change terminology set -> property

algebra/subgroup.hlean

@@ -5,9 +5,9 @@ Authors: Floris van Doorn, Egbert Rijke, Jeremy Avigad
 
 Basic concepts of group theory
 -/
-import algebra.group_theory ..move_to_lib ..set
+import algebra.group_theory ..move_to_lib ..property
 
-open eq algebra is_trunc sigma sigma.ops prod trunc set
+open eq algebra is_trunc sigma sigma.ops prod trunc property
 
 namespace group
 
@@ -17,7 +17,7 @@ namespace group
       group G to be a family of mere propositions over (the underlying
      type of) G, closed under the constants and operations --/
 
-  structure is_subgroup [class] (G : Group) (H : set G) : Type :=
+  structure is_subgroup [class] (G : Group) (H : property G) : Type :=
     (one_mem : 1 ∈ H)
     (mul_mem : Π{g h}, g ∈ H → h ∈ H → g * h ∈ H)
     (inv_mem : Π{g}, g ∈ H → g⁻¹ ∈ H)
@@ -50,14 +50,14 @@ namespace group
     is_subgroup H (image f) :=
     begin
       fapply is_subgroup.mk,
-        -- subset contains 1
+        -- subproperty contains 1
       { fapply image.mk, exact 1, apply respect_one},
-        -- subset is closed under multiplication
+        -- subproperty is closed under multiplication
       { intro h h', intro u v,
         induction u with x p, induction v with y q,
         induction p, induction q,
         apply image.mk (x * y), apply respect_mul},
-        -- subset is closed under inverses
+        -- subproperty is closed under inverses
       { intro g, intro t, induction t with x p, induction p,
         apply image.mk x⁻¹, apply respect_inv }
     end
@@ -66,8 +66,8 @@ namespace group
 
   variables {G₁ G₂ : Group}
 
-  -- TODO: maybe define this in more generality for pointed sets?
+  -- TODO: maybe define this in more generality for pointed propertys?
-  definition kernel [constructor] (φ : G₁ →g G₂) : set G₁ := { g | trunctype.mk (φ g = 1) _}
+  definition kernel [constructor] (φ : G₁ →g G₂) : property G₁ := { g | trunctype.mk (φ g = 1) _}
 
   theorem mul_mem_kernel (φ : G₁ →g G₂) (g h : G₁) (H₁ : g ∈ kernel φ) (H₂ : h ∈ kernel φ) : g * h ∈ kernel φ :=
   calc
@@ -103,10 +103,10 @@ namespace group
   /-- Next, we formalize some aspects of normal subgroups. Recall that a normal subgroup H of a
       group G is a subgroup which is invariant under all inner automorophisms on G. --/
 
-  definition is_normal [constructor] (G : Group) (N : set G) : Prop :=
+  definition is_normal [constructor] (G : Group) (N : property G) : Prop :=
   trunctype.mk (Π{g} h, g ∈ N → h * g * h⁻¹ ∈ N) _
 
-  structure is_normal_subgroup [class] (G : Group) (N : set G) extends is_subgroup G N :=
+  structure is_normal_subgroup [class] (G : Group) (N : property G) extends is_subgroup G N :=
     (is_normal : is_normal G N)
 
   abbreviation subgroup_one_mem   [unfold 2] := @is_subgroup.one_mem
@@ -115,14 +115,14 @@ namespace group
   abbreviation subgroup_is_normal [unfold 2] := @is_normal_subgroup.is_normal
 
 section
-  variables {G G' G₁ G₂ G₃ : Group} {H N : set G} [is_subgroup G H]
+  variables {G G' G₁ G₂ G₃ : Group} {H N : property G} [is_subgroup G H]
             [is_normal_subgroup G N] {g g' h h' k : G}
             {A B : AbGroup}
 
   theorem is_normal_subgroup' (h : G) (r : g ∈ N) : h⁻¹ * g * h ∈ N :=
   inv_inv h ▸ subgroup_is_normal N h⁻¹ r
 
