diff --git a/GroundZero/Algebra/Group/Differential.lean b/GroundZero/Algebra/Group/Differential.lean
index d115cd7..c8d1b3e 100644
--- a/GroundZero/Algebra/Group/Differential.lean
+++ b/GroundZero/Algebra/Group/Differential.lean
@@ -17,21 +17,21 @@ universe u v u' v' w
 namespace Group
   variable {G : Group}
 
-  hott def imImplKer {φ : Hom G G} (H : φ ⋅ φ = 0) : (im φ).set ⊆ (ker φ).set :=
+  hott definition imImplKer {φ : Hom G G} (H : φ ⋅ φ = 0) : (im φ).set ⊆ (ker φ).set :=
   begin
     intro x; fapply HITs.Merely.rec; apply G.hset;
     intro ⟨y, p⟩; change _ = _; transitivity; apply ap _ (Id.inv p);
     apply @idhom _ _ _ _ _ (φ ⋅ φ) 0; apply H
   end
 
-  noncomputable hott def boundaryOfBoundary {φ : Hom G G}
+  noncomputable hott lemma boundaryOfBoundary {φ : Hom G G}
     (p : (im φ).set ⊆ (ker φ).set) : φ ⋅ φ = 0 :=
   begin
     fapply Hom.funext; intro x; apply p;
     apply HITs.Merely.elem; existsi x; reflexivity
   end
 
-  noncomputable hott def boundaryEqv (φ : Hom G G) :
+  noncomputable hott lemma boundaryEqv (φ : Hom G G) :
     (φ ⋅ φ = 0) ≃ ((im φ).set ⊆ (ker φ).set) :=
   begin
     apply Structures.propEquivLemma;
@@ -40,20 +40,20 @@ namespace Group
   end
 end Group
 
-def Diff := Σ (G : Abelian) (δ : Abelian.Hom G G), δ ⋅ δ = 0
+hott definition Diff := Σ (G : Abelian) (δ : Abelian.Hom G G), δ ⋅ δ = 0
 
 -- Accessors
-hott def Diff.abelian (G : Diff) := G.1
-hott def Diff.group (G : Diff) := G.abelian.group
+hott definition Diff.abelian (G : Diff) := G.1
+hott definition Diff.group (G : Diff) := G.abelian.group
 
-hott def Diff.δ   (G : Diff) : Group.Hom G.group G.group := G.2.1
-hott def Diff.sqr (G : Diff) : G.δ ⋅ G.δ = 0 := G.2.2
+hott definition Diff.δ   (G : Diff) : Group.Hom G.group G.group := G.2.1
+hott definition Diff.sqr (G : Diff) : G.δ ⋅ G.δ = 0 := G.2.2
 
 namespace Diff
-  open GroundZero.Algebra.Group (ker)
+  open GroundZero.Algebra.Group (im ker)
   variable (G : Diff)
 
-  hott def univ : (Group.im G.δ).set ⊆ (ker G.δ).set :=
+  hott lemma univ : (Group.im G.δ).set ⊆ (ker G.δ).set :=
   Group.imImplKer G.sqr
 end Diff