ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- InitialSeg.trans_apply
@[simp]
theorem trans_apply (f : r ≼i s) (g : s ≼i t) (a : α) : (f.trans g) a = g (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (f : r ≼i s) (g : s ≼i t) (a : α)
Goal: (f.trans g) a = g (f a)
-- InitialSeg.eq
protected theorem eq [IsWellOrder β s] (f g : r ≼i s) (a) : f a = g a 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder β s] (f g : r ≼i s) (a : α)
Goal: f a = g a
-- InitialSeg.Antisymm.aux
theorem Antisymm.aux [IsWellOrder α r] (f : r ≼i s) (g : s ≼i r) : LeftInverse g f 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder α r] (f : r ≼i s) (g : s ≼i r)
Goal: Function.LeftInverse ⇑g ⇑f
-- InitialSeg.antisymm_toFun
@[simp]
theorem antisymm_toFun [IsWellOrder β s] (f : r ≼i s) (g : s ≼i r) : (antisymm f g : α → β) = f 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder β s] (f : r ≼i s) (g : s ≼i r)
Goal: ⇑(f.antisymm g) = ⇑f
-- InitialSeg.antisymm_symm
@[simp]
theorem antisymm_symm [IsWellOrder α r] [IsWellOrder β s] (f : r ≼i s) (g : s ≼i r) :
    (antisymm f g).symm = antisymm g f 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder α r] [IsWellOrder β s] (f : r ≼i s) (g : s ≼i r)
Goal: (f.antisymm g).symm = g.antisymm f
-- InitialSeg.eq_or_principal
theorem eq_or_principal [IsWellOrder β s] (f : r ≼i s) :
    Surjective f ∨ ∃ b, ∀ x, s x b ↔ ∃ y, f y = x 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder β s] (f : r ≼i s)
Goal: Function.Surjective ⇑f ∨ ∃ b, ∀ (x : β), s x b ↔ ∃ y, f y = x
-- InitialSeg.codRestrict_apply
@[simp]
theorem codRestrict_apply (p) (f : r ≼i s) (H a) : codRestrict p f H a = ⟨f a, H a⟩ 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (p : Set β) (f : r ≼i s) (H : ∀ (a : α), f a ∈ p) (a : α)
Goal: (InitialSeg.codRestrict p f H) a = ⟨f a, ⋯⟩
-- InitialSeg.leAdd_apply
@[simp]
theorem leAdd_apply (r : α → α → Prop) (s : β → β → Prop) (a) : leAdd r s a = Sum.inl a 
Arguments: {α : Type u_1} {β : Type u_2} (r : α → α → Prop) (s : β → β → Prop) (a : α)
Goal: (InitialSeg.leAdd r s) a = Sum.inl a
-- InitialSeg.acc
protected theorem acc (f : r ≼i s) (a : α) : Acc r a ↔ Acc s (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≼i s) (a : α)
Goal: Acc r a ↔ Acc s (f a)
-- PrincipalSeg.down'
structure PrincipalSeg {α β : Type*} (r : α → α → Prop) (s : β → β → Prop) extends r ↪r s where
    top : β
    down' : ∀ b, s b top ↔ ∃ a, toRelEmbedding a = b
Arguments: {α : Type u_4} {β : Type u_5} {r : α → α → Prop} {s : β → β → Prop} (self : r ≺i s) (b : β)
Goal: s b self.top ↔ ∃ a, self.toRelEmbedding a = b
-- PrincipalSeg.coe_fn_mk
@[simp]
theorem coe_fn_mk (f : r ↪r s) (t o) : (@PrincipalSeg.mk _ _ r s f t o : α → β) = f 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ↪r s) (t : β) (o : ∀ (b : β), s b t ↔ ∃ a, f a = b)
Goal: ⇑{ toRelEmbedding := f, top := t, down' := o }.toRelEmbedding = ⇑f
