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Code extracted from "Formalising type-logical grammars in Agda".
Raw
main.agda
-- This file contains the code from the paper "Formalising
-- type-logical grammars in Agda", and was directly extracted from the
-- paper's source.
--
-- Note: because the symbol customarily used for the "left difference"
-- is unavailable in the Unicode character set, and the backslash used
-- for implication is often a reserved symbol, the source file uses a
-- different notation for the connectives: the left and right
-- implications are represented by the double arrows "⇐" and "⇒", and
-- the left and right differences are represented by the triple arrows
-- "⇚" ad "⇛".
module main where
open import Function using (id; flip; _∘_)
open import Data.Bool using (Bool; true; false; _∧_)
open import Data.Empty using (⊥)
open import Data.Product using (_×_; _,_)
open import Data.Unit using (⊤)
open import Reflection using (Term; _≟_)
open import Relation.Binary.PropositionalEquality using (_≡_; refl)
open import Relation.Nullary.Decidable using (⌊_⌋)
module logic (Atom : Set) where
data Type : Set where
el : Atom → Type
_⊗_ _⇒_ _⇐_ : Type → Type → Type
_⊕_ _⇚_ _⇛_ : Type → Type → Type
data Judgement : Set where
_⊢_ : Type → Type → Judgement
infix 1 LG_
infix 2 _⊢_
infixl 3 _[_]
infixl 3 _[_]ᴶ
infixr 4 _⊗_
infixr 5 _⊕_
infixl 6 _⇚_
infixr 6 _⇛_
infixr 7 _⇒_
infixl 7 _⇐_
data LG_ : Judgement → Set where
ax : ∀ {A} → LG el A ⊢ el A
-- residuation and monotonicity for (⇐ , ⊗ , ⇒)
r⇒⊗ : ∀ {A B C} → LG B ⊢ A ⇒ C → LG A ⊗ B ⊢ C
r⊗⇒ : ∀ {A B C} → LG A ⊗ B ⊢ C → LG B ⊢ A ⇒ C
r⇐⊗ : ∀ {A B C} → LG A ⊢ C ⇐ B → LG A ⊗ B ⊢ C
r⊗⇐ : ∀ {A B C} → LG A ⊗ B ⊢ C → LG A ⊢ C ⇐ B
m⊗ : ∀ {A B C D} → LG A ⊢ B → LG C ⊢ D → LG A ⊗ C ⊢ B ⊗ D
m⇒ : ∀ {A B C D} → LG A ⊢ B → LG C ⊢ D → LG B ⇒ C ⊢ A ⇒ D
m⇐ : ∀ {A B C D} → LG A ⊢ B → LG C ⊢ D → LG A ⇐ D ⊢ B ⇐ C
-- residuation and monotonicity for (⇚ , ⊕ , ⇛)
r⇛⊕ : ∀ {A B C} → LG B ⇛ C ⊢ A → LG C ⊢ B ⊕ A
r⊕⇛ : ∀ {A B C} → LG C ⊢ B ⊕ A → LG B ⇛ C ⊢ A
r⊕⇚ : ∀ {A B C} → LG C ⊢ B ⊕ A → LG C ⇚ A ⊢ B
r⇚⊕ : ∀ {A B C} → LG C ⇚ A ⊢ B → LG C ⊢ B ⊕ A
m⊕ : ∀ {A B C D} → LG A ⊢ B → LG C ⊢ D → LG A ⊕ C ⊢ B ⊕ D
m⇛ : ∀ {A B C D} → LG C ⊢ D → LG A ⊢ B → LG D ⇛ A ⊢ C ⇛ B
m⇚ : ∀ {A B C D} → LG A ⊢ B → LG C ⊢ D → LG A ⇚ D ⊢ B ⇚ C
-- grishin distributives
d⇛⇐ : ∀ {A B C D} → LG A ⊗ B ⊢ C ⊕ D → LG C ⇛ A ⊢ D ⇐ B
d⇛⇒ : ∀ {A B C D} → LG A ⊗ B ⊢ C ⊕ D → LG C ⇛ B ⊢ A ⇒ D
d⇚⇒ : ∀ {A B C D} → LG A ⊗ B ⊢ C ⊕ D → LG B ⇚ D ⊢ A ⇒ C
d⇚⇐ : ∀ {A B C D} → LG A ⊗ B ⊢ C ⊕ D → LG A ⇚ D ⊢ C ⇐ B
ax′ : ∀ {A} → LG A ⊢ A
ax′ {A = el _} = ax
ax′ {A = _ ⊗ _} = m⊗ ax′ ax′
ax′ {A = _ ⇐ _} = m⇐ ax′ ax′
ax′ {A = _ ⇒ _} = m⇒ ax′ ax′
ax′ {A = _ ⊕ _} = m⊕ ax′ ax′
ax′ {A = _ ⇚ _} = m⇚ ax′ ax′
ax′ {A = _ ⇛ _} = m⇛ ax′ ax′
appl-⇒′ : ∀ {A B} → LG A ⊗ (A ⇒ B) ⊢ B
appl-⇒′ = r⇒⊗ (m⇒ ax′ ax′)
appl-⇐′ : ∀ {A B} → LG (B ⇐ A) ⊗ A ⊢ B
appl-⇐′ = r⇐⊗ (m⇐ ax′ ax′)
appl-⇛′ : ∀ {A B} → LG B ⊢ A ⊕ (A ⇛ B)
appl-⇛′ = r⇛⊕ (m⇛ ax′ ax′)
appl-⇚′ : ∀ {A B} → LG B ⊢ (B ⇚ A) ⊕ A
appl-⇚′ = r⇚⊕ (m⇚ ax′ ax′)
data Polarity : Set where + - : Polarity
data Context (p : Polarity) : Polarity → Set where
[] : Context p p
_⊗>_ : Type → Context p + → Context p +
_⇒>_ : Type → Context p - → Context p -
_⇐>_ : Type → Context p + → Context p -
_<⊗_ : Context p + → Type → Context p +
_<⇒_ : Context p + → Type → Context p -
_<⇐_ : Context p - → Type → Context p -
_⊕>_ : Type → Context p - → Context p -
_⇚>_ : Type → Context p - → Context p +
_⇛>_ : Type → Context p + → Context p +
_<⊕_ : Context p - → Type → Context p -
_<⇚_ : Context p + → Type → Context p +
_<⇛_ : Context p - → Type → Context p +
data Contextᴶ (p : Polarity) : Set where
_<⊢_ : Context p + → Type → Contextᴶ p
_⊢>_ : Type → Context p - → Contextᴶ p
_[_] : ∀ {p₁ p₂} → Context p₁ p₂ → Type → Type
[] [ A ] = A
(B ⊗> C) [ A ] = B ⊗ (C [ A ])
(C <⊗ B) [ A ] = (C [ A ]) ⊗ B
(B ⇒> C) [ A ] = B ⇒ (C [ A ])
(C <⇒ B) [ A ] = (C [ A ]) ⇒ B
(B ⇐> C) [ A ] = B ⇐ (C [ A ])
(C <⇐ B) [ A ] = (C [ A ]) ⇐ B
(B ⊕> C) [ A ] = B ⊕ (C [ A ])
(C <⊕ B) [ A ] = (C [ A ]) ⊕ B
(B ⇚> C) [ A ] = B ⇚ (C [ A ])
(C <⇚ B) [ A ] = (C [ A ]) ⇚ B
(B ⇛> C) [ A ] = B ⇛ (C [ A ])
(C <⇛ B) [ A ] = (C [ A ]) ⇛ B
_[_]ᴶ : ∀ {p} → Contextᴶ p → Type → Judgement
(A <⊢ B) [ C ]ᴶ = A [ C ] ⊢ B
(A ⊢> B) [ C ]ᴶ = A ⊢ B [ C ]
module ⊗ where
data Origin′ {B C} (J : Contextᴶ -)
(f : LG J [ B ⊗ C ]ᴶ)
: Set where
origin : ∀ {E F} (h₁ : LG E ⊢ B)
(h₂ : LG F ⊢ C)
(f′ : ∀ {G} → LG E ⊗ F ⊢ G → LG J [ G ]ᴶ)
(pr : f ≡ f′ (m⊗ h₁ h₂))
→ Origin′ J f
mutual
find′ : ∀ {B C} (J : Contextᴶ -) (f : LG J [ B ⊗ C ]ᴶ) → Origin′ J f
find′ (._ ⊢> []) (m⊗ f g) = origin f g id refl
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) with find′ (_ ⊢> A) f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (r⊗⇐ ∘ f′) refl
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) with find′ (_ ⊢> B) f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (r⊗⇒ ∘ f′) refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B C}