-  definition is_normal_subgroup_ab.{u} [constructor] {C : set A} (subgrpA : is_subgroup A C)
+  definition is_normal_subgroup_ab.{u} [constructor] {C : property A} (subgrpA : is_subgroup A C)
     : is_normal_subgroup A C :=
   ⦃ is_normal_subgroup, subgrpA,
     is_normal := abstract begin
@@ -164,7 +164,7 @@ section
   end
 
   -- this is just (Σ(g : G), H g), but only defined if (H g) is a prop
-  definition sg {G : Group} (H : set G) : Type := {g : G | g ∈ H}
+  definition sg {G : Group} (H : property G) : Type := {g : G | g ∈ H}
   local attribute sg [reducible]
 
   definition subgroup_one [constructor] : sg H := ⟨one, subgroup_one_mem H⟩
@@ -192,7 +192,7 @@ section
   theorem subgroup_mul_left_inv (g : sg H) : g⁻¹ * g = 1 :=
   subtype_eq !mul.left_inv
 
-  theorem subgroup_mul_comm {G : AbGroup} {H : set G} [subgrpH : is_subgroup G H] (g h : sg H)
+  theorem subgroup_mul_comm {G : AbGroup} {H : property G} [subgrpH : is_subgroup G H] (g h : sg H)
     : subgroup_mul g h = subgroup_mul h g :=
   subtype_eq !mul.comm
 
@@ -209,11 +209,11 @@ section
 
   variable {H}
 
-  definition ab_group_sg [constructor] {G : AbGroup} (H : set G) [is_subgroup G H]
+  definition ab_group_sg [constructor] {G : AbGroup} (H : property G) [is_subgroup G H]
     : ab_group (sg H) :=
   ⦃ab_group, (group_sg H), mul_comm := subgroup_mul_comm⦄
 
-  definition ab_subgroup [constructor] {G : AbGroup} (H : set G) [is_subgroup G H]
+  definition ab_subgroup [constructor] {G : AbGroup} (H : property G) [is_subgroup G H]
     : AbGroup :=
   AbGroup.mk _ (ab_group_sg H)
 
@@ -221,7 +221,7 @@ section
 
   definition ab_kernel {G H : AbGroup} (f : G →g H) : AbGroup := ab_subgroup (kernel f)
 
-  definition incl_of_subgroup [constructor] {G : Group} (H : set G) [is_subgroup G H] :
+  definition incl_of_subgroup [constructor] {G : Group} (H : property G) [is_subgroup G H] :
     subgroup H →g G :=
   begin
     fapply homomorphism.mk,
@@ -231,7 +231,7 @@ section
     intro g h, reflexivity
   end
 
-  definition is_embedding_incl_of_subgroup {G : Group} (H : set G) [is_subgroup G H] :
+  definition is_embedding_incl_of_subgroup {G : Group} (H : property G) [is_subgroup G H] :
     is_embedding (incl_of_subgroup H) :=
   begin
     fapply function.is_embedding_of_is_injective,
@@ -249,20 +249,20 @@ section
     fapply is_embedding_incl_of_subgroup,
   end
 
-  definition is_subgroup_of_subgroup {G : Group} {H1 H2 : set G} [is_subgroup G H1]
+  definition is_subgroup_of_subgroup {G : Group} {H1 H2 : property G} [is_subgroup G H1]
     [is_subgroup G H2] (hyp : Π (g : G), g ∈ H1 → g ∈ H2) :
       is_subgroup (subgroup H2) {h | incl_of_subgroup H2 h ∈ H1} :=
   is_subgroup.mk
     (subgroup_one_mem H1)
     (begin
       intros g h p q,
-      apply mem_set_of,
+      apply mem_property_of,
-      apply subgroup_mul_mem (of_mem_set_of p) (of_mem_set_of q),
+      apply subgroup_mul_mem (of_mem_property_of p) (of_mem_property_of q),
     end)
     (begin
       intros h p,
-      apply mem_set_of,
+      apply mem_property_of,
-      apply subgroup_inv_mem (of_mem_set_of p)
+      apply subgroup_inv_mem (of_mem_property_of p)
     end)
 
