From aa280ca672aff145e811b79f4315ba200a09bad5 Mon Sep 17 00:00:00 2001
From: Hannes Schulze <haschu0103@outlook.de>
Date: Fri, 5 Dec 2025 23:19:57 +0100
Subject: define algebras

---
 Algebra/Semantics.lean | 34 ++++++++++++++++++++++++++++++++++
 1 file changed, 34 insertions(+)
 create mode 100644 Algebra/Semantics.lean

(limited to 'Algebra/Semantics.lean')

diff --git a/Algebra/Semantics.lean b/Algebra/Semantics.lean
new file mode 100644
index 0000000..f5160dd
--- /dev/null
+++ b/Algebra/Semantics.lean
@@ -0,0 +1,34 @@
+import Mathlib
+import Algebra.Terms
+import Algebra.Derivations
+
+namespace NominalAlgebra
+
+variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
+variable (sig : Signature)
+
+structure Algebra where
+  carrier : ℕ → Nom 𝔸
+  pure (f : sig.pure) {m : ℕ} :
+    (sig.pureArgs f → carrier m) →ₑ[𝔸] carrier (m + sig.pureDepth f)
+  free (f : sig.free) {m : ℕ} :
+    (𝔸 × (sig.freeArgs f → carrier m)) →ₑ[𝔸] carrier (m + sig.freeDepth f)
+  bound (f : sig.bound) {m : ℕ} :
+    NameAbstraction 𝔸 (sig.boundArgs f → carrier m) →ₑ[𝔸] carrier (m + sig.boundDepth f)
+
+instance : CoeFun (Algebra 𝔸 sig) (fun _ => ℕ → Nom 𝔸) where
+  coe A := A.carrier
+
+structure Algebra.Hom (A B : Algebra 𝔸 sig) where
+  toFun : (n : ℕ) → (A n →ₑ[𝔸] B n)
+  preserve_pure : ∀ (f : sig.pure) {m : ℕ} (args : sig.pureArgs f → A m),
+    toFun (m + sig.pureDepth f) (A.pure f args) = B.pure f (toFun m ∘ args)
+  preserve_free : ∀ (a : 𝔸) (f : sig.free) {m : ℕ} (args : sig.freeArgs f → A m),
+    toFun (m + sig.freeDepth f) (A.free f ⟨a, args⟩) = B.free f ⟨a, toFun m ∘ args⟩
+  preserve_bound : ∀ (a : 𝔸) (f : sig.bound) {m : ℕ} (args : sig.boundArgs f → A m),
+    toFun (m + sig.boundDepth f) (A.bound f (⟪a⟫ args)) = B.bound f (⟪a⟫ toFun m ∘ args)
+
+instance {A B : Algebra 𝔸 sig} : CoeFun (Algebra.Hom 𝔸 sig A B) (fun _ => (n : ℕ) → (A n →ₑ[𝔸] B n)) where
+  coe h := h.toFun
+
+end NominalAlgebra
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