move more and update after changes

2018-09-11 17:06:46 +02:00 · 2018-09-11 17:06:46 +02:00 · 68345f75ce
commit 68345f75ce
parent f7e0ce0e20
25 changed files with 316 additions and 1486 deletions
--- a/algebra/arrow_group.hlean
+++ b/algebra/arrow_group.hlean
@ -19,35 +19,37 @@ namespace group
  definition ppi_inv [constructor] {A : Type*} {B : A → Type*} (f : Π*a, Ω (B a)) : Π*a, Ω (B a) :=
  proof ppi.mk (λa, (f a)⁻¹ᵖ) (respect_pt f)⁻² qed

-  definition inf_group_ppi [constructor] [instance] {A : Type*} (B : A → Type*) :
-    inf_group (Π*a, Ω (B a)) :=
+  definition inf_pgroup_pppi [constructor] {A : Type*} (B : A → Type*) :
+    inf_pgroup (Π*a, Ω (B a)) :=
  begin
-    fapply inf_group.mk,
+    fapply inf_pgroup.mk,
    { exact ppi_mul },
    { intro f g h, apply eq_of_phomotopy, fapply phomotopy.mk,
      { intro a, exact con.assoc (f a) (g a) (h a) },
      { symmetry, rexact eq_of_square (con2_assoc (respect_pt f) (respect_pt g) (respect_pt h)) }},
-    { apply ppi_const },
+    { exact ppi_inv },
    { intros f, apply eq_of_phomotopy, fapply phomotopy.mk,
      { intro a, exact one_mul (f a) },
      { symmetry, apply eq_of_square, refine _ ⬝vp !ap_id, apply natural_square_tr }},
    { intros f, apply eq_of_phomotopy, fapply phomotopy.mk,
      { intro a, exact mul_one (f a) },
      { reflexivity }},
-    { exact ppi_inv },
    { intro f, apply eq_of_phomotopy, fapply phomotopy.mk,
      { intro a, exact con.left_inv (f a) },
      { exact !con_left_inv_idp }},
  end

-  definition group_trunc_ppi [constructor] [instance] {A : Type*} (B : A → Type*) :
-    group (trunc 0 (Π*a, Ω (B a))) :=
-  !trunc_group
+  definition inf_group_ppi [constructor] {A : Type*} (B : A → Type*) :
+    inf_group (Π*a, Ω (B a)) :=
+  @inf_group_of_inf_pgroup _ (inf_pgroup_pppi B)
+
+  definition gppi_loop [constructor] {A : Type*} (B : A → Type*) : InfGroup :=
+  InfGroup.mk (Π*a, Ω (B a)) (inf_group_ppi B)

  definition Group_trunc_ppi [reducible] [constructor] {A : Type*} (B : A → Type*) : Group :=
-  Group.mk (trunc 0 (Π*a, Ω (B a))) _
+  gtrunc (gppi_loop B)

-  definition ab_inf_group_ppi [constructor] [instance] {A : Type*} (B : A → Type*) :
+  definition ab_inf_group_ppi [constructor] {A : Type*} (B : A → Type*) :
    ab_inf_group (Π*a, Ω (Ω (B a))) :=
  ⦃ab_inf_group, inf_group_ppi (λa, Ω (B a)), mul_comm :=
    begin
@ -56,23 +58,20 @@ namespace group
      { symmetry, rexact eq_of_square (eckmann_hilton_con2 (respect_pt f) (respect_pt g)) }
    end⦄

-  definition ab_group_trunc_ppi [constructor] [instance] {A : Type*} (B : A → Type*) :
-    ab_group (trunc 0 (Π*a, Ω (Ω (B a)))) :=
-  !trunc_ab_group
+  definition agppi_loop [constructor] {A : Type*} (B : A → Type*) : AbInfGroup :=
+  AbInfGroup.mk (Π*a, Ω (Ω (B a))) (ab_inf_group_ppi B)

  definition AbGroup_trunc_ppi [reducible] [constructor] {A : Type*} (B : A → Type*) : AbGroup :=
-  AbGroup.mk (trunc 0 (Π*a, Ω (Ω (B a)))) _
+  agtrunc (agppi_loop B)

-  definition trunc_ppi_isomorphic_pmap (A B : Type*)
-    : Group.mk (trunc 0 (Π*(a : A), Ω B)) !trunc_group
-      ≃g Group.mk (trunc 0 (A →* Ω B)) !trunc_group :=
-  begin
-    reflexivity,
-    -- apply trunc_isomorphism_of_equiv (pppi_equiv_pmap A (Ω B)),
-    -- intro h k, induction h with h h_pt, induction k with k k_pt, reflexivity
-  end
-
-  section
+  -- definition trunc_ppi_isomorphic_pmap (A B : Type*)
+  --   : Group.mk (trunc 0 (Π*(a : A), Ω B)) !group_trunc
+  --     ≃g Group.mk (trunc 0 (A →* Ω B)) !group_trunc :=
+  -- begin
+  --   reflexivity,
+  --   -- apply trunc_isomorphism_of_equiv (pppi_equiv_pmap A (Ω B)),
+  --   -- intro h k, induction h with h h_pt, induction k with k k_pt, reflexivity
+  -- end

  universe variables u v

@ -101,23 +100,22 @@ namespace group

  variable (k)

-  definition trunc_ppi_loop_isomorphism_lemma
-    : isomorphism.{(max u v) (max u v)}
-      (Group.mk (trunc 0 (k = k)) (@trunc_group (k = k) !inf_group_loop))
-      (Group.mk (trunc 0 (Π*(a : A), Ω (pType.mk (B a) (k a)))) !trunc_group) :=
+  definition gloop_ppi_isomorphism :
+    Ωg (pointed.Mk k) ≃∞g gppi_loop (λ a, pointed.Mk (ppi.to_fun k a)) :=
  begin
-    apply @trunc_isomorphism_of_equiv _ _ !inf_group_loop !inf_group_ppi (ppi_loop_equiv k),
+    apply inf_isomorphism_of_equiv (ppi_loop_equiv k),
    intro f g, induction k with k p, induction p,
    apply trans (phomotopy_of_eq_homomorphism f g),
    exact ppi_mul_loop (phomotopy_of_eq f) (phomotopy_of_eq g)
  end

-  end
+  definition trunc_ppi_loop_isomorphism_lemma :
+    gtrunc (gloop (pointed.Mk k)) ≃g gtrunc (gppi_loop (λa, pointed.Mk (k a))) :=
+  gtrunc_isomorphism_gtrunc (gloop_ppi_isomorphism k)

-  definition trunc_ppi_loop_isomorphism {A : Type*} (B : A → Type*)
-    : Group.mk (trunc 0 (Ω (Π*(a : A), B a))) !trunc_group
-      ≃g Group.mk (trunc 0 (Π*(a : A), Ω (B a))) !trunc_group :=
-  trunc_ppi_loop_isomorphism_lemma (ppi_const B)
+  definition trunc_ppi_loop_isomorphism {A : Type*} (B : A → Type*) :
+    gtrunc (gloop (Π*(a : A), B a)) ≃g gtrunc (gppi_loop B) :=
+  proof trunc_ppi_loop_isomorphism_lemma (ppi_const B) qed


  /- We first define the group structure on A →* Ω B (except for truncatedness).
@ -171,21 +169,24 @@ namespace group
    loop_susp_intro (pmap_mul f g) ~* pmap_mul (loop_susp_intro f) (loop_susp_intro g) :=
  pwhisker_right _ !ap1_pmap_mul ⬝* !pmap_mul_pcompose

