ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- hammingNorm_ne_zero_iff
theorem hammingNorm_ne_zero_iff {x : ∀ i, β i} : hammingNorm x ≠ 0 ↔ x ≠ 0 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] [(i : ι) → Zero (β i)] {x : (i : ι) → β i}
Goal: hammingNorm x ≠ 0 ↔ x ≠ 0
-- hammingNorm_pos_iff
@[simp]
theorem hammingNorm_pos_iff {x : ∀ i, β i} : 0 < hammingNorm x ↔ x ≠ 0 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] [(i : ι) → Zero (β i)] {x : (i : ι) → β i}
Goal: 0 < hammingNorm x ↔ x ≠ 0
-- hammingNorm_lt_one
theorem hammingNorm_lt_one {x : ∀ i, β i} : hammingNorm x < 1 ↔ x = 0 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] [(i : ι) → Zero (β i)] {x : (i : ι) → β i}
Goal: hammingNorm x < 1 ↔ x = 0
-- hammingNorm_le_card_fintype
theorem hammingNorm_le_card_fintype {x : ∀ i, β i} : hammingNorm x ≤ Fintype.card ι 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] [(i : ι) → Zero (β i)] {x : (i : ι) → β i}
Goal: hammingNorm x ≤ Fintype.card ι
-- hammingNorm_comp_le_hammingNorm
theorem hammingNorm_comp_le_hammingNorm (f : ∀ i, γ i → β i) {x : ∀ i, γ i} (hf : ∀ i, f i 0 = 0) :
    (hammingNorm fun i => f i (x i)) ≤ hammingNorm x 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] {γ : ι → Type u_4} [(i : ι) → DecidableEq (γ i)] [(i : ι) → Zero (β i)] [(i : ι) → Zero (γ i)] (f : (i : ι) → γ i → β i) {x : (i : ι) → γ i} (hf : ∀ (i : ι), f i 0 = 0)
Goal: (hammingNorm fun i => f i (x i)) ≤ hammingNorm x
-- hammingNorm_comp
theorem hammingNorm_comp (f : ∀ i, γ i → β i) {x : ∀ i, γ i} (hf₁ : ∀ i, Injective (f i))
    (hf₂ : ∀ i, f i 0 = 0) : (hammingNorm fun i => f i (x i)) = hammingNorm x 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] {γ : ι → Type u_4} [(i : ι) → DecidableEq (γ i)] [(i : ι) → Zero (β i)] [(i : ι) → Zero (γ i)] (f : (i : ι) → γ i → β i) {x : (i : ι) → γ i} (hf₁ : ∀ (i : ι), Function.Injective (f i)) (hf₂ : ∀ (i : ι), f i 0 = 0)
Goal: (hammingNorm fun i => f i (x i)) = hammingNorm x
-- hammingNorm_smul_le_hammingNorm
theorem hammingNorm_smul_le_hammingNorm [Zero α] [∀ i, SMulWithZero α (β i)] {k : α}
    {x : ∀ i, β i} : hammingNorm (k • x) ≤ hammingNorm x 
Arguments: {α : Type u_1} {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] [(i : ι) → Zero (β i)] [Zero α] [(i : ι) → SMulWithZero α (β i)] {k : α} {x : (i : ι) → β i}
Goal: hammingNorm (k • x) ≤ hammingNorm x
-- hammingNorm_smul
theorem hammingNorm_smul [Zero α] [∀ i, SMulWithZero α (β i)] {k : α}
    (hk : ∀ i, IsSMulRegular (β i) k) (x : ∀ i, β i) : hammingNorm (k • x) = hammingNorm x 
Arguments: {α : Type u_1} {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] [(i : ι) → Zero (β i)] [Zero α] [(i : ι) → SMulWithZero α (β i)] {k : α} (hk : ∀ (i : ι), IsSMulRegular (β i) k) (x : (i : ι) → β i)
Goal: hammingNorm (k • x) = hammingNorm x
-- hammingDist_eq_hammingNorm
theorem hammingDist_eq_hammingNorm [∀ i, AddGroup (β i)] (x y : ∀ i, β i) :
    hammingDist x y = hammingNorm (x - y) 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] [(i : ι) → AddGroup (β i)] (x y : (i : ι) → β i)
