∈ c, ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub) (x : α) (hxs : x ∈ s)
Goal: ∃ m ∈ s, x ≤ m ∧ ∀ z ∈ s, m ≤ z → z ≤ m
-- zorn_nonempty_Ici₀
theorem zorn_nonempty_Ici₀ (a : α)
    (ih : ∀ c ⊆ Ici a, IsChain (· ≤ ·) c → ∀ y ∈ c, ∃ ub, ∀ z ∈ c, z ≤ ub)
    (x : α) (hax : a ≤ x) : ∃ m, x ≤ m ∧ ∀ z, m ≤ z → z ≤ m 
Arguments: {α : Type u_1} [Preorder α] (a : α) (ih : ∀ c ⊆ Set.Ici a, IsChain (fun x x_1 => x ≤ x_1) c → ∀ y ∈ c, ∃ ub, ∀ z ∈ c, z ≤ ub) (x : α) (hax : a ≤ x)
Goal: ∃ m, x ≤ m ∧ ∀ (z : α), m ≤ z → z ≤ m
-- zorn_partialOrder
theorem zorn_partialOrder (h : ∀ c : Set α, IsChain (· ≤ ·) c → BddAbove c) :
    ∃ m : α, ∀ a, m ≤ a → a = m 
Arguments: {α : Type u_1} [PartialOrder α] (h : ∀ (c : Set α), IsChain (fun x x_1 => x ≤ x_1) c → BddAbove c)
Goal: ∃ m, ∀ (a : α), m ≤ a → a = m
-- zorn_nonempty_partialOrder
theorem zorn_nonempty_partialOrder [Nonempty α]
    (h : ∀ c : Set α, IsChain (· ≤ ·) c → c.Nonempty → BddAbove c) : ∃ m : α, ∀ a, m ≤ a → a = m 
Arguments: {α : Type u_1} [PartialOrder α] [Nonempty α] (h : ∀ (c : Set α), IsChain (fun x x_1 => x ≤ x_1) c → c.Nonempty → BddAbove c)
Goal: ∃ m, ∀ (a : α), m ≤ a → a = m
-- zorn_partialOrder₀
theorem zorn_partialOrder₀ (s : Set α)
    (ih : ∀ c ⊆ s, IsChain (· ≤ ·) c → ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub) :
    ∃ m ∈ s, ∀ z ∈ s, m ≤ z → z = m 
Arguments: {α : Type u_1} [PartialOrder α] (s : Set α) (ih : ∀ c ⊆ s, IsChain (fun x x_1 => x ≤ x_1) c → ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub)
Goal: ∃ m ∈ s, ∀ z ∈ s, m ≤ z → z = m
-- zorn_nonempty_partialOrder₀
theorem zorn_nonempty_partialOrder₀ (s : Set α)
    (ih : ∀ c ⊆ s, IsChain (· ≤ ·) c → ∀ y ∈ c, ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub) (x : α)
    (hxs : x ∈ s) : ∃ m ∈ s, x ≤ m ∧ ∀ z ∈ s, m ≤ z → z = m 
Arguments: {α : Type u_1} [PartialOrder α] (s : Set α) (ih : ∀ c ⊆ s, IsChain (fun x x_1 => x ≤ x_1) c → ∀ y ∈ c, ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub) (x : α) (hxs : x ∈ s)
Goal: ∃ m ∈ s, x ≤ m ∧ ∀ z ∈ s, m ≤ z → z = m
-- zorn_subset
theorem zorn_subset (S : Set (Set α))
    (h : ∀ c ⊆ S, IsChain (· ⊆ ·) c → ∃ ub ∈ S, ∀ s ∈ c, s ⊆ ub) :
    ∃ m ∈ S, ∀ a ∈ S, m ⊆ a → a = m 
Arguments: {α : Type u_1} (S : Set (Set α)) (h : ∀ c ⊆ S, IsChain (fun x x_1 => x ⊆ x_1) c → ∃ ub ∈ S, ∀ s ∈ c, s ⊆ ub)
Goal: ∃ m ∈ S, ∀ a ∈ S, m ⊆ a → a = m
-- zorn_subset_nonempty
theorem zorn_subset_nonempty (S : Set (Set α))
    (H : ∀ c ⊆ S, IsChain (· ⊆ ·) c → c.Nonempty → ∃ ub ∈ S, ∀ s ∈ c, s ⊆ ub) (x)
    (hx : x ∈ S) : ∃ m ∈ S, x ⊆ m ∧ ∀ a ∈ S, m ⊆ a → a = m 
Arguments: {α : Type u_1} (S : Set (Set α)) (H : ∀ c ⊆ S, IsChain (fun x x_1 => x ⊆ x_1) c → c.Nonempty → ∃ ub ∈ S, ∀ s ∈ c, s ⊆ ub) (x : Set α) (hx : x ∈ S)
