ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CompositionSeries.eq_of_head_eq_head_of_last_eq_last_of_length_eq_zero
theorem eq_of_head_eq_head_of_last_eq_last_of_length_eq_zero {s₁ s₂ : CompositionSeries X}
    (hb : s₁.head = s₂.head) (ht : s₁.last = s₂.last) (hs₁0 : s₁.length = 0) : s₁ = s₂ 
Arguments: {X : Type u} [Lattice X] [JordanHolderLattice X] {s₁ s₂ : CompositionSeries X} (hb : RelSeries.head s₁ = RelSeries.head s₂) (ht : RelSeries.last s₁ = RelSeries.last s₂) (hs₁0 : s₁.length = 0)
Goal: s₁ = s₂
-- CompositionSeries.exists_last_eq_snoc_equivalent
theorem exists_last_eq_snoc_equivalent (s : CompositionSeries X) (x : X) (hm : IsMaximal x s.last)
    (hb : s.head ≤ x) :
    ∃ t : CompositionSeries X,
      t.head = s.head ∧ t.length + 1 = s.length ∧
      ∃ htx : t.last = x,
        Equivalent s (snoc t s.last (show IsMaximal t.last _ from htx.symm ▸ hm)) 
Arguments: {X : Type u} [Lattice X] [JordanHolderLattice X] (s : CompositionSeries X) (x : X) (hm : JordanHolderLattice.IsMaximal x (RelSeries.last s)) (hb : RelSeries.head s ≤ x)
Goal: ∃ t,
  RelSeries.head t = RelSeries.head s ∧
    t.length + 1 = s.length ∧ ∃ (htx : RelSeries.last t = x), s.Equivalent (RelSeries.snoc t (RelSeries.last s) ⋯)
-- CompositionSeries.jordan_holder
theorem jordan_holder (s₁ s₂ : CompositionSeries X)
    (hb : s₁.head = s₂.head) (ht : s₁.last = s₂.last) :
    Equivalent s₁ s₂ 
Arguments: {X : Type u} [Lattice X] [JordanHolderLattice X] (s₁ s₂ : CompositionSeries X) (hb : RelSeries.head s₁ = RelSeries.head s₂) (ht : RelSeries.last s₁ = RelSeries.last s₂)
Goal: s₁.Equivalent s₂
-- Set.pi_univ_Ici
@[simp]
theorem pi_univ_Ici : (pi univ fun i ↦ Ici (x i)) = Ici x 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] (x : (i : ι) → α i)
Goal: (Set.univ.pi fun i => Set.Ici (x i)) = Set.Ici x
-- Set.pi_univ_Iic
@[simp]
theorem pi_univ_Iic : (pi univ fun i ↦ Iic (x i)) = Iic x 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] (x : (i : ι) → α i)
Goal: (Set.univ.pi fun i => Set.Iic (x i)) = Set.Iic x
-- Set.pi_univ_Icc
@[simp]
theorem pi_univ_Icc : (pi univ fun i ↦ Icc (x i) (y i)) = Icc x y 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] (x y : (i : ι) → α i)
Goal: (Set.univ.pi fun i => Set.Icc (x i) (y i)) = Set.Icc x y
-- Set.piecewise_mem_Icc
theorem piecewise_mem_Icc {s : Set ι} [∀ j, Decidable (j ∈ s)] {f₁ f₂ g₁ g₂ : ∀ i, α i}
    (h₁ : ∀ i ∈ s, f₁ i ∈ Icc (g₁ i) (g₂ i)) (h₂ : ∀ i ∉ s, f₂ i ∈ Icc (g₁ i) (g₂ i)) :
    s.piecewise f₁ f₂ ∈ Icc g₁ g₂ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] {s : Set ι} [(j : ι) → Decidable (j ∈ s)] {f₁ f₂ g₁ g₂ : (i : ι) → α i} (h₁ : ∀ i ∈ s, f₁ i ∈ Set.Icc (g₁ i) (g₂ i)) (h₂ : ∀ i ∉ s, f₂ i ∈ Set.Icc (g₁ i) (g₂ i))
Goal: s.piecewise f₁ f₂ ∈ Set.Icc g₁ g₂
-- Set.piecewise_mem_Icc'
theorem piecewise_mem_Icc' {s : Set ι} [∀ j, Decidable (j ∈ s)] {f₁ f₂ g₁ g₂ : ∀ i, α i}
    (h₁ : f₁ ∈ Icc g₁ g₂) (h₂ : f₂ ∈ Icc g₁ g₂) : s.piecewise f₁ f₂ ∈ Icc g₁ g₂ 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] {s : Set ι} [(j : ι) → Decidable (j ∈ s)] {f₁ f₂ g₁ g₂ : (i : ι) → α i} (h₁ : f₁ ∈ Set.Icc g₁ g₂) (h₂ : f₂ ∈ Set.Icc g₁ g₂)
