diff --git a/algebra/direct_sum.hlean b/algebra/direct_sum.hlean
index 8596dc3..e4fdeef 100644
--- a/algebra/direct_sum.hlean
+++ b/algebra/direct_sum.hlean
@@ -6,9 +6,9 @@ Authors: Floris van Doorn, Egbert Rijke
 Constructions with groups
 -/
 
-import .quotient_group .free_commutative_group
+import .quotient_group .free_commutative_group .product_group
 
-open eq algebra is_trunc set_quotient relation sigma prod sum list trunc function equiv sigma.ops
+open eq is_equiv algebra is_trunc set_quotient relation sigma prod sum list trunc function equiv sigma.ops
 
 namespace group
 
@@ -73,10 +73,10 @@ namespace group
     definition dirsum_elim [constructor] (f : Πi, Y i →g A') : dirsum →g A' :=
     gqg_elim _ (free_ab_group_elim (λv, f v.1 v.2)) (dirsum_elim_resp_quotient f)
 
-    definition dirsum_elim_compute (f : Πi, Y i →g A') (i : I) :
-      dirsum_elim f ∘g dirsum_incl i ~ f i :=
+    definition dirsum_elim_compute (f : Πi, Y i →g A') (i : I) (y : Y i) :
+      dirsum_elim f (dirsum_incl i y) = f i y :=
     begin
-      intro g, apply one_mul
+      apply one_mul
     end
 
     definition dirsum_elim_unique (f : Πi, Y i →g A') (k : dirsum →g A')
@@ -89,6 +89,35 @@ namespace group
 
   end
 
+  definition binary_dirsum (G H : AbGroup) : dirsum (bool.rec G H) ≃g G ×ag H :=
+  let branch := bool.rec G H in
+  let to_hom := (dirsum_elim (bool.rec (product_inl G H) (product_inr G H))
+                : dirsum (bool.rec G H) →g G ×ag H) in
+  let from_hom := (Group_sum_elim (dirsum (bool.rec G H))
+                    (dirsum_incl branch bool.ff) (dirsum_incl branch bool.tt)
+                  : G ×g H →g dirsum branch) in
+  begin
+    fapply isomorphism.mk,
+    { exact dirsum_elim (bool.rec (product_inl G H) (product_inr G H)) },
+    fapply adjointify,
+    { exact from_hom },
+    { intro gh, induction gh with g h,
+        exact prod_eq (mul_one (1 * g) ⬝ one_mul g) (ap (λ o, o * h) (mul_one 1) ⬝ one_mul h) },
+    { refine dirsum.rec _ _ _,
+      { intro b x,
+        refine ap from_hom (dirsum_elim_compute (bool.rec (product_inl G H) (product_inr G H)) b x) ⬝ _,
+        induction b,
+        { exact ap (λ y, dirsum_incl branch bool.ff x * y) (to_respect_one (dirsum_incl branch bool.tt)) ⬝ mul_one _ },
+        { exact ap (λ y, y * dirsum_incl branch bool.tt x) (to_respect_one (dirsum_incl branch bool.ff)) ⬝ one_mul _ }
+      },
+      { refine ap from_hom (to_respect_one to_hom) ⬝ to_respect_one from_hom },
+      { intro g h gβ hβ,
+        refine ap from_hom (to_respect_mul to_hom _ _) ⬝ to_respect_mul from_hom _ _ ⬝ _,
+        exact ap011 mul gβ hβ
+      }
+    }
+  end
+
   variables {I J : Set} {Y Y' Y'' : I → AbGroup}
 
   definition dirsum_functor [constructor] (f : Πi, Y i →g Y' i) : dirsum Y →g dirsum Y' :=
diff --git a/algebra/product_group.hlean b/algebra/product_group.hlean
index 63fa749..2bd56d1 100644
--- a/algebra/product_group.hlean
+++ b/algebra/product_group.hlean
@@ -8,7 +8,7 @@ Constructions with groups
 
 import algebra.group_theory hit.set_quotient types.list types.sum .subgroup .quotient_group
 
-open eq algebra is_trunc set_quotient relation sigma sigma.ops prod prod.ops sum list trunc function
+open eq algebra is_trunc set_quotient relation sigma prod prod.ops sum list trunc function
      equiv
 namespace group
 
@@ -61,6 +61,19 @@ namespace group
   infix ` ×g `:60 := group.product
   infix ` ×ag `:60 := group.ab_product
 
