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JacquesCarettejamesmckinnaTanebSeiryn21MatthewDaggitt
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Add back accidentally removed lemmas (Issue2788 #2794))
* opposite of a `Ring` [clean version of pr #1900] (#1910) * punchOut preserves ordering (#1913) * Wellfounded proof for sum relations (#1920) * last revert undone by hand * Remove extra line in CHANGELOG * Update CHANGELOG.md Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com> * Update CHANGELOG.md Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com> * Update CHANGELOG.md Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com> * Update CHANGELOG.md This was indeed anachronistic from 2.0 Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com> * Update CHANGELOG.md Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com> * resolve issues pointed out by James. * Update CHANGELOG.md Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com> * remove now obsolete comment * last mistake in CHANGELOG, hopefully fixed now. --------- Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com> Co-authored-by: Nathan van Doorn <nvd1234@gmail.com> Co-authored-by: Alice Laroche <60161310+Seiryn21@users.noreply.github.com> Co-authored-by: matthewdaggitt <matthewdaggitt@gmail.com>
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‎CHANGELOG.md

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@@ -65,6 +65,7 @@ Additions to existing modules
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```
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* In `Data.Fin.Properties`:
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```agda
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≡-irrelevant : Irrelevant {A = Fin n} _≡_
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≟-≡ : (eq : i ≡ j) → (i ≟ j) ≡ yes eq

‎src/Algebra/Construct/Flip/Op.agda

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module Algebra.Construct.Flip.Op where
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open import Algebra
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import Data.Product.Base as Product
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import Data.Sum.Base as Sum
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open import Algebra.Core
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open import Algebra.Bundles
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import Algebra.Definitions as Def
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import Algebra.Structures as Str
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import Algebra.Consequences.Setoid as Consequences
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import Data.Product as Prod
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import Data.Sum as Sum
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open import Function.Base using (flip)
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open import Level using (Level)
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open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_)
@@ -33,136 +37,270 @@ preserves₂ : (∼ ≈ ≋ : Rel A ℓ) →
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∙ Preserves₂ ∼ ⟶ ≈ ⟶ ≋ → (flip ∙) Preserves₂ ≈ ⟶ ∼ ⟶ ≋
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preserves₂ _ _ _ pres = flip pres
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module _ (≈ : Rel A ℓ) (∙ : Op₂ A) where
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module ∙-Properties (≈ : Rel A ℓ) (∙ : Op₂ A) where
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associative : Symmetric ≈ → Associative ≈ ∙ → Associative ≈ (flip ∙)
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open Def ≈
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associative : Symmetric ≈ → Associative ∙ → Associative (flip ∙)
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associative sym assoc x y z = sym (assoc z y x)
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identity : Identity ≈ ε ∙ → Identity ≈ ε (flip ∙)
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identity id = Product.swap id
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identity : Identity ε ∙ → Identity ε (flip ∙)
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identity id = Prod.swap id
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commutative : Commutative ≈ ∙ → Commutative ≈ (flip ∙)
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commutative : Commutative ∙ → Commutative (flip ∙)
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commutative comm = flip comm
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selective : Selective ≈ ∙ → Selective ≈ (flip ∙)
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selective : Selective ∙ → Selective (flip ∙)
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selective sel x y = Sum.swap (sel y x)
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idempotent : Idempotent ≈ ∙ → Idempotent ≈ (flip ∙)
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idempotent : Idempotent ∙ → Idempotent (flip ∙)
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idempotent idem = idem
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inverse : Inverse ≈ ε ⁻¹ ∙ → Inverse ≈ ε ⁻¹ (flip ∙)
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inverse inv = Product.swap inv
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inverse : Inverse ε ⁻¹ ∙ → Inverse ε ⁻¹ (flip ∙)
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inverse inv = Prod.swap inv
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zero : Zero ε ∙ → Zero ε (flip ∙)
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zero zer = Prod.swap zer
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module *-Properties (≈ : Rel A ℓ) (* + : Op₂ A) where
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open Def ≈
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distributes : * DistributesOver + → (flip *) DistributesOver +
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distributes distrib = Prod.swap distrib
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------------------------------------------------------------------------