Goal: hammingDist x y = hammingNorm (x - y)
-- Hamming.toHamming_symm_eq
@[simp]
theorem toHamming_symm_eq : (@toHamming _ β).symm = ofHamming 
Arguments: {ι : Type u_2} {β : ι → Type u_3}
Goal: Hamming.toHamming.symm = Hamming.ofHamming
-- Hamming.ofHamming_symm_eq
@[simp]
theorem ofHamming_symm_eq : (@ofHamming _ β).symm = toHamming 
Arguments: {ι : Type u_2} {β : ι → Type u_3}
Goal: Hamming.ofHamming.symm = Hamming.toHamming
-- Hamming.toHamming_ofHamming
@[simp]
theorem toHamming_ofHamming (x : Hamming β) : toHamming (ofHamming x) = x 
Arguments: {ι : Type u_2} {β : ι → Type u_3} (x : Hamming β)
Goal: Hamming.toHamming (Hamming.ofHamming x) = x
-- Hamming.ofHamming_toHamming
@[simp]
theorem ofHamming_toHamming (x : ∀ i, β i) : ofHamming (toHamming x) = x 
Arguments: {ι : Type u_2} {β : ι → Type u_3} (x : (i : ι) → β i)
Goal: Hamming.ofHamming (Hamming.toHamming x) = x
-- Hamming.toHamming_inj
theorem toHamming_inj {x y : ∀ i, β i} : toHamming x = toHamming y ↔ x = y 
Arguments: {ι : Type u_2} {β : ι → Type u_3} {x y : (i : ι) → β i}
Goal: Hamming.toHamming x = Hamming.toHamming y ↔ x = y
-- Hamming.ofHamming_inj
theorem ofHamming_inj {x y : Hamming β} : ofHamming x = ofHamming y ↔ x = y 
Arguments: {ι : Type u_2} {β : ι → Type u_3} {x y : Hamming β}
Goal: Hamming.ofHamming x = Hamming.ofHamming y ↔ x = y
-- Hamming.toHamming_zero
@[simp]
theorem toHamming_zero [∀ i, Zero (β i)] : toHamming (0 : ∀ i, β i) = 0 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [(i : ι) → Zero (β i)]
Goal: Hamming.toHamming 0 = 0
-- Hamming.ofHamming_zero
@[simp]
theorem ofHamming_zero [∀ i, Zero (β i)] : ofHamming (0 : Hamming β) = 0 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [(i : ι) → Zero (β i)]
Goal: Hamming.ofHamming 0 = 0
-- Hamming.toHamming_neg
@[simp]
theorem toHamming_neg [∀ i, Neg (β i)] {x : ∀ i, β i} : toHamming (-x) = -toHamming x 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [(i : ι) → Neg (β i)] {x : (i : ι) → β i}
Goal: Hamming.toHamming (-x) = -Hamming.toHamming x
-- Hamming.ofHamming_neg
@[simp]
theorem ofHamming_neg [∀ i, Neg (β i)] {x : Hamming β} : ofHamming (-x) = -ofHamming x 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [(i : ι) → Neg (β i)] {x : Hamming β}
Goal: Hamming.ofHamming (-x) = -Hamming.ofHamming x
-- Hamming.toHamming_add
@[simp]
theorem toHamming_add [∀ i, Add (β i)] {x y : ∀ i, β i} :
    toHamming (x + y) = toHamming x + toHamming y 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [(i : ι) → Add (β i)] {x y : (i : ι) → β i}
Goal: Hamming.toHamming (x + y) = Hamming.toHamming x + Hamming.toHamming y
-- Hamming.ofHamming_add
@[simp]
theorem ofHamming_add [∀ i, Add (β i)] {x y : Hamming β} :
    ofHamming (x + y) = ofHamming x + ofHamming y 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [(i : ι) → Add (β i)] {x y : Hamming β}
Goal: Hamming.ofHamming (x + y) = Hamming.ofHamming x + Hamming.ofHamming y