-  definition inf_group_pmap [constructor] [instance] (A B : Type*) : inf_group (A →* Ω B) :=
-  !inf_group_ppi
+  definition InfGroup_pmap [reducible] [constructor] (A B : Type*) : InfGroup :=
+  InfGroup.mk (A →* Ω B) !inf_group_ppi

-  definition group_trunc_pmap [constructor] [instance] (A B : Type*) : group (trunc 0 (A →* Ω B)) :=
-  !trunc_group
+  definition InfGroup_pmap' [reducible] [constructor] (A : Type*) {B C : Type*} (e : Ω C ≃* B) :
+    InfGroup :=
+  InfGroup.mk (A →* B)
+    (@inf_group_of_inf_pgroup _ (inf_pgroup_pequiv_closed (ppmap_pequiv_ppmap_right e)
+      !inf_pgroup_pppi))

  definition Group_trunc_pmap [reducible] [constructor] (A B : Type*) : Group :=
-  Group.mk (trunc 0 (A →* Ω B)) _
+  Group.mk (trunc 0 (A →* Ω B)) (@group_trunc _ !inf_group_ppi)

  definition Group_trunc_pmap_homomorphism [constructor] {A A' B : Type*} (f : A' →* A) :
    Group_trunc_pmap A B →g Group_trunc_pmap A' B :=
  begin
    fapply homomorphism.mk,
    { apply trunc_functor, intro g, exact g ∘* f},
-    { intro g h, induction g with g, induction h with h, apply ap tr,
+    { intro g h, induction g with g, esimp, induction h with h, apply ap tr,
      apply eq_of_phomotopy, fapply phomotopy.mk,
      { intro a, reflexivity },
      { symmetry, refine _ ⬝ !idp_con⁻¹,
@ -242,16 +243,11 @@ namespace group

  definition ab_inf_group_pmap [constructor] [instance] (A B : Type*) :
    ab_inf_group (A →* Ω (Ω B)) :=
-  ⦃ab_inf_group, inf_group_pmap A (Ω B), mul_comm :=
-    begin
-      intro f g, apply eq_of_phomotopy, fapply phomotopy.mk,
-      { intro a, exact eckmann_hilton (f a) (g a) },
-      { symmetry, rexact eq_of_square (eckmann_hilton_con2 (respect_pt f) (respect_pt g)) }
-    end⦄
+  !ab_inf_group_ppi

  definition ab_group_trunc_pmap [constructor] [instance] (A B : Type*) :
    ab_group (trunc 0 (A →* Ω (Ω B))) :=
-  !trunc_ab_group
+  !ab_group_trunc

  definition AbGroup_trunc_pmap [reducible] [constructor] (A B : Type*) : AbGroup :=
  AbGroup.mk (trunc 0 (A →* Ω (Ω B))) _
--- a/algebra/direct_sum.hlean
+++ b/algebra/direct_sum.hlean
@ -165,13 +165,13 @@ namespace group
    fapply is_equiv.adjointify,
    exact dirsum_functor (λ i, (f i)⁻¹ᵍ),
    { intro ds,
-      refine (homomorphism_comp_compute (dirsum_functor (λ i, f i)) (dirsum_functor (λ i, (f i)⁻¹ᵍ)) _)⁻¹ ⬝ _,
+      refine (homomorphism_compose_eq (dirsum_functor (λ i, f i)) (dirsum_functor (λ i, (f i)⁻¹ᵍ)) _)⁻¹ ⬝ _,
      refine dirsum_functor_compose (λ i, f i) (λ i, (f i)⁻¹ᵍ) ds ⬝ _,
      refine dirsum_functor_homotopy _ (λ i, !gid) (λ i, to_right_inv (equiv_of_isomorphism (f i))) ds ⬝ _,
      exact !dirsum_functor_gid
    },
    { intro ds,
-      refine (homomorphism_comp_compute (dirsum_functor (λ i, (f i)⁻¹ᵍ)) (dirsum_functor (λ i, f i)) _)⁻¹ ⬝ _,
+      refine (homomorphism_compose_eq (dirsum_functor (λ i, (f i)⁻¹ᵍ)) (dirsum_functor (λ i, f i)) _)⁻¹ ⬝ _,
      refine dirsum_functor_compose (λ i, (f i)⁻¹ᵍ) (λ i, f i) ds ⬝ _,
      refine dirsum_functor_homotopy _ (λ i, !gid) (λ i x,
        proof
@ -197,19 +197,19 @@ namespace group
    fapply adjointify,
    { exact from_hom },
    { intro ds,
-      refine (homomorphism_comp_compute to_hom from_hom ds)⁻¹ ⬝ _,
+      refine (homomorphism_compose_eq to_hom from_hom ds)⁻¹ ⬝ _,
      refine @dirsum_homotopy I _ Y (dirsum Y) (to_hom ∘g from_hom) !gid _ ds,
      intro i y,
-      refine homomorphism_comp_compute to_hom from_hom _ ⬝ _,
+      refine homomorphism_compose_eq to_hom from_hom _ ⬝ _,
      refine ap to_hom (dirsum_elim_compute (λi, dirsum_incl (Y ∘ down.{u v}) (up.{u v} i)) i y) ⬝ _,
      refine dirsum_elim_compute _ (up.{u v} i) y ⬝ _,
      reflexivity
    },
    { intro ds,
-      refine (homomorphism_comp_compute from_hom to_hom ds)⁻¹ ⬝ _,
+      refine (homomorphism_compose_eq from_hom to_hom ds)⁻¹ ⬝ _,
      refine @dirsum_homotopy _ _ (Y ∘ down.{u v}) (dirsum (Y ∘ down.{u v})) (from_hom ∘g to_hom) !gid _ ds,
      intro i y, induction i with i,
-      refine homomorphism_comp_compute from_hom to_hom _ ⬝ _,
+      refine homomorphism_compose_eq from_hom to_hom _ ⬝ _,
      refine ap from_hom (dirsum_elim_compute (λi, dirsum_incl Y (down.{u v} i)) (up.{u v} i) y) ⬝ _,
      refine dirsum_elim_compute _ i y ⬝ _,
      reflexivity
--- a/algebra/free_group.hlean
+++ b/algebra/free_group.hlean
@ -234,7 +234,7 @@ definition is_trunc_rlist {n : ℕ₋₂} {X : Type} (H : is_trunc (n.+2) X) :
  is_trunc (n.+2) (rlist X) :=
 begin
  apply is_trunc_sigma, { apply is_trunc_list, apply is_trunc_sum },
-  intro l, apply is_trunc_succ_of_is_prop
+  intro l, exact is_trunc_succ_of_is_prop _ _ _
 end

 definition is_reduced_invert (v : X ⊎ X) : is_reduced (v::l) → is_reduced l :=
--- a/algebra/left_module.hlean
+++ b/algebra/left_module.hlean
@ -367,7 +367,7 @@ end
  isomorphism.mk (to_lminv φ) !is_equiv_inv

  definition isomorphism.trans [trans] [constructor] (φ : M₁ ≃lm M₂) (ψ : M₂ ≃lm M₃) : M₁ ≃lm M₃ :=
-  isomorphism.mk (ψ ∘lm φ) !is_equiv_compose
+  isomorphism.mk (ψ ∘lm φ) (is_equiv_compose ψ φ _ _)

  definition isomorphism.eq_trans [trans] [constructor]
     {M₁ M₂ : LeftModule R} {M₃ : LeftModule R} (φ : M₁ = M₂) (ψ : M₂ ≃lm M₃) : M₁ ≃lm M₃ :=
--- a/algebra/product_group.hlean
+++ b/algebra/product_group.hlean
@ -72,7 +72,7 @@ namespace group
      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
+                                      : by exact mul.comm4 (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' :=
--- a/algebra/quotient_group.hlean
+++ b/algebra/quotient_group.hlean
@ -331,7 +331,7 @@ namespace group
    have htpy : Π (a : A), K a ≃ L a,
    begin
      intro a,
-      apply @equiv_of_is_prop (a ∈ K) (a ∈ L) _ _ (K_in_L a) (L_in_K a),
+      exact @equiv_of_is_prop (a ∈ K) (a ∈ L) (K_in_L a) (L_in_K a) _ _,
    end,
    exact subgroup_rel_eq' htpy,
  end
@ -561,11 +561,12 @@ definition ab_group_kernel_quotient_to_image_codomain_triangle {A B : AbGroup} (
    fapply is_prop.elimo
  end

