File: Mathlib/Order/Monotone/Monovary.lean

full_name: Monovary
code: /-- `f` monovaries with `g` if `g i < g j` implies `f i ≤ f j`. -/
def Monovary (f : ι → α) (g : ι → β) : Prop 
goal: Prop

full_name: Antivary
code: /-- `f` antivaries with `g` if `g i < g j` implies `f j ≤ f i`. -/
def Antivary (f : ι → α) (g : ι → β) : Prop 
goal: Prop

full_name: MonovaryOn
code: /-- `f` monovaries with `g` on `s` if `g i < g j` implies `f i ≤ f j` for all `i, j ∈ s`. -/
def MonovaryOn (f : ι → α) (g : ι → β) (s : Set ι) : Prop 
goal: Prop

full_name: AntivaryOn
code: /-- `f` antivaries with `g` on `s` if `g i < g j` implies `f j ≤ f i` for all `i, j ∈ s`. -/
def AntivaryOn (f : ι → α) (g : ι → β) (s : Set ι) : Prop 
goal: Prop

full_name: Monovary.monovaryOn
code: protected theorem Monovary.monovaryOn (h : Monovary f g) (s : Set ι) : MonovaryOn f g s 
goal: MonovaryOn f g s

full_name: Antivary.antivaryOn
code: protected theorem Antivary.antivaryOn (h : Antivary f g) (s : Set ι) : AntivaryOn f g s 
goal: AntivaryOn f g s

full_name: MonovaryOn.empty
code: @[simp]
theorem MonovaryOn.empty : MonovaryOn f g ∅ 
goal: MonovaryOn f g ∅

full_name: AntivaryOn.empty
code: @[simp]
theorem AntivaryOn.empty : AntivaryOn f g ∅ 
goal: AntivaryOn f g ∅

full_name: monovaryOn_univ
code: @[simp]
theorem monovaryOn_univ : MonovaryOn f g univ ↔ Monovary f g 
goal: MonovaryOn f g Set.univ ↔ Monovary f g

full_name: antivaryOn_univ
code: @[simp]
theorem antivaryOn_univ : AntivaryOn f g univ ↔ Antivary f g 
goal: AntivaryOn f g Set.univ ↔ Antivary f g

full_name: MonovaryOn.subset
code: protected theorem MonovaryOn.subset (hst : s ⊆ t) (h : MonovaryOn f g t) : MonovaryOn f g s 
goal: MonovaryOn f g s

full_name: AntivaryOn.subset
code: protected theorem AntivaryOn.subset (hst : s ⊆ t) (h : AntivaryOn f g t) : AntivaryOn f g s 
goal: AntivaryOn f g s

full_name: monovary_const_left
code: theorem monovary_const_left (g : ι → β) (a : α) : Monovary (const ι a) g 
goal: Monovary (Function.const ι a) g

full_name: antivary_const_left
code: theorem antivary_const_left (g : ι → β) (a : α) : Antivary (const ι a) g 
goal: Antivary (Function.const ι a) g

full_name: monovary_const_right
code: theorem monovary_const_right (f : ι → α) (b : β) : Monovary f (const ι b) 
goal: Monovary f (Function.const ι b)

full_name: antivary_const_right
code: theorem antivary_const_right (f : ι → α) (b : β) : Antivary f (const ι b) 
goal: Antivary f (Function.const ι b)

full_name: monovary_self
code: theorem monovary_self (f : ι → α) : Monovary f f 
goal: Monovary f f

full_name: monovaryOn_self
code: theorem monovaryOn_self (f : ι → α) (s : Set ι) : MonovaryOn f f s 
goal: MonovaryOn f f s

full_name: Subsingleton.monovary
code: protected theorem Subsingleton.monovary [Subsingleton ι] (f : ι → α) (g : ι → β) : Monovary f g 
goal: Monovary f g

full_name: Subsingleton.antivary
code: protected theorem Subsingleton.antivary [Subsingleton ι] (f : ι → α) (g : ι → β) : Antivary f g 
goal: Antivary f g

full_name: Subsingleton.monovaryOn
code: protected theorem Subsingleton.monovaryOn [Subsingleton ι] (f : ι → α) (g : ι → β) (s : Set ι) :
    MonovaryOn f g s 
goal: MonovaryOn f g s

full_name: Subsingleton.antivaryOn
code: protected theorem Subsingleton.antivaryOn [Subsingleton ι] (f : ι → α) (g : ι → β) (s : Set ι) :
    AntivaryOn f g s 
goal: AntivaryOn f g s

full_name: monovaryOn_const_left
code: theorem monovaryOn_const_left (g : ι → β) (a : α) (s : Set ι) : MonovaryOn (const ι a) g s 
goal: MonovaryOn (Function.const ι a) g s

