feat(hott): move basic lemmas from the spectral repository to the main repository

hott/algebra/bundled.hlean

@@ -5,7 +5,7 @@ Authors: Jeremy Avigad
 
 Bundled structures
 -/
-import algebra.group
+import algebra.ring
 open algebra pointed is_trunc
 
 namespace algebra
@@ -34,64 +34,82 @@ attribute CommMonoid.carrier [coercion]
 attribute CommMonoid.struct [instance]
 
 structure Group :=
-(carrier : Type) (struct : group carrier)
+(carrier : Type) (struct' : group carrier)
 
-attribute Group.carrier [coercion]
-attribute Group.struct [instance]
+attribute Group.struct' [instance]
 
 section
-  local attribute Group.struct [instance]
-  definition pSet_of_Group [constructor] [reducible] [coercion] (G : Group) : Set* :=
-  ptrunctype.mk G !semigroup.is_set_carrier 1
+local attribute Group.carrier [coercion]
+definition pSet_of_Group [constructor] [reducible] [coercion] (G : Group) : Set* :=
+ptrunctype.mk (Group.carrier G) !semigroup.is_set_carrier 1
 end
 
+definition Group.struct [instance] [priority 2000] (G : Group) : group G :=
+Group.struct' G
+
 attribute algebra._trans_of_pSet_of_Group [unfold 1]
 attribute algebra._trans_of_pSet_of_Group_1 algebra._trans_of_pSet_of_Group_2 [constructor]
 
-definition pType_of_Group [reducible] [constructor] : Group → Type* :=
-algebra._trans_of_pSet_of_Group_1
-definition Set_of_Group [reducible] [constructor] : Group → Set :=
-algebra._trans_of_pSet_of_Group_2
+definition pType_of_Group [reducible] [constructor] (G : Group) : Type* :=
+G
+definition Set_of_Group [reducible] [constructor] (G : Group) : Set :=
+G
 
 definition AddGroup : Type := Group
 
+definition pSet_of_AddGroup [constructor] [reducible] [coercion] (G : AddGroup) : Set* :=
+pSet_of_Group G
+
 definition AddGroup.mk [constructor] [reducible] (G : Type) (H : add_group G) : AddGroup :=
 Group.mk G H
 
-definition AddGroup.struct [reducible] (G : AddGroup) : add_group G :=
+definition AddGroup.struct [reducible] [instance] [priority 2000] (G : AddGroup) : add_group G :=
 Group.struct G
 
-attribute AddGroup.struct Group.struct [instance] [priority 2000]
+attribute algebra._trans_of_pSet_of_AddGroup [unfold 1]
+attribute algebra._trans_of_pSet_of_AddGroup_1 algebra._trans_of_pSet_of_AddGroup_2 [constructor]
+
+definition pType_of_AddGroup [reducible] [constructor] : AddGroup → Type* :=
+algebra._trans_of_pSet_of_AddGroup_1
+definition Set_of_AddGroup [reducible] [constructor] : AddGroup → Set :=
+algebra._trans_of_pSet_of_AddGroup_2
 
 structure AbGroup :=
-(carrier : Type) (struct : ab_group carrier)
+(carrier : Type) (struct' : ab_group carrier)
 
-attribute AbGroup.carrier [coercion]
-
-definition AddAbGroup : Type := AbGroup
-
-definition AddAbGroup.mk [constructor] [reducible] (G : Type) (H : add_ab_group G) :
-  AddAbGroup :=
-AbGroup.mk G H
-
-definition AddAbGroup.struct [reducible] (G : AddAbGroup) : add_ab_group G :=
-AbGroup.struct G
-
-attribute AddAbGroup.struct AbGroup.struct [instance] [priority 2000]
+attribute AbGroup.struct' [instance]
 
+section
+local attribute AbGroup.carrier [coercion]
 definition Group_of_AbGroup [coercion] [constructor] (G : AbGroup) : Group :=
 Group.mk G _
+end
+
+definition AbGroup.struct [instance] [priority 2000] (G : AbGroup) : ab_group G :=
+AbGroup.struct' G
 
 attribute algebra._trans_of_Group_of_AbGroup_1
           algebra._trans_of_Group_of_AbGroup
           algebra._trans_of_Group_of_AbGroup_3 [constructor]
 attribute algebra._trans_of_Group_of_AbGroup_2 [unfold 1]
 
-definition ab_group_AbGroup [instance] (G : AbGroup) : ab_group G :=
+definition AddAbGroup : Type := AbGroup
+
+definition AddGroup_of_AddAbGroup [coercion] [constructor] (G : AddAbGroup) : AddGroup :=
+Group_of_AbGroup G
+
+definition AddAbGroup.struct [reducible] [instance] [priority 2000] (G : AddAbGroup) :
+  add_ab_group G :=
 AbGroup.struct G
 
-definition add_ab_group_AddAbGroup [instance] (G : AddAbGroup) : add_ab_group G :=
-AbGroup.struct G
+definition AddAbGroup.mk [constructor] [reducible] (G : Type) (H : add_ab_group G) :
+  AddAbGroup :=
+AbGroup.mk G H
+
+attribute algebra._trans_of_AddGroup_of_AddAbGroup_1
+          algebra._trans_of_AddGroup_of_AddAbGroup
+          algebra._trans_of_AddGroup_of_AddAbGroup_3 [constructor]
+attribute algebra._trans_of_AddGroup_of_AddAbGroup_2 [unfold 1]
 
 -- structure AddSemigroup :=
 -- (carrier : Type) (struct : add_semigroup carrier)
@@ -132,21 +150,26 @@ AbGroup.struct G
 
 -- some bundled infinity-structures
 structure InfGroup :=
-(carrier : Type) (struct : inf_group carrier)
+(carrier : Type) (struct' : inf_group carrier)
 
-attribute InfGroup.carrier [coercion]
-attribute InfGroup.struct [instance]
+attribute InfGroup.struct' [instance]
 
 section
-  local attribute InfGroup.struct [instance]
+  local attribute InfGroup.carrier [coercion]
   definition pType_of_InfGroup [constructor] [reducible] [coercion] (G : InfGroup) : Type* :=
   pType.mk G 1
 end
 
 attribute algebra._trans_of_pType_of_InfGroup [unfold 1]
 
+definition InfGroup.struct [instance] [priority 2000] (G : InfGroup) : inf_group G :=
+InfGroup.struct' G
+
 definition AddInfGroup : Type := InfGroup
 
+definition pType_of_AddInfGroup [constructor] [reducible] [coercion] (G : AddInfGroup) : Type* :=
+pType_of_InfGroup G
+
 definition AddInfGroup.mk [constructor] [reducible] (G : Type) (H : add_inf_group G) :
   AddInfGroup :=
 InfGroup.mk G H
@@ -154,29 +177,40 @@ InfGroup.mk G H
 definition AddInfGroup.struct [reducible] (G : AddInfGroup) : add_inf_group G :=
 InfGroup.struct G
 
-attribute AddInfGroup.struct InfGroup.struct [instance] [priority 2000]
+attribute algebra._trans_of_pType_of_AddInfGroup [unfold 1]
 
 structure AbInfGroup :=
-(carrier : Type) (struct : ab_inf_group carrier)
+(carrier : Type) (struct' : ab_inf_group carrier)
 
-attribute AbInfGroup.carrier [coercion]
+attribute AbInfGroup.struct' [instance]
+
+section
+local attribute AbInfGroup.carrier [coercion]
+definition InfGroup_of_AbInfGroup [coercion] [constructor] (G : AbInfGroup) : InfGroup :=
+InfGroup.mk G _
+end
+
+definition AbInfGroup.struct [instance] [priority 2000] (G : AbInfGroup) : ab_inf_group G :=
+AbInfGroup.struct' G
+
+attribute algebra._trans_of_InfGroup_of_AbInfGroup_1 [constructor]
+attribute algebra._trans_of_InfGroup_of_AbInfGroup [unfold 1]
 
 definition AddAbInfGroup : Type := AbInfGroup
 
+definition AddInfGroup_of_AddAbInfGroup [coercion] [constructor] (G : AddAbInfGroup) : AddInfGroup :=
+InfGroup_of_AbInfGroup G
+
+definition AddAbInfGroup.struct [reducible] [instance] [priority 2000] (G : AddAbInfGroup) :
+  add_ab_inf_group G :=
+AbInfGroup.struct G
+
 definition AddAbInfGroup.mk [constructor] [reducible] (G : Type) (H : add_ab_inf_group G) :
   AddAbInfGroup :=
 AbInfGroup.mk G H
 
-definition AddAbInfGroup.struct [reducible] (G : AddAbInfGroup) : add_ab_inf_group G :=
-AbInfGroup.struct G
-
-attribute AddAbInfGroup.struct AbInfGroup.struct [instance] [priority 2000]
-
-definition InfGroup_of_AbInfGroup [coercion] [constructor] (G : AbInfGroup) : InfGroup :=
-InfGroup.mk G _
-
-attribute algebra._trans_of_InfGroup_of_AbInfGroup_1 [constructor]
-attribute algebra._trans_of_InfGroup_of_AbInfGroup [unfold 1]
+attribute algebra._trans_of_AddInfGroup_of_AddAbInfGroup_1 [constructor]
+attribute algebra._trans_of_AddInfGroup_of_AddAbInfGroup [unfold 1]
 
 definition InfGroup_of_Group [constructor] (G : Group) : InfGroup :=
 InfGroup.mk G _
@@ -190,4 +224,11 @@ AbInfGroup.mk G _
 definition AddAbInfGroup_of_AddAbGroup [constructor] (G : AddAbGroup) : AddAbInfGroup :=
 AddAbInfGroup.mk G _
 
+/- rings -/
+structure Ring :=
+(carrier : Type) (struct : ring carrier)
+
+attribute Ring.carrier [coercion]
+attribute Ring.struct [instance]
+
 end algebra

hott/algebra/category/constructions/set.hlean

@@ -48,10 +48,10 @@ namespace category
 
     local attribute is_equiv_iso_of_equiv [instance]
 
-    definition iso_of_eq_eq_compose (A B : Set) : @iso_of_eq _ _ A B =
+    definition iso_of_eq_eq_compose (A B : Set) : @iso_of_eq _ _ A B ~
       @iso_of_equiv A B ∘ @equiv_of_eq A B ∘ subtype_eq_inv _ _ ∘
       @ap _ _ (to_fun (trunctype.sigma_char 0)) A B :=
-    eq_of_homotopy (λp, eq.rec_on p idp)
+    λp, eq.rec_on p idp
 
     definition equiv_equiv_iso (A B : set) : (A ≃ B) ≃ (A ≅ B) :=
     equiv.MK (λf, iso_of_equiv f)
@@ -75,11 +75,8 @@ namespace category
            (@is_equiv_subtype_eq_inv _ _ _ _ _))
          !univalence)
        !is_equiv_iso_of_equiv,
-    let H₂ := (iso_of_eq_eq_compose A B)⁻¹ in
-    begin
-      rewrite H₂ at H₁,
-      assumption
-    end
+    is_equiv.homotopy_closed _ (iso_of_eq_eq_compose A B)⁻¹ʰᵗʸ
+
   end set
 
   definition category_Set [instance] [constructor] : category Set :=

hott/algebra/group_theory.hlean

@@ -18,10 +18,6 @@ namespace group
   definition Group.struct' [instance] [reducible] (G : Group) : group G :=
   Group.struct G
 
-  definition ab_group_Group_of_AbGroup [instance] [constructor] [priority 900]
-    (G : AbGroup) : ab_group (Group_of_AbGroup G) :=
-  begin esimp, exact _ end
-
   definition ab_group_pSet_of_Group [instance] (G : AbGroup) : ab_group (pSet_of_Group G) :=
   AbGroup.struct G
 
@@ -29,74 +25,21 @@ namespace group
     group (pSet_of_Group G) :=
   Group.struct G
 
-  /- group homomorphisms -/
-/-
-  definition is_homomorphism [class] [reducible]
-    {G₁ G₂ : Type} [has_mul G₁] [has_mul G₂] (φ : G₁ → G₂) : Type :=
-  Π(g h : G₁), φ (g * h) = φ g * φ h
+  /- left and right actions -/
+  definition is_equiv_mul_right [constructor] {A : Group} (a : A) : is_equiv (λb, b * a) :=
+  adjointify _ (λb : A, b * a⁻¹) (λb, !inv_mul_cancel_right) (λb, !mul_inv_cancel_right)
 
-  section
-  variables {G G₁ G₂ G₃ : Type} {g h : G₁} (ψ : G₂ → G₃) {φ₁ φ₂ : G₁ → G₂} (φ : G₁ → G₂)
-            [group G] [group G₁] [group G₂] [group G₃]
-            [is_homomorphism ψ] [is_homomorphism φ₁] [is_homomorphism φ₂] [is_homomorphism φ]
+  definition right_action [constructor] {A : Group} (a : A) : A ≃ A :=
+  equiv.mk _ (is_equiv_mul_right a)
 
-  definition respect_mul {G₁ G₂ : Type} [has_mul G₁] [has_mul G₂] (φ : G₁ → G₂)
-    [is_homomorphism φ] : Π(g h : G₁), φ (g * h) = φ g * φ h :=
-  by assumption
+  definition is_equiv_add_right [constructor] {A : AddGroup} (a : A) : is_equiv (λb, b + a) :=
+  adjointify _ (λb : A, b - a) (λb, !neg_add_cancel_right) (λb, !add_neg_cancel_right)
 
-  theorem respect_one /- φ -/ : φ 1 = 1 :=
-  mul.right_cancel
-    (calc
-      φ 1 * φ 1 = φ (1 * 1) : respect_mul φ
-            ... = φ 1 : ap φ !one_mul
-            ... = 1 * φ 1 : one_mul)
+  definition add_right_action [constructor] {A : AddGroup} (a : A) : A ≃ A :=
+  equiv.mk _ (is_equiv_add_right a)
 
-  theorem respect_inv /- φ -/ (g : G₁) : φ g⁻¹ = (φ g)⁻¹ :=
-  eq_inv_of_mul_eq_one (!respect_mul⁻¹ ⬝ ap φ !mul.left_inv ⬝ !respect_one)
+  /- homomorphisms -/
 
-  definition is_embedding_homomorphism /- φ -/ (H : Π{g}, φ g = 1 → g = 1) : is_embedding φ :=
-  begin
-    apply function.is_embedding_of_is_injective,
-    intro g g' p,
-    apply eq_of_mul_inv_eq_one,
-    apply H,
-    refine !respect_mul ⬝ _,
-    rewrite [respect_inv φ, p],
-    apply mul.right_inv
-  end
-
-  definition is_homomorphism_compose {ψ : G₂ → G₃} {φ : G₁ → G₂}
-    (H1 : is_homomorphism ψ) (H2 : is_homomorphism φ) : is_homomorphism (ψ ∘ φ) :=
-  λg h, ap ψ !respect_mul ⬝ !respect_mul
-
-  definition is_homomorphism_id (G : Type) [group G] : is_homomorphism (@id G) :=
-  λg h, idp
-
-  end
-
-  section additive
-
-  definition is_add_homomorphism [class] [reducible] {G₁ G₂ : Type} [has_add G₁] [has_add G₂]
-    (φ : G₁ → G₂) : Type :=
-  Π(g h : G₁), φ (g + h) = φ g + φ h
-
-  variables {G₁ G₂ : Type} (φ : G₁ → G₂) [add_group G₁] [add_group G₂] [is_add_homomorphism φ]
-
-  definition respect_add /- φ -/ : Π(g h : G₁), φ (g + h) = φ g + φ h :=
-  by assumption
-
-  theorem respect_zero /- φ -/ : φ 0 = 0 :=
-  add.right_cancel
-    (calc
-      φ 0 + φ 0 = φ (0 + 0) : respect_add φ
-            ... = φ 0 : ap φ !zero_add
-            ... = 0 + φ 0 : zero_add)
-
-  theorem respect_neg /- φ -/ (g : G₁) : φ (-g) = -(φ g) :=
-  eq_neg_of_add_eq_zero (!respect_add⁻¹ ⬝ ap φ !add.left_inv ⬝ !respect_zero)
-
-  end additive
--/
   structure homomorphism (G₁ G₂ : Group) : Type :=
     (φ : G₁ → G₂)
     (p : is_mul_hom φ)
@@ -277,33 +220,102 @@ namespace group
   infixl ` ⬝gp `:75 := isomorphism.trans_eq
   infixl ` ⬝pg `:75 := isomorphism.eq_trans
 
-  definition pmap_of_isomorphism [constructor] (φ : G₁ ≃g G₂) :
-    G₁ →* G₂ :=
+  definition pmap_of_isomorphism [constructor] (φ : G₁ ≃g G₂) : G₁ →* G₂ :=
   pequiv_of_isomorphism φ
 
-  /- category of groups -/
+  definition to_fun_isomorphism_trans {G H K : Group} (φ : G ≃g H) (ψ : H ≃g K) :
+    φ ⬝g ψ ~ ψ ∘ φ :=
+  by reflexivity
 
-  section
-  open category
-  definition precategory_group [constructor] : precategory Group :=
-  precategory.mk homomorphism
-                 @homomorphism_compose
-                 @homomorphism_id
-                 (λG₁ G₂ G₃ G₄ φ₃ φ₂ φ₁, homomorphism_eq (λg, idp))
-                 (λG₁ G₂ φ, homomorphism_eq (λg, idp))
-                 (λG₁ G₂ φ, homomorphism_eq (λg, idp))
+  definition add_homomorphism (G H : AddGroup) : Type := homomorphism G H
+  infix ` →a `:55 := add_homomorphism
+
+  definition agroup_fun [coercion] [unfold 3] [reducible] {G H : AddGroup} (φ : G →a H) : G → H :=
+  φ
+
+  definition add_homomorphism.struct [instance] {G H : AddGroup} (φ : G →a H) : is_add_hom φ :=
+  homomorphism.addstruct φ
+
+  definition add_homomorphism.mk [constructor] {G H : AddGroup} (φ : G → H) (h : is_add_hom φ) : G →g H :=
+  homomorphism.mk φ h
+
+  definition add_homomorphism_compose [constructor] [trans] {G₁ G₂ G₃ : AddGroup}
+    (ψ : G₂ →a G₃) (φ : G₁ →a G₂) : G₁ →a G₃ :=
+  add_homomorphism.mk (ψ ∘ φ) (is_add_hom_compose _ _)
+
+  definition add_homomorphism_id [constructor] [refl] (G : AddGroup) : G →a G :=
+  add_homomorphism.mk (@id G) (is_add_hom_id G)
+
+  abbreviation aid [constructor] := @add_homomorphism_id
+  infixr ` ∘a `:75 := add_homomorphism_compose
+
+  definition to_respect_add' {H₁ H₂ : AddGroup} (χ : H₁ →a H₂) (g h : H₁) : χ (g + h) = χ g + χ h :=
+  respect_add χ g h
+
+  theorem to_respect_zero' {H₁ H₂ : AddGroup} (χ : H₁ →a H₂) : χ 0 = 0 :=
+  respect_zero χ
+
+  theorem to_respect_neg' {H₁ H₂ : AddGroup} (χ : H₁ →a H₂) (g : H₁) : χ (-g) = -(χ g) :=
+  respect_neg χ g
+
+  definition homomorphism_add [constructor] {G H : AddAbGroup} (φ ψ : G →a H) : G →a H :=
+  add_homomorphism.mk (λg, φ g + ψ g)
+    abstract begin
+      intro g g', refine ap011 add !to_respect_add' !to_respect_add' ⬝ _,
+      refine !add.assoc ⬝ ap (add _) (!add.assoc⁻¹ ⬝ ap (λx, x + _) !add.comm ⬝ !add.assoc) ⬝ !add.assoc⁻¹
+    end end
+
+  definition homomorphism_mul [constructor] {G H : AbGroup} (φ ψ : G →g H) : G →g H :=
+  homomorphism.mk (λg, φ g * ψ g) (to_respect_add (homomorphism_add φ ψ))
+
+  definition pmap_of_homomorphism_gid (G : Group) : pmap_of_homomorphism (gid G) ~* pid G :=
+  begin
+    fapply phomotopy_of_homotopy, reflexivity
   end
 
-  -- TODO
-  -- definition category_group : category Group :=
-  -- category.mk precategory_group
-  -- begin
-  --   intro G₁ G₂,
-  --   fapply adjointify,
-  --   { intro φ, fapply Group_eq, },
-  --   { },
-  --   { }
-  -- end
+  definition pmap_of_homomorphism_gcompose {G H K : Group} (ψ : H →g K) (φ : G →g H)
+    : pmap_of_homomorphism (ψ ∘g φ) ~* pmap_of_homomorphism ψ ∘* pmap_of_homomorphism φ :=
+  begin
+    fapply phomotopy_of_homotopy, reflexivity
+  end
+
+  definition pmap_of_homomorphism_phomotopy {G H : Group} {φ ψ : G →g H} (H : φ ~ ψ)
+    : pmap_of_homomorphism φ ~* pmap_of_homomorphism ψ :=
+  begin
+    fapply phomotopy_of_homotopy, exact H
+  end
+
+  definition pequiv_of_isomorphism_trans {G₁ G₂ G₃ : Group} (φ : G₁ ≃g G₂) (ψ : G₂ ≃g G₂) :
+    pequiv_of_isomorphism (φ ⬝g ψ) ~* pequiv_of_isomorphism ψ ∘* pequiv_of_isomorphism φ :=
+  begin
+    apply phomotopy_of_homotopy, reflexivity
+  end
+
+  definition isomorphism_eq {G H : Group} {φ ψ : G ≃g H} (p : φ ~ ψ) : φ = ψ :=
+  begin
+    induction φ with φ φe, induction ψ with ψ ψe,
+    exact apd011 isomorphism.mk (homomorphism_eq p) !is_prop.elimo
+  end
+
+  definition is_set_isomorphism [instance] (G H : Group) : is_set (G ≃g H) :=
+  begin
+    have H : G ≃g H ≃ Σ(f : G →g H), is_equiv f,
+    begin
+      fapply equiv.MK,
+      { intro φ, induction φ, constructor, assumption },
+      { intro v, induction v, constructor, assumption },
+      { intro v, induction v, reflexivity },
+      { intro φ, induction φ, reflexivity }
+    end,
+    apply is_trunc_equiv_closed_rev, exact H
+  end
+
+  definition trivial_homomorphism (A B : Group) : A →g B :=
+  homomorphism.mk (λa, 1) (λa a', (mul_one 1)⁻¹)
+
+  definition trivial_add_homomorphism (A B : AddGroup) : A →a B :=
+  homomorphism.mk (λa, 0) (λa a', (add_zero 0)⁻¹)
+
 
   /- given an equivalence A ≃ B we can transport a group structure on A to a group structure on B -/
 
@@ -347,19 +359,43 @@ namespace group
 
   end
 
+  section
+    variables {A B : Type} (f : A ≃ B) [ab_group A]
+    definition group_equiv_mul_comm (b b' : B) : group_equiv_mul f b b' = group_equiv_mul f b' b :=
+    by rewrite [↑group_equiv_mul, mul.comm]
+
+    definition ab_group_equiv_closed : ab_group B :=
+    ⦃ab_group, group_equiv_closed f,
+      mul_comm := group_equiv_mul_comm f⦄
+  end
+
   variable (G)
 
   /- the trivial group -/
   open unit
-  --rename: group_unit
-  definition trivial_group [constructor] : group unit :=
+  definition group_unit [constructor] : group unit :=
   group.mk _ (λx y, star) (λx y z, idp) star (unit.rec idp) (unit.rec idp) (λx, star) (λx, idp)
 
-  --rename trivial_group
-  definition Trivial_group [constructor] : Group :=
-  Group.mk _ trivial_group
+  definition ab_group_unit [constructor] : ab_group unit :=
+  ⦃ab_group, group_unit, mul_comm := λx y, idp⦄
 
-  abbreviation G0 := Trivial_group
+  definition trivial_group [constructor] : Group :=
+  Group.mk _ group_unit
+
+  abbreviation G0 := trivial_group
+
+  definition AbGroup_of_Group.{u} (G : Group.{u}) (H : Π x y : G, x * y = y * x) : AbGroup.{u} :=
+  begin
+    induction G,
+    fapply AbGroup.mk,
+    assumption,
+    exact ⦃ab_group, struct', mul_comm := H⦄
+  end
+
+  definition trivial_ab_group : AbGroup.{0} :=
+  begin
+    fapply AbGroup_of_Group trivial_group, intro x y, reflexivity
+  end
 
   definition trivial_group_of_is_contr [H : is_contr G] : G ≃g G0 :=
   begin
@@ -368,6 +404,33 @@ namespace group
     { intros, reflexivity}
   end
 