   definition Image {G H : Group} (f : G →g H) : Group :=
@@ -306,7 +306,7 @@ definition iso_surjection_ab_image_incl [constructor] {A B : AbGroup} (f : A →
     exact is_equiv_surjection_ab_image_incl f H
   end
 
-  definition hom_lift [constructor] {G H : Group} (f : G →g H) (K : set H) [is_subgroup H K] (Hyp : Π (g : G), K (f g)) : G →g subgroup K :=
+  definition hom_lift [constructor] {G H : Group} (f : G →g H) (K : property H) [is_subgroup H K] (Hyp : Π (g : G), K (f g)) : G →g subgroup K :=
   begin
     fapply homomorphism.mk,
     intro g,
@@ -316,14 +316,14 @@ definition iso_surjection_ab_image_incl [constructor] {A B : AbGroup} (f : A →
     intro g h, apply subtype_eq, esimp, apply respect_mul
   end
 
-definition hom_factors_through_lift {G H : Group} (f : G →g H) (K : set H) [is_subgroup H K]  (Hyp : Π (g : G), K (f g)) :
+definition hom_factors_through_lift {G H : Group} (f : G →g H) (K : property H) [is_subgroup H K]  (Hyp : Π (g : G), K (f g)) :
 f = incl_of_subgroup K ∘g hom_lift f K Hyp :=
   begin
   fapply homomorphism_eq,
     reflexivity
   end
 
-definition ab_hom_lift [constructor] {G H : AbGroup} (f : G →g H) (K : set H) [is_subgroup H K] (Hyp : Π (g : G), K (f g)) : G →g ab_subgroup K :=
+definition ab_hom_lift [constructor] {G H : AbGroup} (f : G →g H) (K : property H) [is_subgroup H K] (Hyp : Π (g : G), K (f g)) : G →g ab_subgroup K :=
   begin
     fapply homomorphism.mk,
     intro g,
@@ -333,7 +333,7 @@ definition ab_hom_lift [constructor] {G H : AbGroup} (f : G →g H) (K : set H)
     intro g h, apply subtype_eq, apply respect_mul,
   end
 
-definition ab_hom_factors_through_lift {G H : AbGroup} (f : G →g H) (K : set H) [is_subgroup H K] (Hyp : Π (g : G), K (f g)) :
+definition ab_hom_factors_through_lift {G H : AbGroup} (f : G →g H) (K : property H) [is_subgroup H K] (Hyp : Π (g : G), K (f g)) :
 f = incl_of_subgroup K ∘g hom_lift f K Hyp :=
   begin
   fapply homomorphism_eq,
@@ -438,8 +438,8 @@ end
     intro x, induction x with b p, induction p with x, induction p, reflexivity
   end
 
-  variables {G H K : Group} {R : set G} [is_subgroup G R]
+  variables {G H K : Group} {R : property G} [is_subgroup G R]
-            {S : set H} [is_subgroup H S] {T : set K} [is_subgroup K T]
+            {S : property H} [is_subgroup H S] {T : property K} [is_subgroup K T]
   open function
 
   definition subgroup_functor_fun [unfold 7] (φ : G →g H) (h : Πg, g ∈ R → φ g ∈ S)
@@ -460,8 +460,8 @@ end
   end
 
   definition ab_subgroup_functor [constructor] {G H : AbGroup}
-    {R : set G} [is_subgroup G R]
+    {R : property G} [is_subgroup G R]
-    {S : set H} [is_subgroup H S]
+    {S : property H} [is_subgroup H S]
     (φ : G →g H)
     (h : Πg, g ∈ R → φ g ∈ S) : ab_subgroup R →g ab_subgroup S :=
   subgroup_functor φ h
@@ -479,8 +479,8 @@ end
     intro g, induction g with g hg, reflexivity
   end
 