full_name: antivaryOn_const_left
code: theorem antivaryOn_const_left (g : ι → β) (a : α) (s : Set ι) : AntivaryOn (const ι a) g s 
goal: AntivaryOn (Function.const ι a) g s

full_name: monovaryOn_const_right
code: theorem monovaryOn_const_right (f : ι → α) (b : β) (s : Set ι) : MonovaryOn f (const ι b) s 
goal: MonovaryOn f (Function.const ι b) s

full_name: antivaryOn_const_right
code: theorem antivaryOn_const_right (f : ι → α) (b : β) (s : Set ι) : AntivaryOn f (const ι b) s 
goal: AntivaryOn f (Function.const ι b) s

full_name: Monovary.comp_right
code: theorem Monovary.comp_right (h : Monovary f g) (k : ι' → ι) : Monovary (f ∘ k) (g ∘ k) 
goal: Monovary (f ∘ k) (g ∘ k)

full_name: Antivary.comp_right
code: theorem Antivary.comp_right (h : Antivary f g) (k : ι' → ι) : Antivary (f ∘ k) (g ∘ k) 
goal: Antivary (f ∘ k) (g ∘ k)

full_name: MonovaryOn.comp_right
code: theorem MonovaryOn.comp_right (h : MonovaryOn f g s) (k : ι' → ι) :
    MonovaryOn (f ∘ k) (g ∘ k) (k ⁻¹' s) 
goal: MonovaryOn (f ∘ k) (g ∘ k) (k ⁻¹' s)

full_name: AntivaryOn.comp_right
code: theorem AntivaryOn.comp_right (h : AntivaryOn f g s) (k : ι' → ι) :
    AntivaryOn (f ∘ k) (g ∘ k) (k ⁻¹' s) 
goal: AntivaryOn (f ∘ k) (g ∘ k) (k ⁻¹' s)

full_name: Monovary.comp_monotone_left
code: theorem Monovary.comp_monotone_left (h : Monovary f g) (hf : Monotone f') : Monovary (f' ∘ f) g 
goal: Monovary (f' ∘ f) g

full_name: Monovary.comp_antitone_left
code: theorem Monovary.comp_antitone_left (h : Monovary f g) (hf : Antitone f') : Antivary (f' ∘ f) g 
goal: Antivary (f' ∘ f) g

full_name: Antivary.comp_monotone_left
code: theorem Antivary.comp_monotone_left (h : Antivary f g) (hf : Monotone f') : Antivary (f' ∘ f) g 
goal: Antivary (f' ∘ f) g

full_name: Antivary.comp_antitone_left
code: theorem Antivary.comp_antitone_left (h : Antivary f g) (hf : Antitone f') : Monovary (f' ∘ f) g 
goal: Monovary (f' ∘ f) g

full_name: MonovaryOn.comp_monotone_on_left
code: theorem MonovaryOn.comp_monotone_on_left (h : MonovaryOn f g s) (hf : Monotone f') :
    MonovaryOn (f' ∘ f) g s 
goal: MonovaryOn (f' ∘ f) g s

full_name: MonovaryOn.comp_antitone_on_left
code: theorem MonovaryOn.comp_antitone_on_left (h : MonovaryOn f g s) (hf : Antitone f') :
    AntivaryOn (f' ∘ f) g s 
goal: AntivaryOn (f' ∘ f) g s

full_name: AntivaryOn.comp_monotone_on_left
code: theorem AntivaryOn.comp_monotone_on_left (h : AntivaryOn f g s) (hf : Monotone f') :
    AntivaryOn (f' ∘ f) g s 
goal: AntivaryOn (f' ∘ f) g s

full_name: AntivaryOn.comp_antitone_on_left
code: theorem AntivaryOn.comp_antitone_on_left (h : AntivaryOn f g s) (hf : Antitone f') :
    MonovaryOn (f' ∘ f) g s 
goal: MonovaryOn (f' ∘ f) g s

full_name: Monovary.dual
code: theorem Monovary.dual : Monovary f g → Monovary (toDual ∘ f) (toDual ∘ g) 
goal: Monovary f g → Monovary (⇑OrderDual.toDual ∘ f) (⇑OrderDual.toDual ∘ g)

full_name: Antivary.dual
code: theorem Antivary.dual : Antivary f g → Antivary (toDual ∘ f) (toDual ∘ g) 
goal: Antivary f g → Antivary (⇑OrderDual.toDual ∘ f) (⇑OrderDual.toDual ∘ g)

full_name: Monovary.dual_left
code: theorem Monovary.dual_left : Monovary f g → Antivary (toDual ∘ f) g 
goal: Monovary f g → Antivary (⇑OrderDual.toDual ∘ f) g

full_name: Antivary.dual_left
code: theorem Antivary.dual_left : Antivary f g → Monovary (toDual ∘ f) g 
goal: Antivary f g → Monovary (⇑OrderDual.toDual ∘ f) g