+  definition product_inl [constructor] (G H : Group) : G →g G ×g H :=
+    homomorphism.mk (λx, (x, one)) (λx y, prod_eq !refl !one_mul⁻¹)
+
+  definition product_inr [constructor] (G H : Group) : H →g G ×g H :=
+    homomorphism.mk (λx, (one, x)) (λx y, prod_eq !one_mul⁻¹ !refl)
+
+  definition Group_sum_elim [constructor] {G H : Group} (I : AbGroup) (φ : G →g I) (ψ : H →g I) : G ×g H →g I :=
+    homomorphism.mk (λx, φ x.1 * ψ x.2) abstract (λx y, calc
+      φ (x.1 * y.1) * ψ (x.2 * y.2) = (φ x.1 * φ y.1) * (ψ x.2 * ψ y.2)
+                                      : by exact ap011 mul (to_respect_mul φ x.1 y.1) (to_respect_mul ψ x.2 y.2)
+                                ... = (φ x.1 * ψ x.2) * (φ y.1 * ψ y.2)
+                                      : by exact interchange I (φ x.1) (φ y.1) (ψ x.2) (ψ y.2)) end
+
   definition product_functor [constructor] {G G' H H' : Group} (φ : G →g H) (ψ : G' →g H') :
     G ×g G' →g H ×g H' :=
   homomorphism.mk (λx, (φ x.1, ψ x.2)) (λx y, prod_eq !to_respect_mul !to_respect_mul)
@@ -73,4 +86,7 @@ namespace group
 
   infix ` ×≃g `:60 := group.product_isomorphism
 
+  definition product_group_mul_eq {G H : Group} (g h : G ×g H) : g * h = product_mul g h :=
+  idp
+
 end group
diff --git a/algebra/quotient_group.hlean b/algebra/quotient_group.hlean
index ad81334..b2961ae 100644
--- a/algebra/quotient_group.hlean
+++ b/algebra/quotient_group.hlean
@@ -227,10 +227,10 @@ namespace group
       unfold qg_map, esimp, exact to_respect_mul f g h }
   end
 
-  definition quotient_group_compute (f : G →g G') (H : Π⦃g⦄, N g → f g = 1) :
-    quotient_group_elim f H ∘g qg_map N ~ f :=
+  definition quotient_group_compute (f : G →g G') (H : Π⦃g⦄, N g → f g = 1) (g : G) :
+    quotient_group_elim f H (qg_map N g) = f g :=
   begin
-    intro g, reflexivity
+    reflexivity
   end
 
   definition gelim_unique (f : G →g G') (H : Π⦃g⦄, N g → f g = 1) (k : quotient_group N →g G')
@@ -247,7 +247,7 @@ namespace group
   end
 
   definition qg_universal_property (f : G →g G') (H : Π⦃g⦄, N g → f g = 1)  :
-    is_contr (Σ(g : quotient_group N →g G'), g ∘g qg_map N ~ f) :=
+    is_contr (Σ(g : quotient_group N →g G'), g ∘ qg_map N ~ f) :=
   begin
     fapply is_contr.mk,
       -- give center of contraction
@@ -442,7 +442,7 @@ definition kernel_quotient_extension {A B : AbGroup} (f : A →g B) : quotient_a
   end
 
 definition kernel_quotient_extension_triangle {A B : AbGroup} (f : A →g B) :
-  kernel_quotient_extension f ∘g ab_qg_map (kernel_subgroup f) ~ f :=
+  kernel_quotient_extension f ∘ ab_qg_map (kernel_subgroup f) ~ f :=
   begin
     intro a,
     apply quotient_group_compute
diff --git a/algebra/seq_colim.hlean b/algebra/seq_colim.hlean
index 76b1491..b6824cc 100644
--- a/algebra/seq_colim.hlean
+++ b/algebra/seq_colim.hlean
@@ -18,14 +18,21 @@ namespace group
     definition seq_colim_incl [constructor] (i : ℕ) : A i →g seq_colim :=
       qg_map _ ∘g dirsum_incl A i
 
-    definition seq_colim_quotient (h : Πi, A i →g A') (k : Πi a, h i a = h (i + 1) (f i a))
+    definition seq_colim_quotient (h : Πi, A i →g A') (k : Πi a, h i a = h (succ i) (f i a))
                                   (v : seq_colim_carrier) (r : ∥seq_colim_rel v∥) : dirsum_elim h v = 1 :=
       begin
-        induction r with r, induction r, exact sorry
+        induction r with r, induction r, 
+        refine !to_respect_mul ⬝ _,
+        refine ap (λγ, group_fun (dirsum_elim h) (group_fun (dirsum_incl A i) a) * group_fun (dirsum_elim h) γ) (!to_respect_inv)⁻¹ ⬝ _,
+        refine ap (λγ, γ * group_fun (dirsum_elim h) (group_fun (dirsum_incl A (succ i)) (f i a)⁻¹)) !dirsum_elim_compute ⬝ _,
+        refine ap (λγ, (h i a) * γ) !dirsum_elim_compute ⬝ _,
+        refine ap (λγ, γ * group_fun (h (succ i)) (f i a)⁻¹) !k ⬝ _,
+        refine ap (λγ, group_fun (h (succ i)) (f i a) * γ) (!to_respect_inv) ⬝ _,
+        exact !mul.right_inv
       end
 