+  definition ab_group_of_is_contr (A : Type) [is_contr A] : ab_group A :=
+  have ab_group unit, from ab_group_unit,
+  ab_group_equiv_closed (equiv_unit_of_is_contr A)⁻¹ᵉ
+
+  definition group_of_is_contr (A : Type) [is_contr A] : group A :=
+  have ab_group A, from ab_group_of_is_contr A, by apply _
+
+  definition ab_group_lift_unit : ab_group (lift unit) :=
+  ab_group_of_is_contr (lift unit)
+
+  definition trivial_ab_group_lift : AbGroup :=
+  AbGroup.mk _ ab_group_lift_unit
+
+  definition from_trivial_ab_group (A : AbGroup) : trivial_ab_group →g A :=
+  trivial_homomorphism trivial_ab_group A
+
+  definition is_embedding_from_trivial_ab_group (A : AbGroup) :
+    is_embedding (from_trivial_ab_group A) :=
+  begin
+    fapply is_embedding_of_is_injective,
+    intro x y p,
+    induction x, induction y, reflexivity
+  end
+
+  definition to_trivial_ab_group (A : AbGroup) : A →g trivial_ab_group :=
+  trivial_homomorphism A trivial_ab_group
+
   variable {G}
 
   /-

hott/algebra/homomorphism.hlean

@@ -58,11 +58,17 @@ section add_group_A_B
       have x₁ - x₂ = 0, from H _ this,
       eq_of_sub_eq_zero this)
 
-  definition eq_zero_of_is_add_hom [add_group B] {f : A → B} [is_add_hom f]
+  definition eq_zero_of_is_add_hom {f : A → B} [is_add_hom f]
       [is_embedding f] {a : A} (fa0 : f a = 0) :
     a = 0 :=
   have f a = f 0, by rewrite [fa0, respect_zero f],
   show a = 0, from is_injective_of_is_embedding this
+
+  theorem eq_zero_of_eq_zero_of_is_embedding {f : A → B} [is_add_hom f] [is_embedding f]
+    {a : A} (h : f a = 0) : a = 0 :=
+  have f a = f 0, by rewrite [h, respect_zero],
+  show a = 0, from is_injective_of_is_embedding this
+
 end add_group_A_B
 
 /- multiplicative structures -/

hott/algebra/homotopy_group.hlean

@@ -23,6 +23,12 @@ namespace eq
   definition ab_inf_group_loop [constructor] [instance] (A : Type*) : ab_inf_group (Ω (Ω A)) :=
   ⦃ab_inf_group, inf_group_loop _, mul_comm := eckmann_hilton⦄
 
+  definition inf_group_loopn (n : ℕ) (A : Type*) [H : is_succ n] : inf_group (Ω[n] A) :=
+  by induction H; exact _
+
+  definition ab_inf_group_loopn (n : ℕ) (A : Type*) [H : is_at_least_two n] : ab_inf_group (Ω[n] A) :=
+  by induction H; exact _
+
   definition gloop [constructor] (A : Type*) : InfGroup :=
   InfGroup.mk (Ω A) (inf_group_loop A)
 
@@ -124,6 +130,13 @@ namespace eq
     { exact homotopy_group_succ_in_con},
   end
 
+  definition is_contr_homotopy_group_of_is_contr (A : Type*) (n : ℕ) [is_contr A] : is_contr (π[n] A) :=
+  begin
+    apply is_trunc_trunc_of_is_trunc,
+    apply is_contr_loop_of_is_trunc,
+    apply is_trunc_of_is_contr
+  end
+
   definition homotopy_group_functor [constructor] (n : ℕ) {A B : Type*} (f : A →* B)
     : π[n] A →* π[n] B :=
   ptrunc_functor 0 (apn n f)
@@ -195,6 +208,12 @@ namespace eq
 
   notation `π→g[`:95 n:0 `]`:0 := homotopy_group_homomorphism n
 
+  definition homotopy_group_homomorphism_pcompose (n : ℕ) [H : is_succ n] {A B C : Type*} (g : B →* C)
+    (f : A →* B) : π→g[n] (g ∘* f) ~ π→g[n] g ∘ π→g[n] f :=
+  begin
+    induction H with n, exact to_homotopy (homotopy_group_functor_compose (succ n) g f)
+  end
+
   definition homotopy_group_isomorphism_of_pequiv [constructor] (n : ℕ) {A B : Type*} (f : A ≃* B)
     : πg[n+1] A ≃g πg[n+1] B :=
   begin

hott/algebra/inf_group.hlean

@@ -331,6 +331,9 @@ section inf_group
   ⦃ right_cancel_inf_semigroup, s,
     mul_right_cancel := @mul_right_cancel A s ⦄
 
+  definition one_unique {a : A} (H : Πb, a * b = b) : a = 1 :=
+  !mul_one⁻¹ ⬝ H 1
+
 end inf_group
 
 structure ab_inf_group [class] (A : Type) extends inf_group A, comm_inf_monoid A
@@ -533,6 +536,9 @@ section add_inf_group
   theorem add_eq_of_eq_sub {a b c : A} (H : a = c - b) : a + b = c :=
   add_eq_of_eq_add_neg H
 
+  definition zero_unique {a : A} (H : Πb, a + b = b) : a = 0 :=
+  !add_zero⁻¹ ⬝ H 0
+
 end add_inf_group
 
 definition add_ab_inf_group [class] : Type → Type := ab_inf_group

hott/arity.hlean

@@ -137,6 +137,18 @@ namespace eq
     ap010 f (ap g p) a = ap010 (λy, f (g y)) p a :=
   eq.rec_on p idp
 
+  definition ap_eq_ap010 {A B C : Type} (f : A → B → C) {a a' : A} (p : a = a') (b : B) :
+    ap (λa, f a b) p = ap010 f p b :=
+  by reflexivity
+
+  definition ap011_idp {A B C : Type} (f : A → B → C) {a a' : A} (p : a = a') (b : B) :
+    ap011 f p idp = ap010 f p b :=
+  by reflexivity
+
+  definition ap011_flip {A B C : Type} (f : A → B → C) {a a' : A} {b b' : B} (p : a = a') (q : b = b') :
+    ap011 f p q = ap011 (λb a, f a b) q p :=
+  by induction q; induction p; reflexivity
+
   /- the following theorems are function extentionality for functions with multiple arguments -/
 
   definition eq_of_homotopy2 {f g : Πa b, C a b} (H : f ~2 g) : f = g :=

hott/cubical/cube.hlean

@@ -347,4 +347,42 @@ namespace eq
   infixr ` ⬝p3 `:75 := eq_concat3
   infixl ` ⬝3p `:75 := concat3_eq
 
+  definition whisker001 {p₀₀₁' : a₀₀₀ = a₀₀₂} (q : p₀₀₁' = p₀₀₁)
+    (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) : cube (q ⬝ph s₀₁₁) s₂₁₁ (q ⬝ph s₁₀₁) s₁₂₁ s₁₁₀ s₁₁₂ :=
+  by induction q; exact c
+
+  definition whisker021 {p₀₂₁' : a₀₂₀ = a₀₂₂} (q : p₀₂₁' = p₀₂₁)
+    (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
+    cube (s₀₁₁ ⬝hp q⁻¹) s₂₁₁ s₁₀₁ (q ⬝ph s₁₂₁) s₁₁₀ s₁₁₂ :=
+  by induction q; exact c
+
+  definition whisker021' {p₀₂₁' : a₀₂₀ = a₀₂₂} (q : p₀₂₁ = p₀₂₁')
+    (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
+    cube (s₀₁₁ ⬝hp q) s₂₁₁ s₁₀₁ (q⁻¹ ⬝ph s₁₂₁) s₁₁₀ s₁₁₂ :=
+  by induction q; exact c
+
+  definition whisker201 {p₂₀₁' : a₂₀₀ = a₂₀₂} (q : p₂₀₁' = p₂₀₁)
+    (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
+    cube s₀₁₁ (q ⬝ph s₂₁₁) (s₁₀₁ ⬝hp q⁻¹) s₁₂₁ s₁₁₀ s₁₁₂ :=
+  by induction q; exact c
+
+  definition whisker201' {p₂₀₁' : a₂₀₀ = a₂₀₂} (q : p₂₀₁ = p₂₀₁')
+    (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
+    cube s₀₁₁ (q⁻¹ ⬝ph s₂₁₁) (s₁₀₁ ⬝hp q) s₁₂₁ s₁₁₀ s₁₁₂ :=
+  by induction q; exact c
+
+  definition whisker221 {p₂₂₁' : a₂₂₀ = a₂₂₂} (q : p₂₂₁ = p₂₂₁')
+    (c : cube s₀₁₁ s₂₁₁ s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) : cube s₀₁₁ (s₂₁₁ ⬝hp q) s₁₀₁ (s₁₂₁ ⬝hp q) s₁₁₀ s₁₁₂ :=
+  by induction q; exact c
+
+  definition move221 {p₂₂₁' : a₂₂₀ = a₂₂₂} {s₁₂₁ : square p₁₂₀ p₁₂₂ p₀₂₁ p₂₂₁'} (q : p₂₂₁ = p₂₂₁')
+    (c : cube s₀₁₁ (s₂₁₁ ⬝hp q) s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
+    cube s₀₁₁ s₂₁₁ s₁₀₁ (s₁₂₁ ⬝hp q⁻¹) s₁₁₀ s₁₁₂ :=
+  by induction q; exact c
+
+  definition move201 {p₂₀₁' : a₂₀₀ = a₂₀₂} {s₁₀₁ : square p₁₀₀ p₁₀₂ p₀₀₁ p₂₀₁'}  (q : p₂₀₁' = p₂₀₁)
+    (c : cube s₀₁₁ (q ⬝ph s₂₁₁) s₁₀₁ s₁₂₁ s₁₁₀ s₁₁₂) :
+    cube s₀₁₁ s₂₁₁ (s₁₀₁ ⬝hp q) s₁₂₁ s₁₁₀ s₁₁₂ :=
+  by induction q; exact c
+
 end eq

hott/cubical/pathover2.hlean

@@ -117,6 +117,10 @@ namespace eq
     apply ap_compose_ap02_constant
   end
 
+  theorem apd_constant' {A A' : Type} {B : A' → Type} {a₁ a₂ : A} {a' : A'} (b : B a')
+    (p : a₁ = a₂) : apd (λx, b) p = pathover_of_eq p idp :=
+  by induction p; reflexivity
+
   definition apd_change_path {B : A → Type} {a a₂ : A} (f : Πa, B a) {p p' : a = a₂} (s : p = p')
     : apd f p' = change_path s (apd f p) :=
   by induction s; reflexivity

hott/cubical/square.hlean

@@ -10,13 +10,13 @@ open eq equiv is_equiv sigma
 
 namespace eq
 
-  variables {A B : Type} {a a' a'' a₀₀ a₂₀ a₄₀ a₀₂ a₂₂ a₂₄ a₀₄ a₄₂ a₄₄ a₁ a₂ a₃ a₄ : A}
+  variables {A B C : Type} {a a' a'' a₀₀ a₂₀ a₄₀ a₀₂ a₂₂ a₂₄ a₀₄ a₄₂ a₄₄ a₁ a₂ a₃ a₄ : A}
             /-a₀₀-/ {p₁₀ p₁₀' : a₀₀ = a₂₀} /-a₂₀-/ {p₃₀ : a₂₀ = a₄₀} /-a₄₀-/
        {p₀₁ p₀₁' : a₀₀ = a₀₂} /-s₁₁-/ {p₂₁ p₂₁' : a₂₀ = a₂₂} /-s₃₁-/ {p₄₁ : a₄₀ = a₄₂}
             /-a₀₂-/ {p₁₂ p₁₂' : a₀₂ = a₂₂} /-a₂₂-/ {p₃₂ : a₂₂ = a₄₂} /-a₄₂-/
        {p₀₃ : a₀₂ = a₀₄} /-s₁₃-/ {p₂₃ : a₂₂ = a₂₄} /-s₃₃-/ {p₄₃ : a₄₂ = a₄₄}
             /-a₀₄-/ {p₁₄ : a₀₄ = a₂₄} /-a₂₄-/ {p₃₄ : a₂₄ = a₄₄} /-a₄₄-/
-
+            {b : B} {c : C}
 
   inductive square {A : Type} {a₀₀ : A}
     : Π{a₂₀ a₀₂ a₂₂ : A}, a₀₀ = a₂₀ → a₀₂ = a₂₂ → a₀₀ = a₀₂ → a₂₀ = a₂₂ → Type :=
@@ -633,4 +633,93 @@ namespace eq
     induction q, esimp at r, induction r using idp_rec_on, exact hrfl
   end
 
+  /- some higher coherence conditions -/
+
+
+  theorem whisker_bl_whisker_tl_eq (p : a = a')
+    : whisker_bl p (whisker_tl p ids) = con.right_inv p ⬝ph vrfl :=
+  by induction p; reflexivity
+
+  theorem ap_is_constant_natural_square {g : B → C} {f : A → B} (H : Πa, g (f a) = c) (p : a = a') :
+    (ap_is_constant H p)⁻¹ ⬝ph natural_square H p ⬝hp ap_constant p c =
+      whisker_bl (H a') (whisker_tl (H a) ids) :=
+  begin induction p, esimp, rewrite inv_inv, rewrite whisker_bl_whisker_tl_eq end
+
+  definition inv_ph_eq_of_eq_ph {p : a₀₀ = a₀₂} {r : p₀₁ = p} {s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁}
+    {s₁₁' : square p₁₀ p₁₂ p p₂₁} (t : s₁₁ = r ⬝ph s₁₁') : r⁻¹ ⬝ph s₁₁ = s₁₁' :=
+  by induction r; exact t
+
+  -- the following is used for torus.elim_surf
+  theorem whisker_square_aps_eq {f : A → B}
+    {q₁₀ : f a₀₀ = f a₂₀} {q₀₁ : f a₀₀ = f a₀₂} {q₂₁ : f a₂₀ = f a₂₂} {q₁₂ : f a₀₂ = f a₂₂}
+    {r₁₀ : ap f p₁₀ = q₁₀} {r₀₁ : ap f p₀₁ = q₀₁} {r₂₁ : ap f p₂₁ = q₂₁} {r₁₂ : ap f p₁₂ = q₁₂}
+    {s₁₁ : p₁₀ ⬝ p₂₁ = p₀₁ ⬝ p₁₂} {t₁₁ : square q₁₀ q₁₂ q₀₁ q₂₁}
+    (u : square (ap02 f s₁₁) (eq_of_square t₁₁)
+                (ap_con f p₁₀ p₂₁ ⬝ (r₁₀ ◾ r₂₁)) (ap_con f p₀₁ p₁₂ ⬝ (r₀₁ ◾ r₁₂)))
+    : whisker_square r₁₀ r₁₂ r₀₁ r₂₁ (aps f (square_of_eq s₁₁)) = t₁₁ :=
+  begin
+    induction r₁₀, induction r₀₁, induction r₁₂, induction r₂₁,
+    induction p₁₂, induction p₁₀, induction p₂₁, esimp at *, induction s₁₁, esimp at *,
+    esimp [square_of_eq],
+    apply eq_of_fn_eq_fn !square_equiv_eq, esimp,
+    exact (eq_bot_of_square u)⁻¹
+  end
+
+  definition natural_square_eq {A B : Type} {a a' : A} {f g : A → B} (p : f ~ g) (q : a = a')
+    : natural_square p q = square_of_pathover (apd p q) :=
+  idp
+
+  definition eq_of_square_hrfl_hconcat_eq {A : Type} {a a' : A} {p p' : a = a'} (q : p = p')
+    : eq_of_square (hrfl ⬝hp q⁻¹) = !idp_con ⬝ q :=
+  by induction q; induction p; reflexivity
+
+  definition aps_vrfl {A B : Type} {a a' : A} (f : A → B) (p : a = a') :
+    aps f (vrefl p) = vrefl (ap f p) :=
+  by induction p; reflexivity
+
+  definition aps_hrfl {A B : Type} {a a' : A} (f : A → B) (p : a = a') :
+    aps f (hrefl p) = hrefl (ap f p) :=
+  by induction p; reflexivity
+
+  -- should the following two equalities be cubes?
+  definition natural_square_ap_fn {A B C : Type} {a a' : A} {g h : A → B} (f : B → C) (p : g ~ h)
+    (q : a = a') : natural_square (λa, ap f (p a)) q =
+      ap_compose f g q ⬝ph (aps f (natural_square p q) ⬝hp (ap_compose f h q)⁻¹) :=
+  begin
+    induction q, exact !aps_vrfl⁻¹
+  end
+
+  definition natural_square_compose {A B C : Type} {a a' : A} {g g' : B → C}
+    (p : g ~ g') (f : A → B) (q : a = a') : natural_square (λa, p (f a)) q =
+    ap_compose g f q ⬝ph (natural_square p (ap f q) ⬝hp (ap_compose g' f q)⁻¹) :=
+  by induction q; reflexivity
+
+  definition natural_square_eq2 {A B : Type} {a a' : A} {f f' : A → B} (p : f ~ f') {q q' : a = a'}
+    (r : q = q') : natural_square p q = ap02 f r ⬝ph (natural_square p q' ⬝hp (ap02 f' r)⁻¹) :=
+  by induction r; reflexivity
+
+  definition natural_square_refl {A B : Type} {a a' : A} (f : A → B) (q : a = a')
+    : natural_square (homotopy.refl f) q = hrfl :=
+  by induction q; reflexivity
+
+  definition aps_eq_hconcat {p₀₁'} (f : A → B) (q : p₀₁' = p₀₁) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) :
+    aps f (q ⬝ph s₁₁) = ap02 f q ⬝ph aps f s₁₁ :=
+  by induction q; reflexivity
+
+  definition aps_hconcat_eq {p₂₁'} (f : A → B) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₂₁' = p₂₁) :
+    aps f (s₁₁ ⬝hp r⁻¹) = aps f s₁₁ ⬝hp (ap02 f r)⁻¹ :=
+  by induction r; reflexivity
+
+  definition aps_hconcat_eq' {p₂₁'} (f : A → B) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁) (r : p₂₁ = p₂₁') :
+    aps f (s₁₁ ⬝hp r) = aps f s₁₁ ⬝hp ap02 f r :=
+  by induction r; reflexivity
+
+  definition aps_square_of_eq (f : A → B) (s : p₁₀ ⬝ p₂₁ = p₀₁ ⬝ p₁₂) :
+    aps f (square_of_eq s) = square_of_eq ((ap_con f p₁₀ p₂₁)⁻¹ ⬝ ap02 f s ⬝ ap_con f p₀₁ p₁₂) :=
+  by induction p₁₂; esimp at *; induction s; induction p₂₁; induction p₁₀; reflexivity
+
+  definition aps_eq_hconcat_eq {p₀₁' p₂₁'} (f : A → B) (q : p₀₁' = p₀₁) (s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁)
+    (r : p₂₁' = p₂₁) : aps f (q ⬝ph s₁₁ ⬝hp r⁻¹) = ap02 f q ⬝ph aps f s₁₁ ⬝hp (ap02 f r)⁻¹ :=
+  by induction q; induction r; reflexivity
+
 end eq

hott/cubical/square2.hlean

@@ -1,52 +0,0 @@
-/-
-Copyright (c) 2015 Floris van Doorn. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Author: Floris van Doorn
-
-Coherence conditions for operations on squares
--/
-
-import .square
-
-open equiv
-
-namespace eq
-
-  variables {A B C : Type} {a a' a'' a₀₀ a₂₀ a₄₀ a₀₂ a₂₂ a₂₄ a₀₄ a₄₂ a₄₄ a₁ a₂ a₃ a₄ : A}
-            {f : A → B} {b : B} {c : C}
-            /-a₀₀-/ {p₁₀ p₁₀' : a₀₀ = a₂₀} /-a₂₀-/ {p₃₀ : a₂₀ = a₄₀} /-a₄₀-/
-       {p₀₁ p₀₁' : a₀₀ = a₀₂} /-s₁₁-/ {p₂₁ p₂₁' : a₂₀ = a₂₂} /-s₃₁-/ {p₄₁ : a₄₀ = a₄₂}
-            /-a₀₂-/ {p₁₂ p₁₂' : a₀₂ = a₂₂} /-a₂₂-/ {p₃₂ : a₂₂ = a₄₂} /-a₄₂-/
-       {p₀₃ : a₀₂ = a₀₄} /-s₁₃-/ {p₂₃ : a₂₂ = a₂₄} /-s₃₃-/ {p₄₃ : a₄₂ = a₄₄}
-            /-a₀₄-/ {p₁₄ : a₀₄ = a₂₄} /-a₂₄-/ {p₃₄ : a₂₄ = a₄₄} /-a₄₄-/
-
-  theorem whisker_bl_whisker_tl_eq (p : a = a')
-    : whisker_bl p (whisker_tl p ids) = con.right_inv p ⬝ph vrfl :=
-  by induction p; reflexivity
-
-  theorem ap_is_constant_natural_square {g : B → C} {f : A → B} (H : Πa, g (f a) = c) (p : a = a') :
-    (ap_is_constant H p)⁻¹ ⬝ph natural_square H p ⬝hp ap_constant p c =
-      whisker_bl (H a') (whisker_tl (H a) ids) :=
-  begin induction p, esimp, rewrite inv_inv, rewrite whisker_bl_whisker_tl_eq end
-
-  definition inv_ph_eq_of_eq_ph {p : a₀₀ = a₀₂} {r : p₀₁ = p} {s₁₁ : square p₁₀ p₁₂ p₀₁ p₂₁}
-    {s₁₁' : square p₁₀ p₁₂ p p₂₁} (t : s₁₁ = r ⬝ph s₁₁') : r⁻¹ ⬝ph s₁₁ = s₁₁' :=
-  by induction r; exact t
-
-  -- the following is used for torus.elim_surf
-  theorem whisker_square_aps_eq
-    {q₁₀ : f a₀₀ = f a₂₀} {q₀₁ : f a₀₀ = f a₀₂} {q₂₁ : f a₂₀ = f a₂₂} {q₁₂ : f a₀₂ = f a₂₂}
-    {r₁₀ : ap f p₁₀ = q₁₀} {r₀₁ : ap f p₀₁ = q₀₁} {r₂₁ : ap f p₂₁ = q₂₁} {r₁₂ : ap f p₁₂ = q₁₂}
-    {s₁₁ : p₁₀ ⬝ p₂₁ = p₀₁ ⬝ p₁₂} {t₁₁ : square q₁₀ q₁₂ q₀₁ q₂₁}
-    (u : square (ap02 f s₁₁) (eq_of_square t₁₁)
-                (ap_con f p₁₀ p₂₁ ⬝ (r₁₀ ◾ r₂₁)) (ap_con f p₀₁ p₁₂ ⬝ (r₀₁ ◾ r₁₂)))
-    : whisker_square r₁₀ r₁₂ r₀₁ r₂₁ (aps f (square_of_eq s₁₁)) = t₁₁ :=
-  begin
-    induction r₁₀, induction r₀₁, induction r₁₂, induction r₂₁,
-    induction p₁₂, induction p₁₀, induction p₂₁, esimp at *, induction s₁₁, esimp at *,
-    esimp [square_of_eq],
-    apply eq_of_fn_eq_fn !square_equiv_eq, esimp,
-    exact (eq_bot_of_square u)⁻¹
-  end
-
-end eq

hott/eq2.hlean

@@ -6,8 +6,8 @@ Author: Floris van Doorn
 Theorems about 2-dimensional paths
 -/
 
-import .cubical.square
-open function
+import .cubical.square .function
+open function is_equiv equiv
 
 namespace eq
   variables {A B C : Type} {f : A → B} {a a' a₁ a₂ a₃ a₄ : A} {b b' : B}
@@ -141,4 +141,118 @@ namespace eq
     : whisker_left p q⁻² ⬝ q = con.right_inv p :=
   by cases q; reflexivity
 
