diff --git a/hott/algebra/category/functor/basic.hlean b/hott/algebra/category/functor/basic.hlean
index 9a7f6007f..7749d966c 100644
--- a/hott/algebra/category/functor/basic.hlean
+++ b/hott/algebra/category/functor/basic.hlean
@@ -201,7 +201,7 @@ namespace functor
   by (cases F; apply functor_mk_eq'_idp)
 
   definition functor_eq_eta' {F₁ F₂ : C ⇒ D} (p : F₁ = F₂)
-      : functor_eq' (ap to_fun_ob p) (!tr_compose⁻¹ ⬝ apd to_fun_hom p) = p :=
+      : functor_eq' (ap to_fun_ob p) (!tr_compose⁻¹ ⬝ apdt to_fun_hom p) = p :=
   begin
     cases p, cases F₁,
     refine _ ⬝ !functor_eq'_idp,
diff --git a/hott/algebra/category/precategory.hlean b/hott/algebra/category/precategory.hlean
index b3a88b12c..30967ff72 100644
--- a/hott/algebra/category/precategory.hlean
+++ b/hott/algebra/category/precategory.hlean
@@ -247,7 +247,7 @@ namespace category
   definition Precategory_eq_hom [unfold 3] {C D : Precategory} (p : C = D) (a b : C)
     : hom a b = hom (cast (ap carrier p) a) (cast (ap carrier p) b) :=
   by induction p; reflexivity
-  --(ap10 (ap10 (apd (λx, @hom (carrier x) (Precategory.struct x)) p⁻¹ᵖ) a) b)⁻¹ᵖ ⬝ _
+  --(ap10 (ap10 (apdt (λx, @hom (carrier x) (Precategory.struct x)) p⁻¹ᵖ) a) b)⁻¹ᵖ ⬝ _
 
 
   -- beta/eta rules
diff --git a/hott/arity.hlean b/hott/arity.hlean
index 8444a40b6..e611a6eb4 100644
--- a/hott/arity.hlean
+++ b/hott/arity.hlean
@@ -35,7 +35,7 @@ namespace eq
         apdi₀i₁...iₙ.
 
       For versions where only some arguments depend on some other arguments,
-      or for versions with transport in the conclusion (like apd), we don't have a
+      or for versions with transport in the conclusion (like apdt), we don't have a
       consistent naming scheme (yet).
 
       We don't prove each theorem systematically, but prove only the ones which we actually need.
diff --git a/hott/hit/quotient.hlean b/hott/hit/quotient.hlean
index 88f320cd5..1663af705 100644
--- a/hott/hit/quotient.hlean
+++ b/hott/hit/quotient.hlean
@@ -137,7 +137,7 @@ namespace quotient
       induction q,
       { exact Qpt p},
       { apply pi_pathover_left', esimp, intro c,
-        refine _ ⬝op apd Qpt (elim_type_eq_of_rel C f H c)⁻¹ᵖ,
+        refine _ ⬝op apdt Qpt (elim_type_eq_of_rel C f H c)⁻¹ᵖ,
         refine _ ⬝op (tr_compose Q Ppt _ _)⁻¹ ,
         rewrite ap_inv,
         refine pathover_cancel_right _ !tr_pathover⁻¹ᵒ,
diff --git a/hott/init/default.hlean b/hott/init/default.hlean
index 8d3e80475..df475a95a 100644
--- a/hott/init/default.hlean
+++ b/hott/init/default.hlean
@@ -19,7 +19,7 @@ namespace core
   export sigma (hiding pr1 pr2)
   export [notation] prod
   export [notation] nat
-  export eq (idp idpath concat inverse transport ap ap10 cast tr_inv homotopy ap11 apd refl)
+  export eq (idp idpath concat inverse transport ap ap10 cast tr_inv homotopy ap11 apdt refl)
   export [declaration] function
   export equiv (to_inv to_right_inv to_left_inv)
   export is_equiv (inv right_inv left_inv adjointify)
diff --git a/hott/init/equiv.hlean b/hott/init/equiv.hlean
index 79573781b..9eadedf27 100644
--- a/hott/init/equiv.hlean
+++ b/hott/init/equiv.hlean
@@ -186,7 +186,7 @@ namespace is_equiv
           = right_inv f (f x) ▸ df (f⁻¹ (f x))   : by esimp
       ... = ap f (left_inv f x) ▸ df (f⁻¹ (f x)) : by rewrite -adj
       ... = left_inv f x ▸ df (f⁻¹ (f x))        : by rewrite -tr_compose
-      ... = df x                                 : by rewrite (apd df (left_inv f x))
+      ... = df x                                 : by rewrite (apdt df (left_inv f x))
 