     definition seq_colim_elim [constructor] (h : Πi, A i →g A')
-                                            (k : Πi a, h i a = h (i + 1) (f i a)) : seq_colim →g A' :=
+                                            (k : Πi a, h i a = h (succ i) (f i a)) : seq_colim →g A' :=
       gqg_elim _ (dirsum_elim h) (seq_colim_quotient h k)
 
   end
diff --git a/colim.hlean b/colim.hlean
index e772352..7619b49 100644
--- a/colim.hlean
+++ b/colim.hlean
@@ -251,7 +251,7 @@ namespace seq_colim
 
   definition pshift_equiv_pinclusion {A : ℕ → Type*} (f : Πn, A n →* A (succ n)) (n : ℕ) :
     psquare (pinclusion f n) (pinclusion (λn, f (n+1)) n) (f n) (pshift_equiv f)  :=
-  phomotopy.mk homotopy.rfl begin 
+  phomotopy.mk homotopy.rfl begin
     refine !idp_con ⬝ _, esimp,
     induction n with n IH,
     { esimp[inclusion_pt], esimp[shift_diag], exact !idp_con⁻¹ },
@@ -259,7 +259,7 @@ namespace seq_colim
       rewrite ap_con, rewrite ap_con,
       refine _ ⬝ whisker_right _ !con.assoc,
       refine _ ⬝ (con.assoc (_ ⬝ _) _ _)⁻¹,
-      xrewrite [-IH], 
+      xrewrite [-IH],
       esimp[shift_up], rewrite [elim_glue,  ap_inv, -ap_compose'], esimp,
       rewrite [-+con.assoc], apply whisker_right,
       rewrite con.assoc, apply !eq_inv_con_of_con_eq,
@@ -285,7 +285,6 @@ namespace seq_colim
   !elim_glue
 
   omit p
-
   definition is_equiv_seq_colim_functor [constructor] [H : Πn, is_equiv (@g n)]
      : is_equiv (seq_colim_functor @g p) :=
   adjointify _ (seq_colim_functor (λn, (@g _)⁻¹) (λn a, inv_commute' g f f' p a))
@@ -323,29 +322,60 @@ namespace seq_colim
     : Π(x : seq_colim f), P x :=
   by induction v with Pincl Pglue; exact seq_colim.rec f Pincl Pglue
 
-  definition pseq_colim_pequiv [constructor] {A A' : ℕ → Type*} {f : Π{n}, A n →* A (n+1)}
-    {f' : Π{n}, A' n →* A' (n+1)} (g : Π{n}, A n ≃* A' n)
-    (p : Π⦃n⦄, g ∘* f ~ f' ∘* g) : pseq_colim @f ≃* pseq_colim @f' :=
-  pequiv_of_equiv (seq_colim_equiv @g p) (ap (ι _) (respect_pt g))
+  definition pseq_colim_pequiv' [constructor] {A A' : ℕ → Type*} {f : Πn, A n →* A (n+1)}
+    {f' : Πn, A' n →* A' (n+1)} (g : Πn, A n ≃* A' n)
+    (p : Π⦃n⦄, g (n+1) ∘* f n ~ f' n ∘* g n) : pseq_colim @f ≃* pseq_colim @f' :=
+  pequiv_of_equiv (seq_colim_equiv g p) (ap (ι _) (respect_pt (g _)))
+
+  definition pseq_colim_pequiv [constructor] {A A' : ℕ → Type*} {f : Πn, A n →* A (n+1)}
+    {f' : Πn, A' n →* A' (n+1)} (g : Πn, A n ≃* A' n)
+    (p : Πn, g (n+1) ∘* f n ~* f' n ∘* g n) : pseq_colim @f ≃* pseq_colim @f' :=
+  pseq_colim_pequiv' g (λn, @p n)
 
   definition seq_colim_equiv_constant [constructor] {A : ℕ → Type*} {f f' : Π⦃n⦄, A n → A (n+1)}
     (p : Π⦃n⦄ (a : A n), f a = f' a) : seq_colim f ≃ seq_colim f' :=
   seq_colim_equiv (λn, erfl) p
 
-  definition pseq_colim_equiv_constant [constructor] {A : ℕ → Type*} {f f' : Π{n}, A n →* A (n+1)}
-    (p : Π⦃n⦄, f ~ f') : pseq_colim @f ≃* pseq_colim @f' :=
-  pseq_colim_pequiv (λn, pequiv.rfl) p
+  definition pseq_colim_equiv_constant' [constructor] {A : ℕ → Type*} {f f' : Πn, A n →* A (n+1)}
+    (p : Π⦃n⦄, f n ~ f' n) : pseq_colim @f ≃* pseq_colim @f' :=
+  pseq_colim_pequiv' (λn, pequiv.rfl) p
 