Goal: s.piecewise f₁ f₂ ∈ Set.Icc g₁ g₂
-- Set.pi_univ_Ioi_subset
theorem pi_univ_Ioi_subset [Nonempty ι]: (pi univ fun i ↦ Ioi (x i)) ⊆ Ioi x 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] (x : (i : ι) → α i) [Nonempty ι]
Goal: (Set.univ.pi fun i => Set.Ioi (x i)) ⊆ Set.Ioi x
-- Set.pi_univ_Iio_subset
theorem pi_univ_Iio_subset [Nonempty ι]: (pi univ fun i ↦ Iio (x i)) ⊆ Iio x 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] (x : (i : ι) → α i) [Nonempty ι]
Goal: (Set.univ.pi fun i => Set.Iio (x i)) ⊆ Set.Iio x
-- Set.pi_univ_Ioo_subset
theorem pi_univ_Ioo_subset [Nonempty ι]: (pi univ fun i ↦ Ioo (x i) (y i)) ⊆ Ioo x y 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] (x y : (i : ι) → α i) [Nonempty ι]
Goal: (Set.univ.pi fun i => Set.Ioo (x i) (y i)) ⊆ Set.Ioo x y
-- Set.pi_univ_Ioc_subset
theorem pi_univ_Ioc_subset [Nonempty ι]: (pi univ fun i ↦ Ioc (x i) (y i)) ⊆ Ioc x y 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] (x y : (i : ι) → α i) [Nonempty ι]
Goal: (Set.univ.pi fun i => Set.Ioc (x i) (y i)) ⊆ Set.Ioc x y
-- Set.pi_univ_Ico_subset
theorem pi_univ_Ico_subset [Nonempty ι]: (pi univ fun i ↦ Ico (x i) (y i)) ⊆ Ico x y 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] (x y : (i : ι) → α i) [Nonempty ι]
Goal: (Set.univ.pi fun i => Set.Ico (x i) (y i)) ⊆ Set.Ico x y
-- Set.pi_univ_Ioc_update_left
theorem pi_univ_Ioc_update_left {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} (hm : x i₀ ≤ m) :
    (pi univ fun i ↦ Ioc (update x i₀ m i) (y i)) =
      { z | m < z i₀ } ∩ pi univ fun i ↦ Ioc (x i) (y i) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] [DecidableEq ι] {x y : (i : ι) → α i} {i₀ : ι} {m : α i₀} (hm : x i₀ ≤ m)
Goal: (Set.univ.pi fun i => Set.Ioc (Function.update x i₀ m i) (y i)) =
  {z | m < z i₀} ∩ Set.univ.pi fun i => Set.Ioc (x i) (y i)
-- Set.pi_univ_Ioc_update_right
theorem pi_univ_Ioc_update_right {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} (hm : m ≤ y i₀) :
    (pi univ fun i ↦ Ioc (x i) (update y i₀ m i)) =
      { z | z i₀ ≤ m } ∩ pi univ fun i ↦ Ioc (x i) (y i) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] [DecidableEq ι] {x y : (i : ι) → α i} {i₀ : ι} {m : α i₀} (hm : m ≤ y i₀)
Goal: (Set.univ.pi fun i => Set.Ioc (x i) (Function.update y i₀ m i)) =
  {z | z i₀ ≤ m} ∩ Set.univ.pi fun i => Set.Ioc (x i) (y i)
-- Set.disjoint_pi_univ_Ioc_update_left_right
theorem disjoint_pi_univ_Ioc_update_left_right {x y : ∀ i, α i} {i₀ : ι} {m : α i₀} :
    Disjoint (pi univ fun i ↦ Ioc (x i) (update y i₀ m i))
    (pi univ fun i ↦ Ioc (update x i₀ m i) (y i)) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Preorder (α i)] [DecidableEq ι] {x y : (i : ι) → α i} {i₀ : ι} {m : α i₀}
Goal: Disjoint (Set.univ.pi fun i => Set.Ioc (x i) (Function.update y i₀ m i))
  (Set.univ.pi fun i => Set.Ioc (Function.update x i₀ m i) (y i))
-- Set.image_update_Icc
theorem image_update_Icc (f : ∀ i, α i) (i : ι) (a b : α i) :
    update f i '' Icc a b = Icc (update f i a) (update f i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] (f : (i : ι) → α i) (i : ι) (a b : α i)
Goal: Function.update f i '' Set.Icc a b = Set.Icc (Function.update f i a) (Function.update f i b)