→ ( I : Contextᴶ - ) (f : LG I [ B ⊗ C ]ᴶ)
→ { J : Contextᴶ - } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I f {J} g with find′ I f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (g ∘ f′) refl
Origin : ∀ {A B C} (f : LG A ⊢ B ⊗ C) → Set
Origin f = Origin′ (_ ⊢> []) f
find : ∀ {A B C} (f : LG A ⊢ B ⊗ C) → Origin f
find f = find′ (_ ⊢> []) f
module ⇒ where
data Origin′ {B C} (J : Contextᴶ +) (f : LG J [ B ⇒ C ]ᴶ) : Set where
origin : ∀ {E F} (h₁ : LG E ⊢ B)
(h₂ : LG C ⊢ F)
(f′ : ∀ {G} → LG G ⊢ E ⇒ F → LG J [ G ]ᴶ)
(pr : f ≡ f′ (m⇒ h₁ h₂))
→ Origin′ J f
mutual
find′ : ∀ {B C} (J : Contextᴶ +) (f : LG J [ B ⇒ C ]ᴶ) → Origin′ J f
find′ ([] <⊢ ._) (m⇒ f g) = origin f g id refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) = go (_ ⊢> B) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) = go (_ ⊢> A) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B C}
→ ( I : Contextᴶ + ) (f : LG I [ B ⇒ C ]ᴶ)
→ { J : Contextᴶ + } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I f {J} g with find′ I f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (g ∘ f′) refl
Origin : ∀ {A B C} (f : LG A ⇒ B ⊢ C) → Set
Origin f = Origin′ ([] <⊢ _) f
find : ∀ {A B C} (f : LG A ⇒ B ⊢ C) → Origin f
find f = find′ ([] <⊢ _) f
module ⇐ where
data Origin′ {B C} (J : Contextᴶ +) (f : LG J [ B ⇐ C ]ᴶ) : Set where
origin : ∀ {E F} (h₁ : LG B ⊢ E)
(h₂ : LG F ⊢ C)
(f′ : ∀ {G} → LG G ⊢ E ⇐ F → LG J [ G ]ᴶ)
(pr : f ≡ f′ (m⇐ h₁ h₂))
→ Origin′ J f
mutual
find′ : ∀ {B C} (J : Contextᴶ +) (f : LG J [ B ⇐ C ]ᴶ) → Origin′ J f
find′ ([] <⊢ ._) (m⇐ f g) = origin f g id refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) = go (_ ⊢> B) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) = go (_ ⊢> A) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B C}
→ ( I : Contextᴶ + ) (f : LG I [ B ⇐ C ]ᴶ)
→ { J : Contextᴶ + } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I f {J} g with find′ I f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (g ∘ f′) refl
Origin : ∀ {A B C} (f : LG A ⇐ B ⊢ C) → Set
Origin f = Origin′ ([] <⊢ _) f
find : ∀ {A B C} (f : LG A ⇐ B ⊢ C) → Origin f
find f = find′ ([] <⊢ _) f
module ⊕ where
data Origin′ {B C} (J : Contextᴶ +) (f : LG J [ B ⊕ C ]ᴶ) : Set where
origin : ∀ {E F} (h₁ : LG B ⊢ E)
(h₂ : LG C ⊢ F)
(f′ : ∀ {G} → LG G ⊢ E ⊕ F → LG J [ G ]ᴶ)
(pr : f ≡ f′ (m⊕ h₁ h₂))
→ Origin′ J f
mutual
find′ : ∀ {B C} (J : Contextᴶ +) (f : LG J [ B ⊕ C ]ᴶ) → Origin′ J f
find′ ([] <⊢ ._) (m⊕ f g) = origin f g id refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) = go (_ ⊢> B) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) = go (_ ⊢> A) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B C}
→ ( I : Contextᴶ + ) (f : LG I [ B ⊕ C ]ᴶ)
→ { J : Contextᴶ + } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I f {J} g with find′ I f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (g ∘ f′) refl
Origin : ∀ {A B C} (f : LG A ⊕ B ⊢ C) → Set
Origin f = Origin′ ([] <⊢ _) f
find : ∀ {A B C} (f : LG A ⊕ B ⊢ C) → Origin f
find f = find′ ([] <⊢ _) f
module ⇚ where
data Origin′ {B C} (J : Contextᴶ -) (f : LG J [ B ⇚ C ]ᴶ) : Set where
origin : ∀ {E F} (h₁ : LG E ⊢ B)
(h₂ : LG C ⊢ F)
(f′ : ∀ {G} → LG E ⇚ F ⊢ G → LG J [ G ]ᴶ)
(pr : f ≡ f′ (m⇚ h₁ h₂))
→ Origin′ J f
mutual
find′ : ∀ {B C} (J : Contextᴶ -) (f : LG J [ B ⇚ C ]ᴶ) → Origin′ J f
find′ (._ ⊢> []) (m⇚ f g) = origin f g id refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) = go (_ ⊢> B) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) = go (_ ⊢> A) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B C}
→ ( I : Contextᴶ - ) (f : LG I [ B ⇚ C ]ᴶ)
→ { J : Contextᴶ - } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I f {J} g with find′ I f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (g ∘ f′) refl
Origin : ∀ {A B C} (f : LG A ⊢ B ⇚ C) → Set
Origin f = Origin′ (_ ⊢> []) f
find : ∀ {A B C} (f : LG A ⊢ B ⇚ C) → Origin f
find f = find′ (_ ⊢> []) f
module ⇛ where
data Origin′ {B C} (J : Contextᴶ -) (f : LG J [ B ⇛ C ]ᴶ) : Set where
origin : ∀ {E F} (h₁ : LG B ⊢ E)
(h₂ : LG F ⊢ C)
(f′ : ∀ {G} → LG E ⇛ F ⊢ G → LG J [ G ]ᴶ)
(pr : f ≡ f′ (m⇛ h₁ h₂))
→ Origin′ J f
mutual
find′ : ∀ {B C} (J : Contextᴶ -) (f : LG J [ B ⇛ C ]ᴶ) → Origin′ J f
find′ (._ ⊢> []) (m⇛ f g) = origin f g id refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) = go (_ ⊢> B) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) = go (_ ⊢> A) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B C}
→ ( I : Contextᴶ - ) (f : LG I [ B ⇛ C ]ᴶ)
→ { J : Contextᴶ - } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I f {J} g with find′ I f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (g ∘ f′) refl
Origin : ∀ {A B C} (f : LG A ⊢ B ⇛ C) → Set
Origin f = Origin′ (_ ⊢> []) f
find : ∀ {A B C} (f : LG A ⊢ B ⇛ C) → Origin f
find f = find′ (_ ⊢> []) f
module el where
data Origin′ {B} (J : Contextᴶ +) (f : LG J [ el B ]ᴶ) : Set where
origin : (f′ : ∀ {G} → LG G ⊢ el B → LG J [ G ]ᴶ)
(pr : f ≡ f′ ax)
→ Origin′ J f
mutual
find′ : ∀ {B} (J : Contextᴶ +) (f : LG J [ el B ]ᴶ) → Origin′ J f