+  definition cast_fn_cast_square {A : Type} {B C : A → Type} (f : Π⦃a⦄, B a → C a) {a₁ a₂ : A}
+    (p : a₁ = a₂) (q : a₂ = a₁) (r : p ⬝ q = idp) (b : B a₁) :
+    cast (ap C q) (f (cast (ap B p) b)) = f b :=
+  have q⁻¹ = p, from inv_eq_of_idp_eq_con r⁻¹,
+  begin induction this, induction q, reflexivity end
+
+  definition ap011_ap_square_right {A B C : Type} (f : A → B → C) {a a' : A} (p : a = a')
+    {b₁ b₂ b₃ : B} {q₁₂ : b₁ = b₂} {q₂₃ : b₂ = b₃} {q₁₃ : b₁ = b₃} (r : q₁₂ ⬝ q₂₃ = q₁₃) :
+    square (ap011 f p q₁₂) (ap (λx, f x b₃) p) (ap (f a) q₁₃) (ap (f a') q₂₃) :=
+  by induction r; induction q₂₃; induction q₁₂; induction p; exact ids
+
+  definition ap011_ap_square_left {A B C : Type} (f : B → A → C) {a a' : A} (p : a = a')
+    {b₁ b₂ b₃ : B} {q₁₂ : b₁ = b₂} {q₂₃ : b₂ = b₃} {q₁₃ : b₁ = b₃} (r : q₁₂ ⬝ q₂₃ = q₁₃) :
+    square (ap011 f q₁₂ p) (ap (f b₃) p) (ap (λx, f x a) q₁₃) (ap (λx, f x a') q₂₃) :=
+  by induction r; induction q₂₃; induction q₁₂; induction p; exact ids
+
+  definition con2_assoc {A : Type} {x y z t : A} {p p' : x = y} {q q' : y = z} {r r' : z = t}
+    (h : p = p') (h' : q = q') (h'' : r = r') :
+    square ((h ◾ h') ◾ h'') (h ◾ (h' ◾ h'')) (con.assoc p q r) (con.assoc p' q' r') :=
+  by induction h; induction h'; induction h''; exact hrfl
+
+  definition con_left_inv_idp {A : Type} {x : A} {p : x = x} (q : p = idp)
+    : con.left_inv p = q⁻² ◾ q :=
+  by cases q; reflexivity
+
+  definition eckmann_hilton_con2 {A : Type} {x : A} {p p' q q': idp = idp :> x = x}
+    (h : p = p') (h' : q = q') : square (h ◾ h') (h' ◾ h) (eckmann_hilton p q) (eckmann_hilton p' q') :=
+  by induction h; induction h'; exact hrfl
+
+  definition ap_con_fn {A B : Type} {a a' : A} {b : B} (g h : A → b = b) (p : a = a') :
+    ap (λa, g a ⬝ h a) p = ap g p ◾ ap h p :=
+  by induction p; reflexivity
+
+  definition ap_eq_ap011 {A B C X : Type} (f : A → B → C) (g : X → A) (h : X → B) {x x' : X}
+    (p : x = x') : ap (λx, f (g x) (h x)) p = ap011 f (ap g p) (ap h p) :=
+  by induction p; reflexivity
+
+  definition ap_is_weakly_constant {A B : Type} {f : A → B}
+    (h : is_weakly_constant f) {a a' : A} (p : a = a') : ap f p = (h a a)⁻¹ ⬝ h a a' :=
+  by induction p; exact !con.left_inv⁻¹
+
+  definition ap_is_constant_idp {A B : Type} {f : A → B} {b : B} (p : Πa, f a = b) {a : A} (q : a = a)
+    (r : q = idp) : ap_is_constant p q = ap02 f r ⬝ (con.right_inv (p a))⁻¹ :=
+  by cases r; exact !idp_con⁻¹
+
+  definition con_right_inv_natural {A : Type} {a a' : A} {p p' : a = a'} (q : p = p') :
+    con.right_inv p = q ◾ q⁻² ⬝ con.right_inv p' :=
+  by induction q; induction p; reflexivity
+
+  definition whisker_right_ap {A B : Type} {a a' : A}{b₁ b₂ b₃ : B} (q : b₂ = b₃) (f : A → b₁ = b₂)
+    (p : a = a') : whisker_right q (ap f p) = ap (λa, f a ⬝ q) p :=
+  by induction p; reflexivity
+
+  definition ap02_ap_constant {A B C : Type} {a a' : A} (f : B → C) (b : B) (p : a = a') :
+    square (ap_constant p (f b)) (ap02 f (ap_constant p b)) (ap_compose f (λx, b) p) idp :=
+  by induction p; exact ids
+
+  definition ap_constant_compose {A B C : Type} {a a' : A} (c : C) (f : A → B) (p : a = a') :
+    square (ap_constant p c) (ap_constant (ap f p) c) (ap_compose (λx, c) f p) idp :=
+  by induction p; exact ids
+
+  definition ap02_constant {A B : Type} {a a' : A} (b : B) {p p' : a = a'}
+    (q : p = p') : square (ap_constant p b) (ap_constant p' b) (ap02 (λx, b) q) idp :=
+  by induction q; exact vrfl
+
+  section hsquare
+  variables {A₀₀ A₂₀ A₄₀ A₀₂ A₂₂ A₄₂ A₀₄ A₂₄ A₄₄ : Type}
+            {f₁₀ : A₀₀ → A₂₀} {f₃₀ : A₂₀ → A₄₀}
+            {f₀₁ : A₀₀ → A₀₂} {f₂₁ : A₂₀ → A₂₂} {f₄₁ : A₄₀ → A₄₂}
+            {f₁₂ : A₀₂ → A₂₂} {f₃₂ : A₂₂ → A₄₂}
+            {f₀₃ : A₀₂ → A₀₄} {f₂₃ : A₂₂ → A₂₄} {f₄₃ : A₄₂ → A₄₄}
+            {f₁₄ : A₀₄ → A₂₄} {f₃₄ : A₂₄ → A₄₄}
+
+  definition hsquare [reducible] (f₁₀ : A₀₀ → A₂₀) (f₁₂ : A₀₂ → A₂₂)
+                                 (f₀₁ : A₀₀ → A₀₂) (f₂₁ : A₂₀ → A₂₂) : Type :=
+  f₂₁ ∘ f₁₀ ~ f₁₂ ∘ f₀₁
+
+  definition hsquare_of_homotopy (p : f₂₁ ∘ f₁₀ ~ f₁₂ ∘ f₀₁) : hsquare f₁₀ f₁₂ f₀₁ f₂₁ :=
+  p
+
+  definition homotopy_of_hsquare (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) : f₂₁ ∘ f₁₀ ~ f₁₂ ∘ f₀₁ :=
+  p
+
+  definition homotopy_top_of_hsquare {f₂₁ : A₂₀ ≃ A₂₂} (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    f₁₀ ~ f₂₁⁻¹ ∘ f₁₂ ∘ f₀₁ :=
+  homotopy_inv_of_homotopy_post _ _ _ p
+
+  definition homotopy_top_of_hsquare' [is_equiv f₂₁] (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    f₁₀ ~ f₂₁⁻¹ ∘ f₁₂ ∘ f₀₁ :=
+  homotopy_inv_of_homotopy_post _ _ _ p
+
+  definition hhconcat (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) (q : hsquare f₃₀ f₃₂ f₂₁ f₄₁) :
+    hsquare (f₃₀ ∘ f₁₀) (f₃₂ ∘ f₁₂) f₀₁ f₄₁ :=
+  hwhisker_right f₁₀ q ⬝hty hwhisker_left f₃₂ p
+
+  definition hvconcat (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) (q : hsquare f₁₂ f₁₄ f₀₃ f₂₃) :
+    hsquare f₁₀ f₁₄ (f₀₃ ∘ f₀₁) (f₂₃ ∘ f₂₁) :=
+  (hhconcat p⁻¹ʰᵗʸ q⁻¹ʰᵗʸ)⁻¹ʰᵗʸ
+
+  definition hhinverse {f₁₀ : A₀₀ ≃ A₂₀} {f₁₂ : A₀₂ ≃ A₂₂} (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    hsquare f₁₀⁻¹ᵉ f₁₂⁻¹ᵉ f₂₁ f₀₁ :=
+  λb, eq_inv_of_eq ((p (f₁₀⁻¹ᵉ b))⁻¹ ⬝ ap f₂₁ (to_right_inv f₁₀ b))
+
+  definition hvinverse {f₀₁ : A₀₀ ≃ A₀₂} {f₂₁ : A₂₀ ≃ A₂₂} (p : hsquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    hsquare f₁₂ f₁₀ f₀₁⁻¹ᵉ f₂₁⁻¹ᵉ :=
+  (hhinverse p⁻¹ʰᵗʸ)⁻¹ʰᵗʸ
+
+  infix ` ⬝htyh `:73 := hhconcat
+  infix ` ⬝htyv `:73 := hvconcat
+  postfix `⁻¹ʰᵗʸʰ`:(max+1) := hhinverse
+  postfix `⁻¹ʰᵗʸᵛ`:(max+1) := hvinverse
+
+  end hsquare
+
 end eq

hott/function.hlean

@@ -7,24 +7,18 @@ Ported from Coq HoTT
 Theorems about embeddings and surjections
 -/
 
-import hit.trunc types.equiv cubical.square
+import hit.trunc types.equiv cubical.square types.nat
 
-open equiv sigma sigma.ops eq trunc is_trunc pi is_equiv fiber prod
+open equiv sigma sigma.ops eq trunc is_trunc pi is_equiv fiber prod pointed nat
 
-variables {A B C : Type} (f : A → B) {b : B}
+variables {A B C : Type} (f f' : A → B) {b : B}
 
 /- the image of a map is the (-1)-truncated fiber -/
 definition image' [constructor] (f : A → B) (b : B) : Type := ∥ fiber f b ∥
 definition is_prop_image' [instance] (f : A → B) (b : B) : is_prop (image' f b) := !is_trunc_trunc
 definition image [constructor] (f : A → B) (b : B) : Prop := Prop.mk (image' f b) _
 
-definition image.mk [constructor] {f : A → B} {b : B} (a : A) (p : f a = b)
-  : image f b :=
-tr (fiber.mk a p)
-
-protected definition image.rec [unfold 8] [recursor 8] {f : A → B} {b : B} {P : image' f b → Type}
-  [H : Πv, is_prop (P v)] (H : Π(a : A) (p : f a = b), P (image.mk a p)) (v : image' f b) : P v :=
-begin unfold [image'] at *, induction v with v, induction v with a p, exact H a p end
+definition total_image {A B : Type} (f : A → B) : Type := sigma (image f)
 
 definition is_embedding [class] (f : A → B) := Π(a a' : A), is_equiv (ap f : a = a' → f a = f a')
 
@@ -50,6 +44,33 @@ structure is_conditionally_constant [class] (f : A → B) :=
   (g : ∥A∥ → B)
   (eq : Π(a : A), f a = g (tr a))
 
+section image
+protected definition image.mk [constructor] {f : A → B} {b : B} (a : A) (p : f a = b)
+  : image f b :=
+tr (fiber.mk a p)
+
+protected definition image.rec [unfold 8] [recursor 8] {f : A → B} {b : B} {P : image' f b → Type}
+  [H : Πv, is_prop (P v)] (H : Π(a : A) (p : f a = b), P (image.mk a p)) (v : image' f b) : P v :=
+begin unfold [image'] at *, induction v with v, induction v with a p, exact H a p end
+
+definition image.elim {A B : Type} {f : A → B} {C : Type} [is_prop C] {b : B}
+  (H : image f b) (H' : ∀ (a : A), f a = b → C) : C :=
+begin
+  refine (trunc.elim _ H),
+  intro H'', cases H'' with a Ha, exact H' a Ha
+end
+
+definition image.equiv_exists {A B : Type} {f : A → B} {b : B} : image f b ≃ ∃ a, f a = b :=
+trunc_equiv_trunc _ (fiber.sigma_char _ _)
+
+definition image_pathover {f : A → B} {x y : B} (p : x = y) (u : image f x) (v : image f y) :
+  u =[p] v :=
+!is_prop.elimo
+
+/- total_image.elim_set is in hit.prop_trunc to avoid dependency cycle -/
+
+end image
+
 namespace function
 
   abbreviation sect          [unfold 4] := @is_retraction.sect
@@ -304,6 +325,51 @@ namespace function
       : is_embedding (@pr1 A B) :=
   λv v', to_is_equiv (sigma_eq_equiv v v' ⬝e !sigma_equiv_of_is_contr_right)
 
+  variables {f f'}
+  definition is_embedding_homotopy_closed (p : f ~ f') (H : is_embedding f) : is_embedding f' :=
+  begin
+    intro a a', fapply is_equiv_of_equiv_of_homotopy,
+    exact equiv.mk (ap f) _ ⬝e equiv_eq_closed_left _ (p a) ⬝e equiv_eq_closed_right _ (p a'),
+    intro q, esimp, exact (eq_bot_of_square (transpose (natural_square p q)))⁻¹
+  end
+
+  definition is_embedding_homotopy_closed_rev (p : f' ~ f) (H : is_embedding f) : is_embedding f' :=
+  is_embedding_homotopy_closed p⁻¹ʰᵗʸ H
+
+  definition is_surjective_homotopy_closed (p : f ~ f') (H : is_surjective f) : is_surjective f' :=
+  begin
+    intro b, induction H b with a q,
+    exact image.mk a ((p a)⁻¹ ⬝ q)
+  end
+
+  definition is_surjective_homotopy_closed_rev (p : f' ~ f) (H : is_surjective f) :
+    is_surjective f' :=
+  is_surjective_homotopy_closed p⁻¹ʰᵗʸ H
+
+  definition is_equiv_ap1_gen_of_is_embedding {A B : Type} (f : A → B) [is_embedding f]
+    {a a' : A} {b b' : B} (q : f a = b) (q' : f a' = b') : is_equiv (ap1_gen f q q') :=
+  begin
+    induction q, induction q',
+    exact is_equiv.homotopy_closed _ (ap1_gen_idp_left f)⁻¹ʰᵗʸ,
+  end
+
+  definition is_equiv_ap1_of_is_embedding {A B : Type*} (f : A →* B) [is_embedding f] :
+    is_equiv (Ω→ f) :=
+  is_equiv_ap1_gen_of_is_embedding f (respect_pt f) (respect_pt f)
+
+  definition loop_pequiv_loop_of_is_embedding [constructor] {A B : Type*} (f : A →* B)
+    [is_embedding f] : Ω A ≃* Ω B :=
+  pequiv_of_pmap (Ω→ f) (is_equiv_ap1_of_is_embedding f)
+
+  definition loopn_pequiv_loopn_of_is_embedding [constructor] (n : ℕ) [H : is_succ n]
+    {A B : Type*} (f : A →* B) [is_embedding f] : Ω[n] A ≃* Ω[n] B :=
+  begin
+    induction H with n,
+    exact !loopn_succ_in ⬝e*
+      loopn_pequiv_loopn n (loop_pequiv_loop_of_is_embedding f) ⬝e*
+      !loopn_succ_in⁻¹ᵉ*
+  end
+
   /-
     The definitions
       is_surjective_of_is_equiv

hott/hit/prop_trunc.hlean

@@ -1,4 +1,4 @@
-import function types.trunc hit.colimit homotopy.connectedness --types.nat.hott hit.trunc cubical.square
+import types.trunc hit.colimit homotopy.connectedness
 
 open eq is_trunc unit quotient seq_colim pi nat equiv sum algebra is_conn function
 
@@ -409,7 +409,7 @@ open prop_trunc trunc
 -- Corollaries for the actual truncation.
 namespace is_trunc
   local attribute is_prop_trunc_one_step_tr [instance]
-  definition is_prop.elim_set {A : Type} {P : Type} [is_set P] (f : A → P)
+  definition prop_trunc.elim_set [unfold 6] {A : Type} {P : Type} [is_set P] (f : A → P)
     (p : Πa a', f a = f a') (x : trunc -1 A) : P :=
   begin
     have y : trunc 0 (one_step_tr A),
@@ -420,8 +420,32 @@ namespace is_trunc
     { exact p a a'}
   end
 
-  definition is_prop.elim_set_tr {A : Type} {P : Type} {H : is_set P} (f : A → P)
-    (p : Πa a', f a = f a') (a : A) : is_prop.elim_set f p (tr a) = f a :=
+  definition prop_trunc.elim_set_tr {A : Type} {P : Type} {H : is_set P} (f : A → P)
+    (p : Πa a', f a = f a') (a : A) : prop_trunc.elim_set f p (tr a) = f a :=
   by reflexivity
 
+  open sigma
+
+  local attribute prop_trunc.elim_set [recursor 6]
+  definition total_image.elim_set [unfold 8]
+    {A B : Type} {f : A → B} {C : Type} [is_set C]
+    (g : A → C) (h : Πa a', f a = f a' → g a = g a') (x : total_image f) : C :=
+  begin
+    induction x with b v,
+    induction v using prop_trunc.elim_set with x x x',
+    { induction x with a p, exact g a },
+    { induction x with a p, induction x' with a' p', induction p', exact h _ _ p }
+  end
+
+  definition total_image.rec [unfold 7]
+    {A B : Type} {f : A → B} {C : total_image f → Type} [H : Πx, is_prop (C x)]
+    (g : Πa, C ⟨f a, image.mk a idp⟩)
+    (x : total_image f) : C x :=
+  begin
+    induction x with b v,
+    refine @image.rec _ _ _ _ _ (λv, H ⟨b, v⟩) _ v,
+    intro a p,
+    induction p, exact g a
+  end
+
 end is_trunc

hott/hit/pushout.hlean

@@ -112,6 +112,16 @@ namespace pushout
 
   variables {TL BL TR : Type} (f : TL → BL) (g : TL → TR)
 
+  protected theorem elim_inl {P : Type} (Pinl : BL → P) (Pinr : TR → P)
+    (Pglue : Π(x : TL), Pinl (f x) = Pinr (g x)) {b b' : BL} (p : b = b')
+    : ap (pushout.elim Pinl Pinr Pglue) (ap inl p) = ap Pinl p :=
+  !ap_compose⁻¹
+
+  protected theorem elim_inr {P : Type} (Pinl : BL → P) (Pinr : TR → P)
+    (Pglue : Π(x : TL), Pinl (f x) = Pinr (g x)) {b b' : TR} (p : b = b')
+    : ap (pushout.elim Pinl Pinr Pglue) (ap inr p) = ap Pinr p :=
+  !ap_compose⁻¹
+
   /- The non-dependent universal property -/
   definition pushout_arrow_equiv (C : Type)
     : (pushout f g → C) ≃ (Σ(i : BL → C) (j : TR → C), Πc, i (f c) = j (g c)) :=

hott/hit/set_quotient.hlean

@@ -8,7 +8,7 @@ Declaration of set-quotients, i.e. quotient of a mere relation which is then set
 
 import function algebra.relation types.trunc types.eq hit.quotient
 
-open eq is_trunc trunc quotient equiv
+open eq is_trunc trunc quotient equiv is_equiv
 
 namespace set_quotient
 section
@@ -86,6 +86,36 @@ namespace set_quotient
   definition is_surjective_class_of : is_surjective (class_of : A → set_quotient R) :=
   λx, set_quotient.rec_on x (λa, tr (fiber.mk a idp)) (λa a' r, !is_prop.elimo)
 
+  definition is_prop_set_quotient {A : Type} (R : A → A → Prop) [is_prop A] :
+    is_prop (set_quotient R) :=
+  begin
+    apply is_prop.mk, intro x y,
+    induction x using set_quotient.rec_prop, induction y using set_quotient.rec_prop,
+    exact ap class_of !is_prop.elim
+  end
+
+  local attribute is_prop_set_quotient [instance]
+  definition is_trunc_set_quotient [instance] (n : ℕ₋₂) {A : Type} (R : A → A → Prop) [is_trunc n A] :
+    is_trunc n (set_quotient R) :=
+  begin
+    cases n with n, { apply is_contr_of_inhabited_prop, exact class_of !center },
+    cases n with n, { apply _ },
+    apply is_trunc_succ_succ_of_is_set
+  end
+
+  definition is_equiv_class_of [constructor] {A : Type} [is_set A] (R : A → A → Prop)
+    (p : Π⦃a b⦄, R a b → a = b) : is_equiv (@class_of A R) :=
+  begin
+    fapply adjointify,
+    { intro x, induction x, exact a, exact p H },
+    { intro x, induction x using set_quotient.rec_prop, reflexivity },
+    { intro a, reflexivity }
+  end
+
+  definition equiv_set_quotient [constructor] {A : Type} [is_set A] (R : A → A → Prop)
+    (p : Π⦃a b⦄, R a b → a = b) : A ≃ set_quotient R :=
+  equiv.mk _ (is_equiv_class_of R p)
+
   /- non-dependent universal property -/
 
   definition set_quotient_arrow_equiv (B : Type) [H : is_set B] :

hott/hit/trunc.hlean

@@ -126,6 +126,10 @@ namespace trunc
   definition or.intro_left  [reducible] [constructor] (x : X) : X ∨ Y             := tr (inl x)
   definition or.intro_right [reducible] [constructor] (y : Y) : X ∨ Y             := tr (inr y)
 
+  definition exists.elim {A : Type} {p : A → Type} {B : Type} [is_prop B] (H : Exists p)
+    (H' : ∀ (a : A), p a → B) : B :=
+  trunc.elim (sigma.rec H') H
+
   definition is_contr_of_merely_prop [H : is_prop A] (aa : merely A) : is_contr A :=
   is_contr_of_inhabited_prop (trunc.rec_on aa id)
 

hott/hit/two_quotient.hlean

@@ -5,7 +5,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Floris van Doorn
 -/
 
-import homotopy.circle eq2 algebra.e_closure cubical.squareover cubical.cube cubical.square2
+import homotopy.circle eq2 algebra.e_closure cubical.squareover cubical.cube
 
 open quotient eq circle sum sigma equiv function relation e_closure
 

hott/homotopy/circle.hlean

@@ -353,4 +353,27 @@ namespace circle
   definition circle_base_mul [reducible] (x : S¹) : circle_mul base x = x :=
   idp
 
+  /-
+    Suppose for `f, g : A -> B` we prove a homotopy `H : f ~ g` by induction on the element in `A`.
+    And suppose `p : a = a'` is a path constructor in `A`.
+    Then `natural_square_tr H p` has type `square (H a) (H a') (ap f p) (ap g p)` and is equal
+    to the square which defined H on the path constructor
+  -/
+
+  definition natural_square_elim_loop {A : Type} {f g : S¹ → A} (p : f base = g base)
+    (q : square p p (ap f loop) (ap g loop))
+    : natural_square (circle.rec p (eq_pathover q)) loop = q :=
+  begin
+    refine ap square_of_pathover !rec_loop ⬝ _,
+    exact to_right_inv !eq_pathover_equiv_square q
+  end
+
+  definition circle_elim_constant [unfold 5] {A : Type} {a : A} {p : a = a} (r : p = idp) (x : S¹) :
+    circle.elim a p x = a :=
+  begin
+    induction x,
+    { reflexivity },
+    { apply eq_pathover_constant_right, apply hdeg_square, exact !elim_loop ⬝ r }
+  end
+
 end circle

hott/homotopy/connectedness.hlean

@@ -112,7 +112,7 @@ namespace is_conn
           apply eq_equiv_eq_symm
         end,
         apply @is_trunc_equiv_closed _ _ k e, clear e,
-        apply IH (λb : B, (g b = h b)) (λb, @is_trunc_eq (P b) (n +2+ k) (HP b) (g b) (h b))}
+        apply IH (λb : B, (g b = h b)) (λb, @is_trunc_eq (P b) (n +2+ k) (HP b) (g b) (h b)) }
     end
 
   end
@@ -365,6 +365,101 @@ namespace is_conn
     rewrite -of_nat_add_two, exact _
   end
 