   theorem adj_inv (b : B) : left_inv f (f⁻¹ b) = ap f⁻¹ (right_inv f b) :=
   is_equiv_rect f _
@@ -379,7 +379,7 @@ namespace equiv
             = right_inv f (f x) ▸ df (f⁻¹ (f x))   : by esimp
         ... = ap f (left_inv f x) ▸ df (f⁻¹ (f x)) : by rewrite -adj
         ... = left_inv f x ▸ df (f⁻¹ (f x))        : by rewrite -tr_compose
-        ... = df x                                 : by rewrite (apd df (left_inv f x))
+        ... = df x                                 : by rewrite (apdt df (left_inv f x))
   end
 
   section
diff --git a/hott/init/funext.hlean b/hott/init/funext.hlean
index 48e38ce1c..6e0929471 100644
--- a/hott/init/funext.hlean
+++ b/hott/init/funext.hlean
@@ -235,7 +235,7 @@ axiom function_extensionality : funext
 variables {A : Type} {P : A → Type} {f g : Π x, P x}
 
 namespace funext
-  theorem is_equiv_apd [instance] (f g : Π x, P x) : is_equiv (@apd10 A P f g) :=
+  theorem is_equiv_apdt [instance] (f g : Π x, P x) : is_equiv (@apd10 A P f g) :=
   function_extensionality f g
 end funext
 
diff --git a/hott/init/path.hlean b/hott/init/path.hlean
index 86f668f86..b60a7defc 100644
--- a/hott/init/path.hlean
+++ b/hott/init/path.hlean
@@ -257,7 +257,7 @@ namespace eq
   definition ap11 {f g : A → B} (H : f = g) {x y : A} (p : x = y) : f x = g y :=
   by induction H; exact ap f p
 
-  definition apd [unfold 6] (f : Πa, P a) {x y : A} (p : x = y) : p ▸ f x = f y :=
+  definition apdt [unfold 6] (f : Πa, P a) {x y : A} (p : x = y) : p ▸ f x = f y :=
   by induction p; reflexivity
 
   definition ap011 [unfold 9] (f : A → B → C) (Ha : a = a') (Hb : b = b') : f a b = f a' b' :=
@@ -456,16 +456,16 @@ namespace eq
     tr_inv_tr p (transport P p z) = ap (transport P p) (inv_tr_tr p z) :=
   by induction p; reflexivity
 
-  /- some properties for apd -/
+  /- some properties for apdt -/
 
-  definition apd_idp (x : A) (f : Πx, P x) : apd f idp = idp :> (f x = f x) := idp
+  definition apdt_idp (x : A) (f : Πx, P x) : apdt f idp = idp :> (f x = f x) := idp
 
-  definition apd_con (f : Πx, P x) {x y z : A} (p : x = y) (q : y = z)
-    : apd f (p ⬝ q) = con_tr p q (f x) ⬝ ap (transport P q) (apd f p) ⬝ apd f q :=
+  definition apdt_con (f : Πx, P x) {x y z : A} (p : x = y) (q : y = z)
+    : apdt f (p ⬝ q) = con_tr p q (f x) ⬝ ap (transport P q) (apdt f p) ⬝ apdt f q :=
   by cases p; cases q; apply idp
 
-  definition apd_inv (f : Πx, P x) {x y : A} (p : x = y)
-    : apd f p⁻¹ = (eq_inv_tr_of_tr_eq (apd f p))⁻¹ :=
+  definition apdt_inv (f : Πx, P x) {x y : A} (p : x = y)
+    : apdt f p⁻¹ = (eq_inv_tr_of_tr_eq (apdt f p))⁻¹ :=
   by cases p; apply idp
 
 
@@ -562,11 +562,11 @@ namespace eq
   definition tr_eq_cast_ap_fn {P : A → Type} {x y} (p : x = y) : transport P p = cast (ap P p) :=
   by induction p; reflexivity
 