-  definition subgroup_functor_mul {G H : AbGroup} {R : set G} [subgroupR : is_subgroup G R]
+  definition subgroup_functor_mul {G H : AbGroup} {R : property G} [subgroupR : is_subgroup G R]
-    {S : set H} [subgroupS : is_subgroup H S]
+    {S : property H} [subgroupS : is_subgroup H S]
     (ψ φ : G →g H) (hψ : Πg, g ∈ R → ψ g ∈ S) (hφ : Πg, g ∈ R → φ g ∈ S) :
     homomorphism_mul (ab_subgroup_functor ψ hψ) (ab_subgroup_functor φ hφ) ~
     ab_subgroup_functor (homomorphism_mul ψ φ)
@@ -497,12 +497,12 @@ end
     exact subtype_eq (p g)
   end
 
-  definition subgroup_of_subgroup_incl {R S : set G} [is_subgroup G R] [is_subgroup G S]
+  definition subgroup_of_subgroup_incl {R S : property G} [is_subgroup G R] [is_subgroup G S]
     (H : Π (g : G), g ∈ R → g ∈ S) : subgroup R →g subgroup S
   :=
   subgroup_functor (gid G) H
 
-  definition is_embedding_subgroup_of_subgroup_incl {R S : set G} [is_subgroup G R] [is_subgroup G S]
+  definition is_embedding_subgroup_of_subgroup_incl {R S : property G} [is_subgroup G R] [is_subgroup G S]
     (H : Π (g : G), g ∈ R -> g ∈ S) : is_embedding (subgroup_of_subgroup_incl H) :=
   begin
     fapply is_embedding_of_is_injective,
@@ -512,18 +512,18 @@ end
     unfold subgroup_of_subgroup_incl at p, exact ap pr1 p,
   end
 
-  definition ab_subgroup_of_subgroup_incl {A : AbGroup} {R S : set A} [is_subgroup A R] [is_subgroup A S]
+  definition ab_subgroup_of_subgroup_incl {A : AbGroup} {R S : property A} [is_subgroup A R] [is_subgroup A S]
     (H : Π (a : A), a ∈ R → a ∈ S) : ab_subgroup R →g ab_subgroup S
   :=
   ab_subgroup_functor (gid A) H
 
-  definition is_embedding_ab_subgroup_of_subgroup_incl {A : AbGroup} {R S : set A} [is_subgroup A R] [is_subgroup A S]
+  definition is_embedding_ab_subgroup_of_subgroup_incl {A : AbGroup} {R S : property A} [is_subgroup A R] [is_subgroup A S]
     (H : Π (a : A), a ∈ R → a ∈ S) : is_embedding (ab_subgroup_of_subgroup_incl H) :=
   begin
     fapply is_embedding_subgroup_of_subgroup_incl
   end
 
-  definition ab_subgroup_iso {A : AbGroup} {R S : set A} [is_subgroup A R] [is_subgroup A S]
+  definition ab_subgroup_iso {A : AbGroup} {R S : property A} [is_subgroup A R] [is_subgroup A S]
       (H : Π (a : A), R a -> S a) (K : Π (a : A), S a -> R a) :
     ab_subgroup R ≃g ab_subgroup S :=
   begin
@@ -535,7 +535,7 @@ end
     intro r, induction r with a p, fapply subtype_eq, reflexivity
   end
 
