diff --git a/src/runtime/backing.rs b/src/runtime/backing.rs
index f326f4b86..5a5ab133c 100644
--- a/src/runtime/backing.rs
+++ b/src/runtime/backing.rs
@@ -4,7 +4,7 @@ use crate::runtime::{
     module::{ImportName, Module},
     sig_registry::SigRegistry,
     table::{TableBacking, TableElements},
-    types::{GlobalInit, MapIndex, Val},
+    types::{Initializer, MapIndex, Val},
     vm,
 };
 
@@ -149,11 +149,11 @@ impl LocalBacking {
     ) -> Box<[vm::LocalGlobal]> {
         for (to, (_, from)) in globals.iter_mut().zip(module.globals.into_iter()) {
             to.data = match from.init {
-                GlobalInit::Val(Val::I32(x)) => x as u64,
-                GlobalInit::Val(Val::I64(x)) => x as u64,
-                GlobalInit::Val(Val::F32(x)) => x as u64,
-                GlobalInit::Val(Val::F64(x)) => x,
-                GlobalInit::GetGlobal(index) => (imports.globals[index.index()].global).data,
+                Initializer::Const(Val::I32(x)) => x as u64,
+                Initializer::Const(Val::I64(x)) => x as u64,
+                Initializer::Const(Val::F32(x)) => x as u64,
+                Initializer::Const(Val::F64(x)) => x,
+                Initializer::GetGlobal(index) => (imports.globals[index.index()].global).data,
             };
         }
 
diff --git a/src/runtime/module.rs b/src/runtime/module.rs
index 707684682..89d379fcb 100644
--- a/src/runtime/module.rs
+++ b/src/runtime/module.rs
@@ -11,14 +11,14 @@ use std::ptr::NonNull;
 /// This is used to instantiate a new webassembly module.
 pub struct Module {
     pub function_resolver: Box<dyn Fn(&Module, FuncIndex) -> Option<NonNull<vm::Func>>>,
-    pub memories: Map<Memory, MemoryIndex>,
-    pub globals: Map<Global, GlobalIndex>,
-    pub tables: Map<Table, TableIndex>,
+    pub memories: Map<MemoryIndex, Memory>,
+    pub globals: Map<GlobalIndex, Global>,
+    pub tables: Map<TableIndex, Table>,
 
-    pub imported_functions: Map<ImportName, FuncIndex>,
-    pub imported_memories: Map<(ImportName, Memory), MemoryIndex>,
-    pub imported_tables: Map<(ImportName, Table), TableIndex>,
-    pub imported_globals: Map<(ImportName, GlobalDesc), GlobalIndex>,
+    pub imported_functions: Map<FuncIndex, ImportName>,
+    pub imported_memories: Map<MemoryIndex, (ImportName, Memory)>,
+    pub imported_tables: Map<TableIndex, (ImportName, Table)>,
+    pub imported_globals: Map<GlobalIndex, (ImportName, GlobalDesc)>,
 
     pub exports: HashMap<String, Export>,
 
@@ -26,8 +26,8 @@ pub struct Module {
     pub table_initializers: Vec<TableInitializer>,
     pub start_func: Option<FuncIndex>,
 
-    pub signature_assoc: Map<SigIndex, FuncIndex>,
-    pub signatures: Map<FuncSig, SigIndex>,
+    pub signature_assoc: Map<FuncIndex, SigIndex>,
+    pub signatures: Map<SigIndex, FuncSig>,
 }
 
 impl Module {
diff --git a/src/runtime/sig_registry.rs b/src/runtime/sig_registry.rs
index 1912211fb..bc60bae5e 100644
--- a/src/runtime/sig_registry.rs
+++ b/src/runtime/sig_registry.rs
@@ -7,7 +7,7 @@ use hashbrown::HashMap;
 
 pub struct SigRegistry {
     sig_set: HashMap<FuncSig, vm::SigId>,
-    signatures: Map<vm::SigId, SigIndex>,
+    signatures: Map<SigIndex, vm::SigId>,
 }
 
 impl SigRegistry {
@@ -19,7 +19,8 @@ impl SigRegistry {
 
         for (_, &sig_index) in &module.signature_assoc {
             let func_sig = module.signatures[sig_index].clone();
-            registry.register(func_sig);
+            let new_sig_index = registry.register(func_sig);
+            assert_eq!(sig_index, new_sig_index);
         }
 
         registry
@@ -33,12 +34,12 @@ impl SigRegistry {
         self.signatures[sig_index]
     }
 
-    fn register(&mut self, signature: FuncSig) {
+    fn register(&mut self, signature: FuncSig) -> SigIndex {
         let index = self.sig_set.len();
         let vm_sig_id = *self
             .sig_set
             .entry(signature)
             .or_insert_with(|| vm::SigId(index as u32));
-        self.signatures.push(vm_sig_id);
+        self.signatures.push(vm_sig_id)
     }
 }
diff --git a/src/runtime/types.rs b/src/runtime/types.rs
index 34e3e39eb..f41fab26e 100644
--- a/src/runtime/types.rs
+++ b/src/runtime/types.rs
@@ -84,8 +84,10 @@ pub struct Table {
 /// Overtime, this will be able to represent more and more
 /// complex expressions.
 #[derive(Debug, Clone, Copy, PartialEq, Eq)]
-pub enum GlobalInit {
-    Val(Val),
+pub enum Initializer {
+    /// Corresponds to a `const.*` instruction.
+    Const(Val),
+    /// Corresponds to a `get_global` instruction.
     GetGlobal(GlobalIndex),
 }
 
@@ -99,7 +101,7 @@ pub struct GlobalDesc {
 #[derive(Debug, Clone, Copy)]
 pub struct Global {
     pub desc: GlobalDesc,
-    pub init: GlobalInit,
+    pub init: Initializer,
 }
 
 /// A wasm memory.
@@ -144,7 +146,7 @@ pub trait MapIndex {
 
 /// Dense item map
 #[derive(Debug, Clone)]
-pub struct Map<T, I>
+pub struct Map<I, T>
 where
     I: MapIndex,
 {
@@ -152,7 +154,7 @@ where
     _marker: PhantomData<I>,
 }
 
-impl<T, I> Map<T, I>
+impl<I, T> Map<I, T>
 where
     I: MapIndex,
 {
@@ -189,7 +191,7 @@ where
     }
 }
 
-impl<T, I> Index<I> for Map<T, I>
+impl<I, T> Index<I> for Map<I, T>
 where
     I: MapIndex,
 {
@@ -199,7 +201,7 @@ where
     }
 }
 
-impl<T, I> IndexMut<I> for Map<T, I>
+impl<I, T> IndexMut<I> for Map<I, T>
 where
     I: MapIndex,
 {
@@ -208,7 +210,7 @@ where
     }
 }
 
-impl<'a, T, I> IntoIterator for &'a Map<T, I>
+impl<'a, I, T> IntoIterator for &'a Map<I, T>
 where
     I: MapIndex,
 {
@@ -220,7 +222,7 @@ where
     }
 }
 
-impl<'a, T, I> IntoIterator for &'a mut Map<T, I>
+impl<'a, I, T> IntoIterator for &'a mut Map<I, T>
 where
     I: MapIndex,
 {