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/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import data.set_like.basic
import logic.equiv.fintype
import model_theory.semantics
/-!
# Definable Sets
This file defines what it means for a set over a first-order structure to be definable.
## Main Definitions
* `set.definable` is defined so that `A.definable L s` indicates that the
set `s` of a finite cartesian power of `M` is definable with parameters in `A`.
* `set.definable₁` is defined so that `A.definable₁ L s` indicates that
`(s : set M)` is definable with parameters in `A`.
* `set.definable₂` is defined so that `A.definable₂ L s` indicates that
`(s : set (M × M))` is definable with parameters in `A`.
* A `first_order.language.definable_set` is defined so that `L.definable_set A α` is the boolean
algebra of subsets of `α → M` defined by formulas with parameters in `A`.
## Main Results
* `L.definable_set A α` forms a `boolean_algebra`
* `set.definable.image_comp` shows that definability is closed under projections in finite
dimensions.
-/
universes u v w
namespace set
variables {M : Type w} (A : set M) (L : first_order.language.{u v}) [L.Structure M]
open_locale first_order
open first_order.language first_order.language.Structure
variables {α : Type*} {β : Type*}
/-- A subset of a finite Cartesian product of a structure is definable over a set `A` when
membership in the set is given by a first-order formula with parameters from `A`. -/
def definable (s : set (α → M)) : Prop :=
∃ (φ : L[[A]].formula α), s = set_of φ.realize
variables {L} {A} {B : set M} {s : set (α → M)}
lemma definable.map_expansion {L' : first_order.language} [L'.Structure M] (h : A.definable L s)
(φ : L →ᴸ L') [φ.is_expansion_on M] :
A.definable L' s :=
begin
obtain ⟨ψ, rfl⟩ := h,
refine ⟨(φ.add_constants A).on_formula ψ, _⟩,
ext x,
simp only [mem_set_of_eq, Lhom.realize_on_formula],
end
lemma empty_definable_iff :
(∅ : set M).definable L s ↔ ∃ (φ : L.formula α), s = set_of φ.realize :=
begin
rw [definable, equiv.exists_congr_left (Lequiv.add_empty_constants L (∅ : set M)).on_formula],
simp,
end
lemma definable_iff_empty_definable_with_params :
A.definable L s ↔ (∅ : set M).definable (L[[A]]) s :=
empty_definable_iff.symm
lemma definable.mono (hAs : A.definable L s) (hAB : A ⊆ B) :
B.definable L s :=
begin
rw [definable_iff_empty_definable_with_params] at *,
exact hAs.map_expansion (L.Lhom_with_constants_map (set.inclusion hAB)),
end
@[simp]
lemma definable_empty : A.definable L (∅ : set (α → M)) :=
⟨⊥, by {ext, simp} ⟩
@[simp]
lemma definable_univ : A.definable L (univ : set (α → M)) :=
⟨⊤, by {ext, simp} ⟩
@[simp]
lemma definable.inter {f g : set (α → M)} (hf : A.definable L f) (hg : A.definable L g) :
A.definable L (f ∩ g) :=
begin
rcases hf with ⟨φ, rfl⟩,
rcases hg with ⟨θ, rfl⟩,
refine ⟨φ ⊓ θ, _⟩,
ext,
simp,
end
@[simp]
lemma definable.union {f g : set (α → M)} (hf : A.definable L f) (hg : A.definable L g) :
A.definable L (f ∪ g) :=
begin
rcases hf with ⟨φ, hφ⟩,
rcases hg with ⟨θ, hθ⟩,
refine ⟨φ ⊔ θ, _⟩,
ext,
rw [hφ, hθ, mem_set_of_eq, formula.realize_sup, mem_union_eq, mem_set_of_eq,
mem_set_of_eq],
end
lemma definable_finset_inf {ι : Type*} {f : Π (i : ι), set (α → M)}
(hf : ∀ i, A.definable L (f i)) (s : finset ι) :
A.definable L (s.inf f) :=
begin
classical,
refine finset.induction definable_univ (λ i s is h, _) s,
rw finset.inf_insert,
exact (hf i).inter h,
end
lemma definable_finset_sup {ι : Type*} {f : Π (i : ι), set (α → M)}
(hf : ∀ i, A.definable L (f i)) (s : finset ι) :
A.definable L (s.sup f) :=
begin
classical,
refine finset.induction definable_empty (λ i s is h, _) s,
