feat(library/logic/axioms): use diaconescu to prove em

With the new "noncomputable" feature we can use Hilbert's choice without
being concerned it may accidentaly "leak" inside definitions we don't
want to use it.

library/logic/axioms/em.lean

@@ -3,9 +3,53 @@ Copyright (c) 2015 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Author: Leonardo de Moura
 
-Excluded middle
-
-Remark: This axiom can be derived from propext funext and hilbert.
-See examples/diaconescu
+Prove excluded middle using: excluded middle follows from Hilbert's choice operator, function extensionality,
+and Prop extensionality
 -/
-axiom em (a : Prop) : a ∨ ¬a
+import logic.axioms.hilbert logic.eq
+open eq.ops
+
+section
+parameter  p : Prop
+
+private definition U (x : Prop) : Prop := x = true ∨ p
+private definition V (x : Prop) : Prop := x = false ∨ p
+
+private noncomputable definition u := epsilon U
+private noncomputable definition v := epsilon V
+
+private lemma u_def : U u :=
+epsilon_spec (exists.intro true (or.inl rfl))
+
+private lemma v_def : V v :=
+epsilon_spec (exists.intro false (or.inl rfl))
+
+private lemma not_uv_or_p : ¬(u = v) ∨ p :=
+or.elim u_def
+  (assume Hut : u = true,
+    or.elim v_def
+      (assume Hvf : v = false,
+        have Hne : ¬(u = v), from Hvf⁻¹ ▸ Hut⁻¹ ▸ true_ne_false,
+        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 : U = V, from
+  funext (take x : Prop,
+    have Hl : (x = true ∨ p) → (x = false ∨ p), from
+      assume A, or.inr Hp,
+    have Hr : (x = false ∨ p) → (x = true ∨ p), from
+      assume A, or.inr Hp,
+    show (x = true ∨ p) = (x = false ∨ p), from
+      propext (iff.intro Hl Hr)),
+have H' : epsilon U = epsilon V, from Hpred ▸ rfl,
+show u = v, from H'
+
+theorem em : p ∨ ¬p :=
+have H : ¬(u = v) → ¬p, from mt p_implies_uv,
+  or.elim not_uv_or_p
+    (assume Hne : ¬(u = v), or.inr (H Hne))
+    (assume Hp : p, or.inl Hp)
+end

library/logic/axioms/examples/diaconescu.lean

@@ -1,58 +0,0 @@
-/-
-Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-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
-
-/-
-Diaconescu’s theorem: excluded middle follows from Hilbert's choice operator, function extensionality,
-  and Prop extensionality
--/
-
-section
-parameter  p : Prop
-
-private definition U (x : Prop) : Prop := x = true ∨ p
-private definition V (x : Prop) : Prop := x = false ∨ p
-
-private noncomputable definition u := epsilon U
-private noncomputable definition v := epsilon V
-
-private lemma u_def : U u :=
-epsilon_spec (exists.intro true (or.inl rfl))
-
-private lemma v_def : V v :=
-epsilon_spec (exists.intro false (or.inl rfl))
-
-private lemma not_uv_or_p : ¬(u = v) ∨ p :=
-or.elim u_def
-  (assume Hut : u = true,
-    or.elim v_def
-      (assume Hvf : v = false,
-        have Hne : ¬(u = v), from Hvf⁻¹ ▸ Hut⁻¹ ▸ true_ne_false,
-        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 : U = V, from
-  funext (take x : Prop,
-    have Hl : (x = true ∨ p) → (x = false ∨ p), from
-      assume A, or.inr Hp,
-    have Hr : (x = false ∨ p) → (x = true ∨ p), from
-      assume A, or.inr Hp,
-    show (x = true ∨ p) = (x = false ∨ p), from
-      propext (iff.intro Hl Hr)),
-have H' : epsilon U = epsilon V, from Hpred ▸ rfl,
-show u = v, from H'
-
-theorem em : p ∨ ¬p :=
-have H : ¬(u = v) → ¬p, from mt p_implies_uv,
-  or.elim not_uv_or_p
-    (assume Hne : ¬(u = v), or.inr (H Hne))
-    (assume Hp : p, or.inl Hp)
-end

tests/lean/print_ax2.lean.expected.out

@@ -1,4 +1,3 @@
 quot.sound : ∀ {A : Type} [s : setoid A] {a b : A}, setoid.r a b → quot.mk a = quot.mk b
-em : ∀ (a : Prop), a ∨ ¬a
 propext : ∀ {a b : Prop}, (a ↔ b) → a = b
 strong_indefinite_description : Π {A : Type} (P : A → Prop), nonempty A → { (x : A)| Exists P → P x}