-- PrincipalSeg.down
theorem down (f : r ≺i s) : ∀ {b : β}, s b f.top ↔ ∃ a, f a = b 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≺i s) {b : β}
Goal: s b f.top ↔ ∃ a, f.toRelEmbedding a = b
-- PrincipalSeg.lt_top
theorem lt_top (f : r ≺i s) (a : α) : s (f a) f.top 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≺i s) (a : α)
Goal: s (f.toRelEmbedding a) f.top
-- PrincipalSeg.init
theorem init [IsTrans β s] (f : r ≺i s) {a : α} {b : β} (h : s b (f a)) : ∃ a', f a' = b 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsTrans β s] (f : r ≺i s) {a : α} {b : β} (h : s b (f.toRelEmbedding a))
Goal: ∃ a', f.toRelEmbedding a' = b
-- PrincipalSeg.coe_coe_fn'
theorem coe_coe_fn' [IsTrans β s] (f : r ≺i s) : ((f : r ≼i s) : α → β) = f 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsTrans β s] (f : r ≺i s)
Goal: ⇑{ toRelEmbedding := f.toRelEmbedding, init' := ⋯ } = ⇑f.toRelEmbedding
-- PrincipalSeg.init_iff
theorem init_iff [IsTrans β s] (f : r ≺i s) {a : α} {b : β} : s b (f a) ↔ ∃ a', f a' = b ∧ r a' a 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsTrans β s] (f : r ≺i s) {a : α} {b : β}
Goal: s b (f.toRelEmbedding a) ↔ ∃ a', f.toRelEmbedding a' = b ∧ r a' a
-- PrincipalSeg.irrefl
theorem irrefl {r : α → α → Prop} [IsWellOrder α r] (f : r ≺i r) : False 
Arguments: {α : Type u_1} {r : α → α → Prop} [IsWellOrder α r] (f : r ≺i r)
Goal: False
-- PrincipalSeg.lt_le_apply
@[simp]
theorem lt_le_apply (f : r ≺i s) (g : s ≼i t) (a : α) : (f.ltLe g) a = g (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (f : r ≺i s) (g : s ≼i t) (a : α)
Goal: (f.ltLe g).toRelEmbedding a = g (f.toRelEmbedding a)
-- PrincipalSeg.lt_le_top
@[simp]
theorem lt_le_top (f : r ≺i s) (g : s ≼i t) : (f.ltLe g).top = g f.top 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (f : r ≺i s) (g : s ≼i t)
Goal: (f.ltLe g).top = g f.top
-- PrincipalSeg.trans_apply
@[simp]
theorem trans_apply [IsTrans γ t] (f : r ≺i s) (g : s ≺i t) (a : α) : (f.trans g) a = g (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} [IsTrans γ t] (f : r ≺i s) (g : s ≺i t) (a : α)
Goal: (f.trans g).toRelEmbedding a = g.toRelEmbedding (f.toRelEmbedding a)
-- PrincipalSeg.trans_top
@[simp]
theorem trans_top [IsTrans γ t] (f : r ≺i s) (g : s ≺i t) : (f.trans g).top = g f.top 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} [IsTrans γ t] (f : r ≺i s) (g : s ≺i t)
Goal: (f.trans g).top = g.toRelEmbedding f.top
-- PrincipalSeg.equivLT_apply
@[simp]
theorem equivLT_apply (f : r ≃r s) (g : s ≺i t) (a : α) : (equivLT f g) a = g (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (f : r ≃r s) (g : s ≺i t) (a : α)
Goal: (PrincipalSeg.equivLT f g).toRelEmbedding a = g.toRelEmbedding (f a)
-- PrincipalSeg.equivLT_top
@[simp]
theorem equivLT_top (f : r ≃r s) (g : s ≺i t) : (equivLT f g).top = g.top 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} (f : r ≃r s) (g : s ≺i t)