+-- set_option pp.all true
+-- print algebra._trans_of_Group_of_AbGroup_2
 definition is_surjective_kernel_quotient_to_image {A B : AbGroup} (f : A →g B)
           : is_surjective (ab_group_kernel_quotient_to_image f) :=
  begin
-    fapply is_surjective_factor (group_fun (ab_qg_map (kernel f))),
-    exact image_lift f,
+    refine is_surjective_factor (ab_qg_map (kernel f)) (image_lift f) _ _,
    apply @quotient_group_compute _ _ _ (@is_normal_subgroup_ab _ (kernel f) _),
    exact is_surjective_image_lift f
  end
@ -573,7 +574,7 @@ definition is_surjective_kernel_quotient_to_image {A B : AbGroup} (f : A →g B)
 definition is_embedding_kernel_quotient_to_image {A B : AbGroup} (f : A →g B)
           : is_embedding (ab_group_kernel_quotient_to_image f) :=
  begin
-    fapply is_embedding_factor (ab_group_kernel_quotient_to_image f) (image_incl f) (kernel_quotient_extension f),
+    fapply is_embedding_factor (image_incl f) (kernel_quotient_extension f),
    exact ab_group_kernel_quotient_to_image_codomain_triangle f,
    exact is_embedding_kernel_quotient_extension f
  end
--- a/algebra/submodule.hlean
+++ b/algebra/submodule.hlean
@ -164,7 +164,7 @@ definition is_prop_submodule (S : property M) [is_submodule M S] [H : is_prop M]
 begin apply @is_trunc_sigma, exact H end
 local attribute is_prop_submodule [instance]
 definition is_contr_submodule [instance] (S : property M) [is_submodule M S] [is_contr M] : is_contr (submodule S) :=
-is_contr_of_inhabited_prop 0
+is_contr_of_inhabited_prop 0 _

 definition submodule_isomorphism [constructor] (S : property M) [is_submodule M S] (h : Πg, g ∈ S) :
  submodule S ≃lm M :=
@ -244,7 +244,7 @@ local attribute is_prop_quotient_module [instance]

 definition is_contr_quotient_module [instance] (S : property M) [is_submodule M S] [is_contr M] :
  is_contr (quotient_module S) :=
-is_contr_of_inhabited_prop 0
+is_contr_of_inhabited_prop 0 _

 definition rel_of_is_contr_quotient_module (S : property M) [is_submodule M S]
  (H : is_contr (quotient_module S)) (m : M) : S m :=
--- a/choice.hlean
+++ b/choice.hlean
@ -18,7 +18,7 @@ equiv.mk _ (H A)
 definition has_choice_of_succ (X : Type) (H : Πk, has_choice (k.+1) X) (n : ℕ₋₂) : has_choice n X :=
 begin
  cases n with n,
-  { intro A, apply is_equiv_of_is_contr },
+  { intro A, exact is_equiv_of_is_contr _ _ _ },
  { exact H n }
 end

--- a/cohomology/basic.hlean
+++ b/cohomology/basic.hlean
@ -85,7 +85,6 @@ end
 definition cohomology_isomorphism_shomotopy_group_sp_cotensor (X : Type*) (Y : spectrum) {n m : ℤ}
  (p : -m = n) : H^n[X, Y] ≃g πₛ[m] (sp_cotensor X Y) :=
 begin
-  refine !trunc_ppi_isomorphic_pmap⁻¹ᵍ ⬝g _,
  refine parametrized_cohomology_isomorphism_shomotopy_group_spi (λx, Y) p ⬝g _,
  apply shomotopy_group_isomorphism_of_pequiv, intro k,
  apply pppi_pequiv_ppmap
@ -297,7 +296,7 @@ begin
    { exfalso, apply H, reflexivity },
    { apply is_conn_of_le, apply zero_le_of_nat n, exact is_conn_EMadd1 G n, }},
  { apply is_trunc_trunc_of_is_trunc, apply @is_contr_loop_of_is_trunc (n+1) (K G 0),
-    apply is_trunc_of_le _ (zero_le_of_nat n) }
+    apply is_trunc_of_le _ (zero_le_of_nat n) _ }
 end

 theorem EM_dimension (G : AbGroup) (n : ℤ) (H : n ≠ 0) :
--- a/cohomology/serre.hlean
+++ b/cohomology/serre.hlean
@ -85,13 +85,15 @@ begin
  exact pfiber_postnikov_map A n
 end

+
+set_option formatter.hide_full_terms false
 definition pfiber_postnikov_map_pred' (A : spectrum) (n k l : ℤ) (p : n + k = l) :
  pfiber (postnikov_map_pred (A k) (maxm2 l)) ≃* EM_spectrum (πₛ[n] A) l :=
 begin
  cases l with l l,
  { refine pfiber_postnikov_map_pred (A k) l ⬝e* _,
    exact EM_type_pequiv_EM A p },
-  { apply pequiv_of_is_contr, apply is_contr_pfiber_pid,
+  { refine pequiv_of_is_contr _ _ _ _, apply is_contr_pfiber_pid,
    apply is_contr_EM_spectrum_neg }
 end

--- a/colimit/seq_colim.hlean
+++ b/colimit/seq_colim.hlean
@ -322,8 +322,8 @@ variable (f)
 definition is_contr_seq_colim {A : ℕ → Type} (f : seq_diagram A)
  [Πk, is_contr (A k)] : is_contr (seq_colim f) :=
 begin
-  apply @is_trunc_is_equiv_closed (A 0),
-  apply is_equiv_inclusion0, intro n, apply is_equiv_of_is_contr
+  refine is_contr_is_equiv_closed (ι' f 0) _ _,
+  apply is_equiv_inclusion0, intro n, exact is_equiv_of_is_contr _ _ _
 end

 definition seq_colim_equiv_of_is_equiv [constructor] {n : ℕ} (H : Πk, k ≥ n → is_equiv (@f k)) :
@ -882,7 +882,7 @@ theorem is_trunc_fun_inclusion (k : ℕ₋₂) (H : Πn, is_trunc_fun k (@f n))
  is_trunc_fun k (ι' f 0) :=
 begin
  intro x,
-  apply @(is_trunc_equiv_closed_rev k (fiber_inclusion x)),
+  apply is_trunc_equiv_closed_rev k (fiber_inclusion x),
  apply is_trunc_fun_seq_colim_functor,
  intro n, apply is_trunc_fun_lrep, exact H
 end
@ -913,11 +913,11 @@ definition lrep_seq_diagram_fin {n : ℕ} (x : fin n) :
 begin
  induction x with k H, esimp, induction H with n H p,
    reflexivity,
-  exact ap (@lift_succ2 _) p
+  exact ap (@lift_succ _) p
 end

