ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- IsStrictOrder.swap
theorem IsStrictOrder.swap (r) [IsStrictOrder α r] : IsStrictOrder α (swap r) 
Arguments: {α : Type u} (r : α → α → Prop) [IsStrictOrder α r]
Goal: IsStrictOrder α (Function.swap r)
-- IsPartialOrder.swap
theorem IsPartialOrder.swap (r) [IsPartialOrder α r] : IsPartialOrder α (swap r) 
Arguments: {α : Type u} (r : α → α → Prop) [IsPartialOrder α r]
Goal: IsPartialOrder α (Function.swap r)
-- IsTotalPreorder.swap
theorem IsTotalPreorder.swap (r) [IsTotalPreorder α r] : IsTotalPreorder α (swap r) 
Arguments: {α : Type u} (r : α → α → Prop) [IsTotalPreorder α r]
Goal: IsTotalPreorder α (Function.swap r)
-- IsLinearOrder.swap
theorem IsLinearOrder.swap (r) [IsLinearOrder α r] : IsLinearOrder α (swap r) 
Arguments: {α : Type u} (r : α → α → Prop) [IsLinearOrder α r]
Goal: IsLinearOrder α (Function.swap r)
-- IsAsymm.isAntisymm
protected theorem IsAsymm.isAntisymm (r) [IsAsymm α r] : IsAntisymm α r 
Arguments: {α : Type u} (r : α → α → Prop) [IsAsymm α r]
Goal: IsAntisymm α r
-- IsAsymm.isIrrefl
protected theorem IsAsymm.isIrrefl [IsAsymm α r] : IsIrrefl α r 
Arguments: {α : Type u} {r : α → α → Prop} [IsAsymm α r]
Goal: IsIrrefl α r
-- IsTotal.isTrichotomous
protected theorem IsTotal.isTrichotomous (r) [IsTotal α r] : IsTrichotomous α r 
Arguments: {α : Type u} (r : α → α → Prop) [IsTotal α r]
Goal: IsTrichotomous α r
-- ne_of_irrefl
theorem ne_of_irrefl {r} [IsIrrefl α r] : ∀ {x y : α}, r x y → x ≠ y
  
Arguments: {α : Type u} {r : α → α → Prop} [IsIrrefl α r] {x y : α}
Goal: r x y → x ≠ y
-- ne_of_irrefl'
theorem ne_of_irrefl' {r} [IsIrrefl α r] : ∀ {x y : α}, r x y → y ≠ x
  
Arguments: {α : Type u} {r : α → α → Prop} [IsIrrefl α r] {x y : α}
Goal: r x y → y ≠ x
-- not_rel_of_subsingleton
theorem not_rel_of_subsingleton (r) [IsIrrefl α r] [Subsingleton α] (x y) : ¬r x y 
Arguments: {α : Type u} (r : α → α → Prop) [IsIrrefl α r] [Subsingleton α] (x y : α)
Goal: ¬r x y
-- rel_of_subsingleton
theorem rel_of_subsingleton (r) [IsRefl α r] [Subsingleton α] (x y) : r x y 
Arguments: {α : Type u} (r : α → α → Prop) [IsRefl α r] [Subsingleton α] (x y : α)
Goal: r x y
-- empty_relation_apply
@[simp]
theorem empty_relation_apply (a b : α) : EmptyRelation a b ↔ False 
Arguments: {α : Type u} (a b : α)
Goal: EmptyRelation a b ↔ False
-- eq_empty_relation
theorem eq_empty_relation (r) [IsIrrefl α r] [Subsingleton α] : r = EmptyRelation 
Arguments: {α : Type u} (r : α → α → Prop) [IsIrrefl α r] [Subsingleton α]
Goal: r = EmptyRelation
-- trans_trichotomous_left
theorem trans_trichotomous_left [IsTrans α r] [IsTrichotomous α r] {a b c : α} :
    ¬r b a → r b c → r a c 
Arguments: {α : Type u} {r : α → α → Prop} [IsTrans α r] [IsTrichotomous α r] {a b c : α}
Goal: ¬r b a → r b c → r a c
-- trans_trichotomous_right
theorem trans_trichotomous_right [IsTrans α r] [IsTrichotomous α r] {a b c : α} :
    r a b → ¬r c b → r a c 