-  definition ab_subgroup_iso_triangle {A : AbGroup} {R S : set A} [is_subgroup A R] [is_subgroup A S]
+  definition ab_subgroup_iso_triangle {A : AbGroup} {R S : property A} [is_subgroup A R] [is_subgroup A S]
       (H : Π (a : A), R a -> S a) (K : Π (a : A), S a -> R a) :
     incl_of_subgroup R  ~ incl_of_subgroup S ∘g ab_subgroup_iso H K :=
   begin

property.hlean

@@ -0,0 +1,183 @@
+/-
+Copyright (c) 2017 Jeremy Avigad. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jeremy Avigad
+-/
+import types.trunc .logic
+open funext eq trunc is_trunc logic
+
+definition property (X : Type) := X → Prop
+
+namespace property
+
+variable {X : Type}
+
+/- membership and subproperty -/
+
+definition mem (x : X) (a : property X) := a x
+infix ∈ := mem
+notation a ∉ b := ¬ mem a b
+
+/-theorem ext {a b : property X} (H : ∀x, x ∈ a ↔ x ∈ b) : a = b :=
+eq_of_homotopy (take x, propext (H x))
+-/
+
+definition subproperty (a b : property X) : Prop := Prop.mk (∀⦃x⦄, x ∈ a → x ∈ b) _
+infix ⊆ := subproperty
+
+definition superproperty (s t : property X) : Prop := t ⊆ s
+infix ⊇ := superproperty
+
+theorem subproperty.refl (a : property X) : a ⊆ a := take x, assume H, H
+
+theorem subproperty.trans {a b c : property X} (subab : a ⊆ b) (subbc : b ⊆ c) : a ⊆ c :=
+take x, assume ax, subbc (subab ax)
+
+/-
+theorem subproperty.antisymm {a b : property X} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b :=
+ext (λ x, iff.intro (λ ina, h₁ ina) (λ inb, h₂ inb))
+-/
+
+-- an alterantive name
+/-
+theorem eq_of_subproperty_of_subproperty {a b : property X} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b :=
+subproperty.antisymm h₁ h₂
+-/
+
+theorem mem_of_subproperty_of_mem {s₁ s₂ : property X} {a : X} : s₁ ⊆ s₂ → a ∈ s₁ → a ∈ s₂ :=
+assume h₁ h₂, h₁ _ h₂
+
+/- empty property -/
+
+definition empty : property X := λx, false
+notation `∅` := property.empty
+
+theorem not_mem_empty (x : X) : ¬ (x ∈ ∅) :=
+assume H : x ∈ ∅, false.elim H
+
+theorem mem_empty_eq (x : X) : x ∈ ∅ = false := rfl
+
+/-
+theorem eq_empty_of_forall_not_mem {s : property X} (H : ∀ x, x ∉ s) : s = ∅ :=
+ext (take x, iff.intro
+  (assume xs, absurd xs (H x))
+  (assume xe, absurd xe (not_mem_empty x)))
+-/
+
+theorem ne_empty_of_mem {s : property X} {x : X} (H : x ∈ s) : s ≠ ∅ :=
+  begin intro Hs, rewrite Hs at H, apply not_mem_empty x H end
+
+theorem empty_subproperty (s : property X) : ∅ ⊆ s :=
+take x, assume H, false.elim H
+
+/-theorem eq_empty_of_subproperty_empty {s : property X} (H : s ⊆ ∅) : s = ∅ :=
+subproperty.antisymm H (empty_subproperty s)
+
+theorem subproperty_empty_iff (s : property X) : s ⊆ ∅ ↔ s = ∅ :=
+iff.intro eq_empty_of_subproperty_empty (take xeq, by rewrite xeq; apply subproperty.refl ∅)
+-/
+
+/- universal property -/
+
+definition univ : property X := λx, true
+
+theorem mem_univ (x : X) : x ∈ univ := trivial
+
+theorem mem_univ_eq (x : X) : x ∈ univ = true := rfl
+
