ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- map_iInf₂
theorem map_iInf₂ [InfSet α] [InfSet β] [sInfHomClass F α β] (f : F) (g : ∀ i, κ i → α) :
    f (⨅ (i) (j), g i j) = ⨅ (i) (j), f (g i j) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} {ι : Sort u_6} {κ : ι → Sort u_7} [FunLike F α β] [InfSet α] [InfSet β] [sInfHomClass F α β] (f : F) (g : (i : ι) → κ i → α)
Goal: f (⨅ i, ⨅ j, g i j) = ⨅ i, ⨅ j, f (g i j)
-- OrderIso.toCompleteLatticeHom_toFun
@[simps] def OrderIso.toCompleteLatticeHom [CompleteLattice α] [CompleteLattice β]
    (f : OrderIso α β) : CompleteLatticeHom α β 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : α ≃o β) (a : α)
Goal: f.toCompleteLatticeHom.toFun a = f a
-- sSupHom.toFun_eq_coe
@[simp] lemma toFun_eq_coe (f : sSupHom α β) : f.toFun = f 
Arguments: {α : Type u_2} {β : Type u_3} [SupSet α] [SupSet β] (f : sSupHom α β)
Goal: f.toFun = ⇑f
-- sSupHom.coe_mk
@[simp, norm_cast] lemma coe_mk (f : α → β) (hf) : ⇑(mk f hf) = f 
Arguments: {α : Type u_2} {β : Type u_3} [SupSet α] [SupSet β] (f : α → β) (hf : ∀ (s : Set α), f (sSup s) = sSup (f '' s))
Goal: ⇑{ toFun := f, map_sSup' := hf } = f
-- sSupHom.ext
@[ext]
theorem ext {f g : sSupHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [SupSet α] [SupSet β] {f g : sSupHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- sSupHom.coe_copy
@[simp]
theorem coe_copy (f : sSupHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [SupSet α] [SupSet β] (f : sSupHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- sSupHom.copy_eq
theorem copy_eq (f : sSupHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [SupSet α] [SupSet β] (f : sSupHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- sSupHom.coe_id
@[simp]
theorem coe_id : ⇑(sSupHom.id α) = id 
Arguments: (α : Type u_2) [SupSet α]
Goal: ⇑(sSupHom.id α) = id
-- sSupHom.id_apply
@[simp]
theorem id_apply (a : α) : sSupHom.id α a = a 
Arguments: {α : Type u_2} [SupSet α] (a : α)
Goal: (sSupHom.id α) a = a
-- sSupHom.coe_comp
@[simp]
theorem coe_comp (f : sSupHom β γ) (g : sSupHom α β) : ⇑(f.comp g) = f ∘ g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SupSet α] [SupSet β] [SupSet γ] (f : sSupHom β γ) (g : sSupHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- sSupHom.comp_apply
@[simp]
theorem comp_apply (f : sSupHom β γ) (g : sSupHom α β) (a : α) : (f.comp g) a = f (g a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SupSet α] [SupSet β] [SupSet γ] (f : sSupHom β γ) (g : sSupHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- sSupHom.comp_assoc
@[simp]
theorem comp_assoc (f : sSupHom γ δ) (g : sSupHom β γ) (h : sSupHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [SupSet α] [SupSet β] [SupSet γ] [SupSet δ] (f : sSupHom γ δ) (g : sSupHom β γ) (h : sSupHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- sSupHom.comp_id
@[simp]
theorem comp_id (f : sSupHom α β) : f.comp (sSupHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [SupSet α] [SupSet β] (f : sSupHom α β)
Goal: f.comp (sSupHom.id α) = f
-- sSupHom.id_comp
@[simp]
theorem id_comp (f : sSupHom α β) : (sSupHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [SupSet α] [SupSet β] (f : sSupHom α β)
Goal: (sSupHom.id β).comp f = f
-- sSupHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : sSupHom β γ} {f : sSupHom α β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SupSet α] [SupSet β] [SupSet γ] {g₁ g₂ : sSupHom β γ} {f : sSupHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- sSupHom.cancel_left
@[simp]
theorem cancel_left {g : sSupHom β γ} {f₁ f₂ : sSupHom α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SupSet α] [SupSet β] [SupSet γ] {g : sSupHom β γ} {f₁ f₂ : sSupHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- sSupHom.coe_bot
@[simp]
theorem coe_bot : ⇑(⊥ : sSupHom α β) = ⊥ 
Arguments: {α : Type u_2} {β : Type u_3} [SupSet α]
Goal: ∀ {x : CompleteLattice β}, ⇑⊥ = ⊥
-- sSupHom.bot_apply
@[simp]
theorem bot_apply (a : α) : (⊥ : sSupHom α β) a = ⊥ 
Arguments: {α : Type u_2} {β : Type u_3} [SupSet α]
Goal: ∀ {x : CompleteLattice β} (a : α), ⊥ a = ⊥
-- sInfHom.toFun_eq_coe
@[simp] lemma toFun_eq_coe (f : sInfHom α β) : f.toFun = f 
Arguments: {α : Type u_2} {β : Type u_3} [InfSet α] [InfSet β] (f : sInfHom α β)
Goal: f.toFun = ⇑f
-- sInfHom.coe_mk
@[simp] lemma coe_mk (f : α → β) (hf) : ⇑(mk f hf) = f 
Arguments: {α : Type u_2} {β : Type u_3} [InfSet α] [InfSet β] (f : α → β) (hf : ∀ (s : Set α), f (sInf s) = sInf (f '' s))
Goal: ⇑{ toFun := f, map_sInf' := hf } = f
-- sInfHom.ext
@[ext]
theorem ext {f g : sInfHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [InfSet α] [InfSet β] {f g : sInfHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- sInfHom.coe_copy
@[simp]
theorem coe_copy (f : sInfHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [InfSet α] [InfSet β] (f : sInfHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- sInfHom.copy_eq
theorem copy_eq (f : sInfHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [InfSet α] [InfSet β] (f : sInfHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- sInfHom.coe_id
@[simp]
theorem coe_id : ⇑(sInfHom.id α) = id 
Arguments: (α : Type u_2) [InfSet α]
Goal: ⇑(sInfHom.id α) = id
-- sInfHom.id_apply
@[simp]
theorem id_apply (a : α) : sInfHom.id α a = a 
Arguments: {α : Type u_2} [InfSet α] (a : α)
Goal: (sInfHom.id α) a = a
-- sInfHom.coe_comp
@[simp]
theorem coe_comp (f : sInfHom β γ) (g : sInfHom α β) : ⇑(f.comp g) = f ∘ g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [InfSet α] [InfSet β] [InfSet γ] (f : sInfHom β γ) (g : sInfHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- sInfHom.comp_apply
@[simp]
theorem comp_apply (f : sInfHom β γ) (g : sInfHom α β) (a : α) : (f.comp g) a = f (g a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [InfSet α] [InfSet β] [InfSet γ] (f : sInfHom β γ) (g : sInfHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- sInfHom.comp_assoc
@[simp]
theorem comp_assoc (f : sInfHom γ δ) (g : sInfHom β γ) (h : sInfHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [InfSet α] [InfSet β] [InfSet γ] [InfSet δ] (f : sInfHom γ δ) (g : sInfHom β γ) (h : sInfHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- sInfHom.comp_id
@[simp]
theorem comp_id (f : sInfHom α β) : f.comp (sInfHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [InfSet α] [InfSet β] (f : sInfHom α β)
Goal: f.comp (sInfHom.id α) = f
-- sInfHom.id_comp
@[simp]