-- Hamming.toHamming_sub
@[simp]
theorem toHamming_sub [∀ i, Sub (β i)] {x y : ∀ i, β i} :
    toHamming (x - y) = toHamming x - toHamming y 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [(i : ι) → Sub (β i)] {x y : (i : ι) → β i}
Goal: Hamming.toHamming (x - y) = Hamming.toHamming x - Hamming.toHamming y
-- Hamming.ofHamming_sub
@[simp]
theorem ofHamming_sub [∀ i, Sub (β i)] {x y : Hamming β} :
    ofHamming (x - y) = ofHamming x - ofHamming y 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [(i : ι) → Sub (β i)] {x y : Hamming β}
Goal: Hamming.ofHamming (x - y) = Hamming.ofHamming x - Hamming.ofHamming y
-- Hamming.toHamming_smul
@[simp]
theorem toHamming_smul [∀ i, SMul α (β i)] {r : α} {x : ∀ i, β i} :
    toHamming (r • x) = r • toHamming x 
Arguments: {α : Type u_1} {ι : Type u_2} {β : ι → Type u_3} [(i : ι) → SMul α (β i)] {r : α} {x : (i : ι) → β i}
Goal: Hamming.toHamming (r • x) = r • Hamming.toHamming x
-- Hamming.ofHamming_smul
@[simp]
theorem ofHamming_smul [∀ i, SMul α (β i)] {r : α} {x : Hamming β} :
    ofHamming (r • x) = r • ofHamming x 
Arguments: {α : Type u_1} {ι : Type u_2} {β : ι → Type u_3} [(i : ι) → SMul α (β i)] {r : α} {x : Hamming β}
Goal: Hamming.ofHamming (r • x) = r • Hamming.ofHamming x
-- Hamming.dist_eq_hammingDist
@[simp, push_cast]
theorem dist_eq_hammingDist (x y : Hamming β) :
    dist x y = hammingDist (ofHamming x) (ofHamming y) 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] (x y : Hamming β)
Goal: dist x y = ↑(hammingDist (Hamming.ofHamming x) (Hamming.ofHamming y))
-- Hamming.nndist_eq_hammingDist
@[simp, push_cast]
theorem nndist_eq_hammingDist (x y : Hamming β) :
    nndist x y = hammingDist (ofHamming x) (ofHamming y) 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] (x y : Hamming β)
Goal: nndist x y = ↑(hammingDist (Hamming.ofHamming x) (Hamming.ofHamming y))
-- Hamming.norm_eq_hammingNorm
@[simp, push_cast]
theorem norm_eq_hammingNorm [∀ i, Zero (β i)] (x : Hamming β) : ‖x‖ = hammingNorm (ofHamming x) 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] [(i : ι) → Zero (β i)] (x : Hamming β)
Goal: ‖x‖ = ↑(hammingNorm (Hamming.ofHamming x))
-- Hamming.nnnorm_eq_hammingNorm
@[simp, push_cast]
theorem nnnorm_eq_hammingNorm [∀ i, AddCommGroup (β i)] (x : Hamming β) :
    ‖x‖₊ = hammingNorm (ofHamming x) 
Arguments: {ι : Type u_2} {β : ι → Type u_3} [Fintype ι] [(i : ι) → DecidableEq (β i)] [(i : ι) → AddCommGroup (β i)] (x : Hamming β)
Goal: ‖x‖₊ = ↑(hammingNorm (Hamming.ofHamming x))
-- PSigma.lex_wf
theorem lex_wf (ha : WellFounded r) (hb : ∀ x, WellFounded (s x)) : WellFounded (Lex r s) 
Arguments: {α : Sort u} {β : α → Sort v} {r : α → α → Prop} {s : (a : α) → β a → β a → Prop} (ha : WellFounded r) (hb : ∀ (x : α), WellFounded (s x))
Goal: WellFounded (PSigma.Lex r s)
-- PSigma.revLex_wf
theorem revLex_wf (ha : WellFounded r) (hb : WellFounded s) : WellFounded (RevLex r s) 
Arguments: {α : Sort u} {β : Sort v} {r : α → α → Prop} {s : β → β → Prop} (ha : WellFounded r) (hb : WellFounded s)