 definition lrep_seq_diagram_fin_lift_succ {n : ℕ} (x : fin n) :
-  lrep_seq_diagram_fin (lift_succ2 x) = ap (@lift_succ2 _) (lrep_seq_diagram_fin x) :=
+  lrep_seq_diagram_fin (lift_succ x) = ap (@lift_succ _) (lrep_seq_diagram_fin x) :=
 begin
  induction x with k H, reflexivity
 end
--- a/colimit/sequence.hlean
+++ b/colimit/sequence.hlean
@ -6,7 +6,7 @@ Authors: Floris van Doorn, Egbert Rijke

 import ..move_to_lib types.fin types.trunc

-open nat eq equiv sigma sigma.ops is_equiv is_trunc trunc prod fiber function
+open nat eq equiv sigma sigma.ops is_equiv is_trunc trunc prod fiber function is_conn

 namespace seq_colim

@ -77,7 +77,7 @@ namespace seq_colim
  begin
    induction H with m H Hlrepf,
    { apply is_equiv_id },
-    { exact is_equiv_compose (@f _) (lrep f H) },
+    { exact is_equiv_compose (@f _) (lrep f H) _ _ },
  end

  local attribute is_equiv_lrep [instance]
@ -211,7 +211,7 @@ namespace seq_colim

    open fin
    definition seq_diagram_fin [unfold_full] : seq_diagram fin :=
-    lift_succ2
+    lift_succ

    definition id0_seq [unfold_full] (a₁ a₂ : A 0) : ℕ → Type :=
    λ k, rep0 f k a₁ = rep0 f k a₂
--- a/component.hlean
+++ b/component.hlean
@ -0,0 +1,100 @@
+-- Author: Floris van Doorn
+
+import homotopy.connectedness .move_to_lib
+
+open eq equiv pointed is_conn is_trunc sigma prod trunc function group nat fiber
+
+namespace is_conn
+
+  open sigma.ops pointed trunc_index
+
+  /- this is equivalent to pfiber (A → ∥A∥₀) ≡ connect 0 A -/
+  definition component [constructor] (A : Type*) : Type* :=
+  pType.mk (Σ(a : A), merely (pt = a)) ⟨pt, tr idp⟩
+
+  lemma is_conn_component [instance] (A : Type*) : is_conn 0 (component A) :=
+  is_conn_zero_pointed'
+    begin intro x, induction x with a p, induction p with p, induction p, exact tidp end
+
+  definition component_incl [constructor] (A : Type*) : component A →* A :=
+  pmap.mk pr1 idp
+
+  definition is_embedding_component_incl [instance] (A : Type*) : is_embedding (component_incl A) :=
+  is_embedding_pr1 _
+
+  definition component_intro [constructor] {A B : Type*} (f : A →* B) (H : merely_constant f) :
+    A →* component B :=
+  begin
+    fapply pmap.mk,
+    { intro a, refine ⟨f a, _⟩, exact tinverse (merely_constant_pmap H a) },
+    exact subtype_eq !respect_pt
+  end
+
+  definition component_functor [constructor] {A B : Type*} (f : A →* B) :
+    component A →* component B :=
+  component_intro (f ∘* component_incl A) !merely_constant_of_is_conn
+
+  -- definition component_elim [constructor] {A B : Type*} (f : A →* B) (H : merely_constant f) :
+  --   A →* component B :=
+  -- begin
+  --   fapply pmap.mk,
+  --   { intro a, refine ⟨f a, _⟩, exact tinverse (merely_constant_pmap H a) },
+  --   exact subtype_eq !respect_pt
+  -- end
+
+  definition loop_component (A : Type*) : Ω (component A) ≃* Ω A :=
+  loop_pequiv_loop_of_is_embedding (component_incl A)
+
+  lemma loopn_component (n : ℕ) (A : Type*) : Ω[n+1] (component A) ≃* Ω[n+1] A :=
+  !loopn_succ_in ⬝e* loopn_pequiv_loopn n (loop_component A) ⬝e* !loopn_succ_in⁻¹ᵉ*
+
+  -- lemma fundamental_group_component (A : Type*) : π₁ (component A) ≃g π₁ A :=
+  -- isomorphism_of_equiv (trunc_equiv_trunc 0 (loop_component A)) _
+
+  lemma homotopy_group_component (n : ℕ) (A : Type*) : πg[n+1] (component A) ≃g πg[n+1] A :=
+  homotopy_group_isomorphism_of_is_embedding (n+1) (component_incl A)
+
+  definition is_trunc_component [instance] (n : ℕ₋₂) (A : Type*) [is_trunc n A] :
+    is_trunc n (component A) :=
+  begin
+    apply @is_trunc_sigma, intro a, cases n with n,
+    { refine is_contr_of_inhabited_prop _ _, exact tr !is_prop.elim },
+    { exact is_trunc_succ_of_is_prop _ _ _ },
+  end
+
+  definition ptrunc_component' (n : ℕ₋₂) (A : Type*) :
+    ptrunc (n.+2) (component A) ≃* component (ptrunc (n.+2) A) :=
+  begin
+    fapply pequiv.MK',
+    { exact ptrunc.elim (n.+2) (component_functor !ptr) },
+    { intro x, cases x with x p, induction x with a,
+      refine tr ⟨a, _⟩,
+      note q := trunc_functor -1 !tr_eq_tr_equiv p,
+      exact trunc_trunc_equiv_left _ !minus_one_le_succ q },
+    { exact sorry },
+    { exact sorry }
+  end
+
+  definition ptrunc_component (n : ℕ₋₂) (A : Type*) :
+    ptrunc n (component A) ≃* component (ptrunc n A) :=
+  begin
+    cases n with n, exact sorry,
+    cases n with n, exact sorry,
+    exact ptrunc_component' n A
+  end
+
+  definition break_into_components (A : Type) : A ≃ Σ(x : trunc 0 A), Σ(a : A), ∥ tr a = x ∥ :=
+  calc
+    A ≃ Σ(a : A) (x : trunc 0 A), tr a = x :
+        by exact (@sigma_equiv_of_is_contr_right _ _ (λa, !is_contr_sigma_eq))⁻¹ᵉ
+  ... ≃ Σ(x : trunc 0 A) (a : A), tr a = x :
+        by apply sigma_comm_equiv
+  ... ≃ Σ(x : trunc 0 A), Σ(a : A), ∥ tr a = x ∥ :
+        by exact sigma_equiv_sigma_right (λx, sigma_equiv_sigma_right (λa, !trunc_equiv⁻¹ᵉ))
+
+  definition pfiber_pequiv_component_of_is_contr [constructor] {A B : Type*} (f : A →* B)
+    [is_contr B]
+    /- extra condition, something like trunc_functor 0 f is an embedding -/ : pfiber f ≃* component A :=
+  sorry
+
+end is_conn
--- a/higher_groups.hlean
+++ b/higher_groups.hlean
@ -6,7 +6,7 @@ Authors: Ulrik Buchholtz, Egbert Rijke, Floris van Doorn
 Formalization of the higher groups paper
 -/

