N-ary images of sets #
This file defines Set.image2, the binary image of sets.
This is mostly useful to define pointwise operations and Set.seq.
Notes #
This file is very similar to Data.Finset.NAry, to Order.Filter.NAry, and to
Data.Option.NAry. Please keep them in sync.
theorem
Set.mem_image2_iff
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{a : α}
{b : β}
(hf : Function.Injective2 f)
:
theorem
Set.image2_subset
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s s' : Set α}
{t t' : Set β}
(hs : s ⊆ s')
(ht : t ⊆ t')
:
Set.image2 f s t ⊆ Set.image2 f s' t'
image2 is monotone with respect to ⊆.
theorem
Set.image2_subset_left
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t t' : Set β}
(ht : t ⊆ t')
:
Set.image2 f s t ⊆ Set.image2 f s t'
theorem
Set.image2_subset_right
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s s' : Set α}
{t : Set β}
(hs : s ⊆ s')
:
Set.image2 f s t ⊆ Set.image2 f s' t
theorem
Set.forall_mem_image2
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{p : γ → Prop}
:
(∀ z ∈ Set.image2 f s t, p z) ↔ ∀ x ∈ s, ∀ y ∈ t, p (f x y)
theorem
Set.exists_mem_image2
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{p : γ → Prop}
:
(∃ z ∈ Set.image2 f s t, p z) ↔ ∃ x ∈ s, ∃ y ∈ t, p (f x y)
@[deprecated Set.forall_mem_image2]
theorem
Set.forall_image2_iff
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{p : γ → Prop}
:
(∀ z ∈ Set.image2 f s t, p z) ↔ ∀ x ∈ s, ∀ y ∈ t, p (f x y)
Alias of Set.forall_mem_image2.
@[simp]
theorem
Set.image_uncurry_prod
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
(f : α → β → γ)
(s : Set α)
(t : Set β)
:
Function.uncurry f '' s ×ˢ t = Set.image2 f s t
@[simp]
theorem
Set.image2_mk_eq_prod
{α : Type u_1}
{β : Type u_3}
{s : Set α}
{t : Set β}
:
Set.image2 Prod.mk s t = s ×ˢ t
theorem
Set.image2_swap
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
(f : α → β → γ)
(s : Set α)
(t : Set β)
:
Set.image2 f s t = Set.image2 (fun (a : β) (b : α) => f b a) t s
theorem
Set.image2_union_left
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s s' : Set α}
{t : Set β}
:
Set.image2 f (s ∪ s') t = Set.image2 f s t ∪ Set.image2 f s' t
theorem
Set.image2_union_right
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t t' : Set β}
:
Set.image2 f s (t ∪ t') = Set.image2 f s t ∪ Set.image2 f s t'
theorem
Set.image2_inter_left
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s s' : Set α}
{t : Set β}
(hf : Function.Injective2 f)
:
Set.image2 f (s ∩ s') t = Set.image2 f s t ∩ Set.image2 f s' t
theorem
Set.image2_inter_right
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t t' : Set β}
(hf : Function.Injective2 f)
:
Set.image2 f s (t ∩ t') = Set.image2 f s t ∩ Set.image2 f s t'
@[simp]
theorem
Set.image2_empty_left
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{t : Set β}
:
Set.image2 f ∅ t = ∅
@[simp]
theorem
Set.image2_empty_right
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
:
Set.image2 f s ∅ = ∅
theorem
Set.Nonempty.image2
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t : Set β}
:
s.Nonempty → t.Nonempty → (Set.image2 f s t).Nonempty
@[simp]
theorem
Set.image2_nonempty_iff
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t : Set β}
:
(Set.image2 f s t).Nonempty ↔ s.Nonempty ∧ t.Nonempty
theorem
Set.Nonempty.of_image2_left
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t : Set β}
(h : (Set.image2 f s t).Nonempty)
:
s.Nonempty
theorem
Set.Nonempty.of_image2_right
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t : Set β}
(h : (Set.image2 f s t).Nonempty)
:
t.Nonempty
theorem
Set.Subsingleton.image2
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{s : Set α}
{t : Set β}
(hs : s.Subsingleton)