theorem id_comp (f : sInfHom α β) : (sInfHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [InfSet α] [InfSet β] (f : sInfHom α β)
Goal: (sInfHom.id β).comp f = f
-- sInfHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : sInfHom β γ} {f : sInfHom α β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [InfSet α] [InfSet β] [InfSet γ] {g₁ g₂ : sInfHom β γ} {f : sInfHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- sInfHom.cancel_left
@[simp]
theorem cancel_left {g : sInfHom β γ} {f₁ f₂ : sInfHom α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [InfSet α] [InfSet β] [InfSet γ] {g : sInfHom β γ} {f₁ f₂ : sInfHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- sInfHom.coe_top
@[simp]
theorem coe_top : ⇑(⊤ : sInfHom α β) = ⊤ 
Arguments: {α : Type u_2} {β : Type u_3} [InfSet α] [CompleteLattice β]
Goal: ⇑⊤ = ⊤
-- sInfHom.top_apply
@[simp]
theorem top_apply (a : α) : (⊤ : sInfHom α β) a = ⊤ 
Arguments: {α : Type u_2} {β : Type u_3} [InfSet α] [CompleteLattice β] (a : α)
Goal: ⊤ a = ⊤
-- FrameHom.toFun_eq_coe
lemma toFun_eq_coe (f : FrameHom α β) : f.toFun = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : FrameHom α β)
Goal: f.toFun = ⇑f
-- FrameHom.coe_toInfTopHom
@[simp] lemma coe_toInfTopHom (f : FrameHom α β) : ⇑f.toInfTopHom = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : FrameHom α β)
Goal: ⇑f.toInfTopHom = ⇑f
-- FrameHom.coe_toLatticeHom
@[simp] lemma coe_toLatticeHom (f : FrameHom α β) : ⇑f.toLatticeHom = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : FrameHom α β)
Goal: ⇑f.toLatticeHom = ⇑f
-- FrameHom.coe_mk
@[simp] lemma coe_mk (f : InfTopHom α β) (hf) : ⇑(mk f hf) = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : InfTopHom α β) (hf : ∀ (s : Set α), f.toFun (sSup s) = sSup (f.toFun '' s))
Goal: ⇑{ toInfTopHom := f, map_sSup' := hf } = ⇑f
-- FrameHom.ext
@[ext]
theorem ext {f g : FrameHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] {f g : FrameHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- FrameHom.coe_copy
@[simp]
theorem coe_copy (f : FrameHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : FrameHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- FrameHom.copy_eq
theorem copy_eq (f : FrameHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : FrameHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- FrameHom.coe_id
@[simp]
theorem coe_id : ⇑(FrameHom.id α) = id 
Arguments: (α : Type u_2) [CompleteLattice α]
Goal: ⇑(FrameHom.id α) = id
-- FrameHom.id_apply
@[simp]
theorem id_apply (a : α) : FrameHom.id α a = a 
Arguments: {α : Type u_2} [CompleteLattice α] (a : α)
Goal: (FrameHom.id α) a = a
-- FrameHom.coe_comp
@[simp]
theorem coe_comp (f : FrameHom β γ) (g : FrameHom α β) : ⇑(f.comp g) = f ∘ g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice α] [CompleteLattice β] [CompleteLattice γ] (f : FrameHom β γ) (g : FrameHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- FrameHom.comp_apply
@[simp]
theorem comp_apply (f : FrameHom β γ) (g : FrameHom α β) (a : α) : (f.comp g) a = f (g a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice α] [CompleteLattice β] [CompleteLattice γ] (f : FrameHom β γ) (g : FrameHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- FrameHom.comp_assoc
