Move things around, add more to fields (#16)

Rings/RingExamples.agda

@@ -0,0 +1,39 @@
+{-# OPTIONS --safe --warning=error #-}
+
+open import LogicalFormulae
+open import Groups
+open import Functions
+open import Naturals
+open import Integers
+open import IntegersModN
+open import RingExamplesProofs
+open import PrimeNumbers
+
+module RingExamples where
+
+  nToZn : (n : ℕ) (pr : 0 <N n) (x : ℕ) → ℤn n pr
+  nToZn n pr x = nToZn' n pr x
+
+  mod : (n : ℕ) → (pr : 0 <N n) → ℤ → ℤn n pr
+  mod n pr a = mod' n pr a
+
+  modNExampleSurjective : (n : ℕ) → (pr : 0 <N n) → Surjection (mod n pr)
+  modNExampleSurjective n pr = modNExampleSurjective' n pr
+
+{-
+  modNExampleGroupHom : (n : ℕ) → (pr : 0 <N n) → GroupHom ℤGroup (ℤnGroup n pr) (mod n pr)
+  modNExampleGroupHom n pr = modNExampleGroupHom' n pr
+
+  embedZnInZ : {n : ℕ} {pr : 0 <N n} → (a : ℤn n pr) → ℤ
+  embedZnInZ record { x = x } = nonneg x
+
+  modNRoundTrip : (n : ℕ) → (pr : 0 <N n) → (a : ℤn n pr) → mod n pr (embedZnInZ a) ≡ a
+  modNRoundTrip zero ()
+  modNRoundTrip (succ n) pr record { x = x ; xLess = xLess } with divisionAlg (succ n) x
+  modNRoundTrip (succ n) _ record { x = x ; xLess = xLess } | record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = inl remIsSmall ; quotSmall = quotSmall } = equalityZn _ _ p
+    where
+      p : rem ≡ x
+      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) }
+  modNRoundTrip (succ n) _ record { x = x ; xLess = xLess } | record { quot = quot ; rem = rem ; pr = pr ; remIsSmall = inr () }
+
+-}