-  definition pseq_colim_pequiv_pinclusion {A A' : ℕ → Type*} {f : Π(n), A n →* A (n+1)}
-    {f' : Π(n), A' n →* A' (n+1)} (g : Π(n), A n ≃* A' n)
-    (p : Π⦃n⦄, g (n+1) ∘* f n ~ f' n ∘* g n) (n : ℕ) :
+  definition pseq_colim_equiv_constant [constructor] {A : ℕ → Type*} {f f' : Πn, A n →* A (n+1)}
+    (p : Πn, f n ~* f' n) : pseq_colim @f ≃* pseq_colim @f' :=
+  pseq_colim_pequiv (λn, pequiv.rfl) (λn, !pid_pcompose ⬝* p n ⬝* !pcompose_pid⁻¹*)
+
+  definition pseq_colim_pequiv_pinclusion {A A' : ℕ → Type*} {f : Πn, A n →* A (n+1)}
+    {f' : Πn, A' n →* A' (n+1)} (g : Πn, A n ≃* A' n)
+    (p : Π⦃n⦄, g (n+1) ∘* f n ~* f' n ∘* g n) (n : ℕ) :
     psquare (pinclusion f n) (pinclusion f' n) (g n) (pseq_colim_pequiv g p) :=
-  sorry
+  phomotopy.mk homotopy.rfl begin
+    esimp, refine !idp_con ⬝ _,
+    induction n with n IH,
+    { esimp[inclusion_pt], exact !idp_con⁻¹ },
+    { esimp[inclusion_pt], rewrite [+ap_con, -+ap_inv, +con.assoc, +seq_colim_functor_glue],
+      xrewrite[-IH],
+      rewrite[-+ap_compose', -+con.assoc],
+      apply whisker_right, esimp,
+      rewrite[(eq_con_inv_of_con_eq (!to_homotopy_pt))],
+      rewrite[ap_con], esimp,
+      rewrite[-+con.assoc, ap_con, -ap_compose', +ap_inv],
+      rewrite[-+con.assoc],
+      refine _ ⬝ whisker_right _ (whisker_right _ (whisker_right _ (whisker_right _ !con.left_inv⁻¹))),
+      rewrite[idp_con, +con.assoc], apply whisker_left,
+      rewrite[ap_con, -ap_compose', con_inv, +con.assoc], apply whisker_left,
+      refine eq_inv_con_of_con_eq _,
+      symmetry, exact eq_of_square !natural_square
+    }
+  end
 
-  definition seq_colim_equiv_constant_pinclusion {A : ℕ → Type*} {f f' : Π⦃n⦄, A n →* A (n+1)}
-    (p : Π⦃n⦄ (a : A n), f a = f' a) (n : ℕ) :
+  definition seq_colim_equiv_constant_pinclusion {A : ℕ → Type*} {f f' : Πn, A n →* A (n+1)}
+    (p : Πn, f n ~* f' n) (n : ℕ) :
     pseq_colim_equiv_constant p ∘* pinclusion f n ~* pinclusion f' n  :=
-  sorry
+  begin
+    transitivity pinclusion f' n ∘* !pid,
+    refine phomotopy_of_psquare !pseq_colim_pequiv_pinclusion,
+    exact !pcompose_pid
+  end
 
   definition is_equiv_seq_colim_rec (P : seq_colim f → Type) :
     is_equiv (seq_colim_rec_unc :
@@ -369,7 +399,7 @@ namespace seq_colim
   equiv.mk _ !is_equiv_seq_colim_rec
   end functor
 
-  definition pseq_colim.elim [constructor] {A : ℕ → Type*} {B : Type*} {f : Π{n}, A n →* A (n+1)}
+  definition pseq_colim.elim' [constructor] {A : ℕ → Type*} {B : Type*} {f : Π{n}, A n →* A (n+1)}
     (g : Πn, A n →* B) (p : Πn, g (n+1) ∘* f ~ g n) : pseq_colim @f →* B :=
   begin
     fapply pmap.mk,
@@ -379,6 +409,10 @@ namespace seq_colim
     { esimp, apply respect_pt }
   end
 
+  definition pseq_colim.elim [constructor] {A : ℕ → Type*} {B : Type*} {f : Π{n}, A n →* A (n+1)}
+    (g : Πn, A n →* B) (p : Πn, g (n+1) ∘* f ~* g n) : pseq_colim @f →* B :=
+  pseq_colim.elim' g p
+
   definition prep0 [constructor] {A : ℕ → Type*} (f : pseq_diagram A) (k : ℕ) : A 0 →* A k :=
   pmap.mk (rep0 (λn x, f x) k)
           begin induction k with k p, reflexivity, exact ap (@f k) p ⬝ !respect_pt end
diff --git a/homology/homology.hlean b/homology/homology.hlean
index ed8eeff..6fe727b 100644
--- a/homology/homology.hlean
+++ b/homology/homology.hlean
@@ -1,3 +1,11 @@
+/-
+Copyright (c) 2017 Yuri Sulyma, Favonia
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Yuri Sulyma, Favonia
+
+Reduced homology theories
+-/
+
 import ..homotopy.spectrum ..homotopy.EM ..algebra.arrow_group ..algebra.direct_sum ..homotopy.fwedge ..choice ..homotopy.pushout ..move_to_lib
 
