ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Ne.lt_of_le'
theorem Ne.lt_of_le' : b ≠ a → a ≤ b → a < b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α}
Goal: b ≠ a → a ≤ b → a < b
-- lt_self_iff_false
@[simp]
theorem lt_self_iff_false (x : α) : x < x ↔ False 
Arguments: {α : Type u_2} [Preorder α] (x : α)
Goal: x < x ↔ False
-- le_of_le_of_eq'
theorem le_of_le_of_eq' : b ≤ c → a = b → a ≤ c 
Arguments: {α : Type u_2} [Preorder α] {a b c : α}
Goal: b ≤ c → a = b → a ≤ c
-- le_of_eq_of_le'
theorem le_of_eq_of_le' : b = c → a ≤ b → a ≤ c 
Arguments: {α : Type u_2} [Preorder α] {a b c : α}
Goal: b = c → a ≤ b → a ≤ c
-- lt_of_lt_of_eq'
theorem lt_of_lt_of_eq' : b < c → a = b → a < c 
Arguments: {α : Type u_2} [Preorder α] {a b c : α}
Goal: b < c → a = b → a < c
-- lt_of_eq_of_lt'
theorem lt_of_eq_of_lt' : b = c → a < b → a < c 
Arguments: {α : Type u_2} [Preorder α] {a b c : α}
Goal: b = c → a < b → a < c
-- Eq.ge
protected theorem ge (h : x = y) : y ≤ x 
Arguments: {α : Type u_2} [Preorder α] {x y : α} (h : x = y)
Goal: y ≤ x
-- Eq.not_lt
theorem not_lt (h : x = y) : ¬x < y 
Arguments: {α : Type u_2} [Preorder α] {x y : α} (h : x = y)
Goal: ¬x < y
-- Eq.not_gt
theorem not_gt (h : x = y) : ¬y < x 
Arguments: {α : Type u_2} [Preorder α] {x y : α} (h : x = y)
Goal: ¬y < x
-- le_of_subsingleton
@[simp] lemma le_of_subsingleton [Subsingleton α] : a ≤ b 
Arguments: {α : Type u_2} [Preorder α] {a b : α} [Subsingleton α]
Goal: a ≤ b
-- not_lt_of_subsingleton
lemma not_lt_of_subsingleton [Subsingleton α] : ¬a < b 
Arguments: {α : Type u_2} [Preorder α] {a b : α} [Subsingleton α]
Goal: ¬a < b
-- LE.le.ge
protected theorem ge [LE α] {x y : α} (h : x ≤ y) : y ≥ x 
Arguments: {α : Type u_2} [LE α] {x y : α} (h : x ≤ y)
Goal: y ≥ x
-- LE.le.lt_iff_ne
theorem lt_iff_ne (h : a ≤ b) : a < b ↔ a ≠ b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: a < b ↔ a ≠ b
-- LE.le.gt_iff_ne
theorem gt_iff_ne (h : a ≤ b) : a < b ↔ b ≠ a 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: a < b ↔ b ≠ a
-- LE.le.not_lt_iff_eq
theorem not_lt_iff_eq (h : a ≤ b) : ¬a < b ↔ a = b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: ¬a < b ↔ a = b
-- LE.le.not_gt_iff_eq
theorem not_gt_iff_eq (h : a ≤ b) : ¬a < b ↔ b = a 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: ¬a < b ↔ b = a
-- LE.le.le_iff_eq
theorem le_iff_eq (h : a ≤ b) : b ≤ a ↔ b = a 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: b ≤ a ↔ b = a
-- LE.le.ge_iff_eq
theorem ge_iff_eq (h : a ≤ b) : b ≤ a ↔ a = b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: b ≤ a ↔ a = b
-- LE.le.lt_or_le
theorem lt_or_le [LinearOrder α] {a b : α} (h : a ≤ b) (c : α) : a < c ∨ c ≤ b 
Arguments: {α : Type u_2} [LinearOrder α] {a b : α} (h : a ≤ b) (c : α)
Goal: a < c ∨ c ≤ b
-- LE.le.le_or_lt
theorem le_or_lt [LinearOrder α] {a b : α} (h : a ≤ b) (c : α) : a ≤ c ∨ c < b 
Arguments: {α : Type u_2} [LinearOrder α] {a b : α} (h : a ≤ b) (c : α)
Goal: a ≤ c ∨ c < b
-- LE.le.le_or_le