find′ ([] <⊢ ._) ax = origin id refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) = go (_ ⊢> B) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) = go (_ ⊢> A) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B}
→ ( I : Contextᴶ + ) (f : LG I [ el B ]ᴶ)
→ { J : Contextᴶ + } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I x {J} g with find′ I x
... | origin f pr rewrite pr = origin (g ∘ f) refl
Origin : ∀ {A B} (f : LG el A ⊢ B) → Set
Origin f = Origin′ ([] <⊢ _) f
find : ∀ {A B} (f : LG el A ⊢ B) → Origin f
find f = find′ ([] <⊢ _) f
cut′ : ∀ {A B C} (f : LG A ⊢ B) (g : LG B ⊢ C) → LG A ⊢ C
cut′ {B = el _ } f g with el.find g
... | (el.origin g′ _) = g′ f
cut′ {B = _ ⊗ _} f g with ⊗.find f
... | (⊗.origin h₁ h₂ f′ _) = f′ (r⇐⊗ (cut′ h₁ (r⊗⇐ (r⇒⊗ (cut′ h₂ (r⊗⇒ g))))))
cut′ {B = _ ⇐ _} f g with ⇐.find g
... | (⇐.origin h₁ h₂ g′ _) = g′ (r⊗⇐ (r⇒⊗ (cut′ h₂ (r⊗⇒ (cut′ (r⇐⊗ f) h₁)))))
cut′ {B = _ ⇒ _} f g with ⇒.find g
... | (⇒.origin h₁ h₂ g′ _) = g′ (r⊗⇒ (r⇐⊗ (cut′ h₁ (r⊗⇐ (cut′ (r⇒⊗ f) h₂)))))
cut′ {B = _ ⊕ _} f g with ⊕.find g
... | (⊕.origin h₁ h₂ g′ _) = g′ (r⇚⊕ (cut′ (r⊕⇚ (r⇛⊕ (cut′ (r⊕⇛ f) h₂))) h₁))
cut′ {B = _ ⇚ _} f g with ⇚.find f
... | (⇚.origin h₁ h₂ f′ _) = f′ (r⊕⇚ (r⇛⊕ (cut′ (r⊕⇛ (cut′ h₁ (r⇚⊕ g))) h₂)))
cut′ {B = _ ⇛ _} f g with ⇛.find f
... | (⇛.origin h₁ h₂ f′ _) = f′ (r⊕⇛ (r⇚⊕ (cut′ (r⊕⇚ (cut′ h₂ (r⇛⊕ g))) h₁)))
module translation (⌈_⌉ᴬ : Atom → Set) (R : Set) where
infixr 9 ¬_
¬_ : Set → Set
¬ A = A → R
⌈_⌉ : Type → Set
⌈ el A ⌉ = ⌈ A ⌉ᴬ
⌈ A ⊗ B ⌉ = ( ⌈ A ⌉ × ⌈ B ⌉)
⌈ A ⇒ B ⌉ = ¬ ( ⌈ A ⌉ × ¬ ⌈ B ⌉)
⌈ B ⇐ A ⌉ = ¬ (¬ ⌈ B ⌉ × ⌈ A ⌉)
⌈ B ⊕ A ⌉ = ¬ (¬ ⌈ B ⌉ × ¬ ⌈ A ⌉)
⌈ B ⇚ A ⌉ = ( ⌈ B ⌉ × ¬ ⌈ A ⌉)
⌈ A ⇛ B ⌉ = (¬ ⌈ A ⌉ × ⌈ B ⌉)
deMorgan : {A B : Set} → (¬ ¬ A) → (¬ ¬ B) → ¬ ¬ (A × B)
deMorgan c₁ c₂ k = c₁ (λ x → c₂ (λ y → k (x , y)))
mutual
⌈_⌉ᴸ : ∀ {A B} → LG A ⊢ B → ¬ ⌈ B ⌉ → ¬ ⌈ A ⌉
⌈_⌉ᴿ : ∀ {A B} → LG A ⊢ B → ⌈ A ⌉ → ¬ ¬ ⌈ B ⌉
⌈ ax ⌉ᴸ k x = k x
⌈ r⇒⊗ f ⌉ᴸ x = λ{(y , z) → ⌈ f ⌉ᴸ (λ k → k (y , x)) z}
⌈ r⊗⇒ f ⌉ᴸ k x = k (λ{(y , z) → ⌈ f ⌉ᴸ z (y , x)})
⌈ r⇐⊗ f ⌉ᴸ x = λ{(y , z) → ⌈ f ⌉ᴸ (λ k → k (x , z)) y}
⌈ r⊗⇐ f ⌉ᴸ k x = k (λ{(y , z) → ⌈ f ⌉ᴸ y (x , z)})
⌈ m⊗ f g ⌉ᴸ k = λ{(x , y) → deMorgan (⌈ f ⌉ᴿ x) (⌈ g ⌉ᴿ y) k}
⌈ m⇒ f g ⌉ᴸ k k′ = k (λ{(x , y) → deMorgan (⌈ f ⌉ᴿ x) (λ k → k (⌈ g ⌉ᴸ y)) k′})
⌈ m⇐ f g ⌉ᴸ k k′ = k (λ{(x , y) → deMorgan (λ k → k (⌈ f ⌉ᴸ x)) (⌈ g ⌉ᴿ y) k′})
⌈ r⇛⊕ f ⌉ᴸ k x = k (λ{(y , z) → ⌈ f ⌉ᴸ z (y , x)})
⌈ r⊕⇛ f ⌉ᴸ x = λ{(y , z) → ⌈ f ⌉ᴸ (λ k → k (y , x)) z}
⌈ r⇚⊕ f ⌉ᴸ k x = k (λ{(y , z) → ⌈ f ⌉ᴸ y (x , z)})
⌈ r⊕⇚ f ⌉ᴸ x = λ{(y , z) → ⌈ f ⌉ᴸ (λ k → k (x , z)) y}
⌈ m⊕ f g ⌉ᴸ k k′ = k (λ{(x , y) → k′ (⌈ f ⌉ᴸ x , ⌈ g ⌉ᴸ y)})
⌈ m⇛ f g ⌉ᴸ k = λ{(x , y) → deMorgan (λ k → k (⌈ f ⌉ᴸ x)) (⌈ g ⌉ᴿ y) k}
⌈ m⇚ f g ⌉ᴸ k = λ{(x , y) → deMorgan (⌈ f ⌉ᴿ x) (λ k → k (⌈ g ⌉ᴸ y)) k}
⌈ d⇛⇐ f ⌉ᴸ k = λ{(x , y) → k (λ{(z , w) → ⌈ f ⌉ᴸ (λ k → k (x , z)) (y , w)})}
⌈ d⇛⇒ f ⌉ᴸ k = λ{(x , y) → k (λ{(z , w) → ⌈ f ⌉ᴸ (λ k → k (x , w)) (z , y)})}
⌈ d⇚⇒ f ⌉ᴸ k = λ{(x , y) → k (λ{(z , w) → ⌈ f ⌉ᴸ (λ k → k (w , y)) (z , x)})}
⌈ d⇚⇐ f ⌉ᴸ k = λ{(x , y) → k (λ{(z , w) → ⌈ f ⌉ᴸ (λ k → k (z , y)) (x , w)})}
⌈ ax ⌉ᴿ x k = k x
⌈ r⇒⊗ f ⌉ᴿ (x , y) z = ⌈ f ⌉ᴿ y (λ k → k (x , z))
⌈ r⊗⇒ f ⌉ᴿ x k = k (λ{(y , z) → ⌈ f ⌉ᴿ (y , x) z})
⌈ r⇐⊗ f ⌉ᴿ (x , y) z = ⌈ f ⌉ᴿ x (λ k → k (z , y))
⌈ r⊗⇐ f ⌉ᴿ x k = k (λ{(y , z) → ⌈ f ⌉ᴿ (x , z) y})
⌈ m⊗ f g ⌉ᴿ (x , y) k = deMorgan (⌈ f ⌉ᴿ x) (⌈ g ⌉ᴿ y) k
⌈ m⇒ f g ⌉ᴿ k′ k = k (λ{(x , y) → deMorgan (⌈ f ⌉ᴿ x) (λ k → k (⌈ g ⌉ᴸ y)) k′})
⌈ m⇐ f g ⌉ᴿ k′ k = k (λ{(x , y) → deMorgan (λ k → k (⌈ f ⌉ᴸ x)) (⌈ g ⌉ᴿ y) k′})
⌈ r⇛⊕ f ⌉ᴿ x k = k (λ{(y , z) → ⌈ f ⌉ᴿ (y , x) z})
⌈ r⊕⇛ f ⌉ᴿ (x , y) z = ⌈ f ⌉ᴿ y (λ k → k (x , z))
⌈ r⊕⇚ f ⌉ᴿ (x , y) z = ⌈ f ⌉ᴿ x (λ k → k (z , y))
⌈ r⇚⊕ f ⌉ᴿ x k = k (λ{(y , z) → ⌈ f ⌉ᴿ (x , z) y})
⌈ m⊕ f g ⌉ᴿ k′ k = k (λ{(x , y) → k′ (⌈ f ⌉ᴸ x , ⌈ g ⌉ᴸ y)})
⌈ m⇛ f g ⌉ᴿ (x , y) k = deMorgan (λ k → k (⌈ f ⌉ᴸ x)) (⌈ g ⌉ᴿ y) k
⌈ m⇚ f g ⌉ᴿ (x , y) k = deMorgan (⌈ f ⌉ᴿ x) (λ k → k (⌈ g ⌉ᴸ y)) k
⌈ d⇛⇐ f ⌉ᴿ (x , y) k = k (λ{(z , w) → ⌈ f ⌉ᴿ (y , w) (λ k → k (x , z))})
⌈ d⇛⇒ f ⌉ᴿ (x , y) k = k (λ{(z , w) → ⌈ f ⌉ᴿ (z , y) (λ k → k (x , w))})
⌈ d⇚⇒ f ⌉ᴿ (x , y) k = k (λ{(z , w) → ⌈ f ⌉ᴿ (z , x) (λ k → k (w , y))})
⌈ d⇚⇐ f ⌉ᴿ (x , y) k = k (λ{(z , w) → ⌈ f ⌉ᴿ (x , w) (λ k → k (z , y))})
module example where
postulate
Entity : Set
forAll : (Entity → Bool) → Bool
exists : (Entity → Bool) → Bool
ʟᴏᴠᴇs : Entity → Entity → Bool
ᴘᴇʀsᴏɴ : Entity → Bool
_⊃_ : Bool → Bool → Bool
data Atom : Set where N NP S : Atom
⌈_⌉ᴬ : Atom → Set
⌈ N ⌉ᴬ = Entity → Bool
⌈ NP ⌉ᴬ = Entity
⌈ S ⌉ᴬ = Bool
open logic Atom
open logic.translation Atom ⌈_⌉ᴬ Bool
n np s : Type
n = el N
np = el NP
s = el S
someone : ⌈ (np ⇐ n) ⊗ n ⌉
someone = ((λ{(g , f) → exists (λ x → f x ∧ g x)}) , ᴘᴇʀsᴏɴ)
loves : ⌈ (np ⇒ s) ⇐ np ⌉
loves = λ{(k , y) → k (λ{(x , k) → k (ʟᴏᴠᴇs x y)})}
everyone : ⌈ (np ⇐ n) ⊗ n ⌉
everyone = ((λ{(g , f) → forAll (λ x → f x ⊃ g x)}) , ᴘᴇʀsᴏɴ)
sᴇɴᴛ₀ sᴇɴᴛ₁ sᴇɴᴛ₂ sᴇɴᴛ₃ sᴇɴᴛ₄ sᴇɴᴛ₅ sᴇɴᴛ₆ : LG
( ( np ⇐ n ) ⊗ n ) ⊗ ( ( ( np ⇒ s ) ⇐ np ) ⊗ ( ( np ⇐ n ) ⊗ n ) ) ⊢ s