Goal: ∃ m ∈ S, x ⊆ m ∧ ∀ a ∈ S, m ⊆ a → a = m
-- zorn_superset
theorem zorn_superset (S : Set (Set α))
    (h : ∀ c ⊆ S, IsChain (· ⊆ ·) c → ∃ lb ∈ S, ∀ s ∈ c, lb ⊆ s) :
    ∃ m ∈ S, ∀ a ∈ S, a ⊆ m → a = m 
Arguments: {α : Type u_1} (S : Set (Set α)) (h : ∀ c ⊆ S, IsChain (fun x x_1 => x ⊆ x_1) c → ∃ lb ∈ S, ∀ s ∈ c, lb ⊆ s)
Goal: ∃ m ∈ S, ∀ a ∈ S, a ⊆ m → a = m
-- zorn_superset_nonempty
theorem zorn_superset_nonempty (S : Set (Set α))
    (H : ∀ c ⊆ S, IsChain (· ⊆ ·) c → c.Nonempty → ∃ lb ∈ S, ∀ s ∈ c, lb ⊆ s) (x)
    (hx : x ∈ S) : ∃ m ∈ S, m ⊆ x ∧ ∀ a ∈ S, a ⊆ m → a = m 
Arguments: {α : Type u_1} (S : Set (Set α)) (H : ∀ c ⊆ S, IsChain (fun x x_1 => x ⊆ x_1) c → c.Nonempty → ∃ lb ∈ S, ∀ s ∈ c, lb ⊆ s) (x : Set α) (hx : x ∈ S)
Goal: ∃ m ∈ S, m ⊆ x ∧ ∀ a ∈ S, a ⊆ m → a = m
-- IsChain.exists_maxChain
theorem IsChain.exists_maxChain (hc : IsChain r c) : ∃ M, @IsMaxChain _ r M ∧ c ⊆ M 
Arguments: {α : Type u_1} {r : α → α → Prop} {c : Set α} (hc : IsChain r c)
Goal: ∃ M, IsMaxChain r M ∧ c ⊆ M
-- SupIrred.not_isMin
theorem SupIrred.not_isMin (ha : SupIrred a) : ¬IsMin a 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α} (ha : SupIrred a)
Goal: ¬IsMin a
-- SupPrime.not_isMin
theorem SupPrime.not_isMin (ha : SupPrime a) : ¬IsMin a 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α} (ha : SupPrime a)
Goal: ¬IsMin a
-- IsMin.not_supIrred
theorem IsMin.not_supIrred (ha : IsMin a) : ¬SupIrred a 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α} (ha : IsMin a)
Goal: ¬SupIrred a
-- IsMin.not_supPrime
theorem IsMin.not_supPrime (ha : IsMin a) : ¬SupPrime a 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α} (ha : IsMin a)
Goal: ¬SupPrime a
-- not_supIrred
@[simp]
theorem not_supIrred : ¬SupIrred a ↔ IsMin a ∨ ∃ b c, b ⊔ c = a ∧ b < a ∧ c < a 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α}
Goal: ¬SupIrred a ↔ IsMin a ∨ ∃ b c, b ⊔ c = a ∧ b < a ∧ c < a
-- not_supPrime
@[simp]
theorem not_supPrime : ¬SupPrime a ↔ IsMin a ∨ ∃ b c, a ≤ b ⊔ c ∧ ¬a ≤ b ∧ ¬a ≤ c 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α}
Goal: ¬SupPrime a ↔ IsMin a ∨ ∃ b c, a ≤ b ⊔ c ∧ ¬a ≤ b ∧ ¬a ≤ c
-- SupPrime.supIrred
protected theorem SupPrime.supIrred : SupPrime a → SupIrred a 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α}
Goal: SupPrime a → SupIrred a
-- SupPrime.le_sup
theorem SupPrime.le_sup (ha : SupPrime a) : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c 
Arguments: {α : Type u_2} [SemilatticeSup α] {a b c : α} (ha : SupPrime a)
Goal: a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c
-- not_supIrred_bot
@[simp]
theorem not_supIrred_bot : ¬SupIrred (⊥ : α) 
Arguments: {α : Type u_2} [SemilatticeSup α] [OrderBot α]
Goal: ¬SupIrred ⊥