full_name: Monovary.dual_right
code: theorem Monovary.dual_right : Monovary f g → Antivary f (toDual ∘ g) 
goal: Monovary f g → Antivary f (⇑OrderDual.toDual ∘ g)

full_name: Antivary.dual_right
code: theorem Antivary.dual_right : Antivary f g → Monovary f (toDual ∘ g) 
goal: Antivary f g → Monovary f (⇑OrderDual.toDual ∘ g)

full_name: MonovaryOn.dual
code: theorem MonovaryOn.dual : MonovaryOn f g s → MonovaryOn (toDual ∘ f) (toDual ∘ g) s 
goal: MonovaryOn f g s → MonovaryOn (⇑OrderDual.toDual ∘ f) (⇑OrderDual.toDual ∘ g) s

full_name: AntivaryOn.dual
code: theorem AntivaryOn.dual : AntivaryOn f g s → AntivaryOn (toDual ∘ f) (toDual ∘ g) s 
goal: AntivaryOn f g s → AntivaryOn (⇑OrderDual.toDual ∘ f) (⇑OrderDual.toDual ∘ g) s

full_name: MonovaryOn.dual_left
code: theorem MonovaryOn.dual_left : MonovaryOn f g s → AntivaryOn (toDual ∘ f) g s 
goal: MonovaryOn f g s → AntivaryOn (⇑OrderDual.toDual ∘ f) g s

full_name: AntivaryOn.dual_left
code: theorem AntivaryOn.dual_left : AntivaryOn f g s → MonovaryOn (toDual ∘ f) g s 
goal: AntivaryOn f g s → MonovaryOn (⇑OrderDual.toDual ∘ f) g s

full_name: MonovaryOn.dual_right
code: theorem MonovaryOn.dual_right : MonovaryOn f g s → AntivaryOn f (toDual ∘ g) s 
goal: MonovaryOn f g s → AntivaryOn f (⇑OrderDual.toDual ∘ g) s

full_name: AntivaryOn.dual_right
code: theorem AntivaryOn.dual_right : AntivaryOn f g s → MonovaryOn f (toDual ∘ g) s 
goal: AntivaryOn f g s → MonovaryOn f (⇑OrderDual.toDual ∘ g) s

full_name: monovary_toDual_left
code: @[simp]
theorem monovary_toDual_left : Monovary (toDual ∘ f) g ↔ Antivary f g 
goal: Monovary (⇑OrderDual.toDual ∘ f) g ↔ Antivary f g

full_name: monovary_toDual_right
code: @[simp]
theorem monovary_toDual_right : Monovary f (toDual ∘ g) ↔ Antivary f g 
goal: Monovary f (⇑OrderDual.toDual ∘ g) ↔ Antivary f g

full_name: antivary_toDual_left
code: @[simp]
theorem antivary_toDual_left : Antivary (toDual ∘ f) g ↔ Monovary f g 
goal: Antivary (⇑OrderDual.toDual ∘ f) g ↔ Monovary f g

full_name: antivary_toDual_right
code: @[simp]
theorem antivary_toDual_right : Antivary f (toDual ∘ g) ↔ Monovary f g 
goal: Antivary f (⇑OrderDual.toDual ∘ g) ↔ Monovary f g

full_name: monovaryOn_toDual_left
code: @[simp]
theorem monovaryOn_toDual_left : MonovaryOn (toDual ∘ f) g s ↔ AntivaryOn f g s 
goal: MonovaryOn (⇑OrderDual.toDual ∘ f) g s ↔ AntivaryOn f g s

full_name: monovaryOn_toDual_right
code: @[simp]
theorem monovaryOn_toDual_right : MonovaryOn f (toDual ∘ g) s ↔ AntivaryOn f g s 
goal: MonovaryOn f (⇑OrderDual.toDual ∘ g) s ↔ AntivaryOn f g s

full_name: antivaryOn_toDual_left
code: @[simp]
theorem antivaryOn_toDual_left : AntivaryOn (toDual ∘ f) g s ↔ MonovaryOn f g s 
goal: AntivaryOn (⇑OrderDual.toDual ∘ f) g s ↔ MonovaryOn f g s

full_name: antivaryOn_toDual_right
code: @[simp]
theorem antivaryOn_toDual_right : AntivaryOn f (toDual ∘ g) s ↔ MonovaryOn f g s 
goal: AntivaryOn f (⇑OrderDual.toDual ∘ g) s ↔ MonovaryOn f g s

full_name: monovary_id_iff
code: @[simp]
theorem monovary_id_iff : Monovary f id ↔ Monotone f 
goal: Monovary f id ↔ Monotone f

full_name: antivary_id_iff
code: @[simp]
theorem antivary_id_iff : Antivary f id ↔ Antitone f 
goal: Antivary f id ↔ Antitone f