+  definition is_conn_equiv_closed_rev (n : ℕ₋₂) {A B : Type} (f : A ≃ B) (H : is_conn n B) :
+    is_conn n A :=
+  is_conn_equiv_closed n f⁻¹ᵉ _
+
+  definition is_conn_succ_intro {n : ℕ₋₂} {A : Type} (a : trunc (n.+1) A)
+    (H2 : Π(a a' : A), is_conn n (a = a')) : is_conn (n.+1) A :=
+  begin
+    apply @is_contr_of_inhabited_prop,
+    { apply is_trunc_succ_intro,
+      refine trunc.rec _, intro a, refine trunc.rec _, intro a',
+      apply is_contr_equiv_closed !tr_eq_tr_equiv⁻¹ᵉ },
+    exact a
+  end
+
+  definition is_conn_pathover (n : ℕ₋₂) {A : Type} {B : A → Type} {a a' : A} (p : a = a') (b : B a)
+    (b' : B a') [is_conn (n.+1) (B a')] : is_conn n (b =[p] b') :=
+  is_conn_equiv_closed_rev n !pathover_equiv_tr_eq _
+
+  open sigma
+  lemma is_conn_sigma [instance] {A : Type} (B : A → Type) (n : ℕ₋₂)
+    [HA : is_conn n A] [HB : Πa, is_conn n (B a)] : is_conn n (Σa, B a) :=
+  begin
+    revert A B HA HB, induction n with n IH: intro A B HA HB,
+    { apply is_conn_minus_two },
+    apply is_conn_succ_intro,
+    { induction center (trunc (n.+1) A) with a, induction center (trunc (n.+1) (B a)) with b,
+      exact tr ⟨a, b⟩ },
+    intro a a', refine is_conn_equiv_closed_rev n !sigma_eq_equiv _,
+    apply IH, apply is_conn_eq, intro p, apply is_conn_pathover
+    /- an alternative proof of the successor case -/
+    -- induction center (trunc (n.+1) A) with a₀,
+    -- induction center (trunc (n.+1) (B a₀)) with b₀,
+    -- apply is_contr.mk (tr ⟨a₀, b₀⟩),
+    -- intro ab, induction ab with ab, induction ab with a b,
+    -- induction tr_eq_tr_equiv n a₀ a !is_prop.elim with p, induction p,
+    -- induction tr_eq_tr_equiv n b₀ b !is_prop.elim with q, induction q,
+    -- reflexivity
+  end
+
+  lemma is_conn_prod [instance] (A B : Type) (n : ℕ₋₂) [is_conn n A] [is_conn n B] :
+    is_conn n (A × B) :=
+  is_conn_equiv_closed n !sigma.equiv_prod _
+
+  lemma is_conn_fun_of_is_conn {A B : Type} (n : ℕ₋₂) (f : A → B)
+    [HA : is_conn n A] [HB : is_conn (n.+1) B] : is_conn_fun n f :=
+  λb, is_conn_equiv_closed_rev n !fiber.sigma_char _
+
+  lemma is_conn_pfiber {A B : Type*} (n : ℕ₋₂) (f : A →* B)
+    [HA : is_conn n A] [HB : is_conn (n.+1) B] : is_conn n (pfiber f) :=
+  is_conn_fun_of_is_conn n f pt
+
+  definition is_conn_fun_trunc_elim_of_le {n k : ℕ₋₂} {A B : Type} [is_trunc n B] (f : A → B)
+    (H : k ≤ n) [H2 : is_conn_fun k f] : is_conn_fun k (trunc.elim f : trunc n A → B) :=
+  begin
+    apply is_conn_fun.intro,
+    intro P, have Πb, is_trunc n (P b), from (λb, is_trunc_of_le _ H),
+    fconstructor,
+    { intro f' b,
+      refine is_conn_fun.elim k H2 _ _ b, intro a, exact f' (tr a) },
+    { intro f', apply eq_of_homotopy, intro a,
+      induction a with a, esimp, rewrite [is_conn_fun.elim_β] }
+  end
+
+  definition is_conn_fun_trunc_elim_of_ge {n k : ℕ₋₂} {A B : Type} [is_trunc n B] (f : A → B)
+    (H : n ≤ k) [H2 : is_conn_fun k f] : is_conn_fun k (trunc.elim f : trunc n A → B) :=
+  begin
+   apply is_conn_fun_of_is_equiv,
+   have H3 : is_equiv (trunc_functor k f), from !is_equiv_trunc_functor_of_is_conn_fun,
+   have H4 : is_equiv (trunc_functor n f), from is_equiv_trunc_functor_of_le _ H,
+   apply is_equiv_of_equiv_of_homotopy (equiv.mk (trunc_functor n f) _ ⬝e !trunc_equiv),
+   intro x, induction x, reflexivity
+  end
+
+  definition is_conn_fun_trunc_elim {n k : ℕ₋₂} {A B : Type} [is_trunc n B] (f : A → B)
+    [H2 : is_conn_fun k f] : is_conn_fun k (trunc.elim f : trunc n A → B) :=
+  begin
+    eapply algebra.le_by_cases k n: intro H,
+    { exact is_conn_fun_trunc_elim_of_le f H },
+    { exact is_conn_fun_trunc_elim_of_ge f H }
+  end
+
+  lemma is_conn_fun_tr (n : ℕ₋₂) (A : Type) : is_conn_fun n (tr : A → trunc n A) :=
+  begin
+    apply is_conn_fun.intro,
+    intro P,
+    fconstructor,
+    { intro f' b, induction b with a, exact f' a },
+    { intro f', reflexivity }
+  end
+
+
+  definition is_contr_of_is_conn_of_is_trunc {n : ℕ₋₂} {A : Type} (H : is_trunc n A)
+    (K : is_conn n A) : is_contr A :=
+  is_contr_equiv_closed (trunc_equiv n A)
+
 end is_conn
 
 /-

hott/homotopy/homotopy_group.hlean

@@ -223,6 +223,52 @@ namespace is_trunc
     cases A with A a, exact H k H'
   end
 
+
+
+  definition ab_group_homotopy_group_of_is_conn (n : ℕ) (A : Type*) [H : is_conn 1 A] :
+    ab_group (π[n] A) :=
+  begin
+    have is_conn 0 A, from !is_conn_of_is_conn_succ,
+    cases n with n,
+    { unfold [homotopy_group, ptrunc], apply ab_group_of_is_contr },
+    cases n with n,
+    { unfold [homotopy_group, ptrunc], apply ab_group_of_is_contr },
+    exact ab_group_homotopy_group n A
+  end
+
+  definition is_contr_of_trivial_homotopy' (n : ℕ₋₂) (A : Type) [is_trunc n A] [is_conn -1 A]
+    (H : Πk a, is_contr (π[k] (pointed.MK A a))) : is_contr A :=
+  begin
+    assert aa : trunc -1 A,
+    { apply center },
+    assert H3 : is_conn 0 A,
+    { induction aa with a, exact H 0 a },
+    exact is_contr_of_trivial_homotopy n A H
+  end
+
+  definition is_conn_of_trivial_homotopy (n : ℕ₋₂) (m : ℕ) (A : Type) [is_trunc n A] [is_conn 0 A]
+    (H : Π(k : ℕ) a, k ≤ m → is_contr (π[k] (pointed.MK A a))) : is_conn m A :=
+  begin
+    apply is_contr_of_trivial_homotopy_nat m (trunc m A),
+    intro k a H2,
+    induction a with a,
+    apply is_trunc_equiv_closed_rev,
+      exact equiv_of_pequiv (homotopy_group_trunc_of_le (pointed.MK A a) _ _ H2),
+    exact H k a H2
+  end
+
+  definition is_conn_of_trivial_homotopy_pointed (n : ℕ₋₂) (m : ℕ) (A : Type*) [is_trunc n A]
+    (H : Π(k : ℕ), k ≤ m → is_contr (π[k] A)) : is_conn m A :=
+  begin
+    have is_conn 0 A, proof H 0 !zero_le qed,
+    apply is_conn_of_trivial_homotopy n m A,
+    intro k a H2, revert a, apply is_conn.elim -1,
+    cases A with A a, exact H k H2
+  end
+
+
+
+
   definition is_conn_fun_of_equiv_on_homotopy_groups.{u} (n : ℕ) {A B : Type.{u}} (f : A → B)
     [is_equiv (trunc_functor 0 f)]
     (H1 : Πa k, k ≤ n → is_equiv (homotopy_group_functor k (pmap_of_map f a)))

hott/homotopy/sphere.hlean

@@ -302,10 +302,17 @@ namespace sphere
   definition sphere_eq_pbool : S* 0 = pbool :=
   pType_eq sphere_equiv_bool idp
 
+  definition psphere_pequiv_iterate_psusp (n : ℕ) : psphere n ≃* iterate_psusp n pbool :=
+  begin
+    induction n with n e,
+    { exact psphere_pequiv_pbool },
+    { exact psusp_pequiv e }
+  end
+
   definition psphere_pmap_pequiv' (A : Type*) (n : ℕ) : ppmap (S* n) A ≃* Ω[n] A :=
   begin
     revert A, induction n with n IH: intro A,
-    { refine _ ⬝e* !pmap_pbool_pequiv, exact pequiv_ppcompose_right psphere_pequiv_pbool⁻¹ᵉ* },
+    { refine _ ⬝e* !ppmap_pbool_pequiv, exact pequiv_ppcompose_right psphere_pequiv_pbool⁻¹ᵉ* },
     { refine psusp_adjoint_loop (S* n) A ⬝e* IH (Ω A) ⬝e* !loopn_succ_in⁻¹ᵉ*  }
   end
 

hott/homotopy/susp.hlean

@@ -226,20 +226,43 @@ namespace susp
   definition psusp_pequiv [constructor] (f : X ≃* Y) : psusp X ≃* psusp Y :=
   pequiv_of_equiv (susp.equiv f) idp
 
-  definition psusp_functor_compose (g : Y →* Z) (f : X →* Y)
-    : psusp_functor (g ∘* f) ~* psusp_functor g ∘* psusp_functor f :=
+  definition psusp_functor_pcompose (g : Y →* Z) (f : X →* Y) :
+    psusp_functor (g ∘* f) ~* psusp_functor g ∘* psusp_functor f :=
   begin
-    fconstructor,
-    { intro a, induction a,
+    fapply phomotopy.mk,
+    { intro x, induction x,
       { reflexivity },
       { reflexivity },
-      { apply eq_pathover, apply hdeg_square,
-        rewrite [▸*,ap_compose' _ (psusp_functor f)],
-        krewrite +susp.elim_merid } },
-    { reflexivity }
+      { apply eq_pathover, apply hdeg_square, esimp,
+        refine !elim_merid ⬝ _ ⬝ (ap_compose (psusp_functor g) _ _)⁻¹ᵖ,
+        refine _ ⬝ ap02 _ !elim_merid⁻¹, exact !elim_merid⁻¹ }},
+    { reflexivity },
   end
 
-  -- adjunction from Coq-HoTT
+  definition psusp_functor_phomotopy {f g : X →* Y} (p : f ~* g) :
+    psusp_functor f ~* psusp_functor g :=
+  begin
+    fapply phomotopy.mk,
+    { intro x, induction x,
+      { reflexivity },
+      { reflexivity },
+      { apply eq_pathover, apply hdeg_square, esimp, refine !elim_merid ⬝ _ ⬝ !elim_merid⁻¹ᵖ,
+        exact ap merid (p a), }},
+    { reflexivity },
+  end
+
+  definition psusp_functor_pid (A : Type*) : psusp_functor (pid A) ~* pid (psusp A) :=
+  begin
+    fapply phomotopy.mk,
+    { intro x, induction x,
+      { reflexivity },
+      { reflexivity },
+      { apply eq_pathover_id_right, apply hdeg_square, apply elim_merid }},
+    { reflexivity },
+  end
+
+  /- adjunction originally  ported from Coq-HoTT,
+     but we proved some additional naturality conditions -/
 
   definition loop_psusp_unit [constructor] (X : Type*) : X →* Ω(psusp X) :=
   begin
@@ -324,6 +347,28 @@ namespace susp
   definition loop_psusp_intro [constructor] {X Y : Type*} (f : psusp X →* Y) : X →* Ω Y :=
   ap1 f ∘* loop_psusp_unit X
 
+  definition psusp_elim_psusp_functor {A B C : Type*} (g : B →* Ω C) (f : A →* B) :
+    psusp.elim g ∘* psusp_functor f ~* psusp.elim (g ∘* f) :=
+  begin
+    refine !passoc ⬝* _, exact pwhisker_left _ !psusp_functor_pcompose⁻¹*
+  end
+
+  definition psusp_elim_phomotopy {A B : Type*} {f g : A →* Ω B} (p : f ~* g) : psusp.elim f ~* psusp.elim g :=
+  pwhisker_left _ (psusp_functor_phomotopy p)
+
+  definition psusp_elim_natural {X Y Z : Type*} (g : Y →* Z) (f : X →* Ω Y)
+    : g ∘* psusp.elim f ~* psusp.elim (Ω→ g ∘* f) :=
+  begin
+    refine _ ⬝* pwhisker_left _ !psusp_functor_pcompose⁻¹*,
+    refine !passoc⁻¹* ⬝* _ ⬝* !passoc,
+    exact pwhisker_right _ !loop_psusp_counit_natural
+  end
+
+  definition loop_psusp_intro_natural {X Y Z : Type*} (g : psusp Y →* Z) (f : X →* Y) :
+    loop_psusp_intro (g ∘* psusp_functor f) ~* loop_psusp_intro g ∘* f :=
+  pwhisker_right _ !ap1_pcompose ⬝* !passoc ⬝* pwhisker_left _ !loop_psusp_unit_natural⁻¹* ⬝*
+  !passoc⁻¹*
+
   definition psusp_adjoint_loop_right_inv {X Y : Type*} (g : X →* Ω Y) :
     loop_psusp_intro (psusp.elim g) ~* g :=
   begin
@@ -338,7 +383,7 @@ namespace susp
   definition psusp_adjoint_loop_left_inv {X Y : Type*} (f : psusp X →* Y) :
     psusp.elim (loop_psusp_intro f) ~* f :=
   begin
-    refine !pwhisker_left !psusp_functor_compose ⬝* _,
+    refine !pwhisker_left !psusp_functor_pcompose ⬝* _,
     refine !passoc⁻¹* ⬝* _,
     refine !pwhisker_right !loop_psusp_counit_natural⁻¹* ⬝* _,
     refine !passoc ⬝* _,
@@ -388,7 +433,7 @@ namespace susp
     esimp [psusp_adjoint_loop],
     refine _ ⬝* !passoc⁻¹*,
     apply pwhisker_left,
-    apply psusp_functor_compose
+    apply psusp_functor_pcompose
   end
 
   /- iterated suspension -/
@@ -442,4 +487,6 @@ namespace susp
       symmetry, apply loopn_succ_in }
   end
 
+
+
 end susp

hott/homotopy/vankampen.hlean

@@ -59,7 +59,7 @@ namespace pushout
   protected definition code_equiv (x : BL + TR) (y : TL) :
     @hom C _ x (sum.inl (f y)) ≃ @hom C _ x (sum.inr (g y)) :=
   begin
-    refine @is_prop.elim_set _ _ _ _ _ (ksurj y), { apply @is_trunc_equiv: apply is_set_hom},
+    refine @prop_trunc.elim_set _ _ _ _ _ (ksurj y), { apply @is_trunc_equiv: apply is_set_hom},
     { intro v, cases v with s p,
       exact code_equiv_pt x p},
     intro v v', cases v with s p, cases v' with s' p',
@@ -74,7 +74,7 @@ namespace pushout
     refine @set_quotient.rec_prop _ _ _ _ _ h, {intro l, apply is_trunc_eq, apply is_set_hom},
     intro l,
     have ksurj (k s) = tr (fiber.mk s idp), from !is_prop.elim,
-    refine ap (λz, to_fun (@is_prop.elim_set _ _ _ _ _ z) (class_of l)) this ⬝ _,
+    refine ap (λz, to_fun (@prop_trunc.elim_set _ _ _ _ _ z) (class_of l)) this ⬝ _,
     change class_of ([iE k F G (tr idp), DE k F G s, iD k F G (tr idp)] ++ l) =
            class_of (DE k F G s :: l) :> @hom C _ _ _,
     refine eq_of_rel (tr _) ⬝ (eq_of_rel (tr _)),

hott/homotopy/wedge.hlean

@@ -16,8 +16,11 @@ infixr ` ∨ ` := pwedge
 
 namespace wedge
 
+  protected definition glue {A B : Type*} : inl pt = inr pt :> wedge A B :=
+  pushout.glue ⋆
+
   protected definition rec {A B : Type*} {P : wedge A B → Type} (Pinl : Π(x : A), P (inl x))
-    (Pinr : Π(x : B), P (inr x)) (Pglue : pathover P (Pinl pt) (glue ⋆) (Pinr pt))
+    (Pinr : Π(x : B), P (inr x)) (Pglue : pathover P (Pinl pt) wedge.glue (Pinr pt))
     (y : wedge A B) : P y :=
   by induction y; apply Pinl; apply Pinr; induction x; exact Pglue
 
@@ -25,6 +28,16 @@ namespace wedge
     (Pinr : B → P) (Pglue : Pinl pt = Pinr pt) (y : wedge A B) : P :=
   by induction y with a b x; exact Pinl a; exact Pinr b; induction x; exact Pglue
 
+  protected definition rec_glue {A B : Type*} {P : wedge A B → Type} (Pinl : Π(x : A), P (inl x))
+    (Pinr : Π(x : B), P (inr x)) (Pglue : pathover P (Pinl pt) wedge.glue (Pinr pt)) :
+    apd (wedge.rec Pinl Pinr Pglue) wedge.glue = Pglue :=
+  !pushout.rec_glue
+
+  protected definition elim_glue {A B : Type*} {P : Type} (Pinl : A → P)
+    (Pinr : B → P) (Pglue : Pinl pt = Pinr pt) : ap (wedge.elim Pinl Pinr Pglue) wedge.glue = Pglue :=
+  !pushout.elim_glue
+
+
 end wedge
 
 attribute wedge.rec wedge.elim [recursor 7] [unfold 7]
@@ -38,7 +51,7 @@ namespace wedge
     { fapply pmap.mk, intro a, apply pinr a, apply respect_pt },
     { fapply is_equiv.adjointify, intro x, fapply pushout.elim_on x,
       exact λ x, Point A, exact id, intro u, reflexivity,
-      intro x, fapply pushout.rec_on x, intro u, cases u, esimp, apply (glue unit.star)⁻¹,
+      intro x, fapply pushout.rec_on x, intro u, cases u, esimp, apply wedge.glue⁻¹,
       intro a, reflexivity,
       intro u, cases u, esimp, apply eq_pathover,
       refine _ ⬝hp !ap_id⁻¹, fapply eq_hconcat, apply ap_compose inr,

hott/init/equiv.hlean

@@ -119,6 +119,10 @@ namespace is_equiv
              (λ b, ap f !Hty⁻¹ ⬝ right_inv f b)
              (λ a, !Hty⁻¹ ⬝ left_inv f a)
 
+  definition inv_homotopy_inv {A B : Type} {f g : A → B} [is_equiv f] [is_equiv g] (p : f ~ g)
+    : f⁻¹ ~ g⁻¹ :=
+  λb, (left_inv g (f⁻¹ b))⁻¹ ⬝ ap g⁻¹ ((p (f⁻¹ b))⁻¹ ⬝ right_inv f b)
+
   definition is_equiv_up [instance] [constructor] (A : Type)
     : is_equiv (up : A → lift A) :=
   adjointify up down (λa, by induction a;reflexivity) (λa, idp)
@@ -376,6 +380,9 @@ namespace equiv
   definition eq_of_fn_eq_fn_ap (f : A ≃ B) {x y : A} (q : x = y) : eq_of_fn_eq_fn' f (ap f q) = q :=
   eq_of_fn_eq_fn'_ap f q
 
+  definition to_inv_homotopy_inv {f g : A ≃ B} (p : f ~ g) : f⁻¹ᵉ ~ g⁻¹ᵉ :=
+  inv_homotopy_inv p
+
   --we need this theorem for the funext_of_ua proof
   theorem inv_eq {A B : Type} (eqf eqg : A ≃ B) (p : eqf = eqg) : (to_fun eqf)⁻¹ = (to_fun eqg)⁻¹ :=
   eq.rec_on p idp

hott/init/funext.hlean

@@ -278,6 +278,13 @@ namespace eq
     refine homotopy.rec_on' p _, intro q, induction q, exact H
   end
 
+  protected definition homotopy.rec_on_idp_left {A : Type} {P : A → Type} {g : Πa, P a}
+    {Q : Πf, (f ~ g) → Type} {f : Π x, P x}
+    (p : f ~ g) (H : Q g (homotopy.refl g)) : Q f p :=
+  begin
+    induction p using homotopy.rec_on, induction q, exact H
+  end
+
   definition eq_of_homotopy_inv {f g : Π x, P x} (H : f ~ g)
     : eq_of_homotopy (λx, (H x)⁻¹) = (eq_of_homotopy H)⁻¹ :=
   begin

hott/init/path.hlean

@@ -67,11 +67,11 @@ namespace eq
     p₁ ⬝ (p₂ ⬝ p₃ ⬝ p₄) ⬝ p₅ = (p₁ ⬝ p₂) ⬝ p₃ ⬝ (p₄ ⬝ p₅) :=
   by induction p₅; induction p₄; induction p₃; reflexivity
 
-  -- The left inverse law.
+  -- The right inverse law.
   definition con.right_inv [unfold 4] (p : x = y) : p ⬝ p⁻¹ = idp :=
   by induction p; reflexivity
 
-  -- The right inverse law.
+  -- The left inverse law.
   definition con.left_inv [unfold 4] (p : x = y) : p⁻¹ ⬝ p = idp :=
   by induction p; reflexivity
 
@@ -112,6 +112,12 @@ namespace eq
     (H₁ : a = b) (H₂ : C (H₁⁻¹⁻¹)) : C H₁ :=
   eq.rec_on (inv_inv H₁) H₂
 
+  definition eq.rec_symm {A : Type} {a₀ : A} {P : Π⦃a₁⦄, a₁ = a₀ → Type}
+    (H : P idp) ⦃a₁ : A⦄ (p : a₁ = a₀) : P p :=
+  begin
+    cases p, exact H
+  end
+
   /- Theorems for moving things around in equations -/
 
   definition con_eq_of_eq_inv_con {p : x = z} {q : y = z} {r : y = x} :
@@ -234,6 +240,9 @@ namespace eq
   protected definition homotopy.refl [refl] [reducible] [unfold_full] (f : Πx, P x) : f ~ f :=
   λ x, idp
 
+  protected definition homotopy.rfl [reducible] [unfold_full] {f : Πx, P x} : f ~ f :=
+  homotopy.refl f
+
   protected definition homotopy.symm [symm] [reducible] [unfold_full] {f g : Πx, P x} (H : f ~ g)
     : g ~ f :=
   λ x, (H x)⁻¹
@@ -242,6 +251,9 @@ namespace eq
     (H1 : f ~ g) (H2 : g ~ h) : f ~ h :=
   λ x, H1 x ⬝ H2 x
 
+  infix ` ⬝hty `:75 := homotopy.trans
+  postfix `⁻¹ʰᵗʸ`:(max+1) := homotopy.symm
+
   definition hwhisker_left [unfold_full] (g : B → C) {f f' : A → B} (H : f ~ f') :
     g ∘ f ~ g ∘ f' :=
   λa, ap g (H a)
@@ -250,6 +262,19 @@ namespace eq
     g ∘ f ~ g' ∘ f :=
   λa, H (f a)
 
+  definition compose_id (f : A → B) : f ∘ id ~ f :=
+  by reflexivity
+
+  definition id_compose (f : A → B) : id ∘ f ~ f :=
+  by reflexivity
+
+  definition compose2 {A B C : Type} {g g' : B → C} {f f' : A → B}
+    (p : g ~ g') (q : f ~ f') : g ∘ f ~ g' ∘ f' :=
+  hwhisker_right f p ⬝hty hwhisker_left g' q
+
+  definition hassoc {A B C D : Type} (h : C → D) (g : B → C) (f : A → B) : (h ∘ g) ∘ f ~ h ∘ (g ∘ f) :=
+  λa, idp
+
   definition homotopy_of_eq {f g : Πx, P x} (H1 : f = g) : f ~ g :=
   H1 ▸ homotopy.refl f
 
@@ -277,14 +302,6 @@ namespace eq
   definition ap011 [unfold 9] (f : A → B → C) (Ha : a = a') (Hb : b = b') : f a b = f a' b' :=
   by cases Ha; exact ap (f a) Hb
 
-  definition ap_eq_ap011_left (f : A → B → C) (Ha : a = a') (b : B) :
-    ap (λa, f a b) Ha = ap011 f Ha idp :=
-  by induction Ha; reflexivity
-
-  definition ap_eq_ap011_right (f : A → B → C) (a : A) (Hb : b = b') :
-    ap (f a) Hb = ap011 f idp Hb :=
-  by reflexivity
-
   /- More theorems for moving things around in equations -/
 
   definition tr_eq_of_eq_inv_tr {P : A → Type} {x y : A} {p : x = y} {u : P x} {v : P y} :
@@ -453,6 +470,22 @@ namespace eq
     ap h (ap10 p a) = ap10 (ap (λ f', h ∘ f') p) a:=
   by induction p; reflexivity
 
+  /- some lemma's about ap011 -/
+
+  definition ap_eq_ap011_left (f : A → B → C) (Ha : a = a') (b : B) :
+    ap (λa, f a b) Ha = ap011 f Ha idp :=
+  by induction Ha; reflexivity
+
+  definition ap_eq_ap011_right (f : A → B → C) (a : A) (Hb : b = b') :
+    ap (f a) Hb = ap011 f idp Hb :=
+  by reflexivity
+
+  definition ap_ap011 {A B C D : Type} (g : C → D) (f : A → B → C) {a a' : A} {b b' : B}
+    (p : a = a') (q : b = b') : ap g (ap011 f p q) = ap011 (λa b, g (f a b)) p q :=
+  begin
+    induction p, exact (ap_compose g (f a) q)⁻¹
+  end
+
 
   /- Transport and the groupoid structure of paths -/
 

hott/init/pathover.hlean

@@ -281,6 +281,11 @@ namespace eq
     (q : b =[p] b₂) : f b =[p] g b₂ :=
   by induction q; exact apo10 r b
 
