mirror of
https://github.com/Smaug123/agdaproofs
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42 lines
1.9 KiB
Agda
42 lines
1.9 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Groups.Groups
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open import Functions
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open import Numbers.Naturals.Semiring
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open import Numbers.Naturals.Order
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open import Numbers.Integers.Integers
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open import Numbers.Modulo.Group
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open import Numbers.Modulo.Definition
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open import Rings.Examples.Proofs
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open import Numbers.Primes.PrimeNumbers
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module Rings.Examples.Examples where
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nToZn : (n : ℕ) (pr : 0 <N n) (x : ℕ) → ℤn n pr
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nToZn n pr x = nToZn' n pr x
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mod : (n : ℕ) → (pr : 0 <N n) → ℤ → ℤn n pr
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mod n pr a = mod' n pr a
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modNExampleSurjective : (n : ℕ) → (pr : 0 <N n) → Surjection (mod n pr)
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modNExampleSurjective n pr = modNExampleSurjective' n pr
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{-
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modNExampleGroupHom : (n : ℕ) → (pr : 0 <N n) → GroupHom ℤGroup (ℤnGroup n pr) (mod n pr)
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modNExampleGroupHom n pr = modNExampleGroupHom' n pr
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embedZnInZ : {n : ℕ} {pr : 0 <N n} → (a : ℤn n pr) → ℤ
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embedZnInZ record { x = x } = nonneg x
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modNRoundTrip : (n : ℕ) → (pr : 0 <N n) → (a : ℤn n pr) → mod n pr (embedZnInZ a) ≡ a
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modNRoundTrip zero ()
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modNRoundTrip (succ n) pr record { x = x ; xLess = xLess } with divisionAlg (succ n) x
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modNRoundTrip (succ n) _ record { x = x ; xLess = xLess } | record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = inl remIsSmall ; quotSmall = quotSmall } = equalityZn _ _ p
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where
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p : rem ≡ x
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p = modIsUnique record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = inl remIsSmall ; quotSmall = quotSmall } record { quot = 0 ; rem = x ; pr = identityOfIndiscernablesLeft _ _ _ _≡_ refl (applyEquality (λ i → i +N x) (multiplicationNIsCommutative 0 n)) ; remIsSmall = inl xLess ; quotSmall = inl (succIsPositive n) }
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modNRoundTrip (succ n) _ record { x = x ; xLess = xLess } | record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = inr () }
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-}
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