@[simp]
theorem comp_assoc (f : FrameHom γ δ) (g : FrameHom β γ) (h : FrameHom α β) :
    (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [CompleteLattice α] [CompleteLattice β] [CompleteLattice γ] [CompleteLattice δ] (f : FrameHom γ δ) (g : FrameHom β γ) (h : FrameHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- FrameHom.comp_id
@[simp]
theorem comp_id (f : FrameHom α β) : f.comp (FrameHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : FrameHom α β)
Goal: f.comp (FrameHom.id α) = f
-- FrameHom.id_comp
@[simp]
theorem id_comp (f : FrameHom α β) : (FrameHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : FrameHom α β)
Goal: (FrameHom.id β).comp f = f
-- FrameHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : FrameHom β γ} {f : FrameHom α β} (hf : Surjective f) :
    g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice α] [CompleteLattice β] [CompleteLattice γ] {g₁ g₂ : FrameHom β γ} {f : FrameHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- FrameHom.cancel_left
@[simp]
theorem cancel_left {g : FrameHom β γ} {f₁ f₂ : FrameHom α β} (hg : Injective g) :
    g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice α] [CompleteLattice β] [CompleteLattice γ] {g : FrameHom β γ} {f₁ f₂ : FrameHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- CompleteLatticeHom.toFun_eq_coe
lemma toFun_eq_coe (f : CompleteLatticeHom α β) : f.toFun = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : CompleteLatticeHom α β)
Goal: f.toFun = ⇑f
-- CompleteLatticeHom.coe_tosInfHom
@[simp] lemma coe_tosInfHom (f : CompleteLatticeHom α β) : ⇑f.tosInfHom = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : CompleteLatticeHom α β)
Goal: ⇑f.tosInfHom = ⇑f
-- CompleteLatticeHom.coe_tosSupHom
@[simp] lemma coe_tosSupHom (f : CompleteLatticeHom α β) : ⇑f.tosSupHom = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : CompleteLatticeHom α β)
Goal: ⇑f.tosSupHom = ⇑f
-- CompleteLatticeHom.coe_toBoundedLatticeHom
@[simp] lemma coe_toBoundedLatticeHom (f : CompleteLatticeHom α β) : ⇑f.toBoundedLatticeHom = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : CompleteLatticeHom α β)
Goal: ⇑f.toBoundedLatticeHom = ⇑f
-- CompleteLatticeHom.coe_mk
@[simp] lemma coe_mk (f : sInfHom α β) (hf) : ⇑(mk f hf) = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : sInfHom α β) (hf : ∀ (s : Set α), f.toFun (sSup s) = sSup (f.toFun '' s))
Goal: ⇑{ tosInfHom := f, map_sSup' := hf } = ⇑f
-- CompleteLatticeHom.ext
@[ext]
theorem ext {f g : CompleteLatticeHom α β} (h : ∀ a, f a = g a) : f = g 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] {f g : CompleteLatticeHom α β} (h : ∀ (a : α), f a = g a)
Goal: f = g
-- CompleteLatticeHom.coe_copy
@[simp]
theorem coe_copy (f : CompleteLatticeHom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : CompleteLatticeHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: ⇑(f.copy f' h) = f'
-- CompleteLatticeHom.copy_eq
theorem copy_eq (f : CompleteLatticeHom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : CompleteLatticeHom α β) (f' : α → β) (h : f' = ⇑f)
Goal: f.copy f' h = f
-- CompleteLatticeHom.coe_id
@[simp]
theorem coe_id : ⇑(CompleteLatticeHom.id α) = id 