rw finset.sup_insert,
exact (hf i).union h,
end
lemma definable_finset_bInter {ι : Type*} {f : Π (i : ι), set (α → M)}
(hf : ∀ i, A.definable L (f i)) (s : finset ι) :
A.definable L (⋂ i ∈ s, f i) :=
begin
rw ← finset.inf_set_eq_bInter,
exact definable_finset_inf hf s,
end
lemma definable_finset_bUnion {ι : Type*} {f : Π (i : ι), set (α → M)}
(hf : ∀ i, A.definable L (f i)) (s : finset ι) :
A.definable L (⋃ i ∈ s, f i) :=
begin
rw ← finset.sup_set_eq_bUnion,
exact definable_finset_sup hf s,
end
@[simp]
lemma definable.compl {s : set (α → M)} (hf : A.definable L s) :
A.definable L sᶜ :=
begin
rcases hf with ⟨φ, hφ⟩,
refine ⟨φ.not, _⟩,
rw hφ,
refl,
end
@[simp]
lemma definable.sdiff {s t : set (α → M)} (hs : A.definable L s)
(ht : A.definable L t) :
A.definable L (s \ t) :=
hs.inter ht.compl
lemma definable.preimage_comp (f : α → β) {s : set (α → M)}
(h : A.definable L s) :
A.definable L ((λ g : β → M, g ∘ f) ⁻¹' s) :=
begin
obtain ⟨φ, rfl⟩ := h,
refine ⟨(φ.relabel f), _⟩,
ext,
simp only [set.preimage_set_of_eq, mem_set_of_eq, formula.realize_relabel],
end
lemma definable.image_comp_equiv {s : set (β → M)}
(h : A.definable L s) (f : α ≃ β) :
A.definable L ((λ g : β → M, g ∘ f) '' s) :=
begin
refine (congr rfl _).mp (h.preimage_comp f.symm),
rw image_eq_preimage_of_inverse,
{ intro i,
ext b,
simp only [function.comp_app, equiv.apply_symm_apply], },
{ intro i,
ext a,
simp }
end
lemma fin.coe_cast_add_zero {m : ℕ} : (fin.cast_add 0 : fin m → fin (m + 0)) = id :=
funext (λ _, fin.ext rfl)
/-- This lemma is only intended as a helper for `definable.image_comp. -/
lemma definable.image_comp_sum_inl_fin (m : ℕ) {s : set ((α ⊕ fin m) → M)}
(h : A.definable L s) :
A.definable L ((λ g : (α ⊕ fin m) → M, g ∘ sum.inl) '' s) :=
begin
obtain ⟨φ, rfl⟩ := h,
refine ⟨(bounded_formula.relabel id φ).exs, _⟩,
ext x,
simp only [set.mem_image, mem_set_of_eq, bounded_formula.realize_exs,
bounded_formula.realize_relabel, function.comp.right_id, fin.coe_cast_add_zero],
split,
{ rintro ⟨y, hy, rfl⟩,
exact ⟨y ∘ sum.inr,
(congr (congr rfl (sum.elim_comp_inl_inr y).symm) (funext fin_zero_elim)).mp hy⟩ },
{ rintro ⟨y, hy⟩,
exact ⟨sum.elim x y, (congr rfl (funext fin_zero_elim)).mp hy, sum.elim_comp_inl _ _⟩, },
end
/-- Shows that definability is closed under finite projections. -/
lemma definable.image_comp_embedding {s : set (β → M)} (h : A.definable L s)
(f : α ↪ β) [fintype β] :
A.definable L ((λ g : β → M, g ∘ f) '' s) :=
begin
classical,
refine (congr rfl (ext (λ x, _))).mp (((h.image_comp_equiv
(equiv.set.sum_compl (range f))).image_comp_equiv (equiv.sum_congr
(equiv.of_injective f f.injective) (fintype.equiv_fin _).symm)).image_comp_sum_inl_fin _),
simp only [mem_preimage, mem_image, exists_exists_and_eq_and],
refine exists_congr (λ y, and_congr_right (λ ys, eq.congr_left (funext (λ a, _)))),
simp,
end
/-- Shows that definability is closed under finite projections. -/
lemma definable.image_comp {s : set (β → M)} (h : A.definable L s)
(f : α → β) [fintype α] [fintype β] :
A.definable L ((λ g : β → M, g ∘ f) '' s) :=
begin
classical,
have h := (((h.image_comp_equiv (equiv.set.sum_compl (range f))).image_comp_equiv
(equiv.sum_congr (_root_.equiv.refl _)
(fintype.equiv_fin _).symm)).image_comp_sum_inl_fin _).preimage_comp (range_splitting f),
have h' : A.definable L ({ x : α → M |
∀ a, x a = x (range_splitting f (range_factorization f a))}),
{ have h' : ∀ a, A.definable L {x : α → M | x a =
x (range_splitting f (range_factorization f a))},
{ refine λ a, ⟨(var a).equal (var (range_splitting f (range_factorization f a))), ext _⟩,
simp, },
refine (congr rfl (ext _)).mp (definable_finset_bInter h' finset.univ),
