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/- | |
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved. | |
Released under Apache 2.0 license as described in the file LICENSE. | |
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn | |
-/ | |
import model_theory.basic | |
/-! | |
# Language Maps | |
Maps between first-order languages in the style of the | |
[Flypitch project](https://flypitch.github.io/), as well as several important maps between | |
structures. | |
## Main Definitions | |
* A `first_order.language.Lhom`, denoted `L →ᴸ L'`, is a map between languages, sending the symbols | |
of one to symbols of the same kind and arity in the other. | |
* A `first_order.language.Lequiv`, denoted `L ≃ᴸ L'`, is an invertible language homomorphism. | |
* `first_order.language.with_constants` is defined so that if `M` is an `L.Structure` and | |
`A : set M`, `L.with_constants A`, denoted `L[[A]]`, is a language which adds constant symbols for | |
elements of `A` to `L`. | |
## References | |
For the Flypitch project: | |
- [J. Han, F. van Doorn, *A formal proof of the independence of the continuum hypothesis*] | |
[flypitch_cpp] | |
- [J. Han, F. van Doorn, *A formalization of forcing and the unprovability of | |
the continuum hypothesis*][flypitch_itp] | |
-/ | |
universes u v u' v' w w' | |
namespace first_order | |
namespace language | |
open Structure cardinal | |
open_locale cardinal | |
variables (L : language.{u v}) (L' : language.{u' v'}) {M : Type w} [L.Structure M] | |
/-- A language homomorphism maps the symbols of one language to symbols of another. -/ | |
structure Lhom := | |
(on_function : ∀{n}, L.functions n → L'.functions n) | |
(on_relation : ∀{n}, L.relations n → L'.relations n) | |
infix ` →ᴸ `:10 := Lhom -- \^L | |
variables {L L'} | |
namespace Lhom | |
/-- Defines a map between languages defined with `language.mk₂`. -/ | |
protected def mk₂ {c f₁ f₂ : Type u} {r₁ r₂ : Type v} | |
(φ₀ : c → L'.constants) (φ₁ : f₁ → L'.functions 1) (φ₂ : f₂ → L'.functions 2) | |
(φ₁' : r₁ → L'.relations 1) (φ₂' : r₂ → L'.relations 2) : | |
language.mk₂ c f₁ f₂ r₁ r₂ →ᴸ L' := | |
⟨λ n, nat.cases_on n φ₀ (λ n, nat.cases_on n φ₁ (λ n, nat.cases_on n φ₂ (λ _, pempty.elim))), | |
λ n, nat.cases_on n pempty.elim | |
(λ n, nat.cases_on n φ₁' (λ n, nat.cases_on n φ₂' (λ _, pempty.elim)))⟩ | |
variables (ϕ : L →ᴸ L') | |
/-- Pulls a structure back along a language map. -/ | |
def reduct (M : Type*) [L'.Structure M] : L.Structure M := | |
{ fun_map := λ n f xs, fun_map (ϕ.on_function f) xs, | |
rel_map := λ n r xs, rel_map (ϕ.on_relation r) xs } | |
/-- The identity language homomorphism. -/ | |
@[simps] protected def id (L : language) : L →ᴸ L := | |
⟨λn, id, λ n, id⟩ | |
instance : inhabited (L →ᴸ L) := ⟨Lhom.id L⟩ | |
/-- The inclusion of the left factor into the sum of two languages. -/ | |
