chore(hott): standardize names of homotopy_of_inv_homotopy_post and friends

hott/homotopy/susp.hlean

@@ -225,11 +225,12 @@ namespace susp
   begin
     fconstructor,
     { intro a, induction a,
-      { reflexivity},
+      { reflexivity },
-      { reflexivity},
+      { reflexivity },
       { apply eq_pathover, apply hdeg_square,
-        rewrite [▸*,ap_compose' _ (psusp_functor f),↑psusp_functor,+elim_merid]}},
+        rewrite [▸*,ap_compose' _ (psusp_functor f),↑psusp_functor],
-    { reflexivity}
+        krewrite +susp.elim_merid } },
+    { reflexivity }
   end
 
   -- adjunction from Coq-HoTT

hott/init/equiv.hlean

@@ -198,34 +198,57 @@ namespace is_equiv
   end
 
   section
-  variables {A B C : Type} {f : A → B} [Hf : is_equiv f] {a : A} {b : B} {g : B → C} {h : A → C}
+  variables {A B C : Type} {f : A → B} [Hf : is_equiv f]
   include Hf
 
-  --Rewrite rules
+  section rewrite_rules
-  definition eq_of_eq_inv (p : a = f⁻¹ b) : f a = b :=
+    variables {a : A} {b : B}
-  ap f p ⬝ right_inv f b
+    definition eq_of_eq_inv (p : a = f⁻¹ b) : f a = b :=
+    ap f p ⬝ right_inv f b
 
-  definition eq_of_inv_eq (p : f⁻¹ b = a) : b = f a :=
+    definition eq_of_inv_eq (p : f⁻¹ b = a) : b = f a :=
-  (eq_of_eq_inv p⁻¹)⁻¹
+    (eq_of_eq_inv p⁻¹)⁻¹
 
-  definition inv_eq_of_eq (p : b = f a) : f⁻¹ b = a :=
+    definition inv_eq_of_eq (p : b = f a) : f⁻¹ b = a :=
-  ap f⁻¹ p ⬝ left_inv f a
+    ap f⁻¹ p ⬝ left_inv f a
 
-  definition eq_inv_of_eq (p : f a = b) : a = f⁻¹ b :=
+    definition eq_inv_of_eq (p : f a = b) : a = f⁻¹ b :=
-  (inv_eq_of_eq p⁻¹)⁻¹
+    (inv_eq_of_eq p⁻¹)⁻¹
+  end rewrite_rules
 
   variable (f)
-  definition homotopy_of_homotopy_inv' (p : g ~ h ∘ f⁻¹) : g ∘ f ~ h :=
-  λa, p (f a) ⬝ ap h (left_inv f a)
 
-  definition homotopy_of_inv_homotopy' (p : h ∘ f⁻¹ ~ g) : h ~ g ∘ f :=
+  section pre_compose
-  λa, ap h (left_inv f a)⁻¹ ⬝ p (f a)
+    variables (α : A → C) (β : B → C)
 
-  definition inv_homotopy_of_homotopy' (p : h ~ g ∘ f) : h ∘ f⁻¹ ~ g :=
+    definition homotopy_of_homotopy_inv_pre (p : β ~ α ∘ f⁻¹) : β ∘ f ~ α :=
-  λb, p (f⁻¹ b) ⬝ ap g (right_inv f b)
+    λ a, p (f a) ⬝ ap α (left_inv f a)
 
-  definition homotopy_inv_of_homotopy' (p : g ∘ f ~ h) : g ~ h ∘ f⁻¹  :=
+    definition homotopy_of_inv_homotopy_pre (p : α ∘ f⁻¹ ~ β) : α ~ β ∘ f :=
-  λb, ap g (right_inv f b)⁻¹ ⬝ p (f⁻¹ b)
+    λ a, (ap α (left_inv f a))⁻¹ ⬝ p (f a)
+
+    definition inv_homotopy_of_homotopy_pre (p : α ~ β ∘ f) : α ∘ f⁻¹ ~ β :=
+    λ b, p (f⁻¹ b) ⬝ ap β (right_inv f b)
+
+    definition homotopy_inv_of_homotopy_pre (p : β ∘ f ~ α) : β ~ α ∘ f⁻¹  :=
+    λ b, (ap β (right_inv f b))⁻¹ ⬝ p (f⁻¹ b)
+  end pre_compose
+
+  section post_compose
+    variables (α : C → A) (β : C → B)
+
+    definition homotopy_of_homotopy_inv_post (p : α ~ f⁻¹ ∘ β) : f ∘ α ~ β :=
+    λ c, ap f (p c) ⬝ (right_inv f (β c))
+
+    definition homotopy_of_inv_homotopy_post (p : f⁻¹ ∘ β ~ α) : β ~ f ∘ α :=
+    λ c, (right_inv f (β c))⁻¹ ⬝ ap f (p c)
+
+    definition inv_homotopy_of_homotopy_post (p : β ~ f ∘ α) : f⁻¹ ∘ β ~ α :=
+    λ c, ap f⁻¹ (p c) ⬝ (left_inv f (α c))
+
+    definition homotopy_inv_of_homotopy_post (p : f ∘ α ~ β) : α ~ f⁻¹ ∘ β :=
+    λ c, (left_inv f (α c))⁻¹ ⬝ ap f⁻¹ (p c)
+  end post_compose
 