 open eq spectrum int pointed group algebra sphere nat equiv susp is_trunc
@@ -10,36 +18,39 @@ namespace homology
   structure homology_theory.{u} : Type.{u+1} :=
     (HH : ℤ → pType.{u} → AbGroup.{u})
     (Hh : Π(n : ℤ) {X Y : Type*} (f : X →* Y), HH n X →g HH n Y)
-    (Hid : Π(n : ℤ) {X : Type*} (x : HH n X), Hh n (pid X) x = x)
-    (Hcompose : Π(n : ℤ) {X Y Z : Type*} (f : Y →* Z) (g : X →* Y),
+    (Hpid : Π(n : ℤ) {X : Type*} (x : HH n X), Hh n (pid X) x = x)
+    (Hpcompose : Π(n : ℤ) {X Y Z : Type*} (f : Y →* Z) (g : X →* Y),
       Hh n (f ∘* g) ~ Hh n f ∘ Hh n g)
-    (Hsusp : Π(n : ℤ) (X : Type*), HH (succ n) (psusp X) ≃g HH n X)
-    (Hsusp_natural : Π(n : ℤ) {X Y : Type*} (f : X →* Y),
-      Hsusp n Y ∘ Hh (succ n) (psusp_functor f) ~ Hh n f ∘ Hsusp n X)
+    (Hpsusp : Π(n : ℤ) (X : Type*), HH (succ n) (psusp X) ≃g HH n X)
+    (Hpsusp_natural : Π(n : ℤ) {X Y : Type*} (f : X →* Y),
+      Hpsusp n Y ∘ Hh (succ n) (psusp_functor f) ~ Hh n f ∘ Hpsusp n X)
     (Hexact : Π(n : ℤ) {X Y : Type*} (f : X →* Y), is_exact_g (Hh n f) (Hh n (pcod f)))
     (Hadditive : Π(n : ℤ) {I : Set.{u}} (X : I → Type*), is_equiv
       (dirsum_elim (λi, Hh n (pinl i)) : dirsum (λi, HH n (X i)) → HH n (⋁ X)))
 
+  structure ordinary_homology_theory.{u} extends homology_theory.{u} : Type.{u+1} :=
+    (Hdimension : Π(n : ℤ), n ≠ 0 → is_contr (HH n (plift (psphere 0))))
+
   section
     parameter (theory : homology_theory)
     open homology_theory
 
     theorem HH_base_indep (n : ℤ) {A : Type} (a b : A)
       : HH theory n (pType.mk A a) ≃g HH theory n (pType.mk A b) :=
-      calc HH theory n (pType.mk A a) ≃g HH theory (int.succ n) (psusp A) : by exact (Hsusp theory n (pType.mk A a)) ⁻¹ᵍ
-                                  ... ≃g HH theory n (pType.mk A b)       : by exact Hsusp theory n (pType.mk A b)
+      calc HH theory n (pType.mk A a) ≃g HH theory (int.succ n) (psusp A) : by exact (Hpsusp theory n (pType.mk A a)) ⁻¹ᵍ
+                                  ... ≃g HH theory n (pType.mk A b)       : by exact Hpsusp theory n (pType.mk A b)
 
     theorem Hh_homotopy' (n : ℤ) {A B : Type*} (f : A → B) (p q : f pt = pt)
       : Hh theory n (pmap.mk f p) ~ Hh theory n (pmap.mk f q) := λ x,
     calc       Hh theory n (pmap.mk f p) x
-             = Hh theory n (pmap.mk f p) (Hsusp theory n A ((Hsusp theory n A)⁻¹ᵍ x))
-               : by exact ap (Hh theory n (pmap.mk f p)) (equiv.to_right_inv (equiv_of_isomorphism (Hsusp theory n A)) x)⁻¹
-         ... = Hsusp theory n B (Hh theory (succ n) (pmap.mk (susp.functor f) !refl) ((Hsusp theory n A)⁻¹ x))
-               : by exact (Hsusp_natural theory n (pmap.mk f p) ((Hsusp theory n A)⁻¹ᵍ x))⁻¹
-         ... = Hh theory n (pmap.mk f q) (Hsusp theory n A ((Hsusp theory n A)⁻¹ x))
-               : by exact Hsusp_natural theory n (pmap.mk f q) ((Hsusp theory n A)⁻¹ x)
+             = Hh theory n (pmap.mk f p) (Hpsusp theory n A ((Hpsusp theory n A)⁻¹ᵍ x))
+               : by exact ap (Hh theory n (pmap.mk f p)) (equiv.to_right_inv (equiv_of_isomorphism (Hpsusp theory n A)) x)⁻¹
+         ... = Hpsusp theory n B (Hh theory (succ n) (pmap.mk (susp.functor f) !refl) ((Hpsusp theory n A)⁻¹ x))
+               : by exact (Hpsusp_natural theory n (pmap.mk f p) ((Hpsusp theory n A)⁻¹ᵍ x))⁻¹
+         ... = Hh theory n (pmap.mk f q) (Hpsusp theory n A ((Hpsusp theory n A)⁻¹ x))
+               : by exact Hpsusp_natural theory n (pmap.mk f q) ((Hpsusp theory n A)⁻¹ x)
          ... = Hh theory n (pmap.mk f q) x
-               : by exact ap (Hh theory n (pmap.mk f q)) (equiv.to_right_inv (equiv_of_isomorphism (Hsusp theory n A)) x)
+               : by exact ap (Hh theory n (pmap.mk f q)) (equiv.to_right_inv (equiv_of_isomorphism (Hpsusp theory n A)) x)
 