Arguments: {α : Type u} {r : α → α → Prop} [IsTrans α r] [IsTrichotomous α r] {a b c : α}
Goal: r a b → ¬r c b → r a c
-- transitive_of_trans
theorem transitive_of_trans (r : α → α → Prop) [IsTrans α r] : Transitive r 
Arguments: {α : Type u} (r : α → α → Prop) [IsTrans α r]
Goal: Transitive r
-- extensional_of_trichotomous_of_irrefl
theorem extensional_of_trichotomous_of_irrefl (r : α → α → Prop) [IsTrichotomous α r] [IsIrrefl α r]
    {a b : α} (H : ∀ x, r x a ↔ r x b) : a = b 
Arguments: {α : Type u} (r : α → α → Prop) [IsTrichotomous α r] [IsIrrefl α r] {a b : α} (H : ∀ (x : α), r x a ↔ r x b)
Goal: a = b
-- IsStrictTotalOrder.swap
theorem IsStrictTotalOrder.swap (r) [IsStrictTotalOrder α r] : IsStrictTotalOrder α (swap r) 
Arguments: {α : Type u} (r : α → α → Prop) [IsStrictTotalOrder α r]
Goal: IsStrictTotalOrder α (Function.swap r)
-- IsOrderConnected.conn
class IsOrderConnected (α : Type u) (lt : α → α → Prop) : Prop where
    conn : ∀ a b c, lt a c → lt a b ∨ lt b c
Arguments: {α : Type u} {lt : α → α → Prop} [self : IsOrderConnected α lt] (a b c : α)
Goal: lt a c → lt a b ∨ lt b c
-- IsOrderConnected.neg_trans
theorem IsOrderConnected.neg_trans {r : α → α → Prop} [IsOrderConnected α r] {a b c}
    (h₁ : ¬r a b) (h₂ : ¬r b c) : ¬r a c 
Arguments: {α : Type u} {r : α → α → Prop} [IsOrderConnected α r] {a b c : α} (h₁ : ¬r a b) (h₂ : ¬r b c)
Goal: ¬r a c
-- isStrictWeakOrder_of_isOrderConnected
theorem isStrictWeakOrder_of_isOrderConnected [IsAsymm α r] [IsOrderConnected α r] :
    IsStrictWeakOrder α r 
Arguments: {α : Type u} {r : α → α → Prop} [IsAsymm α r] [IsOrderConnected α r]
Goal: IsStrictWeakOrder α r
-- IsWellFounded.wf
@[mk_iff] class IsWellFounded (α : Type u) (r : α → α → Prop) : Prop where
    wf : WellFounded r
Arguments: {α : Type u} {r : α → α → Prop} [self : IsWellFounded α r]
Goal: WellFounded r
-- isWellFounded_iff
@[mk_iff] class IsWellFounded (α : Type u) (r : α → α → Prop) : Prop where
    wf : WellFounded r
Arguments: (α : Type u) (r : α → α → Prop)
Goal: IsWellFounded α r ↔ WellFounded r
-- WellFoundedRelation.asymmetric
theorem WellFoundedRelation.asymmetric {α : Sort*} [WellFoundedRelation α] {a b : α} :
    WellFoundedRelation.rel a b → ¬ WellFoundedRelation.rel b a 
Arguments: {α : Sort u_1} [WellFoundedRelation α] {a b : α}
Goal: WellFoundedRelation.rel a b → ¬WellFoundedRelation.rel b a
-- WellFounded.prod_lex
lemma WellFounded.prod_lex {ra : α → α → Prop} {rb : β → β → Prop} (ha : WellFounded ra)
    (hb : WellFounded rb) : WellFounded (Prod.Lex ra rb) 
Arguments: {α : Type u} {β : Type v} {ra : α → α → Prop} {rb : β → β → Prop} (ha : WellFounded ra) (hb : WellFounded rb)
Goal: WellFounded (Prod.Lex ra rb)
-- IsWellFounded.induction
theorem induction {C : α → Prop} : ∀ a, (∀ x, (∀ y, r y x → C y) → C x) → C a 
Arguments: {α : Type u} (r : α → α → Prop) [IsWellFounded α r] {C : α → Prop} (a : α)
