```------------------------------------------------------------------------
-- Properties relating Any to various list functions
------------------------------------------------------------------------

module Data.List.Any.Properties where

open import Data.Bool
open import Data.Bool.Properties
open import Data.Function
open import Data.List as List
open import Data.List.Any as Any using (Any; here; there)
open import Data.Product as Prod hiding (map)
open import Data.Sum as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
open import Relation.Unary using (Pred; _⟨×⟩_; _⟨→⟩_)
open import Relation.Binary
import Relation.Binary.EqReasoning as EqReasoning
open import Relation.Binary.FunctionSetoid
import Relation.Binary.List.Pointwise as ListEq
open import Relation.Binary.Product.Pointwise
open import Relation.Binary.PropositionalEquality as PropEq
using (_≡_; refl; inspect; _with-≡_)

------------------------------------------------------------------------
-- Lemmas related to Any

-- Introduction (⁺) and elimination (⁻) rules for map.

map⁺ : ∀ {A B} {P : Pred B} {f : A → B} {xs} →
Any (P ∘₀ f) xs → Any P (map f xs)
map⁺ (here p)  = here p
map⁺ (there p) = there \$ map⁺ p

map⁻ : ∀ {A B} {P : Pred B} {f : A → B} {xs} →
Any P (map f xs) → Any (P ∘₀ f) xs
map⁻ {xs = []}     ()
map⁻ {xs = x ∷ xs} (here p)  = here p
map⁻ {xs = x ∷ xs} (there p) = there \$ map⁻ p

-- Introduction and elimination rules for _++_.

++⁺ˡ : ∀ {A} {P : Pred A} {xs ys} →
Any P xs → Any P (xs ++ ys)
++⁺ˡ (here p)  = here p
++⁺ˡ (there p) = there (++⁺ˡ p)

++⁺ʳ : ∀ {A} {P : Pred A} xs {ys} →
Any P ys → Any P (xs ++ ys)
++⁺ʳ []       p = p
++⁺ʳ (x ∷ xs) p = there (++⁺ʳ xs p)

++⁻ : ∀ {A} {P : Pred A} xs {ys} →
Any P (xs ++ ys) → Any P xs ⊎ Any P ys
++⁻ []       p         = inj₂ p
++⁻ (x ∷ xs) (here p)  = inj₁ (here p)
++⁻ (x ∷ xs) (there p) = Sum.map there id (++⁻ xs p)

-- Introduction and elimination rules for return.

return⁺ : ∀ {A} {P : Pred A} {x} →
P x → Any P (return x)
return⁺ = here

return⁻ : ∀ {A} {P : Pred A} {x} →
Any P (return x) → P x
return⁻ (here p)   = p
return⁻ (there ())

-- Introduction and elimination rules for concat.

concat⁺ : ∀ {A} {P : Pred A} {xss} →
Any (Any P) xss → Any P (concat xss)
concat⁺ (here p)           = ++⁺ˡ p
concat⁺ (there {x = xs} p) = ++⁺ʳ xs (concat⁺ p)

concat⁻ : ∀ {A} {P : Pred A} xss →
Any P (concat xss) → Any (Any P) xss
concat⁻ []               ()
concat⁻ ([]       ∷ xss) p         = there \$ concat⁻ xss p
concat⁻ ((x ∷ xs) ∷ xss) (here p)  = here (here p)
concat⁻ ((x ∷ xs) ∷ xss) (there p)
with concat⁻ (xs ∷ xss) p
... | here  p′ = here (there p′)
... | there p′ = there p′

-- Introduction and elimination rules for _>>=_.

>>=⁺ : ∀ {A B P xs} {f : A → List B} →
Any (Any P ∘₀ f) xs → Any P (xs >>= f)
>>=⁺ = concat⁺ ∘ map⁺

>>=⁻ : ∀ {A B P} xs {f : A → List B} →
Any P (xs >>= f) → Any (Any P ∘₀ f) xs
>>=⁻ _ = map⁻ ∘ concat⁻ _

-- Introduction and elimination rules for _⊛_.

