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https://github.com/Smaug123/agdaproofs
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36 lines
2.3 KiB
Agda
36 lines
2.3 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import Setoids.Setoids
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open import Rings.Definition
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open import Groups.Definition
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open import Fields.Fields
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open import Sets.EquivalenceRelations
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open import LogicalFormulae
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open import Rings.IntegralDomains.Definition
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module Fields.Lemmas {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {_*_ : A → A → A} {R : Ring S _+_ _*_} (F : Field R) where
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open Setoid S
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open Field F
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open Ring R
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open Group additiveGroup
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halve : (charNot2 : ((1R + 1R) ∼ 0R) → False) → (a : A) → Sg A (λ i → i + i ∼ a)
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halve charNot2 a with allInvertible (1R + 1R) charNot2
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... | 1/2 , pr1/2 = (a * 1/2) , Equivalence.transitive eq (+WellDefined *Commutative *Commutative) (Equivalence.transitive eq (Equivalence.symmetric eq (*DistributesOver+ {1/2} {a} {a})) (Equivalence.transitive eq (*WellDefined (Equivalence.reflexive eq) r) (Equivalence.transitive eq (*Associative) (Equivalence.transitive eq (*WellDefined pr1/2 (Equivalence.reflexive eq)) identIsIdent))))
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where
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r : a + a ∼ (1R + 1R) * a
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r = Equivalence.symmetric eq (Equivalence.transitive eq *Commutative (Equivalence.transitive eq *DistributesOver+ (+WellDefined (Equivalence.transitive eq *Commutative identIsIdent) (Equivalence.transitive eq *Commutative identIsIdent))))
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abstract
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halfHalves : {x : A} (1/2 : A) (pr : 1/2 + 1/2 ∼ 1R) → (x + x) * 1/2 ∼ x
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halfHalves {x} 1/2 pr = Equivalence.transitive eq (Equivalence.transitive eq (Equivalence.transitive eq *Commutative (Equivalence.transitive eq (Equivalence.transitive eq *DistributesOver+ (Equivalence.transitive eq (+WellDefined *Commutative *Commutative) (Equivalence.symmetric eq *DistributesOver+))) *Commutative)) (*WellDefined pr (Equivalence.reflexive eq))) identIsIdent
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fieldIsIntDom : (Setoid._∼_ S (Ring.1R R) (Ring.0R R) → False) → IntegralDomain R
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IntegralDomain.intDom (fieldIsIntDom 1!=0) {a} {b} ab=0 a!=0 with Field.allInvertible F a a!=0
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IntegralDomain.intDom (fieldIsIntDom _) {a} {b} ab=0 a!=0 | 1/a , prA = transitive (symmetric identIsIdent) (transitive (*WellDefined (symmetric prA) reflexive) (transitive (symmetric *Associative) (transitive (*WellDefined reflexive ab=0) (Ring.timesZero R))))
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where
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open Equivalence eq
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IntegralDomain.nontrivial (fieldIsIntDom 1!=0) = 1!=0
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