open import Data.Bool using (Bool; true; false)
open import Data.Empty using (⊥)
open import Data.List using (List; _∷_; [])
open import Data.String using (String; _++_)
open import Data.Unit using (⊤)
open import Relation.Nullary.Decidable using (⌊_⌋)
open import Reflection
mutual
dropAbs : Term → Term
dropAbs (var x args) = var x (dropAbsArgs args)
dropAbs (con c args) = con c (dropAbsArgs args)
dropAbs (def f args) = def f (dropAbsArgs args)
dropAbs (lam v (abs _ x)) = lam v (abs "_" (dropAbs x))
dropAbs (pi (arg i (el s₁ t₁)) (abs _ (el s₂ t₂))) = pi (arg i (el s₁ (dropAbs t₁))) (abs "_" (el s₂ (dropAbs t₂)))
dropAbs t = t
dropAbsArgs : List (Arg Term) → List (Arg Term)
dropAbsArgs [] = []
dropAbsArgs (arg i x ∷ args) = arg i (dropAbs x) ∷ dropAbsArgs args
assert : Bool → Term
assert true = def (quote ⊤) []
assert false = def (quote ⊥) []
infix 0 _↦_
macro
_↦_ : Term → Term → Term
act ↦ exp = assert ⌊ dropAbs act ≟ dropAbs exp ⌋
sᴇɴᴛ₀ = r⇒⊗ (r⇐⊗ (m⇐ (m⇒ (r⇐⊗ ax′) ax) (r⇐⊗ ax′)))
sᴇɴᴛ₁ = r⇐⊗ (r⇐⊗ (m⇐ (r⊗⇐ (r⇒⊗ (r⇐⊗ (m⇐ ax′ (r⇐⊗ ax′))))) ax))
sᴇɴᴛ₂ = r⇒⊗ (r⇒⊗ (r⇐⊗ (m⇐ (r⊗⇒ (r⇐⊗ (m⇐ (m⇒ (r⇐⊗ ax′) ax) ax))) ax)))
sᴇɴᴛ₃ = r⇐⊗ (r⇐⊗ (m⇐ (r⊗⇐ (r⇒⊗ (r⇒⊗ (r⇐⊗ (m⇐ (r⊗⇒ (r⇐⊗ ax′)) ax))))) ax))
sᴇɴᴛ₄ = r⇒⊗ (r⇐⊗ (m⇐ (r⊗⇒ (r⇐⊗ (r⇐⊗ (m⇐ (r⊗⇐ (r⇒⊗ ax′)) ax)))) (r⇐⊗ ax′)))
sᴇɴᴛ₅ = r⇒⊗ (r⇒⊗ (r⇐⊗ (m⇐ (r⊗⇒ (r⇐⊗ (m⇐ (r⊗⇒ (r⇐⊗ (r⇐⊗ (m⇐ (r⊗⇐ (r⇒⊗ ax′)) ax)))) ax))) ax)))
sᴇɴᴛ₆ = r⇒⊗ (r⇒⊗ (r⇐⊗ (m⇐ (r⊗⇒ (r⊗⇒ (r⇐⊗ (r⇐⊗ (m⇐ (r⊗⇐ (r⇒⊗ (r⇐⊗ ax′))) ax))))) ax)))
sent₀ : ⌈ sᴇɴᴛ₀ ⌉ᴿ (someone , loves , everyone) id ↦ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ exists (λ x → ᴘᴇʀsᴏɴ x ∧ ʟᴏᴠᴇs x y))
sent₁ : ⌈ sᴇɴᴛ₁ ⌉ᴿ (someone , loves , everyone) id ↦ exists (λ x → ᴘᴇʀsᴏɴ x ∧ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ ʟᴏᴠᴇs x y))
sent₂ : ⌈ sᴇɴᴛ₂ ⌉ᴿ (someone , loves , everyone) id ↦ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ exists (λ x → ᴘᴇʀsᴏɴ x ∧ ʟᴏᴠᴇs x y))
sent₃ : ⌈ sᴇɴᴛ₃ ⌉ᴿ (someone , loves , everyone) id ↦ exists (λ x → ᴘᴇʀsᴏɴ x ∧ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ ʟᴏᴠᴇs x y))
sent₄ : ⌈ sᴇɴᴛ₄ ⌉ᴿ (someone , loves , everyone) id ↦ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ exists (λ x → ᴘᴇʀsᴏɴ x ∧ ʟᴏᴠᴇs x y))
sent₅ : ⌈ sᴇɴᴛ₅ ⌉ᴿ (someone , loves , everyone) id ↦ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ exists (λ x → ᴘᴇʀsᴏɴ x ∧ ʟᴏᴠᴇs x y))
sent₆ : ⌈ sᴇɴᴛ₆ ⌉ᴿ (someone , loves , everyone) id ↦ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ exists (λ x → ᴘᴇʀsᴏɴ x ∧ ʟᴏᴠᴇs x y))
sent₀ = _
sent₁ = _
sent₂ = _
sent₃ = _
sent₄ = _
sent₅ = _
sent₆ = _
Raw
my.agda
-- This file contains the code from the paper "Formalising
-- type-logical grammars in Agda", and was directly extracted from the
-- paper's source.
--
-- Note: because the symbol customarily used for the "left difference"
-- is unavailable in the Unicode character set, and the backslash used
-- for implication is often a reserved symbol, the source file uses a
-- different notation for the connectives: the left and right
-- implications are represented by the double arrows "⇐" and "⇒", and
-- the left and right differences are represented by the triple arrows
-- "⇚" ad "⇛".
module main where
open import Function using (id; flip; _∘_)
open import Data.Bool using (Bool; true; false; _∧_)
open import Data.Empty using (⊥)
open import Data.Product using (_×_; _,_)
open import Data.Unit using (⊤)
open import Reflection using (Term; _≟_)
open import Relation.Binary.PropositionalEquality using (_≡_; refl)
open import Relation.Nullary.Decidable using (⌊_⌋)
module logic (Atom : Set) where
data Type : Set where
el : Atom → Type
_⊗_ _⇒_ _⇐_ : Type → Type → Type
_⊕_ _⇚_ _⇛_ : Type → Type → Type
data Judgement : Set where
_⊢_ : Type → Type → Judgement
infix 1 LG_
infix 2 _⊢_
infixl 3 _[_]
infixl 3 _[_]ᴶ
infixr 4 _⊗_
infixr 5 _⊕_
infixl 6 _⇚_
infixr 6 _⇛_
infixr 7 _⇒_
infixl 7 _⇐_
data LG_ : Judgement → Set where
ax : ∀ {A} → LG el A ⊢ el A
-- residuation and monotonicity for (⇐ , ⊗ , ⇒)
r⇒⊗ : ∀ {A B C} → LG B ⊢ A ⇒ C → LG A ⊗ B ⊢ C
r⊗⇒ : ∀ {A B C} → LG A ⊗ B ⊢ C → LG B ⊢ A ⇒ C
r⇐⊗ : ∀ {A B C} → LG A ⊢ C ⇐ B → LG A ⊗ B ⊢ C
r⊗⇐ : ∀ {A B C} → LG A ⊗ B ⊢ C → LG A ⊢ C ⇐ B
m⊗ : ∀ {A B C D} → LG A ⊢ B → LG C ⊢ D → LG A ⊗ C ⊢ B ⊗ D
m⇒ : ∀ {A B C D} → LG A ⊢ B → LG C ⊢ D → LG B ⇒ C ⊢ A ⇒ D
m⇐ : ∀ {A B C D} → LG A ⊢ B → LG C ⊢ D → LG A ⇐ D ⊢ B ⇐ C
-- residuation and monotonicity for (⇚ , ⊕ , ⇛)
r⇛⊕ : ∀ {A B C} → LG B ⇛ C ⊢ A → LG C ⊢ B ⊕ A
r⊕⇛ : ∀ {A B C} → LG C ⊢ B ⊕ A → LG B ⇛ C ⊢ A
r⊕⇚ : ∀ {A B C} → LG C ⊢ B ⊕ A → LG C ⇚ A ⊢ B
r⇚⊕ : ∀ {A B C} → LG C ⇚ A ⊢ B → LG C ⊢ B ⊕ A
m⊕ : ∀ {A B C D} → LG A ⊢ B → LG C ⊢ D → LG A ⊕ C ⊢ B ⊕ D
m⇛ : ∀ {A B C D} → LG C ⊢ D → LG A ⊢ B → LG D ⇛ A ⊢ C ⇛ B
m⇚ : ∀ {A B C D} → LG A ⊢ B → LG C ⊢ D → LG A ⇚ D ⊢ B ⇚ C
-- grishin distributives
d⇛⇐ : ∀ {A B C D} → LG A ⊗ B ⊢ C ⊕ D → LG C ⇛ A ⊢ D ⇐ B
d⇛⇒ : ∀ {A B C D} → LG A ⊗ B ⊢ C ⊕ D → LG C ⇛ B ⊢ A ⇒ D
d⇚⇒ : ∀ {A B C D} → LG A ⊗ B ⊢ C ⊕ D → LG B ⇚ D ⊢ A ⇒ C
d⇚⇐ : ∀ {A B C D} → LG A ⊗ B ⊢ C ⊕ D → LG A ⇚ D ⊢ C ⇐ B