Goal: (∀ (x : α), (∀ (y : α), r y x → C y) → C x) → C a
-- IsWellFounded.apply
theorem apply : ∀ a, Acc r a 
Arguments: {α : Type u} (r : α → α → Prop) [IsWellFounded α r] (a : α)
Goal: Acc r a
-- IsWellFounded.fix_eq
theorem fix_eq {C : α → Sort*} (F : ∀ x : α, (∀ y : α, r y x → C y) → C x) :
    ∀ x, fix r F x = F x fun y _ => fix r F y 
Arguments: {α : Type u} (r : α → α → Prop) [IsWellFounded α r] {C : α → Sort u_1} (F : (x : α) → ((y : α) → r y x → C y) → C x) (x : α)
Goal: IsWellFounded.fix r F x = F x fun y x => IsWellFounded.fix r F y
-- WellFounded.asymmetric
theorem WellFounded.asymmetric {α : Sort*} {r : α → α → Prop} (h : WellFounded r) (a b) :
    r a b → ¬r b a 
Arguments: {α : Sort u_1} {r : α → α → Prop} (h : WellFounded r) (a b : α)
Goal: r a b → ¬r b a
-- wellFounded_lt
lemma wellFounded_lt [LT α] [WellFoundedLT α] : @WellFounded α (· < ·) 
Arguments: {α : Type u} [LT α] [WellFoundedLT α]
Goal: WellFounded fun x x_1 => x < x_1
-- wellFounded_gt
lemma wellFounded_gt [LT α] [WellFoundedGT α] : @WellFounded α (· > ·) 
Arguments: {α : Type u} [LT α] [WellFoundedGT α]
Goal: WellFounded fun x x_1 => x > x_1
-- wellFoundedGT_dual_iff
theorem wellFoundedGT_dual_iff (α : Type*) [LT α] : WellFoundedGT αᵒᵈ ↔ WellFoundedLT α 
Arguments: (α : Type u_1) [LT α]
Goal: WellFoundedGT αᵒᵈ ↔ WellFoundedLT α
-- wellFoundedLT_dual_iff
theorem wellFoundedLT_dual_iff (α : Type*) [LT α] : WellFoundedLT αᵒᵈ ↔ WellFoundedGT α 
Arguments: (α : Type u_1) [LT α]
Goal: WellFoundedLT αᵒᵈ ↔ WellFoundedGT α
-- WellFoundedLT.induction
theorem induction {C : α → Prop} : ∀ a, (∀ x, (∀ y, y < x → C y) → C x) → C a 
Arguments: {α : Type u} [LT α] [WellFoundedLT α] {C : α → Prop} (a : α)
Goal: (∀ (x : α), (∀ (y : α), y < x → C y) → C x) → C a
-- WellFoundedLT.apply
theorem apply : ∀ a : α, Acc (· < ·) a 
Arguments: {α : Type u} [LT α] [WellFoundedLT α] (a : α)
Goal: Acc (fun x x_1 => x < x_1) a
-- WellFoundedLT.fix_eq
theorem fix_eq {C : α → Sort*} (F : ∀ x : α, (∀ y : α, y < x → C y) → C x) :
    ∀ x, fix F x = F x fun y _ => fix F y 
Arguments: {α : Type u} [LT α] [WellFoundedLT α] {C : α → Sort u_1} (F : (x : α) → ((y : α) → y < x → C y) → C x) (x : α)
Goal: WellFoundedLT.fix F x = F x fun y x => WellFoundedLT.fix F y
-- WellFoundedGT.induction
theorem induction {C : α → Prop} : ∀ a, (∀ x, (∀ y, x < y → C y) → C x) → C a 
Arguments: {α : Type u} [LT α] [WellFoundedGT α] {C : α → Prop} (a : α)
Goal: (∀ (x : α), (∀ (y : α), x < y → C y) → C x) → C a
-- WellFoundedGT.apply
theorem apply : ∀ a : α, Acc (· > ·) a 
Arguments: {α : Type u} [LT α] [WellFoundedGT α] (a : α)
Goal: Acc (fun x x_1 => x > x_1) a
-- WellFoundedGT.fix_eq
theorem fix_eq {C : α → Sort*} (F : ∀ x : α, (∀ y : α, x < y → C y) → C x) :
    ∀ x, fix F x = F x fun y _ => fix F y 
Arguments: {α : Type u} [LT α] [WellFoundedGT α] {C : α → Sort u_1} (F : (x : α) → ((y : α) → x < y → C y) → C x) (x : α)
Goal: WellFoundedGT.fix F x = F x fun y x => WellFoundedGT.fix F y