-import .homotopy.EM
+import .homotopy.EM algebra.category.constructions.pullback
 open eq is_conn pointed is_trunc trunc equiv is_equiv trunc_index susp nat algebra
     prod.ops sigma sigma.ops category EM
 namespace higher_group
--- a/homology/basic.hlean
+++ b/homology/basic.hlean
@ -102,8 +102,8 @@ namespace homology
      fapply is_equiv_of_equiv_of_homotopy,
      { exact equiv_of_isomorphism iso  },
      { refine dirsum_homotopy _, intro i y,
-        refine homomorphism_comp_compute iso3 (iso2 ∘g iso1) _ ⬝ _,
-        refine ap iso3 (homomorphism_comp_compute iso2 iso1 _) ⬝ _,
+        refine homomorphism_compose_eq iso3 (iso2 ∘g iso1) _ ⬝ _,
+        refine ap iso3 (homomorphism_compose_eq iso2 iso1 _) ⬝ _,
        refine ap (iso3 ∘ iso2) _ ⬝ _,
        { exact dirsum_incl (λ i, HH theory n (X (down i))) (up i) y },
        { refine _ ⬝ dirsum_elim_compute (λi, dirsum_incl (λ i, HH theory n (X (down.{0 u} i))) (up i)) i y,
--- a/homotopy/EM.hlean
+++ b/homotopy/EM.hlean
@ -197,35 +197,35 @@ namespace EM
  definition homotopy_group_functor_EMadd1_functor' {G H : AbGroup} (φ : G →g H) (n : ℕ) :
    hsquare (ghomotopy_group_EMadd1' G n)⁻¹ᵍ (ghomotopy_group_EMadd1' H n)⁻¹ᵍ
            (π→g[n+1] (EMadd1_functor φ n)) φ :=
-  (homotopy_group_functor_EMadd1_functor φ n)⁻¹ʰᵗʸʰ
+  begin apply hhinverse, exact (homotopy_group_functor_EMadd1_functor φ n) end

-  definition EM1_pmap_natural {G H : Group} {X Y : Type*} (f : X →* Y) (eX : G → Ω X)
-    (eY : H → Ω Y) (rX : Πg h, eX (g * h) = eX g ⬝ eX h) (rY : Πg h, eY (g * h) = eY g ⬝ eY h)
-    [H2 : is_trunc 1 X] [is_trunc 1 Y] (φ : G →g H) (p : hsquare eX eY φ (Ω→ f)) :
-    psquare (EM1_pmap eX rX) (EM1_pmap eY rY) (EM1_functor φ) f :=
+  section infgroup
+  open infgroup
+  definition EM1_pmap_natural {G H : Group} {X Y : Type*} (f : X →* Y) (eX : G →∞g Ωg X)
+    (eY : H →∞g Ωg Y) [H2 : is_trunc 1 X] [is_trunc 1 Y] (φ : G →g H)
+    (p : hsquare eX eY φ (Ωg→ f)) : psquare (EM1_pmap eX) (EM1_pmap eY) (EM1_functor φ) f :=
  begin
    fapply phomotopy.mk,
    { intro x, induction x using EM.set_rec,
      { exact respect_pt f },
      { apply eq_pathover,
-        refine ap_compose f _ _ ⬝ph _ ⬝hp (ap_compose (EM1_pmap eY rY) _ _)⁻¹,
+        refine ap_compose f _ _ ⬝ph _ ⬝hp (ap_compose (EM1_pmap eY) _ _)⁻¹,
        refine ap02 _ !elim_pth ⬝ph _ ⬝hp ap02 _ !elim_pth⁻¹,
        refine _ ⬝hp !elim_pth⁻¹, apply transpose, exact square_of_eq_bot (p g) }},
    { exact !idp_con⁻¹ }
  end

-  definition EM1_pequiv'_natural {G H : Group} {X Y : Type*} (f : X →* Y) (eX : G ≃* Ω X)
-    (eY : H ≃* Ω Y) (rX : Πg h, eX (g * h) = eX g ⬝ eX h)  (rY : Πg h, eY (g * h) = eY g ⬝ eY h)
-    [H1 : is_conn 0 X] [H2 : is_trunc 1 X] [is_conn 0 Y] [is_trunc 1 Y]
+  definition EM1_pequiv'_natural {G H : Group} {X Y : Type*} (f : X →* Y) (eX : G ≃∞g Ωg X)
+    (eY : H ≃∞g Ωg Y) [H1 : is_conn 0 X] [H2 : is_trunc 1 X] [is_conn 0 Y] [is_trunc 1 Y]
    (φ : G →g H) (p : hsquare eX eY φ (Ω→ f)) :
-    psquare (EM1_pequiv' eX rX) (EM1_pequiv' eY rY) (EM1_functor φ) f :=
-  EM1_pmap_natural f eX eY rX rY φ p
+    psquare (EM1_pequiv' eX) (EM1_pequiv' eY) (EM1_functor φ) f :=
+  EM1_pmap_natural f eX eY φ p

  definition EM1_pequiv_natural {G H : Group} {X Y : Type*} (f : X →* Y) (eX : G ≃g π₁ X)
    (eY : H ≃g π₁ Y) [H1 : is_conn 0 X] [H2 : is_trunc 1 X] [is_conn 0 Y] [is_trunc 1 Y]
    (φ : G →g H) (p : hsquare eX eY φ (π→g[1] f)) :
    psquare (EM1_pequiv eX) (EM1_pequiv eY) (EM1_functor φ) f :=
-  EM1_pequiv'_natural f _ _ _ _ φ
+  EM1_pequiv'_natural f _ _ φ
    begin
      assert p' : ptrunc_functor 0 (Ω→ f) ∘* pequiv_of_isomorphism eX ~*
                  pequiv_of_isomorphism eY ∘* pmap_of_homomorphism φ, exact phomotopy_of_homotopy p,
@ -238,38 +238,35 @@ namespace EM
  begin refine EM1_pequiv_natural f _ _ _ _, exact homotopy.rfl end

  definition EM_up_natural {G H : AbGroup} (φ : G →g H) {X Y : Type*} (f : X →* Y) {n : ℕ}
-    (eX : G →* Ω[succ (succ n)] X) (eY : H →* Ω[succ (succ n)] Y)
-    (p : hsquare eX eY φ (Ω→[succ (succ n)] f))
+    (eX : G →∞g Ωg[succ (succ n)] X) (eY : H →∞g Ωg[succ (succ n)] Y)
+    (p : hsquare eX eY φ (Ωg→[succ (succ n)] f))
    : hsquare (EM_up eX) (EM_up eY) φ (Ω→[succ n] (Ω→ f)) :=
-  begin
-    refine p ⬝htyh hsquare_of_psquare (loopn_succ_in_natural (succ n) f),
-  end
+  p ⬝htyh hsquare_of_psquare (loopn_succ_in_natural (succ n) f)

  definition EMadd1_pmap_natural {G H : AbGroup} {X Y : Type*} (f : X →* Y) (n : ℕ)
-    (eX : G →* Ω[succ n] X) (eY : H →* Ω[succ n] Y) (rX : Π(g h : G), eX (g * h) = eX g ⬝ eX h)
-    (rY : Π(g h : H), eY (g * h) = eY g ⬝ eY h) [HX : is_trunc (n.+1) X] [HY : is_trunc (n.+1) Y]
-    (φ : G →g H) (p : hsquare eX eY φ (Ω→[succ n] f)) :
-    psquare (EMadd1_pmap n eX rX) (EMadd1_pmap n eY rY) (EMadd1_functor φ n) f :=
+    (eX : G →∞g Ωg[succ n] X) (eY : H →∞g Ωg[succ n] Y)
+    [HX : is_trunc (n.+1) X] [HY : is_trunc (n.+1) Y]
+    (φ : G →g H) (p : hsquare eX eY φ (Ωg→[succ n] f)) :
+    psquare (EMadd1_pmap n eX) (EMadd1_pmap n eY) (EMadd1_functor φ n) f :=
  begin
-    revert X Y f eX eY rX rY HX HY p, induction n with n IH: intros,
+    revert X Y f eX eY HX HY p, induction n with n IH: intros,
    { apply EM1_pmap_natural, exact p },
    { do 2 rewrite [EMadd1_pmap_succ], refine _ ⬝* pwhisker_left _ !EMadd1_functor_succ⁻¹*,
      refine (ptrunc_elim_pcompose ((succ n).+1) _ _)⁻¹* ⬝* _ ⬝*
             (ptrunc_elim_ptrunc_functor ((succ n).+1) _ _)⁻¹*,
      apply ptrunc_elim_phomotopy,
      refine _ ⬝* !susp_elim_susp_functor⁻¹*,
-      refine _ ⬝* susp_elim_phomotopy (IH _ _ _ _ _ (is_homomorphism_EM_up eX rX) _
+      refine _ ⬝* susp_elim_phomotopy (IH _ _ _ _ _
                                           (@is_trunc_loop _ _ HX) _ (EM_up_natural φ f eX eY p)),
      apply susp_elim_natural }
  end