+theorem empty_ne_univ [h : inhabited X] : (empty : property X) ≠ univ :=
+assume H : empty = univ,
+absurd (mem_univ (inhabited.value h)) (eq.rec_on H (not_mem_empty (arbitrary X)))
+
+theorem subproperty_univ (s : property X) : s ⊆ univ := λ x H, trivial
+
+/-
+theorem eq_univ_of_univ_subproperty {s : property X} (H : univ ⊆ s) : s = univ :=
+eq_of_subproperty_of_subproperty (subproperty_univ s) H
+-/
+
+/-
+theorem eq_univ_of_forall {s : property X} (H : ∀ x, x ∈ s) : s = univ :=
+ext (take x, iff.intro (assume H', trivial) (assume H', H x))
+-/
+
+/- property-builder notation -/
+
+-- {x : X | P}
+definition property_of (P : X → Prop) : property X := P
+notation `{` binder ` | ` r:(scoped:1 P, property_of P) `}` := r
+
+theorem mem_property_of {P : X → Prop} {a : X} (h : P a) : a ∈ {x | P x} := h
+
+theorem of_mem_property_of {P : X → Prop} {a : X} (h : a ∈ {x | P x}) : P a := h
+
+-- {x ∈ s | P}
+definition sep (P : X → Prop) (s : property X) : property X := λx, x ∈ s ∧ P x
+notation `{` binder ` ∈ ` s ` | ` r:(scoped:1 p, sep p s) `}` := r
+
+/- insert -/
+
+definition insert (x : X) (a : property X) : property X := {y : X | y = x ∨ y ∈ a}
+
+abbreviation insert_same_level.{u} := @insert.{u u}
+
+-- '{x, y, z}
+notation `'{`:max a:(foldr `, ` (x b, insert_same_level x b) ∅) `}`:0 := a
+
+theorem subproperty_insert (x : X) (a : property X) : a ⊆ insert x a :=
+take y, assume ys, or.inr ys
+
+theorem mem_insert (x : X) (s : property X) : x ∈ insert x s :=
+or.inl rfl
+
+theorem mem_insert_of_mem {x : X} {s : property X} (y : X) : x ∈ s → x ∈ insert y s :=
+assume h, or.inr h
+
+theorem eq_or_mem_of_mem_insert {x a : X} {s : property X} : x ∈ insert a s → x = a ∨ x ∈ s :=
+assume h, h
+
+/- singleton -/
+
+open trunc_index
+
+theorem mem_singleton_iff {X : Type} [is_set X] (a b : X) : a ∈ '{b} ↔ a = b :=
+iff.intro
+  (assume ainb, or.elim ainb (λ aeqb, aeqb) (λ f, false.elim f))
+  (assume aeqb, or.inl aeqb)
+
+theorem mem_singleton (a : X) : a ∈ '{a} := !mem_insert
+
+theorem eq_of_mem_singleton {X : Type} [is_set X] {x y : X} (h : x ∈ '{y}) : x = y :=
+or.elim (eq_or_mem_of_mem_insert h)
+  (suppose x = y, this)
+  (suppose x ∈ ∅, absurd this (not_mem_empty x))
+
+theorem mem_singleton_of_eq {x y : X} (H : x = y) : x ∈ '{y} :=
+eq.symm H ▸ mem_singleton y
+
+/-
+theorem insert_eq (x : X) (s : property X) : insert x s = '{x} ∪ s :=
+ext (take y, iff.intro
+  (suppose y ∈ insert x s,
+    or.elim this (suppose y = x, or.inl (or.inl this)) (suppose y ∈ s, or.inr this))
+  (suppose y ∈ '{x} ∪ s,
+    or.elim this
+      (suppose y ∈ '{x}, or.inl (eq_of_mem_singleton this))
+      (suppose y ∈ s, or.inr this)))
+-/
+
+/-
+theorem pair_eq_singleton (a : X) : '{a, a} = '{a} :=
+by rewrite [insert_eq_of_mem !mem_singleton]
+-/
+/-
+theorem singleton_ne_empty (a : X) : '{a} ≠ ∅ :=
+begin
+  intro H,
+  apply not_mem_empty a,
+  rewrite -H,
+  apply mem_insert
+end
+-/
+
+end property