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-- Structures
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module _ {≈ : Rel A ℓ} {∙ : Op₂ A} where
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isMagma : IsMagma ≈ ∙ → IsMagma ≈ (flip ∙)
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open Def ≈
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open Str ≈
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open ∙-Properties ≈ ∙
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isMagma : IsMagma ∙ → IsMagma (flip ∙)
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isMagma m = record
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{ isEquivalence = isEquivalence
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; ∙-cong = preserves₂ ≈ ≈ ≈ ∙-cong
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}
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where open IsMagma m
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isSelectiveMagma : IsSelectiveMagma ≈ ∙ → IsSelectiveMagma ≈ (flip ∙)
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isSelectiveMagma : IsSelectiveMagma ∙ → IsSelectiveMagma (flip ∙)
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isSelectiveMagma m = record
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{ isMagma = isMagma m.isMagma
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; sel = selective ≈ ∙ m.sel
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; sel = selective m.sel
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}
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where module m = IsSelectiveMagma m
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isCommutativeMagma : IsCommutativeMagma ≈ ∙ → IsCommutativeMagma ≈ (flip ∙)
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isCommutativeMagma : IsCommutativeMagma ∙ → IsCommutativeMagma (flip ∙)
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isCommutativeMagma m = record
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{ isMagma = isMagma m.isMagma
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; comm = commutative ≈ ∙ m.comm
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; comm = commutative m.comm
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}
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where module m = IsCommutativeMagma m
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isSemigroup : IsSemigroup ≈ ∙ → IsSemigroup ≈ (flip ∙)
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isSemigroup : IsSemigroup ∙ → IsSemigroup (flip ∙)
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isSemigroup s = record
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{ isMagma = isMagma s.isMagma
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; assoc = associative ≈ ∙ s.sym s.assoc
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; assoc = associative s.sym s.assoc
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}
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where module s = IsSemigroup s
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isBand : IsBand ≈ ∙ → IsBand ≈ (flip ∙)
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isBand : IsBand ∙ → IsBand (flip ∙)
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isBand b = record
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{ isSemigroup = isSemigroup b.isSemigroup
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; idem = b.idem
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}
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where module b = IsBand b
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isCommutativeSemigroup : IsCommutativeSemigroup ≈ ∙ →
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IsCommutativeSemigroup ≈ (flip ∙)
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isCommutativeSemigroup : IsCommutativeSemigroup ∙ →
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IsCommutativeSemigroup (flip ∙)
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isCommutativeSemigroup s = record
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{ isSemigroup = isSemigroup s.isSemigroup
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; comm = commutative ≈ ∙ s.comm
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; comm = commutative s.comm
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}
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where module s = IsCommutativeSemigroup s
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isUnitalMagma : IsUnitalMagma ≈ ∙ ε → IsUnitalMagma ≈ (flip ∙) ε
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isUnitalMagma : IsUnitalMagma ∙ ε → IsUnitalMagma (flip ∙) ε
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isUnitalMagma m = record
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{ isMagma = isMagma m.isMagma
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; identity = identity ≈ ∙ m.identity
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; identity = identity m.identity
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}
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where module m = IsUnitalMagma m
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isMonoid : IsMonoid ≈ ∙ ε → IsMonoid ≈ (flip ∙) ε
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isMonoid : IsMonoid ∙ ε → IsMonoid (flip ∙) ε
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isMonoid m = record
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{ isSemigroup = isSemigroup m.isSemigroup
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; identity = identity ≈ ∙ m.identity
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; identity = identity m.identity
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}
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where module m = IsMonoid m
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isCommutativeMonoid : IsCommutativeMonoid ≈ ∙ ε →
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IsCommutativeMonoid ≈ (flip ∙) ε
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isCommutativeMonoid : IsCommutativeMonoid ∙ ε →