theorem le_or_le [LinearOrder α] {a b : α} (h : a ≤ b) (c : α) : a ≤ c ∨ c ≤ b 
Arguments: {α : Type u_2} [LinearOrder α] {a b : α} (h : a ≤ b) (c : α)
Goal: a ≤ c ∨ c ≤ b
-- LT.lt.gt
protected theorem gt [LT α] {x y : α} (h : x < y) : y > x 
Arguments: {α : Type u_2} [LT α] {x y : α} (h : x < y)
Goal: y > x
-- LT.lt.false
protected theorem false [Preorder α] {x : α} : x < x → False 
Arguments: {α : Type u_2} [Preorder α] {x : α}
Goal: x < x → False
-- LT.lt.ne'
theorem ne' [Preorder α] {x y : α} (h : x < y) : y ≠ x 
Arguments: {α : Type u_2} [Preorder α] {x y : α} (h : x < y)
Goal: y ≠ x
-- LT.lt.lt_or_lt
theorem lt_or_lt [LinearOrder α] {x y : α} (h : x < y) (z : α) : x < z ∨ z < y 
Arguments: {α : Type u_2} [LinearOrder α] {x y : α} (h : x < y) (z : α)
Goal: x < z ∨ z < y
-- GE.ge.le
protected theorem GE.ge.le [LE α] {x y : α} (h : x ≥ y) : y ≤ x 
Arguments: {α : Type u_2} [LE α] {x y : α} (h : x ≥ y)
Goal: y ≤ x
-- GT.gt.lt
protected theorem GT.gt.lt [LT α] {x y : α} (h : x > y) : y < x 
Arguments: {α : Type u_2} [LT α] {x y : α} (h : x > y)
Goal: y < x
-- ge_of_eq
theorem ge_of_eq [Preorder α] {a b : α} (h : a = b) : a ≥ b 
Arguments: {α : Type u_2} [Preorder α] {a b : α} (h : a = b)
Goal: a ≥ b
-- not_le_of_lt
theorem not_le_of_lt [Preorder α] {a b : α} (h : a < b) : ¬b ≤ a 
Arguments: {α : Type u_2} [Preorder α] {a b : α} (h : a < b)
Goal: ¬b ≤ a
-- not_lt_of_le
theorem not_lt_of_le [Preorder α] {a b : α} (h : a ≤ b) : ¬b < a 
Arguments: {α : Type u_2} [Preorder α] {a b : α} (h : a ≤ b)
Goal: ¬b < a
-- ne_of_not_le
theorem ne_of_not_le [Preorder α] {a b : α} (h : ¬a ≤ b) : a ≠ b 
Arguments: {α : Type u_2} [Preorder α] {a b : α} (h : ¬a ≤ b)
Goal: a ≠ b
-- Decidable.le_iff_eq_or_lt
protected theorem Decidable.le_iff_eq_or_lt [PartialOrder α] [@DecidableRel α (· ≤ ·)] {a b : α} :
    a ≤ b ↔ a = b ∨ a < b 
Arguments: {α : Type u_2} [PartialOrder α] [DecidableRel fun x x_1 => x ≤ x_1] {a b : α}
Goal: a ≤ b ↔ a = b ∨ a < b
-- le_iff_eq_or_lt
theorem le_iff_eq_or_lt [PartialOrder α] {a b : α} : a ≤ b ↔ a = b ∨ a < b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α}
Goal: a ≤ b ↔ a = b ∨ a < b
-- lt_iff_le_and_ne
theorem lt_iff_le_and_ne [PartialOrder α] {a b : α} : a < b ↔ a ≤ b ∧ a ≠ b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α}
Goal: a < b ↔ a ≤ b ∧ a ≠ b
-- eq_iff_not_lt_of_le
theorem eq_iff_not_lt_of_le [PartialOrder α] {x y : α} : x ≤ y → y = x ↔ ¬x < y 
Arguments: {α : Type u_2} [PartialOrder α] {x y : α}
Goal: x ≤ y → y = x ↔ ¬x < y
-- Decidable.eq_iff_le_not_lt
protected theorem Decidable.eq_iff_le_not_lt [PartialOrder α] [@DecidableRel α (· ≤ ·)] {a b : α} :
    a = b ↔ a ≤ b ∧ ¬a < b 
Arguments: {α : Type u_2} [PartialOrder α] [DecidableRel fun x x_1 => x ≤ x_1] {a b : α}
Goal: a = b ↔ a ≤ b ∧ ¬a < b
-- eq_iff_le_not_lt
theorem eq_iff_le_not_lt [PartialOrder α] {a b : α} : a = b ↔ a ≤ b ∧ ¬a < b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α}
Goal: a = b ↔ a ≤ b ∧ ¬a < b
-- eq_or_lt_of_le