-- Set.image_update_Ico
theorem image_update_Ico (f : ∀ i, α i) (i : ι) (a b : α i) :
    update f i '' Ico a b = Ico (update f i a) (update f i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] (f : (i : ι) → α i) (i : ι) (a b : α i)
Goal: Function.update f i '' Set.Ico a b = Set.Ico (Function.update f i a) (Function.update f i b)
-- Set.image_update_Ioc
theorem image_update_Ioc (f : ∀ i, α i) (i : ι) (a b : α i) :
    update f i '' Ioc a b = Ioc (update f i a) (update f i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] (f : (i : ι) → α i) (i : ι) (a b : α i)
Goal: Function.update f i '' Set.Ioc a b = Set.Ioc (Function.update f i a) (Function.update f i b)
-- Set.image_update_Ioo
theorem image_update_Ioo (f : ∀ i, α i) (i : ι) (a b : α i) :
    update f i '' Ioo a b = Ioo (update f i a) (update f i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] (f : (i : ι) → α i) (i : ι) (a b : α i)
Goal: Function.update f i '' Set.Ioo a b = Set.Ioo (Function.update f i a) (Function.update f i b)
-- Set.image_update_Icc_left
theorem image_update_Icc_left (f : ∀ i, α i) (i : ι) (a : α i) :
    update f i '' Icc a (f i) = Icc (update f i a) f 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] (f : (i : ι) → α i) (i : ι) (a : α i)
Goal: Function.update f i '' Set.Icc a (f i) = Set.Icc (Function.update f i a) f
-- Set.image_update_Ico_left
theorem image_update_Ico_left (f : ∀ i, α i) (i : ι) (a : α i) :
    update f i '' Ico a (f i) = Ico (update f i a) f 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] (f : (i : ι) → α i) (i : ι) (a : α i)
Goal: Function.update f i '' Set.Ico a (f i) = Set.Ico (Function.update f i a) f
-- Set.image_update_Ioc_left
theorem image_update_Ioc_left (f : ∀ i, α i) (i : ι) (a : α i) :
    update f i '' Ioc a (f i) = Ioc (update f i a) f 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] (f : (i : ι) → α i) (i : ι) (a : α i)
Goal: Function.update f i '' Set.Ioc a (f i) = Set.Ioc (Function.update f i a) f
-- Set.image_update_Ioo_left
theorem image_update_Ioo_left (f : ∀ i, α i) (i : ι) (a : α i) :
    update f i '' Ioo a (f i) = Ioo (update f i a) f 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] (f : (i : ι) → α i) (i : ι) (a : α i)
Goal: Function.update f i '' Set.Ioo a (f i) = Set.Ioo (Function.update f i a) f
-- Set.image_update_Icc_right
theorem image_update_Icc_right (f : ∀ i, α i) (i : ι) (b : α i) :
    update f i '' Icc (f i) b = Icc f (update f i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] (f : (i : ι) → α i) (i : ι) (b : α i)
Goal: Function.update f i '' Set.Icc (f i) b = Set.Icc f (Function.update f i b)
-- Set.image_update_Ico_right
theorem image_update_Ico_right (f : ∀ i, α i) (i : ι) (b : α i) :
    update f i '' Ico (f i) b = Ico f (update f i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] (f : (i : ι) → α i) (i : ι) (b : α i)
Goal: Function.update f i '' Set.Ico (f i) b = Set.Ico f (Function.update f i b)
-- Set.image_update_Ioc_right
theorem image_update_Ioc_right (f : ∀ i, α i) (i : ι) (b : α i) :
    update f i '' Ioc (f i) b = Ioc f (update f i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] (f : (i : ι) → α i) (i : ι) (b : α i)