-- Subsingleton.isWellOrder
theorem Subsingleton.isWellOrder [Subsingleton α] (r : α → α → Prop) [hr : IsIrrefl α r] :
    IsWellOrder α r 
Arguments: {α : Type u} [Subsingleton α] (r : α → α → Prop) [hr : IsIrrefl α r]
Goal: IsWellOrder α r
-- Subrelation.isWellFounded
theorem Subrelation.isWellFounded (r : α → α → Prop) [IsWellFounded α r] {s : α → α → Prop}
    (h : Subrelation s r) : IsWellFounded α s 
Arguments: {α : Type u} (r : α → α → Prop) [IsWellFounded α r] {s : α → α → Prop} (h : Subrelation s r)
Goal: IsWellFounded α s
-- Set.not_bounded_iff
@[simp]
theorem not_bounded_iff {r : α → α → Prop} (s : Set α) : ¬Bounded r s ↔ Unbounded r s 
Arguments: {α : Type u} {r : α → α → Prop} (s : Set α)
Goal: ¬Set.Bounded r s ↔ Set.Unbounded r s
-- Set.not_unbounded_iff
@[simp]
theorem not_unbounded_iff {r : α → α → Prop} (s : Set α) : ¬Unbounded r s ↔ Bounded r s 
Arguments: {α : Type u} {r : α → α → Prop} (s : Set α)
Goal: ¬Set.Unbounded r s ↔ Set.Bounded r s
-- Set.unbounded_of_isEmpty
theorem unbounded_of_isEmpty [IsEmpty α] {r : α → α → Prop} (s : Set α) : Unbounded r s 
Arguments: {α : Type u} [IsEmpty α] {r : α → α → Prop} (s : Set α)
Goal: Set.Unbounded r s
-- IsNonstrictStrictOrder.right_iff_left_not_left
class IsNonstrictStrictOrder (α : Type*) (r : semiOutParam (α → α → Prop)) (s : α → α → Prop) :
    Prop where
    right_iff_left_not_left (a b : α) : s a b ↔ r a b ∧ ¬r b a
Arguments: {α : Type u_1} {r : semiOutParam (α → α → Prop)} {s : α → α → Prop} [self : IsNonstrictStrictOrder α r s] (a b : α)
Goal: s a b ↔ r a b ∧ ¬r b a
-- right_iff_left_not_left
theorem right_iff_left_not_left {r s : α → α → Prop} [IsNonstrictStrictOrder α r s] {a b : α} :
    s a b ↔ r a b ∧ ¬r b a 
Arguments: {α : Type u} {r s : α → α → Prop} [IsNonstrictStrictOrder α r s] {a b : α}
Goal: s a b ↔ r a b ∧ ¬r b a
-- right_iff_left_not_left_of
theorem right_iff_left_not_left_of (r s : α → α → Prop) [IsNonstrictStrictOrder α r s] {a b : α} :
    s a b ↔ r a b ∧ ¬r b a 
Arguments: {α : Type u} (r s : α → α → Prop) [IsNonstrictStrictOrder α r s] {a b : α}
Goal: s a b ↔ r a b ∧ ¬r b a
-- subset_of_eq_of_subset
lemma subset_of_eq_of_subset (hab : a = b) (hbc : b ⊆ c) : a ⊆ c 
Arguments: {α : Type u} [HasSubset α] {a b c : α} (hab : a = b) (hbc : b ⊆ c)
Goal: a ⊆ c
-- subset_of_subset_of_eq
lemma subset_of_subset_of_eq (hab : a ⊆ b) (hbc : b = c) : a ⊆ c 
Arguments: {α : Type u} [HasSubset α] {a b c : α} (hab : a ⊆ b) (hbc : b = c)
Goal: a ⊆ c
-- subset_refl
@[refl, simp]
lemma subset_refl [IsRefl α (· ⊆ ·)] (a : α) : a ⊆ a 
Arguments: {α : Type u} [HasSubset α] [IsRefl α fun x x_1 => x ⊆ x_1] (a : α)
Goal: a ⊆ a
-- subset_rfl
lemma subset_rfl [IsRefl α (· ⊆ ·)] : a ⊆ a 
Arguments: {α : Type u} [HasSubset α] {a : α} [IsRefl α fun x x_1 => x ⊆ x_1]
Goal: a ⊆ a
-- subset_of_eq
lemma subset_of_eq [IsRefl α (· ⊆ ·)] : a = b → a ⊆ b 