Goal: WellFounded (PSigma.RevLex r s)
-- PSigma.skipLeft_wf
theorem skipLeft_wf (α : Type u) {β : Type v} {s : β → β → Prop} (hb : WellFounded s) :
    WellFounded (SkipLeft α s) 
Arguments: (α : Type u) {β : Type v} {s : β → β → Prop} (hb : WellFounded s)
Goal: WellFounded (PSigma.SkipLeft α s)
-- Set.Ioo_def
theorem Ioo_def (a b : α) : { x | a < x ∧ x < b } = Ioo a b 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: {x | a < x ∧ x < b} = Set.Ioo a b
-- Set.Ico_def
theorem Ico_def (a b : α) : { x | a ≤ x ∧ x < b } = Ico a b 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: {x | a ≤ x ∧ x < b} = Set.Ico a b
-- Set.Iio_def
theorem Iio_def (a : α) : { x | x < a } = Iio a 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: {x | x < a} = Set.Iio a
-- Set.Icc_def
theorem Icc_def (a b : α) : { x | a ≤ x ∧ x ≤ b } = Icc a b 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: {x | a ≤ x ∧ x ≤ b} = Set.Icc a b
-- Set.Iic_def
theorem Iic_def (b : α) : { x | x ≤ b } = Iic b 
Arguments: {α : Type u_1} [Preorder α] (b : α)
Goal: {x | x ≤ b} = Set.Iic b
-- Set.Ioc_def
theorem Ioc_def (a b : α) : { x | a < x ∧ x ≤ b } = Ioc a b 
Arguments: {α : Type u_1} [Preorder α] (a b : α)
Goal: {x | a < x ∧ x ≤ b} = Set.Ioc a b
-- Set.Ici_def
theorem Ici_def (a : α) : { x | a ≤ x } = Ici a 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: {x | a ≤ x} = Set.Ici a
-- Set.Ioi_def
theorem Ioi_def (a : α) : { x | a < x } = Ioi a 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: {x | a < x} = Set.Ioi a
-- Set.mem_Ioo
@[simp]
theorem mem_Ioo : x ∈ Ioo a b ↔ a < x ∧ x < b 
Arguments: {α : Type u_1} [Preorder α] {a b x : α}
Goal: x ∈ Set.Ioo a b ↔ a < x ∧ x < b
-- Set.mem_Ico
@[simp]
theorem mem_Ico : x ∈ Ico a b ↔ a ≤ x ∧ x < b 
Arguments: {α : Type u_1} [Preorder α] {a b x : α}
Goal: x ∈ Set.Ico a b ↔ a ≤ x ∧ x < b
-- Set.mem_Iio
@[simp]
theorem mem_Iio : x ∈ Iio b ↔ x < b 
Arguments: {α : Type u_1} [Preorder α] {b x : α}
Goal: x ∈ Set.Iio b ↔ x < b
-- Set.mem_Icc
@[simp]
theorem mem_Icc : x ∈ Icc a b ↔ a ≤ x ∧ x ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b x : α}
Goal: x ∈ Set.Icc a b ↔ a ≤ x ∧ x ≤ b
-- Set.mem_Iic
@[simp]
theorem mem_Iic : x ∈ Iic b ↔ x ≤ b 
Arguments: {α : Type u_1} [Preorder α] {b x : α}
Goal: x ∈ Set.Iic b ↔ x ≤ b
-- Set.mem_Ioc
@[simp]
theorem mem_Ioc : x ∈ Ioc a b ↔ a < x ∧ x ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b x : α}
Goal: x ∈ Set.Ioc a b ↔ a < x ∧ x ≤ b
-- Set.mem_Ici
@[simp]
theorem mem_Ici : x ∈ Ici a ↔ a ≤ x 
Arguments: {α : Type u_1} [Preorder α] {a x : α}
Goal: x ∈ Set.Ici a ↔ a ≤ x
-- Set.mem_Ioi
@[simp]
theorem mem_Ioi : x ∈ Ioi a ↔ a < x 
Arguments: {α : Type u_1} [Preorder α] {a x : α}
Goal: x ∈ Set.Ioi a ↔ a < x
-- Set.left_mem_Ioo
theorem left_mem_Ioo : a ∈ Ioo a b ↔ False 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: a ∈ Set.Ioo a b ↔ False
-- Set.left_mem_Ico
theorem left_mem_Ico : a ∈ Ico a b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: a ∈ Set.Ico a b ↔ a < b