Goal: Function.update f i '' Set.Ioc (f i) b = Set.Ioc f (Function.update f i b)
-- Set.image_update_Ioo_right
theorem image_update_Ioo_right (f : ∀ i, α i) (i : ι) (b : α i) :
    update f i '' Ioo (f i) b = Ioo f (update f i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] (f : (i : ι) → α i) (i : ι) (b : α i)
Goal: Function.update f i '' Set.Ioo (f i) b = Set.Ioo f (Function.update f i b)
-- Set.image_mulSingle_Icc
@[to_additive]
theorem image_mulSingle_Icc (i : ι) (a b : α i) :
    Pi.mulSingle i '' Icc a b = Icc (Pi.mulSingle i a) (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → One (α i)] (i : ι) (a b : α i)
Goal: Pi.mulSingle i '' Set.Icc a b = Set.Icc (Pi.mulSingle i a) (Pi.mulSingle i b)
-- Set.image_single_Icc
@[to_additive]
theorem image_mulSingle_Icc (i : ι) (a b : α i) :
    Pi.mulSingle i '' Icc a b = Icc (Pi.mulSingle i a) (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] (i : ι) (a b : α i)
Goal: Pi.single i '' Set.Icc a b = Set.Icc (Pi.single i a) (Pi.single i b)
-- Set.image_single_Ico
@[to_additive]
theorem image_mulSingle_Ico (i : ι) (a b : α i) :
    Pi.mulSingle i '' Ico a b = Ico (Pi.mulSingle i a) (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] (i : ι) (a b : α i)
Goal: Pi.single i '' Set.Ico a b = Set.Ico (Pi.single i a) (Pi.single i b)
-- Set.image_mulSingle_Ico
@[to_additive]
theorem image_mulSingle_Ico (i : ι) (a b : α i) :
    Pi.mulSingle i '' Ico a b = Ico (Pi.mulSingle i a) (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → One (α i)] (i : ι) (a b : α i)
Goal: Pi.mulSingle i '' Set.Ico a b = Set.Ico (Pi.mulSingle i a) (Pi.mulSingle i b)
-- Set.image_mulSingle_Ioc
@[to_additive]
theorem image_mulSingle_Ioc (i : ι) (a b : α i) :
    Pi.mulSingle i '' Ioc a b = Ioc (Pi.mulSingle i a) (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → One (α i)] (i : ι) (a b : α i)
Goal: Pi.mulSingle i '' Set.Ioc a b = Set.Ioc (Pi.mulSingle i a) (Pi.mulSingle i b)
-- Set.image_single_Ioc
@[to_additive]
theorem image_mulSingle_Ioc (i : ι) (a b : α i) :
    Pi.mulSingle i '' Ioc a b = Ioc (Pi.mulSingle i a) (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] (i : ι) (a b : α i)
Goal: Pi.single i '' Set.Ioc a b = Set.Ioc (Pi.single i a) (Pi.single i b)
-- Set.image_single_Ioo
@[to_additive]
theorem image_mulSingle_Ioo (i : ι) (a b : α i) :
    Pi.mulSingle i '' Ioo a b = Ioo (Pi.mulSingle i a) (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] (i : ι) (a b : α i)
Goal: Pi.single i '' Set.Ioo a b = Set.Ioo (Pi.single i a) (Pi.single i b)
-- Set.image_mulSingle_Ioo
@[to_additive]
theorem image_mulSingle_Ioo (i : ι) (a b : α i) :
    Pi.mulSingle i '' Ioo a b = Ioo (Pi.mulSingle i a) (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → One (α i)] (i : ι) (a b : α i)
Goal: Pi.mulSingle i '' Set.Ioo a b = Set.Ioo (Pi.mulSingle i a) (Pi.mulSingle i b)
-- Set.image_mulSingle_Icc_left
@[to_additive]
theorem image_mulSingle_Icc_left (i : ι) (a : α i) :
    Pi.mulSingle i '' Icc a 1 = Icc (Pi.mulSingle i a) 1 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → One (α i)] (i : ι) (a : α i)
