refactor(library/init/funext.lean): break out definition of equivalence, and hide auxiliary theorems

library/init/funext.lean

@@ -5,40 +5,47 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Module: init.funext
 Author: Jeremy Avigad
 
-Function extensionality follows from quotients.
+Extensional equality for functions, and a proof of function extensionality from quotients.
 -/
 prelude
 import init.quot init.logic
 
-section
+namespace function
+  variables {A : Type} {B : A → Type}
+
+  protected definition equiv (f₁ f₂ : Πx : A, B x) : Prop := ∀x, f₁ x = f₂ x
+
+  namespace equiv_notation
+    infix `~` := function.equiv
+  end equiv_notation
+  open equiv_notation
+
+  protected theorem equiv.refl (f : Πx : A, B x) : f ~ f := take x, rfl
+
+  protected theorem equiv.symm {f₁ f₂ : Πx: A, B x} : f₁ ~ f₂ → f₂ ~ f₁ :=
+  λH x, eq.symm (H x)
+
+  protected theorem equiv.trans {f₁ f₂ f₃ : Πx: A, B x} : f₁ ~ f₂ → f₂ ~ f₃ → f₁ ~ f₃ :=
+  λH₁ H₂ x, eq.trans (H₁ x) (H₂ x)
+
+  protected theorem equiv.is_equivalence (A : Type) (B : A → Type) : equivalence (@equiv A B) :=
+  mk_equivalence (@equiv A B) (@equiv.refl A B) (@equiv.symm A B) (@equiv.trans A B)
+end function
+
+context
   open quot
   variables {A : Type} {B : A → Type}
 
-  private definition fun_eqv (f₁ f₂ : Πx : A, B x) : Prop := ∀x, f₁ x = f₂ x
+  private definition fun_setoid [instance] (A : Type) (B : A → Type) : setoid (Πx : A, B x) :=
+  setoid.mk (@function.equiv A B) (function.equiv.is_equivalence A B)
 
-  infix `~` := fun_eqv
+  private definition extfun (A : Type) (B : A → Type) : Type :=
-
-  private theorem fun_eqv.refl (f : Πx : A, B x) : f ~ f := take x, rfl
-
-  private theorem fun_eqv.symm {f₁ f₂ : Πx: A, B x} : f₁ ~ f₂ → f₂ ~ f₁ :=
-  λH x, eq.symm (H x)
-
-  private theorem fun_eqv.trans {f₁ f₂ f₃ : Πx: A, B x} : f₁ ~ f₂ → f₂ ~ f₃ → f₁ ~ f₃ :=
-  λH₁ H₂ x, eq.trans (H₁ x) (H₂ x)
-
-  private theorem fun_eqv.is_equivalence (A : Type) (B : A → Type) : equivalence (@fun_eqv A B) :=
-  mk_equivalence (@fun_eqv A B) (@fun_eqv.refl A B) (@fun_eqv.symm A B) (@fun_eqv.trans A B)
-
-  definition fun_setoid [instance] (A : Type) (B : A → Type) : setoid (Πx : A, B x) :=
-  setoid.mk (@fun_eqv A B) (fun_eqv.is_equivalence A B)
-
-  definition extfun (A : Type) (B : A → Type) : Type :=
   quot (fun_setoid A B)
 
-  definition fun_to_extfun (f : Πx : A, B x) : extfun A B :=
+  private definition fun_to_extfun (f : Πx : A, B x) : extfun A B :=
   ⟦f⟧
 
-  definition extfun_app (f : extfun A B) : Πx : A, B x :=
+  private definition extfun_app (f : extfun A B) : Πx : A, B x :=
   take x,
   quot.lift_on f
     (λf : Πx : A, B x, f x)
@@ -46,12 +53,15 @@ section
 
   theorem funext {f₁ f₂ : Πx : A, B x} : (∀x, f₁ x = f₂ x) → f₁ = f₂ :=
   assume H, calc
-     f₁ = extfun_app ⟦f₁⟧      : rfl
+     f₁ = extfun_app ⟦f₁⟧ : rfl
-    ... = extfun_app ⟦f₂⟧      : {sound H}
+    ... = extfun_app ⟦f₂⟧ : {sound H}
-    ... = f₂                   : rfl
+    ... = f₂             : rfl
 end
 
-definition subsingleton_pi [instance] {A : Type} {B : A → Type} (H : ∀ a, subsingleton (B a)) : subsingleton (Π a, B a) :=
+open function.equiv_notation
+
+definition subsingleton_pi [instance] {A : Type} {B : A → Type} (H : ∀ a, subsingleton (B a)) :
+  subsingleton (Π a, B a) :=
 subsingleton.intro (take f₁ f₂,
   have eqv : f₁ ~ f₂, from
     take a, subsingleton.elim (f₁ a) (f₂ a),