-- Set.left_mem_Icc
theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: a ∈ Set.Icc a b ↔ a ≤ b
-- Set.left_mem_Ioc
theorem left_mem_Ioc : a ∈ Ioc a b ↔ False 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: a ∈ Set.Ioc a b ↔ False
-- Set.left_mem_Ici
theorem left_mem_Ici : a ∈ Ici a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: a ∈ Set.Ici a
-- Set.right_mem_Ioo
theorem right_mem_Ioo : b ∈ Ioo a b ↔ False 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: b ∈ Set.Ioo a b ↔ False
-- Set.right_mem_Ico
theorem right_mem_Ico : b ∈ Ico a b ↔ False 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: b ∈ Set.Ico a b ↔ False
-- Set.right_mem_Icc
theorem right_mem_Icc : b ∈ Icc a b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: b ∈ Set.Icc a b ↔ a ≤ b
-- Set.right_mem_Ioc
theorem right_mem_Ioc : b ∈ Ioc a b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: b ∈ Set.Ioc a b ↔ a < b
-- Set.right_mem_Iic
theorem right_mem_Iic : a ∈ Iic a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: a ∈ Set.Iic a
-- Set.dual_Ici
@[simp]
theorem dual_Ici : Ici (toDual a) = ofDual ⁻¹' Iic a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: Set.Ici (OrderDual.toDual a) = ⇑OrderDual.ofDual ⁻¹' Set.Iic a
-- Set.dual_Iic
@[simp]
theorem dual_Iic : Iic (toDual a) = ofDual ⁻¹' Ici a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: Set.Iic (OrderDual.toDual a) = ⇑OrderDual.ofDual ⁻¹' Set.Ici a
-- Set.dual_Ioi
@[simp]
theorem dual_Ioi : Ioi (toDual a) = ofDual ⁻¹' Iio a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: Set.Ioi (OrderDual.toDual a) = ⇑OrderDual.ofDual ⁻¹' Set.Iio a
-- Set.dual_Iio
@[simp]
theorem dual_Iio : Iio (toDual a) = ofDual ⁻¹' Ioi a 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: Set.Iio (OrderDual.toDual a) = ⇑OrderDual.ofDual ⁻¹' Set.Ioi a
-- Set.dual_Icc
@[simp]
theorem dual_Icc : Icc (toDual a) (toDual b) = ofDual ⁻¹' Icc b a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Icc (OrderDual.toDual a) (OrderDual.toDual b) = ⇑OrderDual.ofDual ⁻¹' Set.Icc b a
-- Set.dual_Ioc
@[simp]
theorem dual_Ioc : Ioc (toDual a) (toDual b) = ofDual ⁻¹' Ico b a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioc (OrderDual.toDual a) (OrderDual.toDual b) = ⇑OrderDual.ofDual ⁻¹' Set.Ico b a
-- Set.dual_Ico
@[simp]
theorem dual_Ico : Ico (toDual a) (toDual b) = ofDual ⁻¹' Ioc b a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ico (OrderDual.toDual a) (OrderDual.toDual b) = ⇑OrderDual.ofDual ⁻¹' Set.Ioc b a
-- Set.dual_Ioo
@[simp]
theorem dual_Ioo : Ioo (toDual a) (toDual b) = ofDual ⁻¹' Ioo b a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioo (OrderDual.toDual a) (OrderDual.toDual b) = ⇑OrderDual.ofDual ⁻¹' Set.Ioo b a
-- Set.nonempty_Icc
@[simp]
theorem nonempty_Icc : (Icc a b).Nonempty ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: (Set.Icc a b).Nonempty ↔ a ≤ b
-- Set.nonempty_Ico
@[simp]