   end
 
@@ -374,23 +397,6 @@ namespace equiv
 
   end
 
-  section
-  variables {A B C : Type} (f : A ≃ B) {a : A} {b : B} {g : B → C} {h : A → C}
-
-  definition homotopy_of_homotopy_inv (p : g ~ h ∘ f⁻¹) : g ∘ f ~ h :=
-  homotopy_of_homotopy_inv' f p
-
-  definition homotopy_of_inv_homotopy (p : h ∘ f⁻¹ ~ g) : h ~ g ∘ f :=
-  homotopy_of_inv_homotopy' f p
-
-  definition inv_homotopy_of_homotopy (p : h ~ g ∘ f) : h ∘ f⁻¹ ~ g :=
-  inv_homotopy_of_homotopy' f p
-
-  definition homotopy_inv_of_homotopy (p : g ∘ f ~ h) : g ~ h ∘ f⁻¹  :=
-  homotopy_inv_of_homotopy' f p
-
-  end
-
   infixl ` ⬝pe `:75 := equiv_of_equiv_of_eq
   infixl ` ⬝ep `:75 := equiv_of_eq_of_equiv
 

hott/types/arrow_2.hlean

@@ -124,9 +124,9 @@ namespace arrow
   open function
   definition inv_commute_of_commute (p : f' ∘ α ~ β ∘ f) : f'⁻¹ ∘ β ~ α ∘ f⁻¹ :=
   begin
-    apply homotopy_inv_of_homotopy_post f' β (α ∘ f⁻¹),
+    apply inv_homotopy_of_homotopy_post f' (α ∘ f⁻¹) β,
     apply homotopy.symm,
-    apply homotopy_inv_of_homotopy_pre f (f' ∘ α) β,
+    apply inv_homotopy_of_homotopy_pre f (f' ∘ α) β,
     apply p
   end
 
@@ -135,8 +135,8 @@ namespace arrow
     =  (ap f'⁻¹ (p a))⁻¹ ⬝ left_inv f' (α a) ⬝ ap α (left_inv f a)⁻¹ :=
   begin
     unfold inv_commute_of_commute,
-    unfold homotopy_inv_of_homotopy_post,
+    unfold inv_homotopy_of_homotopy_post,
-    unfold homotopy_inv_of_homotopy_pre,
+    unfold inv_homotopy_of_homotopy_pre,
     rewrite [adj f a,-(ap_compose β f)],
     rewrite [eq_of_square (natural_square_tr p (left_inv f a))],
     rewrite [ap_inv f'⁻¹,ap_con f'⁻¹,con_inv,con.assoc,con.assoc],
@@ -152,8 +152,8 @@ namespace arrow
     = right_inv f' (β b) ⬝ ap β (right_inv f b)⁻¹ ⬝ (p (f⁻¹ b))⁻¹ :=
   begin
     unfold inv_commute_of_commute,
-    unfold homotopy_inv_of_homotopy_post,
+    unfold inv_homotopy_of_homotopy_post,
-    unfold homotopy_inv_of_homotopy_pre,
+    unfold inv_homotopy_of_homotopy_pre,
     rewrite [ap_con,-(ap_compose f' f'⁻¹),-(adj f' (α (f⁻¹ b)))],
     rewrite [con.assoc (right_inv f' (β b)) (ap β (right_inv f b)⁻¹)
                        (p (f⁻¹ b))⁻¹],
@@ -173,13 +173,13 @@ namespace arrow
   begin
     unfold inv_commute_of_commute,
     apply @is_equiv_compose _ _ _
-      (homotopy.symm ∘ (homotopy_inv_of_homotopy_pre f (f' ∘ α) β))
+      (homotopy.symm ∘ (inv_homotopy_of_homotopy_pre f (f' ∘ α) β))
-      (homotopy_inv_of_homotopy_post f' β (α ∘ f⁻¹)),
+      (inv_homotopy_of_homotopy_post f' (α ∘ f⁻¹) β),
     { apply @is_equiv_compose _ _ _
-            (homotopy_inv_of_homotopy_pre f (f' ∘ α) β) homotopy.symm,
+            (inv_homotopy_of_homotopy_pre f (f' ∘ α) β) homotopy.symm,
-      { apply homotopy_inv_of_homotopy_pre.is_equiv },
+      { apply inv_homotopy_of_homotopy_pre.is_equiv },
       { apply pi.is_equiv_homotopy_symm }
     },
-    { apply homotopy_inv_of_homotopy_post.is_equiv }
+    { apply inv_homotopy_of_homotopy_post.is_equiv }
   end
 end arrow