-- not_supPrime_bot
@[simp]
theorem not_supPrime_bot : ¬SupPrime (⊥ : α) 
Arguments: {α : Type u_2} [SemilatticeSup α] [OrderBot α]
Goal: ¬SupPrime ⊥
-- SupIrred.ne_bot
theorem SupIrred.ne_bot (ha : SupIrred a) : a ≠ ⊥ 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α} [OrderBot α] (ha : SupIrred a)
Goal: a ≠ ⊥
-- SupPrime.ne_bot
theorem SupPrime.ne_bot (ha : SupPrime a) : a ≠ ⊥ 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α} [OrderBot α] (ha : SupPrime a)
Goal: a ≠ ⊥
-- SupIrred.finset_sup_eq
theorem SupIrred.finset_sup_eq (ha : SupIrred a) (h : s.sup f = a) : ∃ i ∈ s, f i = a 
Arguments: {ι : Type u_1} {α : Type u_2} [SemilatticeSup α] {a : α} [OrderBot α] {s : Finset ι} {f : ι → α} (ha : SupIrred a) (h : s.sup f = a)
Goal: ∃ i ∈ s, f i = a
-- SupPrime.le_finset_sup
theorem SupPrime.le_finset_sup (ha : SupPrime a) : a ≤ s.sup f ↔ ∃ i ∈ s, a ≤ f i 
Arguments: {ι : Type u_1} {α : Type u_2} [SemilatticeSup α] {a : α} [OrderBot α] {s : Finset ι} {f : ι → α} (ha : SupPrime a)
Goal: a ≤ s.sup f ↔ ∃ i ∈ s, a ≤ f i
-- exists_supIrred_decomposition
theorem exists_supIrred_decomposition (a : α) :
    ∃ s : Finset α, s.sup id = a ∧ ∀ ⦃b⦄, b ∈ s → SupIrred b 
Arguments: {α : Type u_2} [SemilatticeSup α] [OrderBot α] [WellFoundedLT α] (a : α)
Goal: ∃ s, s.sup id = a ∧ ∀ ⦃b : α⦄, b ∈ s → SupIrred b
-- IsMax.not_infIrred
@[simp]
theorem IsMax.not_infIrred (ha : IsMax a) : ¬InfIrred a 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : α} (ha : IsMax a)
Goal: ¬InfIrred a
-- IsMax.not_infPrime
@[simp]
theorem IsMax.not_infPrime (ha : IsMax a) : ¬InfPrime a 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : α} (ha : IsMax a)
Goal: ¬InfPrime a
-- not_infIrred
@[simp]
theorem not_infIrred : ¬InfIrred a ↔ IsMax a ∨ ∃ b c, b ⊓ c = a ∧ a < b ∧ a < c 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : α}
Goal: ¬InfIrred a ↔ IsMax a ∨ ∃ b c, b ⊓ c = a ∧ a < b ∧ a < c
-- not_infPrime
@[simp]
theorem not_infPrime : ¬InfPrime a ↔ IsMax a ∨ ∃ b c, b ⊓ c ≤ a ∧ ¬b ≤ a ∧ ¬c ≤ a 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : α}
Goal: ¬InfPrime a ↔ IsMax a ∨ ∃ b c, b ⊓ c ≤ a ∧ ¬b ≤ a ∧ ¬c ≤ a
-- InfPrime.infIrred
protected theorem InfPrime.infIrred : InfPrime a → InfIrred a 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : α}
Goal: InfPrime a → InfIrred a
-- InfPrime.inf_le
theorem InfPrime.inf_le (ha : InfPrime a) : b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a 
Arguments: {α : Type u_2} [SemilatticeInf α] {a b c : α} (ha : InfPrime a)
Goal: b ⊓ c ≤ a ↔ b ≤ a ∨ c ≤ a
-- not_infIrred_top
theorem not_infIrred_top : ¬InfIrred (⊤ : α) 
Arguments: {α : Type u_2} [SemilatticeInf α] [OrderTop α]
Goal: ¬InfIrred ⊤
-- not_infPrime_top