Arguments: {α : Type u} [HasSubset α] {a b : α} [IsRefl α fun x x_1 => x ⊆ x_1]
Goal: a = b → a ⊆ b
-- superset_of_eq
lemma superset_of_eq [IsRefl α (· ⊆ ·)] : a = b → b ⊆ a 
Arguments: {α : Type u} [HasSubset α] {a b : α} [IsRefl α fun x x_1 => x ⊆ x_1]
Goal: a = b → b ⊆ a
-- ne_of_not_subset
lemma ne_of_not_subset [IsRefl α (· ⊆ ·)] : ¬a ⊆ b → a ≠ b 
Arguments: {α : Type u} [HasSubset α] {a b : α} [IsRefl α fun x x_1 => x ⊆ x_1]
Goal: ¬a ⊆ b → a ≠ b
-- ne_of_not_superset
lemma ne_of_not_superset [IsRefl α (· ⊆ ·)] : ¬a ⊆ b → b ≠ a 
Arguments: {α : Type u} [HasSubset α] {a b : α} [IsRefl α fun x x_1 => x ⊆ x_1]
Goal: ¬a ⊆ b → b ≠ a
-- subset_trans
@[trans]
lemma subset_trans [IsTrans α (· ⊆ ·)] {a b c : α} : a ⊆ b → b ⊆ c → a ⊆ c 
Arguments: {α : Type u} [HasSubset α] [IsTrans α fun x x_1 => x ⊆ x_1] {a b c : α}
Goal: a ⊆ b → b ⊆ c → a ⊆ c
-- subset_antisymm
lemma subset_antisymm [IsAntisymm α (· ⊆ ·)] : a ⊆ b → b ⊆ a → a = b 
Arguments: {α : Type u} [HasSubset α] {a b : α} [IsAntisymm α fun x x_1 => x ⊆ x_1]
Goal: a ⊆ b → b ⊆ a → a = b
-- superset_antisymm
lemma superset_antisymm [IsAntisymm α (· ⊆ ·)] : a ⊆ b → b ⊆ a → b = a 
Arguments: {α : Type u} [HasSubset α] {a b : α} [IsAntisymm α fun x x_1 => x ⊆ x_1]
Goal: a ⊆ b → b ⊆ a → b = a
-- subset_antisymm_iff
theorem subset_antisymm_iff [IsRefl α (· ⊆ ·)] [IsAntisymm α (· ⊆ ·)] : a = b ↔ a ⊆ b ∧ b ⊆ a 
Arguments: {α : Type u} [HasSubset α] {a b : α} [IsRefl α fun x x_1 => x ⊆ x_1] [IsAntisymm α fun x x_1 => x ⊆ x_1]
Goal: a = b ↔ a ⊆ b ∧ b ⊆ a
-- superset_antisymm_iff
theorem superset_antisymm_iff [IsRefl α (· ⊆ ·)] [IsAntisymm α (· ⊆ ·)] : a = b ↔ b ⊆ a ∧ a ⊆ b 
Arguments: {α : Type u} [HasSubset α] {a b : α} [IsRefl α fun x x_1 => x ⊆ x_1] [IsAntisymm α fun x x_1 => x ⊆ x_1]
Goal: a = b ↔ b ⊆ a ∧ a ⊆ b
-- ssubset_of_eq_of_ssubset
lemma ssubset_of_eq_of_ssubset (hab : a = b) (hbc : b ⊂ c) : a ⊂ c 
Arguments: {α : Type u} [HasSSubset α] {a b c : α} (hab : a = b) (hbc : b ⊂ c)
Goal: a ⊂ c
-- ssubset_of_ssubset_of_eq
lemma ssubset_of_ssubset_of_eq (hab : a ⊂ b) (hbc : b = c) : a ⊂ c 
Arguments: {α : Type u} [HasSSubset α] {a b c : α} (hab : a ⊂ b) (hbc : b = c)
Goal: a ⊂ c
-- ssubset_irrefl
lemma ssubset_irrefl [IsIrrefl α (· ⊂ ·)] (a : α) : ¬a ⊂ a 
Arguments: {α : Type u} [HasSSubset α] [IsIrrefl α fun x x_1 => x ⊂ x_1] (a : α)
Goal: ¬a ⊂ a
-- ssubset_irrfl
lemma ssubset_irrfl [IsIrrefl α (· ⊂ ·)] {a : α} : ¬a ⊂ a 
Arguments: {α : Type u} [HasSSubset α] [IsIrrefl α fun x x_1 => x ⊂ x_1] {a : α}
Goal: ¬a ⊂ a
-- ne_of_ssubset
lemma ne_of_ssubset [IsIrrefl α (· ⊂ ·)] {a b : α} : a ⊂ b → a ≠ b 
Arguments: {α : Type u} [HasSSubset α] [IsIrrefl α fun x x_1 => x ⊂ x_1] {a b : α}
Goal: a ⊂ b → a ≠ b
-- ne_of_ssuperset
lemma ne_of_ssuperset [IsIrrefl α (· ⊂ ·)] {a b : α} : a ⊂ b → b ≠ a 
Arguments: {α : Type u} [HasSSubset α] [IsIrrefl α fun x x_1 => x ⊂ x_1] {a b : α}
Goal: a ⊂ b → b ≠ a
-- ssubset_trans
@[trans]
lemma ssubset_trans [IsTrans α (· ⊂ ·)] {a b c : α} : a ⊂ b → b ⊂ c → a ⊂ c 