+  definition apo011 {A : Type} {B C D : A → Type} {a a' : A} {p : a = a'} {b : B a} {b' : B a'}
+    {c : C a} {c' : C a'} (f : Π⦃a⦄, B a → C a → D a) (q : b =[p] b') (r : c =[p] c') :
+    f b c =[p] f b' c' :=
+  begin induction q, induction r using idp_rec_on, exact idpo end
+
   definition apdo011 {A : Type} {B : A → Type} {C : Π⦃a⦄, B a → Type}
     (f : Π⦃a⦄ (b : B a), C b) {a a' : A} (p : a = a') {b : B a} {b' : B a'} (q : b =[p] b')
       : f b =[apd011 C p q] f b' :=
@@ -445,5 +450,4 @@ namespace eq
       apd0111 f (ap k p) (pathover_ap B k (apo l q)) (pathover_tro _ (m c)) :=
   by induction q; reflexivity
 
-
 end eq

hott/init/trunc.hlean

@@ -145,6 +145,8 @@ namespace is_trunc
   definition center (A : Type) [H : is_contr A] : A :=
   contr_internal.center (is_trunc.to_internal -2 A)
 
+  definition center' {A : Type} (H : is_contr A) : A := center A
+
   definition center_eq [H : is_contr A] (a : A) : !center = a :=
   contr_internal.center_eq (is_trunc.to_internal -2 A) a
 
@@ -337,13 +339,17 @@ namespace is_trunc
             {a a₂ : A} (p : a = a₂)
             (c : C a) (c₂ : C a₂)
 
-  definition is_prop.elimo [H : is_prop (C a)] : c =[p] c₂ :=
-  pathover_of_eq_tr !is_prop.elim
-
   definition is_trunc_pathover [instance]
     (n : ℕ₋₂) [H : is_trunc (n.+1) (C a)] : is_trunc n (c =[p] c₂) :=
   is_trunc_equiv_closed_rev n !pathover_equiv_eq_tr
 
+  definition is_prop.elimo [H : is_prop (C a)] : c =[p] c₂ :=
+  pathover_of_eq_tr !is_prop.elim
+
+  definition is_prop_elimo_self {A : Type} (B : A → Type) {a : A} (b : B a) {H : is_prop (B a)} :
+    @is_prop.elimo A B a a idp b b H = idpo :=
+  !is_prop.elim
+
   variables {p c c₂}
   theorem is_set.elimo (q q' : c =[p] c₂) [H : is_set (C a)] : q = q' :=
   !is_prop.elim

hott/prop_trunc.hlean

@@ -35,7 +35,7 @@ namespace is_trunc
       induction (P a b), apply idp},
   end
 
-  definition is_prop_is_trunc [instance] (n : trunc_index) :
+  definition is_prop_is_trunc (n : trunc_index) :
     Π (A : Type), is_prop (is_trunc n A) :=
   begin
     induction n,
@@ -50,4 +50,10 @@ namespace is_trunc
       apply equiv.to_is_equiv,
       apply is_trunc.pi_char},
   end
+
+  local attribute is_prop_is_trunc [instance]
+  definition is_trunc_succ_is_trunc [instance] (n m : ℕ₋₂) (A : Type) :
+    is_trunc (n.+1) (is_trunc m A) :=
+  !is_trunc_succ_of_is_prop
+
 end is_trunc

hott/types/fiber.hlean

@@ -146,26 +146,7 @@ namespace fiber
     : ppoint f ~* pmap.mk pr1 idp ∘* pfiber.sigma_char f :=
   !phomotopy.refl
 
-  definition pfiber_loop_space {A B : Type*} (f : A →* B) : pfiber (Ω→ f) ≃* Ω (pfiber f) :=
-    pequiv_of_equiv
-    (calc pfiber (Ω→ f) ≃ Σ(p : Point A = Point A), ap1 f p = rfl
-                          : (fiber.sigma_char (ap1 f) (Point (Ω B)))
-                    ... ≃ Σ(p : Point A = Point A), (respect_pt f) = ap f p ⬝ (respect_pt f)
-                          : (sigma_equiv_sigma_right (λp,
-                              calc (ap1 f p = rfl) ≃ !respect_pt⁻¹ ⬝ (ap f p ⬝ !respect_pt) = rfl
-                                                     : equiv_eq_closed_left _ (con.assoc _ _ _)
-                                               ... ≃ ap f p ⬝ (respect_pt f) = (respect_pt f)
-                                                     : eq_equiv_inv_con_eq_idp
-                                               ... ≃ (respect_pt f) = ap f p ⬝ (respect_pt f)
-                                                     : eq_equiv_eq_symm))
-                    ... ≃ fiber.mk (Point A) (respect_pt f) = fiber.mk pt (respect_pt f)
-                          : fiber_eq_equiv
-                    ... ≃ Ω (pfiber f)
-                          : erfl)
-    (begin cases f with f p, cases A with A a, cases B with B b, esimp at p, esimp at f,
-           induction p, reflexivity end)
-
-  definition pfiber_equiv_of_phomotopy {A B : Type*} {f g : A →* B} (h : f ~* g)
+  definition pfiber_pequiv_of_phomotopy {A B : Type*} {f g : A →* B} (h : f ~* g)
     : pfiber f ≃* pfiber g :=
   begin
     fapply pequiv_of_equiv,
@@ -202,12 +183,125 @@ namespace fiber
     { apply eq_pathover_Fl' }
   end
 
-  definition pfiber_equiv_of_square {A B C D : Type*} {f : A →* B} {g : C →* D} (h : A ≃* C)
+  definition pfiber_pequiv_of_square {A B C D : Type*} {f : A →* B} {g : C →* D} (h : A ≃* C)
     (k : B ≃* D) (s : k ∘* f ~* g ∘* h) : pfiber f ≃* pfiber g :=
     calc pfiber f ≃* pfiber (k ∘* f) : pequiv_postcompose
-              ... ≃* pfiber (g ∘* h) : pfiber_equiv_of_phomotopy s
+              ... ≃* pfiber (g ∘* h) : pfiber_pequiv_of_phomotopy s
               ... ≃* pfiber g : pequiv_precompose
 
+  definition pcompose_ppoint {A B : Type*} (f : A →* B) : f ∘* ppoint f ~* pconst (pfiber f) B :=
+  begin
+    fapply phomotopy.mk,
+    { exact point_eq },
+    { exact !idp_con⁻¹ }
+  end
+
+  definition point_fiber_eq {A B : Type} {f : A → B} {b : B} {x y : fiber f b}
+    (p : point x = point y) (q : point_eq x = ap f p ⬝ point_eq y) :
+    ap point (fiber_eq p q) = p :=
+  begin
+    induction x with a r, induction y with a' s, esimp at *, induction p,
+    induction q using eq.rec_symm, induction s, reflexivity
+  end
+
+  definition fiber_eq_equiv_fiber {A B : Type} {f : A → B} {b : B} (x y : fiber f b) :
+    x = y ≃ fiber (ap1_gen f (point_eq x) (point_eq y)) (idpath b) :=
+  calc
+    x = y ≃ fiber.sigma_char f b x = fiber.sigma_char f b y :
+      eq_equiv_fn_eq_of_equiv (fiber.sigma_char f b) x y
+      ... ≃ Σ(p : point x = point y), point_eq x =[p] point_eq y : sigma_eq_equiv
+      ... ≃ Σ(p : point x = point y), (point_eq x)⁻¹ ⬝ ap f p ⬝ point_eq y = idp :
+      sigma_equiv_sigma_right (λp,
+      calc point_eq x =[p] point_eq y ≃ point_eq x = ap f p ⬝ point_eq y : eq_pathover_equiv_Fl
+           ... ≃ ap f p ⬝ point_eq y = point_eq x : eq_equiv_eq_symm
+           ... ≃ (point_eq x)⁻¹ ⬝ (ap f p ⬝ point_eq y) = idp : eq_equiv_inv_con_eq_idp
+           ... ≃ (point_eq x)⁻¹ ⬝ ap f p ⬝ point_eq y = idp : equiv_eq_closed_left _ !con.assoc⁻¹)
+      ... ≃ fiber (ap1_gen f (point_eq x) (point_eq y)) (idpath b) : fiber.sigma_char
+
+  definition loop_pfiber [constructor] {A B : Type*} (f : A →* B) : Ω (pfiber f) ≃* pfiber (Ω→ f) :=
+  pequiv_of_equiv (fiber_eq_equiv_fiber pt pt)
+    begin
+      induction f with f f₀, induction B with B b₀, esimp at (f,f₀), induction f₀, reflexivity
+    end
+
+  definition pfiber_loop_space {A B : Type*} (f : A →* B) : pfiber (Ω→ f) ≃* Ω (pfiber f) :=
+  (loop_pfiber f)⁻¹ᵉ*
+
+  definition point_fiber_eq_equiv_fiber {A B : Type} {f : A → B} {b : B} {x y : fiber f b}
+    (p : x = y) : point (fiber_eq_equiv_fiber x y p) = ap1_gen point idp idp p :=
+  by induction p; reflexivity
+
+  lemma ppoint_loop_pfiber {A B : Type*} (f : A →* B) :
+    ppoint (Ω→ f) ∘* loop_pfiber f ~* Ω→ (ppoint f) :=
+  phomotopy.mk (point_fiber_eq_equiv_fiber)
+    begin
+     induction f with f f₀, induction B with B b₀, esimp at (f,f₀), induction f₀, reflexivity
+    end
+
+  lemma ppoint_loop_pfiber_inv {A B : Type*} (f : A →* B) :
+    Ω→ (ppoint f) ∘* (loop_pfiber f)⁻¹ᵉ* ~* ppoint (Ω→ f) :=
+  (phomotopy_pinv_right_of_phomotopy (ppoint_loop_pfiber f))⁻¹*
+
+  lemma pfiber_pequiv_of_phomotopy_ppoint {A B : Type*} {f g : A →* B} (h : f ~* g)
+    : ppoint g ∘* pfiber_pequiv_of_phomotopy h ~* ppoint f :=
+  begin
+    induction f with f f₀, induction g with g g₀, induction h with h h₀, induction B with B b₀,
+    esimp at *, induction h₀, induction g₀,
+    fapply phomotopy.mk,
+    { reflexivity },
+    { esimp [pfiber_pequiv_of_phomotopy], exact !point_fiber_eq⁻¹ }
+  end
+
+  lemma pequiv_postcompose_ppoint {A B B' : Type*} (f : A →* B) (g : B ≃* B')
+    : ppoint f ∘* fiber.pequiv_postcompose f g ~* ppoint (g ∘* f) :=
+  begin
+    induction f with f f₀, induction g with g hg g₀, induction B with B b₀,
+    induction B' with B' b₀', esimp at *, induction g₀, induction f₀,
+    fapply phomotopy.mk,
+    { reflexivity },
+    { esimp [pequiv_postcompose], symmetry,
+      refine !ap_compose⁻¹ ⬝ _, apply ap_constant }
+  end
+
+  lemma pequiv_precompose_ppoint {A A' B : Type*} (f : A →* B) (g : A' ≃* A)
+    : ppoint f ∘* fiber.pequiv_precompose f g ~* g ∘* ppoint (f ∘* g) :=
+  begin
+    induction f with f f₀, induction g with g hg g₀, induction B with B b₀,
+    induction A with A a₀', esimp at *, induction g₀, induction f₀,
+    reflexivity,
+  end
+
+  definition pfiber_pequiv_of_square_ppoint {A B C D : Type*} {f : A →* B} {g : C →* D}
+    (h : A ≃* C) (k : B ≃* D) (s : k ∘* f ~* g ∘* h)
+    : ppoint g ∘* pfiber_pequiv_of_square h k s ~* h ∘* ppoint f :=
+  begin
+    refine !passoc⁻¹* ⬝* _,
+    refine pwhisker_right _ !pequiv_precompose_ppoint ⬝* _,
+    refine !passoc ⬝* _,
+    apply pwhisker_left,
+    refine !passoc⁻¹* ⬝* _,
+    refine pwhisker_right _ !pfiber_pequiv_of_phomotopy_ppoint ⬝* _,
+    apply pinv_right_phomotopy_of_phomotopy,
+    refine !pequiv_postcompose_ppoint⁻¹*,
+  end
+
+  -- this breaks certain proofs if it is an instance
+  definition is_trunc_fiber (n : ℕ₋₂) {A B : Type} (f : A → B) (b : B)
+    [is_trunc n A] [is_trunc (n.+1) B] : is_trunc n (fiber f b) :=
+  is_trunc_equiv_closed_rev n !fiber.sigma_char
+
+  definition is_trunc_pfiber (n : ℕ₋₂) {A B : Type*} (f : A →* B)
+    [is_trunc n A] [is_trunc (n.+1) B] : is_trunc n (pfiber f) :=
+  is_trunc_fiber n f pt
+
+  definition fiber_equiv_of_is_contr [constructor] {A B : Type} (f : A → B) (b : B) [is_contr B] :
+    fiber f b ≃ A :=
+  !fiber.sigma_char ⬝e !sigma_equiv_of_is_contr_right
+
+  definition pfiber_pequiv_of_is_contr [constructor] {A B : Type*} (f : A →* B) [is_contr B] :
+    pfiber f ≃* A :=
+  pequiv_of_equiv (fiber_equiv_of_is_contr f pt) idp
+
 end fiber
 
 open function is_equiv

hott/types/int/hott.hlean

@@ -6,7 +6,7 @@ Author: Floris van Doorn
 Theorems about the integers specific to HoTT
 -/
 
-import .basic types.eq arity algebra.bundled
+import .order types.eq arity algebra.bundled
 open core eq is_equiv equiv algebra is_trunc
 open nat (hiding pred)
 
@@ -28,6 +28,12 @@ namespace int
   AddAbGroup.mk ℤ _
 
   notation `agℤ` := AbGroup_int
+
+  definition ring_int : Ring :=
+  Ring.mk ℤ _
+
+  notation `rℤ` := ring_int
+
   end
 
   definition is_equiv_succ [constructor] [instance] : is_equiv succ :=
@@ -43,6 +49,17 @@ namespace int
                (λb g, f ⬝e g)
                (λb g, g ⬝e f⁻¹ᵉ)
 
+  definition max0 : ℤ → ℕ
+  | (of_nat n) := n
+  | (-[1+ n])  := 0
+
+  lemma le_max0 : Π(n : ℤ), n ≤ of_nat (max0 n)
+  | (of_nat n) := proof le.refl n qed
+  | (-[1+ n])  := proof unit.star qed
+
+  lemma le_of_max0_le {n : ℤ} {m : ℕ} (h : max0 n ≤ m) : n ≤ of_nat m :=
+  le.trans (le_max0 n) (of_nat_le_of_nat_of_le h)
+
   -- definition iterate_trans {A : Type} (f : A ≃ A) (a : ℤ)
   --   : iterate f a ⬝e f = iterate f (a + 1) :=
   -- sorry

hott/types/nat/hott.hlean

@@ -6,9 +6,9 @@ Author: Floris van Doorn
 Theorems about the natural numbers specific to HoTT
 -/
 
-import .order types.pointed
+import .order types.pointed .sub
 
-open is_trunc unit empty eq equiv algebra pointed
+open is_trunc unit empty eq equiv algebra pointed is_equiv equiv function
 
 namespace nat
   definition is_prop_le [instance] (n m : ℕ) : is_prop (n ≤ m) :=
@@ -194,5 +194,44 @@ namespace nat
   definition is_at_least_two_bit1 [constructor] (n : ℕ) [H : is_succ n] : is_at_least_two (bit1 n) :=
   by exact _
 
+  /- some facts about iterate -/
+
+  definition iterate_succ {A : Type} (f : A → A) (n : ℕ) (x : A) :
+    f^[succ n] x = f^[n] (f x) :=
+  by induction n with n p; reflexivity; exact ap f p
+
+  lemma iterate_sub {A : Type} (f : A ≃ A) {n m : ℕ} (h : n ≥ m) (a : A) :
+    iterate f (n - m) a = iterate f n (iterate f⁻¹ m a) :=
+  begin
+    revert n h, induction m with m p: intro n h,
+    { reflexivity },
+    { cases n with n, exfalso, apply not_succ_le_zero _ h,
+      rewrite [succ_sub_succ], refine p n (le_of_succ_le_succ h) ⬝ _,
+      refine ap (f^[n]) _ ⬝ !iterate_succ⁻¹, exact !to_right_inv⁻¹ }
+  end
+
+  definition iterate_commute {A : Type} {f g : A → A} (n : ℕ) (h : f ∘ g ~ g ∘ f) :
+    iterate f n ∘ g ~ g ∘ iterate f n :=
+  by induction n with n IH; reflexivity; exact λx, ap f (IH x) ⬝ !h
+
+  definition iterate_equiv {A : Type} (f : A ≃ A) (n : ℕ) : A ≃ A :=
+  equiv.mk (iterate f n)
+           (by induction n with n IH; apply is_equiv_id; exact is_equiv_compose f (iterate f n))
+
+  definition iterate_inv {A : Type} (f : A ≃ A) (n : ℕ) :
+    (iterate_equiv f n)⁻¹ ~ iterate f⁻¹ n :=
+  begin
+    induction n with n p: intro a,
+      reflexivity,
+      exact p (f⁻¹ a) ⬝ !iterate_succ⁻¹
+  end
+
+  definition iterate_left_inv {A : Type} (f : A ≃ A) (n : ℕ) (a : A) : f⁻¹ᵉ^[n] (f^[n] a) = a :=
+  (iterate_inv f n (f^[n] a))⁻¹ ⬝ to_left_inv (iterate_equiv f n) a
+
+  definition iterate_right_inv {A : Type} (f : A ≃ A) (n : ℕ) (a : A) : f^[n] (f⁻¹ᵉ^[n] a) = a :=
+  ap (f^[n]) (iterate_inv f n a)⁻¹ ⬝ to_right_inv (iterate_equiv f n) a
+
+
 
 end nat

hott/types/pi.hlean

@@ -326,6 +326,11 @@ namespace pi
   local attribute ne [reducible]
   theorem is_prop_ne [instance] {A : Type} (a b : A) : is_prop (a ≠ b) := _
 
+  definition is_contr_pi_of_neg {A : Type} (B : A → Type) (H : ¬ A) : is_contr (Πa, B a) :=
+  begin
+    apply is_contr.mk (λa, empty.elim (H a)), intro f, apply eq_of_homotopy, intro x, contradiction
+  end
+
   /- Symmetry of Π -/
   definition is_equiv_flip [instance] {P : A → A' → Type}
     : is_equiv (@function.flip A A' P) :=

hott/types/pointed.hlean

@@ -3,8 +3,10 @@ Copyright (c) 2014-2016 Jakob von Raumer. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Jakob von Raumer, Floris van Doorn
 
-Ported from Coq HoTT
+Early library ported from Coq HoTT, but greatly extended since.
 The basic definitions are in init.pointed
+
+See also .pointed2
 -/
 
 import .nat.basic ..arity ..prop_trunc
@@ -126,6 +128,10 @@ namespace pointed
 
   infixr ` ∘* `:60 := pcompose
 
+  definition respect_pt_pcompose {A B C : Type*} (g : B →* C) (f : A →* B)
+    : respect_pt (g ∘* f) = ap g (respect_pt f) ⬝ respect_pt g :=
+  idp
+
   definition passoc [constructor] (h : C →* D) (g : B →* C) (f : A →* B) : (h ∘* g) ∘* f ~* h ∘* (g ∘* f) :=
   phomotopy.mk (λa, idp)
     abstract !idp_con ⬝ whisker_right _ (!ap_con ⬝ whisker_right _ !ap_compose'⁻¹) ⬝ !con.assoc end
@@ -348,6 +354,10 @@ namespace pointed
   definition phomotopy_of_eq [constructor] {A B : Type*} {f g : A →* B} (p : f = g) : f ~* g :=
   phomotopy.mk (ap010 pmap.to_fun p) begin induction p, apply idp_con end
 
+  definition phomotopy_of_eq_idp {A B : Type*} (f : A →* B) :
+    phomotopy_of_eq idp = phomotopy.refl f :=
+  idp
+
   definition pconcat_eq [constructor] {A B : Type*} {f g h : A →* B} (p : f ~* g) (q : g = h)
     : f ~* h :=
   p ⬝* phomotopy_of_eq q
@@ -359,6 +369,10 @@ namespace pointed
   infix ` ⬝*p `:75 := pconcat_eq
   infix ` ⬝p* `:75 := eq_pconcat
 
+  definition pr1_phomotopy_eq {A B : Type*} {f g : A →* B} {p q : f ~* g} (r : p = q) (a : A) :
+    p a = q a :=
+  ap010 to_homotopy r a
+
   definition pmap_eq_equiv_internal {A B : Type*} (f g : A →* B) : (f = g) ≃ (f ~* g) :=
   calc (f = g) ≃ pmap.sigma_char f = pmap.sigma_char g
                  : eq_equiv_fn_eq pmap.sigma_char f g
@@ -397,6 +411,19 @@ namespace pointed
   definition eq_of_phomotopy (p : f ~* g) : f = g :=
   to_inv (pmap_eq_equiv f g) p
 
+  definition eq_of_phomotopy_refl {X Y : Type*} (f : X →* Y) :
+    eq_of_phomotopy (phomotopy.refl f) = idpath f :=
+  begin
+    apply to_inv_eq_of_eq, reflexivity
+  end
+
+  definition phomotopy_of_homotopy {X Y : Type*} {f g : X →* Y} (h : f ~ g) [is_set Y] : f ~* g :=
+  begin
+    fapply phomotopy.mk,
+    { exact h },
+    { apply is_set.elim }
+  end
+
   -- TODO: flip arguments in s
   definition pmap_eq (r : Πa, f a = g a) (s : respect_pt f = (r pt) ⬝ respect_pt g) : f = g :=
   eq_of_phomotopy (phomotopy.mk r s⁻¹)
@@ -404,6 +431,37 @@ namespace pointed
   definition pmap_eq_of_homotopy {A B : Type*} {f g : A →* B} [is_set B] (p : f ~ g) : f = g :=
   pmap_eq p !is_set.elim
 
+  definition phomotopy_of_eq_of_phomotopy {A B : Type*} {f g : A →* B} (p : f ~* g) :
+    phomotopy_of_eq (eq_of_phomotopy p) = p :=
+  to_right_inv (pmap_eq_equiv f g) p
+
+  definition phomotopy_rec_on_eq [recursor] {A B : Type*} {f g : A →* B}
+    {Q : (f ~* g) → Type} (p : f ~* g) (H : Π(q : f = g), Q (phomotopy_of_eq q)) : Q p :=
+  phomotopy_of_eq_of_phomotopy p ▸ H (eq_of_phomotopy p)
+
+  definition phomotopy_rec_on_idp [recursor] {A B : Type*} {f : A →* B}
+    {Q : Π{g}, (f ~* g) → Type} {g : A →* B} (p : f ~* g) (H : Q (phomotopy.refl f)) : Q p :=
+  begin
+    induction p using phomotopy_rec_on_eq,
+    induction q, exact H
+  end
+
+  definition phomotopy_rec_on_eq_phomotopy_of_eq {A B : Type*} {f g: A →* B}
+    {Q : (f ~* g) → Type} (p : f = g) (H : Π(q : f = g), Q (phomotopy_of_eq q)) :
+    phomotopy_rec_on_eq (phomotopy_of_eq p) H = H p :=
+  begin
+    unfold phomotopy_rec_on_eq,
+    refine ap (λp, p ▸ _) !adj ⬝ _,
+    refine !tr_compose⁻¹ ⬝ _,
+    apply apdt
+  end
+
+  definition phomotopy_rec_on_idp_refl {A B : Type*} (f : A →* B)
+    {Q : Π{g}, (f ~* g) → Type} (H : Q (phomotopy.refl f)) :
+    phomotopy_rec_on_idp phomotopy.rfl H = H :=
+  !phomotopy_rec_on_eq_phomotopy_of_eq
+
+  /- adjunction between (-)₊ : Type → Type* and pType.carrier : Type* → Type  -/
   definition pmap_equiv_left (A : Type) (B : Type*) : A₊ →* B ≃ (A → B) :=
   begin
     fapply equiv.MK,
@@ -440,12 +498,21 @@ namespace pointed
     resulting pointed homotopy is reflexivity
   -/
   definition pap (F : (A →* B) → (C →* D)) {f g : A →* B} (p : f ~* g) : F f ~* F g :=
-  phomotopy.mk (ap010 (λf, pmap.to_fun (F f)) (eq_of_phomotopy p))
-               begin cases eq_of_phomotopy p, apply idp_con end
+  begin
+    induction p using phomotopy_rec_on_idp, reflexivity
+  end
+
+  definition pap_refl (F : (A →* B) → (C →* D)) (f : A →* B) :
+    pap F (phomotopy.refl f) = phomotopy.refl (F f) :=
+  !phomotopy_rec_on_idp_refl
 
   definition ap1_phomotopy {f g : A →* B} (p : f ~* g) : Ω→ f ~* Ω→ g :=
   pap Ω→ p
 
