refactor(library/logic/axioms): rename files, import logic.axioms.classical now imports all classical axioms

library/classical.lean

@@ -7,5 +7,4 @@ Author: Jeremy Avigad
 
 The standard library together with the classical axioms.
 -/
-
-import standard logic.axioms
+import standard logic.axioms.classical

library/logic/axioms/axioms.md

@@ -4,7 +4,8 @@ logic.axioms
 Axioms that extend the Calculus of Constructions.
 
 * [funext](funext.lean) : function extensionality
-* [classical](classical.lean) : the law of the excluded middle
+* [prop_complete](classical.lean) : the law of the excluded middle
 * [hilbert](hilbert.lean) : choice functions
 * [prop_decidable](prop_decidable.lean) : the decidable class is trivial with excluded middle
+* [classical](classical.lean) : imports all of the above
 * [examples](examples/examples.md)

library/logic/axioms/classical.lean

@@ -3,57 +3,7 @@ Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 
 Module: logic.axioms.classical
-Author: Leonardo de Moura
+Author: Jeremy Avigad
 -/
-
-import logic.connectives logic.quantifiers logic.cast algebra.relation
-
-open eq.ops
-
-axiom prop_complete (a : Prop) : a = true ∨ a = false
-
-definition eq_true_or_eq_false := prop_complete
-
-theorem cases (P : Prop → Prop) (H1 : P true) (H2 : P false) (a : Prop) : P a :=
-or.elim (prop_complete a)
-  (assume Ht : a = true,  Ht⁻¹ ▸ H1)
-  (assume Hf : a = false, Hf⁻¹ ▸ H2)
-
-theorem cases_on (a : Prop) {P : Prop → Prop} (H1 : P true) (H2 : P false) : P a :=
-cases P H1 H2 a
-
--- this supercedes the em in decidable
-theorem em (a : Prop) : a ∨ ¬a :=
-or.elim (prop_complete a)
-  (assume Ht : a = true,  or.inl (of_eq_true Ht))
-  (assume Hf : a = false, or.inr (not_of_eq_false Hf))
-
-theorem eq_false_or_eq_true (a : Prop) : a = false ∨ a = true :=
-cases (λ x, x = false ∨ x = true)
-  (or.inr rfl)
-  (or.inl rfl)
-  a
-
-theorem propext {a b : Prop} (Hab : a → b) (Hba : b → a) : a = b :=
-or.elim (prop_complete a)
-  (assume Hat,  or.elim (prop_complete b)
-    (assume Hbt,  Hat ⬝ Hbt⁻¹)
-    (assume Hbf, false.elim (Hbf ▸ (Hab (of_eq_true Hat)))))
-  (assume Haf, or.elim (prop_complete b)
-    (assume Hbt,  false.elim (Haf ▸ (Hba (of_eq_true Hbt))))
-    (assume Hbf, Haf ⬝ Hbf⁻¹))
-
-theorem eq.of_iff {a b : Prop} (H : a ↔ b) : a = b :=
-iff.elim (assume H1 H2, propext H1 H2) H
-
-theorem iff_eq_eq {a b : Prop} : (a ↔ b) = (a = b) :=
-propext
-  (assume H, eq.of_iff H)
-  (assume H, iff.of_eq H)
-
-open relation
-theorem iff_congruence [instance] (P : Prop → Prop) : is_congruence iff iff P :=
-is_congruence.mk
-  (take (a b : Prop),
-    assume H : a ↔ b,
-    show P a ↔ P b, from iff.of_eq (eq.of_iff H ▸ eq.refl (P a)))
+import logic.axioms.prop_complete logic.axioms.funext logic.axioms.hilbert
+import logic.axioms.prop_decidable

library/logic/axioms/default.lean

@@ -1,10 +0,0 @@
-/-
-Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-
-Module: logic.axioms.default
-Author: Jeremy Avigad
--/
-
-import logic.axioms.classical logic.axioms.funext logic.axioms.hilbert
-import logic.axioms.prop_decidable

library/logic/axioms/funext.lean

@@ -1,14 +1,15 @@
--- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
--- Released under Apache 2.0 license as described in the file LICENSE.
--- Author: Leonardo de Moura
+/-
+Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
 