theorem nonempty_Ico : (Ico a b).Nonempty ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: (Set.Ico a b).Nonempty ↔ a < b
-- Set.nonempty_Ioc
@[simp]
theorem nonempty_Ioc : (Ioc a b).Nonempty ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: (Set.Ioc a b).Nonempty ↔ a < b
-- Set.nonempty_Ici
@[simp]
theorem nonempty_Ici : (Ici a).Nonempty 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: (Set.Ici a).Nonempty
-- Set.nonempty_Iic
@[simp]
theorem nonempty_Iic : (Iic a).Nonempty 
Arguments: {α : Type u_1} [Preorder α] {a : α}
Goal: (Set.Iic a).Nonempty
-- Set.nonempty_Ioo
@[simp]
theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α} [DenselyOrdered α]
Goal: (Set.Ioo a b).Nonempty ↔ a < b
-- Set.nonempty_Ioi
@[simp]
theorem nonempty_Ioi [NoMaxOrder α] : (Ioi a).Nonempty 
Arguments: {α : Type u_1} [Preorder α] {a : α} [NoMaxOrder α]
Goal: (Set.Ioi a).Nonempty
-- Set.nonempty_Iio
@[simp]
theorem nonempty_Iio [NoMinOrder α] : (Iio a).Nonempty 
Arguments: {α : Type u_1} [Preorder α] {a : α} [NoMinOrder α]
Goal: (Set.Iio a).Nonempty
-- Set.nonempty_Icc_subtype
theorem nonempty_Icc_subtype (h : a ≤ b) : Nonempty (Icc a b) 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a ≤ b)
Goal: Nonempty ↑(Set.Icc a b)
-- Set.nonempty_Ico_subtype
theorem nonempty_Ico_subtype (h : a < b) : Nonempty (Ico a b) 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a < b)
Goal: Nonempty ↑(Set.Ico a b)
-- Set.nonempty_Ioc_subtype
theorem nonempty_Ioc_subtype (h : a < b) : Nonempty (Ioc a b) 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : a < b)
Goal: Nonempty ↑(Set.Ioc a b)
-- Set.nonempty_Ioo_subtype
theorem nonempty_Ioo_subtype [DenselyOrdered α] (h : a < b) : Nonempty (Ioo a b) 
Arguments: {α : Type u_1} [Preorder α] {a b : α} [DenselyOrdered α] (h : a < b)
Goal: Nonempty ↑(Set.Ioo a b)
-- Set.Icc_eq_empty
@[simp]
theorem Icc_eq_empty (h : ¬a ≤ b) : Icc a b = ∅ 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : ¬a ≤ b)
Goal: Set.Icc a b = ∅
-- Set.Ico_eq_empty
@[simp]
theorem Ico_eq_empty (h : ¬a < b) : Ico a b = ∅ 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : ¬a < b)
Goal: Set.Ico a b = ∅
-- Set.Ioc_eq_empty
@[simp]
theorem Ioc_eq_empty (h : ¬a < b) : Ioc a b = ∅ 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : ¬a < b)
Goal: Set.Ioc a b = ∅
-- Set.Ioo_eq_empty
@[simp]
theorem Ioo_eq_empty (h : ¬a < b) : Ioo a b = ∅ 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : ¬a < b)
Goal: Set.Ioo a b = ∅
-- Set.Icc_eq_empty_of_lt
@[simp]
theorem Icc_eq_empty_of_lt (h : b < a) : Icc a b = ∅ 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : b < a)
Goal: Set.Icc a b = ∅
-- Set.Ico_eq_empty_of_le
@[simp]
theorem Ico_eq_empty_of_le (h : b ≤ a) : Ico a b = ∅ 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : b ≤ a)
Goal: Set.Ico a b = ∅
-- Set.Ioc_eq_empty_of_le
@[simp]
theorem Ioc_eq_empty_of_le (h : b ≤ a) : Ioc a b = ∅ 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : b ≤ a)
Goal: Set.Ioc a b = ∅
-- Set.Ioo_eq_empty_of_le
@[simp]