ax′ : ∀ {A} → LG A ⊢ A
ax′ {A = el _} = ax
ax′ {A = _ ⊗ _} = m⊗ ax′ ax′
ax′ {A = _ ⇐ _} = m⇐ ax′ ax′
ax′ {A = _ ⇒ _} = m⇒ ax′ ax′
ax′ {A = _ ⊕ _} = m⊕ ax′ ax′
ax′ {A = _ ⇚ _} = m⇚ ax′ ax′
ax′ {A = _ ⇛ _} = m⇛ ax′ ax′
appl-⇒′ : ∀ {A B} → LG A ⊗ (A ⇒ B) ⊢ B
appl-⇒′ = r⇒⊗ (m⇒ ax′ ax′)
appl-⇐′ : ∀ {A B} → LG (B ⇐ A) ⊗ A ⊢ B
appl-⇐′ = r⇐⊗ (m⇐ ax′ ax′)
appl-⇛′ : ∀ {A B} → LG B ⊢ A ⊕ (A ⇛ B)
appl-⇛′ = r⇛⊕ (m⇛ ax′ ax′)
appl-⇚′ : ∀ {A B} → LG B ⊢ (B ⇚ A) ⊕ A
appl-⇚′ = r⇚⊕ (m⇚ ax′ ax′)
data Polarity : Set where + - : Polarity
data Context (p : Polarity) : Polarity → Set where
[] : Context p p
_⊗>_ : Type → Context p + → Context p +
_⇒>_ : Type → Context p - → Context p -
_⇐>_ : Type → Context p + → Context p -
_<⊗_ : Context p + → Type → Context p +
_<⇒_ : Context p + → Type → Context p -
_<⇐_ : Context p - → Type → Context p -
_⊕>_ : Type → Context p - → Context p -
_⇚>_ : Type → Context p - → Context p +
_⇛>_ : Type → Context p + → Context p +
_<⊕_ : Context p - → Type → Context p -
_<⇚_ : Context p + → Type → Context p +
_<⇛_ : Context p - → Type → Context p +
data Contextᴶ (p : Polarity) : Set where
_<⊢_ : Context p + → Type → Contextᴶ p
_⊢>_ : Type → Context p - → Contextᴶ p
_[_] : ∀ {p₁ p₂} → Context p₁ p₂ → Type → Type
[] [ A ] = A
(B ⊗> C) [ A ] = B ⊗ (C [ A ])
(C <⊗ B) [ A ] = (C [ A ]) ⊗ B
(B ⇒> C) [ A ] = B ⇒ (C [ A ])
(C <⇒ B) [ A ] = (C [ A ]) ⇒ B
(B ⇐> C) [ A ] = B ⇐ (C [ A ])
(C <⇐ B) [ A ] = (C [ A ]) ⇐ B
(B ⊕> C) [ A ] = B ⊕ (C [ A ])
(C <⊕ B) [ A ] = (C [ A ]) ⊕ B
(B ⇚> C) [ A ] = B ⇚ (C [ A ])
(C <⇚ B) [ A ] = (C [ A ]) ⇚ B
(B ⇛> C) [ A ] = B ⇛ (C [ A ])
(C <⇛ B) [ A ] = (C [ A ]) ⇛ B
_[_]ᴶ : ∀ {p} → Contextᴶ p → Type → Judgement
(A <⊢ B) [ C ]ᴶ = A [ C ] ⊢ B
(A ⊢> B) [ C ]ᴶ = A ⊢ B [ C ]
module ⊗ where
data Origin′ {B C} (J : Contextᴶ -)
(f : LG J [ B ⊗ C ]ᴶ)
: Set where
origin : ∀ {E F} (h₁ : LG E ⊢ B)
(h₂ : LG F ⊢ C)
(f′ : ∀ {G} → LG E ⊗ F ⊢ G → LG J [ G ]ᴶ)
(pr : f ≡ f′ (m⊗ h₁ h₂))
→ Origin′ J f
mutual
find′ : ∀ {B C} (J : Contextᴶ -) (f : LG J [ B ⊗ C ]ᴶ) → Origin′ J f
find′ (._ ⊢> []) (m⊗ f g) = origin f g id refl
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) with find′ (_ ⊢> A) f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (r⊗⇐ ∘ f′) refl
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) with find′ (_ ⊢> B) f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (r⊗⇒ ∘ f′) refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B C}
→ ( I : Contextᴶ - ) (f : LG I [ B ⊗ C ]ᴶ)
→ { J : Contextᴶ - } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I f {J} g with find′ I f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (g ∘ f′) refl
Origin : ∀ {A B C} (f : LG A ⊢ B ⊗ C) → Set
Origin f = Origin′ (_ ⊢> []) f
find : ∀ {A B C} (f : LG A ⊢ B ⊗ C) → Origin f
find f = find′ (_ ⊢> []) f
module ⇒ where
data Origin′ {B C} (J : Contextᴶ +) (f : LG J [ B ⇒ C ]ᴶ) : Set where
origin : ∀ {E F} (h₁ : LG E ⊢ B)
(h₂ : LG C ⊢ F)
(f′ : ∀ {G} → LG G ⊢ E ⇒ F → LG J [ G ]ᴶ)
(pr : f ≡ f′ (m⇒ h₁ h₂))
→ Origin′ J f
mutual
find′ : ∀ {B C} (J : Contextᴶ +) (f : LG J [ B ⇒ C ]ᴶ) → Origin′ J f
find′ ([] <⊢ ._) (m⇒ f g) = origin f g id refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) = go (_ ⊢> B) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) = go (_ ⊢> A) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B C}
→ ( I : Contextᴶ + ) (f : LG I [ B ⇒ C ]ᴶ)
→ { J : Contextᴶ + } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I f {J} g with find′ I f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (g ∘ f′) refl
Origin : ∀ {A B C} (f : LG A ⇒ B ⊢ C) → Set
Origin f = Origin′ ([] <⊢ _) f
find : ∀ {A B C} (f : LG A ⇒ B ⊢ C) → Origin f
find f = find′ ([] <⊢ _) f
module ⇐ where
data Origin′ {B C} (J : Contextᴶ +) (f : LG J [ B ⇐ C ]ᴶ) : Set where
origin : ∀ {E F} (h₁ : LG B ⊢ E)
(h₂ : LG F ⊢ C)
(f′ : ∀ {G} → LG G ⊢ E ⇐ F → LG J [ G ]ᴶ)
(pr : f ≡ f′ (m⇐ h₁ h₂))
→ Origin′ J f
mutual
find′ : ∀ {B C} (J : Contextᴶ +) (f : LG J [ B ⇐ C ]ᴶ) → Origin′ J f
find′ ([] <⊢ ._) (m⇐ f g) = origin f g id refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) = go (_ ⊢> B) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) = go (_ ⊢> A) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B C}
→ ( I : Contextᴶ + ) (f : LG I [ B ⇐ C ]ᴶ)
→ { J : Contextᴶ + } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I f {J} g with find′ I f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (g ∘ f′) refl
Origin : ∀ {A B C} (f : LG A ⇐ B ⊢ C) → Set
Origin f = Origin′ ([] <⊢ _) f
find : ∀ {A B C} (f : LG A ⇐ B ⊢ C) → Origin f
find f = find′ ([] <⊢ _) f
module ⊕ where
data Origin′ {B C} (J : Contextᴶ +) (f : LG J [ B ⊕ C ]ᴶ) : Set where
origin : ∀ {E F} (h₁ : LG B ⊢ E)
(h₂ : LG C ⊢ F)
(f′ : ∀ {G} → LG G ⊢ E ⊕ F → LG J [ G ]ᴶ)
(pr : f ≡ f′ (m⊕ h₁ h₂))
→ Origin′ J f
mutual
find′ : ∀ {B C} (J : Contextᴶ +) (f : LG J [ B ⊕ C ]ᴶ) → Origin′ J f
find′ ([] <⊢ ._) (m⊕ f g) = origin f g id refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) = go (_ ⊢> B) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) = go (_ ⊢> A) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B C}