--- logic.axioms.funext
--- ===================
+Module: logic.axioms.funext
+Author: Leonardo de Moura
 
+Function extensionality.
+-/
 import logic.cast algebra.function data.sigma
 open function eq.ops
 
--- Function extensionality
 axiom funext : ∀ {A : Type} {B : A → Type} {f g : Π a, B a} (H : ∀ a, f a = g a), f = g
 
 namespace function
@@ -30,5 +31,4 @@ namespace function
       (H : ∀ a, f a == g a) : f == g :=
   let HH : B = B' := (funext (λ x, heq.type_eq (H x))) in
   cast_to_heq (funext (λ a, heq.to_eq (heq.trans (cast_app HH f a) (H a))))
-
 end function

library/logic/axioms/hilbert.lean

@@ -7,10 +7,8 @@ Authors: Leonardo de Moura, Jeremy Avigad
 
 Follows Coq.Logic.ClassicalEpsilon (but our definition of "inhabited" is the constructive one).
 -/
-
 import logic.quantifiers
 import data.subtype data.sum
-
 open subtype inhabited nonempty
 
 /- the axiom -/

library/logic/axioms/prop_complete.lean

@@ -0,0 +1,59 @@
+/-
+Copyright (c) 2014 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+
+Module: logic.axioms.classical
+Author: Leonardo de Moura
+-/
+
+import logic.connectives logic.quantifiers logic.cast algebra.relation
+
+open eq.ops
+
+axiom prop_complete (a : Prop) : a = true ∨ a = false
+
+definition eq_true_or_eq_false := prop_complete
+
+theorem cases (P : Prop → Prop) (H1 : P true) (H2 : P false) (a : Prop) : P a :=
+or.elim (prop_complete a)
+  (assume Ht : a = true,  Ht⁻¹ ▸ H1)
+  (assume Hf : a = false, Hf⁻¹ ▸ H2)
+
+theorem cases_on (a : Prop) {P : Prop → Prop} (H1 : P true) (H2 : P false) : P a :=
+cases P H1 H2 a
+
+-- this supercedes the em in decidable
+theorem em (a : Prop) : a ∨ ¬a :=
+or.elim (prop_complete a)
+  (assume Ht : a = true,  or.inl (of_eq_true Ht))
+  (assume Hf : a = false, or.inr (not_of_eq_false Hf))
+
+theorem eq_false_or_eq_true (a : Prop) : a = false ∨ a = true :=
+cases (λ x, x = false ∨ x = true)
+  (or.inr rfl)
+  (or.inl rfl)
+  a
+
+theorem propext {a b : Prop} (Hab : a → b) (Hba : b → a) : a = b :=
+or.elim (prop_complete a)
+  (assume Hat,  or.elim (prop_complete b)
+    (assume Hbt,  Hat ⬝ Hbt⁻¹)
+    (assume Hbf, false.elim (Hbf ▸ (Hab (of_eq_true Hat)))))
+  (assume Haf, or.elim (prop_complete b)
+    (assume Hbt,  false.elim (Haf ▸ (Hba (of_eq_true Hbt))))
+    (assume Hbf, Haf ⬝ Hbf⁻¹))
+
+theorem eq.of_iff {a b : Prop} (H : a ↔ b) : a = b :=
+iff.elim (assume H1 H2, propext H1 H2) H
+
+theorem iff_eq_eq {a b : Prop} : (a ↔ b) = (a = b) :=
+propext
+  (assume H, eq.of_iff H)
+  (assume H, iff.of_eq H)
+
+open relation
+theorem iff_congruence [instance] (P : Prop → Prop) : is_congruence iff iff P :=
+is_congruence.mk
+  (take (a b : Prop),
+    assume H : a ↔ b,
+    show P a ↔ P b, from iff.of_eq (eq.of_iff H ▸ eq.refl (P a)))

library/logic/axioms/prop_decidable.lean

@@ -5,8 +5,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Module: logic.axioms.prop_decidable
 Author: Leonardo de Moura
 -/
-
-import logic.axioms.classical logic.axioms.hilbert
+import logic.axioms.prop_complete logic.axioms.hilbert
 open decidable inhabited nonempty
 
 -- Excluded middle + Hilbert implies every proposition is decidable