theorem eq_or_lt_of_le [PartialOrder α] {a b : α} (h : a ≤ b) : a = b ∨ a < b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: a = b ∨ a < b
-- eq_or_gt_of_le
theorem eq_or_gt_of_le [PartialOrder α] {a b : α} (h : a ≤ b) : b = a ∨ a < b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: b = a ∨ a < b
-- gt_or_eq_of_le
theorem gt_or_eq_of_le [PartialOrder α] {a b : α} (h : a ≤ b) : a < b ∨ b = a 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≤ b)
Goal: a < b ∨ b = a
-- eq_of_le_of_not_lt
theorem eq_of_le_of_not_lt [PartialOrder α] {a b : α} (hab : a ≤ b) (hba : ¬a < b) : a = b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (hab : a ≤ b) (hba : ¬a < b)
Goal: a = b
-- eq_of_ge_of_not_gt
theorem eq_of_ge_of_not_gt [PartialOrder α] {a b : α} (hab : a ≤ b) (hba : ¬a < b) : b = a 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (hab : a ≤ b) (hba : ¬a < b)
Goal: b = a
-- Ne.le_iff_lt
theorem Ne.le_iff_lt [PartialOrder α] {a b : α} (h : a ≠ b) : a ≤ b ↔ a < b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≠ b)
Goal: a ≤ b ↔ a < b
-- Ne.not_le_or_not_le
theorem Ne.not_le_or_not_le [PartialOrder α] {a b : α} (h : a ≠ b) : ¬a ≤ b ∨ ¬b ≤ a 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (h : a ≠ b)
Goal: ¬a ≤ b ∨ ¬b ≤ a
-- Decidable.ne_iff_lt_iff_le
protected theorem Decidable.ne_iff_lt_iff_le [PartialOrder α] [DecidableEq α] {a b : α} :
    (a ≠ b ↔ a < b) ↔ a ≤ b 
Arguments: {α : Type u_2} [PartialOrder α] [DecidableEq α] {a b : α}
Goal: (a ≠ b ↔ a < b) ↔ a ≤ b
-- ne_iff_lt_iff_le
@[simp]
theorem ne_iff_lt_iff_le [PartialOrder α] {a b : α} : (a ≠ b ↔ a < b) ↔ a ≤ b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α}
Goal: (a ≠ b ↔ a < b) ↔ a ≤ b
-- min_def'
theorem min_def' [LinearOrder α] (a b : α) : min a b = if b ≤ a then b else a 
Arguments: {α : Type u_2} [LinearOrder α] (a b : α)
Goal: min a b = if b ≤ a then b else a
-- max_def'
theorem max_def' [LinearOrder α] (a b : α) : max a b = if b ≤ a then a else b 
Arguments: {α : Type u_2} [LinearOrder α] (a b : α)
Goal: max a b = if b ≤ a then a else b
-- lt_of_not_le
theorem lt_of_not_le [LinearOrder α] {a b : α} (h : ¬b ≤ a) : a < b 
Arguments: {α : Type u_2} [LinearOrder α] {a b : α} (h : ¬b ≤ a)
Goal: a < b
-- lt_iff_not_le
theorem lt_iff_not_le [LinearOrder α] {x y : α} : x < y ↔ ¬y ≤ x 
Arguments: {α : Type u_2} [LinearOrder α] {x y : α}
Goal: x < y ↔ ¬y ≤ x
-- Ne.lt_or_lt
theorem Ne.lt_or_lt [LinearOrder α] {x y : α} (h : x ≠ y) : x < y ∨ y < x 
Arguments: {α : Type u_2} [LinearOrder α] {x y : α} (h : x ≠ y)
Goal: x < y ∨ y < x
-- lt_or_lt_iff_ne
@[simp]
theorem lt_or_lt_iff_ne [LinearOrder α] {x y : α} : x < y ∨ y < x ↔ x ≠ y 
Arguments: {α : Type u_2} [LinearOrder α] {x y : α}
Goal: x < y ∨ y < x ↔ x ≠ y
-- not_lt_iff_eq_or_lt
theorem not_lt_iff_eq_or_lt [LinearOrder α] {a b : α} : ¬a < b ↔ a = b ∨ b < a 
Arguments: {α : Type u_2} [LinearOrder α] {a b : α}
Goal: ¬a < b ↔ a = b ∨ b < a
-- exists_ge_of_linear
theorem exists_ge_of_linear [LinearOrder α] (a b : α) : ∃ c, a ≤ c ∧ b ≤ c 