⊛⁺ : ∀ {A B P} {fs : List (A → B)} {xs} →
Any (λ f → Any (P ∘₀ f) xs) fs → Any P (fs ⊛ xs)
⊛⁺ = >>=⁺ ∘ Any.map (>>=⁺ ∘ Any.map return⁺)

⊛⁺′ : ∀ {A B P Q} {fs : List (A → B)} {xs} →
Any (P ⟨→⟩ Q) fs → Any P xs → Any Q (fs ⊛ xs)
⊛⁺′ pq p = ⊛⁺ (Any.map (λ pq → Any.map (λ {x} → pq {x}) p) pq)

⊛⁻ : ∀ {A B P} (fs : List (A → B)) xs →
Any P (fs ⊛ xs) → Any (λ f → Any (P ∘₀ f) xs) fs
⊛⁻ fs xs = Any.map (Any.map return⁻ ∘ >>=⁻ xs) ∘ >>=⁻ fs

-- Introduction and elimination rules for _⊗_.

⊗⁺ : ∀ {A B P} {xs : List A} {ys : List B} →
Any (λ x → Any (λ y → P (x , y)) ys) xs → Any P (xs ⊗ ys)
⊗⁺ = ⊛⁺ ∘′ ⊛⁺ ∘′ return⁺

⊗⁺′ : ∀ {A B} {P : Pred A} {Q : Pred B} {xs ys} →
Any P xs → Any Q ys → Any (P ⟨×⟩ Q) (xs ⊗ ys)
⊗⁺′ p q = ⊗⁺ (Any.map (λ p → Any.map (λ q → (p , q)) q) p)

⊗⁻ : ∀ {A B P} (xs : List A) (ys : List B) →
Any P (xs ⊗ ys) → Any (λ x → Any (λ y → P (x , y)) ys) xs
⊗⁻ _ _ = return⁻ ∘ ⊛⁻ _ _ ∘ ⊛⁻ _ _

⊗⁻′ : ∀ {A B} {P : Pred A} {Q : Pred B} xs ys →
Any (P ⟨×⟩ Q) (xs ⊗ ys) → Any P xs × Any Q ys
⊗⁻′ _ _ pq =
(Any.map (proj₁ ∘ proj₂ ∘ Any.satisfied) lem ,
(Any.map proj₂ \$ proj₂ (Any.satisfied lem)))
where lem = ⊗⁻ _ _ pq

-- Any and any are related via T.

any⁺ : ∀ {A} (p : A → Bool) {xs} →
Any (T ∘₀ p) xs → T (any p xs)
any⁺ p (here  px)          = proj₂ T-∨ (inj₁ px)
any⁺ p (there {x = x} pxs) with p x
... | true  = _
... | false = any⁺ p pxs

any⁻ : ∀ {A} (p : A → Bool) xs →
T (any p xs) → Any (T ∘₀ p) xs
any⁻ p []       ()
any⁻ p (x ∷ xs) px∷xs with inspect (p x)
any⁻ p (x ∷ xs) px∷xs | true  with-≡ eq = here (proj₂ T-≡ \$
PropEq.sym eq)
any⁻ p (x ∷ xs) px∷xs | false with-≡ eq with p x
any⁻ p (x ∷ xs) pxs   | false with-≡ refl | .false =
there (any⁻ p xs pxs)

------------------------------------------------------------------------
-- Lemmas related to _∈_, parameterised on underlying equalities

module Membership₁ (S : Setoid) where

open Any.Membership S
private
open module S = Setoid S using (_≈_)
open module [M] = Any.Membership (ListEq.setoid S)
using () renaming (_∈_ to _[∈]_; _⊆_ to _[⊆]_)
open module M≡ = Any.Membership-≡
using () renaming (_∈_ to _∈≡_; _⊆_ to _⊆≡_)

-- Any is monotone.

mono : ∀ {P xs ys} →
P Respects _≈_ → xs ⊆ ys → Any P xs → Any P ys
mono resp xs⊆ys pxs with find pxs
... | (x , x∈xs , px) = lose resp (xs⊆ys x∈xs) px

-- _++_ is monotone.

_++-mono_ : ∀ {xs₁ xs₂ ys₁ ys₂} →
xs₁ ⊆ ys₁ → xs₂ ⊆ ys₂ → xs₁ ++ xs₂ ⊆ ys₁ ++ ys₂
_++-mono_ {ys₁ = ys₁} xs₁⊆ys₁ xs₂⊆ys₂ =
[ ++⁺ˡ ∘ xs₁⊆ys₁ , ++⁺ʳ ys₁ ∘ xs₂⊆ys₂ ]′ ∘ ++⁻ _

-- _++_ is idempotent.