Arguments: (α : Type u_2) [CompleteLattice α]
Goal: ⇑(CompleteLatticeHom.id α) = id
-- CompleteLatticeHom.id_apply
@[simp]
theorem id_apply (a : α) : CompleteLatticeHom.id α a = a 
Arguments: {α : Type u_2} [CompleteLattice α] (a : α)
Goal: (CompleteLatticeHom.id α) a = a
-- CompleteLatticeHom.coe_comp
@[simp]
theorem coe_comp (f : CompleteLatticeHom β γ) (g : CompleteLatticeHom α β) : ⇑(f.comp g) = f ∘ g 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice α] [CompleteLattice β] [CompleteLattice γ] (f : CompleteLatticeHom β γ) (g : CompleteLatticeHom α β)
Goal: ⇑(f.comp g) = ⇑f ∘ ⇑g
-- CompleteLatticeHom.comp_apply
@[simp]
theorem comp_apply (f : CompleteLatticeHom β γ) (g : CompleteLatticeHom α β) (a : α) :
    (f.comp g) a = f (g a) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice α] [CompleteLattice β] [CompleteLattice γ] (f : CompleteLatticeHom β γ) (g : CompleteLatticeHom α β) (a : α)
Goal: (f.comp g) a = f (g a)
-- CompleteLatticeHom.comp_assoc
@[simp]
theorem comp_assoc (f : CompleteLatticeHom γ δ) (g : CompleteLatticeHom β γ)
    (h : CompleteLatticeHom α β) : (f.comp g).comp h = f.comp (g.comp h) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} {δ : Type u_5} [CompleteLattice α] [CompleteLattice β] [CompleteLattice γ] [CompleteLattice δ] (f : CompleteLatticeHom γ δ) (g : CompleteLatticeHom β γ) (h : CompleteLatticeHom α β)
Goal: (f.comp g).comp h = f.comp (g.comp h)
-- CompleteLatticeHom.comp_id
@[simp]
theorem comp_id (f : CompleteLatticeHom α β) : f.comp (CompleteLatticeHom.id α) = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : CompleteLatticeHom α β)
Goal: f.comp (CompleteLatticeHom.id α) = f
-- CompleteLatticeHom.id_comp
@[simp]
theorem id_comp (f : CompleteLatticeHom α β) : (CompleteLatticeHom.id β).comp f = f 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : CompleteLatticeHom α β)
Goal: (CompleteLatticeHom.id β).comp f = f
-- CompleteLatticeHom.cancel_right
@[simp]
theorem cancel_right {g₁ g₂ : CompleteLatticeHom β γ} {f : CompleteLatticeHom α β}
    (hf : Surjective f) : g₁.comp f = g₂.comp f ↔ g₁ = g₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice α] [CompleteLattice β] [CompleteLattice γ] {g₁ g₂ : CompleteLatticeHom β γ} {f : CompleteLatticeHom α β} (hf : Function.Surjective ⇑f)
Goal: g₁.comp f = g₂.comp f ↔ g₁ = g₂
-- CompleteLatticeHom.cancel_left
@[simp]
theorem cancel_left {g : CompleteLatticeHom β γ} {f₁ f₂ : CompleteLatticeHom α β}
    (hg : Injective g) : g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice α] [CompleteLattice β] [CompleteLattice γ] {g : CompleteLatticeHom β γ} {f₁ f₂ : CompleteLatticeHom α β} (hg : Function.Injective ⇑g)
Goal: g.comp f₁ = g.comp f₂ ↔ f₁ = f₂
-- sSupHom.dual_symm_apply_toFun
@[simps]
protected def dual : sSupHom α β ≃ sInfHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_2} {β : Type u_3} [SupSet α] [SupSet β] (f : sInfHom αᵒᵈ βᵒᵈ)
Goal: ∀ (a : α), (sSupHom.dual.symm f) a = (⇑OrderDual.ofDual ∘ ⇑f ∘ ⇑OrderDual.toDual) a
-- sSupHom.dual_apply_toFun
@[simps]
protected def dual : sSupHom α β ≃ sInfHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_2} {β : Type u_3} [SupSet α] [SupSet β] (f : sSupHom α β)
Goal: ∀ (a : αᵒᵈ), (sSupHom.dual f).toFun a = (⇑OrderDual.toDual ∘ ⇑f ∘ ⇑OrderDual.ofDual) a