Goal: (PrincipalSeg.equivLT f g).top = g.top
-- PrincipalSeg.top_eq
theorem top_eq [IsWellOrder γ t] (e : r ≃r s) (f : r ≺i t) (g : s ≺i t) : f.top = g.top 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} [IsWellOrder γ t] (e : r ≃r s) (f : r ≺i t) (g : s ≺i t)
Goal: f.top = g.top
-- PrincipalSeg.topLTTop
theorem topLTTop {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} [IsWellOrder γ t]
    (f : PrincipalSeg r s) (g : PrincipalSeg s t) (h : PrincipalSeg r t) : t h.top g.top 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} [IsWellOrder γ t] (f : r ≺i s) (g : s ≺i t) (h : r ≺i t)
Goal: t h.top g.top
-- PrincipalSeg.ofElement_apply
@[simp, nolint simpNF]
theorem ofElement_apply {α : Type*} (r : α → α → Prop) (a : α) (b) : ofElement r a b = b.1 
Arguments: {α : Type u_4} (r : α → α → Prop) (a : α) (b : ↑{b | r b a})
Goal: (PrincipalSeg.ofElement r a).toRelEmbedding b = ↑b
-- PrincipalSeg.ofElement_top
@[simp]
theorem ofElement_top {α : Type*} (r : α → α → Prop) (a : α) : (ofElement r a).top = a 
Arguments: {α : Type u_4} (r : α → α → Prop) (a : α)
Goal: (PrincipalSeg.ofElement r a).top = a
-- PrincipalSeg.subrelIso_symm_apply
@[simps! symm_apply]
noncomputable def subrelIso (f : r ≺i s) : Subrel s {b | s b f.top} ≃r r 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≺i s)
Goal: ∀ (a : α), f.subrelIso.symm a = (Equiv.setCongr ⋯) ⟨f.toRelEmbedding a, ⋯⟩
-- PrincipalSeg.apply_subrelIso
@[simp, nolint simpNF]
theorem apply_subrelIso (f : r ≺i s) (b : {b | s b f.top}) :
    f (f.subrelIso b) = b 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≺i s) (b : ↑{b | s b f.top})
Goal: f.toRelEmbedding (f.subrelIso b) = ↑b
-- PrincipalSeg.subrelIso_apply
@[simp, nolint simpNF]
theorem subrelIso_apply (f : r ≺i s) (a : α) :
    f.subrelIso ⟨f a, f.down.mpr ⟨a, rfl⟩⟩ = a 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (f : r ≺i s) (a : α)
Goal: f.subrelIso ⟨f.toRelEmbedding a, ⋯⟩ = a
-- PrincipalSeg.codRestrict_apply
@[simp]
theorem codRestrict_apply (p) (f : r ≺i s) (H H₂ a) : codRestrict p f H H₂ a = ⟨f a, H a⟩ 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (p : Set β) (f : r ≺i s) (H : ∀ (a : α), f.toRelEmbedding a ∈ p) (H₂ : f.top ∈ p) (a : α)
Goal: (PrincipalSeg.codRestrict p f H H₂).toRelEmbedding a = ⟨f.toRelEmbedding a, ⋯⟩
-- PrincipalSeg.codRestrict_top
@[simp]
theorem codRestrict_top (p) (f : r ≺i s) (H H₂) : (codRestrict p f H H₂).top = ⟨f.top, H₂⟩ 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} (p : Set β) (f : r ≺i s) (H : ∀ (a : α), f.toRelEmbedding a ∈ p) (H₂ : f.top ∈ p)
Goal: (PrincipalSeg.codRestrict p f H H₂).top = ⟨f.top, H₂⟩
-- PrincipalSeg.ofIsEmpty_top
@[simp]
theorem ofIsEmpty_top (r : α → α → Prop) [IsEmpty α] {b : β} (H : ∀ b', ¬s b' b) :
    (ofIsEmpty r H).top = b 