Arguments: {α : Type u} [HasSSubset α] [IsTrans α fun x x_1 => x ⊂ x_1] {a b c : α}
Goal: a ⊂ b → b ⊂ c → a ⊂ c
-- ssubset_asymm
lemma ssubset_asymm [IsAsymm α (· ⊂ ·)] {a b : α} : a ⊂ b → ¬b ⊂ a 
Arguments: {α : Type u} [HasSSubset α] [IsAsymm α fun x x_1 => x ⊂ x_1] {a b : α}
Goal: a ⊂ b → ¬b ⊂ a
-- ssubset_iff_subset_not_subset
theorem ssubset_iff_subset_not_subset : a ⊂ b ↔ a ⊆ b ∧ ¬b ⊆ a 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α}
Goal: a ⊂ b ↔ a ⊆ b ∧ ¬b ⊆ a
-- subset_of_ssubset
theorem subset_of_ssubset (h : a ⊂ b) : a ⊆ b 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α} (h : a ⊂ b)
Goal: a ⊆ b
-- not_subset_of_ssubset
theorem not_subset_of_ssubset (h : a ⊂ b) : ¬b ⊆ a 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α} (h : a ⊂ b)
Goal: ¬b ⊆ a
-- not_ssubset_of_subset
theorem not_ssubset_of_subset (h : a ⊆ b) : ¬b ⊂ a 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α} (h : a ⊆ b)
Goal: ¬b ⊂ a
-- ssubset_of_subset_not_subset
theorem ssubset_of_subset_not_subset (h₁ : a ⊆ b) (h₂ : ¬b ⊆ a) : a ⊂ b 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α} (h₁ : a ⊆ b) (h₂ : ¬b ⊆ a)
Goal: a ⊂ b
-- ssubset_of_subset_of_ssubset
theorem ssubset_of_subset_of_ssubset [IsTrans α (· ⊆ ·)] (h₁ : a ⊆ b) (h₂ : b ⊂ c) : a ⊂ c 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b c : α} [IsTrans α fun x x_1 => x ⊆ x_1] (h₁ : a ⊆ b) (h₂ : b ⊂ c)
Goal: a ⊂ c
-- ssubset_of_ssubset_of_subset
theorem ssubset_of_ssubset_of_subset [IsTrans α (· ⊆ ·)] (h₁ : a ⊂ b) (h₂ : b ⊆ c) : a ⊂ c 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b c : α} [IsTrans α fun x x_1 => x ⊆ x_1] (h₁ : a ⊂ b) (h₂ : b ⊆ c)
Goal: a ⊂ c
-- ssubset_of_subset_of_ne
theorem ssubset_of_subset_of_ne [IsAntisymm α (· ⊆ ·)] (h₁ : a ⊆ b) (h₂ : a ≠ b) : a ⊂ b 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α} [IsAntisymm α fun x x_1 => x ⊆ x_1] (h₁ : a ⊆ b) (h₂ : a ≠ b)
Goal: a ⊂ b
-- ssubset_of_ne_of_subset
theorem ssubset_of_ne_of_subset [IsAntisymm α (· ⊆ ·)] (h₁ : a ≠ b) (h₂ : a ⊆ b) : a ⊂ b 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α} [IsAntisymm α fun x x_1 => x ⊆ x_1] (h₁ : a ≠ b) (h₂ : a ⊆ b)
Goal: a ⊂ b
-- eq_or_ssubset_of_subset
theorem eq_or_ssubset_of_subset [IsAntisymm α (· ⊆ ·)] (h : a ⊆ b) : a = b ∨ a ⊂ b 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α} [IsAntisymm α fun x x_1 => x ⊆ x_1] (h : a ⊆ b)
Goal: a = b ∨ a ⊂ b
-- ssubset_or_eq_of_subset
theorem ssubset_or_eq_of_subset [IsAntisymm α (· ⊆ ·)] (h : a ⊆ b) : a ⊂ b ∨ a = b 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α} [IsAntisymm α fun x x_1 => x ⊆ x_1] (h : a ⊆ b)
Goal: a ⊂ b ∨ a = b
-- eq_of_subset_of_not_ssubset
lemma eq_of_subset_of_not_ssubset [IsAntisymm α (· ⊆ ·)] (hab : a ⊆ b) (hba : ¬ a ⊂ b) : a = b 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α} [IsAntisymm α fun x x_1 => x ⊆ x_1] (hab : a ⊆ b) (hba : ¬a ⊂ b)
Goal: a = b
-- eq_of_superset_of_not_ssuperset