++-idempotent : ∀ {xs} → xs ++ xs ⊆ xs
++-idempotent = [ id , id ]′ ∘ ++⁻ _

-- Introduction and elimination rules for concat.

concat-∈⁺ : ∀ {x xs xss} →
x ∈ xs → xs [∈] xss → x ∈ concat xss
concat-∈⁺ x∈xs xs∈xss =
concat⁺ (Any.map (λ xs≈ys → P.reflexive xs≈ys x∈xs) xs∈xss)
where module P = Preorder ⊆-preorder

concat-∈⁻ : ∀ {x} xss → x ∈ concat xss →
∃ λ xs → x ∈ xs × xs [∈] xss
concat-∈⁻ xss x∈ = Prod.map id swap \$ [M].find (concat⁻ xss x∈)

-- concat is monotone.

concat-mono : ∀ {xss yss} →
xss [⊆] yss → concat xss ⊆ concat yss
concat-mono {xss = xss} xss⊆yss x∈ with concat-∈⁻ xss x∈
... | (xs , x∈xs , xs∈xss) = concat-∈⁺ x∈xs (xss⊆yss xs∈xss)

-- any is monotone.

any-mono : ∀ p → (T ∘₀ p) Respects _≈_ →
∀ {xs ys} → xs ⊆ ys → T (any p xs) → T (any p ys)
any-mono p resp xs⊆ys = any⁺ p ∘ mono resp xs⊆ys ∘ any⁻ p _

-- Introduction and elimination rules for map-with-∈.

map-with-∈-∈⁺ : ∀ {A} {xs : List A}
(f : ∀ {x} → x ∈≡ xs → S.carrier) {x} →
(x∈xs : x ∈≡ xs) → f x∈xs ∈ M≡.map-with-∈ xs f
map-with-∈-∈⁺ f (here refl)  = here S.refl
map-with-∈-∈⁺ f (there x∈xs) = there \$ map-with-∈-∈⁺ (f ∘ there) x∈xs

map-with-∈-∈⁻ : ∀ {A} {xs : List A}
(f : ∀ {x} → x ∈≡ xs → S.carrier) {fx∈xs} →
fx∈xs ∈ M≡.map-with-∈ xs f →
∃ λ x → Σ (x ∈≡ xs) λ x∈xs → fx∈xs ≈ f x∈xs
map-with-∈-∈⁻ {xs = []}     f ()
map-with-∈-∈⁻ {xs = y ∷ xs} f (here fx≈)   = (y , here refl , fx≈)
map-with-∈-∈⁻ {xs = y ∷ xs} f (there x∈xs) =
Prod.map id (Prod.map there id) \$ map-with-∈-∈⁻ (f ∘ there) x∈xs

-- map-with-∈ is monotone.

map-with-∈-mono :
∀ {A} {xs : List A} {f : ∀ {x} → x ∈≡ xs → S.carrier}
{ys : List A} {g : ∀ {x} → x ∈≡ ys → S.carrier} →
(xs⊆ys : xs ⊆≡ ys) →
(∀ {x} (x∈xs : x ∈≡ xs) → f x∈xs ≈ g (xs⊆ys x∈xs)) →
M≡.map-with-∈ xs f ⊆ M≡.map-with-∈ ys g
map-with-∈-mono {f = f} {g = g} xs⊆ys f≈g {fx∈xs} fx∈xs∈
with map-with-∈-∈⁻ f fx∈xs∈
... | (x , x∈xs , fx∈xs≈) =
Any.map (λ {y} g[xs⊆ys-x∈xs]≈y → begin
fx∈xs           ≈⟨ fx∈xs≈ ⟩
f x∈xs          ≈⟨ f≈g x∈xs ⟩
g (xs⊆ys x∈xs)  ≈⟨ g[xs⊆ys-x∈xs]≈y ⟩
y               ∎) \$
map-with-∈-∈⁺ g (xs⊆ys x∈xs)
where open EqReasoning S

module Membership₂ (S₁ S₂ : Setoid) where

private
open module S₁ = Setoid S₁ using () renaming (_≈_ to _≈₁_)
open module S₂ = Setoid S₂ using () renaming (_≈_ to _≈₂_)
LS₂            = ListEq.setoid S₂
open module M₁ = Any.Membership S₁
using () renaming (_∈_ to _∈₁_; _⊆_ to _⊆₁_)
open module M₂ = Any.Membership S₂
using () renaming (_∈_ to _∈₂_; _⊆_ to _⊆₂_)
open module M₁₂ = Any.Membership (S₁ ⇨ S₂)
using () renaming (_∈_ to _∈₁₂_; _⊆_ to _⊆₁₂_)
open Any.Membership (S₁ ×-setoid S₂)
using () renaming (_⊆_ to _⊆₁,₂_)

-- Introduction and elimination rules for map.