Arguments: {α : Type u_2} [LinearOrder α] (a b : α)
Goal: ∃ c, a ≤ c ∧ b ≤ c
-- exists_forall_ge_and
lemma exists_forall_ge_and [LinearOrder α] {p q : α → Prop} :
    (∃ i, ∀ j ≥ i, p j) → (∃ i, ∀ j ≥ i, q j) → ∃ i, ∀ j ≥ i, p j ∧ q j
  
Arguments: {α : Type u_2} [LinearOrder α] {p q : α → Prop}
Goal: (∃ i, ∀ (j : α), j ≥ i → p j) → (∃ i, ∀ (j : α), j ≥ i → q j) → ∃ i, ∀ (j : α), j ≥ i → p j ∧ q j
-- lt_imp_lt_of_le_imp_le
theorem lt_imp_lt_of_le_imp_le {β} [LinearOrder α] [Preorder β] {a b : α} {c d : β}
    (H : a ≤ b → c ≤ d) (h : d < c) : b < a 
Arguments: {α : Type u_2} {β : Type u_5} [LinearOrder α] [Preorder β] {a b : α} {c d : β} (H : a ≤ b → c ≤ d) (h : d < c)
Goal: b < a
-- le_imp_le_iff_lt_imp_lt
theorem le_imp_le_iff_lt_imp_lt {β} [LinearOrder α] [LinearOrder β] {a b : α} {c d : β} :
    a ≤ b → c ≤ d ↔ d < c → b < a 
Arguments: {α : Type u_2} {β : Type u_5} [LinearOrder α] [LinearOrder β] {a b : α} {c d : β}
Goal: a ≤ b → c ≤ d ↔ d < c → b < a
-- lt_iff_lt_of_le_iff_le'
theorem lt_iff_lt_of_le_iff_le' {β} [Preorder α] [Preorder β] {a b : α} {c d : β}
    (H : a ≤ b ↔ c ≤ d) (H' : b ≤ a ↔ d ≤ c) : b < a ↔ d < c 
Arguments: {α : Type u_2} {β : Type u_5} [Preorder α] [Preorder β] {a b : α} {c d : β} (H : a ≤ b ↔ c ≤ d) (H' : b ≤ a ↔ d ≤ c)
Goal: b < a ↔ d < c
-- lt_iff_lt_of_le_iff_le
theorem lt_iff_lt_of_le_iff_le {β} [LinearOrder α] [LinearOrder β] {a b : α} {c d : β}
    (H : a ≤ b ↔ c ≤ d) : b < a ↔ d < c 
Arguments: {α : Type u_2} {β : Type u_5} [LinearOrder α] [LinearOrder β] {a b : α} {c d : β} (H : a ≤ b ↔ c ≤ d)
Goal: b < a ↔ d < c
-- le_iff_le_iff_lt_iff_lt
theorem le_iff_le_iff_lt_iff_lt {β} [LinearOrder α] [LinearOrder β] {a b : α} {c d : β} :
    (a ≤ b ↔ c ≤ d) ↔ (b < a ↔ d < c) 
Arguments: {α : Type u_2} {β : Type u_5} [LinearOrder α] [LinearOrder β] {a b : α} {c d : β}
Goal: (a ≤ b ↔ c ≤ d) ↔ (b < a ↔ d < c)
-- eq_of_forall_le_iff
theorem eq_of_forall_le_iff [PartialOrder α] {a b : α} (H : ∀ c, c ≤ a ↔ c ≤ b) : a = b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (H : ∀ (c : α), c ≤ a ↔ c ≤ b)
Goal: a = b
-- le_of_forall_le
theorem le_of_forall_le [Preorder α] {a b : α} (H : ∀ c, c ≤ a → c ≤ b) : a ≤ b 
Arguments: {α : Type u_2} [Preorder α] {a b : α} (H : ∀ (c : α), c ≤ a → c ≤ b)
Goal: a ≤ b
-- le_of_forall_le'
theorem le_of_forall_le' [Preorder α] {a b : α} (H : ∀ c, a ≤ c → b ≤ c) : b ≤ a 
Arguments: {α : Type u_2} [Preorder α] {a b : α} (H : ∀ (c : α), a ≤ c → b ≤ c)
Goal: b ≤ a
-- le_of_forall_lt
theorem le_of_forall_lt [LinearOrder α] {a b : α} (H : ∀ c, c < a → c < b) : a ≤ b 
Arguments: {α : Type u_2} [LinearOrder α] {a b : α} (H : ∀ (c : α), c < a → c < b)
Goal: a ≤ b
-- forall_lt_iff_le
theorem forall_lt_iff_le [LinearOrder α] {a b : α} : (∀ ⦃c⦄, c < a → c < b) ↔ a ≤ b 
Arguments: {α : Type u_2} [LinearOrder α] {a b : α}
Goal: (∀ ⦃c : α⦄, c < a → c < b) ↔ a ≤ b
-- le_of_forall_lt'
theorem le_of_forall_lt' [LinearOrder α] {a b : α} (H : ∀ c, a < c → b < c) : b ≤ a 
