diff --git a/hott/algebra/category/groupoid.hlean b/hott/algebra/category/groupoid.hlean
index 50b3ff5c1..815c07ba8 100644
--- a/hott/algebra/category/groupoid.hlean
+++ b/hott/algebra/category/groupoid.hlean
@@ -6,7 +6,7 @@ Authors: Jakob von Raumer, Floris van Doorn
 Ported from Coq HoTT
 -/
 
-import .iso ..group
+import .iso algebra.group
 
 open eq is_trunc iso category algebra nat unit
 
diff --git a/hott/hit/set_quotient.hlean b/hott/hit/set_quotient.hlean
index e9eb6b1a6..46bc6cb25 100644
--- a/hott/hit/set_quotient.hlean
+++ b/hott/hit/set_quotient.hlean
@@ -6,7 +6,7 @@ Authors: Floris van Doorn
 Declaration of set-quotients, i.e. quotient of a mere relation which is then set-truncated.
 -/
 
-import function algebra.relation types.trunc types.eq
+import function algebra.relation types.trunc types.eq hit.quotient
 
 open eq is_trunc trunc quotient equiv
 
@@ -32,9 +32,9 @@ section
   begin
     apply (@trunc.rec_on _ _ P x),
     { intro x', apply Pt},
-    { intro y, fapply (quotient.rec_on y),
-      { exact Pc},
-      { intros, exact pathover_of_pathover_ap P tr (Pp H)}}
+    { intro y, induction y,
+      { apply Pc},
+      { exact pathover_of_pathover_ap P tr (Pp H)}}
   end
 
   protected definition rec_on [reducible] {P : set_quotient → Type} (x : set_quotient)
diff --git a/hott/types/trunc.hlean b/hott/types/trunc.hlean
index e2113c372..818473643 100644
--- a/hott/types/trunc.hlean
+++ b/hott/types/trunc.hlean
@@ -186,15 +186,14 @@ namespace trunc
 
   protected definition encode (n : trunc_index) (aa aa' : trunc n.+1 A) : aa = aa' → trunc.code n aa aa' :=
   begin
-    intro p, induction p, apply (trunc.rec_on aa),
-    intro a, esimp [trunc.code,trunc.rec_on], exact (tr idp)
+    intro p, induction p, induction aa with a, esimp [trunc.code,trunc.rec_on], exact (tr idp)
   end
 
   protected definition decode (n : trunc_index) (aa aa' : trunc n.+1 A) : trunc.code n aa aa' → aa = aa' :=
   begin
-    eapply (trunc.rec_on aa'), eapply (trunc.rec_on aa),
-    intro a a' x, esimp [trunc.code, trunc.rec_on] at x,
-    apply (trunc.rec_on x), intro p, exact (ap tr p)
+    induction aa' with a', induction aa with a,
+    esimp [trunc.code, trunc.rec_on], intro x,
+    induction x with p, exact ap tr p,
   end
 
   definition trunc_eq_equiv [constructor] (n : trunc_index) (aa aa' : trunc n.+1 A)