-  /- The behavior of [ap] and [apd] -/
+  /- The behavior of [ap] and [apdt] -/
 
-  -- In a constant fibration, [apd] reduces to [ap], modulo [transport_const].
-  definition apd_eq_tr_constant_con_ap (f : A → B) (p : x = y) :
-    apd f p = tr_constant p (f x) ⬝ ap f p :=
+  -- In a constant fibration, [apdt] reduces to [ap], modulo [transport_const].
+  definition apdt_eq_tr_constant_con_ap (f : A → B) (p : x = y) :
+    apdt f p = tr_constant p (f x) ⬝ ap f p :=
   by induction p; reflexivity
 
 
@@ -708,17 +708,17 @@ namespace eq
                         ⬝ (ap_con f p' q')⁻¹ :=
   by induction r; induction s; induction q; induction p; reflexivity
 
-  definition apd02 [unfold 8] {p q : x = y} (f : Π x, P x) (r : p = q) :
-    apd f p = transport2 P r (f x) ⬝ apd f q :=
+  definition apdt02 [unfold 8] {p q : x = y} (f : Π x, P x) (r : p = q) :
+    apdt f p = transport2 P r (f x) ⬝ apdt f q :=
   by induction r; exact !idp_con⁻¹
 
   -- And now for a lemma whose statement is much longer than its proof.
-  definition apd02_con {P : A → Type} (f : Π x:A, P x) {x y : A}
+  definition apdt02_con {P : A → Type} (f : Π x:A, P x) {x y : A}
       {p1 p2 p3 : x = y} (r1 : p1 = p2) (r2 : p2 = p3) :
-    apd02 f (r1 ⬝ r2) = apd02 f r1
-      ⬝ whisker_left (transport2 P r1 (f x)) (apd02 f r2)
+    apdt02 f (r1 ⬝ r2) = apdt02 f r1
+      ⬝ whisker_left (transport2 P r1 (f x)) (apdt02 f r2)
       ⬝ con.assoc' _ _ _
-      ⬝ (whisker_right (tr2_con r1 r2 (f x))⁻¹ (apd f p3)) :=
+      ⬝ (whisker_right (tr2_con r1 r2 (f x))⁻¹ (apdt f p3)) :=
   by induction r2; induction r1; induction p1; reflexivity
 
 end eq
diff --git a/hott/types/eq.hlean b/hott/types/eq.hlean
index 5c9827b61..ec8c91372 100644
--- a/hott/types/eq.hlean
+++ b/hott/types/eq.hlean
@@ -151,7 +151,7 @@ namespace eq
 
   definition transport_eq_FlFr_D {B : A → Type} {f g : Πa, B a}
     (p : a₁ = a₂) (q : f a₁ = g a₁)
-      : transport (λx, f x = g x) p q = (apd f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apd g p) :=
+      : transport (λx, f x = g x) p q = (apdt f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apdt g p) :=
   by induction p; rewrite [▸*,idp_con,ap_id]
 
   definition transport_eq_FFlr (p : a₁ = a₂) (q : h (f a₁) = a₁)
@@ -189,7 +189,7 @@ namespace eq
   by induction p; rewrite [▸*,idp_con]; exact idpo
 
   definition pathover_eq_FlFr_D {B : A → Type} {f g : Πa, B a} (p : a₁ = a₂) (q : f a₁ = g a₁)
-    : q =[p] (apd f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apd g p) := /-(λx, f x = g x)-/
+    : q =[p] (apdt f p)⁻¹ ⬝ ap (transport B p) q ⬝ (apdt g p) := /-(λx, f x = g x)-/
   by induction p; rewrite [▸*,idp_con,ap_id];exact idpo
 
   definition pathover_eq_FFlr (p : a₁ = a₂) (q : h (f a₁) = a₁) : q =[p] (ap h (ap f p))⁻¹ ⬝ q ⬝ p :=
diff --git a/hott/types/pi.hlean b/hott/types/pi.hlean
index e2da9cffb..19dd30710 100644
--- a/hott/types/pi.hlean
+++ b/hott/types/pi.hlean
@@ -215,12 +215,12 @@ namespace pi
     begin
       intro h, apply eq_of_homotopy, intro a', esimp,
       rewrite [adj f0 a',-tr_compose,fn_tr_eq_tr_fn _ f1,right_inv (f1 _)],
-      apply apd
+      apply apdt
     end,
     begin
       intro h, apply eq_of_homotopy, intro a, esimp,
       rewrite [left_inv (f1 _)],
-      apply apd
+      apply apdt
     end
   end
 