map-∈⁺ : ∀ (f : S₁ ⟶ S₂) {x xs} →
x ∈₁ xs → f ⟨\$⟩ x ∈₂ map (_⟨\$⟩_ f) xs
map-∈⁺ f = map⁺ ∘ Any.map (pres f)

map-∈⁻ : ∀ {f fx} xs →
fx ∈₂ map f xs → ∃ λ x → x ∈₁ xs × fx ≈₂ f x
map-∈⁻ _ fx∈ = M₁.find (map⁻ fx∈)

-- map is monotone.

map-mono : ∀ (f : S₁ ⟶ S₂) {xs ys} →
xs ⊆₁ ys → map (_⟨\$⟩_ f) xs ⊆₂ map (_⟨\$⟩_ f) ys
map-mono f xs⊆ys fx∈ with map-∈⁻ _ fx∈
... | (x , x∈ , eq) = Any.map (S₂.trans eq) (map-∈⁺ f (xs⊆ys x∈))

-- Introduction and elimination rules for _>>=_.

>>=-∈⁺ : ∀ (f : S₁ ⟶ LS₂) {x y xs} →
x ∈₁ xs → y ∈₂ f ⟨\$⟩ x → y ∈₂ (xs >>= _⟨\$⟩_ f)
>>=-∈⁺ f x∈xs y∈fx =
>>=⁺ (Any.map (flip M₂.∈-resp-list-≈ y∈fx ∘ pres f) x∈xs)

>>=-∈⁻ : ∀ (f : S₁ ⟶ LS₂) {y} xs →
y ∈₂ (xs >>= _⟨\$⟩_ f) → ∃ λ x → x ∈₁ xs × y ∈₂ f ⟨\$⟩ x
>>=-∈⁻ f xs y∈ = M₁.find (>>=⁻ xs y∈)

-- _>>=_ is monotone.

>>=-mono : ∀ (f g : S₁ ⟶ LS₂) {xs ys} →
xs ⊆₁ ys → (∀ {x} → f ⟨\$⟩ x ⊆₂ g ⟨\$⟩ x) →
(xs >>= _⟨\$⟩_ f) ⊆₂ (ys >>= _⟨\$⟩_ g)
>>=-mono f g {xs} xs⊆ys f⊆g z∈ with >>=-∈⁻ f xs z∈
... | (x , x∈xs , z∈fx) = >>=-∈⁺ g (xs⊆ys x∈xs) (f⊆g z∈fx)

-- Introduction and elimination rules for _⊛_.

private

infixl 4 _⟨⊛⟩_

_⟨⊛⟩_ : List (S₁ ⟶ S₂) → List S₁.carrier → List S₂.carrier
fs ⟨⊛⟩ xs = map _⟨\$⟩_ fs ⊛ xs

⊛-∈⁺ : ∀ f {fs x xs} →
f ∈₁₂ fs → x ∈₁ xs → f ⟨\$⟩ x ∈₂ fs ⟨⊛⟩ xs
⊛-∈⁺ _ f∈fs x∈xs =
⊛⁺′ (map⁺ (Any.map (λ f≈g x≈y → f≈g x≈y) f∈fs)) x∈xs

⊛-∈⁻ : ∀ fs xs {fx} → fx ∈₂ fs ⟨⊛⟩ xs →
∃₂ λ f x → f ∈₁₂ fs × x ∈₁ xs × fx ≈₂ f ⟨\$⟩ x
⊛-∈⁻ fs xs fx∈ with M₁₂.find \$ map⁻ (⊛⁻ (map _⟨\$⟩_ fs) xs fx∈)
... | (f , f∈fs , x∈) with M₁.find x∈
...   | (x , x∈xs , fx≈fx) = (f , x , f∈fs , x∈xs , fx≈fx)

-- _⊛_ is monotone.