set.hlean

@@ -1,186 +0,0 @@
-/-
-Copyright (c) 2017 Jeremy Avigad. All rights reserved.
-Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Jeremy Avigad
--/
-import types.trunc .logic
-open funext eq trunc is_trunc logic
-
-definition set (X : Type) := X → Prop
-
-namespace set
-
-variable {X : Type}
-
-/- membership and subset -/
-
-definition mem (x : X) (a : set X) := a x
-infix ∈ := mem
-notation a ∉ b := ¬ mem a b
-
-/-theorem ext {a b : set X} (H : ∀x, x ∈ a ↔ x ∈ b) : a = b :=
-eq_of_homotopy (take x, propext (H x))
--/
-
-definition subset (a b : set X) : Prop := Prop.mk (∀⦃x⦄, x ∈ a → x ∈ b) _
-infix ⊆ := subset
-
-definition superset (s t : set X) : Prop := t ⊆ s
-infix ⊇ := superset
-
-theorem subset.refl (a : set X) : a ⊆ a := take x, assume H, H
-
-theorem subset.trans {a b c : set X} (subab : a ⊆ b) (subbc : b ⊆ c) : a ⊆ c :=
-take x, assume ax, subbc (subab ax)
-
-/-
-theorem subset.antisymm {a b : set X} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b :=
-ext (λ x, iff.intro (λ ina, h₁ ina) (λ inb, h₂ inb))
--/
-
--- an alterantive name
-/-
-theorem eq_of_subset_of_subset {a b : set X} (h₁ : a ⊆ b) (h₂ : b ⊆ a) : a = b :=
-subset.antisymm h₁ h₂
--/
-
-theorem mem_of_subset_of_mem {s₁ s₂ : set X} {a : X} : s₁ ⊆ s₂ → a ∈ s₁ → a ∈ s₂ :=
-assume h₁ h₂, h₁ _ h₂
-
-/- empty set -/
-
-definition empty : set X := λx, false
-notation `∅` := set.empty
-
-theorem not_mem_empty (x : X) : ¬ (x ∈ ∅) :=
-assume H : x ∈ ∅, false.elim H
-
-theorem mem_empty_eq (x : X) : x ∈ ∅ = false := rfl
-
-/-
-theorem eq_empty_of_forall_not_mem {s : set X} (H : ∀ x, x ∉ s) : s = ∅ :=
-ext (take x, iff.intro
-  (assume xs, absurd xs (H x))
-  (assume xe, absurd xe (not_mem_empty x)))
--/
-
-set_option formatter.hide_full_terms false
-
-theorem ne_empty_of_mem {s : set X} {x : X} (H : x ∈ s) : s ≠ ∅ :=
-  begin intro Hs, rewrite Hs at H, apply not_mem_empty x H end
-
-
-theorem empty_subset (s : set X) : ∅ ⊆ s :=
-take x, assume H, false.elim H
-
-/-theorem eq_empty_of_subset_empty {s : set X} (H : s ⊆ ∅) : s = ∅ :=
-subset.antisymm H (empty_subset s)
-
-theorem subset_empty_iff (s : set X) : s ⊆ ∅ ↔ s = ∅ :=
-iff.intro eq_empty_of_subset_empty (take xeq, by rewrite xeq; apply subset.refl ∅)
--/
-
-/- universal set -/
-
-definition univ : set X := λx, true
-
-theorem mem_univ (x : X) : x ∈ univ := trivial
-
-theorem mem_univ_eq (x : X) : x ∈ univ = true := rfl
-
-theorem empty_ne_univ [h : inhabited X] : (empty : set X) ≠ univ :=
-assume H : empty = univ,
-absurd (mem_univ (inhabited.value h)) (eq.rec_on H (not_mem_empty (arbitrary X)))
-