full_name: monovaryOn_id_iff
code: @[simp]
theorem monovaryOn_id_iff : MonovaryOn f id s ↔ MonotoneOn f s 
goal: MonovaryOn f id s ↔ MonotoneOn f s

full_name: antivaryOn_id_iff
code: @[simp]
theorem antivaryOn_id_iff : AntivaryOn f id s ↔ AntitoneOn f s 
goal: AntivaryOn f id s ↔ AntitoneOn f s

full_name: Monotone.monovary
code: protected theorem Monotone.monovary (hf : Monotone f) (hg : Monotone g) : Monovary f g 
goal: Monovary f g

full_name: Monotone.antivary
code: protected theorem Monotone.antivary (hf : Monotone f) (hg : Antitone g) : Antivary f g 
goal: Antivary f g

full_name: Antitone.monovary
code: protected theorem Antitone.monovary (hf : Antitone f) (hg : Antitone g) : Monovary f g 
goal: Monovary f g

full_name: Antitone.antivary
code: protected theorem Antitone.antivary (hf : Antitone f) (hg : Monotone g) : Antivary f g 
goal: Antivary f g

full_name: MonotoneOn.monovaryOn
code: protected theorem MonotoneOn.monovaryOn (hf : MonotoneOn f s) (hg : MonotoneOn g s) :
    MonovaryOn f g s 
goal: MonovaryOn f g s

full_name: MonotoneOn.antivaryOn
code: protected theorem MonotoneOn.antivaryOn (hf : MonotoneOn f s) (hg : AntitoneOn g s) :
    AntivaryOn f g s 
goal: AntivaryOn f g s

full_name: AntitoneOn.monovaryOn
code: protected theorem AntitoneOn.monovaryOn (hf : AntitoneOn f s) (hg : AntitoneOn g s) :
    MonovaryOn f g s 
goal: MonovaryOn f g s

full_name: AntitoneOn.antivaryOn
code: protected theorem AntitoneOn.antivaryOn (hf : AntitoneOn f s) (hg : MonotoneOn g s) :
    AntivaryOn f g s 
goal: AntivaryOn f g s

full_name: MonovaryOn.comp_monotoneOn_right
code: theorem MonovaryOn.comp_monotoneOn_right (h : MonovaryOn f g s) (hg : MonotoneOn g' (g '' s)) :
    MonovaryOn f (g' ∘ g) s 
goal: MonovaryOn f (g' ∘ g) s

full_name: MonovaryOn.comp_antitoneOn_right
code: theorem MonovaryOn.comp_antitoneOn_right (h : MonovaryOn f g s) (hg : AntitoneOn g' (g '' s)) :
    AntivaryOn f (g' ∘ g) s 
goal: AntivaryOn f (g' ∘ g) s

full_name: AntivaryOn.comp_monotoneOn_right
code: theorem AntivaryOn.comp_monotoneOn_right (h : AntivaryOn f g s) (hg : MonotoneOn g' (g '' s)) :
    AntivaryOn f (g' ∘ g) s 
goal: AntivaryOn f (g' ∘ g) s

full_name: AntivaryOn.comp_antitoneOn_right
code: theorem AntivaryOn.comp_antitoneOn_right (h : AntivaryOn f g s) (hg : AntitoneOn g' (g '' s)) :
    MonovaryOn f (g' ∘ g) s 
goal: MonovaryOn f (g' ∘ g) s

full_name: Monovary.symm
code: @[symm]
protected theorem Monovary.symm (h : Monovary f g) : Monovary g f 
goal: Monovary g f

full_name: Antivary.symm
code: @[symm]
protected theorem Antivary.symm (h : Antivary f g) : Antivary g f 
goal: Antivary g f

full_name: MonovaryOn.symm
code: @[symm]
protected theorem MonovaryOn.symm (h : MonovaryOn f g s) : MonovaryOn g f s 
goal: MonovaryOn g f s

full_name: AntivaryOn.symm
code: @[symm]
protected theorem AntivaryOn.symm (h : AntivaryOn f g s) : AntivaryOn g f s 
goal: AntivaryOn g f s

full_name: monovary_comm
code: theorem monovary_comm : Monovary f g ↔ Monovary g f 
goal: Monovary f g ↔ Monovary g f

full_name: antivary_comm
code: theorem antivary_comm : Antivary f g ↔ Antivary g f 
goal: Antivary f g ↔ Antivary g f

full_name: monovaryOn_comm
code: theorem monovaryOn_comm : MonovaryOn f g s ↔ MonovaryOn g f s 
goal: MonovaryOn f g s ↔ MonovaryOn g f s

full_name: antivaryOn_comm
code: theorem antivaryOn_comm : AntivaryOn f g s ↔ AntivaryOn g f s 
goal: AntivaryOn f g s ↔ AntivaryOn g f s