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8 | 8 |
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9 | 9 | module Data.Bool.Properties where
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10 | 10 | open import Algebra.Bundles
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11 |
| -open import Algebra.Lattice.Bundles using |
12 |
| - (BooleanAlgebra; DistributiveLattice; Lattice; Semilattice) |
| 11 | + using (Magma; Semigroup; Band; CommutativeMonoid |
| 12 | + ; IdempotentCommutativeMonoid; CommutativeSemiring; CommutativeRing) |
| 13 | +open import Algebra.Lattice.Bundles |
| 14 | + using (Lattice; DistributiveLattice; BooleanAlgebra; Semilattice) |
13 | 15 | import Algebra.Lattice.Properties.BooleanAlgebra as BooleanAlgebraProperties
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14 |
| -open import Data.Bool.Base using (Bool; T; _<_; _≤_; _∧_; _∨_; _xor_; b≤b; f<t; |
15 |
| - f≤t; false; if_then_else_; not; true) |
| 16 | +open import Data.Bool.Base |
| 17 | + using (Bool; true; false; not; _∧_; _∨_; _xor_ ; if_then_else_; T; _≤_; _<_ |
| 18 | + ; b≤b; f≤t; f<t) |
16 | 19 | open import Data.Empty using (⊥; ⊥-elim)
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17 |
| -open import Data.Product.Base using (_,_; _×_; proj₁; proj₂) |
18 |
| -open import Data.Sum.Base using ([_,_]; _⊎_; inj₁; inj₂) |
| 20 | +open import Data.Product.Base using (_×_; _,_; proj₁; proj₂) |
| 21 | +open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]) |
19 | 22 | open import Function.Base using (_⟨_⟩_; const; id)
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20 | 23 | open import Function.Bundles hiding (Inverse; LeftInverse; RightInverse)
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21 | 24 | open import Induction.WellFounded using (Acc; WellFounded; acc)
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22 | 25 | open import Level using (0ℓ; Level)
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23 | 26 | open import Relation.Binary.Bundles using (DecSetoid; DecTotalOrder; Poset;
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24 | 27 | Preorder; Setoid; StrictPartialOrder; StrictTotalOrder; TotalOrder)
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25 | 28 | open import Relation.Binary.Core using (_⇒_)
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26 |
| -open import Relation.Binary.Definitions using (Antisymmetric; Asymmetric; |
27 |
| - Decidable; DecidableEquality; Irreflexive; Irrelevant; Maximum; Minimum; |
28 |
| - Reflexive; Total; Trans; Transitive; Trichotomous; _Respects₂_; |
29 |
| - tri<; tri>; tri≈) |
30 |
| -open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; |
31 |
| - cong; cong₂; refl; subst; sym; trans) |
32 |
| -open import Relation.Binary.PropositionalEquality.Properties using |
33 |
| - (decSetoid; isEquivalence; module ≡-Reasoning; setoid) |
34 |
| -open import Relation.Binary.Structures using (IsDecTotalOrder; |
35 |
| - IsPartialOrder; IsPreorder; IsStrictPartialOrder; IsStrictTotalOrder; |
36 |
| - IsTotalOrder) |
37 |
| -open import Relation.Nullary.Decidable.Core using (True; fromWitness; no; yes) |
38 |
| -import Relation.Unary as U using (Irrelevant) |
| 29 | +open import Relation.Binary.Structures |
| 30 | + using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder |
| 31 | + ; IsStrictPartialOrder; IsStrictTotalOrder) |
| 32 | +open import Relation.Binary.Bundles |
| 33 | + using (Setoid; DecSetoid; Poset; Preorder; TotalOrder; DecTotalOrder |
| 34 | + ; StrictPartialOrder; StrictTotalOrder) |
| 35 | +open import Relation.Binary.Definitions |
| 36 | + using (Decidable; DecidableEquality ; Reflexive; Transitive; Antisymmetric |
| 37 | + ; Minimum; Maximum; Total; Irrelevant ; Irreflexive; Asymmetric; Trans |
| 38 | + ; Trichotomous; tri≈; tri<; tri>; _Respects₂_) |
| 39 | +open import Relation.Binary.PropositionalEquality.Core |
| 40 | + using (_≡_; refl; sym; cong; cong₂; subst; trans; _≢_) |
| 41 | +open import Relation.Binary.PropositionalEquality.Properties |
| 42 | + using (module ≡-Reasoning; setoid; decSetoid; isEquivalence) |
| 43 | +open import Relation.Nullary.Decidable.Core |
| 44 | + using (True; yes; no; fromWitness ; toWitness) |
| 45 | +import Relation.Unary as U |
39 | 46 |
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40 | 47 | open import Algebra.Definitions {A = Bool} _≡_
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41 | 48 | open import Algebra.Structures {A = Bool} _≡_
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