chore(library/hott) move theorem about precomposition to its own file

library/hott/equiv.lean

@@ -212,41 +212,6 @@ namespace IsEquiv
 
   end
 
-  --If pre- or post-composing with a function is always an equivalence,
-  --then that function is also an equivalence.  It's convenient to know
-  --that we only need to assume the equivalence when the other type is
-  --the domain or the codomain.
-  section
-
-  definition precomp (C : Type) (h : B → C) : A → C := h ∘ f
-
-  definition inv_precomp (C D : Type) (Ceq : IsEquiv (precomp C))
-      (Deq : IsEquiv (@precomp A B f D)) (k : C → D) (h : A → C) :
-      k ∘ (inv (precomp C)) h ≈ (inv (precomp D)) (k ∘ h) :=
-    let invD := inv (precomp D) in
-    let invC := inv (precomp C) in
-    have eq1 : invD (k ∘ h) ≈ k ∘ (invC h),
-      from calc invD (k ∘ h) ≈ invD (k ∘ (precomp C (invC h))) : retr (precomp C) h
-                ... ≈ k ∘ (invC h) : !sect,
-      eq1⁻¹
-
-  definition isequiv_precompose (Aeq : IsEquiv (@precomp A B f A))
-      (Beq : IsEquiv (@precomp A B f B)) : (IsEquiv f) :=
-    let invA := inv (precomp A) in
-    let invB := inv (precomp B) in
-    let sect' : Sect (invA id) f := (λx,
-      calc f (invA id x) ≈ (f ∘ invA id) x : idp
-        ... ≈ invB (f ∘ id) x : apD10 (!inv_precomp)
-        ... ≈ invB (@precomp A B f B id) x : idp
-        ... ≈ x : apD10 (sect (precomp B) id))
-      in
-    let retr' : Sect f (invA id) := (λx,
-      calc invA id (f x) ≈ @precomp A B f A (invA id) x : idp
-        ... ≈ x : apD10 (retr (precomp A) id)) in
-    adjointify f (invA id) sect' retr'
-
-  end
-
 end IsEquiv
 
 namespace Equiv

library/hott/equiv_precomp.lean

@@ -0,0 +1,46 @@
+-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+-- Released under Apache 2.0 license as described in the file LICENSE.
+-- Author: Jeremy Avigad, Jakob von Raumer
+-- Ported from Coq HoTT
+import .equiv .funext
+open path function
+
+namespace IsEquiv
+
+  --If pre- or post-composing with a function is always an equivalence,
+  --then that function is also an equivalence.  It's convenient to know
+  --that we only need to assume the equivalence when the other type is
+  --the domain or the codomain.
+  context
+  parameters {A B : Type} (f : A → B)
+
+  definition precomp (C : Type) (h : B → C) : A → C := h ∘ f
+
+  definition inv_precomp (C D : Type) (Ceq : IsEquiv (precomp C))
+      (Deq : IsEquiv (precomp D)) (k : C → D) (h : A → C) :
+      k ∘ (inv (precomp C)) h ≈ (inv (precomp D)) (k ∘ h) :=
+    let invD := inv (precomp D) in
+    let invC := inv (precomp C) in
+    have eq1 : invD (k ∘ h) ≈ k ∘ (invC h),
+      from calc invD (k ∘ h) ≈ invD (k ∘ (precomp C (invC h))) : retr (precomp C) h
+		... ≈ k ∘ (invC h) : !sect,
+      eq1⁻¹
+
+  definition isequiv_precompose (Aeq : IsEquiv (precomp A))
+      (Beq : IsEquiv (precomp B)) : (IsEquiv f) :=
+    let invA := inv (precomp A) in
+    let invB := inv (precomp B) in
+    let sect' : Sect (invA id) f := (λx,
+      calc f (invA id x) ≈ (f ∘ invA id) x : idp
+	... ≈ invB (f ∘ id) x : apD10 (!inv_precomp)
+	... ≈ invB (precomp B id) x : idp
+	... ≈ x : apD10 (sect (precomp B) id))
+      in
+    let retr' : Sect f (invA id) := (λx,
+      calc invA id (f x) ≈ precomp A (invA id) x : idp
+	... ≈ x : apD10 (retr (precomp A) id)) in
+    adjointify f (invA id) sect' retr'
+
+  end
+
+end IsEquiv