(ht : t.Subsingleton)
(f : α → β → γ)
:
(Set.image2 f s t).Subsingleton
theorem
Set.image2_inter_subset_left
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s s' : Set α}
{t : Set β}
:
Set.image2 f (s ∩ s') t ⊆ Set.image2 f s t ∩ Set.image2 f s' t
theorem
Set.image2_inter_subset_right
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t t' : Set β}
:
Set.image2 f s (t ∩ t') ⊆ Set.image2 f s t ∩ Set.image2 f s t'
@[simp]
theorem
Set.image2_singleton_left
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{t : Set β}
{a : α}
:
Set.image2 f {a} t = f a '' t
@[simp]
theorem
Set.image2_singleton_right
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{b : β}
:
Set.image2 f s {b} = (fun (a : α) => f a b) '' s
theorem
Set.image2_singleton
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{a : α}
{b : β}
:
Set.image2 f {a} {b} = {f a b}
@[simp]
theorem
Set.image2_insert_left
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{a : α}
:
Set.image2 f (insert a s) t = (fun (b : β) => f a b) '' t ∪ Set.image2 f s t
@[simp]
theorem
Set.image2_insert_right
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{b : β}
:
Set.image2 f s (insert b t) = (fun (a : α) => f a b) '' s ∪ Set.image2 f s t
theorem
Set.image2_congr
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f f' : α → β → γ}
{s : Set α}
{t : Set β}
(h : ∀ a ∈ s, ∀ b ∈ t, f a b = f' a b)
:
Set.image2 f s t = Set.image2 f' s t
theorem
Set.image2_congr'
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f f' : α → β → γ}
{s : Set α}
{t : Set β}
(h : ∀ (a : α) (b : β), f a b = f' a b)
:
Set.image2 f s t = Set.image2 f' s t
A common special case of image2_congr
theorem
Set.image_image2
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{δ : Type u_7}
{s : Set α}
{t : Set β}
(f : α → β → γ)
(g : γ → δ)
:
g '' Set.image2 f s t = Set.image2 (fun (a : α) (b : β) => g (f a b)) s t
theorem
Set.image2_image_left
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{δ : Type u_7}
{s : Set α}
{t : Set β}
(f : γ → β → δ)
(g : α → γ)
:
Set.image2 f (g '' s) t = Set.image2 (fun (a : α) (b : β) => f (g a) b) s t
theorem
Set.image2_image_right
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{δ : Type u_7}
{s : Set α}
{t : Set β}
(f : α → γ → δ)
(g : β → γ)
:
Set.image2 f s (g '' t) = Set.image2 (fun (a : α) (b : β) => f a (g b)) s t
@[simp]
theorem
Set.image2_left
{α : Type u_1}
{β : Type u_3}
{s : Set α}
{t : Set β}
(h : t.Nonempty)
:
Set.image2 (fun (x : α) (x_1 : β) => x) s t = s
@[simp]
theorem
Set.image2_right
{α : Type u_1}
{β : Type u_3}
{s : Set α}
{t : Set β}
(h : s.Nonempty)
:
Set.image2 (fun (x : α) (y : β) => y) s t = t
theorem
Set.image2_assoc
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{δ : Type u_7}
{ε : Type u_9}
{ε' : Type u_10}
{s : Set α}
{t : Set β}
{u : Set γ}
{f : δ → γ → ε}
{g : α → β → δ}
{f' : α → ε' → ε}
{g' : β → γ → ε'}
(h_assoc : ∀ (a : α) (b : β) (c : γ), f (g a b) c = f' a (g' b c))
:
Set.image2 f (Set.image2 g s t) u = Set.image2 f' s (Set.image2 g' t u)
theorem
Set.image2_comm
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{g : β → α → γ}
(h_comm : ∀ (a : α) (b : β), f a b = g b a)
:
Set.image2 f s t = Set.image2 g t s
theorem
Set.image2_left_comm
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{δ : Type u_7}
{δ' : Type u_8}
{ε : Type u_9}
{s : Set α}
{t : Set β}
{u : Set γ}
{f : α → δ → ε}
{g : β → γ → δ}
{f' : α → γ → δ'}
{g' : β → δ' → ε}
(h_left_comm : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' b (f' a c))
:
Set.image2 f s (Set.image2 g t u) = Set.image2 g' t (Set.image2 f' s u)
theorem
Set.image2_right_comm
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{δ : Type u_7}