lemma eq_of_superset_of_not_ssuperset [IsAntisymm α (· ⊆ ·)] (hab : a ⊆ b) (hba : ¬ a ⊂ b) :
    b = a 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α} [IsAntisymm α fun x x_1 => x ⊆ x_1] (hab : a ⊆ b) (hba : ¬a ⊂ b)
Goal: b = a
-- ssubset_iff_subset_ne
theorem ssubset_iff_subset_ne [IsAntisymm α (· ⊆ ·)] : a ⊂ b ↔ a ⊆ b ∧ a ≠ b 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α} [IsAntisymm α fun x x_1 => x ⊆ x_1]
Goal: a ⊂ b ↔ a ⊆ b ∧ a ≠ b
-- subset_iff_ssubset_or_eq
theorem subset_iff_ssubset_or_eq [IsRefl α (· ⊆ ·)] [IsAntisymm α (· ⊆ ·)] :
    a ⊆ b ↔ a ⊂ b ∨ a = b 
Arguments: {α : Type u} [HasSubset α] [HasSSubset α] [IsNonstrictStrictOrder α (fun x x_1 => x ⊆ x_1) fun x x_1 => x ⊂ x_1] {a b : α} [IsRefl α fun x x_1 => x ⊆ x_1] [IsAntisymm α fun x x_1 => x ⊆ x_1]
Goal: a ⊆ b ↔ a ⊂ b ∨ a = b
-- transitive_le
theorem transitive_le [Preorder α] : Transitive (@LE.le α _) 
Arguments: {α : Type u} [Preorder α]
Goal: Transitive LE.le
-- transitive_lt
theorem transitive_lt [Preorder α] : Transitive (@LT.lt α _) 
Arguments: {α : Type u} [Preorder α]
Goal: Transitive LT.lt
-- transitive_ge
theorem transitive_ge [Preorder α] : Transitive (@GE.ge α _) 
Arguments: {α : Type u} [Preorder α]
Goal: Transitive GE.ge
-- transitive_gt
theorem transitive_gt [Preorder α] : Transitive (@GT.gt α _) 
Arguments: {α : Type u} [Preorder α]
Goal: Transitive GT.gt
-- not_of_eq_false
theorem not_of_eq_false {p : Prop} (h : p = False) : ¬p 
Arguments: {p : Prop} (h : p = False)
Goal: ¬p
-- cast_proof_irrel
theorem cast_proof_irrel {β : Sort u} (h₁ h₂ : α = β) (a : α) : cast h₁ a = cast h₂ a 
Arguments: {α β : Sort u} (h₁ h₂ : α = β) (a : α)
Goal: cast h₁ a = cast h₂ a
-- heq_of_eq_rec_left
theorem heq_of_eq_rec_left {φ : α → Sort v} {a a' : α} {p₁ : φ a} {p₂ : φ a'} :
    (e : a = a') → (h₂ : Eq.rec (motive := fun a _ ↦ φ a) p₁ e = p₂) → HEq p₁ p₂
  
Arguments: {α : Sort u} {φ : α → Sort v} {a a' : α} {p₁ : φ a} {p₂ : φ a'} (e : a = a') (h₂ : e ▸ p₁ = p₂)
Goal: HEq p₁ p₂
-- heq_of_eq_rec_right
theorem heq_of_eq_rec_right {φ : α → Sort v} {a a' : α} {p₁ : φ a} {p₂ : φ a'} :
    (e : a' = a) → (h₂ : p₁ = Eq.rec (motive := fun a _ ↦ φ a) p₂ e) → HEq p₁ p₂
  
Arguments: {α : Sort u} {φ : α → Sort v} {a a' : α} {p₁ : φ a} {p₂ : φ a'} (e : a' = a) (h₂ : p₁ = e ▸ p₂)
Goal: HEq p₁ p₂
-- of_heq_true
theorem of_heq_true {a : Prop} (h : HEq a True) : a 
Arguments: {a : Prop} (h : HEq a True)
Goal: a
-- eq_rec_compose
theorem eq_rec_compose {α β φ : Sort u} :
    ∀ (p₁ : β = φ) (p₂ : α = β) (a : α),
      (Eq.recOn p₁ (Eq.recOn p₂ a : β) : φ) = Eq.recOn (Eq.trans p₂ p₁) a
  
Arguments: {α β φ : Sort u} (p₁ : β = φ) (p₂ : α = β) (a : α)
Goal: Eq.recOn p₁ (Eq.recOn p₂ a) = Eq.recOn ⋯ a
-- heq_prop
theorem heq_prop {P Q : Prop} (p : P) (q : Q) : HEq p q 
Arguments: {P Q : Prop} (p : P) (q : Q)
Goal: HEq p q
-- and_true_iff
theorem and_true_iff : p ∧ True ↔ p 
Arguments: (p : Prop)