Arguments: {α : Type u_2} [LinearOrder α] {a b : α} (H : ∀ (c : α), a < c → b < c)
Goal: b ≤ a
-- forall_lt_iff_le'
theorem forall_lt_iff_le' [LinearOrder α] {a b : α} : (∀ ⦃c⦄, a < c → b < c) ↔ b ≤ a 
Arguments: {α : Type u_2} [LinearOrder α] {a b : α}
Goal: (∀ ⦃c : α⦄, a < c → b < c) ↔ b ≤ a
-- eq_of_forall_ge_iff
theorem eq_of_forall_ge_iff [PartialOrder α] {a b : α} (H : ∀ c, a ≤ c ↔ b ≤ c) : a = b 
Arguments: {α : Type u_2} [PartialOrder α] {a b : α} (H : ∀ (c : α), a ≤ c ↔ b ≤ c)
Goal: a = b
-- eq_of_forall_lt_iff
theorem eq_of_forall_lt_iff [LinearOrder α] {a b : α} (h : ∀ c, c < a ↔ c < b) : a = b 
Arguments: {α : Type u_2} [LinearOrder α] {a b : α} (h : ∀ (c : α), c < a ↔ c < b)
Goal: a = b
-- eq_of_forall_gt_iff
theorem eq_of_forall_gt_iff [LinearOrder α] {a b : α} (h : ∀ c, a < c ↔ b < c) : a = b 
Arguments: {α : Type u_2} [LinearOrder α] {a b : α} (h : ∀ (c : α), a < c ↔ b < c)
Goal: a = b
-- rel_imp_eq_of_rel_imp_le
theorem rel_imp_eq_of_rel_imp_le [PartialOrder β] (r : α → α → Prop) [IsSymm α r] {f : α → β}
    (h : ∀ a b, r a b → f a ≤ f b) {a b : α} : r a b → f a = f b 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder β] (r : α → α → Prop) [IsSymm α r] {f : α → β} (h : ∀ (a b : α), r a b → f a ≤ f b) {a b : α}
Goal: r a b → f a = f b
-- le_implies_le_of_le_of_le
theorem le_implies_le_of_le_of_le {a b c d : α} [Preorder α] (hca : c ≤ a) (hbd : b ≤ d) :
    a ≤ b → c ≤ d 
Arguments: {α : Type u_2} {a b c d : α} [Preorder α] (hca : c ≤ a) (hbd : b ≤ d)
Goal: a ≤ b → c ≤ d
-- commutative_of_le
theorem commutative_of_le {f : β → β → α} (comm : ∀ a b, f a b ≤ f b a) : ∀ a b, f a b = f b a 
Arguments: {α : Type u_2} {β : Type u_3} [PartialOrder α] {f : β → β → α} (comm : ∀ (a b : β), f a b ≤ f b a) (a b : β)
Goal: f a b = f b a
-- associative_of_commutative_of_le
theorem associative_of_commutative_of_le {f : α → α → α} (comm : Commutative f)
    (assoc : ∀ a b c, f (f a b) c ≤ f a (f b c)) : Associative f 
Arguments: {α : Type u_2} [PartialOrder α] {f : α → α → α} (comm : Commutative f) (assoc : ∀ (a b c : α), f (f a b) c ≤ f a (f b c))
Goal: Associative f
-- Preorder.toLE_injective
@[ext]
theorem Preorder.toLE_injective : Function.Injective (@Preorder.toLE α) 
Arguments: {α : Type u_2}
Goal: Function.Injective (@Preorder.toLE α)
-- PartialOrder.toPreorder_injective
@[ext]
theorem PartialOrder.toPreorder_injective :
    Function.Injective (@PartialOrder.toPreorder α) 
Arguments: {α : Type u_2}
Goal: Function.Injective (@PartialOrder.toPreorder α)
-- LinearOrder.toPartialOrder_injective
@[ext]
theorem LinearOrder.toPartialOrder_injective :
    Function.Injective (@LinearOrder.toPartialOrder α) 
Arguments: {α : Type u_2}
Goal: Function.Injective (@LinearOrder.toPartialOrder α)
-- Preorder.ext
theorem Preorder.ext {A B : Preorder α}
    (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) : A = B 
Arguments: {α : Type u_2} {A B : Preorder α} (H : ∀ (x y : α), x ≤ y ↔ x ≤ y)
Goal: A = B
-- PartialOrder.ext
theorem PartialOrder.ext {A B : PartialOrder α}