  definition EMadd1_pequiv'_natural {G H : AbGroup} {X Y : Type*} (f : X →* Y) (n : ℕ)
-    (eX : G ≃* Ω[succ n] X) (eY : H ≃* Ω[succ n] Y) (rX : Π(g h : G), eX (g * h) = eX g ⬝ eX h)
-    (rY : Π(g h : H), eY (g * h) = eY g ⬝ eY h)
+    (eX : G ≃∞g Ωg[succ n] X) (eY : H ≃∞g Ωg[succ n] Y)
    [H1 : is_conn n X] [H2 : is_trunc (n.+1) X] [is_conn n Y] [is_trunc (n.+1) Y]
-    (φ : G →g H) (p : hsquare eX eY φ (Ω→[succ n] f)) :
-     psquare (EMadd1_pequiv' n eX rX) (EMadd1_pequiv' n eY rY) (EMadd1_functor φ n) f :=
-  EMadd1_pmap_natural f n eX eY rX rY φ p
+    (φ : G →g H) (p : hsquare eX eY φ (Ωg→[succ n] f)) :
+     psquare (EMadd1_pequiv' n eX) (EMadd1_pequiv' n eY) (EMadd1_functor φ n) f :=
+  EMadd1_pmap_natural f n eX eY φ p

  definition EMadd1_pequiv_natural_local_instance {X : Type*} (n : ℕ) [H : is_trunc (n.+1) X] :
    is_set (Ω[succ n] X) :=
@ -277,14 +274,17 @@ namespace EM

  local attribute EMadd1_pequiv_natural_local_instance [instance]

-  definition EMadd1_pequiv_natural {G H : AbGroup} {X Y : Type*} (f : X →* Y) (n : ℕ) (eX : G ≃g πg[n+1] X)
-    (eY : H ≃g πg[n+1] Y) [H1 : is_conn n X] [H2 : is_trunc (n.+1) X] [H3 : is_conn n Y]
+  definition EMadd1_pequiv_natural {G H : AbGroup} {X Y : Type*} (f : X →* Y) (n : ℕ)
+    (eX : G ≃g πg[n+1] X) (eY : H ≃g πg[n+1] Y)
+    [H1 : is_conn n X] [H2 : is_trunc (n.+1) X] [H3 : is_conn n Y]
    [H4 : is_trunc (n.+1) Y] (φ : G →g H) (p : hsquare eX eY φ (π→g[n+1] f)) :
    psquare (EMadd1_pequiv n eX) (EMadd1_pequiv n eY) (EMadd1_functor φ n) f :=
  EMadd1_pequiv'_natural f n
-    (pequiv_of_isomorphism eX ⬝e* ptrunc_pequiv 0 (Ω[succ n] X))
-    (pequiv_of_isomorphism eY ⬝e* ptrunc_pequiv 0 (Ω[succ n] Y))
-    _ _ φ (p ⬝htyh hsquare_of_psquare (ptrunc_pequiv_natural 0 (Ω→[succ n] f)))
+    _ --(inf_isomorphism_of_isomorphism eX ⬝∞g gtrunc_isomorphism (Ω[succ n] X))
+    _ --(inf_isomorphism_of_isomorphism eY ⬝∞g gtrunc_isomorphism (Ω[succ n] Y))
+    φ (p ⬝htyh hsquare_of_psquare (ptrunc_pequiv_natural 0 (Ω→[succ n] f)))
+
+  end infgroup

  definition EMadd1_pequiv_succ_natural {G H : AbGroup} {X Y : Type*} (f : X →* Y) (n : ℕ)
    (eX : G ≃g πag[n+2] X) (eY : H ≃g πag[n+2] Y) [is_conn (n.+1) X] [is_trunc (n.+2) X]
@ -343,6 +343,24 @@ namespace EM
    { refine !loopn_succ_in ⬝e* Ω≃[n] (loop_EMadd1 G (m + n))⁻¹ᵉ* ⬝e* e }
  end

+  /- properties about EM -/
+
+  definition gEM (G : AbGroup) (n : ℕ) : InfGroup :=
+  InfGroup.mk (EM G n) (inf_group_equiv_closed (loop_EM G n) _)
+
+  definition gEM_functor {G H : AbGroup} (φ : G →g H) (n : ℕ) : gEM G n →∞g gEM H n :=
+  inf_homomorphism.mk (EM_functor φ n) sorry
+
+  definition EM_pmap [unfold 8] {G : AbGroup} (X : InfGroup) (n : ℕ)
+    (e : AbInfGroup_of_AbGroup G →∞g Ωg'[n] X) [H : is_trunc n X] : EM G n →* X :=
+  begin
+    cases n with n,
+    { exact pmap_of_inf_homomorphism e },
+    { have is_trunc (n.+1) X, from H, exact EMadd1_pmap n e }
+  end
+
+  -- definition gEM_gfunctor {G H : AbGroup} (n : ℕ) : (G →gg H) →∞g (gEM G n →∞g gEM H n) :=
+  -- inf_homomorphism.mk (EM_functor _ n) sorry

  /- The Eilenberg-MacLane space K(G,n) with the same homotopy group as X on level n.
     On paper this is written K(πₙ(X), n). The problem is that for
@ -508,14 +526,14 @@ namespace EM
  definition homotopy_group_cfunctor : cType*[0] ⇒ Grp :=
  functor.mk
    (λX, πg[1] X)
-    (λX Y f, π→g[1] f)
+    (λX Y (f : X →* Y), π→g[1] f)
    begin intro, apply homomorphism_eq, exact to_homotopy !homotopy_group_functor_pid end
    begin intros, apply homomorphism_eq, exact to_homotopy !homotopy_group_functor_pcompose end

  definition ab_homotopy_group_cfunctor (n : ℕ) : cType*[n.+1] ⇒ AbGrp :=
  functor.mk
    (λX, πag[n+2] X)
-    (λX Y f, π→g[n+2] f)
+    (λX Y (f : X →* Y), by rexact π→g[n+2] f)
    begin intro, apply homomorphism_eq, exact to_homotopy !homotopy_group_functor_pid end
    begin intros, apply homomorphism_eq, exact to_homotopy !homotopy_group_functor_pcompose end