theorem Ioo_eq_empty_of_le (h : b ≤ a) : Ioo a b = ∅ 
Arguments: {α : Type u_1} [Preorder α] {a b : α} (h : b ≤ a)
Goal: Set.Ioo a b = ∅
-- Set.Ico_self
theorem Ico_self (a : α) : Ico a a = ∅ 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: Set.Ico a a = ∅
-- Set.Ioc_self
theorem Ioc_self (a : α) : Ioc a a = ∅ 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: Set.Ioc a a = ∅
-- Set.Ioo_self
theorem Ioo_self (a : α) : Ioo a a = ∅ 
Arguments: {α : Type u_1} [Preorder α] (a : α)
Goal: Set.Ioo a a = ∅
-- Set.Ici_subset_Ici
theorem Ici_subset_Ici : Ici a ⊆ Ici b ↔ b ≤ a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ici a ⊆ Set.Ici b ↔ b ≤ a
-- Set.Iic_subset_Iic
theorem Iic_subset_Iic : Iic a ⊆ Iic b ↔ a ≤ b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Iic a ⊆ Set.Iic b ↔ a ≤ b
-- Set.Ici_subset_Ioi
theorem Ici_subset_Ioi : Ici a ⊆ Ioi b ↔ b < a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ici a ⊆ Set.Ioi b ↔ b < a
-- Set.Iic_subset_Iio
theorem Iic_subset_Iio : Iic a ⊆ Iio b ↔ a < b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Iic a ⊆ Set.Iio b ↔ a < b
-- Set.Ioo_subset_Ioo
@[gcongr]
theorem Ioo_subset_Ioo (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioo a₁ b₁ ⊆ Ioo a₂ b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂)
Goal: Set.Ioo a₁ b₁ ⊆ Set.Ioo a₂ b₂
-- Set.Ioo_subset_Ioo_left
@[gcongr]
theorem Ioo_subset_Ioo_left (h : a₁ ≤ a₂) : Ioo a₂ b ⊆ Ioo a₁ b 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b : α} (h : a₁ ≤ a₂)
Goal: Set.Ioo a₂ b ⊆ Set.Ioo a₁ b
-- Set.Ioo_subset_Ioo_right
@[gcongr]
theorem Ioo_subset_Ioo_right (h : b₁ ≤ b₂) : Ioo a b₁ ⊆ Ioo a b₂ 
Arguments: {α : Type u_1} [Preorder α] {a b₁ b₂ : α} (h : b₁ ≤ b₂)
Goal: Set.Ioo a b₁ ⊆ Set.Ioo a b₂
-- Set.Ico_subset_Ico
@[gcongr]
theorem Ico_subset_Ico (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ico a₁ b₁ ⊆ Ico a₂ b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂)
Goal: Set.Ico a₁ b₁ ⊆ Set.Ico a₂ b₂
-- Set.Ico_subset_Ico_left
@[gcongr]
theorem Ico_subset_Ico_left (h : a₁ ≤ a₂) : Ico a₂ b ⊆ Ico a₁ b 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b : α} (h : a₁ ≤ a₂)
Goal: Set.Ico a₂ b ⊆ Set.Ico a₁ b
-- Set.Ico_subset_Ico_right
@[gcongr]
theorem Ico_subset_Ico_right (h : b₁ ≤ b₂) : Ico a b₁ ⊆ Ico a b₂ 
Arguments: {α : Type u_1} [Preorder α] {a b₁ b₂ : α} (h : b₁ ≤ b₂)
Goal: Set.Ico a b₁ ⊆ Set.Ico a b₂
-- Set.Icc_subset_Icc
@[gcongr]
theorem Icc_subset_Icc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Icc a₁ b₁ ⊆ Icc a₂ b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂)
Goal: Set.Icc a₁ b₁ ⊆ Set.Icc a₂ b₂
-- Set.Icc_subset_Icc_left
@[gcongr]
theorem Icc_subset_Icc_left (h : a₁ ≤ a₂) : Icc a₂ b ⊆ Icc a₁ b 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b : α} (h : a₁ ≤ a₂)
Goal: Set.Icc a₂ b ⊆ Set.Icc a₁ b
-- Set.Icc_subset_Icc_right
@[gcongr]
theorem Icc_subset_Icc_right (h : b₁ ≤ b₂) : Icc a b₁ ⊆ Icc a b₂ 
Arguments: {α : Type u_1} [Preorder α] {a b₁ b₂ : α} (h : b₁ ≤ b₂)