Arguments: {α : Type u_1} {β : Type u_2} {s : β → β → Prop} (r : α → α → Prop) [IsEmpty α] {b : β} (H : ∀ (b' : β), ¬s b' b)
Goal: (PrincipalSeg.ofIsEmpty r H).top = b
-- PrincipalSeg.acc
protected theorem acc [IsTrans β s] (f : r ≺i s) (a : α) : Acc r a ↔ Acc s (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsTrans β s] (f : r ≺i s) (a : α)
Goal: Acc r a ↔ Acc s (f.toRelEmbedding a)
-- wellFounded_iff_wellFounded_subrel
theorem wellFounded_iff_wellFounded_subrel {β : Type*} {s : β → β → Prop} [IsTrans β s] :
    WellFounded s ↔ ∀ b, WellFounded (Subrel s { b' | s b' b }) 
Arguments: {β : Type u_4} {s : β → β → Prop} [IsTrans β s]
Goal: WellFounded s ↔ ∀ (b : β), WellFounded (Subrel s {b' | s b' b})
-- wellFounded_iff_principalSeg
theorem wellFounded_iff_principalSeg.{u} {β : Type u} {s : β → β → Prop} [IsTrans β s] :
    WellFounded s ↔ ∀ (α : Type u) (r : α → α → Prop) (_ : r ≺i s), WellFounded r 
Arguments: {β : Type u} {s : β → β → Prop} [IsTrans β s]
Goal: WellFounded s ↔ ∀ (α : Type u) (r : α → α → Prop), r ≺i s → WellFounded r
-- InitialSeg.ltOrEq_apply_left
theorem InitialSeg.ltOrEq_apply_left [IsWellOrder β s] (f : r ≼i s) (g : r ≺i s) (a : α) :
    g a = f a 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder β s] (f : r ≼i s) (g : r ≺i s) (a : α)
Goal: g.toRelEmbedding a = f a
-- InitialSeg.ltOrEq_apply_right
theorem InitialSeg.ltOrEq_apply_right [IsWellOrder β s] (f : r ≼i s) (g : r ≃r s) (a : α) :
    g a = f a 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder β s] (f : r ≼i s) (g : r ≃r s) (a : α)
Goal: g a = f a
-- InitialSeg.leLT_apply
@[simp]
theorem InitialSeg.leLT_apply [IsWellOrder β s] [IsTrans γ t] (f : r ≼i s) (g : s ≺i t) (a : α) :
    (f.leLT g) a = g (f a) 
Arguments: {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → α → Prop} {s : β → β → Prop} {t : γ → γ → Prop} [IsWellOrder β s] [IsTrans γ t] (f : r ≼i s) (g : s ≺i t) (a : α)
Goal: (f.leLT g).toRelEmbedding a = g.toRelEmbedding (f a)
-- RelEmbedding.collapseF.lt
theorem collapseF.lt [IsWellOrder β s] (f : r ↪r s) {a : α} :
    ∀ {a'}, r a' a → s (collapseF f a').1 (collapseF f a).1 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder β s] (f : r ↪r s) {a a' : α}
Goal: r a' a → s ↑(f.collapseF a') ↑(f.collapseF a)
-- RelEmbedding.collapseF.not_lt
theorem collapseF.not_lt [IsWellOrder β s] (f : r ↪r s) (a : α) {b}
    (h : ∀ a' (_ : r a' a), s (collapseF f a').1 b) : ¬s b (collapseF f a).1 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder β s] (f : r ↪r s) (a : α) {b : β} (h : ∀ (a' : α), r a' a → s (↑(f.collapseF a')) b)
Goal: ¬s b ↑(f.collapseF a)
-- RelEmbedding.collapse_apply
theorem collapse_apply [IsWellOrder β s] (f : r ↪r s) (a) : collapse f a = (collapseF f a).1 
Arguments: {α : Type u_1} {β : Type u_2} {r : α → α → Prop} {s : β → β → Prop} [IsWellOrder β s] (f : r ↪r s) (a : α)
Goal: f.collapse a = ↑(f.collapseF a)
-- Coheyting.inf_hnot_self