+  definition ap1_phomotopy_refl {X Y : Type*} (f : X →* Y) :
+    ap1_phomotopy (phomotopy.refl f) = phomotopy.refl (Ω→ f) :=
+  !pap_refl
+
   --a proof not using function extensionality:
   definition ap1_phomotopy_explicit {f g : A →* B} (p : f ~* g) : Ω→ f ~* Ω→ g :=
   begin
@@ -465,6 +532,26 @@ namespace pointed
     { exact ap1_phomotopy IH}
   end
 
+  -- the following two definitiongs are mostly the same, maybe we should remove one
+  definition ap_eq_of_phomotopy {A B : Type*} {f g : A →* B} (p : f ~* g) (a : A) :
+    ap (λf : A →* B, f a) (eq_of_phomotopy p) = p a :=
+  ap010 to_homotopy (phomotopy_of_eq_of_phomotopy p) a
+
+  definition to_fun_eq_of_phomotopy {A B : Type*} {f g : A →* B} (p : f ~* g) (a : A) :
+    ap010 pmap.to_fun (eq_of_phomotopy p) a = p a :=
+  begin
+    induction p using phomotopy_rec_on_idp,
+    exact ap (λx, ap010 pmap.to_fun x a) !eq_of_phomotopy_refl
+  end
+
+  definition ap1_eq_of_phomotopy {A B : Type*} {f g : A →* B} (p : f ~* g) :
+    ap Ω→ (eq_of_phomotopy p) = eq_of_phomotopy (ap1_phomotopy p) :=
+  begin
+    induction p using phomotopy_rec_on_idp,
+    refine ap02 _ !eq_of_phomotopy_refl ⬝ !eq_of_phomotopy_refl⁻¹ ⬝ ap eq_of_phomotopy _,
+    exact !ap1_phomotopy_refl⁻¹
+  end
+
   /- pointed homotopies between the given pointed maps -/
 
   definition ap1_pid [constructor] {A : Type*} : ap1 (pid A) ~* pid (Ω A) :=
@@ -509,6 +596,20 @@ namespace pointed
   definition ap1_pconst [constructor] (A B : Type*) : Ω→(pconst A B) ~* pconst (Ω A) (Ω B) :=
   phomotopy.mk (λp, ap1_gen_idp_left (const A pt) p ⬝ ap_constant p pt) rfl
 
+  definition ap1_gen_con_left {A B : Type} {a a' : A} {b₀ b₁ b₂ : B}
+    {f : A → b₀ = b₁} {f' : A → b₁ = b₂} {q₀ q₁ : b₀ = b₁} {q₀' q₁' : b₁ = b₂}
+    (r₀ : f a = q₀) (r₁ : f a' = q₁) (r₀' : f' a = q₀') (r₁' : f' a' = q₁') (p : a = a') :
+      ap1_gen (λa, f a ⬝ f' a) (r₀ ◾ r₀') (r₁ ◾ r₁') p =
+      whisker_right q₀' (ap1_gen f r₀ r₁ p) ⬝ whisker_left q₁ (ap1_gen f' r₀' r₁' p) :=
+  begin induction r₀, induction r₁, induction r₀', induction r₁', induction p, reflexivity end
+
+  definition ap1_gen_con_left_idp {A B : Type} {a : A} {b₀ b₁ b₂ : B}
+    {f : A → b₀ = b₁} {f' : A → b₁ = b₂} {q₀ : b₀ = b₁} {q₁ : b₁ = b₂}
+    (r₀ : f a = q₀) (r₁ : f' a = q₁) :
+      ap1_gen_con_left r₀ r₀ r₁ r₁ idp =
+      !con.left_inv ⬝ (ap (whisker_right q₁) !con.left_inv ◾ ap (whisker_left _) !con.left_inv)⁻¹ :=
+  begin induction r₀, induction r₁, reflexivity end
+
   definition ptransport_change_eq [constructor] {A : Type} (B : A → Type*) {a a' : A} {p q : a = a'}
     (r : p = q) : ptransport B p ~* ptransport B q :=
   phomotopy.mk (λb, ap (λp, transport B p b) r) begin induction r, apply idp_con end
@@ -655,6 +756,9 @@ namespace pointed
     : pequiv.to_pmap (f ⬝e* g) = g ∘* f :=
   !to_pmap_pequiv_of_pmap
 
+  definition to_fun_pequiv_trans {X Y Z : Type*} (f : X ≃* Y) (g :Y ≃* Z) : f ⬝e* g ~ g ∘ f :=
+  λx, idp
+
   definition pequiv_change_fun [constructor] (f : A ≃* B) (f' : A →* B) (Heq : f ~ f') : A ≃* B :=
   pequiv_of_pmap f' (is_equiv.homotopy_closed f Heq)
 
@@ -892,6 +996,10 @@ namespace pointed
   definition loop_pequiv_loop [constructor] (f : A ≃* B) : Ω A ≃* Ω B :=
   loopn_pequiv_loopn 1 f
 
+  definition loop_pequiv_eq_closed [constructor] {A : Type} {a a' : A} (p : a = a')
+    : pointed.MK (a = a) idp ≃* pointed.MK (a' = a') idp :=
+  pequiv_of_equiv (loop_equiv_eq_closed p) (con.left_inv p)
+
   definition to_pmap_loopn_pequiv_loopn [constructor] (n : ℕ) (f : A ≃* B)
     : loopn_pequiv_loopn n f ~* apn n f :=
   !to_pmap_pequiv_MK2
@@ -919,6 +1027,12 @@ namespace pointed
     loop_pequiv_loop (pequiv.refl A) ~* pequiv.refl (Ω A) :=
   loopn_pequiv_loopn_rfl 1 A
 
+  definition apn_pinv (n : ℕ) {A B : Type*} (f : A ≃* B) :
+    Ω→[n] f⁻¹ᵉ* ~* (loopn_pequiv_loopn n f)⁻¹ᵉ* :=
+  begin
+    refine !to_pinv_pequiv_MK2⁻¹*
+  end
+
   definition pmap_functor [constructor] {A A' B B' : Type*} (f : A' →* A) (g : B →* B') :
     ppmap A B →* ppmap A' B' :=
   pmap.mk (λh, g ∘* h ∘* f)
@@ -950,20 +1064,6 @@ namespace pointed
       exact !con.right_inv⁻¹ ⬝ ((!idp_con⁻¹ ⬝ !ap_id⁻¹) ◾ (!ap_id⁻¹⁻² ⬝ !idp_con⁻¹)), }
   end
 
-/- -- TODO
-  definition pmap_pequiv_pmap {A A' B B' : Type*} (f : A ≃* A') (g : B ≃* B') :
-    ppmap A B ≃* ppmap A' B' :=
-  pequiv.MK (pmap_functor f⁻¹ᵉ* g) (pmap_functor f g⁻¹ᵉ*)
-    abstract begin
-      intro a, esimp, apply pmap_eq,
-      { esimp, },
-      { }
-    end end
-    abstract begin
-
-    end end
--/
-
   /- properties of iterated loop space -/
   variable (A)
   definition loopn_succ_in (n : ℕ) : Ω[succ n] A ≃* Ω[n] (Ω A) :=
@@ -1071,7 +1171,7 @@ namespace pointed
       refine pwhisker_left g !pleft_inv ⬝* !pcompose_pid, },
   end
 
-  definition loop_pmap_commute (A B : Type*) : Ω(ppmap A B) ≃* (ppmap A (Ω B)) :=
+  definition loop_ppmap_commute (A B : Type*) : Ω(ppmap A B) ≃* (ppmap A (Ω B)) :=
     pequiv_of_equiv
       (calc Ω(ppmap A B) ≃ (pconst A B ~* pconst A B)                       : pmap_eq_equiv _ _
                      ... ≃ Σ(p : pconst A B ~ pconst A B), p pt ⬝ rfl = rfl : phomotopy.sigma_char
@@ -1084,7 +1184,7 @@ namespace pointed
   definition papply_pcompose [constructor] {A : Type*} (B : Type*) (a : A) : ppmap A B →* B :=
   pmap.mk (λ(f : A →* B), f a) idp
 
-  definition pmap_pbool_pequiv [constructor] (B : Type*) : ppmap pbool B ≃* B :=
+  definition ppmap_pbool_pequiv [constructor] (B : Type*) : ppmap pbool B ≃* B :=
   begin
     fapply pequiv.MK,
     { exact papply B tt },