Goal: Pi.mulSingle i '' Set.Icc a 1 = Set.Icc (Pi.mulSingle i a) 1
-- Set.image_single_Icc_left
@[to_additive]
theorem image_mulSingle_Icc_left (i : ι) (a : α i) :
    Pi.mulSingle i '' Icc a 1 = Icc (Pi.mulSingle i a) 1 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] (i : ι) (a : α i)
Goal: Pi.single i '' Set.Icc a 0 = Set.Icc (Pi.single i a) 0
-- Set.image_mulSingle_Ico_left
@[to_additive]
theorem image_mulSingle_Ico_left (i : ι) (a : α i) :
    Pi.mulSingle i '' Ico a 1 = Ico (Pi.mulSingle i a) 1 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → One (α i)] (i : ι) (a : α i)
Goal: Pi.mulSingle i '' Set.Ico a 1 = Set.Ico (Pi.mulSingle i a) 1
-- Set.image_single_Ico_left
@[to_additive]
theorem image_mulSingle_Ico_left (i : ι) (a : α i) :
    Pi.mulSingle i '' Ico a 1 = Ico (Pi.mulSingle i a) 1 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] (i : ι) (a : α i)
Goal: Pi.single i '' Set.Ico a 0 = Set.Ico (Pi.single i a) 0
-- Set.image_single_Ioc_left
@[to_additive]
theorem image_mulSingle_Ioc_left (i : ι) (a : α i) :
    Pi.mulSingle i '' Ioc a 1 = Ioc (Pi.mulSingle i a) 1 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] (i : ι) (a : α i)
Goal: Pi.single i '' Set.Ioc a 0 = Set.Ioc (Pi.single i a) 0
-- Set.image_mulSingle_Ioc_left
@[to_additive]
theorem image_mulSingle_Ioc_left (i : ι) (a : α i) :
    Pi.mulSingle i '' Ioc a 1 = Ioc (Pi.mulSingle i a) 1 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → One (α i)] (i : ι) (a : α i)
Goal: Pi.mulSingle i '' Set.Ioc a 1 = Set.Ioc (Pi.mulSingle i a) 1
-- Set.image_mulSingle_Ioo_left
@[to_additive]
theorem image_mulSingle_Ioo_left (i : ι) (a : α i) :
    Pi.mulSingle i '' Ioo a 1 = Ioo (Pi.mulSingle i a) 1 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → One (α i)] (i : ι) (a : α i)
Goal: Pi.mulSingle i '' Set.Ioo a 1 = Set.Ioo (Pi.mulSingle i a) 1
-- Set.image_single_Ioo_left
@[to_additive]
theorem image_mulSingle_Ioo_left (i : ι) (a : α i) :
    Pi.mulSingle i '' Ioo a 1 = Ioo (Pi.mulSingle i a) 1 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] (i : ι) (a : α i)
Goal: Pi.single i '' Set.Ioo a 0 = Set.Ioo (Pi.single i a) 0
-- Set.image_mulSingle_Icc_right
@[to_additive]
theorem image_mulSingle_Icc_right (i : ι) (b : α i) :
    Pi.mulSingle i '' Icc 1 b = Icc 1 (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → One (α i)] (i : ι) (b : α i)
Goal: Pi.mulSingle i '' Set.Icc 1 b = Set.Icc 1 (Pi.mulSingle i b)
-- Set.image_single_Icc_right
@[to_additive]
theorem image_mulSingle_Icc_right (i : ι) (b : α i) :
    Pi.mulSingle i '' Icc 1 b = Icc 1 (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] (i : ι) (b : α i)
Goal: Pi.single i '' Set.Icc 0 b = Set.Icc 0 (Pi.single i b)
-- Set.image_mulSingle_Ico_right
@[to_additive]
theorem image_mulSingle_Ico_right (i : ι) (b : α i) :
    Pi.mulSingle i '' Ico 1 b = Ico 1 (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → One (α i)] (i : ι) (b : α i)
Goal: Pi.mulSingle i '' Set.Ico 1 b = Set.Ico 1 (Pi.mulSingle i b)
-- Set.image_single_Ico_right
@[to_additive]
theorem image_mulSingle_Ico_right (i : ι) (b : α i) :