hott/types/equiv.hlean

@@ -111,15 +111,14 @@ namespace is_equiv
   section pre_compose
     variables (α : A → C) (β : B → C)
 
-    definition homotopy_inv_of_homotopy_pre (p : α ~ β ∘ f) : α ∘ f⁻¹ ~ β :=
+    -- homotopy_inv_of_homotopy_pre is in init.equiv
-    λb, p (f⁻¹ b) ⬝ ap β (right_inv f b)
+    protected definition inv_homotopy_of_homotopy_pre.is_equiv
-
+      : is_equiv (inv_homotopy_of_homotopy_pre f α β) :=
-    protected definition homotopy_inv_of_homotopy_pre.is_equiv
+    adjointify _ (homotopy_of_inv_homotopy_pre f α β)
-      : is_equiv (homotopy_inv_of_homotopy_pre f α β) :=
-    adjointify _ (λq a, (ap α (left_inv f a))⁻¹ ⬝ q (f a))
     abstract begin
       intro q, apply eq_of_homotopy, intro b,
-      unfold homotopy_inv_of_homotopy_pre,
+      unfold inv_homotopy_of_homotopy_pre,
+      unfold homotopy_of_inv_homotopy_pre,
       apply inverse, apply eq_bot_of_square,
       apply eq_hconcat (ap02 α (adj_inv f b)),
       apply eq_hconcat (ap_compose α f⁻¹ (right_inv f b))⁻¹,
@@ -127,7 +126,8 @@ namespace is_equiv
     end end
     abstract begin
       intro p, apply eq_of_homotopy, intro a,
-      unfold homotopy_inv_of_homotopy_pre,
+      unfold inv_homotopy_of_homotopy_pre,
+      unfold homotopy_of_inv_homotopy_pre,
       apply trans (con.assoc
         (ap α (left_inv f a))⁻¹
         (p (f⁻¹ (f a)))
@@ -140,17 +140,16 @@ namespace is_equiv
   end pre_compose
 
   section post_compose
-    variables (β : C → B) (α : C → A)
+    variables (α : C → A) (β : C → B)
 
-    definition homotopy_inv_of_homotopy_post (p : β ~ f ∘ α) : f⁻¹ ∘ β ~ α :=
+    -- homotopy_inv_of_homotopy_post is in init.equiv
-    λc, ap f⁻¹ (p c) ⬝ (left_inv f (α c))
+    protected definition inv_homotopy_of_homotopy_post.is_equiv
-
+      : is_equiv (inv_homotopy_of_homotopy_post f α β) :=
-    protected definition homotopy_inv_of_homotopy_post.is_equiv
+    adjointify _ (homotopy_of_inv_homotopy_post f α β)
-      : is_equiv (homotopy_inv_of_homotopy_post f β α) :=
-    adjointify _ (λq c, (right_inv f (β c))⁻¹ ⬝ ap f (q c))
     abstract begin
       intro q, apply eq_of_homotopy, intro c,
-      unfold homotopy_inv_of_homotopy_post,
+      unfold inv_homotopy_of_homotopy_post,
+      unfold homotopy_of_inv_homotopy_post,
       apply trans (whisker_right
        (ap_con f⁻¹ (right_inv f (β c))⁻¹ (ap f (q c))
        ⬝ whisker_right (ap_inv f⁻¹ (right_inv f (β c)))
@@ -163,7 +162,8 @@ namespace is_equiv
     end end
     abstract begin
       intro p, apply eq_of_homotopy, intro c,
-      unfold homotopy_inv_of_homotopy_post,
+      unfold inv_homotopy_of_homotopy_post,
+      unfold homotopy_of_inv_homotopy_post,
       apply trans (whisker_left (right_inv f (β c))⁻¹
         (ap_con f (ap f⁻¹ (p c)) (left_inv f (α c)))),
       apply trans (con.assoc (right_inv f (β c))⁻¹ (ap f (ap f⁻¹ (p c)))