    (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) : A = B 
Arguments: {α : Type u_2} {A B : PartialOrder α} (H : ∀ (x y : α), x ≤ y ↔ x ≤ y)
Goal: A = B
-- LinearOrder.ext
theorem LinearOrder.ext {A B : LinearOrder α}
    (H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) : A = B 
Arguments: {α : Type u_2} {A B : LinearOrder α} (H : ∀ (x y : α), x ≤ y ↔ x ≤ y)
Goal: A = B
-- ltByCases_lt
@[simp]
lemma ltByCases_lt (h : x < y) {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} :
    ltByCases x y h₁ h₂ h₃ = h₁ h 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} (h : x < y) {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P}
Goal: ltByCases x y h₁ h₂ h₃ = h₁ h
-- ltByCases_gt
@[simp]
lemma ltByCases_gt (h : y < x) {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} :
    ltByCases x y h₁ h₂ h₃ = h₃ h 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} (h : y < x) {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P}
Goal: ltByCases x y h₁ h₂ h₃ = h₃ h
-- ltByCases_eq
@[simp]
lemma ltByCases_eq (h : x = y) {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} :
    ltByCases x y h₁ h₂ h₃ = h₂ h 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} (h : x = y) {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P}
Goal: ltByCases x y h₁ h₂ h₃ = h₂ h
-- ltByCases_not_lt
lemma ltByCases_not_lt (h : ¬ x < y) {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P}
    (p : ¬ y < x → x = y := fun h' => (le_antisymm (le_of_not_gt h') (le_of_not_gt h))) :
    ltByCases x y h₁ h₂ h₃ = if h' : y < x then h₃ h' else h₂ (p h') 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} (h : ¬x < y) {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} (p : ¬y < x → x = y := ⋯)
Goal: ltByCases x y h₁ h₂ h₃ = if h' : y < x then h₃ h' else h₂ ⋯
-- ltByCases_not_gt
lemma ltByCases_not_gt (h : ¬ y < x) {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P}
    (p : ¬ x < y → x = y := fun h' => (le_antisymm (le_of_not_gt h) (le_of_not_gt h'))) :
    ltByCases x y h₁ h₂ h₃ = if h' : x < y then h₁ h' else h₂ (p h') 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} (h : ¬y < x) {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} (p : ¬x < y → x = y := ⋯)
Goal: ltByCases x y h₁ h₂ h₃ = if h' : x < y then h₁ h' else h₂ ⋯
-- ltByCases_ne
lemma ltByCases_ne (h : x ≠ y) {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P}
    (p : ¬ x < y → y < x := fun h' => h.lt_or_lt.resolve_left h') :
    ltByCases x y h₁ h₂ h₃ = if h' : x < y then h₁ h' else h₃ (p h') 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} (h : x ≠ y) {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} (p : ¬x < y → y < x := ⋯)
Goal: ltByCases x y h₁ h₂ h₃ = if h' : x < y then h₁ h' else h₃ ⋯
-- ltByCases_comm
lemma ltByCases_comm {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P}
    (p : y = x → x = y := fun h' => h'.symm) :
    ltByCases x y h₁ h₂ h₃ = ltByCases y x h₃ (h₂ ∘ p) h₁ 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} (p : y = x → x = y := ⋯)
