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https://github.com/Smaug123/agdaproofs
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31 lines
1.3 KiB
Agda
31 lines
1.3 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Setoids.Setoids
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open import Functions
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open import Sets.EquivalenceRelations
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open import Rings.Definition
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module Rings.Divisible.Lemmas {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
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open Setoid S
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open Equivalence eq
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open Ring R
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open import Rings.Divisible.Definition R
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open import Rings.Units.Definition R
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divisionTransitive : (x y z : A) → x ∣ y → y ∣ z → x ∣ z
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divisionTransitive x y z (a , pr) (b , pr2) = (a * b) , transitive (transitive *Associative (*WellDefined pr reflexive)) pr2
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divisionReflexive : (x : A) → x ∣ x
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divisionReflexive x = 1R , transitive *Commutative identIsIdent
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everythingDividesZero : (r : A) → r ∣ 0R
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everythingDividesZero r = 0R , timesZero
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nonzeroInherits : {x y : A} (nz : (x ∼ 0R) → False) → y ∣ x → (y ∼ 0R) → False
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nonzeroInherits {x} {y} nz (c , pr) y=0 = nz (transitive (symmetric pr) (transitive (*WellDefined y=0 reflexive) (transitive *Commutative timesZero)))
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nonunitInherits : {x y : A} (nonunit : Unit x → False) → x ∣ y → Unit y → False
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nonunitInherits nu (s , pr) (a , b) = nu ((s * a) , transitive (transitive *Associative (*WellDefined pr reflexive)) b)
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