theorem not_infPrime_top : ¬InfPrime (⊤ : α) 
Arguments: {α : Type u_2} [SemilatticeInf α] [OrderTop α]
Goal: ¬InfPrime ⊤
-- InfIrred.ne_top
theorem InfIrred.ne_top (ha : InfIrred a) : a ≠ ⊤ 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : α} [OrderTop α] (ha : InfIrred a)
Goal: a ≠ ⊤
-- InfPrime.ne_top
theorem InfPrime.ne_top (ha : InfPrime a) : a ≠ ⊤ 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : α} [OrderTop α] (ha : InfPrime a)
Goal: a ≠ ⊤
-- InfIrred.finset_inf_eq
theorem InfIrred.finset_inf_eq : InfIrred a → s.inf f = a → ∃ i ∈ s, f i = a 
Arguments: {ι : Type u_1} {α : Type u_2} [SemilatticeInf α] {a : α} [OrderTop α] {s : Finset ι} {f : ι → α}
Goal: InfIrred a → s.inf f = a → ∃ i ∈ s, f i = a
-- InfPrime.finset_inf_le
theorem InfPrime.finset_inf_le (ha : InfPrime a) : s.inf f ≤ a ↔ ∃ i ∈ s, f i ≤ a 
Arguments: {ι : Type u_1} {α : Type u_2} [SemilatticeInf α] {a : α} [OrderTop α] {s : Finset ι} {f : ι → α} (ha : InfPrime a)
Goal: s.inf f ≤ a ↔ ∃ i ∈ s, f i ≤ a
-- exists_infIrred_decomposition
theorem exists_infIrred_decomposition (a : α) :
    ∃ s : Finset α, s.inf id = a ∧ ∀ ⦃b⦄, b ∈ s → InfIrred b 
Arguments: {α : Type u_2} [SemilatticeInf α] [OrderTop α] [WellFoundedGT α] (a : α)
Goal: ∃ s, s.inf id = a ∧ ∀ ⦃b : α⦄, b ∈ s → InfIrred b
-- infIrred_toDual
@[simp]
theorem infIrred_toDual {a : α} : InfIrred (toDual a) ↔ SupIrred a 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α}
Goal: InfIrred (OrderDual.toDual a) ↔ SupIrred a
-- infPrime_toDual
@[simp]
theorem infPrime_toDual {a : α} : InfPrime (toDual a) ↔ SupPrime a 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : α}
Goal: InfPrime (OrderDual.toDual a) ↔ SupPrime a
-- supIrred_ofDual
@[simp]
theorem supIrred_ofDual {a : αᵒᵈ} : SupIrred (ofDual a) ↔ InfIrred a 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : αᵒᵈ}
Goal: SupIrred (OrderDual.ofDual a) ↔ InfIrred a
-- supPrime_ofDual
@[simp]
theorem supPrime_ofDual {a : αᵒᵈ} : SupPrime (ofDual a) ↔ InfPrime a 
Arguments: {α : Type u_2} [SemilatticeSup α] {a : αᵒᵈ}
Goal: SupPrime (OrderDual.ofDual a) ↔ InfPrime a
-- supIrred_toDual
@[simp]
theorem supIrred_toDual {a : α} : SupIrred (toDual a) ↔ InfIrred a 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : α}
Goal: SupIrred (OrderDual.toDual a) ↔ InfIrred a
-- supPrime_toDual
@[simp]
theorem supPrime_toDual {a : α} : SupPrime (toDual a) ↔ InfPrime a 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : α}
Goal: SupPrime (OrderDual.toDual a) ↔ InfPrime a
-- infIrred_ofDual
@[simp]
theorem infIrred_ofDual {a : αᵒᵈ} : InfIrred (ofDual a) ↔ SupIrred a 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : αᵒᵈ}
Goal: InfIrred (OrderDual.ofDual a) ↔ SupIrred a
-- infPrime_ofDual
@[simp]
