Free-group lemmas (#106)

Setoids/Cardinality/Infinite/Definition.agda

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+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+open import LogicalFormulae
+open import Functions
+open import Numbers.Naturals.Definition
+open import Sets.FinSet.Definition
+open import Setoids.Setoids
+
+module Setoids.Cardinality.Infinite.Definition where
+
+InfiniteSetoid : {a b : _} {A : Set a} (S : Setoid {a} {b} A) → Set (a ⊔ b)
+InfiniteSetoid {A = A} S = (n : ℕ) → (f : FinSet n → A) → (SetoidBijection (reflSetoid (FinSet n)) S f) → False
+
+record DedekindInfiniteSetoid {a b : _} {A : Set a} (S : Setoid {a} {b} A) : Set (a ⊔ b) where
+  field
+    inj : ℕ → A
+    isInjection : SetoidInjection (reflSetoid ℕ) S inj