Goal: Set.Icc a b₁ ⊆ Set.Icc a b₂
-- Set.Icc_subset_Ioo
theorem Icc_subset_Ioo (ha : a₂ < a₁) (hb : b₁ < b₂) : Icc a₁ b₁ ⊆ Ioo a₂ b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ b₂ : α} (ha : a₂ < a₁) (hb : b₁ < b₂)
Goal: Set.Icc a₁ b₁ ⊆ Set.Ioo a₂ b₂
-- Set.Icc_subset_Ici_self
theorem Icc_subset_Ici_self : Icc a b ⊆ Ici a 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Icc a b ⊆ Set.Ici a
-- Set.Icc_subset_Iic_self
theorem Icc_subset_Iic_self : Icc a b ⊆ Iic b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Icc a b ⊆ Set.Iic b
-- Set.Ioc_subset_Iic_self
theorem Ioc_subset_Iic_self : Ioc a b ⊆ Iic b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioc a b ⊆ Set.Iic b
-- Set.Ioc_subset_Ioc
@[gcongr]
theorem Ioc_subset_Ioc (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂) : Ioc a₁ b₁ ⊆ Ioc a₂ b₂ 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b₁ b₂ : α} (h₁ : a₂ ≤ a₁) (h₂ : b₁ ≤ b₂)
Goal: Set.Ioc a₁ b₁ ⊆ Set.Ioc a₂ b₂
-- Set.Ioc_subset_Ioc_left
@[gcongr]
theorem Ioc_subset_Ioc_left (h : a₁ ≤ a₂) : Ioc a₂ b ⊆ Ioc a₁ b 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b : α} (h : a₁ ≤ a₂)
Goal: Set.Ioc a₂ b ⊆ Set.Ioc a₁ b
-- Set.Ioc_subset_Ioc_right
@[gcongr]
theorem Ioc_subset_Ioc_right (h : b₁ ≤ b₂) : Ioc a b₁ ⊆ Ioc a b₂ 
Arguments: {α : Type u_1} [Preorder α] {a b₁ b₂ : α} (h : b₁ ≤ b₂)
Goal: Set.Ioc a b₁ ⊆ Set.Ioc a b₂
-- Set.Ico_subset_Ioo_left
theorem Ico_subset_Ioo_left (h₁ : a₁ < a₂) : Ico a₂ b ⊆ Ioo a₁ b 
Arguments: {α : Type u_1} [Preorder α] {a₁ a₂ b : α} (h₁ : a₁ < a₂)
Goal: Set.Ico a₂ b ⊆ Set.Ioo a₁ b
-- Set.Ioc_subset_Ioo_right
theorem Ioc_subset_Ioo_right (h : b₁ < b₂) : Ioc a b₁ ⊆ Ioo a b₂ 
Arguments: {α : Type u_1} [Preorder α] {a b₁ b₂ : α} (h : b₁ < b₂)
Goal: Set.Ioc a b₁ ⊆ Set.Ioo a b₂
-- Set.Icc_subset_Ico_right
theorem Icc_subset_Ico_right (h₁ : b₁ < b₂) : Icc a b₁ ⊆ Ico a b₂ 
Arguments: {α : Type u_1} [Preorder α] {a b₁ b₂ : α} (h₁ : b₁ < b₂)
Goal: Set.Icc a b₁ ⊆ Set.Ico a b₂
-- Set.Ioo_subset_Ico_self
theorem Ioo_subset_Ico_self : Ioo a b ⊆ Ico a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioo a b ⊆ Set.Ico a b
-- Set.Ioo_subset_Ioc_self
theorem Ioo_subset_Ioc_self : Ioo a b ⊆ Ioc a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioo a b ⊆ Set.Ioc a b
-- Set.Ico_subset_Icc_self
theorem Ico_subset_Icc_self : Ico a b ⊆ Icc a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ico a b ⊆ Set.Icc a b
-- Set.Ioc_subset_Icc_self
theorem Ioc_subset_Icc_self : Ioc a b ⊆ Icc a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioc a b ⊆ Set.Icc a b
-- Set.Ioo_subset_Icc_self
theorem Ioo_subset_Icc_self : Ioo a b ⊆ Icc a b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioo a b ⊆ Set.Icc a b
-- Set.Ico_subset_Iio_self
theorem Ico_subset_Iio_self : Ico a b ⊆ Iio b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ico a b ⊆ Set.Iio b
-- Set.Ioo_subset_Iio_self
theorem Ioo_subset_Iio_self : Ioo a b ⊆ Iio b 
Arguments: {α : Type u_1} [Preorder α] {a b : α}
Goal: Set.Ioo a b ⊆ Set.Iio b