theorem infPrime_ofDual {a : αᵒᵈ} : InfPrime (ofDual a) ↔ SupPrime a 
Arguments: {α : Type u_2} [SemilatticeInf α] {a : αᵒᵈ}
Goal: InfPrime (OrderDual.ofDual a) ↔ SupPrime a
-- supPrime_iff_supIrred
@[simp]
theorem supPrime_iff_supIrred : SupPrime a ↔ SupIrred a 
Arguments: {α : Type u_2} [DistribLattice α] {a : α}
Goal: SupPrime a ↔ SupIrred a
-- infPrime_iff_infIrred
@[simp]
theorem infPrime_iff_infIrred : InfPrime a ↔ InfIrred a 
Arguments: {α : Type u_2} [DistribLattice α] {a : α}
Goal: InfPrime a ↔ InfIrred a
-- supPrime_iff_not_isMin
theorem supPrime_iff_not_isMin : SupPrime a ↔ ¬IsMin a 
Arguments: {α : Type u_2} [LinearOrder α] {a : α}
Goal: SupPrime a ↔ ¬IsMin a
-- infPrime_iff_not_isMax
theorem infPrime_iff_not_isMax : InfPrime a ↔ ¬IsMax a 
Arguments: {α : Type u_2} [LinearOrder α] {a : α}
Goal: InfPrime a ↔ ¬IsMax a
-- supIrred_iff_not_isMin
@[simp]
theorem supIrred_iff_not_isMin : SupIrred a ↔ ¬IsMin a 
Arguments: {α : Type u_2} [LinearOrder α] {a : α}
Goal: SupIrred a ↔ ¬IsMin a
-- infIrred_iff_not_isMax
@[simp]
theorem infIrred_iff_not_isMax : InfIrred a ↔ ¬IsMax a 
Arguments: {α : Type u_2} [LinearOrder α] {a : α}
Goal: InfIrred a ↔ ¬IsMax a
-- krullDim_nonneg_of_nonempty
lemma krullDim_nonneg_of_nonempty [Nonempty α] : 0 ≤ krullDim α 
Arguments: {α : Type u_1} [Preorder α] [Nonempty α]
Goal: 0 ≤ krullDim α
-- krullDim_eq_bot_of_isEmpty
lemma krullDim_eq_bot_of_isEmpty [IsEmpty α] : krullDim α = ⊥ 
Arguments: {α : Type u_1} [Preorder α] [IsEmpty α]
Goal: krullDim α = ⊥
-- krullDim_eq_top_of_infiniteDimensionalOrder
lemma krullDim_eq_top_of_infiniteDimensionalOrder [InfiniteDimensionalOrder α] :
    krullDim α = ⊤ 
Arguments: {α : Type u_1} [Preorder α] [InfiniteDimensionalOrder α]
Goal: krullDim α = ⊤
-- krullDim_le_of_strictMono
lemma krullDim_le_of_strictMono (f : α → β) (hf : StrictMono f) : krullDim α ≤ krullDim β 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α → β) (hf : StrictMono f)
Goal: krullDim α ≤ krullDim β
-- height_mono
lemma height_mono {a b : α} (h : a ≤ b) : height α a ≤ height α b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ≤ b)
Goal: height α a ≤ height α b
-- krullDim_eq_length_of_finiteDimensionalOrder
lemma krullDim_eq_length_of_finiteDimensionalOrder [FiniteDimensionalOrder α] :
    krullDim α = (LTSeries.longestOf α).length 
Arguments: {α : Type u_1} [Preorder α] [FiniteDimensionalOrder α]
Goal: krullDim α = ↑(LTSeries.longestOf α).length
-- krullDim_eq_zero_of_unique
lemma krullDim_eq_zero_of_unique [Unique α] : krullDim α = 0 
Arguments: {α : Type u_1} [Preorder α] [Unique α]
Goal: krullDim α = 0
-- krullDim_le_of_strictComono_and_surj
lemma krullDim_le_of_strictComono_and_surj
    (f : α → β) (hf : ∀ ⦃a b⦄, f a < f b → a < b) (hf' : Function.Surjective f) :