theorem inf_hnot_self (a : α) : a ⊓ ￢a = ∂ a 
Arguments: {α : Type u_1} [CoheytingAlgebra α] (a : α)
Goal: a ⊓ ￢a = ∂ a
-- Coheyting.boundary_le
theorem boundary_le : ∂ a ≤ a 
Arguments: {α : Type u_1} [CoheytingAlgebra α] {a : α}
Goal: ∂ a ≤ a
-- Coheyting.boundary_le_hnot
theorem boundary_le_hnot : ∂ a ≤ ￢a 
Arguments: {α : Type u_1} [CoheytingAlgebra α] {a : α}
Goal: ∂ a ≤ ￢a
-- Coheyting.boundary_bot
@[simp]
theorem boundary_bot : ∂ (⊥ : α) = ⊥ 
Arguments: {α : Type u_1} [CoheytingAlgebra α]
Goal: ∂ ⊥ = ⊥
-- Coheyting.boundary_top
@[simp]
theorem boundary_top : ∂ (⊤ : α) = ⊥ 
Arguments: {α : Type u_1} [CoheytingAlgebra α]
Goal: ∂ ⊤ = ⊥
-- Coheyting.boundary_hnot_le
theorem boundary_hnot_le (a : α) : ∂ (￢a) ≤ ∂ a 
Arguments: {α : Type u_1} [CoheytingAlgebra α] (a : α)
Goal: ∂ (￢a) ≤ ∂ a
-- Coheyting.boundary_hnot_hnot
@[simp]
theorem boundary_hnot_hnot (a : α) : ∂ (￢￢a) = ∂ (￢a) 
Arguments: {α : Type u_1} [CoheytingAlgebra α] (a : α)
Goal: ∂ (￢￢a) = ∂ (￢a)
-- Coheyting.hnot_boundary
@[simp]
theorem hnot_boundary (a : α) : ￢∂ a = ⊤ 
Arguments: {α : Type u_1} [CoheytingAlgebra α] (a : α)
Goal: ￢∂ a = ⊤
-- Coheyting.boundary_inf
theorem boundary_inf (a b : α) : ∂ (a ⊓ b) = ∂ a ⊓ b ⊔ a ⊓ ∂ b 
Arguments: {α : Type u_1} [CoheytingAlgebra α] (a b : α)
Goal: ∂ (a ⊓ b) = ∂ a ⊓ b ⊔ a ⊓ ∂ b
-- Coheyting.boundary_inf_le
theorem boundary_inf_le : ∂ (a ⊓ b) ≤ ∂ a ⊔ ∂ b 
Arguments: {α : Type u_1} [CoheytingAlgebra α] {a b : α}
Goal: ∂ (a ⊓ b) ≤ ∂ a ⊔ ∂ b
-- Coheyting.boundary_sup_le
theorem boundary_sup_le : ∂ (a ⊔ b) ≤ ∂ a ⊔ ∂ b 
Arguments: {α : Type u_1} [CoheytingAlgebra α] {a b : α}
Goal: ∂ (a ⊔ b) ≤ ∂ a ⊔ ∂ b
-- Coheyting.boundary_le_boundary_sup_sup_boundary_inf_left
theorem boundary_le_boundary_sup_sup_boundary_inf_left : ∂ a ≤ ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b) 
Arguments: {α : Type u_1} [CoheytingAlgebra α] {a b : α}
Goal: ∂ a ≤ ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b)
-- Coheyting.boundary_le_boundary_sup_sup_boundary_inf_right
theorem boundary_le_boundary_sup_sup_boundary_inf_right : ∂ b ≤ ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b) 
Arguments: {α : Type u_1} [CoheytingAlgebra α] {a b : α}
Goal: ∂ b ≤ ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b)
-- Coheyting.boundary_sup_sup_boundary_inf
theorem boundary_sup_sup_boundary_inf (a b : α) : ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b) = ∂ a ⊔ ∂ b 
Arguments: {α : Type u_1} [CoheytingAlgebra α] (a b : α)
Goal: ∂ (a ⊔ b) ⊔ ∂ (a ⊓ b) = ∂ a ⊔ ∂ b
-- Coheyting.boundary_idem
@[simp]
theorem boundary_idem (a : α) : ∂ ∂ a = ∂ a 
Arguments: {α : Type u_1} [CoheytingAlgebra α] (a : α)
Goal: ∂ ∂ a = ∂ a
-- Coheyting.hnot_hnot_sup_boundary
theorem hnot_hnot_sup_boundary (a : α) : ￢￢a ⊔ ∂ a = a 
Arguments: {α : Type u_1} [CoheytingAlgebra α] (a : α)
Goal: ￢￢a ⊔ ∂ a = a
-- Coheyting.hnot_eq_top_iff_exists_boundary