-- sSupHom.dual_id
@[simp]
theorem dual_id : sSupHom.dual (sSupHom.id α) = sInfHom.id _ 
Arguments: {α : Type u_2} [SupSet α]
Goal: sSupHom.dual (sSupHom.id α) = sInfHom.id αᵒᵈ
-- sSupHom.dual_comp
@[simp]
theorem dual_comp (g : sSupHom β γ) (f : sSupHom α β) :
    sSupHom.dual (g.comp f) = (sSupHom.dual g).comp (sSupHom.dual f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SupSet α] [SupSet β] [SupSet γ] (g : sSupHom β γ) (f : sSupHom α β)
Goal: sSupHom.dual (g.comp f) = (sSupHom.dual g).comp (sSupHom.dual f)
-- sSupHom.symm_dual_id
@[simp]
theorem symm_dual_id : sSupHom.dual.symm (sInfHom.id _) = sSupHom.id α 
Arguments: {α : Type u_2} [SupSet α]
Goal: sSupHom.dual.symm (sInfHom.id αᵒᵈ) = sSupHom.id α
-- sSupHom.symm_dual_comp
@[simp]
theorem symm_dual_comp (g : sInfHom βᵒᵈ γᵒᵈ) (f : sInfHom αᵒᵈ βᵒᵈ) :
    sSupHom.dual.symm (g.comp f) = (sSupHom.dual.symm g).comp (sSupHom.dual.symm f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SupSet α] [SupSet β] [SupSet γ] (g : sInfHom βᵒᵈ γᵒᵈ) (f : sInfHom αᵒᵈ βᵒᵈ)
Goal: sSupHom.dual.symm (g.comp f) = (sSupHom.dual.symm g).comp (sSupHom.dual.symm f)
-- sInfHom.dual_apply_toFun
@[simps]
protected def dual : sInfHom α β ≃ sSupHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_2} {β : Type u_3} [InfSet α] [InfSet β] (f : sInfHom α β)
Goal: ∀ (a : αᵒᵈ), (sInfHom.dual f) a = (⇑OrderDual.toDual ∘ ⇑f ∘ ⇑OrderDual.ofDual) a
-- sInfHom.dual_symm_apply_toFun
@[simps]
protected def dual : sInfHom α β ≃ sSupHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_2} {β : Type u_3} [InfSet α] [InfSet β] (f : sSupHom αᵒᵈ βᵒᵈ)
Goal: ∀ (a : α), (sInfHom.dual.symm f).toFun a = (⇑OrderDual.ofDual ∘ ⇑f ∘ ⇑OrderDual.toDual) a
-- sInfHom.dual_id
@[simp]
theorem dual_id : sInfHom.dual (sInfHom.id α) = sSupHom.id _ 
Arguments: {α : Type u_2} [InfSet α]
Goal: sInfHom.dual (sInfHom.id α) = sSupHom.id αᵒᵈ
-- sInfHom.dual_comp
@[simp]
theorem dual_comp (g : sInfHom β γ) (f : sInfHom α β) :
    sInfHom.dual (g.comp f) = (sInfHom.dual g).comp (sInfHom.dual f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [InfSet α] [InfSet β] [InfSet γ] (g : sInfHom β γ) (f : sInfHom α β)
Goal: sInfHom.dual (g.comp f) = (sInfHom.dual g).comp (sInfHom.dual f)
-- sInfHom.symm_dual_id
@[simp]
theorem symm_dual_id : sInfHom.dual.symm (sSupHom.id _) = sInfHom.id α 
Arguments: {α : Type u_2} [InfSet α]
Goal: sInfHom.dual.symm (sSupHom.id αᵒᵈ) = sInfHom.id α
-- sInfHom.symm_dual_comp
@[simp]
theorem symm_dual_comp (g : sSupHom βᵒᵈ γᵒᵈ) (f : sSupHom αᵒᵈ βᵒᵈ) :
    sInfHom.dual.symm (g.comp f) = (sInfHom.dual.symm g).comp (sInfHom.dual.symm f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [InfSet α] [InfSet β] [InfSet γ] (g : sSupHom βᵒᵈ γᵒᵈ) (f : sSupHom αᵒᵈ βᵒᵈ)
Goal: sInfHom.dual.symm (g.comp f) = (sInfHom.dual.symm g).comp (sInfHom.dual.symm f)
-- CompleteLatticeHom.dual_symm_apply_toFun
@[simps!]
protected def dual : CompleteLatticeHom α β ≃ CompleteLatticeHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : CompleteLatticeHom αᵒᵈ βᵒᵈ)
Goal: ∀ (a : αᵒᵈᵒᵈ), (CompleteLatticeHom.dual.symm f).toFun a = OrderDual.toDual (f (OrderDual.ofDual a))
-- CompleteLatticeHom.dual_apply_toFun
@[simps!]