Goal: ltByCases x y h₁ h₂ h₃ = ltByCases y x h₃ (h₂ ∘ p) h₁
-- eq_iff_eq_of_lt_iff_lt_of_gt_iff_gt
lemma eq_iff_eq_of_lt_iff_lt_of_gt_iff_gt {x' y' : α}
    (ltc : (x < y) ↔ (x' < y')) (gtc : (y < x) ↔ (y' < x')) :
    x = y ↔ x' = y' 
Arguments: {α : Type u_2} [LinearOrder α] {x y x' y' : α} (ltc : x < y ↔ x' < y') (gtc : y < x ↔ y' < x')
Goal: x = y ↔ x' = y'
-- ltByCases_rec
lemma ltByCases_rec {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} (p : P)
    (hlt : (h : x < y) → h₁ h = p) (heq : (h : x = y) → h₂ h = p)
    (hgt : (h : y < x) → h₃ h = p) :
    ltByCases x y h₁ h₂ h₃ = p 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} (p : P) (hlt : ∀ (h : x < y), h₁ h = p) (heq : ∀ (h : x = y), h₂ h = p) (hgt : ∀ (h : y < x), h₃ h = p)
Goal: ltByCases x y h₁ h₂ h₃ = p
-- ltByCases_eq_iff
lemma ltByCases_eq_iff {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} {p : P} :
    ltByCases x y h₁ h₂ h₃ = p ↔ (∃ h, h₁ h = p) ∨ (∃ h, h₂ h = p) ∨ (∃ h, h₃ h = p) 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} {p : P}
Goal: ltByCases x y h₁ h₂ h₃ = p ↔ (∃ h, h₁ h = p) ∨ (∃ h, h₂ h = p) ∨ ∃ h, h₃ h = p
-- ltByCases_congr
lemma ltByCases_congr {x' y' : α} {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P}
    {h₁' : x' < y' → P} {h₂' : x' = y' → P} {h₃' : y' < x' → P} (ltc : (x < y) ↔ (x' < y'))
    (gtc : (y < x) ↔ (y' < x')) (hh'₁ : ∀ (h : x' < y'), h₁ (ltc.mpr h) = h₁' h)
    (hh'₂ : ∀ (h : x' = y'), h₂ ((eq_iff_eq_of_lt_iff_lt_of_gt_iff_gt ltc gtc).mpr h) = h₂' h)
    (hh'₃ : ∀ (h : y' < x'), h₃ (gtc.mpr h) = h₃' h) :
    ltByCases x y h₁ h₂ h₃ = ltByCases x' y' h₁' h₂' h₃' 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y x' y' : α} {h₁ : x < y → P} {h₂ : x = y → P} {h₃ : y < x → P} {h₁' : x' < y' → P} {h₂' : x' = y' → P} {h₃' : y' < x' → P} (ltc : x < y ↔ x' < y') (gtc : y < x ↔ y' < x') (hh'₁ : ∀ (h : x' < y'), h₁ ⋯ = h₁' h) (hh'₂ : ∀ (h : x' = y'), h₂ ⋯ = h₂' h) (hh'₃ : ∀ (h : y' < x'), h₃ ⋯ = h₃' h)
Goal: ltByCases x y h₁ h₂ h₃ = ltByCases x' y' h₁' h₂' h₃'
-- ltTrichotomy_lt
@[simp]
lemma ltTrichotomy_lt (h : x < y) : ltTrichotomy x y p q r = p 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {p q r : P} (h : x < y)
Goal: ltTrichotomy x y p q r = p
-- ltTrichotomy_gt
@[simp]
lemma ltTrichotomy_gt (h : y < x) : ltTrichotomy x y p q r = r 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {p q r : P} (h : y < x)
Goal: ltTrichotomy x y p q r = r
-- ltTrichotomy_eq
@[simp]
lemma ltTrichotomy_eq (h : x = y) : ltTrichotomy x y p q r = q 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {p q r : P} (h : x = y)
Goal: ltTrichotomy x y p q r = q
-- ltTrichotomy_not_lt
lemma ltTrichotomy_not_lt (h : ¬ x < y) :
    ltTrichotomy x y p q r = if y < x then r else q 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {p q r : P} (h : ¬x < y)
Goal: ltTrichotomy x y p q r = if y < x then r else q
-- ltTrichotomy_not_gt
lemma ltTrichotomy_not_gt (h : ¬ y < x) :
