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N-ary expansions (#113)
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@@ -1,6 +1,7 @@
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{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Numbers.Naturals.Definition
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open import Numbers.Naturals.Naturals
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open import Numbers.Integers.Integers
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open import Groups.Groups
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@@ -22,6 +23,7 @@ open import Fields.FieldOfFractions.Addition ℤIntDom
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open import Fields.FieldOfFractions.Multiplication ℤIntDom
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open import Fields.FieldOfFractions.Ring ℤIntDom
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open import Fields.FieldOfFractions.Field ℤIntDom
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open import Fields.FieldOfFractions.Lemmas ℤIntDom
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open import Fields.FieldOfFractions.Order ℤIntDom ℤOrderedRing
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ℚ : Set
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@@ -43,7 +45,10 @@ a *Q b = fieldOfFractionsTimes a b
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0Q = Ring.0R ℚRing
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injectionQ : ℤ → ℚ
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injectionQ z = z ,, (nonneg 1 , λ ())
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injectionQ = embedIntoFieldOfFractions
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injectionNQ : ℕ → ℚ
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injectionNQ n = injectionQ (nonneg n)
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injectionQInjective : Injection injectionQ
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injectionQInjective pr = equalityLeft pr
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@@ -78,6 +83,9 @@ a-A a = Group.invRight (Ring.additiveGroup ℚRing) {a}
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ℚTotalOrder : SetoidTotalOrder fieldOfFractionsOrder
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ℚTotalOrder = fieldOfFractionsTotalOrder
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ℚOrderInherited : (a b : ℤ) → a <Z b → injectionQ a <Q injectionQ b
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ℚOrderInherited a b a<b = fieldOfFractionsOrderInherited a<b
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open SetoidTotalOrder fieldOfFractionsTotalOrder
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open SetoidPartialOrder partial
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open Setoid fieldOfFractionsSetoid
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