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change terminology set -> property

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Jeremy Avigad 2017-06-26 17:39:40 +01:00
parent c74779959a
commit 1cb3e5c658
3 changed files with 222 additions and 225 deletions
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78
algebra/subgroup.hlean
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@ -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?
definition kernel [constructor] (φ : G₁ →g G₂) : set G₁ := { g | trunctype.mk (φ g = 1) _}
-- TODO: maybe define this in more generality for pointed propertys?
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 subgroup_mul_mem (of_mem_set_of p) (of_mem_set_of q),
apply mem_property_of,
apply subgroup_mul_mem (of_mem_property_of p) (of_mem_property_of q),
end)
(begin
intros h p,
apply mem_set_of,
apply subgroup_inv_mem (of_mem_set_of p)
apply mem_property_of,
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]
{S : set H} [is_subgroup H S] {T : set K} [is_subgroup K T]
variables {G H K : Group} {R : property G} [is_subgroup G R]
{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]
{S : set H} [is_subgroup H S]
{R : property G} [is_subgroup G R]
{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]
{S : set H} [subgroupS : is_subgroup H S]
definition subgroup_functor_mul {G H : AbGroup} {R : property G} [subgroupR : is_subgroup G R]
{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

183
property.hlean Normal file
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@ -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

186
set.hlean
View file

@ -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