theorem hnot_eq_top_iff_exists_boundary : ￢a = ⊤ ↔ ∃ b, ∂ b = a 
Arguments: {α : Type u_1} [CoheytingAlgebra α] {a : α}
Goal: ￢a = ⊤ ↔ ∃ b, ∂ b = a
-- Coheyting.boundary_eq_bot
@[simp]
theorem Coheyting.boundary_eq_bot (a : α) : ∂ a = ⊥ 
Arguments: {α : Type u_1} [BooleanAlgebra α] (a : α)
Goal: ∂ a = ⊥
-- disjoint_of_subsingleton
@[simp]
theorem disjoint_of_subsingleton [Subsingleton α] : Disjoint a b 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} [Subsingleton α]
Goal: Disjoint a b
-- disjoint_comm
theorem disjoint_comm : Disjoint a b ↔ Disjoint b a 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α}
Goal: Disjoint a b ↔ Disjoint b a
-- Disjoint.symm
@[symm]
theorem Disjoint.symm ⦃a b : α⦄ : Disjoint a b → Disjoint b a 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] ⦃a b : α⦄
Goal: Disjoint a b → Disjoint b a
-- symmetric_disjoint
theorem symmetric_disjoint : Symmetric (Disjoint : α → α → Prop) 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α]
Goal: Symmetric Disjoint
-- disjoint_bot_left
@[simp]
theorem disjoint_bot_left : Disjoint ⊥ a 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a : α}
Goal: Disjoint ⊥ a
-- disjoint_bot_right
@[simp]
theorem disjoint_bot_right : Disjoint a ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a : α}
Goal: Disjoint a ⊥
-- Disjoint.mono
theorem Disjoint.mono (h₁ : a ≤ b) (h₂ : c ≤ d) : Disjoint b d → Disjoint a c 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a b c d : α} (h₁ : a ≤ b) (h₂ : c ≤ d)
Goal: Disjoint b d → Disjoint a c
-- Disjoint.mono_left
theorem Disjoint.mono_left (h : a ≤ b) : Disjoint b c → Disjoint a c 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a b c : α} (h : a ≤ b)
Goal: Disjoint b c → Disjoint a c
-- Disjoint.mono_right
theorem Disjoint.mono_right : b ≤ c → Disjoint a c → Disjoint a b 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a b c : α}
Goal: b ≤ c → Disjoint a c → Disjoint a b
-- disjoint_self
@[simp]
theorem disjoint_self : Disjoint a a ↔ a = ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a : α}
Goal: Disjoint a a ↔ a = ⊥
-- Disjoint.ne
theorem Disjoint.ne (ha : a ≠ ⊥) (hab : Disjoint a b) : a ≠ b 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} (ha : a ≠ ⊥) (hab : Disjoint a b)
Goal: a ≠ b
-- Disjoint.eq_bot_of_le
theorem Disjoint.eq_bot_of_le (hab : Disjoint a b) (h : a ≤ b) : a = ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} (hab : Disjoint a b) (h : a ≤ b)
Goal: a = ⊥
-- Disjoint.eq_bot_of_ge
theorem Disjoint.eq_bot_of_ge (hab : Disjoint a b) : b ≤ a → b = ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} (hab : Disjoint a b)
Goal: b ≤ a → b = ⊥
-- Disjoint.eq_iff
lemma Disjoint.eq_iff (hab : Disjoint a b) : a = b ↔ a = ⊥ ∧ b = ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} (hab : Disjoint a b)
Goal: a = b ↔ a = ⊥ ∧ b = ⊥
-- Disjoint.ne_iff