@ -593,8 +611,12 @@ namespace EM
             EM1_functor_gid H end

  definition is_contr_EM1 {G : Group} (H : is_contr G) : is_contr (EM1 G) :=
-  is_contr_of_is_conn_of_is_trunc
-    (is_trunc_succ_succ_of_is_trunc_loop _ _ (is_trunc_equiv_closed _ !loop_EM1 _) _) !is_conn_EM1
+  begin
+    refine is_contr_of_is_conn_of_is_trunc _ !is_conn_EM1,
+    refine is_trunc_succ_succ_of_is_trunc_loop _ _ _ _,
+    refine is_trunc_equiv_closed _ !loop_EM1 _,
+    apply is_trunc_succ, exact H
+  end

  definition is_contr_EMadd1 (n : ℕ) {G : AbGroup} (H : is_contr G) : is_contr (EMadd1 G n) :=
  begin
@ -642,7 +664,7 @@ namespace EM
  definition is_conn_fiber_EM1_functor {G H : Group} (φ : G →g H) :
    is_conn -1 (pfiber (EM1_functor φ)) :=
  begin
-    apply is_conn_fiber
+    apply is_conn_fiber_of_is_conn
  end

  definition is_trunc_fiber_EMadd1_functor {G H : AbGroup} (φ : G →g H) (n : ℕ) :
@ -654,7 +676,7 @@ namespace EM
  definition is_conn_fiber_EMadd1_functor {G H : AbGroup} (φ : G →g H) (n : ℕ) :
    is_conn (n.-1) (pfiber (EMadd1_functor φ n)) :=
  begin
-    apply is_conn_fiber, apply is_conn_of_is_conn_succ, apply is_conn_EMadd1,
+    apply is_conn_fiber_of_is_conn, apply is_conn_of_is_conn_succ, apply is_conn_EMadd1,
    apply is_conn_EMadd1
  end

@ -667,7 +689,7 @@ namespace EM
  definition is_conn_fiber_EM_functor {G H : AbGroup} (φ : G →g H) (n : ℕ) :
    is_conn (n.-2) (pfiber (EM_functor φ n)) :=
  begin
-    apply is_conn_fiber, apply is_conn_of_is_conn_succ
+    apply is_conn_fiber_of_is_conn, apply is_conn_of_is_conn_succ
  end

  section
--- a/homotopy/EMRing.hlean
+++ b/homotopy/EMRing.hlean
@ -1,6 +1,6 @@
 -- Authors: Floris van Doorn

-import .EM .smash_adjoint ..algebra.ring
+import .EM .smash_adjoint ..algebra.ring ..algebra.arrow_group

 open algebra eq EM is_equiv equiv is_trunc is_conn pointed trunc susp smash group nat
 namespace EM
@ -10,44 +10,58 @@ definition EM1product_adj {R : Ring} :
 begin
  have is_trunc 1 (ppmap (EM1 (AbGroup_of_Ring R)) (EMadd1 (AbGroup_of_Ring R) 1)),
    from is_trunc_pmap_of_is_conn _ _ _ _ _ _ (le.refl 2) !is_trunc_EMadd1,
-  fapply EM1_pmap,
+  apply EM1_pmap, fapply inf_homomorphism.mk,
  { intro r, refine pfunext _ _, exact !loop_EM2⁻¹ᵉ* ∘* EM1_functor (ring_right_action r), },
  { intro r r', exact sorry }
 end

-definition EMproduct_map {G H K : AbGroup} (φ : G → H →g K) (n m : ℕ) (g : G) :
-  EMadd1 H n →* EMadd1 K n :=
+definition EMproduct_map {A B C : AbGroup} (φ : A → B →g C) (n m : ℕ) (a : A) :
+  EMadd1 B n →* EMadd1 C n :=
 begin
-  fapply EMadd1_functor (φ g) n
+  fapply EMadd1_functor (φ a) n
 end

-definition EM0EMadd1product {G H K : AbGroup} (φ : G →g H →gg K) (n : ℕ) :
-  G →* EMadd1 H n →** EMadd1 K n :=
-EMadd1_pfunctor H K n ∘* pmap_of_homomorphism φ
+definition EM0EMadd1product {A B C : AbGroup} (φ : A →g B →gg C) (n : ℕ) :
+  A →* EMadd1 B n →** EMadd1 C n :=
+EMadd1_pfunctor B C n ∘* pmap_of_homomorphism φ

-definition EMadd1product {G H K : AbGroup} (φ : G →g H →gg K) (n m : ℕ) :
-  EMadd1 G n →* EMadd1 H m →** EMadd1 K (m + succ n) :=
+definition EMadd1product {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
+  EMadd1 A n →* EMadd1 B m →** EMadd1 C (m + succ n) :=
 begin
-  assert H1 : is_trunc n.+1 (EMadd1 H m →** EMadd1 K (m + succ n)),
+  assert H1 : is_trunc n.+1 (EMadd1 B m →** EMadd1 C (m + succ n)),
  { refine is_trunc_pmap_of_is_conn _ (m.-1) !is_conn_EMadd1 _ _ _ _ !is_trunc_EMadd1,
    exact le_of_eq (trunc_index.of_nat_add_plus_two_of_nat m n)⁻¹ᵖ },
-  fapply EMadd1_pmap,
-  { refine (loopn_ppmap_pequiv _ _ _)⁻¹ᵉ* ∘* ppcompose_left !loopn_EMadd1_add⁻¹ᵉ* ∘*
-      EM0EMadd1product φ m },
-  { exact sorry }
+  apply EMadd1_pmap,
+  exact sorry
+  /- the underlying pointed map is: -/
+  -- refine (loopn_ppmap_pequiv _ _ _)⁻¹ᵉ* ∘* ppcompose_left !loopn_EMadd1_add⁻¹ᵉ* ∘*
+  --     EM0EMadd1product φ m
 end

-definition EMproduct {G H K : AbGroup} (φ : G →g H →gg K) (n m : ℕ) :
-  EM G n →* EM H m →** EM K (m + n) :=
+definition EMproduct {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
+  EM A n →* EM B m →** EM C (m + n) :=
 begin
  cases n with n,
  { cases m with m,
    { exact pmap_of_homomorphism2 φ },
    { exact EM0EMadd1product φ m }},
  { cases m with m,
-    { exact ppcompose_left (ptransport (EMadd1 K) (zero_add n)⁻¹) ∘*
+    { exact ppcompose_left (ptransport (EMadd1 C) (zero_add n)⁻¹) ∘*
        pmap_swap_map (EM0EMadd1product (homomorphism_swap φ) n) },
-    { exact ppcompose_left (ptransport (EMadd1 K) !succ_add⁻¹) ∘* EMadd1product φ n m }}
+    { exact ppcompose_left (ptransport (EMadd1 C) !succ_add⁻¹) ∘* EMadd1product φ n m }}
 end

+definition EMproduct' {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
+  EM A n →* EM B m →** EM C (m + n) :=
+begin
+  assert H1 : is_trunc n (InfGroup_pmap' (EM B m) (loop_EM C (m + n))),
+  { exact is_trunc_pmap_of_is_conn_nat _ m !is_conn_EM _ _ _ !le.refl !is_trunc_EM },
+  apply EM_pmap (InfGroup_pmap' (EM B m) (loop_EM C (m + n))) n,
+  exact sorry
+  -- exact _ /- (loopn_ppmap_pequiv _ _ _)⁻¹ᵉ* -/ ∘∞g _ /-ppcompose_left !loopn_EMadd1_add⁻¹ᵉ*-/ ∘∞g
+  --      _ ∘∞g inf_homomorphism_of_homomorphism φ
+
+end
+
+
 end EM
--- a/homotopy/smash.hlean
+++ b/homotopy/smash.hlean
@ -472,7 +472,7 @@ namespace smash
  definition smash_pequiv [constructor] (f : A ≃* C) (g : B ≃* D) : A ∧ B ≃* C ∧ D :=
  begin
    fapply pequiv_of_pmap (f ∧→ g),
-    refine @homotopy_closed _ _ _ _ _ (smash_functor_homotopy_pushout_functor f g)⁻¹ʰᵗʸ,
+    refine homotopy_closed _ (smash_functor_homotopy_pushout_functor f g)⁻¹ʰᵗʸ _,
    apply pushout.is_equiv_functor
  end