{δ' : Type u_8}
{ε : Type u_9}
{s : Set α}
{t : Set β}
{u : Set γ}
{f : δ → γ → ε}
{g : α → β → δ}
{f' : α → γ → δ'}
{g' : δ' → β → ε}
(h_right_comm : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f' a c) b)
:
Set.image2 f (Set.image2 g s t) u = Set.image2 g' (Set.image2 f' s u) t
theorem
Set.image2_image2_image2_comm
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{δ : Type u_7}
{ε : Type u_9}
{ε' : Type u_10}
{ζ : Type u_11}
{ζ' : Type u_12}
{ν : Type u_13}
{s : Set α}
{t : Set β}
{u : Set γ}
{v : Set δ}
{f : ε → ζ → ν}
{g : α → β → ε}
{h : γ → δ → ζ}
{f' : ε' → ζ' → ν}
{g' : α → γ → ε'}
{h' : β → δ → ζ'}
(h_comm : ∀ (a : α) (b : β) (c : γ) (d : δ), f (g a b) (h c d) = f' (g' a c) (h' b d))
:
Set.image2 f (Set.image2 g s t) (Set.image2 h u v) = Set.image2 f' (Set.image2 g' s u) (Set.image2 h' t v)
theorem
Set.image_image2_distrib
{α : Type u_1}
{α' : Type u_2}
{β : Type u_3}
{β' : Type u_4}
{γ : Type u_5}
{δ : Type u_7}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{g : γ → δ}
{f' : α' → β' → δ}
{g₁ : α → α'}
{g₂ : β → β'}
(h_distrib : ∀ (a : α) (b : β), g (f a b) = f' (g₁ a) (g₂ b))
:
g '' Set.image2 f s t = Set.image2 f' (g₁ '' s) (g₂ '' t)
theorem
Set.image_image2_distrib_left
{α : Type u_1}
{α' : Type u_2}
{β : Type u_3}
{γ : Type u_5}
{δ : Type u_7}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{g : γ → δ}
{f' : α' → β → δ}
{g' : α → α'}
(h_distrib : ∀ (a : α) (b : β), g (f a b) = f' (g' a) b)
:
g '' Set.image2 f s t = Set.image2 f' (g' '' s) t
Symmetric statement to Set.image2_image_left_comm.
theorem
Set.image_image2_distrib_right
{α : Type u_1}
{β : Type u_3}
{β' : Type u_4}
{γ : Type u_5}
{δ : Type u_7}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{g : γ → δ}
{f' : α → β' → δ}
{g' : β → β'}
(h_distrib : ∀ (a : α) (b : β), g (f a b) = f' a (g' b))
:
g '' Set.image2 f s t = Set.image2 f' s (g' '' t)
Symmetric statement to Set.image_image2_right_comm.
theorem
Set.image2_image_left_comm
{α : Type u_1}
{α' : Type u_2}
{β : Type u_3}
{γ : Type u_5}
{δ : Type u_7}
{s : Set α}
{t : Set β}
{f : α' → β → γ}
{g : α → α'}
{f' : α → β → δ}
{g' : δ → γ}
(h_left_comm : ∀ (a : α) (b : β), f (g a) b = g' (f' a b))
:
Set.image2 f (g '' s) t = g' '' Set.image2 f' s t
Symmetric statement to Set.image_image2_distrib_left.
theorem
Set.image_image2_right_comm
{α : Type u_1}
{β : Type u_3}
{β' : Type u_4}
{γ : Type u_5}
{δ : Type u_7}
{s : Set α}
{t : Set β}
{f : α → β' → γ}
{g : β → β'}
{f' : α → β → δ}
{g' : δ → γ}
(h_right_comm : ∀ (a : α) (b : β), f a (g b) = g' (f' a b))
:
Set.image2 f s (g '' t) = g' '' Set.image2 f' s t
Symmetric statement to Set.image_image2_distrib_right.
theorem
Set.image2_distrib_subset_left
{α : Type u_1}
{β : Type u_3}
{β' : Type u_4}
{γ : Type u_5}
{γ' : Type u_6}
{δ : Type u_7}
{ε : Type u_9}
{s : Set α}
{t : Set β}
{u : Set γ}
{f : α → δ → ε}
{g : β → γ → δ}
{f₁ : α → β → β'}
{f₂ : α → γ → γ'}
{g' : β' → γ' → ε}
(h_distrib : ∀ (a : α) (b : β) (c : γ), f a (g b c) = g' (f₁ a b) (f₂ a c))
:
Set.image2 f s (Set.image2 g t u) ⊆ Set.image2 g' (Set.image2 f₁ s t) (Set.image2 f₂ s u)
The other direction does not hold because of the s-s cross terms on the RHS.
theorem
Set.image2_distrib_subset_right
{α : Type u_1}
{α' : Type u_2}
{β : Type u_3}
{β' : Type u_4}
{γ : Type u_5}
{δ : Type u_7}
{ε : Type u_9}
{s : Set α}
{t : Set β}
{u : Set γ}
{f : δ → γ → ε}
{g : α → β → δ}
{f₁ : α → γ → α'}
{f₂ : β → γ → β'}
{g' : α' → β' → ε}
(h_distrib : ∀ (a : α) (b : β) (c : γ), f (g a b) c = g' (f₁ a c) (f₂ b c))
:
Set.image2 f (Set.image2 g s t) u ⊆ Set.image2 g' (Set.image2 f₁ s u) (Set.image2 f₂ t u)
The other direction does not hold because of the u-u cross terms on the RHS.