Goal: p ∧ True ↔ p
-- true_and_iff
theorem true_and_iff : True ∧ p ↔ p 
Arguments: (p : Prop)
Goal: True ∧ p ↔ p
-- and_false_iff
theorem and_false_iff : p ∧ False ↔ False 
Arguments: (p : Prop)
Goal: p ∧ False ↔ False
-- false_and_iff
theorem false_and_iff : False ∧ p ↔ False 
Arguments: (p : Prop)
Goal: False ∧ p ↔ False
-- true_or_iff
theorem true_or_iff : True ∨ p ↔ True 
Arguments: (p : Prop)
Goal: True ∨ p ↔ True
-- or_true_iff
theorem or_true_iff : p ∨ True ↔ True 
Arguments: (p : Prop)
Goal: p ∨ True ↔ True
-- false_or_iff
theorem false_or_iff : False ∨ p ↔ p 
Arguments: (p : Prop)
Goal: False ∨ p ↔ p
-- or_false_iff
theorem or_false_iff : p ∨ False ↔ p 
Arguments: (p : Prop)
Goal: p ∨ False ↔ p
-- not_or_of_not
theorem not_or_of_not : ¬a → ¬b → ¬(a ∨ b) 
Arguments: {a b : Prop}
Goal: ¬a → ¬b → ¬(a ∨ b)
-- iff_true_iff
theorem iff_true_iff : (a ↔ True) ↔ a 
Arguments: {a : Prop}
Goal: (a ↔ True) ↔ a
-- true_iff_iff
theorem true_iff_iff : (True ↔ a) ↔ a 
Arguments: {a : Prop}
Goal: (True ↔ a) ↔ a
-- iff_false_iff
theorem iff_false_iff : (a ↔ False) ↔ ¬a 
Arguments: {a : Prop}
Goal: (a ↔ False) ↔ ¬a
-- false_iff_iff
theorem false_iff_iff : (False ↔ a) ↔ ¬a 
Arguments: {a : Prop}
Goal: (False ↔ a) ↔ ¬a
-- iff_self_iff
theorem iff_self_iff (a : Prop) : (a ↔ a) ↔ True 
Arguments: (a : Prop)
Goal: (a ↔ a) ↔ True
-- ExistsUnique.intro
theorem ExistsUnique.intro {p : α → Prop} (w : α)
    (h₁ : p w) (h₂ : ∀ y, p y → y = w) : ∃! x, p x 
Arguments: {α : Sort u} {p : α → Prop} (w : α) (h₁ : p w) (h₂ : ∀ (y : α), p y → y = w)
Goal: ∃! x, p x
-- ExistsUnique.elim
theorem ExistsUnique.elim {α : Sort u} {p : α → Prop} {b : Prop}
    (h₂ : ∃! x, p x) (h₁ : ∀ x, p x → (∀ y, p y → y = x) → b) : b 
Arguments: {α : Sort u} {p : α → Prop} {b : Prop} (h₂ : ∃! x, p x) (h₁ : ∀ (x : α), p x → (∀ (y : α), p y → y = x) → b)
Goal: b
-- exists_unique_of_exists_of_unique
theorem exists_unique_of_exists_of_unique {α : Sort u} {p : α → Prop}
    (hex : ∃ x, p x) (hunique : ∀ y₁ y₂, p y₁ → p y₂ → y₁ = y₂) : ∃! x, p x 
Arguments: {α : Sort u} {p : α → Prop} (hex : ∃ x, p x) (hunique : ∀ (y₁ y₂ : α), p y₁ → p y₂ → y₁ = y₂)
Goal: ∃! x, p x
-- ExistsUnique.exists
theorem ExistsUnique.exists {p : α → Prop} : (∃! x, p x) → ∃ x, p x 
Arguments: {α : Sort u} {p : α → Prop}
Goal: (∃! x, p x) → ∃ x, p x
-- ExistsUnique.unique
theorem ExistsUnique.unique {α : Sort u} {p : α → Prop}
    (h : ∃! x, p x) {y₁ y₂ : α} (py₁ : p y₁) (py₂ : p y₂) : y₁ = y₂ 
Arguments: {α : Sort u} {p : α → Prop} (h : ∃! x, p x) {y₁ y₂ : α} (py₁ : p y₁) (py₂ : p y₂)
Goal: y₁ = y₂
-- existsUnique_congr
theorem existsUnique_congr {p q : α → Prop} (h : ∀ a, p a ↔ q a) : (∃! a, p a) ↔ ∃! a, q a 
Arguments: {α : Sort u} {p q : α → Prop} (h : ∀ (a : α), p a ↔ q a)
Goal: (∃! a, p a) ↔ ∃! a, q a
-- decide_True'
theorem decide_True' (h : Decidable True) : decide True = true 
Arguments: (h : Decidable True)
Goal: decide True = true