→ ( I : Contextᴶ + ) (f : LG I [ B ⊕ C ]ᴶ)
→ { J : Contextᴶ + } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I f {J} g with find′ I f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (g ∘ f′) refl
Origin : ∀ {A B C} (f : LG A ⊕ B ⊢ C) → Set
Origin f = Origin′ ([] <⊢ _) f
find : ∀ {A B C} (f : LG A ⊕ B ⊢ C) → Origin f
find f = find′ ([] <⊢ _) f
module ⇚ where
data Origin′ {B C} (J : Contextᴶ -) (f : LG J [ B ⇚ C ]ᴶ) : Set where
origin : ∀ {E F} (h₁ : LG E ⊢ B)
(h₂ : LG C ⊢ F)
(f′ : ∀ {G} → LG E ⇚ F ⊢ G → LG J [ G ]ᴶ)
(pr : f ≡ f′ (m⇚ h₁ h₂))
→ Origin′ J f
mutual
find′ : ∀ {B C} (J : Contextᴶ -) (f : LG J [ B ⇚ C ]ᴶ) → Origin′ J f
find′ (._ ⊢> []) (m⇚ f g) = origin f g id refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) = go (_ ⊢> B) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) = go (_ ⊢> A) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B C}
→ ( I : Contextᴶ - ) (f : LG I [ B ⇚ C ]ᴶ)
→ { J : Contextᴶ - } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I f {J} g with find′ I f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (g ∘ f′) refl
Origin : ∀ {A B C} (f : LG A ⊢ B ⇚ C) → Set
Origin f = Origin′ (_ ⊢> []) f
find : ∀ {A B C} (f : LG A ⊢ B ⇚ C) → Origin f
find f = find′ (_ ⊢> []) f
module ⇛ where
data Origin′ {B C} (J : Contextᴶ -) (f : LG J [ B ⇛ C ]ᴶ) : Set where
origin : ∀ {E F} (h₁ : LG B ⊢ E)
(h₂ : LG F ⊢ C)
(f′ : ∀ {G} → LG E ⇛ F ⊢ G → LG J [ G ]ᴶ)
(pr : f ≡ f′ (m⇛ h₁ h₂))
→ Origin′ J f
mutual
find′ : ∀ {B C} (J : Contextᴶ -) (f : LG J [ B ⇛ C ]ᴶ) → Origin′ J f
find′ (._ ⊢> []) (m⇛ f g) = origin f g id refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) = go (_ ⊢> B) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) = go (_ ⊢> A) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B C}
→ ( I : Contextᴶ - ) (f : LG I [ B ⇛ C ]ᴶ)
→ { J : Contextᴶ - } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I f {J} g with find′ I f
... | origin h₁ h₂ f′ pr rewrite pr = origin h₁ h₂ (g ∘ f′) refl
Origin : ∀ {A B C} (f : LG A ⊢ B ⇛ C) → Set
Origin f = Origin′ (_ ⊢> []) f
find : ∀ {A B C} (f : LG A ⊢ B ⇛ C) → Origin f
find f = find′ (_ ⊢> []) f
module el where
data Origin′ {B} (J : Contextᴶ +) (f : LG J [ el B ]ᴶ) : Set where
origin : (f′ : ∀ {G} → LG G ⊢ el B → LG J [ G ]ᴶ)
(pr : f ≡ f′ ax)
→ Origin′ J f
mutual
find′ : ∀ {B} (J : Contextᴶ +) (f : LG J [ el B ]ᴶ) → Origin′ J f
find′ ([] <⊢ ._) ax = origin id refl
find′ ((A <⊗ _) <⊢ ._) (m⊗ f g) = go (A <⊢ _) f (flip m⊗ g)
find′ ((_ ⊗> B) <⊢ ._) (m⊗ f g) = go (B <⊢ _) g (m⊗ f)
find′ (._ ⊢> (_ ⇒> B)) (m⇒ f g) = go (_ ⊢> B) g (m⇒ f)
find′ (._ ⊢> (A <⇒ _)) (m⇒ f g) = go (A <⊢ _) f (flip m⇒ g)
find′ (._ ⊢> (_ ⇐> B)) (m⇐ f g) = go (B <⊢ _) g (m⇐ f)
find′ (._ ⊢> (A <⇐ _)) (m⇐ f g) = go (_ ⊢> A) f (flip m⇐ g)
find′ (A <⊢ ._) (r⊗⇒ f) = go ((_ ⊗> A) <⊢ _) f r⊗⇒
find′ (._ ⊢> (_ ⇒> B)) (r⊗⇒ f) = go (_ ⊢> B) f r⊗⇒
find′ (._ ⊢> (A <⇒ _)) (r⊗⇒ f) = go ((A <⊗ _) <⊢ _) f r⊗⇒
find′ ((_ ⊗> B) <⊢ ._) (r⇒⊗ f) = go (B <⊢ _) f r⇒⊗
find′ ((A <⊗ _) <⊢ ._) (r⇒⊗ f) = go (_ ⊢> (A <⇒ _)) f r⇒⊗
find′ (._ ⊢> B) (r⇒⊗ f) = go (_ ⊢> (_ ⇒> B)) f r⇒⊗
find′ (A <⊢ ._) (r⊗⇐ f) = go ((A <⊗ _) <⊢ _) f r⊗⇐
find′ (._ ⊢> (_ ⇐> B)) (r⊗⇐ f) = go ((_ ⊗> B) <⊢ _) f r⊗⇐
find′ (._ ⊢> (A <⇐ _)) (r⊗⇐ f) = go (_ ⊢> A) f r⊗⇐
find′ ((_ ⊗> B) <⊢ ._) (r⇐⊗ f) = go (_ ⊢> (_ ⇐> B)) f r⇐⊗
find′ ((A <⊗ _) <⊢ ._) (r⇐⊗ f) = go (A <⊢ _) f r⇐⊗
find′ (._ ⊢> B) (r⇐⊗ f) = go (_ ⊢> (B <⇐ _)) f r⇐⊗
find′ ((A <⇚ _) <⊢ ._) (m⇚ f g) = go (A <⊢ _) f (flip m⇚ g)
find′ ((_ ⇚> B) <⊢ ._) (m⇚ f g) = go (_ ⊢> B) g (m⇚ f)
find′ ((A <⇛ _) <⊢ ._) (m⇛ f g) = go (_ ⊢> A) f (flip m⇛ g)
find′ ((_ ⇛> B) <⊢ ._) (m⇛ f g) = go (B <⊢ _) g (m⇛ f)
find′ (._ ⊢> (A <⊕ _)) (m⊕ f g) = go (_ ⊢> A) f (flip m⊕ g)
find′ (._ ⊢> (_ ⊕> B)) (m⊕ f g) = go (_ ⊢> B) g (m⊕ f)
find′ (A <⊢ ._) (r⇚⊕ f) = go ((A <⇚ _) <⊢ _) f r⇚⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇚⊕ f) = go ((_ ⇚> B) <⊢ _) f r⇚⊕
find′ (._ ⊢> (A <⊕ _)) (r⇚⊕ f) = go (_ ⊢> A) f r⇚⊕
find′ ((A <⇚ _) <⊢ ._) (r⊕⇚ f) = go (A <⊢ _) f r⊕⇚
find′ ((_ ⇚> B) <⊢ ._) (r⊕⇚ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇚
find′ (._ ⊢> B) (r⊕⇚ f) = go (_ ⊢> (B <⊕ _)) f r⊕⇚
find′ (A <⊢ ._) (r⇛⊕ f) = go ((_ ⇛> A) <⊢ _) f r⇛⊕
find′ (._ ⊢> (_ ⊕> B)) (r⇛⊕ f) = go (_ ⊢> B) f r⇛⊕
find′ (._ ⊢> (A <⊕ _)) (r⇛⊕ f) = go ((A <⇛ _) <⊢ _) f r⇛⊕
find′ ((A <⇛ _) <⊢ ._) (r⊕⇛ f) = go (_ ⊢> (A <⊕ _)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (r⊕⇛ f) = go (B <⊢ _) f r⊕⇛
find′ (._ ⊢> B) (r⊕⇛ f) = go (_ ⊢> (_ ⊕> B)) f r⊕⇛
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇐ f) = go ((B <⊗ _) <⊢ _) f d⇛⇐
find′ ((A <⇛ _) <⊢ ._) (d⇛⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇐
find′ (._ ⊢> (A <⇐ _)) (d⇛⇐ f) = go (_ ⊢> (_ ⊕> A)) f d⇛⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇛⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇐
find′ ((_ ⇛> B) <⊢ ._) (d⇛⇒ f) = go ((_ ⊗> B) <⊢ _) f d⇛⇒