     theorem Hh_homotopy (n : ℤ) {A B : Type*} (f g : A →* B) (h : f ~ g) : Hh theory n f ~ Hh theory n g := λ x,
     calc       Hh theory n f x
@@ -56,15 +67,15 @@ namespace homology
       { exact Hh theory n e⁻¹ᵉ* },
       { intro x, exact calc
               Hh theory n e (Hh theory n e⁻¹ᵉ* x)
-            = Hh theory n (e ∘* e⁻¹ᵉ*) x : by exact (Hcompose theory n e e⁻¹ᵉ* x)⁻¹
+            = Hh theory n (e ∘* e⁻¹ᵉ*) x : by exact (Hpcompose theory n e e⁻¹ᵉ* x)⁻¹
         ... = Hh theory n !pid x : by exact Hh_homotopy n (e ∘* e⁻¹ᵉ*) !pid (to_right_inv e) x
-        ... = x : by exact Hid theory n x
+        ... = x : by exact Hpid theory n x
       },
       { intro x, exact calc
               Hh theory n e⁻¹ᵉ* (Hh theory n e x)
-            = Hh theory n (e⁻¹ᵉ* ∘* e) x : by exact (Hcompose theory n e⁻¹ᵉ* e x)⁻¹
+            = Hh theory n (e⁻¹ᵉ* ∘* e) x : by exact (Hpcompose theory n e⁻¹ᵉ* e x)⁻¹
         ... = Hh theory n !pid x : by exact Hh_homotopy n (e⁻¹ᵉ* ∘* e) !pid (to_left_inv e) x
-        ... = x : by exact Hid theory n x
+        ... = x : by exact Hpid theory n x
       }
     end
 
diff --git a/homology/sphere.hlean b/homology/sphere.hlean
index e5eb677..d6f789f 100644
--- a/homology/sphere.hlean
+++ b/homology/sphere.hlean
@@ -1,6 +1,10 @@
--- Author: Kuen-Bang Hou (Favonia)
+/-
+Copyright (c) 2017 Kuen-Bang Hou (Favonia).
+Released under Apache 2.0 license as described in the file LICENSE.
+
+Author: Kuen-Bang Hou (Favonia)
+-/
 
-import core
 import .homology
 
 open eq pointed group algebra circle sphere nat equiv susp
@@ -13,7 +17,7 @@ section
 
   open homology_theory
 
-  theorem Hsphere : Π(n : ℤ)(m : ℕ), HH theory n (plift (psphere m)) ≃g HH theory (n - m) (plift (psphere 0)) :=
+  theorem Hpsphere : Π(n : ℤ)(m : ℕ), HH theory n (plift (psphere m)) ≃g HH theory (n - m) (plift (psphere 0)) :=
     begin
       intros n m, revert n, induction m with m,
       { exact λ n, isomorphism_ap (λ n, HH theory n (plift (psphere 0))) (sub_zero n)⁻¹ },
@@ -22,7 +26,7 @@ section
                ≃g HH theory n (psusp (plift (psphere m))) : by exact HH_isomorphism theory n (plift_psusp (psphere m))
            ... ≃g HH theory (succ (pred n)) (psusp (plift (psphere m)))
                   : by exact isomorphism_ap (λ n, HH theory n (psusp (plift (psphere m)))) (succ_pred n)⁻¹
-           ... ≃g HH theory (pred n) (plift (psphere m)) : by exact Hsusp theory (pred n) (plift (psphere m))
+           ... ≃g HH theory (pred n) (plift (psphere m)) : by exact Hpsusp theory (pred n) (plift (psphere m))
            ... ≃g HH theory (pred n - m) (plift (psphere 0)) : by exact v_0 (pred n)
            ... ≃g HH theory (n - succ m) (plift (psphere 0))
                   : by exact isomorphism_ap (λ n, HH theory n (plift (psphere 0))) (sub_sub n 1 m ⬝ ap (λ m, n - m) (add.comm 1 m))
diff --git a/homotopy/cohomology.hlean b/homotopy/cohomology.hlean
index 38a7e4c..8c25b70 100644
--- a/homotopy/cohomology.hlean
+++ b/homotopy/cohomology.hlean
@@ -189,7 +189,7 @@ structure cohomology_theory.{u} : Type.{u+1} :=
   (Hadditive : Π(n : ℤ) {I : Type.{u}} (X : I → Type*), has_choice 0 I →
     is_equiv (Group_pi_intro (λi, Hh n (pinl i)) : HH n (⋁ X) → Πᵍ i, HH n (X i)))
 
