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https://github.com/Smaug123/agdaproofs
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24 lines
1.0 KiB
Agda
24 lines
1.0 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import Functions
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import LogicalFormulae
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open import Setoids.Subset
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open import Setoids.Setoids
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open import Setoids.Orders.Partial.Definition
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open import Fields.Fields
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open import Rings.Orders.Total.Definition
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open import Rings.Orders.Total.Lemmas
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open import Rings.Orders.Partial.Definition
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open import Rings.Definition
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module Fields.Orders.LeastUpperBounds.Definition {a b : _} {A : Set a} {S : Setoid {a} {b} A} {c : _} {_<_ : Rel {_} {c} A} (pOrder : SetoidPartialOrder S _<_) where
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UpperBound : {d : _} {pred : A → Set d} (sub : subset S pred) (x : A) → Set _
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UpperBound {pred = pred} sub x = (y : A) → pred y → (y < x) || (Setoid._∼_ S y x)
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record LeastUpperBound {d : _} {pred : A → Set d} (sub : subset S pred) (x : A) : Set (a ⊔ b ⊔ c ⊔ d) where
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field
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upperBound : UpperBound sub x
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leastUpperBound : (y : A) → UpperBound sub y → (x < y) || (Setoid._∼_ S x y)
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