protected def dual : CompleteLatticeHom α β ≃ CompleteLatticeHom αᵒᵈ βᵒᵈ 
Arguments: {α : Type u_2} {β : Type u_3} [CompleteLattice α] [CompleteLattice β] (f : CompleteLatticeHom α β)
Goal: ∀ (a : αᵒᵈ), (CompleteLatticeHom.dual f).toFun a = OrderDual.toDual (f (OrderDual.ofDual a))
-- CompleteLatticeHom.dual_id
@[simp]
theorem dual_id : CompleteLatticeHom.dual (CompleteLatticeHom.id α) = CompleteLatticeHom.id _ 
Arguments: {α : Type u_2} [CompleteLattice α]
Goal: CompleteLatticeHom.dual (CompleteLatticeHom.id α) = CompleteLatticeHom.id αᵒᵈ
-- CompleteLatticeHom.dual_comp
@[simp]
theorem dual_comp (g : CompleteLatticeHom β γ) (f : CompleteLatticeHom α β) :
    CompleteLatticeHom.dual (g.comp f) =
      (CompleteLatticeHom.dual g).comp (CompleteLatticeHom.dual f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice α] [CompleteLattice β] [CompleteLattice γ] (g : CompleteLatticeHom β γ) (f : CompleteLatticeHom α β)
Goal: CompleteLatticeHom.dual (g.comp f) = (CompleteLatticeHom.dual g).comp (CompleteLatticeHom.dual f)
-- CompleteLatticeHom.symm_dual_id
@[simp]
theorem symm_dual_id :
    CompleteLatticeHom.dual.symm (CompleteLatticeHom.id _) = CompleteLatticeHom.id α 
Arguments: {α : Type u_2} [CompleteLattice α]
Goal: CompleteLatticeHom.dual.symm (CompleteLatticeHom.id αᵒᵈ) = CompleteLatticeHom.id α
-- CompleteLatticeHom.symm_dual_comp
@[simp]
theorem symm_dual_comp (g : CompleteLatticeHom βᵒᵈ γᵒᵈ) (f : CompleteLatticeHom αᵒᵈ βᵒᵈ) :
    CompleteLatticeHom.dual.symm (g.comp f) =
      (CompleteLatticeHom.dual.symm g).comp (CompleteLatticeHom.dual.symm f) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} [CompleteLattice α] [CompleteLattice β] [CompleteLattice γ] (g : CompleteLatticeHom βᵒᵈ γᵒᵈ) (f : CompleteLatticeHom αᵒᵈ βᵒᵈ)
Goal: CompleteLatticeHom.dual.symm (g.comp f) = (CompleteLatticeHom.dual.symm g).comp (CompleteLatticeHom.dual.symm f)
-- CompleteLatticeHom.coe_setPreimage
@[simp]
theorem coe_setPreimage (f : α → β) : ⇑(setPreimage f) = preimage f 
Arguments: {α : Type u_2} {β : Type u_3} (f : α → β)
Goal: ⇑(CompleteLatticeHom.setPreimage f) = Set.preimage f
-- CompleteLatticeHom.setPreimage_apply
@[simp]
theorem setPreimage_apply (f : α → β) (s : Set β) : setPreimage f s = s.preimage f 
Arguments: {α : Type u_2} {β : Type u_3} (f : α → β) (s : Set β)
Goal: (CompleteLatticeHom.setPreimage f) s = f ⁻¹' s
-- CompleteLatticeHom.setPreimage_id
@[simp]
theorem setPreimage_id : setPreimage (id : α → α) = CompleteLatticeHom.id _ 
Arguments: {α : Type u_2}
Goal: CompleteLatticeHom.setPreimage id = CompleteLatticeHom.id (Set α)
-- CompleteLatticeHom.setPreimage_comp
theorem setPreimage_comp (g : β → γ) (f : α → β) :
    setPreimage (g ∘ f) = (setPreimage f).comp (setPreimage g) 
Arguments: {α : Type u_2} {β : Type u_3} {γ : Type u_4} (g : β → γ) (f : α → β)
Goal: CompleteLatticeHom.setPreimage (g ∘ f) = (CompleteLatticeHom.setPreimage f).comp (CompleteLatticeHom.setPreimage g)
-- Set.image_sSup
theorem Set.image_sSup {f : α → β} (s : Set (Set α)) : f '' sSup s = sSup (image f '' s) 
Arguments: {α : Type u_2} {β : Type u_3} {f : α → β} (s : Set (Set α))
Goal: f '' sSup s = sSup (Set.image f '' s)