find′ ((A <⇛ _) <⊢ ._) (d⇛⇒ f) = go (_ ⊢> (A <⊕ _)) f d⇛⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇛⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇛⇒
find′ (._ ⊢> (A <⇒ _)) (d⇛⇒ f) = go ((A <⊗ _) <⊢ _) f d⇛⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇒ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇒
find′ ((A <⇚ _) <⊢ ._) (d⇚⇒ f) = go ((_ ⊗> A) <⊢ _) f d⇚⇒
find′ (._ ⊢> (_ ⇒> B)) (d⇚⇒ f) = go (_ ⊢> (B <⊕ _)) f d⇚⇒
find′ (._ ⊢> (A <⇒ _)) (d⇚⇒ f) = go ((A <⊗ _) <⊢ _) f d⇚⇒
find′ ((_ ⇚> B) <⊢ ._) (d⇚⇐ f) = go (_ ⊢> (_ ⊕> B)) f d⇚⇐
find′ ((A <⇚ _) <⊢ ._) (d⇚⇐ f) = go ((A <⊗ _) <⊢ _) f d⇚⇐
find′ (._ ⊢> (_ ⇐> B)) (d⇚⇐ f) = go ((_ ⊗> B) <⊢ _) f d⇚⇐
find′ (._ ⊢> (A <⇐ _)) (d⇚⇐ f) = go (_ ⊢> (A <⊕ _)) f d⇚⇐
private
go : ∀ {B}
→ ( I : Contextᴶ + ) (f : LG I [ el B ]ᴶ)
→ { J : Contextᴶ + } (g : ∀ {G} → LG I [ G ]ᴶ → LG J [ G ]ᴶ)
→ Origin′ J (g f)
go I x {J} g with find′ I x
... | origin f pr rewrite pr = origin (g ∘ f) refl
Origin : ∀ {A B} (f : LG el A ⊢ B) → Set
Origin f = Origin′ ([] <⊢ _) f
find : ∀ {A B} (f : LG el A ⊢ B) → Origin f
find f = find′ ([] <⊢ _) f
cut′ : ∀ {A B C} (f : LG A ⊢ B) (g : LG B ⊢ C) → LG A ⊢ C
cut′ {B = el _ } f g with el.find g
... | (el.origin g′ _) = g′ f
cut′ {B = _ ⊗ _} f g with ⊗.find f
... | (⊗.origin h₁ h₂ f′ _) = f′ (r⇐⊗ (cut′ h₁ (r⊗⇐ (r⇒⊗ (cut′ h₂ (r⊗⇒ g))))))
cut′ {B = _ ⇐ _} f g with ⇐.find g
... | (⇐.origin h₁ h₂ g′ _) = g′ (r⊗⇐ (r⇒⊗ (cut′ h₂ (r⊗⇒ (cut′ (r⇐⊗ f) h₁)))))
cut′ {B = _ ⇒ _} f g with ⇒.find g
... | (⇒.origin h₁ h₂ g′ _) = g′ (r⊗⇒ (r⇐⊗ (cut′ h₁ (r⊗⇐ (cut′ (r⇒⊗ f) h₂)))))
cut′ {B = _ ⊕ _} f g with ⊕.find g
... | (⊕.origin h₁ h₂ g′ _) = g′ (r⇚⊕ (cut′ (r⊕⇚ (r⇛⊕ (cut′ (r⊕⇛ f) h₂))) h₁))
cut′ {B = _ ⇚ _} f g with ⇚.find f
... | (⇚.origin h₁ h₂ f′ _) = f′ (r⊕⇚ (r⇛⊕ (cut′ (r⊕⇛ (cut′ h₁ (r⇚⊕ g))) h₂)))
cut′ {B = _ ⇛ _} f g with ⇛.find f
... | (⇛.origin h₁ h₂ f′ _) = f′ (r⊕⇛ (r⇚⊕ (cut′ (r⊕⇚ (cut′ h₂ (r⇛⊕ g))) h₁)))
module translation (⌈_⌉ᴬ : Atom → Set) (R : Set) where
infixr 9 ¬_
¬_ : Set → Set
¬ A = A → R
⌈_⌉ : Type → Set
⌈ el A ⌉ = ⌈ A ⌉ᴬ
⌈ A ⊗ B ⌉ = ( ⌈ A ⌉ × ⌈ B ⌉)
⌈ A ⇒ B ⌉ = ¬ ( ⌈ A ⌉ × ¬ ⌈ B ⌉)
⌈ B ⇐ A ⌉ = ¬ (¬ ⌈ B ⌉ × ⌈ A ⌉)
⌈ B ⊕ A ⌉ = ¬ (¬ ⌈ B ⌉ × ¬ ⌈ A ⌉)
⌈ B ⇚ A ⌉ = ( ⌈ B ⌉ × ¬ ⌈ A ⌉)
⌈ A ⇛ B ⌉ = (¬ ⌈ A ⌉ × ⌈ B ⌉)
deMorgan : {A B : Set} → (¬ ¬ A) → (¬ ¬ B) → ¬ ¬ (A × B)
deMorgan c₁ c₂ k = c₁ (λ x → c₂ (λ y → k (x , y)))
mutual
⌈_⌉ᴸ : ∀ {A B} → LG A ⊢ B → ¬ ⌈ B ⌉ → ¬ ⌈ A ⌉
⌈_⌉ᴿ : ∀ {A B} → LG A ⊢ B → ⌈ A ⌉ → ¬ ¬ ⌈ B ⌉
⌈ ax ⌉ᴸ k x = k x
⌈ r⇒⊗ f ⌉ᴸ x = λ{(y , z) → ⌈ f ⌉ᴸ (λ k → k (y , x)) z}
⌈ r⊗⇒ f ⌉ᴸ k x = k (λ{(y , z) → ⌈ f ⌉ᴸ z (y , x)})
⌈ r⇐⊗ f ⌉ᴸ x = λ{(y , z) → ⌈ f ⌉ᴸ (λ k → k (x , z)) y}
⌈ r⊗⇐ f ⌉ᴸ k x = k (λ{(y , z) → ⌈ f ⌉ᴸ y (x , z)})
⌈ m⊗ f g ⌉ᴸ k = λ{(x , y) → deMorgan (⌈ f ⌉ᴿ x) (⌈ g ⌉ᴿ y) k}
⌈ m⇒ f g ⌉ᴸ k k′ = k (λ{(x , y) → deMorgan (⌈ f ⌉ᴿ x) (λ k → k (⌈ g ⌉ᴸ y)) k′})
⌈ m⇐ f g ⌉ᴸ k k′ = k (λ{(x , y) → deMorgan (λ k → k (⌈ f ⌉ᴸ x)) (⌈ g ⌉ᴿ y) k′})
⌈ r⇛⊕ f ⌉ᴸ k x = k (λ{(y , z) → ⌈ f ⌉ᴸ z (y , x)})
⌈ r⊕⇛ f ⌉ᴸ x = λ{(y , z) → ⌈ f ⌉ᴸ (λ k → k (y , x)) z}
⌈ r⇚⊕ f ⌉ᴸ k x = k (λ{(y , z) → ⌈ f ⌉ᴸ y (x , z)})
⌈ r⊕⇚ f ⌉ᴸ x = λ{(y , z) → ⌈ f ⌉ᴸ (λ k → k (x , z)) y}
⌈ m⊕ f g ⌉ᴸ k k′ = k (λ{(x , y) → k′ (⌈ f ⌉ᴸ x , ⌈ g ⌉ᴸ y)})
⌈ m⇛ f g ⌉ᴸ k = λ{(x , y) → deMorgan (λ k → k (⌈ f ⌉ᴸ x)) (⌈ g ⌉ᴿ y) k}
⌈ m⇚ f g ⌉ᴸ k = λ{(x , y) → deMorgan (⌈ f ⌉ᴿ x) (λ k → k (⌈ g ⌉ᴸ y)) k}
⌈ d⇛⇐ f ⌉ᴸ k = λ{(x , y) → k (λ{(z , w) → ⌈ f ⌉ᴸ (λ k → k (x , z)) (y , w)})}
⌈ d⇛⇒ f ⌉ᴸ k = λ{(x , y) → k (λ{(z , w) → ⌈ f ⌉ᴸ (λ k → k (x , w)) (z , y)})}
⌈ d⇚⇒ f ⌉ᴸ k = λ{(x , y) → k (λ{(z , w) → ⌈ f ⌉ᴸ (λ k → k (w , y)) (z , x)})}
⌈ d⇚⇐ f ⌉ᴸ k = λ{(x , y) → k (λ{(z , w) → ⌈ f ⌉ᴸ (λ k → k (z , y)) (x , w)})}
⌈ ax ⌉ᴿ x k = k x
⌈ r⇒⊗ f ⌉ᴿ (x , y) z = ⌈ f ⌉ᴿ y (λ k → k (x , z))
⌈ r⊗⇒ f ⌉ᴿ x k = k (λ{(y , z) → ⌈ f ⌉ᴿ (y , x) z})
⌈ r⇐⊗ f ⌉ᴿ (x , y) z = ⌈ f ⌉ᴿ x (λ k → k (z , y))
⌈ r⊗⇐ f ⌉ᴿ x k = k (λ{(y , z) → ⌈ f ⌉ᴿ (x , z) y})
⌈ m⊗ f g ⌉ᴿ (x , y) k = deMorgan (⌈ f ⌉ᴿ x) (⌈ g ⌉ᴿ y) k
⌈ m⇒ f g ⌉ᴿ k′ k = k (λ{(x , y) → deMorgan (⌈ f ⌉ᴿ x) (λ k → k (⌈ g ⌉ᴸ y)) k′})
⌈ m⇐ f g ⌉ᴿ k′ k = k (λ{(x , y) → deMorgan (λ k → k (⌈ f ⌉ᴸ x)) (⌈ g ⌉ᴿ y) k′})
⌈ r⇛⊕ f ⌉ᴿ x k = k (λ{(y , z) → ⌈ f ⌉ᴿ (y , x) z})
⌈ r⊕⇛ f ⌉ᴿ (x , y) z = ⌈ f ⌉ᴿ y (λ k → k (x , z))
⌈ r⊕⇚ f ⌉ᴿ (x , y) z = ⌈ f ⌉ᴿ x (λ k → k (z , y))
⌈ r⇚⊕ f ⌉ᴿ x k = k (λ{(y , z) → ⌈ f ⌉ᴿ (x , z) y})
⌈ m⊕ f g ⌉ᴿ k′ k = k (λ{(x , y) → k′ (⌈ f ⌉ᴸ x , ⌈ g ⌉ᴸ y)})
⌈ m⇛ f g ⌉ᴿ (x , y) k = deMorgan (λ k → k (⌈ f ⌉ᴸ x)) (⌈ g ⌉ᴿ y) k
⌈ m⇚ f g ⌉ᴿ (x , y) k = deMorgan (⌈ f ⌉ᴿ x) (λ k → k (⌈ g ⌉ᴸ y)) k
⌈ d⇛⇐ f ⌉ᴿ (x , y) k = k (λ{(z , w) → ⌈ f ⌉ᴿ (y , w) (λ k → k (x , z))})
⌈ d⇛⇒ f ⌉ᴿ (x , y) k = k (λ{(z , w) → ⌈ f ⌉ᴿ (z , y) (λ k → k (x , w))})
⌈ d⇚⇒ f ⌉ᴿ (x , y) k = k (λ{(z , w) → ⌈ f ⌉ᴿ (z , x) (λ k → k (w , y))})
⌈ d⇚⇐ f ⌉ᴿ (x , y) k = k (λ{(z , w) → ⌈ f ⌉ᴿ (x , w) (λ k → k (z , y))})
module example where
postulate
Entity : Set
forAll : (Entity → Bool) → Bool
exists : (Entity → Bool) → Bool
ʟᴏᴠᴇs : Entity → Entity → Bool
ᴘᴇʀsᴏɴ : Entity → Bool
_⊃_ : Bool → Bool → Bool
data Atom : Set where N NP S : Atom
⌈_⌉ᴬ : Atom → Set
⌈ N ⌉ᴬ = Entity → Bool