diff --git a/hott/types/trunc.hlean b/hott/types/trunc.hlean
index ec16ff10e..f532ee034 100644
--- a/hott/types/trunc.hlean
+++ b/hott/types/trunc.hlean
@@ -281,7 +281,7 @@ namespace is_trunc
     (imp : Π{a b : A}, R a b → a = b) : is_set A :=
   is_set_of_axiom_K
     (λa p,
-      have H2 : transport (λx, R a x → a = x) p (@imp a a) = @imp a a, from !apd,
+      have H2 : transport (λx, R a x → a = x) p (@imp a a) = @imp a a, from !apdt,
       have H3 : Π(r : R a a), transport (λx, a = x) p (imp r)
                               = imp (transport (λx, R a x) p r), from
         to_fun (equiv.symm !heq_pi) H2,
diff --git a/tests/lean/531.hlean b/tests/lean/531.hlean
index e867d51a0..1325896d0 100644
--- a/tests/lean/531.hlean
+++ b/tests/lean/531.hlean
@@ -52,7 +52,7 @@ namespace colimit
   definition comp_cglue [D : diagram] {P : colimit D → Type}
     (Pincl : Π⦃i : Iob⦄ (x : ob i), P (ι x))
     (Pglue : Π(j : Ihom) (x : ob (dom j)), cglue j x ▸ Pincl (hom j x) = Pincl x)
-      {j : Ihom} (x : ob (dom j)) : apd (rec Pincl Pglue) (cglue j x) = sorry ⬝ Pglue j x ⬝ sorry :=
+      {j : Ihom} (x : ob (dom j)) : apdt (rec Pincl Pglue) (cglue j x) = sorry ⬝ Pglue j x ⬝ sorry :=
   --the sorry's in the statement can be removed when comp_incl is definitional
   sorry --idp
 
diff --git a/tests/lean/hott/433.hlean b/tests/lean/hott/433.hlean
index 5cfd8ece8..a92e36460 100644
--- a/tests/lean/hott/433.hlean
+++ b/tests/lean/hott/433.hlean
@@ -38,7 +38,7 @@ namespace pi
   /- The identification of the path space of a dependent function space, up to equivalence, is of course just funext. -/
 
   definition path_equiv_homotopy (H : funext) (f g : Πx, B x) : (f = g) ≃ (f ~ g) :=
-  equiv.mk _ !is_equiv_apd
+  equiv.mk _ !is_equiv_apdt
 
   definition is_equiv_path_pi [instance] (H : funext) (f g : Πx, B x)
       : is_equiv (@eq_of_homotopy _ _ f g) :=
@@ -111,11 +111,11 @@ namespace pi
   rewrite -tr_compose,
   rewrite {f1 a' _}(fn_tr_eq_tr_fn _ f1 _),
   rewrite (right_inv (f1 _) _),
-  apply apd,
+  apply apdt,
   intro h,
   apply eq_of_homotopy, intro a, esimp,
   apply (transport_V (λx, right_inv f0 a ▸ x = h a) (left_inv (f1 _) _)),
-  apply apd
+  apply apdt
   end
 
 end pi
diff --git a/tests/lean/hott/531b.hlean b/tests/lean/hott/531b.hlean
index 1d85fa49b..872f00902 100644
--- a/tests/lean/hott/531b.hlean
+++ b/tests/lean/hott/531b.hlean
@@ -52,7 +52,7 @@ namespace colimit
   definition comp_cglue [D : diagram] {P : colimit D → Type}
     (Pincl : Π⦃i : Iob⦄ (x : ob i), P (ι x))
     (Pglue : Π(j : Ihom) (x : ob (dom j)), cglue j x ▸ Pincl (hom j x) = Pincl x)
-      {j : Ihom} (x : ob (dom j)) : apd (rec Pincl Pglue) (cglue j x) = sorry ⬝ Pglue j x ⬝ sorry :=
+      {j : Ihom} (x : ob (dom j)) : apdt (rec Pincl Pglue) (cglue j x) = sorry ⬝ Pglue j x ⬝ sorry :=
   --the sorry's in the statement can be removed when comp_incl is definitional
   sorry --idp