refactor(library/logic/axioms/examples/diaconescu.lean): mild reformatting, to match tutorial

library/logic/axioms/examples/diaconescu.lean

@@ -1,51 +1,58 @@
--- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+/-
--- Released under Apache 2.0 license as described in the file LICENSE.
+Copyright (c) 2014 Microsoft Corporation. All rights reserved.
--- Author: Leonardo de Moura
+Released under Apache 2.0 license as described in the file LICENSE.
+Author: Leonardo de Moura
+-/
 
 import logic.axioms.hilbert logic.eq
-open eq.ops nonempty inhabited
+open eq.ops
+
+/-
+Diaconescu’s theorem: excluded middle follows from Hilbert's choice operator, function extensionality,
+  and Prop extensionality
+-/
 
--- Diaconescu’s theorem
--- Show that Excluded middle follows from
---   Hilbert's choice operator, function extensionality and Prop extensionality
 section
 parameter  p : Prop
 
-private definition u [reducible] := epsilon (λx, x = true ∨ p)
+private definition U (x : Prop) : Prop := x = true ∨ p
+private definition V (x : Prop) : Prop := x = false ∨ p
 
-private definition v [reducible] := epsilon (λx, x = false ∨ p)
+private definition u := epsilon U
+private definition v := epsilon V
 
-private lemma u_def : u = true ∨ p :=
+private lemma u_def : U u :=
 epsilon_spec (exists.intro true (or.inl rfl))
 
-private lemma v_def : v = false ∨ p :=
+private lemma v_def : V v :=
 epsilon_spec (exists.intro false (or.inl rfl))
 
-private lemma uv_implies_p : ¬(u = v) ∨ p :=
+private lemma not_uv_or_p : ¬(u = v) ∨ p :=
 or.elim u_def
-  (assume Hut : u = true, or.elim v_def
+  (assume Hut : u = true,
-    (assume Hvf : v = false,
+    or.elim v_def
-      have Hne : ¬(u = v), from Hvf⁻¹ ▸ Hut⁻¹ ▸ true_ne_false,
+      (assume Hvf : v = false,
-      or.inl Hne)
+        have Hne : ¬(u = v), from Hvf⁻¹ ▸ Hut⁻¹ ▸ true_ne_false,
-    (assume Hp : p, or.inr Hp))
+        or.inl Hne)
+      (assume Hp : p, or.inr Hp))
   (assume Hp : p, or.inr Hp)
 
 private lemma p_implies_uv : p → u = v :=
 assume Hp : p,
-  have Hpred : (λ x, x = true ∨ p) = (λ x, x = false ∨ p), from
+have Hpred : U = V, from
-    funext (take x : Prop,
+  funext (take x : Prop,
-      have Hl : (x = true ∨ p) → (x = false ∨ p), from
+    have Hl : (x = true ∨ p) → (x = false ∨ p), from
-        assume A, or.inr Hp,
+      assume A, or.inr Hp,
-      have Hr : (x = false ∨ p) → (x = true ∨ p), from
+    have Hr : (x = false ∨ p) → (x = true ∨ p), from
-        assume A, or.inr Hp,
+      assume A, or.inr Hp,
-      show (x = true ∨ p) = (x = false ∨ p), from
+    show (x = true ∨ p) = (x = false ∨ p), from
-        propext (iff.intro Hl Hr)),
+      propext (iff.intro Hl Hr)),
-  show u = v, from
+have H' : epsilon U = epsilon V, from Hpred ▸ rfl,
-    Hpred ▸ (eq.refl (epsilon (λ x, x = true ∨ p)))
+show u = v, from H'
 
 theorem em : p ∨ ¬p :=
 have H : ¬(u = v) → ¬p, from mt p_implies_uv,
-  or.elim uv_implies_p
+  or.elim not_uv_or_p
     (assume Hne : ¬(u = v), or.inr (H Hne))
     (assume Hp : p, or.inl Hp)
 end