⌈ NP ⌉ᴬ = Entity
⌈ S ⌉ᴬ = Bool
open logic Atom
open logic.translation Atom ⌈_⌉ᴬ Bool
n np s : Type
n = el N
np = el NP
s = el S
someone : ⌈ (np ⇐ n) ⊗ n ⌉
someone = ((λ{(g , f) → exists (λ x → f x ∧ g x)}) , ᴘᴇʀsᴏɴ)
loves : ⌈ (np ⇒ s) ⇐ np ⌉
loves = λ{(k , y) → k (λ{(x , k) → k (ʟᴏᴠᴇs x y)})}
everyone : ⌈ (np ⇐ n) ⊗ n ⌉
everyone = ((λ{(g , f) → forAll (λ x → f x ⊃ g x)}) , ᴘᴇʀsᴏɴ)
-- open import Data.Bool using (Bool; true; false)
-- open import Data.Empty using (⊥)
-- open import Data.List using (List; _∷_; [])
-- open import Data.String using (String; _++_)
-- open import Data.Unit using (⊤)
-- open import Relation.Nullary.Decidable using (⌊_⌋)
-- open import Reflection
-- mutual
-- dropAbs : Term → Term
-- dropAbs (var x args) = var x (dropAbsArgs args)
-- dropAbs (con c args) = con c (dropAbsArgs args)
-- dropAbs (def f args) = def f (dropAbsArgs args)
-- dropAbs (lam v (abs _ x)) = lam v (abs "_" (dropAbs x))
-- dropAbs (pi (arg i (el s₁ t₁)) (abs _ (el s₂ t₂))) = pi (arg i (el s₁ (dropAbs t₁))) (abs "_" (el s₂ (dropAbs t₂)))
-- dropAbs t = t
-- dropAbsArgs : List (Arg Term) → List (Arg Term)
-- dropAbsArgs [] = []
-- dropAbsArgs (arg i x ∷ args) = arg i (dropAbs x) ∷ dropAbsArgs args
-- assert : Bool → Term
-- assert true = def (quote ⊤) []
-- assert false = def (quote ⊥) []
-- infix 0 _↦_
-- macro
-- _↦_ : Term → Term → Term
-- act ↦ exp = assert ⌊ dropAbs act ≟ dropAbs exp ⌋
sᴇɴᴛ₀ sᴇɴᴛ₁ sᴇɴᴛ₂ sᴇɴᴛ₃ sᴇɴᴛ₄ sᴇɴᴛ₅ sᴇɴᴛ₆ : LG
( ( np ⇐ n ) ⊗ n ) ⊗ ( ( ( np ⇒ s ) ⇐ np ) ⊗ ( ( np ⇐ n ) ⊗ n ) ) ⊢ s
sᴇɴᴛ₀ = r⇒⊗ (r⇐⊗ (m⇐ (m⇒ (r⇐⊗ ax′) ax) (r⇐⊗ ax′)))
sᴇɴᴛ₁ = r⇐⊗ (r⇐⊗ (m⇐ (r⊗⇐ (r⇒⊗ (r⇐⊗ (m⇐ ax′ (r⇐⊗ ax′))))) ax))
sᴇɴᴛ₂ = r⇒⊗ (r⇒⊗ (r⇐⊗ (m⇐ (r⊗⇒ (r⇐⊗ (m⇐ (m⇒ (r⇐⊗ ax′) ax) ax))) ax)))
sᴇɴᴛ₃ = r⇐⊗ (r⇐⊗ (m⇐ (r⊗⇐ (r⇒⊗ (r⇒⊗ (r⇐⊗ (m⇐ (r⊗⇒ (r⇐⊗ ax′)) ax))))) ax))
sᴇɴᴛ₄ = r⇒⊗ (r⇐⊗ (m⇐ (r⊗⇒ (r⇐⊗ (r⇐⊗ (m⇐ (r⊗⇐ (r⇒⊗ ax′)) ax)))) (r⇐⊗ ax′)))
sᴇɴᴛ₅ = r⇒⊗ (r⇒⊗ (r⇐⊗ (m⇐ (r⊗⇒ (r⇐⊗ (m⇐ (r⊗⇒ (r⇐⊗ (r⇐⊗ (m⇐ (r⊗⇐ (r⇒⊗ ax′)) ax)))) ax))) ax)))
sᴇɴᴛ₆ = r⇒⊗ (r⇒⊗ (r⇐⊗ (m⇐ (r⊗⇒ (r⊗⇒ (r⇐⊗ (r⇐⊗ (m⇐ (r⊗⇐ (r⇒⊗ (r⇐⊗ ax′))) ax))))) ax)))
s1 = ⌈ sᴇɴᴛ₀ ⌉ᴿ (someone , loves , everyone) id
-- test : LG
-- ( ( np ⇐ n ) ⊗ n ) ⊗ ( ( ( np ⇒ s ) ⇐ np ) ⊗ ( ( np ⇐ n ) ⊗ n ) ) ⊢ s
-- test = {! !}
-- sent₀ : ⌈ sᴇɴᴛ₀ ⌉ᴿ (someone , loves , everyone) id ↦ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ exists (λ x → ᴘᴇʀsᴏɴ x ∧ ʟᴏᴠᴇs x y))
-- sent₁ : ⌈ sᴇɴᴛ₁ ⌉ᴿ (someone , loves , everyone) id ↦ exists (λ x → ᴘᴇʀsᴏɴ x ∧ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ ʟᴏᴠᴇs x y))
-- sent₂ : ⌈ sᴇɴᴛ₂ ⌉ᴿ (someone , loves , everyone) id ↦ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ exists (λ x → ᴘᴇʀsᴏɴ x ∧ ʟᴏᴠᴇs x y))
-- sent₃ : ⌈ sᴇɴᴛ₃ ⌉ᴿ (someone , loves , everyone) id ↦ exists (λ x → ᴘᴇʀsᴏɴ x ∧ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ ʟᴏᴠᴇs x y))
-- sent₄ : ⌈ sᴇɴᴛ₄ ⌉ᴿ (someone , loves , everyone) id ↦ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ exists (λ x → ᴘᴇʀsᴏɴ x ∧ ʟᴏᴠᴇs x y))
-- sent₅ : ⌈ sᴇɴᴛ₅ ⌉ᴿ (someone , loves , everyone) id ↦ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ exists (λ x → ᴘᴇʀsᴏɴ x ∧ ʟᴏᴠᴇs x y))
-- sent₆ : ⌈ sᴇɴᴛ₆ ⌉ᴿ (someone , loves , everyone) id ↦ forAll (λ y → ᴘᴇʀsᴏɴ y ⊃ exists (λ x → ᴘᴇʀsᴏɴ x ∧ ʟᴏᴠᴇs x y))
-- sent₀ = _
-- sent₁ = _
-- sent₂ = _
-- sent₃ = _
-- sent₄ = _
-- sent₅ = _
-- sent₆ = _
module tense where
open import Data.Nat as Nat using (ℕ; _+_)
import Data.List
open import Data.List as List using (List; _∷_; [])
postulate
Entity : Set
PassP : Entity → Entity → (Bool)
They : Entity
Exams : Entity
data ReiAtom : Set where
E R S : ReiAtom
data Rei : Set where
NA : Rei
ReiD : (List ReiAtom) -> Rei
data Atom : Set where
NP T : Atom
S : Rei -> Atom
⌈_⌉ᴬ : Atom → Set
⌈ NP ⌉ᴬ = Entity
⌈ S _ ⌉ᴬ = Bool × ℕ
⌈ T ⌉ᴬ = ℕ
open logic Atom
open logic.translation Atom ⌈_⌉ᴬ (Bool × ℕ)
np t : Type
np = el NP
s : Rei -> Type
s tt = el (S tt)
t = el T
anteriorize : Rei -> Rei
anteriorize (ReiD (a ∷ b ∷ [])) = ReiD (a ∷ R ∷ b ∷ [])
anteriorize _ = NA
passed : ⌈ (np ⇒ (s (ReiD (E ∷ S ∷ [])))) ⇐ np ⌉
passed = λ{(k , y) → k λ{(x , kk) → kk ((PassP x y) , 1)}}
had : {r : Rei} → ⌈ (np ⇒ (((s (anteriorize r)) ⇐ np) ⇐ ((np ⇒ (s r)) ⇐ np))) ⌉
had = λ{(x , k) → k λ{(k , y) → k λ{(t , k) → (y ((((λ z → z ((x , (λ{(b , n) → t (b , (_+_ n 1))}))))) , k)))}}}
by : {r : Rei} → ⌈ ((s r) ⇒ (s r)) ⇐ t ⌉
by = λ{(k , t) → k (λ{((b , n) , s) → s (b , (_+_ t n))})}
then : ⌈ t ⌉
then = 10 -- an arbitrary time index
they : ⌈ np ⌉
they = They
exams : ⌈ np ⌉
exams = Exams
sentence : {r : Rei} → LG ((
(((np ⊗
(np ⇒ (((s (anteriorize r)) ⇐ np) ⇐ ((np ⇒ (s r)) ⇐ np)))) ⊗
((np ⇒ (s r)) ⇐ np)) ⊗
np) ⊗
((((s (anteriorize r)) ⇒ (s (anteriorize r))) ⇐ t) ⊗
t)) ⊢ s (anteriorize r))
sentence = {! !}
-- sentence = r⇒⊗ (r⇐⊗ (m⇐ (m⇒ (r⇐⊗ (r⇐⊗ (r⇒⊗ (m⇒ ax (m⇐ (m⇐ ax ax) (m⇐ (m⇒ ax ax) ax)))))) ax) ax))
out = ⌈ sentence ⌉ᴿ ((((they , had) , passed) , exams) , (by , then)) id
-- C-c C-n tense.out -> tense.PassP tense.They tense.Exams , 12
-- 12 = 1 + 1 + 10
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