    krullDim β ≤ krullDim α 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α → β) (hf : ∀ ⦃a b : α⦄, f a < f b → a < b) (hf' : Function.Surjective f)
Goal: krullDim β ≤ krullDim α
-- krullDim_eq_of_orderIso
lemma krullDim_eq_of_orderIso (f : α ≃o β) : krullDim α = krullDim β 
Arguments: {α : Type u_1} {β : Type u_2} [Preorder α] [Preorder β] (f : α ≃o β)
Goal: krullDim α = krullDim β
-- krullDim_eq_iSup_height
lemma krullDim_eq_iSup_height : krullDim α = ⨆ (a : α), height α a 
Arguments: {α : Type u_1} [Preorder α]
Goal: krullDim α = ⨆ a, height α a
-- krullDim_orderDual
@[simp] lemma krullDim_orderDual : krullDim αᵒᵈ = krullDim α 
Arguments: {α : Type u_1} [Preorder α]
Goal: krullDim αᵒᵈ = krullDim α
-- Heyting.IsRegular.eq
protected theorem IsRegular.eq : IsRegular a → aᶜᶜ = a 
Arguments: {α : Type u_1} [HasCompl α] {a : α}
Goal: Heyting.IsRegular a → aᶜᶜ = a
-- Heyting.isRegular_bot
theorem isRegular_bot : IsRegular (⊥ : α) 
Arguments: {α : Type u_1} [HeytingAlgebra α]
Goal: Heyting.IsRegular ⊥
-- Heyting.isRegular_top
theorem isRegular_top : IsRegular (⊤ : α) 
Arguments: {α : Type u_1} [HeytingAlgebra α]
Goal: Heyting.IsRegular ⊤
-- Heyting.IsRegular.inf
theorem IsRegular.inf (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a ⊓ b) 
Arguments: {α : Type u_1} [HeytingAlgebra α] {a b : α} (ha : Heyting.IsRegular a) (hb : Heyting.IsRegular b)
Goal: Heyting.IsRegular (a ⊓ b)
-- Heyting.IsRegular.himp
theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a ⇨ b) 
Arguments: {α : Type u_1} [HeytingAlgebra α] {a b : α} (ha : Heyting.IsRegular a) (hb : Heyting.IsRegular b)
Goal: Heyting.IsRegular (a ⇨ b)
-- Heyting.isRegular_compl
theorem isRegular_compl (a : α) : IsRegular aᶜ 
Arguments: {α : Type u_1} [HeytingAlgebra α] (a : α)
Goal: Heyting.IsRegular aᶜ
-- Heyting.IsRegular.disjoint_compl_left_iff
protected theorem IsRegular.disjoint_compl_left_iff (ha : IsRegular a) :
    Disjoint aᶜ b ↔ b ≤ a 
Arguments: {α : Type u_1} [HeytingAlgebra α] {a b : α} (ha : Heyting.IsRegular a)
Goal: Disjoint aᶜ b ↔ b ≤ a
-- Heyting.IsRegular.disjoint_compl_right_iff
protected theorem IsRegular.disjoint_compl_right_iff (hb : IsRegular b) :
    Disjoint a bᶜ ↔ a ≤ b 
Arguments: {α : Type u_1} [HeytingAlgebra α] {a b : α} (hb : Heyting.IsRegular b)
Goal: Disjoint a bᶜ ↔ a ≤ b
-- Heyting.Regular.prop
theorem prop : ∀ a : Regular α, IsRegular a.val 
Arguments: {α : Type u_1} [HeytingAlgebra α] (a : Heyting.Regular α)
Goal: Heyting.IsRegular ↑a
-- Heyting.Regular.coe_injective
theorem coe_injective : Injective ((↑) : Regular α → α) 
Arguments: {α : Type u_1} [HeytingAlgebra α]
Goal: Function.Injective Heyting.Regular.val
-- Heyting.Regular.coe_inj
