Cleanup finset and modulo (#92)

Sets/Cardinality/Finite/Definition.agda

@@ -0,0 +1,18 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Functions
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+open import LogicalFormulae
+open import Numbers.Naturals.Naturals
+open import Numbers.Naturals.Definition
+open import Numbers.Naturals.Order
+open import Sets.FinSet.Definition
+
+module Sets.Cardinality.Finite.Definition where
+
+record FiniteSet {a : _} (A : Set a) : Set a where
+  field
+    size : ℕ
+    mapping : FinSet size → A
+    bij : Bijection mapping
+

Sets/Cardinality/Finite/Lemmas.agda

@@ -0,0 +1,26 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Functions
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+open import LogicalFormulae
+open import Numbers.Naturals.Naturals
+open import Numbers.Naturals.Definition
+open import Numbers.Naturals.Order
+open import Sets.FinSet.Definition
+open import Sets.FinSet.Lemmas
+
+module Sets.Cardinality.Finite.Lemmas where
+
+finsetInjectIntoℕ : {n : ℕ} → Injection (toNat {n})
+finsetInjectIntoℕ {zero} {()}
+finsetInjectIntoℕ {succ n} = ans
+  where
+    ans : {n : ℕ} → {x y : FinSet (succ n)} → (toNat x ≡ toNat y) → x ≡ y
+    ans {zero} {fzero} {fzero} _ = refl
+    ans {zero} {fzero} {fsucc ()}
+    ans {zero} {fsucc ()} {y}
+    ans {succ n} {fzero} {fzero} pr = refl
+    ans {succ n} {fzero} {fsucc y} ()
+    ans {succ n} {fsucc x} {fzero} ()
+    ans {succ n} {fsucc x} {fsucc y} pr with succInjective pr
+    ... | pr' = applyEquality fsucc (ans {n} {x} {y} pr')