-theorem subset_univ (s : set X) : s ⊆ univ := λ x H, trivial
-
-/-
-theorem eq_univ_of_univ_subset {s : set X} (H : univ ⊆ s) : s = univ :=
-eq_of_subset_of_subset (subset_univ s) H
--/
-
-/-
-theorem eq_univ_of_forall {s : set X} (H : ∀ x, x ∈ s) : s = univ :=
-ext (take x, iff.intro (assume H', trivial) (assume H', H x))
--/
-
-/- set-builder notation -/
-
--- {x : X | P}
-definition set_of (P : X → Prop) : set X := P
-notation `{` binder ` | ` r:(scoped:1 P, set_of P) `}` := r
-
-theorem mem_set_of {P : X → Prop} {a : X} (h : P a) : a ∈ {x | P x} := h
-
-theorem of_mem_set_of {P : X → Prop} {a : X} (h : a ∈ {x | P x}) : P a := h
-
--- {x ∈ s | P}
-definition sep (P : X → Prop) (s : set X) : set X := λx, x ∈ s ∧ P x
-notation `{` binder ` ∈ ` s ` | ` r:(scoped:1 p, sep p s) `}` := r
-
-/- insert -/
-
-definition insert (x : X) (a : set X) : set X := {y : X | y = x ∨ y ∈ a}
-
-abbreviation insert_same_level.{u} := @insert.{u u}
-
--- '{x, y, z}
-notation `'{`:max a:(foldr `, ` (x b, insert_same_level x b) ∅) `}`:0 := a
-
-theorem subset_insert (x : X) (a : set X) : a ⊆ insert x a :=
-take y, assume ys, or.inr ys
-
-theorem mem_insert (x : X) (s : set X) : x ∈ insert x s :=
-or.inl rfl
-
-theorem mem_insert_of_mem {x : X} {s : set X} (y : X) : x ∈ s → x ∈ insert y s :=
-assume h, or.inr h
-
-theorem eq_or_mem_of_mem_insert {x a : X} {s : set X} : x ∈ insert a s → x = a ∨ x ∈ s :=
-assume h, h
-
-/- singleton -/
-
-open trunc_index
-
-theorem mem_singleton_iff {X : Type} [is_set X] (a b : X) : a ∈ '{b} ↔ a = b :=
-iff.intro
-  (assume ainb, or.elim ainb (λ aeqb, aeqb) (λ f, false.elim f))
-  (assume aeqb, or.inl aeqb)
-
-theorem mem_singleton (a : X) : a ∈ '{a} := !mem_insert
-
-theorem eq_of_mem_singleton {X : Type} [is_set X] {x y : X} (h : x ∈ '{y}) : x = y :=
-or.elim (eq_or_mem_of_mem_insert h)
-  (suppose x = y, this)
-  (suppose x ∈ ∅, absurd this (not_mem_empty x))
-
-theorem mem_singleton_of_eq {x y : X} (H : x = y) : x ∈ '{y} :=
-eq.symm H ▸ mem_singleton y
-
-/-
-theorem insert_eq (x : X) (s : set X) : insert x s = '{x} ∪ s :=
-ext (take y, iff.intro
-  (suppose y ∈ insert x s,
-    or.elim this (suppose y = x, or.inl (or.inl this)) (suppose y ∈ s, or.inr this))
-  (suppose y ∈ '{x} ∪ s,
-    or.elim this
-      (suppose y ∈ '{x}, or.inl (eq_of_mem_singleton this))
-      (suppose y ∈ s, or.inr this)))
--/
-
-/-
-theorem pair_eq_singleton (a : X) : '{a, a} = '{a} :=
-by rewrite [insert_eq_of_mem !mem_singleton]
--/
-/-
-theorem singleton_ne_empty (a : X) : '{a} ≠ ∅ :=
-begin
-  intro H,
-  apply not_mem_empty a,
-  rewrite -H,
-  apply mem_insert
-end
--/
-
-end set