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IsCommutativeMonoid (flip ∙) ε
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isCommutativeMonoid m = record
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{ isMonoid = isMonoid m.isMonoid
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; comm = commutative ≈ ∙ m.comm
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; comm = commutative m.comm
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}
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where module m = IsCommutativeMonoid m
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isIdempotentCommutativeMonoid : IsIdempotentCommutativeMonoid ≈ ∙ ε →
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IsIdempotentCommutativeMonoid ≈ (flip ∙) ε
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isIdempotentCommutativeMonoid : IsIdempotentCommutativeMonoid ∙ ε →
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IsIdempotentCommutativeMonoid (flip ∙) ε
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isIdempotentCommutativeMonoid m = record
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{ isCommutativeMonoid = isCommutativeMonoid m.isCommutativeMonoid
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; idem = idempotent ≈ ∙ m.idem
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; idem = idempotent m.idem
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}
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where module m = IsIdempotentCommutativeMonoid m
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isInvertibleMagma : IsInvertibleMagma ≈ ∙ ε ⁻¹ →
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IsInvertibleMagma ≈ (flip ∙) ε ⁻¹
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isInvertibleMagma : IsInvertibleMagma ∙ ε ⁻¹ →
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IsInvertibleMagma (flip ∙) ε ⁻¹
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isInvertibleMagma m = record
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{ isMagma = isMagma m.isMagma
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; inverse = inverse ≈ ∙ m.inverse
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; inverse = inverse m.inverse
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; ⁻¹-cong = m.⁻¹-cong
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}
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where module m = IsInvertibleMagma m
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isInvertibleUnitalMagma : IsInvertibleUnitalMagma ≈ ∙ ε ⁻¹ →
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IsInvertibleUnitalMagma ≈ (flip ∙) ε ⁻¹
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isInvertibleUnitalMagma : IsInvertibleUnitalMagma ∙ ε ⁻¹ →
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IsInvertibleUnitalMagma (flip ∙) ε ⁻¹
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isInvertibleUnitalMagma m = record
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{ isInvertibleMagma = isInvertibleMagma m.isInvertibleMagma
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; identity = identity ≈ ∙ m.identity
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; identity = identity m.identity
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}
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where module m = IsInvertibleUnitalMagma m
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isGroup : IsGroup ≈ ∙ ε ⁻¹ → IsGroup ≈ (flip ∙) ε ⁻¹
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isGroup : IsGroup ∙ ε ⁻¹ → IsGroup (flip ∙) ε ⁻¹
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isGroup g = record
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{ isMonoid = isMonoid g.isMonoid
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; inverse = inverse ≈ ∙ g.inverse
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; inverse = inverse g.inverse
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; ⁻¹-cong = g.⁻¹-cong
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}
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where module g = IsGroup g
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isAbelianGroup : IsAbelianGroup ≈ ∙ ε ⁻¹ → IsAbelianGroup ≈ (flip ∙) ε ⁻¹
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isAbelianGroup : IsAbelianGroup ∙ ε ⁻¹ → IsAbelianGroup (flip ∙) ε ⁻¹
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isAbelianGroup g = record
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{ isGroup = isGroup g.isGroup
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; comm = commutative ≈ ∙ g.comm
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; comm = commutative g.comm
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}
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where module g = IsAbelianGroup g
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module _ {≈ : Rel A ℓ} {+ * : Op₂ A} {0# 1# : A} where
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open Str ≈
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open ∙-Properties ≈ *
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open *-Properties ≈ * +
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isSemiringWithoutAnnihilatingZero : IsSemiringWithoutAnnihilatingZero + * 0# 1# →
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IsSemiringWithoutAnnihilatingZero + (flip *) 0# 1#
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isSemiringWithoutAnnihilatingZero r = record
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{ +-isCommutativeMonoid = r.+-isCommutativeMonoid
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; *-cong = preserves₂ ≈ ≈ ≈ r.*-cong
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; *-assoc = associative r.sym r.*-assoc
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; *-identity = identity r.*-identity
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; distrib = distributes r.distrib
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}
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where module r = IsSemiringWithoutAnnihilatingZero r
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isSemiring : IsSemiring + * 0# 1# → IsSemiring + (flip *) 0# 1#
