cat/src/Cat/Prelude.agda

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-- | Custom prelude for this module
module Cat.Prelude where
open import Agda.Primitive public
-- FIXME Use:
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open import Agda.Builtin.Sigma public
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-- Rather than
open import Data.Product public
renaming (∃! to ∃!≈)
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using (_×_ ; Σ-syntax ; swap)
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-- TODO Import Data.Function under appropriate names.
open import Cubical public
-- FIXME rename `gradLemma` to `fromIsomorphism` - perhaps just use wrapper
-- module.
open import Cubical.GradLemma
using (gradLemma)
public
open import Cubical.NType
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using (⟨-2⟩ ; ⟨-1⟩ ; ⟨0⟩ ; TLevel ; HasLevel)
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public
open import Cubical.NType.Properties
using
( lemPropF ; lemSig ; lemSigP ; isSetIsProp
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; propPi ; propPiImpl ; propHasLevel ; setPi ; propSet
; propSig)
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public
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propIsContr : { : Level} {A : Set } isProp (isContr A)
propIsContr = propHasLevel ⟨-2⟩
open import Cubical.Sigma using (setSig ; sigPresSet ; sigPresNType) public
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module _ ( : Level) where
-- FIXME Ask if we can push upstream.
-- A redefinition of `Cubical.Universe` with an explicit parameter
_-type : TLevel Set (lsuc )
n -type = Σ (Set ) (HasLevel n)
hSet : Set (lsuc )
hSet = ⟨0⟩ -type
Prop : Set (lsuc )
Prop = ⟨-1⟩ -type
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-----------------
-- * Utilities --
-----------------
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-- | Unique existentials.
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∃! : {a b} {A : Set a}
(A Set b) Set (a b)
∃! = ∃!≈ _≡_
∃!-syntax : {a b} {A : Set a} (A Set b) Set (a b)
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∃!-syntax = ∃!
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syntax ∃!-syntax (λ x B) = ∃![ x ] B
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module _ {a b} {A : Set a} {P : A Set b} (f g : ∃! P) where
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open Σ (snd f) renaming (snd to u)
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∃-unique : fst f fst g
∃-unique = u (fst (snd g))
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module _ {a b : Level} {A : Set a} {B : A Set b} {a b : Σ A B}
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(fst≡ : (λ _ A) [ fst a fst b ])
(snd≡ : (λ i B (fst≡ i)) [ snd a snd b ]) where
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Σ≡ : a b
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fst (Σ≡ i) = fst≡ i
snd (Σ≡ i) = snd≡ i
import Relation.Binary
open Relation.Binary public using (Preorder ; Transitive ; IsEquivalence ; Rel)
equalityIsEquivalence : { : Level} {A : Set } IsEquivalence {A = A} _≡_
IsEquivalence.refl equalityIsEquivalence = refl
IsEquivalence.sym equalityIsEquivalence = sym
IsEquivalence.trans equalityIsEquivalence = trans
IsPreorder
: {a : Level} {A : Set a}
(__ : Rel A ) -- The relation.
Set _
IsPreorder _~_ = Relation.Binary.IsPreorder _≡_ _~_