theorem
Set.image_image2_antidistrib
{α : Type u_1}
{α' : Type u_2}
{β : Type u_3}
{β' : Type u_4}
{γ : Type u_5}
{δ : Type u_7}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{g : γ → δ}
{f' : β' → α' → δ}
{g₁ : β → β'}
{g₂ : α → α'}
(h_antidistrib : ∀ (a : α) (b : β), g (f a b) = f' (g₁ b) (g₂ a))
:
g '' Set.image2 f s t = Set.image2 f' (g₁ '' t) (g₂ '' s)
theorem
Set.image_image2_antidistrib_left
{α : Type u_1}
{β : Type u_3}
{β' : Type u_4}
{γ : Type u_5}
{δ : Type u_7}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{g : γ → δ}
{f' : β' → α → δ}
{g' : β → β'}
(h_antidistrib : ∀ (a : α) (b : β), g (f a b) = f' (g' b) a)
:
g '' Set.image2 f s t = Set.image2 f' (g' '' t) s
Symmetric statement to Set.image2_image_left_anticomm.
theorem
Set.image_image2_antidistrib_right
{α : Type u_1}
{α' : Type u_2}
{β : Type u_3}
{γ : Type u_5}
{δ : Type u_7}
{f : α → β → γ}
{s : Set α}
{t : Set β}
{g : γ → δ}
{f' : β → α' → δ}
{g' : α → α'}
(h_antidistrib : ∀ (a : α) (b : β), g (f a b) = f' b (g' a))
:
g '' Set.image2 f s t = Set.image2 f' t (g' '' s)
Symmetric statement to Set.image_image2_right_anticomm.
theorem
Set.image2_image_left_anticomm
{α : Type u_1}
{α' : Type u_2}
{β : Type u_3}
{γ : Type u_5}
{δ : Type u_7}
{s : Set α}
{t : Set β}
{f : α' → β → γ}
{g : α → α'}
{f' : β → α → δ}
{g' : δ → γ}
(h_left_anticomm : ∀ (a : α) (b : β), f (g a) b = g' (f' b a))
:
Set.image2 f (g '' s) t = g' '' Set.image2 f' t s
Symmetric statement to Set.image_image2_antidistrib_left.
theorem
Set.image_image2_right_anticomm
{α : Type u_1}
{β : Type u_3}
{β' : Type u_4}
{γ : Type u_5}
{δ : Type u_7}
{s : Set α}
{t : Set β}
{f : α → β' → γ}
{g : β → β'}
{f' : β → α → δ}
{g' : δ → γ}
(h_right_anticomm : ∀ (a : α) (b : β), f a (g b) = g' (f' b a))
:
Set.image2 f s (g '' t) = g' '' Set.image2 f' t s
Symmetric statement to Set.image_image2_antidistrib_right.
theorem
Set.image2_left_identity
{α : Type u_1}
{β : Type u_3}
{f : α → β → β}
{a : α}
(h : ∀ (b : β), f a b = b)
(t : Set β)
:
Set.image2 f {a} t = t
If a is a left identity for f : α → β → β, then {a} is a left identity for
Set.image2 f.
theorem
Set.image2_right_identity
{α : Type u_1}
{β : Type u_3}
{f : α → β → α}
{b : β}
(h : ∀ (a : α), f a b = a)
(s : Set α)
:
Set.image2 f s {b} = s
If b is a right identity for f : α → β → α, then {b} is a right identity for
Set.image2 f.
theorem
Set.image2_inter_union_subset_union
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s s' : Set α}
{t t' : Set β}
:
Set.image2 f (s ∩ s') (t ∪ t') ⊆ Set.image2 f s t ∪ Set.image2 f s' t'
theorem
Set.image2_union_inter_subset_union
{α : Type u_1}
{β : Type u_3}
{γ : Type u_5}
{f : α → β → γ}
{s s' : Set α}
{t t' : Set β}
:
Set.image2 f (s ∪ s') (t ∩ t') ⊆ Set.image2 f s t ∪ Set.image2 f s' t'
theorem
Set.image2_inter_union_subset
{α : Type u_1}
{β : Type u_3}
{f : α → α → β}
{s t : Set α}
(hf : ∀ (a b : α), f a b = f b a)
:
Set.image2 f (s ∩ t) (s ∪ t) ⊆ Set.image2 f s t
theorem
Set.image2_union_inter_subset
{α : Type u_1}
{β : Type u_3}
{f : α → α → β}
{s t : Set α}
(hf : ∀ (a b : α), f a b = f b a)
:
Set.image2 f (s ∪ t) (s ∩ t) ⊆ Set.image2 f s t