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isSemiring r = record
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{ isSemiringWithoutAnnihilatingZero = isSemiringWithoutAnnihilatingZero r.isSemiringWithoutAnnihilatingZero
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; zero = zero r.zero
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}
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where module r = IsSemiring r
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isCommutativeSemiring : IsCommutativeSemiring + * 0# 1# →
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IsCommutativeSemiring + (flip *) 0# 1#
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isCommutativeSemiring r = record
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{ isSemiring = isSemiring r.isSemiring
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; *-comm = commutative r.*-comm
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}
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where module r = IsCommutativeSemiring r
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isCancellativeCommutativeSemiring : IsCancellativeCommutativeSemiring + * 0# 1# →
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IsCancellativeCommutativeSemiring + (flip *) 0# 1#
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isCancellativeCommutativeSemiring r = record
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{ isCommutativeSemiring = isCommutativeSemiring r.isCommutativeSemiring
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; *-cancelˡ-nonZero = Consequences.comm∧almostCancelˡ⇒almostCancelʳ r.setoid r.*-comm r.*-cancelˡ-nonZero
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}
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where module r = IsCancellativeCommutativeSemiring r
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isIdempotentSemiring : IsIdempotentSemiring + * 0# 1# →
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IsIdempotentSemiring + (flip *) 0# 1#
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isIdempotentSemiring r = record
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{ isSemiring = isSemiring r.isSemiring
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; +-idem = r.+-idem
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}
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where module r = IsIdempotentSemiring r
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isQuasiring : IsQuasiring + * 0# 1# → IsQuasiring + (flip *) 0# 1#
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isQuasiring r = record
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{ +-isMonoid = r.+-isMonoid
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; *-cong = preserves₂ ≈ ≈ ≈ r.*-cong
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; *-assoc = associative r.sym r.*-assoc
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; *-identity = identity r.*-identity
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; distrib = distributes r.distrib
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; zero = zero r.zero
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}
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where module r = IsQuasiring r
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module _ {≈ : Rel A ℓ} {+ * : Op₂ A} { - : Op₁ A} {0# : A} where
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open Str ≈
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open ∙-Properties ≈ *
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open *-Properties ≈ * +
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isRingWithoutOne : IsRingWithoutOne + * - 0# → IsRingWithoutOne + (flip *) - 0#
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isRingWithoutOne r = record
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{ +-isAbelianGroup = r.+-isAbelianGroup
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; *-cong = preserves₂ ≈ ≈ ≈ r.*-cong
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; *-assoc = associative r.sym r.*-assoc
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; distrib = distributes r.distrib
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}
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where module r = IsRingWithoutOne r
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module _ {≈ : Rel A ℓ} {+ * : Op₂ A} { - : Op₁ A} {0# 1# : A} where
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open Str ≈
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open ∙-Properties ≈ *
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open *-Properties ≈ * +
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isNonAssociativeRing : IsNonAssociativeRing + * - 0# 1# →
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IsNonAssociativeRing + (flip *) - 0# 1#
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isNonAssociativeRing r = record
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{ +-isAbelianGroup = r.+-isAbelianGroup
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; *-cong = preserves₂ ≈ ≈ ≈ r.*-cong
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; *-identity = identity r.*-identity
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; distrib = distributes r.distrib
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; zero = zero r.zero
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}
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where module r = IsNonAssociativeRing r
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isNearring : IsNearring + * 0# 1# - → IsNearring + (flip *) 0# 1# -
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isNearring r = record
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{ isQuasiring = isQuasiring r.isQuasiring
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; +-inverse = r.+-inverse
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; ⁻¹-cong = r.⁻¹-cong
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}
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where module r = IsNearring r