_⊛-mono_ : ∀ {fs gs xs ys} →
fs ⊆₁₂ gs → xs ⊆₁ ys → fs ⟨⊛⟩ xs ⊆₂ gs ⟨⊛⟩ ys
_⊛-mono_ {fs = fs} {xs = xs} fs⊆gs xs⊆ys fx∈ with ⊛-∈⁻ fs xs fx∈
... | (f , x , f∈fs , x∈xs , fx≈fx) =
Any.map (S₂.trans fx≈fx) \$ ⊛-∈⁺ f (fs⊆gs {f} f∈fs) (xs⊆ys x∈xs)

-- _⊗_ is monotone.

_⊗-mono_ : ∀ {xs₁ xs₂ ys₁ ys₂} →
xs₁ ⊆₁ ys₁ → xs₂ ⊆₂ ys₂ → (xs₁ ⊗ xs₂) ⊆₁,₂ (ys₁ ⊗ ys₂)
_⊗-mono_ {xs₁ = xs₁} {xs₂} xs₁⊆ys₁ xs₂⊆ys₂ {x , y} x,y∈
with ⊗⁻′ {P = _≈₁_ x} {Q = _≈₂_ y} xs₁ xs₂ x,y∈
... | (x∈ , y∈) = ⊗⁺′ (xs₁⊆ys₁ x∈) (xs₂⊆ys₂ y∈)

------------------------------------------------------------------------
-- Lemmas related to the variant of _∈_ which is defined using
-- propositional equality

module Membership-≡ where

open Any.Membership-≡
private
module S {A} = Setoid (ListEq.setoid (PropEq.setoid A))
open module M₁ {A} = Membership₁ (PropEq.setoid A) public
using (_++-mono_; ++-idempotent;
map-with-∈-∈⁺; map-with-∈-∈⁻; map-with-∈-mono)
open module M₂ {A} {B} =
Membership₂ (PropEq.setoid A) (PropEq.setoid B) public
using (map-∈⁻)

-- Any is monotone.

mono : ∀ {A xs ys} {P : Pred A} → xs ⊆ ys → Any P xs → Any P ys
mono {P = P} = M₁.mono (PropEq.subst P)

-- Introduction and elimination rules for concat.

concat-∈⁺ : ∀ {A} {x : A} {xs xss} →
x ∈ xs → xs ∈ xss → x ∈ concat xss
concat-∈⁺ x∈xs = M₁.concat-∈⁺ x∈xs ∘ Any.map S.reflexive

concat-∈⁻ : ∀ {A} {x : A} xss →
x ∈ concat xss → ∃ λ xs → x ∈ xs × xs ∈ xss
concat-∈⁻ xss x∈ =
Prod.map id (Prod.map id (Any.map ListEq.Rel≡⇒≡)) \$
M₁.concat-∈⁻ xss x∈

-- concat is monotone.

concat-mono : ∀ {A} {xss yss : List (List A)} →
xss ⊆ yss → concat xss ⊆ concat yss
concat-mono xss⊆yss =
M₁.concat-mono (Any.map S.reflexive ∘ xss⊆yss ∘
Any.map ListEq.Rel≡⇒≡)

-- any is monotone.

any-mono : ∀ {A} (p : A → Bool) {xs ys} →
xs ⊆ ys → T (any p xs) → T (any p ys)
any-mono p = M₁.any-mono p (PropEq.subst (T ∘₀ p))

-- Introduction rule for map.

map-∈⁺ : ∀ {A B} {f : A → B} {x xs} →
x ∈ xs → f x ∈ map f xs
map-∈⁺ {f = f} = M₂.map-∈⁺ (PropEq.→-to-⟶ f)

-- map is monotone.

map-mono : ∀ {A B} {f : A → B} {xs ys} →
xs ⊆ ys → map f xs ⊆ map f ys
map-mono {f = f} = M₂.map-mono (PropEq.→-to-⟶ f)

-- Introduction and elimination rules for _>>=_.