lemma Disjoint.ne_iff (hab : Disjoint a b) : a ≠ b ↔ a ≠ ⊥ ∨ b ≠ ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} (hab : Disjoint a b)
Goal: a ≠ b ↔ a ≠ ⊥ ∨ b ≠ ⊥
-- disjoint_top
@[simp]
theorem disjoint_top : Disjoint a ⊤ ↔ a = ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a : α}
Goal: Disjoint a ⊤ ↔ a = ⊥
-- top_disjoint
@[simp]
theorem top_disjoint : Disjoint ⊤ a ↔ a = ⊥ 
Arguments: {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a : α}
Goal: Disjoint ⊤ a ↔ a = ⊥
-- disjoint_iff_inf_le
theorem disjoint_iff_inf_le : Disjoint a b ↔ a ⊓ b ≤ ⊥ 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α}
Goal: Disjoint a b ↔ a ⊓ b ≤ ⊥
-- disjoint_iff
theorem disjoint_iff : Disjoint a b ↔ a ⊓ b = ⊥ 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α}
Goal: Disjoint a b ↔ a ⊓ b = ⊥
-- Disjoint.le_bot
theorem Disjoint.le_bot : Disjoint a b → a ⊓ b ≤ ⊥ 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α}
Goal: Disjoint a b → a ⊓ b ≤ ⊥
-- Disjoint.eq_bot
theorem Disjoint.eq_bot : Disjoint a b → a ⊓ b = ⊥ 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α}
Goal: Disjoint a b → a ⊓ b = ⊥
-- disjoint_assoc
theorem disjoint_assoc : Disjoint (a ⊓ b) c ↔ Disjoint a (b ⊓ c) 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b c : α}
Goal: Disjoint (a ⊓ b) c ↔ Disjoint a (b ⊓ c)
-- disjoint_left_comm
theorem disjoint_left_comm : Disjoint a (b ⊓ c) ↔ Disjoint b (a ⊓ c) 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b c : α}
Goal: Disjoint a (b ⊓ c) ↔ Disjoint b (a ⊓ c)
-- disjoint_right_comm
theorem disjoint_right_comm : Disjoint (a ⊓ b) c ↔ Disjoint (a ⊓ c) b 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b c : α}
Goal: Disjoint (a ⊓ b) c ↔ Disjoint (a ⊓ c) b
-- Disjoint.inf_left
theorem Disjoint.inf_left (h : Disjoint a b) : Disjoint (a ⊓ c) b 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} (c : α) (h : Disjoint a b)
Goal: Disjoint (a ⊓ c) b
-- Disjoint.inf_left'
theorem Disjoint.inf_left' (h : Disjoint a b) : Disjoint (c ⊓ a) b 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} (c : α) (h : Disjoint a b)
Goal: Disjoint (c ⊓ a) b
-- Disjoint.inf_right
theorem Disjoint.inf_right (h : Disjoint a b) : Disjoint a (b ⊓ c) 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} (c : α) (h : Disjoint a b)
Goal: Disjoint a (b ⊓ c)
-- Disjoint.inf_right'
theorem Disjoint.inf_right' (h : Disjoint a b) : Disjoint a (c ⊓ b) 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} (c : α) (h : Disjoint a b)
Goal: Disjoint a (c ⊓ b)
-- Disjoint.of_disjoint_inf_of_le
theorem Disjoint.of_disjoint_inf_of_le (h : Disjoint (a ⊓ b) c) (hle : a ≤ c) : Disjoint a b 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b c : α} (h : Disjoint (a ⊓ b) c) (hle : a ≤ c)
Goal: Disjoint a b
-- Disjoint.of_disjoint_inf_of_le'
theorem Disjoint.of_disjoint_inf_of_le' (h : Disjoint (a ⊓ b) c) (hle : b ≤ c) : Disjoint a b 