    Pi.mulSingle i '' Ico 1 b = Ico 1 (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] (i : ι) (b : α i)
Goal: Pi.single i '' Set.Ico 0 b = Set.Ico 0 (Pi.single i b)
-- Set.image_mulSingle_Ioc_right
@[to_additive]
theorem image_mulSingle_Ioc_right (i : ι) (b : α i) :
    Pi.mulSingle i '' Ioc 1 b = Ioc 1 (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → One (α i)] (i : ι) (b : α i)
Goal: Pi.mulSingle i '' Set.Ioc 1 b = Set.Ioc 1 (Pi.mulSingle i b)
-- Set.image_single_Ioc_right
@[to_additive]
theorem image_mulSingle_Ioc_right (i : ι) (b : α i) :
    Pi.mulSingle i '' Ioc 1 b = Ioc 1 (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] (i : ι) (b : α i)
Goal: Pi.single i '' Set.Ioc 0 b = Set.Ioc 0 (Pi.single i b)
-- Set.image_mulSingle_Ioo_right
@[to_additive]
theorem image_mulSingle_Ioo_right (i : ι) (b : α i) :
    Pi.mulSingle i '' Ioo 1 b = Ioo 1 (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → One (α i)] (i : ι) (b : α i)
Goal: Pi.mulSingle i '' Set.Ioo 1 b = Set.Ioo 1 (Pi.mulSingle i b)
-- Set.image_single_Ioo_right
@[to_additive]
theorem image_mulSingle_Ioo_right (i : ι) (b : α i) :
    Pi.mulSingle i '' Ioo 1 b = Ioo 1 (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → PartialOrder (α i)] [(i : ι) → Zero (α i)] (i : ι) (b : α i)
Goal: Pi.single i '' Set.Ioo 0 b = Set.Ioo 0 (Pi.single i b)
-- Set.pi_univ_uIcc
@[simp]
theorem pi_univ_uIcc (a b : ∀ i, α i) : (pi univ fun i => uIcc (a i) (b i)) = uIcc a b 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Lattice (α i)] (a b : (i : ι) → α i)
Goal: (Set.univ.pi fun i => Set.uIcc (a i) (b i)) = Set.uIcc a b
-- Set.image_update_uIcc
theorem image_update_uIcc (f : ∀ i, α i) (i : ι) (a b : α i) :
    update f i '' uIcc a b = uIcc (update f i a) (update f i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Lattice (α i)] [DecidableEq ι] (f : (i : ι) → α i) (i : ι) (a b : α i)
Goal: Function.update f i '' Set.uIcc a b = Set.uIcc (Function.update f i a) (Function.update f i b)
-- Set.image_update_uIcc_left
theorem image_update_uIcc_left (f : ∀ i, α i) (i : ι) (a : α i) :
    update f i '' uIcc a (f i) = uIcc (update f i a) f 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Lattice (α i)] [DecidableEq ι] (f : (i : ι) → α i) (i : ι) (a : α i)
Goal: Function.update f i '' Set.uIcc a (f i) = Set.uIcc (Function.update f i a) f
-- Set.image_update_uIcc_right
theorem image_update_uIcc_right (f : ∀ i, α i) (i : ι) (b : α i) :
    update f i '' uIcc (f i) b = uIcc f (update f i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Lattice (α i)] [DecidableEq ι] (f : (i : ι) → α i) (i : ι) (b : α i)
Goal: Function.update f i '' Set.uIcc (f i) b = Set.uIcc f (Function.update f i b)
-- Set.image_single_uIcc
@[to_additive]
theorem image_mulSingle_uIcc (i : ι) (a b : α i) :
    Pi.mulSingle i '' uIcc a b = uIcc (Pi.mulSingle i a) (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Lattice (α i)] [DecidableEq ι] [(i : ι) → Zero (α i)] (i : ι) (a b : α i)
Goal: Pi.single i '' Set.uIcc a b = Set.uIcc (Pi.single i a) (Pi.single i b)
-- Set.image_mulSingle_uIcc
@[to_additive]
theorem image_mulSingle_uIcc (i : ι) (a b : α i) :