hott/types/pointed2.hlean

@@ -0,0 +1,847 @@
+/-
+Copyright (c) 2017 Floris van Doorn. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Floris van Doorn
+
+More results about pointed types.
+
+Contains
+- squares of pointed maps,
+- equalities between pointed homotopies and
+- squares between pointed homotopies
+- pointed maps into and out of (ppmap A B), the pointed type of pointed maps from A to B
+-/
+
+
+import algebra.homotopy_group eq2
+
+open pointed eq unit is_trunc trunc nat group is_equiv equiv sigma function
+
+namespace pointed
+
+
+  section psquare
+  /-
+    Squares of pointed maps
+
+    We treat expressions of the form
+      psquare f g h k :≡ k ∘* f ~* g ∘* h
+    as squares, where f is the top, g is the bottom, h is the left face and k is the right face.
+    Then the following are operations on squares
+  -/
+
+  variables {A A' A₀₀ A₂₀ A₄₀ A₀₂ A₂₂ A₄₂ A₀₄ A₂₄ A₄₄ : Type*}
+            {f₁₀ f₁₀' : A₀₀ →* A₂₀} {f₃₀ : A₂₀ →* A₄₀}
+            {f₀₁ f₀₁' : A₀₀ →* A₀₂} {f₂₁ f₂₁' : A₂₀ →* A₂₂} {f₄₁ : A₄₀ →* A₄₂}
+            {f₁₂ f₁₂' : A₀₂ →* A₂₂} {f₃₂ : A₂₂ →* A₄₂}
+            {f₀₃ : A₀₂ →* A₀₄} {f₂₃ : A₂₂ →* A₂₄} {f₄₃ : A₄₂ →* A₄₄}
+            {f₁₄ : A₀₄ →* A₂₄} {f₃₄ : A₂₄ →* A₄₄}
+
+  definition psquare [reducible] (f₁₀ : A₀₀ →* A₂₀) (f₁₂ : A₀₂ →* A₂₂)
+                                 (f₀₁ : A₀₀ →* A₀₂) (f₂₁ : A₂₀ →* A₂₂) : Type :=
+  f₂₁ ∘* f₁₀ ~* f₁₂ ∘* f₀₁
+
+  definition psquare_of_phomotopy (p : f₂₁ ∘* f₁₀ ~* f₁₂ ∘* f₀₁) : psquare f₁₀ f₁₂ f₀₁ f₂₁ :=
+  p
+
+  definition phomotopy_of_psquare (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : f₂₁ ∘* f₁₀ ~* f₁₂ ∘* f₀₁ :=
+  p
+
+  definition phdeg_square {f f' : A →* A'} (p : f ~* f') : psquare !pid !pid f f' :=
+  !pcompose_pid ⬝* p⁻¹* ⬝* !pid_pcompose⁻¹*
+  definition pvdeg_square {f f' : A →* A'} (p : f ~* f') : psquare f f' !pid !pid :=
+  !pid_pcompose ⬝* p ⬝* !pcompose_pid⁻¹*
+
+  variables (f₀₁ f₁₀)
+  definition phrefl : psquare !pid !pid f₀₁ f₀₁ := phdeg_square phomotopy.rfl
+  definition pvrefl : psquare f₁₀ f₁₀ !pid !pid := pvdeg_square phomotopy.rfl
+  variables {f₀₁ f₁₀}
+  definition phrfl : psquare !pid !pid f₀₁ f₀₁ := phrefl f₀₁
+  definition pvrfl : psquare f₁₀ f₁₀ !pid !pid := pvrefl f₁₀
+
+  definition phconcat (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) (q : psquare f₃₀ f₃₂ f₂₁ f₄₁) :
+    psquare (f₃₀ ∘* f₁₀) (f₃₂ ∘* f₁₂) f₀₁ f₄₁ :=
+  !passoc⁻¹* ⬝* pwhisker_right f₁₀ q ⬝* !passoc ⬝* pwhisker_left f₃₂ p ⬝* !passoc⁻¹*
+
+  definition pvconcat (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) (q : psquare f₁₂ f₁₄ f₀₃ f₂₃) :
+    psquare f₁₀ f₁₄ (f₀₃ ∘* f₀₁) (f₂₃ ∘* f₂₁) :=
+  !passoc ⬝* pwhisker_left _ p ⬝* !passoc⁻¹* ⬝* pwhisker_right _ q ⬝* !passoc
+
+  definition phinverse {f₁₀ : A₀₀ ≃* A₂₀} {f₁₂ : A₀₂ ≃* A₂₂} (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    psquare f₁₀⁻¹ᵉ* f₁₂⁻¹ᵉ* f₂₁ f₀₁ :=
+  !pid_pcompose⁻¹* ⬝* pwhisker_right _ (pleft_inv f₁₂)⁻¹* ⬝* !passoc ⬝*
+  pwhisker_left _
+    (!passoc⁻¹* ⬝* pwhisker_right _ p⁻¹* ⬝* !passoc ⬝* pwhisker_left _ !pright_inv ⬝* !pcompose_pid)
+
+  definition pvinverse {f₀₁ : A₀₀ ≃* A₀₂} {f₂₁ : A₂₀ ≃* A₂₂} (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    psquare f₁₂ f₁₀ f₀₁⁻¹ᵉ* f₂₁⁻¹ᵉ* :=
+  (phinverse p⁻¹*)⁻¹*
+
+  definition phomotopy_hconcat (q : f₀₁' ~* f₀₁) (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    psquare f₁₀ f₁₂ f₀₁' f₂₁ :=
+  p ⬝* pwhisker_left f₁₂ q⁻¹*
+
+  definition hconcat_phomotopy (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) (q : f₂₁' ~* f₂₁) :
+    psquare f₁₀ f₁₂ f₀₁ f₂₁' :=
+  pwhisker_right f₁₀ q ⬝* p
+
+  definition phomotopy_vconcat (q : f₁₀' ~* f₁₀) (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    psquare f₁₀' f₁₂ f₀₁ f₂₁ :=
+  pwhisker_left f₂₁ q ⬝* p
+
+  definition vconcat_phomotopy (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) (q : f₁₂' ~* f₁₂) :
+    psquare f₁₀ f₁₂' f₀₁ f₂₁ :=
+  p ⬝* pwhisker_right f₀₁ q⁻¹*
+
+  infix ` ⬝h* `:73 := phconcat
+  infix ` ⬝v* `:73 := pvconcat
+  infixl ` ⬝hp* `:72 := hconcat_phomotopy
+  infixr ` ⬝ph* `:72 := phomotopy_hconcat
+  infixl ` ⬝vp* `:72 := vconcat_phomotopy
+  infixr ` ⬝pv* `:72 := phomotopy_vconcat
+  postfix `⁻¹ʰ*`:(max+1) := phinverse
+  postfix `⁻¹ᵛ*`:(max+1) := pvinverse
+
+  definition pwhisker_tl (f : A →* A₀₀) (q : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    psquare (f₁₀ ∘* f) f₁₂ (f₀₁ ∘* f) f₂₁ :=
+  !passoc⁻¹* ⬝* pwhisker_right f q ⬝* !passoc
+
+  definition ap1_psquare (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    psquare (Ω→ f₁₀) (Ω→ f₁₂) (Ω→ f₀₁) (Ω→ f₂₁) :=
+  !ap1_pcompose⁻¹* ⬝* ap1_phomotopy p ⬝* !ap1_pcompose
+
+  definition apn_psquare (n : ℕ) (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    psquare (Ω→[n] f₁₀) (Ω→[n] f₁₂) (Ω→[n] f₀₁) (Ω→[n] f₂₁) :=
+  !apn_pcompose⁻¹* ⬝* apn_phomotopy n p ⬝* !apn_pcompose
+
+  definition ptrunc_functor_psquare (n : ℕ₋₂) (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    psquare (ptrunc_functor n f₁₀) (ptrunc_functor n f₁₂)
+            (ptrunc_functor n f₀₁) (ptrunc_functor n f₂₁) :=
+  !ptrunc_functor_pcompose⁻¹* ⬝* ptrunc_functor_phomotopy n p ⬝* !ptrunc_functor_pcompose
+
+  definition homotopy_group_functor_psquare (n : ℕ) (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
+        psquare (π→[n] f₁₀) (π→[n] f₁₂) (π→[n] f₀₁) (π→[n] f₂₁) :=
+  !homotopy_group_functor_compose⁻¹* ⬝* homotopy_group_functor_phomotopy n p ⬝*
+  !homotopy_group_functor_compose
+
+  definition homotopy_group_homomorphism_psquare (n : ℕ) [H : is_succ n]
+    (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : hsquare (π→g[n] f₁₀) (π→g[n] f₁₂) (π→g[n] f₀₁) (π→g[n] f₂₁) :=
+  begin
+    induction H with n, exact to_homotopy (ptrunc_functor_psquare 0 (apn_psquare (succ n) p))
+  end
+
+  end psquare
+
+  definition punit_pmap_phomotopy [constructor] {A : Type*} (f : punit →* A) :
+    f ~* pconst punit A :=
+  begin
+    fapply phomotopy.mk,
+    { intro u, induction u, exact respect_pt f },
+    { reflexivity }
+  end
+
+  definition is_contr_punit_pmap (A : Type*) : is_contr (punit →* A) :=
+  is_contr.mk (pconst punit A) (λf, eq_of_phomotopy (punit_pmap_phomotopy f)⁻¹*)
+
+  definition phomotopy_eq_equiv {A B : Type*} {f g : A →* B} (h k : f ~* g) :
+    (h = k) ≃ Σ(p : to_homotopy h ~ to_homotopy k),
+      whisker_right (respect_pt g) (p pt) ⬝ to_homotopy_pt k = to_homotopy_pt h :=
+  calc
+    h = k ≃ phomotopy.sigma_char _ _ h = phomotopy.sigma_char _ _ k
+      : eq_equiv_fn_eq (phomotopy.sigma_char f g) h k
+      ... ≃ Σ(p : to_homotopy h = to_homotopy k),
+              pathover (λp, p pt ⬝ respect_pt g = respect_pt f) (to_homotopy_pt h) p (to_homotopy_pt k)
+      : sigma_eq_equiv _ _
+      ... ≃ Σ(p : to_homotopy h = to_homotopy k),
+              to_homotopy_pt h = ap (λq, q pt ⬝ respect_pt g) p ⬝ to_homotopy_pt k
+      : sigma_equiv_sigma_right (λp, eq_pathover_equiv_Fl p (to_homotopy_pt h) (to_homotopy_pt k))
+      ... ≃ Σ(p : to_homotopy h = to_homotopy k),
+              ap (λq, q pt ⬝ respect_pt g) p ⬝ to_homotopy_pt k = to_homotopy_pt h
+      : sigma_equiv_sigma_right (λp, eq_equiv_eq_symm _ _)
+      ... ≃ Σ(p : to_homotopy h = to_homotopy k),
+      whisker_right (respect_pt g) (apd10 p pt) ⬝ to_homotopy_pt k = to_homotopy_pt h
+      : sigma_equiv_sigma_right (λp, equiv_eq_closed_left _ (whisker_right _ !whisker_right_ap⁻¹))
+      ... ≃ Σ(p : to_homotopy h ~ to_homotopy k),
+      whisker_right (respect_pt g) (p pt) ⬝ to_homotopy_pt k = to_homotopy_pt h
+      : sigma_equiv_sigma_left' eq_equiv_homotopy
+
+  definition phomotopy_eq {A B : Type*} {f g : A →* B} {h k : f ~* g} (p : to_homotopy h ~ to_homotopy k)
+    (q : whisker_right (respect_pt g) (p pt) ⬝ to_homotopy_pt k = to_homotopy_pt h) : h = k :=
+  to_inv (phomotopy_eq_equiv h k) ⟨p, q⟩
+
+  definition phomotopy_eq' {A B : Type*} {f g : A →* B} {h k : f ~* g} (p : to_homotopy h ~ to_homotopy k)
+    (q : square (to_homotopy_pt h) (to_homotopy_pt k) (whisker_right (respect_pt g) (p pt)) idp) : h = k :=
+  phomotopy_eq p (eq_of_square q)⁻¹
+
+  definition trans_refl {A B : Type*} {f g : A →* B} (p : f ~* g) : p ⬝* phomotopy.refl g = p :=
+  begin
+    induction A with A a₀, induction B with B b₀,
+    induction f with f f₀, induction g with g g₀, induction p with p p₀,
+    esimp at *, induction g₀, induction p₀,
+    reflexivity
+  end
+
+  definition eq_of_phomotopy_trans {X Y : Type*} {f g h : X →* Y} (p : f ~* g) (q : g ~* h) :
+    eq_of_phomotopy (p ⬝* q) = eq_of_phomotopy p ⬝ eq_of_phomotopy q :=
+  begin
+    induction p using phomotopy_rec_on_idp, induction q using phomotopy_rec_on_idp,
+    exact ap eq_of_phomotopy !trans_refl ⬝ whisker_left _ !eq_of_phomotopy_refl⁻¹
+  end
+
+  definition refl_trans {A B : Type*} {f g : A →* B} (p : f ~* g) : phomotopy.refl f ⬝* p = p :=
+  begin
+    induction p using phomotopy_rec_on_idp,
+    induction A with A a₀, induction B with B b₀,
+    induction f with f f₀, esimp at *, induction f₀,
+    reflexivity
+  end
+
+  definition trans_assoc {A B : Type*} {f g h i : A →* B} (p : f ~* g) (q : g ~* h)
+    (r : h ~* i) : p ⬝* q ⬝* r = p ⬝* (q ⬝* r) :=
+  begin
+    induction r using phomotopy_rec_on_idp,
+    induction q using phomotopy_rec_on_idp,
+    induction p using phomotopy_rec_on_idp,
+    induction B with B b₀,
+    induction f with f f₀, esimp at *, induction f₀,
+    reflexivity
+  end
+
+  definition refl_symm {A B : Type*} (f : A →* B) : phomotopy.rfl⁻¹* = phomotopy.refl f :=
+  begin
+    induction B with B b₀,
+    induction f with f f₀, esimp at *, induction f₀,
+    reflexivity
+  end
+
+  definition symm_symm {A B : Type*} {f g : A →* B} (p : f ~* g) : p⁻¹*⁻¹* = p :=
+  phomotopy_eq (λa, !inv_inv)
+    begin
+      induction p using phomotopy_rec_on_idp, induction f with f f₀, induction B with B b₀,
+      esimp at *, induction f₀, reflexivity
+    end
+
+  definition trans_right_inv {A B : Type*} {f g : A →* B} (p : f ~* g) : p ⬝* p⁻¹* = phomotopy.rfl :=
+  begin
+    induction p using phomotopy_rec_on_idp, exact !refl_trans ⬝ !refl_symm
+  end
+
+  definition trans_left_inv {A B : Type*} {f g : A →* B} (p : f ~* g) : p⁻¹* ⬝* p = phomotopy.rfl :=
+  begin
+    induction p using phomotopy_rec_on_idp, exact !trans_refl ⬝ !refl_symm
+  end
+
+  definition trans2 {A B : Type*} {f g h : A →* B} {p p' : f ~* g} {q q' : g ~* h}
+    (r : p = p') (s : q = q') : p ⬝* q = p' ⬝* q' :=
+  ap011 phomotopy.trans r s
+
+  definition pcompose3 {A B C : Type*} {g g' : B →* C} {f f' : A →* B}
+  {p p' : g ~* g'} {q q' : f ~* f'} (r : p = p') (s : q = q') : p ◾* q = p' ◾* q' :=
+  ap011 pcompose2 r s
+
+  definition symm2 {A B : Type*} {f g : A →* B} {p p' : f ~* g} (r : p = p') : p⁻¹* = p'⁻¹* :=
+  ap phomotopy.symm r
+
+  infixl ` ◾** `:80 := pointed.trans2
+  infixl ` ◽* `:81 := pointed.pcompose3
+  postfix `⁻²**`:(max+1) := pointed.symm2
+
+  definition trans_symm {A B : Type*} {f g h : A →* B} (p : f ~* g) (q : g ~* h) :
+    (p ⬝* q)⁻¹* = q⁻¹* ⬝* p⁻¹* :=
+  begin
+    induction p using phomotopy_rec_on_idp, induction q using phomotopy_rec_on_idp,
+    exact !trans_refl⁻²** ⬝ !trans_refl⁻¹ ⬝ idp ◾** !refl_symm⁻¹
+  end
+
+  definition phwhisker_left {A B : Type*} {f g h : A →* B} (p : f ~* g) {q q' : g ~* h}
+    (s : q = q') : p ⬝* q = p ⬝* q' :=
+  idp ◾** s
+
+  definition phwhisker_right {A B : Type*} {f g h : A →* B} {p p' : f ~* g} (q : g ~* h)
+    (r : p = p') : p ⬝* q = p' ⬝* q :=
+  r ◾** idp
+
+  definition pwhisker_left_refl {A B C : Type*} (g : B →* C) (f : A →* B) :
+    pwhisker_left g (phomotopy.refl f) = phomotopy.refl (g ∘* f) :=
+  begin
+    induction A with A a₀, induction B with B b₀, induction C with C c₀,
+    induction f with f f₀, induction g with g g₀,
+    esimp at *, induction g₀, induction f₀, reflexivity
+  end
+
+  definition pwhisker_right_refl {A B C : Type*} (f : A →* B) (g : B →* C) :
+    pwhisker_right f (phomotopy.refl g) = phomotopy.refl (g ∘* f) :=
+  begin
+    induction A with A a₀, induction B with B b₀, induction C with C c₀,
+    induction f with f f₀, induction g with g g₀,
+    esimp at *, induction g₀, induction f₀, reflexivity
+  end
+
+  definition pcompose2_refl {A B C : Type*} (g : B →* C) (f : A →* B) :
+    phomotopy.refl g ◾* phomotopy.refl f = phomotopy.rfl :=
+  !pwhisker_right_refl ◾** !pwhisker_left_refl ⬝ !refl_trans
+
+  definition pcompose2_refl_left {A B C : Type*} (g : B →* C) {f f' : A →* B} (p : f ~* f') :
+    phomotopy.rfl ◾* p = pwhisker_left g p :=
+  !pwhisker_right_refl ◾** idp ⬝ !refl_trans
+
+  definition pcompose2_refl_right {A B C : Type*} {g g' : B →* C} (f : A →* B) (p : g ~* g') :
+    p ◾* phomotopy.rfl = pwhisker_right f p :=
+  idp ◾** !pwhisker_left_refl ⬝ !trans_refl
+
+  definition pwhisker_left_trans {A B C : Type*} (g : B →* C) {f₁ f₂ f₃ : A →* B}
+    (p : f₁ ~* f₂) (q : f₂ ~* f₃) :
+    pwhisker_left g (p ⬝* q) = pwhisker_left g p ⬝* pwhisker_left g q :=
+  begin
+    induction p using phomotopy_rec_on_idp,
+    induction q using phomotopy_rec_on_idp,
+    refine _ ⬝ !pwhisker_left_refl⁻¹ ◾** !pwhisker_left_refl⁻¹,
+    refine ap (pwhisker_left g) !trans_refl ⬝ !pwhisker_left_refl ⬝ !trans_refl⁻¹
+  end
+
+  definition pwhisker_right_trans {A B C : Type*} (f : A →* B) {g₁ g₂ g₃ : B →* C}
+    (p : g₁ ~* g₂) (q : g₂ ~* g₃) :
+    pwhisker_right f (p ⬝* q) = pwhisker_right f p ⬝* pwhisker_right f q :=
+  begin
+    induction p using phomotopy_rec_on_idp,
+    induction q using phomotopy_rec_on_idp,
+    refine _ ⬝ !pwhisker_right_refl⁻¹ ◾** !pwhisker_right_refl⁻¹,
+    refine ap (pwhisker_right f) !trans_refl ⬝ !pwhisker_right_refl ⬝ !trans_refl⁻¹
+  end
+
+  definition pwhisker_left_symm {A B C : Type*} (g : B →* C) {f₁ f₂ : A →* B} (p : f₁ ~* f₂) :
+    pwhisker_left g p⁻¹* = (pwhisker_left g p)⁻¹* :=
+  begin
+    induction p using phomotopy_rec_on_idp,
+    refine _ ⬝ ap phomotopy.symm !pwhisker_left_refl⁻¹,
+    refine ap (pwhisker_left g) !refl_symm ⬝ !pwhisker_left_refl ⬝ !refl_symm⁻¹
+  end
+
+  definition pwhisker_right_symm {A B C : Type*} (f : A →* B) {g₁ g₂ : B →* C} (p : g₁ ~* g₂) :
+    pwhisker_right f p⁻¹* = (pwhisker_right f p)⁻¹* :=
+  begin
+    induction p using phomotopy_rec_on_idp,
+    refine _ ⬝ ap phomotopy.symm !pwhisker_right_refl⁻¹,
+    refine ap (pwhisker_right f) !refl_symm ⬝ !pwhisker_right_refl ⬝ !refl_symm⁻¹
+  end
+
+  definition trans_eq_of_eq_symm_trans {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
+    {r : f ~* h} (s : q = p⁻¹* ⬝* r) : p ⬝* q = r :=
+  idp ◾** s ⬝ !trans_assoc⁻¹ ⬝ trans_right_inv p ◾** idp ⬝ !refl_trans
+
+  definition eq_symm_trans_of_trans_eq {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
+    {r : f ~* h} (s : p ⬝* q = r) : q = p⁻¹* ⬝* r :=
+  !refl_trans⁻¹ ⬝ !trans_left_inv⁻¹ ◾** idp ⬝ !trans_assoc ⬝ idp ◾** s
+
+  definition trans_eq_of_eq_trans_symm {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
+    {r : f ~* h} (s : p = r ⬝* q⁻¹*) : p ⬝* q = r :=
+  s ◾** idp ⬝ !trans_assoc ⬝ idp ◾** trans_left_inv q ⬝ !trans_refl
+
+  definition eq_trans_symm_of_trans_eq {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
+    {r : f ~* h} (s : p ⬝* q = r) : p = r ⬝* q⁻¹* :=
+  !trans_refl⁻¹ ⬝ idp ◾** !trans_right_inv⁻¹ ⬝ !trans_assoc⁻¹ ⬝ s ◾** idp
+
+  definition eq_trans_of_symm_trans_eq {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
+    {r : f ~* h} (s : p⁻¹* ⬝* r = q) : r = p ⬝* q :=
+  !refl_trans⁻¹ ⬝ !trans_right_inv⁻¹ ◾** idp ⬝ !trans_assoc ⬝ idp ◾** s
+
+  definition symm_trans_eq_of_eq_trans {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
+    {r : f ~* h} (s : r = p ⬝* q) : p⁻¹* ⬝* r = q :=
+  idp ◾** s ⬝ !trans_assoc⁻¹ ⬝ trans_left_inv p ◾** idp ⬝ !refl_trans
+
+  definition eq_trans_of_trans_symm_eq {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
+    {r : f ~* h} (s : r ⬝* q⁻¹* = p) : r = p ⬝* q :=
+  !trans_refl⁻¹ ⬝ idp ◾** !trans_left_inv⁻¹ ⬝ !trans_assoc⁻¹ ⬝ s ◾** idp
+
+  definition trans_symm_eq_of_eq_trans {A B : Type*} {f g h : A →* B} {p : f ~* g} {q : g ~* h}
+    {r : f ~* h} (s : r = p ⬝* q) : r ⬝* q⁻¹* = p :=
+  s ◾** idp ⬝ !trans_assoc ⬝ idp ◾** trans_right_inv q ⬝ !trans_refl
+
+  section phsquare
+  /-
+    Squares of pointed homotopies
+  -/
+
+  variables {A B C : Type*} {f f' f₀₀ f₂₀ f₄₀ f₀₂ f₂₂ f₄₂ f₀₄ f₂₄ f₄₄ : A →* B}
+            {p₁₀ : f₀₀ ~* f₂₀} {p₃₀ : f₂₀ ~* f₄₀}
+            {p₀₁ : f₀₀ ~* f₀₂} {p₂₁ : f₂₀ ~* f₂₂} {p₄₁ : f₄₀ ~* f₄₂}
+            {p₁₂ : f₀₂ ~* f₂₂} {p₃₂ : f₂₂ ~* f₄₂}
+            {p₀₃ : f₀₂ ~* f₀₄} {p₂₃ : f₂₂ ~* f₂₄} {p₄₃ : f₄₂ ~* f₄₄}
+            {p₁₄ : f₀₄ ~* f₂₄} {p₃₄ : f₂₄ ~* f₄₄}
+
+  definition phsquare [reducible] (p₁₀ : f₀₀ ~* f₂₀) (p₁₂ : f₀₂ ~* f₂₂)
+                                  (p₀₁ : f₀₀ ~* f₀₂) (p₂₁ : f₂₀ ~* f₂₂) : Type :=
+  p₁₀ ⬝* p₂₁ = p₀₁ ⬝* p₁₂
+
+  definition phsquare_of_eq (p : p₁₀ ⬝* p₂₁ = p₀₁ ⬝* p₁₂) : phsquare p₁₀ p₁₂ p₀₁ p₂₁ := p
+  definition eq_of_phsquare (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) : p₁₀ ⬝* p₂₁ = p₀₁ ⬝* p₁₂ := p
+
+  -- definition phsquare.mk (p : Πx, square (p₁₀ x) (p₁₂ x) (p₀₁ x) (p₂₁ x))
+  --   (q : cube (square_of_eq (to_homotopy_pt p₁₀)) (square_of_eq (to_homotopy_pt p₁₂))
+  --             (square_of_eq (to_homotopy_pt p₀₁)) (square_of_eq (to_homotopy_pt p₂₁))
+  --             (p pt) ids) : phsquare p₁₀ p₁₂ p₀₁ p₂₁ :=
+  -- begin
+  --   fapply phomotopy_eq,
+  --   { intro x, apply eq_of_square (p x) },
+  --   { generalize p pt, intro r, exact sorry }
+  -- end
+
+
+  definition phhconcat (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) (q : phsquare p₃₀ p₃₂ p₂₁ p₄₁) :
+    phsquare (p₁₀ ⬝* p₃₀) (p₁₂ ⬝* p₃₂) p₀₁ p₄₁ :=
+  !trans_assoc ⬝ idp ◾** q ⬝ !trans_assoc⁻¹ ⬝ p ◾** idp ⬝ !trans_assoc
+
+  definition phvconcat (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) (q : phsquare p₁₂ p₁₄ p₀₃ p₂₃) :
+    phsquare p₁₀ p₁₄ (p₀₁ ⬝* p₀₃) (p₂₁ ⬝* p₂₃) :=
+  (phhconcat p⁻¹ q⁻¹)⁻¹
+
+  definition phhdeg_square {p₁ p₂ : f ~* f'} (q : p₁ = p₂) : phsquare phomotopy.rfl phomotopy.rfl p₁ p₂ :=
+  !refl_trans ⬝ q⁻¹ ⬝ !trans_refl⁻¹
+  definition phvdeg_square {p₁ p₂ : f ~* f'} (q : p₁ = p₂) : phsquare p₁ p₂ phomotopy.rfl phomotopy.rfl :=
+  !trans_refl ⬝ q ⬝ !refl_trans⁻¹
+
+  variables (p₀₁ p₁₀)
+  definition phhrefl : phsquare phomotopy.rfl phomotopy.rfl p₀₁ p₀₁ := phhdeg_square idp
+  definition phvrefl : phsquare p₁₀ p₁₀ phomotopy.rfl phomotopy.rfl := phvdeg_square idp
+  variables {p₀₁ p₁₀}
+  definition phhrfl : phsquare phomotopy.rfl phomotopy.rfl p₀₁ p₀₁ := phhrefl p₀₁
+  definition phvrfl : phsquare p₁₀ p₁₀ phomotopy.rfl phomotopy.rfl := phvrefl p₁₀
+
+  /-
+    The names are very baroque. The following stands for
+    "pointed homotopy path-horizontal composition" (i.e. composition on the left with a path)
+    The names are obtained by using the ones for squares, and putting "ph" in front of it.
+    In practice, use the notation ⬝ph** defined below, which might be easier to remember
+  -/
+  definition phphconcat {p₀₁'} (p : p₀₁' = p₀₁) (q : phsquare p₁₀ p₁₂ p₀₁ p₂₁) :
+    phsquare p₁₀ p₁₂ p₀₁' p₂₁ :=
+  by induction p; exact q
+
+  definition phhpconcat {p₂₁'} (q : phsquare p₁₀ p₁₂ p₀₁ p₂₁) (p : p₂₁ = p₂₁') :
+    phsquare p₁₀ p₁₂ p₀₁ p₂₁' :=
+  by induction p; exact q
+
+  definition phpvconcat {p₁₀'} (p : p₁₀' = p₁₀) (q : phsquare p₁₀ p₁₂ p₀₁ p₂₁) :
+    phsquare p₁₀' p₁₂ p₀₁ p₂₁ :=
+  by induction p; exact q
+
+  definition phvpconcat {p₁₂'} (q : phsquare p₁₀ p₁₂ p₀₁ p₂₁) (p : p₁₂ = p₁₂') :
+    phsquare p₁₀ p₁₂' p₀₁ p₂₁ :=
+  by induction p; exact q
+
+  definition phhinverse (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) : phsquare p₁₀⁻¹* p₁₂⁻¹* p₂₁ p₀₁ :=
+  begin
+    refine (eq_symm_trans_of_trans_eq _)⁻¹,
+    refine !trans_assoc⁻¹ ⬝ _,
+    refine (eq_trans_symm_of_trans_eq _)⁻¹,
+    exact (eq_of_phsquare p)⁻¹
+  end
+
+  definition phvinverse (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) : phsquare p₁₂ p₁₀ p₀₁⁻¹* p₂₁⁻¹* :=
+  (phhinverse p⁻¹)⁻¹
+
+  infix ` ⬝h** `:78 := phhconcat
+  infix ` ⬝v** `:78 := phvconcat
+  infixr ` ⬝ph** `:77 := phphconcat
+  infixl ` ⬝hp** `:77 := phhpconcat
+  infixr ` ⬝pv** `:77 := phpvconcat
+  infixl ` ⬝vp** `:77 := phvpconcat
+  postfix `⁻¹ʰ**`:(max+1) := phhinverse
+  postfix `⁻¹ᵛ**`:(max+1) := phvinverse
+
+  definition phwhisker_rt (p : f ~* f₂₀) (q : phsquare p₁₀ p₁₂ p₀₁ p₂₁) :
+    phsquare (p₁₀ ⬝* p⁻¹*) p₁₂ p₀₁ (p ⬝* p₂₁) :=
+  !trans_assoc ⬝ idp ◾** (!trans_assoc⁻¹ ⬝ !trans_left_inv ◾** idp ⬝ !refl_trans) ⬝ q
+
+  definition phwhisker_br (p : f₂₂ ~* f) (q : phsquare p₁₀ p₁₂ p₀₁ p₂₁) :
+    phsquare p₁₀ (p₁₂ ⬝* p) p₀₁ (p₂₁ ⬝* p) :=
+  !trans_assoc⁻¹ ⬝ q ◾** idp ⬝ !trans_assoc
+
+  definition phmove_top_of_left' {p₀₁ : f ~* f₀₂} (p : f₀₀ ~* f)
+    (q : phsquare p₁₀ p₁₂ (p ⬝* p₀₁) p₂₁) : phsquare (p⁻¹* ⬝* p₁₀) p₁₂ p₀₁ p₂₁ :=
+  !trans_assoc ⬝ (eq_symm_trans_of_trans_eq (q ⬝ !trans_assoc)⁻¹)⁻¹
+
+  definition phmove_bot_of_left {p₀₁ : f₀₀ ~* f} (p : f ~* f₀₂)
+    (q : phsquare p₁₀ p₁₂ (p₀₁ ⬝* p) p₂₁) : phsquare p₁₀ (p ⬝* p₁₂) p₀₁ p₂₁ :=
+  q ⬝ !trans_assoc
+
+  definition passoc_phomotopy_right {A B C D : Type*} (h : C →* D) (g : B →* C) {f f' : A →* B}
+    (p : f ~* f') : phsquare (passoc h g f) (passoc h g f')
+      (pwhisker_left (h ∘* g) p) (pwhisker_left h (pwhisker_left g p)) :=
+  begin
+    induction p using phomotopy_rec_on_idp,
+    refine idp ◾** (ap (pwhisker_left h) !pwhisker_left_refl ⬝ !pwhisker_left_refl) ⬝ _ ⬝
+          !pwhisker_left_refl⁻¹ ◾** idp,
+    exact !trans_refl ⬝ !refl_trans⁻¹
+  end
+
+  theorem passoc_phomotopy_middle {A B C D : Type*} (h : C →* D) {g g' : B →* C} (f : A →* B)
+    (p : g ~* g') : phsquare (passoc h g f) (passoc h g' f)
+      (pwhisker_right f (pwhisker_left h p)) (pwhisker_left h (pwhisker_right f p)) :=
+  begin
+    induction p using phomotopy_rec_on_idp,
+    rewrite [pwhisker_right_refl, pwhisker_left_refl],
+    rewrite [pwhisker_right_refl, pwhisker_left_refl],
+    exact phvrfl
+  end
+
+  definition pwhisker_right_pwhisker_left {A B C : Type*} {g g' : B →* C} {f f' : A →* B}
+    (p : g ~* g') (q : f ~* f') :
+    phsquare (pwhisker_right f p) (pwhisker_right f' p) (pwhisker_left g q) (pwhisker_left g' q) :=
+  begin
+    induction p using phomotopy_rec_on_idp,
+    induction q using phomotopy_rec_on_idp,
+    exact !pwhisker_right_refl ◾** !pwhisker_left_refl ⬝
+          !pwhisker_left_refl⁻¹ ◾** !pwhisker_right_refl⁻¹
+  end
+
+  definition pwhisker_left_phsquare (f : B →* C) (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) :
+    phsquare (pwhisker_left f p₁₀) (pwhisker_left f p₁₂)
+             (pwhisker_left f p₀₁) (pwhisker_left f p₂₁) :=
+  !pwhisker_left_trans⁻¹ ⬝ ap (pwhisker_left f) p ⬝ !pwhisker_left_trans
+
+  definition pwhisker_right_phsquare (f : C →* A) (p : phsquare p₁₀ p₁₂ p₀₁ p₂₁) :
+    phsquare (pwhisker_right f p₁₀) (pwhisker_right f p₁₂)
+             (pwhisker_right f p₀₁) (pwhisker_right f p₂₁) :=
+  !pwhisker_right_trans⁻¹ ⬝ ap (pwhisker_right f) p ⬝ !pwhisker_right_trans
+
+  end phsquare
+
+  definition phomotopy_of_eq_con {A B : Type*} {f g h : A →* B} (p : f = g) (q : g = h) :
+    phomotopy_of_eq (p ⬝ q) = phomotopy_of_eq p ⬝* phomotopy_of_eq q :=
+  begin induction q, induction p, exact !trans_refl⁻¹ end
+
+  definition pcompose_left_eq_of_phomotopy {A B C : Type*} (g : B →* C) {f f' : A →* B}
+    (H : f ~* f') : ap (λf, g ∘* f) (eq_of_phomotopy H) = eq_of_phomotopy (pwhisker_left g H) :=
+  begin
+    induction H using phomotopy_rec_on_idp,
+    refine ap02 _ !eq_of_phomotopy_refl ⬝ !eq_of_phomotopy_refl⁻¹ ⬝ ap eq_of_phomotopy _,
+    exact !pwhisker_left_refl⁻¹
+  end
+
+  definition pcompose_right_eq_of_phomotopy {A B C : Type*} {g g' : B →* C} (f : A →* B)
+    (H : g ~* g') : ap (λg, g ∘* f) (eq_of_phomotopy H) = eq_of_phomotopy (pwhisker_right f H) :=
+  begin
+    induction H using phomotopy_rec_on_idp,
+    refine ap02 _ !eq_of_phomotopy_refl ⬝ !eq_of_phomotopy_refl⁻¹ ⬝ ap eq_of_phomotopy _,
+    exact !pwhisker_right_refl⁻¹
+  end
+
+  definition phomotopy_of_eq_pcompose_left {A B C : Type*} (g : B →* C) {f f' : A →* B}
+    (p : f = f') : phomotopy_of_eq (ap (λf, g ∘* f) p) = pwhisker_left g (phomotopy_of_eq p) :=
+  begin
+    induction p, exact !pwhisker_left_refl⁻¹
+  end
+
+  definition phomotopy_of_eq_pcompose_right {A B C : Type*} {g g' : B →* C} (f : A →* B)
+    (p : g = g') : phomotopy_of_eq (ap (λg, g ∘* f) p) = pwhisker_right f (phomotopy_of_eq p) :=
+  begin
+    induction p, exact !pwhisker_right_refl⁻¹
+  end
+
+  definition phomotopy_mk_ppmap [constructor] {A B C : Type*} {f g : A →* ppmap B C} (p : Πa, f a ~* g a)
+    (q : p pt ⬝* phomotopy_of_eq (respect_pt g) = phomotopy_of_eq (respect_pt f))
+    : f ~* g :=
+  begin
+    apply phomotopy.mk (λa, eq_of_phomotopy (p a)),
+    apply eq_of_fn_eq_fn (pmap_eq_equiv _ _), esimp [pmap_eq_equiv],
+    refine !phomotopy_of_eq_con ⬝ _,
+    refine !phomotopy_of_eq_of_phomotopy ◾** idp ⬝ q,
+  end
+
+  definition pconst_pcompose_pconst (A B C : Type*) :
+    pconst_pcompose (pconst A B) = pcompose_pconst (pconst B C) :=
+  idp
+
+  definition pconst_pcompose_phomotopy_pconst {A B C : Type*} {f : A →* B} (p : f ~* pconst A B) :
+    pconst_pcompose f = pwhisker_left (pconst B C) p ⬝* pcompose_pconst (pconst B C) :=
+  begin
+    assert H : Π(p : pconst A B ~* f),
+      pconst_pcompose f = pwhisker_left (pconst B C) p⁻¹* ⬝* pcompose_pconst (pconst B C),
+    { intro p, induction p using phomotopy_rec_on_idp, reflexivity },
+    refine H p⁻¹* ⬝ ap (pwhisker_left _) !symm_symm ◾** idp,
+  end
+
+  definition passoc_pconst_right {A B C D : Type*} (h : C →* D) (g : B →* C) :
+    passoc h g (pconst A B) ⬝* (pwhisker_left h (pcompose_pconst g) ⬝* pcompose_pconst h) =
+    pcompose_pconst (h ∘* g) :=
+  begin
+    fapply phomotopy_eq,
+    { intro a, exact !idp_con },
+    { induction h with h h₀, induction g with g g₀, induction D with D d₀, induction C with C c₀,
+      esimp at *, induction g₀, induction h₀, reflexivity }
+  end
+
+  definition passoc_pconst_middle {A A' B B' : Type*} (g : B →* B') (f : A' →* A) :
+    passoc g (pconst A B) f ⬝* (pwhisker_left g (pconst_pcompose f) ⬝* pcompose_pconst g) =
+    pwhisker_right f (pcompose_pconst g) ⬝* pconst_pcompose f :=
+  begin
+    fapply phomotopy_eq,
+    { intro a, exact !idp_con ⬝ !idp_con },
+    { induction g with g g₀, induction f with f f₀, induction B' with D d₀, induction A with C c₀,
+      esimp at *, induction g₀, induction f₀, reflexivity }
+  end
+
+  definition passoc_pconst_left {A B C D : Type*} (g : B →* C) (f : A →* B) :
+    phsquare (passoc (pconst C D) g f) (pconst_pcompose f)
+             (pwhisker_right f (pconst_pcompose g)) (pconst_pcompose (g ∘* f)) :=
+  begin
+    fapply phomotopy_eq,
+    { intro a, exact !idp_con },
+    { induction g with g g₀, induction f with f f₀, induction C with C c₀, induction B with B b₀,
+      esimp at *, induction g₀, induction f₀, reflexivity }
+  end
+
+  definition ppcompose_left_pcompose [constructor] {A B C D : Type*} (h : C →* D) (g : B →* C) :
+    @ppcompose_left A _ _ (h ∘* g) ~* ppcompose_left h ∘* ppcompose_left g :=
+  begin
+    fapply phomotopy_mk_ppmap,
+    { exact passoc h g },
+    { refine idp ◾** (!phomotopy_of_eq_con ⬝
+        (ap phomotopy_of_eq !pcompose_left_eq_of_phomotopy ⬝ !phomotopy_of_eq_of_phomotopy) ◾**
+        !phomotopy_of_eq_of_phomotopy) ⬝ _ ⬝ !phomotopy_of_eq_of_phomotopy⁻¹,
+      exact passoc_pconst_right h g }
+  end
+
+  definition ppcompose_right_pcompose [constructor] {A B C D : Type*} (g : B →* C) (f : A →* B) :
+    @ppcompose_right _ _ D (g ∘* f) ~* ppcompose_right f ∘* ppcompose_right g :=
+  begin
+    symmetry,
+    fapply phomotopy_mk_ppmap,
+    { intro h, exact passoc h g f },
+    { refine idp ◾** !phomotopy_of_eq_of_phomotopy ⬝ _ ⬝ (!phomotopy_of_eq_con ⬝
+        (ap phomotopy_of_eq !pcompose_right_eq_of_phomotopy ⬝ !phomotopy_of_eq_of_phomotopy) ◾** !phomotopy_of_eq_of_phomotopy)⁻¹,
+      exact passoc_pconst_left g f }
+  end
+
+  definition ppcompose_left_ppcompose_right {A A' B B' : Type*} (g : B →* B') (f : A' →* A) :
+    psquare (ppcompose_left g) (ppcompose_left g) (ppcompose_right f) (ppcompose_right f) :=
+  begin
+    fapply phomotopy_mk_ppmap,
+    { intro h, exact passoc g h f },
+    { refine idp ◾** (!phomotopy_of_eq_con ⬝
+        (ap phomotopy_of_eq !pcompose_left_eq_of_phomotopy ⬝ !phomotopy_of_eq_of_phomotopy) ◾**
+        !phomotopy_of_eq_of_phomotopy) ⬝ _ ⬝ (!phomotopy_of_eq_con ⬝
+        (ap phomotopy_of_eq !pcompose_right_eq_of_phomotopy ⬝ !phomotopy_of_eq_of_phomotopy) ◾**
+        !phomotopy_of_eq_of_phomotopy)⁻¹,
+      apply passoc_pconst_middle }
+  end
+
+  definition pcompose_pconst_phomotopy {A B C : Type*} {f f' : B →* C} (p : f ~* f') :
+    pwhisker_right (pconst A B) p ⬝* pcompose_pconst f' = pcompose_pconst f :=
+  begin
+    fapply phomotopy_eq,
+    { intro a, exact to_homotopy_pt p },
+    { induction p using phomotopy_rec_on_idp, induction C with C c₀, induction f with f f₀,
+      esimp at *, induction f₀, reflexivity }
+  end
+
+  definition pid_pconst (A B : Type*) : pcompose_pconst (pid B) = pid_pcompose (pconst A B) :=
+  by reflexivity
+
+  definition pid_pconst_pcompose {A B C : Type*} (f : A →* B) :
+    phsquare (pid_pcompose (pconst B C ∘* f))
+             (pcompose_pconst (pid C))
+             (pwhisker_left (pid C) (pconst_pcompose f))
+             (pconst_pcompose f) :=
+  begin
+    fapply phomotopy_eq,
+    { reflexivity },
+    { induction f with f f₀, induction B with B b₀, esimp at *, induction f₀, reflexivity }
+  end
+
+  definition ppcompose_left_pconst [constructor] (A B C : Type*) :
+    @ppcompose_left A _ _ (pconst B C) ~* pconst (ppmap A B) (ppmap A C) :=
+  begin
+    fapply phomotopy_mk_ppmap,
+    { exact pconst_pcompose },
+    { refine idp ◾** !phomotopy_of_eq_idp ⬝ !phomotopy_of_eq_of_phomotopy⁻¹ }
+  end
+
+  definition ppcompose_left_phomotopy [constructor] {A B C : Type*} {g g' : B →* C} (p : g ~* g') :
+    @ppcompose_left A _ _ g ~* ppcompose_left g' :=
+  begin
+    induction p using phomotopy_rec_on_idp,
+    reflexivity
+  end
+
+  definition ppcompose_left_phomotopy_refl {A B C : Type*} (g : B →* C) :
+    ppcompose_left_phomotopy (phomotopy.refl g) = phomotopy.refl (@ppcompose_left A _ _ g) :=
+  !phomotopy_rec_on_idp_refl
+
+    /- a more explicit proof of ppcompose_left_phomotopy, which might be useful if we need to prove properties about it
+    -/
+    -- fapply phomotopy_mk_ppmap,
+    -- { intro f, exact pwhisker_right f p },
+    -- { refine ap (λx, _ ⬝* x) !phomotopy_of_eq_of_phomotopy ⬝ _ ⬝ !phomotopy_of_eq_of_phomotopy⁻¹,
+    --   exact pcompose_pconst_phomotopy p }
+
+  definition ppcompose_right_phomotopy [constructor] {A B C : Type*} {f f' : A →* B} (p : f ~* f') :
+    @ppcompose_right _ _ C f ~* ppcompose_right f' :=
+  begin
+    induction p using phomotopy_rec_on_idp,
+    reflexivity
+  end
+
+  definition pppcompose [constructor] (A B C : Type*) : ppmap B C →* ppmap (ppmap A B) (ppmap A C) :=
+  pmap.mk ppcompose_left (eq_of_phomotopy !ppcompose_left_pconst)
+
+  section psquare
+
+  variables {A A' A₀₀ A₂₀ A₄₀ A₀₂ A₂₂ A₄₂ A₀₄ A₂₄ A₄₄ : Type*}
+            {f₁₀ f₁₀' : A₀₀ →* A₂₀} {f₃₀ : A₂₀ →* A₄₀}
+            {f₀₁ f₀₁' : A₀₀ →* A₀₂} {f₂₁ f₂₁' : A₂₀ →* A₂₂} {f₄₁ : A₄₀ →* A₄₂}
+            {f₁₂ f₁₂' : A₀₂ →* A₂₂} {f₃₂ : A₂₂ →* A₄₂}
+            {f₀₃ : A₀₂ →* A₀₄} {f₂₃ : A₂₂ →* A₂₄} {f₄₃ : A₄₂ →* A₄₄}
+            {f₁₄ : A₀₄ →* A₂₄} {f₃₄ : A₂₄ →* A₄₄}
+
+  definition ppcompose_left_psquare {A : Type*} (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    psquare (@ppcompose_left A _ _ f₁₀) (ppcompose_left f₁₂)
+            (ppcompose_left f₀₁) (ppcompose_left f₂₁) :=
+  !ppcompose_left_pcompose⁻¹* ⬝* ppcompose_left_phomotopy p ⬝* !ppcompose_left_pcompose
+
+  definition ppcompose_right_psquare {A : Type*} (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    psquare (@ppcompose_right _ _ A f₁₂) (ppcompose_right f₁₀)
+            (ppcompose_right f₂₁) (ppcompose_right f₀₁) :=
+  !ppcompose_right_pcompose⁻¹* ⬝* ppcompose_right_phomotopy p⁻¹* ⬝* !ppcompose_right_pcompose
+
+  definition trans_phomotopy_hconcat {f₀₁' f₀₁''}
+    (q₂ : f₀₁'' ~* f₀₁') (q₁ : f₀₁' ~* f₀₁) (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) :
+    (q₂ ⬝* q₁) ⬝ph* p = q₂ ⬝ph* q₁ ⬝ph* p :=
+  idp ◾** (ap (pwhisker_left f₁₂) !trans_symm ⬝ !pwhisker_left_trans) ⬝ !trans_assoc⁻¹
+
+  definition symm_phomotopy_hconcat {f₀₁'} (q : f₀₁ ~* f₀₁')
+    (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : q⁻¹* ⬝ph* p = p ⬝* pwhisker_left f₁₂ q :=
+  idp ◾** ap (pwhisker_left f₁₂) !symm_symm
+
+  definition refl_phomotopy_hconcat (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : phomotopy.rfl ⬝ph* p = p :=
+  idp ◾** (ap (pwhisker_left _) !refl_symm ⬝ !pwhisker_left_refl) ⬝ !trans_refl
+
+  local attribute phomotopy.rfl [reducible]
+  theorem pwhisker_left_phomotopy_hconcat {f₀₁'} (r : f₀₁' ~* f₀₁)
+    (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) (q : psquare f₁₂ f₁₄ f₀₃ f₂₃) :
+    pwhisker_left f₀₃ r ⬝ph* (p ⬝v* q) = (r ⬝ph* p) ⬝v* q :=
+  by induction r using phomotopy_rec_on_idp; rewrite [pwhisker_left_refl, +refl_phomotopy_hconcat]
+
+  theorem pvcompose_pwhisker_left {f₀₁'} (r : f₀₁ ~* f₀₁')
+    (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) (q : psquare f₁₂ f₁₄ f₀₃ f₂₃) :
+    (p ⬝v* q) ⬝* (pwhisker_left f₁₄ (pwhisker_left f₀₃ r)) = (p ⬝* pwhisker_left f₁₂ r) ⬝v* q :=
+  by induction r using phomotopy_rec_on_idp; rewrite [+pwhisker_left_refl, + trans_refl]
+
+  definition phconcat2 {p p' : psquare f₁₀ f₁₂ f₀₁ f₂₁} {q q' : psquare f₃₀ f₃₂ f₂₁ f₄₁}
+    (r : p = p') (s : q = q') : p ⬝h* q = p' ⬝h* q' :=
+  ap011 phconcat r s
+
+  definition pvconcat2 {p p' : psquare f₁₀ f₁₂ f₀₁ f₂₁} {q q' : psquare f₁₂ f₁₄ f₀₃ f₂₃}
+    (r : p = p') (s : q = q') : p ⬝v* q = p' ⬝v* q' :=
+  ap011 pvconcat r s
+
+  definition phinverse2 {f₁₀ : A₀₀ ≃* A₂₀} {f₁₂ : A₀₂ ≃* A₂₂} {p p' : psquare f₁₀ f₁₂ f₀₁ f₂₁}
+    (r : p = p') : p⁻¹ʰ* = p'⁻¹ʰ* :=
+  ap phinverse r
+
+  definition pvinverse2 {f₀₁ : A₀₀ ≃* A₀₂} {f₂₁ : A₂₀ ≃* A₂₂} {p p' : psquare f₁₀ f₁₂ f₀₁ f₂₁}
+    (r : p = p') : p⁻¹ᵛ* = p'⁻¹ᵛ* :=
+  ap pvinverse r
+
+  definition phomotopy_hconcat2 {q q' : f₀₁' ~* f₀₁} {p p' : psquare f₁₀ f₁₂ f₀₁ f₂₁}
+    (r : q = q') (s : p = p') : q ⬝ph* p = q' ⬝ph* p' :=
+  ap011 phomotopy_hconcat r s
+
+  definition hconcat_phomotopy2 {p p' : psquare f₁₀ f₁₂ f₀₁ f₂₁} {q q' : f₂₁' ~* f₂₁}
+    (r : p = p') (s : q = q') : p ⬝hp* q = p' ⬝hp* q' :=
+  ap011 hconcat_phomotopy r s
+
+  definition phomotopy_vconcat2 {q q' : f₁₀' ~* f₁₀} {p p' : psquare f₁₀ f₁₂ f₀₁ f₂₁}
+    (r : q = q') (s : p = p') : q ⬝pv* p = q' ⬝pv* p' :=
+  ap011 phomotopy_vconcat r s
+
+  definition vconcat_phomotopy2 {p p' : psquare f₁₀ f₁₂ f₀₁ f₂₁} {q q' : f₁₂' ~* f₁₂}
+    (r : p = p') (s : q = q') : p ⬝vp* q = p' ⬝vp* q' :=
+  ap011 vconcat_phomotopy r s
+
+  -- for consistency, should there be a second star here?
+  infix ` ◾h* `:79 := phconcat2
+  infix ` ◾v* `:79 := pvconcat2
+  infixl ` ◾hp* `:79 := hconcat_phomotopy2
+  infixr ` ◾ph* `:79 := phomotopy_hconcat2
+  infixl ` ◾vp* `:79 := vconcat_phomotopy2
+  infixr ` ◾pv* `:79 := phomotopy_vconcat2
+  postfix `⁻²ʰ*`:(max+1) := phinverse2
+  postfix `⁻²ᵛ*`:(max+1) := pvinverse2
+
+  end psquare
+
+  variables {X X' Y Y' Z : Type*}
+  definition pap1 [constructor] (X Y : Type*) : ppmap X Y →* ppmap (Ω X) (Ω Y) :=
+  pmap.mk ap1 (eq_of_phomotopy !ap1_pconst)
+
+  definition ap1_gen_const {A B : Type} {a₁ a₂ : A} (b : B) (p : a₁ = a₂) :
+    ap1_gen (const A b) idp idp p = idp :=
+  ap1_gen_idp_left (const A b) p ⬝ ap_constant p b
+
+  definition ap1_gen_compose_const_left
+    {A B C : Type} (c : C) (f : A → B) {a₁ a₂ : A} (p : a₁ = a₂) :
+    ap1_gen_compose (const B c) f idp idp idp idp p ⬝
+    ap1_gen_const c (ap1_gen f idp idp p) =
+    ap1_gen_const c p :=
+  begin induction p, reflexivity end
+
+  definition ap1_gen_compose_const_right
+    {A B C : Type} (g : B → C) (b : B) {a₁ a₂ : A} (p : a₁ = a₂) :
+    ap1_gen_compose g (const A b) idp idp idp idp p ⬝
+    ap (ap1_gen g idp idp) (ap1_gen_const b p) =
+    ap1_gen_const (g b) p :=
+  begin induction p, reflexivity end
+
+  definition ap1_pcompose_pconst_left {A B C : Type*} (f : A →* B) :
+    phsquare (ap1_pcompose (pconst B C) f)
+             (ap1_pconst A C)
+             (ap1_phomotopy (pconst_pcompose f))
+             (pwhisker_right (Ω→ f) (ap1_pconst B C) ⬝* pconst_pcompose (Ω→ f)) :=
+  begin
+    induction A with A a₀, induction B with B b₀, induction C with C c₀, induction f with f f₀,
+    esimp at *, induction f₀,
+    refine idp ◾** !trans_refl ⬝ _ ⬝ !refl_trans⁻¹ ⬝ !ap1_phomotopy_refl⁻¹ ◾** idp,
+    fapply phomotopy_eq,
+    { exact ap1_gen_compose_const_left c₀ f },
+    { reflexivity }
+  end
+
+  definition ap1_pcompose_pconst_right {A B C : Type*} (g : B →* C) :
+    phsquare (ap1_pcompose g (pconst A B))
+             (ap1_pconst A C)
+             (ap1_phomotopy (pcompose_pconst g))
+             (pwhisker_left (Ω→ g) (ap1_pconst A B) ⬝* pcompose_pconst (Ω→ g)) :=
+  begin
+    induction A with A a₀, induction B with B b₀, induction C with C c₀, induction g with g g₀,
+    esimp at *, induction g₀,
+    refine idp ◾** !trans_refl ⬝ _ ⬝ !refl_trans⁻¹ ⬝ !ap1_phomotopy_refl⁻¹ ◾** idp,
+    fapply phomotopy_eq,
+    { exact ap1_gen_compose_const_right g b₀ },
+    { reflexivity }
+  end
+
+  definition pap1_natural_left [constructor] (f : X' →* X) :
+    psquare (pap1 X Y) (pap1 X' Y) (ppcompose_right f) (ppcompose_right (Ω→ f)) :=
+  begin
+    fapply phomotopy_mk_ppmap,
+    { intro g, exact !ap1_pcompose⁻¹* },
+    { refine idp ◾** (ap phomotopy_of_eq (!ap1_eq_of_phomotopy  ◾ idp ⬝ !eq_of_phomotopy_trans⁻¹) ⬝
+      !phomotopy_of_eq_of_phomotopy)  ⬝ _ ⬝ (ap phomotopy_of_eq (!pcompose_right_eq_of_phomotopy ◾
+      idp ⬝ !eq_of_phomotopy_trans⁻¹) ⬝ !phomotopy_of_eq_of_phomotopy)⁻¹,
+      apply symm_trans_eq_of_eq_trans, exact (ap1_pcompose_pconst_left f)⁻¹ }
+  end
+
+  definition pap1_natural_right [constructor] (f : Y →* Y') :
+    psquare (pap1 X Y) (pap1 X Y') (ppcompose_left f) (ppcompose_left (Ω→ f)) :=
+  begin
+    fapply phomotopy_mk_ppmap,
+    { intro g, exact !ap1_pcompose⁻¹* },
+    { refine idp ◾** (ap phomotopy_of_eq (!ap1_eq_of_phomotopy  ◾ idp ⬝ !eq_of_phomotopy_trans⁻¹) ⬝
+      !phomotopy_of_eq_of_phomotopy)  ⬝ _ ⬝ (ap phomotopy_of_eq (!pcompose_left_eq_of_phomotopy ◾
+      idp ⬝ !eq_of_phomotopy_trans⁻¹) ⬝ !phomotopy_of_eq_of_phomotopy)⁻¹,
+      apply symm_trans_eq_of_eq_trans, exact (ap1_pcompose_pconst_right f)⁻¹ }
+  end
+
+end pointed