Arguments: {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b c : α} (h : Disjoint (a ⊓ b) c) (hle : b ≤ c)
Goal: Disjoint a b
-- disjoint_sup_left
@[simp]
theorem disjoint_sup_left : Disjoint (a ⊔ b) c ↔ Disjoint a c ∧ Disjoint b c 
Arguments: {α : Type u_1} [DistribLattice α] [OrderBot α] {a b c : α}
Goal: Disjoint (a ⊔ b) c ↔ Disjoint a c ∧ Disjoint b c
-- disjoint_sup_right
@[simp]
theorem disjoint_sup_right : Disjoint a (b ⊔ c) ↔ Disjoint a b ∧ Disjoint a c 
Arguments: {α : Type u_1} [DistribLattice α] [OrderBot α] {a b c : α}
Goal: Disjoint a (b ⊔ c) ↔ Disjoint a b ∧ Disjoint a c
-- Disjoint.sup_left
theorem Disjoint.sup_left (ha : Disjoint a c) (hb : Disjoint b c) : Disjoint (a ⊔ b) c 
Arguments: {α : Type u_1} [DistribLattice α] [OrderBot α] {a b c : α} (ha : Disjoint a c) (hb : Disjoint b c)
Goal: Disjoint (a ⊔ b) c
-- Disjoint.sup_right
theorem Disjoint.sup_right (hb : Disjoint a b) (hc : Disjoint a c) : Disjoint a (b ⊔ c) 
Arguments: {α : Type u_1} [DistribLattice α] [OrderBot α] {a b c : α} (hb : Disjoint a b) (hc : Disjoint a c)
Goal: Disjoint a (b ⊔ c)
-- Disjoint.left_le_of_le_sup_right
theorem Disjoint.left_le_of_le_sup_right (h : a ≤ b ⊔ c) (hd : Disjoint a c) : a ≤ b 
Arguments: {α : Type u_1} [DistribLattice α] [OrderBot α] {a b c : α} (h : a ≤ b ⊔ c) (hd : Disjoint a c)
Goal: a ≤ b
-- Disjoint.left_le_of_le_sup_left
theorem Disjoint.left_le_of_le_sup_left (h : a ≤ c ⊔ b) (hd : Disjoint a c) : a ≤ b 
Arguments: {α : Type u_1} [DistribLattice α] [OrderBot α] {a b c : α} (h : a ≤ c ⊔ b) (hd : Disjoint a c)
Goal: a ≤ b
-- Codisjoint_comm
theorem Codisjoint_comm : Codisjoint a b ↔ Codisjoint b a 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α}
Goal: Codisjoint a b ↔ Codisjoint b a
-- Codisjoint.symm
@[symm]
theorem Codisjoint.symm ⦃a b : α⦄ : Codisjoint a b → Codisjoint b a 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] ⦃a b : α⦄
Goal: Codisjoint a b → Codisjoint b a
-- symmetric_codisjoint
theorem symmetric_codisjoint : Symmetric (Codisjoint : α → α → Prop) 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α]
Goal: Symmetric Codisjoint
-- codisjoint_top_left
@[simp]
theorem codisjoint_top_left : Codisjoint ⊤ a 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] {a : α}
Goal: Codisjoint ⊤ a
-- codisjoint_top_right
@[simp]
theorem codisjoint_top_right : Codisjoint a ⊤ 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] {a : α}
Goal: Codisjoint a ⊤
-- Codisjoint.mono
theorem Codisjoint.mono (h₁ : a ≤ b) (h₂ : c ≤ d) : Codisjoint a c → Codisjoint b d 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] {a b c d : α} (h₁ : a ≤ b) (h₂ : c ≤ d)
Goal: Codisjoint a c → Codisjoint b d
-- Codisjoint.mono_left
theorem Codisjoint.mono_left (h : a ≤ b) : Codisjoint a c → Codisjoint b c 
Arguments: {α : Type u_1} [PartialOrder α] [OrderTop α] {a b c : α} (h : a ≤ b)
Goal: Codisjoint a c → Codisjoint b c