@[simp]
theorem coe_inj {a b : Regular α} : (a : α) = b ↔ a = b 
Arguments: {α : Type u_1} [HeytingAlgebra α] {a b : Heyting.Regular α}
Goal: ↑a = ↑b ↔ a = b
-- Heyting.Regular.coe_top
@[simp, norm_cast]
theorem coe_top : ((⊤ : Regular α) : α) = ⊤ 
Arguments: {α : Type u_1} [HeytingAlgebra α]
Goal: ↑⊤ = ⊤
-- Heyting.Regular.coe_bot
@[simp, norm_cast]
theorem coe_bot : ((⊥ : Regular α) : α) = ⊥ 
Arguments: {α : Type u_1} [HeytingAlgebra α]
Goal: ↑⊥ = ⊥
-- Heyting.Regular.coe_inf
@[simp, norm_cast]
theorem coe_inf (a b : Regular α) : (↑(a ⊓ b) : α) = (a : α) ⊓ b 
Arguments: {α : Type u_1} [HeytingAlgebra α] (a b : Heyting.Regular α)
Goal: ↑(a ⊓ b) = ↑a ⊓ ↑b
-- Heyting.Regular.coe_himp
@[simp, norm_cast]
theorem coe_himp (a b : Regular α) : (↑(a ⇨ b) : α) = (a : α) ⇨ b 
Arguments: {α : Type u_1} [HeytingAlgebra α] (a b : Heyting.Regular α)
Goal: ↑(a ⇨ b) = ↑a ⇨ ↑b
-- Heyting.Regular.coe_compl
@[simp, norm_cast]
theorem coe_compl (a : Regular α) : (↑aᶜ : α) = (a : α)ᶜ 
Arguments: {α : Type u_1} [HeytingAlgebra α] (a : Heyting.Regular α)
Goal: ↑aᶜ = (↑a)ᶜ
-- Heyting.Regular.coe_le_coe
@[simp, norm_cast]
theorem coe_le_coe {a b : Regular α} : (a : α) ≤ b ↔ a ≤ b 
Arguments: {α : Type u_1} [HeytingAlgebra α] {a b : Heyting.Regular α}
Goal: ↑a ≤ ↑b ↔ a ≤ b
-- Heyting.Regular.coe_lt_coe
@[simp, norm_cast]
theorem coe_lt_coe {a b : Regular α} : (a : α) < b ↔ a < b 
Arguments: {α : Type u_1} [HeytingAlgebra α] {a b : Heyting.Regular α}
Goal: ↑a < ↑b ↔ a < b
-- Heyting.Regular.coe_toRegular
@[simp, norm_cast]
theorem coe_toRegular (a : α) : (toRegular a : α) = aᶜᶜ 
Arguments: {α : Type u_1} [HeytingAlgebra α] (a : α)
Goal: ↑(Heyting.Regular.toRegular a) = aᶜᶜ
-- Heyting.Regular.toRegular_coe
@[simp]
theorem toRegular_coe (a : Regular α) : toRegular (a : α) = a 
Arguments: {α : Type u_1} [HeytingAlgebra α] (a : Heyting.Regular α)
Goal: Heyting.Regular.toRegular ↑a = a
-- Heyting.Regular.coe_sup
@[simp, norm_cast]
theorem coe_sup (a b : Regular α) : (↑(a ⊔ b) : α) = ((a : α) ⊔ b)ᶜᶜ 
Arguments: {α : Type u_1} [HeytingAlgebra α] (a b : Heyting.Regular α)
Goal: ↑(a ⊔ b) = (↑a ⊔ ↑b)ᶜᶜ
-- Heyting.Regular.coe_sdiff
@[simp, norm_cast]
theorem coe_sdiff (a b : Regular α) : (↑(a \ b) : α) = (a : α) ⊓ bᶜ 
Arguments: {α : Type u_1} [HeytingAlgebra α] (a b : Heyting.Regular α)
Goal: ↑(a \ b) = ↑a ⊓ (↑b)ᶜ
-- Heyting.isRegular_of_boolean
theorem isRegular_of_boolean : ∀ a : α, IsRegular a 
Arguments: {α : Type u_1} [BooleanAlgebra α] (a : α)
Goal: Heyting.IsRegular a
-- Heyting.isRegular_of_decidable
-- Porting note: removed @[nolint decidable_classical]
theorem isRegular_of_decidable (p : Prop) [Decidable p] : IsRegular p 
Arguments: (p : Prop) [Decidable p]
Goal: Heyting.IsRegular p