    Pi.mulSingle i '' uIcc a b = uIcc (Pi.mulSingle i a) (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Lattice (α i)] [DecidableEq ι] [(i : ι) → One (α i)] (i : ι) (a b : α i)
Goal: Pi.mulSingle i '' Set.uIcc a b = Set.uIcc (Pi.mulSingle i a) (Pi.mulSingle i b)
-- Set.image_mulSingle_uIcc_left
@[to_additive]
theorem image_mulSingle_uIcc_left (i : ι) (a : α i) :
    Pi.mulSingle i '' uIcc a 1 = uIcc (Pi.mulSingle i a) 1 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Lattice (α i)] [DecidableEq ι] [(i : ι) → One (α i)] (i : ι) (a : α i)
Goal: Pi.mulSingle i '' Set.uIcc a 1 = Set.uIcc (Pi.mulSingle i a) 1
-- Set.image_single_uIcc_left
@[to_additive]
theorem image_mulSingle_uIcc_left (i : ι) (a : α i) :
    Pi.mulSingle i '' uIcc a 1 = uIcc (Pi.mulSingle i a) 1 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Lattice (α i)] [DecidableEq ι] [(i : ι) → Zero (α i)] (i : ι) (a : α i)
Goal: Pi.single i '' Set.uIcc a 0 = Set.uIcc (Pi.single i a) 0
-- Set.image_mulSingle_uIcc_right
@[to_additive]
theorem image_mulSingle_uIcc_right (i : ι) (b : α i) :
    Pi.mulSingle i '' uIcc 1 b = uIcc 1 (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Lattice (α i)] [DecidableEq ι] [(i : ι) → One (α i)] (i : ι) (b : α i)
Goal: Pi.mulSingle i '' Set.uIcc 1 b = Set.uIcc 1 (Pi.mulSingle i b)
-- Set.image_single_uIcc_right
@[to_additive]
theorem image_mulSingle_uIcc_right (i : ι) (b : α i) :
    Pi.mulSingle i '' uIcc 1 b = uIcc 1 (Pi.mulSingle i b) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Lattice (α i)] [DecidableEq ι] [(i : ι) → Zero (α i)] (i : ι) (b : α i)
Goal: Pi.single i '' Set.uIcc 0 b = Set.uIcc 0 (Pi.single i b)
-- Set.pi_univ_Ioc_update_union
theorem pi_univ_Ioc_update_union (x y : ∀ i, α i) (i₀ : ι) (m : α i₀) (hm : m ∈ Icc (x i₀) (y i₀)) :
    ((pi univ fun i ↦ Ioc (x i) (update y i₀ m i)) ∪
        pi univ fun i ↦ Ioc (update x i₀ m i) (y i)) =
      pi univ fun i ↦ Ioc (x i) (y i) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → LinearOrder (α i)] (x y : (i : ι) → α i) (i₀ : ι) (m : α i₀) (hm : m ∈ Set.Icc (x i₀) (y i₀))
Goal: ((Set.univ.pi fun i => Set.Ioc (x i) (Function.update y i₀ m i)) ∪
    Set.univ.pi fun i => Set.Ioc (Function.update x i₀ m i) (y i)) =
  Set.univ.pi fun i => Set.Ioc (x i) (y i)
-- Set.Icc_diff_pi_univ_Ioo_subset
theorem Icc_diff_pi_univ_Ioo_subset (x y x' y' : ∀ i, α i) :
    (Icc x y \ pi univ fun i ↦ Ioo (x' i) (y' i)) ⊆
    (⋃ i : ι, Icc x (update y i (x' i))) ∪ ⋃ i : ι, Icc (update x i (y' i)) y 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → LinearOrder (α i)] (x y x' y' : (i : ι) → α i)
Goal: (Set.Icc x y \ Set.univ.pi fun i => Set.Ioo (x' i) (y' i)) ⊆
  (⋃ i, Set.Icc x (Function.update y i (x' i))) ∪ ⋃ i, Set.Icc (Function.update x i (y' i)) y
-- Set.Icc_diff_pi_univ_Ioc_subset
theorem Icc_diff_pi_univ_Ioc_subset (x y z : ∀ i, α i) :
    (Icc x z \ pi univ fun i ↦ Ioc (y i) (z i)) ⊆ ⋃ i : ι, Icc x (update z i (y i)) 
Arguments: {ι : Type u_1} {α : ι → Type u_2} [DecidableEq ι] [(i : ι) → LinearOrder (α i)] (x y z : (i : ι) → α i)
Goal: (Set.Icc x z \ Set.univ.pi fun i => Set.Ioc (y i) (z i)) ⊆ ⋃ i, Set.Icc x (Function.update z i (y i))