-structure ordinary_theory.{u} extends cohomology_theory.{u} : Type.{u+1} :=
+structure ordinary_cohomology_theory.{u} extends cohomology_theory.{u} : Type.{u+1} :=
  (Hdimension : Π(n : ℤ), n ≠ 0 → is_contr (HH n (plift pbool)))
 
 attribute cohomology_theory.HH [coercion]
@@ -270,7 +270,7 @@ cohomology_theory.mk
 -- print has_choice_lift
 -- print equiv_lift
 -- print has_choice_equiv_closed
-definition ordinary_theory_EM [constructor] (G : AbGroup) : ordinary_theory :=
-⦃ordinary_theory, cohomology_theory_spectrum (EM_spectrum G), Hdimension := EM_dimension G ⦄
+definition ordinary_cohomology_theory_EM [constructor] (G : AbGroup) : ordinary_cohomology_theory :=
+⦃ordinary_cohomology_theory, cohomology_theory_spectrum (EM_spectrum G), Hdimension := EM_dimension G ⦄
 
 end cohomology
diff --git a/homotopy/spectrum.hlean b/homotopy/spectrum.hlean
index 03e3e39..9ed3f7d 100644
--- a/homotopy/spectrum.hlean
+++ b/homotopy/spectrum.hlean
@@ -373,13 +373,17 @@ namespace spectrum
   definition spectrify_type_term {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) : Type* :=
   Ω[k] (X (n +' k))
 
-  definition spectrify_type_fun' {N : succ_str} (X : gen_prespectrum N) (k : ℕ) (n : N) :
+  definition spectrify_type_fun' {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) :
     Ω[k] (X n) →* Ω[k+1] (X (S n)) :=
   !loopn_succ_in⁻¹ᵉ* ∘* Ω→[k] (glue X n)
 
   definition spectrify_type_fun {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) :
     spectrify_type_term X n k →* spectrify_type_term X n (k+1) :=
-  spectrify_type_fun' X k (n +' k)
+  spectrify_type_fun' X (n +' k) k
+
+  definition spectrify_type_fun_zero {N : succ_str} (X : gen_prespectrum N) (n : N) :
+    spectrify_type_fun X n 0 ~* glue X n :=
+  !pid_pcompose
 
   definition spectrify_type {N : succ_str} (X : gen_prespectrum N) (n : N) : Type* :=
   pseq_colim (spectrify_type_fun X n)
@@ -393,20 +397,28 @@ namespace spectrum
           ... ≡ Y n
   -/
 
+  definition spectrify_type_fun'_succ {N : succ_str} (X : gen_prespectrum N) (n : N) (k : ℕ) :
+    spectrify_type_fun' X n (succ k) ~* Ω→ (spectrify_type_fun' X n k) :=
+  begin
+    refine _ ⬝* !ap1_pcompose⁻¹*,
+    apply !pwhisker_right, 
+    refine !to_pinv_pequiv_MK2
+  end
+
   definition spectrify_pequiv {N : succ_str} (X : gen_prespectrum N) (n : N) :
     spectrify_type X n ≃* Ω (spectrify_type X (S n)) :=
   begin
     refine !pshift_equiv ⬝e* _,
-    transitivity pseq_colim (λk, spectrify_type_fun' X (succ k) (S n +' k)),
+    transitivity pseq_colim (λk, spectrify_type_fun' X (S n +' k) (succ k)),
     fapply pseq_colim_pequiv,
     { intro n, apply loopn_pequiv_loopn, apply pequiv_ap X, apply succ_str.add_succ },
-    { intro k, apply to_homotopy,
+    { exact abstract begin intro k,
       refine !passoc⁻¹* ⬝* _, refine pwhisker_right _ (loopn_succ_in_inv_natural (succ k) _) ⬝* _,
       refine !passoc ⬝* _ ⬝* !passoc⁻¹*, apply pwhisker_left,
       refine !apn_pcompose⁻¹* ⬝* _ ⬝* !apn_pcompose, apply apn_phomotopy,
-      exact !glue_ptransport⁻¹* },
+      exact !glue_ptransport⁻¹* end end },
     refine _ ⬝e* !pseq_colim_loop⁻¹ᵉ*,
-    refine pseq_colim_equiv_constant (λn, !ap1_pcompose⁻¹*),
+    exact pseq_colim_equiv_constant (λn, !spectrify_type_fun'_succ),
   end
 