    ltTrichotomy x y p q r = if x < y then p else q 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {p q r : P} (h : ¬y < x)
Goal: ltTrichotomy x y p q r = if x < y then p else q
-- ltTrichotomy_ne
lemma ltTrichotomy_ne (h : x ≠ y) :
    ltTrichotomy x y p q r = if x < y then p else r 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {p q r : P} (h : x ≠ y)
Goal: ltTrichotomy x y p q r = if x < y then p else r
-- ltTrichotomy_comm
lemma ltTrichotomy_comm : ltTrichotomy x y p q r = ltTrichotomy y x r q p 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {p q r : P}
Goal: ltTrichotomy x y p q r = ltTrichotomy y x r q p
-- ltTrichotomy_self
lemma ltTrichotomy_self {p : P} : ltTrichotomy x y p p p = p 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {p : P}
Goal: ltTrichotomy x y p p p = p
-- ltTrichotomy_eq_iff
lemma ltTrichotomy_eq_iff : ltTrichotomy x y p q r = s ↔
    (x < y ∧ p = s) ∨ (x = y ∧ q = s) ∨ (y < x ∧ r = s) 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {p q r s : P}
Goal: ltTrichotomy x y p q r = s ↔ x < y ∧ p = s ∨ x = y ∧ q = s ∨ y < x ∧ r = s
-- ltTrichotomy_congr
lemma ltTrichotomy_congr {x' y' : α} {p' q' r' : P} (ltc : (x < y) ↔ (x' < y'))
    (gtc : (y < x) ↔ (y' < x')) (hh'₁ : x' < y' → p = p')
    (hh'₂ : x' = y' → q = q') (hh'₃ : y' < x' → r = r') :
    ltTrichotomy x y p q r = ltTrichotomy x' y' p' q' r' 
Arguments: {α : Type u_2} [LinearOrder α] {P : Sort u_5} {x y : α} {p q r : P} {x' y' : α} {p' q' r' : P} (ltc : x < y ↔ x' < y') (gtc : y < x ↔ y' < x') (hh'₁ : x' < y' → p = p') (hh'₂ : x' = y' → q = q') (hh'₃ : y' < x' → r = r')
Goal: ltTrichotomy x y p q r = ltTrichotomy x' y' p' q' r'
-- OrderDual.Preorder.dual_dual
theorem Preorder.dual_dual (α : Type*) [H : Preorder α] : OrderDual.instPreorder αᵒᵈ = H 
Arguments: (α : Type u_5) [H : Preorder α]
Goal: OrderDual.instPreorder αᵒᵈ = H
-- OrderDual.instPartialOrder.dual_dual
theorem instPartialOrder.dual_dual (α : Type*) [H : PartialOrder α] :
    OrderDual.instPartialOrder αᵒᵈ = H 
Arguments: (α : Type u_5) [H : PartialOrder α]
Goal: OrderDual.instPartialOrder αᵒᵈ = H
-- OrderDual.instLinearOrder.dual_dual
theorem instLinearOrder.dual_dual (α : Type*) [H : LinearOrder α] :
    OrderDual.instLinearOrder αᵒᵈ = H 
Arguments: (α : Type u_5) [H : LinearOrder α]
Goal: OrderDual.instLinearOrder αᵒᵈ = H
-- Pi.compl_def
theorem Pi.compl_def [∀ i, HasCompl (π i)] (x : ∀ i, π i) :
    xᶜ = fun i ↦ (x i)ᶜ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [(i : ι) → HasCompl (π i)] (x : (i : ι) → π i)
Goal: xᶜ = fun i => (x i)ᶜ
-- Pi.compl_apply
@[simp]
theorem Pi.compl_apply [∀ i, HasCompl (π i)] (x : ∀ i, π i) (i : ι) :
    xᶜ i = (x i)ᶜ 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [(i : ι) → HasCompl (π i)] (x : (i : ι) → π i) (i : ι)
Goal: xᶜ i = (x i)ᶜ
-- Pi.le_def
theorem Pi.le_def [∀ i, LE (π i)] {x y : ∀ i, π i} :
    x ≤ y ↔ ∀ i, x i ≤ y i 
Arguments: {ι : Type u_1} {π : ι → Type u_4} [(i : ι) → LE (π i)] {x y : (i : ι) → π i}
Goal: x ≤ y ↔ ∀ (i : ι), x i ≤ y i