private

[→-to-⟶] : ∀ {A B} → (A → List B) →
PropEq.setoid A ⟶
ListEq.setoid (PropEq.setoid B)
[→-to-⟶] f =
record { _⟨\$⟩_ = f; pres = S.reflexive ∘ PropEq.cong f }

>>=-∈⁺ : ∀ {A B} (f : A → List B) {x y xs} →
x ∈ xs → y ∈ f x → y ∈ (xs >>= f)
>>=-∈⁺ f = M₂.>>=-∈⁺ ([→-to-⟶] f)

>>=-∈⁻ : ∀ {A B} (f : A → List B) {y} xs →
y ∈ (xs >>= f) → ∃ λ x → x ∈ xs × y ∈ f x
>>=-∈⁻ f = M₂.>>=-∈⁻ ([→-to-⟶] f)

-- _>>=_ is monotone.

_>>=-mono_ : ∀ {A B} {f g : A → List B} {xs ys} →
xs ⊆ ys → (∀ {x} → f x ⊆ g x) →
(xs >>= f) ⊆ (ys >>= g)
_>>=-mono_ {f = f} {g} = M₂.>>=-mono ([→-to-⟶] f) ([→-to-⟶] g)

-- Introduction and elimination rules for _⊛_.

⊛-∈⁺ : ∀ {A B} {fs : List (A → B)} {xs f x} →
f ∈ fs → x ∈ xs → f x ∈ fs ⊛ xs
⊛-∈⁺ f∈fs x∈xs =
⊛⁺′ (Any.map (λ f≡g x≡y → PropEq.cong₂ _\$_ f≡g x≡y) f∈fs) x∈xs

⊛-∈⁻ : ∀ {A B} (fs : List (A → B)) xs {fx} →
fx ∈ fs ⊛ xs → ∃₂ λ f x → f ∈ fs × x ∈ xs × fx ≡ f x
⊛-∈⁻ fs xs fx∈ with find \$ ⊛⁻ fs xs fx∈
... | (f , f∈fs , x∈) with find x∈
...   | (x , x∈xs , fx≡fx) = (f , x , f∈fs , x∈xs , fx≡fx)

-- _⊛_ is monotone.

_⊛-mono_ : ∀ {A B} {fs gs : List (A → B)} {xs ys} →
fs ⊆ gs → xs ⊆ ys → fs ⊛ xs ⊆ gs ⊛ ys
_⊛-mono_ {fs = fs} {xs = xs} fs⊆gs xs⊆ys fx∈ with ⊛-∈⁻ fs xs fx∈
... | (f , x , f∈fs , x∈xs , refl) =
⊛-∈⁺ (fs⊆gs f∈fs) (xs⊆ys x∈xs)

-- Introduction and elimination rules for _⊗_.

private

lemma₁ : ∀ {A B} {p q : A × B} →
(p ⟨ _≡_ on₁ proj₁ ⟩₁ q) × (p ⟨ _≡_ on₁ proj₂ ⟩₁ q) → p ≡ q
lemma₁ {p = (x , y)} {q = (.x , .y)} (refl , refl) = refl

lemma₂ : ∀ {A B} {p q : A × B} →
p ≡ q → (p ⟨ _≡_ on₁ proj₁ ⟩₁ q) × (p ⟨ _≡_ on₁ proj₂ ⟩₁ q)
lemma₂ = < PropEq.cong proj₁ , PropEq.cong proj₂ >

⊗-∈⁺ : ∀ {A B} {xs : List A} {ys : List B} {x y} →
x ∈ xs → y ∈ ys → (x , y) ∈ (xs ⊗ ys)
⊗-∈⁺ x∈xs y∈ys = Any.map lemma₁ (⊗⁺′ x∈xs y∈ys)

⊗-∈⁻ : ∀ {A B} (xs : List A) (ys : List B) {p} →
p ∈ (xs ⊗ ys) → proj₁ p ∈ xs × proj₂ p ∈ ys
⊗-∈⁻ xs ys = ⊗⁻′ xs ys ∘ Any.map lemma₂

-- _⊗_ is monotone.

_⊗-mono_ : ∀ {A B} {xs₁ ys₁ : List A} {xs₂ ys₂ : List B} →
xs₁ ⊆ ys₁ → xs₂ ⊆ ys₂ → (xs₁ ⊗ xs₂) ⊆ (ys₁ ⊗ ys₂)
_⊗-mono_ xs₁⊆ys₁ xs₂⊆ys₂ {p} =
Any.map lemma₁ ∘ M₂._⊗-mono_ xs₁⊆ys₁ xs₂⊆ys₂ {p} ∘ Any.map lemma₂
```