   definition spectrify [constructor] {N : succ_str} (X : gen_prespectrum N) : gen_spectrum N :=
@@ -419,21 +431,31 @@ namespace spectrum
   -- note: the forward map is (currently) not definitionally equal to gluen. Is that a problem?
   definition equiv_gluen {N : succ_str} (X : gen_spectrum N) (n : N) (k : ℕ)
     : X n ≃* Ω[k] (X (n +' k)) :=
-  by induction k with k f; reflexivity; exact f ⬝e* loopn_pequiv_loopn k (equiv_glue X (n +' k))
-                                                ⬝e* !loopn_succ_in⁻¹ᵉ*
+  by induction k with k f; reflexivity; exact f ⬝e* (loopn_pequiv_loopn k (equiv_glue X (n +' k))
+                                                ⬝e* !loopn_succ_in⁻¹ᵉ*)
+
+  definition equiv_gluen_inv_succ {N : succ_str} (X : gen_spectrum N) (n : N) (k : ℕ) :
+    (equiv_gluen X n (k+1))⁻¹ᵉ* ~*
+    (equiv_gluen X n k)⁻¹ᵉ* ∘* Ω→[k] (equiv_glue X (n +' k))⁻¹ᵉ* ∘* !loopn_succ_in :=
+  begin
+    refine !trans_pinv ⬝* pwhisker_left _ _, refine !trans_pinv ⬝* _, refine !to_pinv_pequiv_MK2 ◾* !pinv_pinv
+  end
 
   definition spectrify_map {N : succ_str} {X : gen_prespectrum N} : X →ₛ spectrify X :=
   begin
     fapply smap.mk,
     { intro n, exact pinclusion _ 0 },
-    { intro n, apply phomotopy_of_psquare, refine !pid_pcompose⁻¹* ⬝ph* _,
+    { intro n, apply phomotopy_of_psquare,
       refine !pid_pcompose⁻¹* ⬝ph* _,
-      --pshift_equiv_pinclusion (spectrify_type_fun X n) 0
-      refine _ ⬝v* _,
-      rotate 1, exact pshift_equiv_pinclusion (spectrify_type_fun X n) 0,
---      refine !passoc⁻¹* ⬝* pwhisker_left _ _ ⬝* _,
-      exact sorry
-}
+      refine !passoc ⬝* pwhisker_left _ (pshift_equiv_pinclusion (spectrify_type_fun X n) 0) ⬝* _,
+      refine !passoc⁻¹* ⬝* _,
+      refine _ ◾* (spectrify_type_fun_zero X n ⬝* !pid_pcompose⁻¹*),
+      refine !passoc ⬝* pwhisker_left _ !pseq_colim_pequiv_pinclusion ⬝* _,
+      refine pwhisker_left _ (pwhisker_left _ (ap1_pid) ⬝* !pcompose_pid) ⬝* _,
+      refine !passoc ⬝* pwhisker_left _ !seq_colim_equiv_constant_pinclusion ⬝* _, 
+      apply pinv_left_phomotopy_of_phomotopy,
+      exact !pseq_colim_loop_pinclusion⁻¹*
+    }
   end
 
   definition spectrify.elim {N : succ_str} {X : gen_prespectrum N} {Y : gen_spectrum N}
@@ -442,7 +464,9 @@ namespace spectrum
     fapply smap.mk,
     { intro n, fapply pseq_colim.elim,
       { intro k, refine !equiv_gluen⁻¹ᵉ* ∘* apn k (f (n +' k)) },
-      { intro k, apply to_homotopy, exact sorry }},
+      { intro k, refine !passoc ⬝* pwhisker_right _ !equiv_gluen_inv_succ ⬝* _,
+        refine !passoc ⬝* _, apply pwhisker_left,
+        refine !passoc ⬝* _, exact sorry }},
     { intro n, exact sorry }
   end
 
diff --git a/move_to_lib.hlean b/move_to_lib.hlean
index 340eb82..14bf911 100644
--- a/move_to_lib.hlean
+++ b/move_to_lib.hlean
@@ -180,6 +180,13 @@ namespace group
   definition isomorphism_ap {A : Type} (F : A → Group) {a b : A} (p : a = b) : F a ≃g F b :=
     isomorphism_of_eq (ap F p)
 
+  definition interchange (G : AbGroup) (a b c d : G) : (a * b) * (c * d) = (a * c) * (b * d) :=
+  calc (a * b) * (c * d) = a * (b * (c * d)) : by exact mul.assoc a b (c * d)
+                     ... = a * ((b * c) * d) : by exact ap (λ bcd, a * bcd) (mul.assoc b c d)⁻¹
+                     ... = a * ((c * b) * d) : by exact ap (λ bc, a * (bc * d)) (mul.comm b c)
+                     ... = a * (c * (b * d)) : by exact ap (λ bcd, a * bcd) (mul.assoc c b d)
+                     ... = (a * c) * (b * d) : by exact (mul.assoc a c (b * d))⁻¹
+
 end group open group
 
 namespace function