simp },
refine (congr rfl (ext (λ x, _))).mp (h.inter h'),
simp only [equiv.coe_trans, mem_inter_eq, mem_preimage, mem_image,
exists_exists_and_eq_and, mem_set_of_eq],
split,
{ rintro ⟨⟨y, ys, hy⟩, hx⟩,
refine ⟨y, ys, _⟩,
ext a,
rw [hx a, ← function.comp_apply x, ← hy],
simp, },
{ rintro ⟨y, ys, rfl⟩,
refine ⟨⟨y, ys, _⟩, λ a, _⟩,
{ ext,
simp [set.apply_range_splitting f] },
{ rw [function.comp_apply, function.comp_apply, apply_range_splitting f,
range_factorization_coe], }}
end
variables (L) {M} (A)
/-- A 1-dimensional version of `definable`, for `set M`. -/
def definable₁ (s : set M) : Prop := A.definable L { x : fin 1 → M | x 0 ∈ s }
/-- A 2-dimensional version of `definable`, for `set (M × M)`. -/
def definable₂ (s : set (M × M)) : Prop := A.definable L { x : fin 2 → M | (x 0, x 1) ∈ s }
end set
namespace first_order
namespace language
open set
variables (L : first_order.language.{u v}) {M : Type w} [L.Structure M] (A : set M) (α : Type*)
/-- Definable sets are subsets of finite Cartesian products of a structure such that membership is
given by a first-order formula. -/
def definable_set := { s : set (α → M) // A.definable L s}
namespace definable_set
variables {L} {A} {α}
instance : has_top (L.definable_set A α) := ⟨⟨⊤, definable_univ⟩⟩
instance : has_bot (L.definable_set A α) := ⟨⟨⊥, definable_empty⟩⟩
instance : inhabited (L.definable_set A α) := ⟨⊥⟩
instance : set_like (L.definable_set A α) (α → M) :=
{ coe := subtype.val,
coe_injective' := subtype.val_injective }
@[simp]
lemma mem_top {x : α → M} : x ∈ (⊤ : L.definable_set A α) := mem_univ x
@[simp]
lemma coe_top : ((⊤ : L.definable_set A α) : set (α → M)) = ⊤ := rfl
@[simp]
lemma not_mem_bot {x : α → M} : ¬ x ∈ (⊥ : L.definable_set A α) := not_mem_empty x
@[simp]
lemma coe_bot : ((⊥ : L.definable_set A α) : set (α → M)) = ⊥ := rfl
instance : lattice (L.definable_set A α) :=
subtype.lattice (λ _ _, definable.union) (λ _ _, definable.inter)
lemma le_iff {s t : L.definable_set A α} : s ≤ t ↔ (s : set (α → M)) ≤ (t : set (α → M)) := iff.rfl
@[simp]
lemma coe_sup {s t : L.definable_set A α} : ((s ⊔ t : L.definable_set A α) : set (α → M)) = s ∪ t :=
rfl
@[simp]
lemma mem_sup {s t : L.definable_set A α} {x : α → M} : x ∈ s ⊔ t ↔ x ∈ s ∨ x ∈ t := iff.rfl
@[simp]
lemma coe_inf {s t : L.definable_set A α} : ((s ⊓ t : L.definable_set A α) : set (α → M)) = s ∩ t :=
rfl
@[simp]
lemma mem_inf {s t : L.definable_set A α} {x : α → M} : x ∈ s ⊓ t ↔ x ∈ s ∧ x ∈ t := iff.rfl
instance : bounded_order (L.definable_set A α) :=
{ bot_le := λ s x hx, false.elim hx,
le_top := λ s x hx, mem_univ x,
.. definable_set.has_top,
.. definable_set.has_bot }
instance : distrib_lattice (L.definable_set A α) :=
{ le_sup_inf := begin
intros s t u x,
simp only [and_imp, mem_inter_eq, set_like.mem_coe, coe_sup, coe_inf, mem_union_eq,
subtype.val_eq_coe],
tauto,
end,
.. definable_set.lattice }
/-- The complement of a definable set is also definable. -/
@[reducible] instance : has_compl (L.definable_set A α) :=
⟨λ ⟨s, hs⟩, ⟨sᶜ, hs.compl⟩⟩
@[simp]
lemma mem_compl {s : L.definable_set A α} {x : α → M} : x ∈ sᶜ ↔ ¬ x ∈ s :=
begin
cases s with s hs,
refl,
end
@[simp]
lemma coe_compl {s : L.definable_set A α} : ((sᶜ : L.definable_set A α) : set (α → M)) = sᶜ :=
begin
ext,
simp,
end
instance : boolean_algebra (L.definable_set A α) :=
{ sdiff := λ s t, s ⊓ tᶜ,
sdiff_eq := λ s t, rfl,
inf_compl_le_bot := λ ⟨s, hs⟩, by simp [le_iff],
top_le_sup_compl := λ ⟨s, hs⟩, by simp [le_iff],
.. definable_set.has_compl,
.. definable_set.bounded_order,
.. definable_set.distrib_lattice }
end definable_set
end language
end first_order