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isRing : IsRing + * - 0# 1# → IsRing + (flip *) - 0# 1#
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isRing r = record
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{ +-isAbelianGroup = r.+-isAbelianGroup
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; *-cong = preserves₂ ≈ ≈ ≈ r.*-cong
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; *-assoc = associative r.sym r.*-assoc
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; *-identity = identity r.*-identity
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; distrib = distributes r.distrib
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}
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where module r = IsRing r
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isCommutativeRing : IsCommutativeRing + * - 0# 1# →
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IsCommutativeRing + (flip *) - 0# 1#
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isCommutativeRing r = record
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{ isRing = isRing r.isRing
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; *-comm = commutative r.*-comm
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}
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where module r = IsCommutativeRing r
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------------------------------------------------------------------------
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-- Bundles
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@@ -239,3 +377,54 @@ group g = record { isGroup = isGroup g.isGroup }
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abelianGroup : AbelianGroup a ℓ → AbelianGroup a ℓ
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abelianGroup g = record { isAbelianGroup = isAbelianGroup g.isAbelianGroup }
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where module g = AbelianGroup g
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semiringWithoutAnnihilatingZero : SemiringWithoutAnnihilatingZero a ℓ →
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SemiringWithoutAnnihilatingZero a ℓ
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semiringWithoutAnnihilatingZero r = record
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{ isSemiringWithoutAnnihilatingZero = isSemiringWithoutAnnihilatingZero r.isSemiringWithoutAnnihilatingZero }
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where module r = SemiringWithoutAnnihilatingZero r
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semiring : Semiring a ℓ → Semiring a ℓ
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semiring r = record { isSemiring = isSemiring r.isSemiring }
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where module r = Semiring r
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commutativeSemiring : CommutativeSemiring a ℓ → CommutativeSemiring a ℓ
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commutativeSemiring r = record
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{ isCommutativeSemiring = isCommutativeSemiring r.isCommutativeSemiring }
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where module r = CommutativeSemiring r
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cancellativeCommutativeSemiring : CancellativeCommutativeSemiring a ℓ →
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CancellativeCommutativeSemiring a ℓ
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cancellativeCommutativeSemiring r = record
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{ isCancellativeCommutativeSemiring = isCancellativeCommutativeSemiring r.isCancellativeCommutativeSemiring }
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where module r = CancellativeCommutativeSemiring r
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idempotentSemiring : IdempotentSemiring a ℓ → IdempotentSemiring a ℓ
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idempotentSemiring r = record
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{ isIdempotentSemiring = isIdempotentSemiring r.isIdempotentSemiring }
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where module r = IdempotentSemiring r
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quasiring : Quasiring a ℓ → Quasiring a ℓ
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quasiring r = record { isQuasiring = isQuasiring r.isQuasiring }
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where module r = Quasiring r
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ringWithoutOne : RingWithoutOne a ℓ → RingWithoutOne a ℓ
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ringWithoutOne r = record { isRingWithoutOne = isRingWithoutOne r.isRingWithoutOne }
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where module r = RingWithoutOne r
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nonAssociativeRing : NonAssociativeRing a ℓ → NonAssociativeRing a ℓ
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nonAssociativeRing r = record
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{ isNonAssociativeRing = isNonAssociativeRing r.isNonAssociativeRing }
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where module r = NonAssociativeRing r
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nearring : Nearring a ℓ → Nearring a ℓ
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nearring r = record { isNearring = isNearring r.isNearring }
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where module r = Nearring r
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424+
ring : Ring a ℓ → Ring a ℓ
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ring r = record { isRing = isRing r.isRing }
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where module r = Ring r
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commutativeRing : CommutativeRing a ℓ → CommutativeRing a ℓ
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commutativeRing r = record { isCommutativeRing = isCommutativeRing r.isCommutativeRing }
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where module r = CommutativeRing r

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