hott/types/prod.hlean

@@ -90,6 +90,29 @@ namespace prod
   definition ap_prod_mk_right (p : b = b') : ap (λb, prod.mk a b) p = prod_eq idp p :=
   ap_eq_ap011_right prod.mk a p
 
+  definition pair_eq_eta {A B : Type} {u v : A × B}
+    (p : u = v) : pair_eq (p..1) (p..2) = prod.eta u ⬝ p ⬝ (prod.eta v)⁻¹ :=
+  by induction p; induction u; reflexivity
+
+  definition prod_eq_eq {A B : Type} {u v : A × B}
+    {p₁ q₁ : u.1 = v.1} {p₂ q₂ : u.2 = v.2} (α₁ : p₁ = q₁) (α₂ : p₂ = q₂)
+    : prod_eq p₁ p₂ = prod_eq q₁ q₂ :=
+  by cases α₁; cases α₂; reflexivity
+
+  definition prod_eq_assemble {A B : Type} {u v : A × B}
+    {p q : u = v} (α₁ : p..1 = q..1) (α₂ : p..2 = q..2) : p = q :=
+  (prod_eq_eta p)⁻¹ ⬝ prod.prod_eq_eq α₁ α₂ ⬝ prod_eq_eta q
+
+  definition eq_pr1_concat {A B : Type} {u v w : A × B}
+    (p : u = v) (q : v = w)
+    : (p ⬝ q)..1 = p..1 ⬝ q..1 :=
+  by cases q; reflexivity
+
+  definition eq_pr2_concat {A B : Type} {u v w : A × B}
+    (p : u = v) (q : v = w)
+    : (p ⬝ q)..2 = p..2 ⬝ q..2 :=
+  by cases q; reflexivity
+
   /- Groupoid structure -/
   definition prod_eq_inv (p : a = a') (q : b = b') : (prod_eq p q)⁻¹ = prod_eq p⁻¹ q⁻¹ :=
   by cases p; cases q; reflexivity
@@ -125,6 +148,19 @@ namespace prod
     apply idpo
   end
 
+  open prod.ops
+  definition prod_pathover_equiv {A : Type} {B C : A → Type} {a a' : A} (p : a = a')
+    (x : B a × C a) (x' : B a' × C a') : x =[p] x' ≃ x.1 =[p] x'.1 × x.2 =[p] x'.2 :=
+  begin
+    fapply equiv.MK,
+    { intro q, induction q, constructor: constructor },
+    { intro v, induction v with q r, exact prod_pathover _ _ _ q r },
+    { intro v, induction v with q r, induction x with b c, induction x' with b' c',
+      esimp at *, induction q, refine idp_rec_on r _, reflexivity },
+    { intro q, induction q, induction x with b c, reflexivity }
+  end
+
+
   /-
     TODO:
     * define the projections from the type u =[p] v
@@ -301,6 +337,8 @@ namespace prod
   definition ptprod [constructor] {n : ℕ₋₂} (A B : n-Type*) : n-Type* :=
   ptrunctype.mk' n (A × B)
 
+  definition pprod_functor [constructor] {A B C D : Type*} (f : A →* C) (g : B →* D) : A ×* B →* C ×* D :=
+  pmap.mk (prod_functor f g) (prod_eq (respect_pt f) (respect_pt g))
 
 
 end prod

hott/types/sigma.hlean

@@ -214,6 +214,25 @@ namespace sigma
     induction s using idp_rec_on, apply idpo
   end
 
+  definition pathover_pr1 [unfold 9] {A : Type} {B : A → Type} {C : Πa, B a → Type}
+    {a a' : A} {p : a = a'} {x : Σb, C a b} {x' : Σb', C a' b'}
+    (q : x =[p] x') : x.1 =[p] x'.1 :=
+  begin induction q, constructor end
+
+  definition sigma_pathover_equiv_of_is_prop {A : Type} {B : A → Type} (C : Πa, B a → Type)
+    {a a' : A} (p : a = a') (x : Σb, C a b) (x' : Σb', C a' b')
+    [Πa b, is_prop (C a b)] : x =[p] x' ≃ x.1 =[p] x'.1 :=
+  begin
+    fapply equiv.MK,
+    { exact pathover_pr1 },
+    { intro q, induction x with b c, induction x' with b' c', esimp at q, induction q,
+      apply pathover_idp_of_eq, exact sigma_eq idp !is_prop.elimo },
+    { intro q, induction x with b c, induction x' with b' c', esimp at q, induction q,
+      have c = c', from !is_prop.elim, induction this,
+      rewrite [▸*, is_prop_elimo_self (C a) c] },
+    { intro q, induction q, induction x with b c, rewrite [▸*, is_prop_elimo_self (C a) c] }
+  end
+
   /-
     TODO:
     * define the projections from the type u =[p] v
@@ -274,6 +293,10 @@ namespace sigma
     ap (sigma_functor f g) (sigma_eq p q) = sigma_eq (ap f p) (pathover.rec_on q idpo) :=
   by induction q; reflexivity
 
+  definition sigma_ua {A B : Type} (C : A ≃ B → Type) :
+    (Σ(p : A = B), C (equiv_of_eq p)) ≃ Σ(e : A ≃ B), C e :=
+  sigma_equiv_sigma_left' !eq_equiv_equiv
+
   -- definition ap_sigma_functor_eq (p : u.1 = v.1) (q : u.2 =[p] v.2)
   --   : ap (sigma_functor f g) (sigma_eq p q) =
   --     sigma_eq (ap f p)
@@ -511,7 +534,6 @@ namespace sigma
     (b : B a) (b' : B a') : a = a' :=
   (is_prop.elim ⟨a, b⟩ ⟨a', b'⟩)..1
 
-
 end sigma
 
 attribute sigma.is_trunc_sigma [instance] [priority 1490]

hott/types/trunc.hlean

@@ -8,10 +8,10 @@ Properties of trunc_index, is_trunc, trunctype, trunc, and the pointed versions
 
 -- NOTE: the fact that (is_trunc n A) is a mere proposition is proved in .prop_trunc
 
-import .pointed ..function algebra.order types.nat.order
+import .pointed ..function algebra.order types.nat.order types.unit
 
 open eq sigma sigma.ops pi function equiv trunctype
-     is_equiv prod pointed nat is_trunc algebra sum
+     is_equiv prod pointed nat is_trunc algebra sum unit
 
   /- basic computation with ℕ₋₂, its operations and its order -/
 namespace trunc_index
@@ -471,6 +471,22 @@ namespace is_trunc
   have is_trunc (0+[ℕ₋₂]n) A, by rewrite [trunc_index.zero_add]; exact _,
   is_trunc_loopn 0 n A
 
+  definition pequiv_punit_of_is_contr [constructor] (A : Type*) (H : is_contr A) : A ≃* punit :=
+  pequiv_of_equiv (equiv_unit_of_is_contr A) (@is_prop.elim unit _ _ _)
+
+  definition pequiv_punit_of_is_contr' [constructor] (A : Type) (H : is_contr A)
+    : pointed.MK A (center A) ≃* punit :=
+  pequiv_punit_of_is_contr (pointed.MK A (center A)) H
+
+  definition is_trunc_is_contr_fiber (n : ℕ₋₂) {A B : Type} (f : A → B)
+    (b : B) [is_trunc n A] [is_trunc n B] : is_trunc n (is_contr (fiber f b)) :=
+  begin
+    cases n,
+    { apply is_contr_of_inhabited_prop, apply is_contr_fun_of_is_equiv,
+      apply is_equiv_of_is_contr },
+    { apply is_trunc_succ_of_is_prop }
+  end
+
 end is_trunc open is_trunc
 
 namespace trunc
@@ -735,9 +751,10 @@ namespace trunc
     revert n, induction k with k IH: intro n,
     { reflexivity},
     { refine _ ⬝e* loop_ptrunc_pequiv n (Ω[k] A),
-      rewrite [loopn_succ_eq], apply loop_pequiv_loop,
+      change Ω (Ω[k] (ptrunc (n + succ k) A)) ≃* Ω (ptrunc (n + 1) (Ω[k] A)),
+      apply loop_pequiv_loop,
       refine _ ⬝e* IH (n.+1),
-      rewrite succ_add_nat}
+      exact loopn_pequiv_loopn k (pequiv_of_eq (ap (λn, ptrunc n A) !succ_add_nat⁻¹)) }
   end
 
   definition loopn_ptrunc_pequiv_con {n : ℕ₋₂} {k : ℕ} {A : Type*}

hott/types/types.md

@@ -31,3 +31,4 @@ Types in HoTT:
 * [pullback](pullback.hlean)
 * [univ](univ.hlean)
 * [type_functor](type_functor.hlean)
+* [pointed2](pointed2.hlean): equalities between pointed homotopies, squares of poitned maps and pointed homotopies, and pointed maps in or out of `ppmap A B`