@ -1054,7 +1054,7 @@ namespace smash
  begin
    unfold [smash, cofiber, smash'], symmetry,
    fapply pushout_vcompose_equiv wedge_of_sum,
-    { symmetry, apply equiv_unit_of_is_contr, apply is_contr_pushout_wedge_of_sum },
+    { symmetry, refine equiv_unit_of_is_contr _ _, apply is_contr_pushout_wedge_of_sum },
    { intro x, reflexivity },
    { apply prod_of_wedge_of_sum }
  end
--- a/homotopy/susp.hlean
+++ b/homotopy/susp.hlean
@ -71,7 +71,7 @@ namespace susp
  begin
    intro p,
    refine susp_functor_psquare p ⬝v* _,
-    exact psquare_transpose (loop_susp_counit_natural f₁₂)
+    exact ptranspose (loop_susp_counit_natural f₁₂)
  end

  open pushout unit prod sigma sigma.ops
--- a/logic.hlean
+++ b/logic.hlean
@ -21,7 +21,7 @@ section
 open trunc_index

 definition propext {p q : Prop} (h : p ↔ q) : p = q :=
-tua (equiv_of_iff_of_is_prop h)
+tua (equiv_of_iff_of_is_prop h _ _)

 end

--- a/move_to_lib.hlean
+++ b/move_to_lib.hlean
--- a/pointed_pi.hlean
+++ b/pointed_pi.hlean
@ -269,7 +269,7 @@ namespace pointed
    fapply phomotopy.mk,
    { intro x, reflexivity },
    { refine !idp_con ⬝ _, esimp,
-      refine whisker_right _ !ap_sigma_functor_eq_dpair ⬝ _,
+      refine whisker_right _ !ap_sigma_functor_sigma_eq ⬝ _,
      induction f₁ with f₁ f₁₀, induction f₂ with f₂ f₂₀, induction A₂ with A₂ a₂₀,
      induction A₃ with A₃ a₃₀, esimp at * ⊢, induction p₁, induction p₂, reflexivity }
  end
@ -977,10 +977,11 @@ open is_trunc is_conn
 namespace is_conn
  section

+  /- todo: reorder arguments and make some implicit -/
  variables (A : Type*) (n : ℕ₋₂) (H : is_conn (n.+1) A)
  include H

-  definition is_contr_ppi_match (P : A → Type*) (H : Πa, is_trunc (n.+1) (P a))
+  definition is_contr_ppi_match (P : A → Type*) (H2 : Πa, is_trunc (n.+1) (P a))
    : is_contr (Π*(a : A), P a) :=
  begin
    apply is_contr.mk pt,
@ -997,7 +998,7 @@ namespace is_conn
    (P : A → Type*) (H3 : Πa, is_trunc l (P a)) :
    is_trunc k.+1 (Π*(a : A), P a) :=
  begin
-    have H4 : Πa, is_trunc (n.+1+2+k) (P a), from λa, is_trunc_of_le (P a) H2,
+    have H4 : Πa, is_trunc (n.+1+2+k) (P a), from λa, is_trunc_of_le (P a) H2 _,
    clear H2 H3, revert P H4,
    induction k with k IH: intro P H4,
    { apply is_prop_of_imp_is_contr, intro f,
@ -1012,11 +1013,24 @@ namespace is_conn

  definition is_trunc_pmap_of_is_conn (k l : ℕ₋₂) (B : Type*) (H2 : l ≤ n.+1+2+k)
    (H3 : is_trunc l B) : is_trunc k.+1 (A →* B) :=
-  is_trunc_equiv_closed k.+1 (pppi_equiv_pmap A B)
-    (is_trunc_ppi_of_is_conn A n H k l H2 (λ a, B) _)
+  is_trunc_ppi_of_is_conn A n H k l H2 (λ a, B) _

  end

+  open trunc_index algebra nat
+  definition is_trunc_ppi_of_is_conn_nat
+    (A : Type*) (n : ℕ) (H : is_conn (n.-1) A) (k l : ℕ) (H2 : l ≤ n + k)
+    (P : A → Type*) (H3 : Πa, is_trunc l (P a)) :
+    is_trunc k (Π*(a : A), P a) :=
+  begin
+    refine is_trunc_ppi_of_is_conn A (n.-2) H (k.-1) l _ P H3,
+    refine le.trans (of_nat_le_of_nat H2) (le_of_eq !sub_one_add_plus_two_sub_one⁻¹)
+  end
+
+  definition is_trunc_pmap_of_is_conn_nat (A : Type*) (n : ℕ) (H : is_conn (n.-1) A) (k l : ℕ)
+    (B : Type*) (H2 : l ≤ n + k) (H3 : is_trunc l B) : is_trunc k (A →* B) :=
+  is_trunc_ppi_of_is_conn_nat A n H k l H2 (λ a, B) _
+
  -- this is probably much easier to prove directly
  definition is_trunc_ppi (A : Type*) (n k : ℕ₋₂) (H : n ≤ k) (P : A → Type*)
    (H2 : Πa, is_trunc n (P a)) : is_trunc k (Π*(a : A), P a) :=
--- a/spectrum/smash.hlean
+++ b/spectrum/smash.hlean
@ -19,7 +19,7 @@ definition smash_prespectrum_fun {X X' : Type*} {Y Y' : prespectrum} (f : X →*
 smap.mk (λn, smash_functor f (g n)) begin
  intro n,
  refine susp_to_loop_psquare _ _ _ _ _,
-  refine pvconcat (psquare_transpose (phinverse (smash_susp_natural f (g n)))) _,
+  refine pvconcat (ptranspose (phinverse (smash_susp_natural f (g n)))) _,
  refine vconcat_phomotopy _ (smash_functor_split f (g (S n))),
  refine phomotopy_vconcat (smash_functor_split f (susp_functor (g n))) _,
  refine phconcat _ _,
--- a/spectrum/trunc.hlean
+++ b/spectrum/trunc.hlean
@ -107,8 +107,7 @@ begin induction p, exact id end
 definition is_strunc_of_le {k l : ℤ} (E : spectrum) (H : k ≤ l)
  : is_strunc k E → is_strunc l E :=
 begin
-  intro T, intro n, exact is_trunc_of_le (E n)
-    (maxm2_monotone (algebra.add_le_add_right H n))
+  intro T, intro n, exact is_trunc_of_le (E n) (maxm2_monotone (algebra.add_le_add_right H n)) _
 end

 definition is_strunc_pequiv_closed {k : ℤ} {E F : spectrum} (H : Πn, E n ≃* F n)
--- a/univalent_subcategory.hlean
+++ b/univalent_subcategory.hlean
@ -28,7 +28,7 @@ section univ_subcat
  definition is_univalent_domain_of_fully_faithful_embedding : is_univalent C :=
  begin
    intros,
-    apply homotopy_closed eq_equiv_iso_of_fully_faithful eq_equiv_iso_of_fully_faithful_homot
+    exact homotopy_closed eq_equiv_iso_of_fully_faithful eq_equiv_iso_of_fully_faithful_homot _
  end
 end univ_subcat

@ -216,7 +216,7 @@ end
    (m, (i, o)) = (m', (i', o')) ≃ (m ~2 m') :=
  begin
    have is_set A, from pr1 H,
-    apply equiv_of_is_prop,
+    refine equiv_of_is_prop _ _ _ _,
    { intro p, exact apd100 (eq_pr1 p)},
    { intro p, apply prod_eq (eq_of_homotopy2 p),
      apply prod_eq: esimp [Group_props] at *; esimp,