@[simps] protected def sum_inl : L →ᴸ L.sum L' := | |
⟨λn, sum.inl, λ n, sum.inl⟩ | |
/-- The inclusion of the right factor into the sum of two languages. -/ | |
@[simps] protected def sum_inr : L' →ᴸ L.sum L' := | |
⟨λn, sum.inr, λ n, sum.inr⟩ | |
variables (L L') | |
/-- The inclusion of an empty language into any other language. -/ | |
@[simps] protected def of_is_empty [L.is_algebraic] [L.is_relational] : L →ᴸ L' := | |
⟨λ n, (is_relational.empty_functions n).elim, λ n, (is_algebraic.empty_relations n).elim⟩ | |
variables {L L'} {L'' : language} | |
@[ext] protected lemma funext {F G : L →ᴸ L'} (h_fun : F.on_function = G.on_function ) | |
(h_rel : F.on_relation = G.on_relation ) : F = G := | |
by {cases F with Ff Fr, cases G with Gf Gr, simp only *, exact and.intro h_fun h_rel} | |
instance [L.is_algebraic] [L.is_relational] : unique (L →ᴸ L') := | |
⟨⟨Lhom.of_is_empty L L'⟩, λ _, Lhom.funext (subsingleton.elim _ _) (subsingleton.elim _ _)⟩ | |
lemma mk₂_funext {c f₁ f₂ : Type u} {r₁ r₂ : Type v} {F G : language.mk₂ c f₁ f₂ r₁ r₂ →ᴸ L'} | |
(h0 : ∀ (c : (language.mk₂ c f₁ f₂ r₁ r₂).constants), F.on_function c = G.on_function c) | |
(h1 : ∀ (f : (language.mk₂ c f₁ f₂ r₁ r₂).functions 1), F.on_function f = G.on_function f) | |
(h2 : ∀ (f : (language.mk₂ c f₁ f₂ r₁ r₂).functions 2), F.on_function f = G.on_function f) | |
(h1' : ∀ (r : (language.mk₂ c f₁ f₂ r₁ r₂).relations 1), F.on_relation r = G.on_relation r) | |
(h2' : ∀ (r : (language.mk₂ c f₁ f₂ r₁ r₂).relations 2), F.on_relation r = G.on_relation r) : | |
F = G := | |
Lhom.funext (funext (λ n, nat.cases_on n (funext h0) (λ n, nat.cases_on n (funext h1) | |
(λ n, nat.cases_on n (funext h2) (λ n, funext (λ f, pempty.elim f)))))) | |
(funext (λ n, nat.cases_on n (funext (λ r, pempty.elim r)) (λ n, nat.cases_on n (funext h1') | |
(λ n, nat.cases_on n (funext h2') (λ n, funext (λ r, pempty.elim r)))))) | |
/-- The composition of two language homomorphisms. -/ | |
@[simps] def comp (g : L' →ᴸ L'') (f : L →ᴸ L') : L →ᴸ L'' := | |
⟨λ n F, g.1 (f.1 F), λ _ R, g.2 (f.2 R)⟩ | |
local infix ` ∘ `:60 := Lhom.comp | |
@[simp] lemma id_comp (F : L →ᴸ L') : (Lhom.id L') ∘ F = F := | |
by {cases F, refl} | |
@[simp] lemma comp_id (F : L →ᴸ L') : F ∘ (Lhom.id L) = F := | |
by {cases F, refl} | |
lemma comp_assoc {L3 : language} (F: L'' →ᴸ L3) (G : L' →ᴸ L'') (H : L →ᴸ L') : | |
(F ∘ G) ∘ H = F ∘ (G ∘ H) := | |
rfl | |
section sum_elim | |
variables (ψ : L'' →ᴸ L') | |
/-- A language map defined on two factors of a sum. -/ | |
@[simps] protected def sum_elim : L.sum L'' →ᴸ L' := | |
{ on_function := λ n, sum.elim (λ f, ϕ.on_function f) (λ f, ψ.on_function f), | |
on_relation := λ n, sum.elim (λ f, ϕ.on_relation f) (λ f, ψ.on_relation f) } | |
lemma sum_elim_comp_inl (ψ : L'' →ᴸ L') : | |
(ϕ.sum_elim ψ) ∘ Lhom.sum_inl = ϕ := | |
Lhom.funext (funext (λ _, rfl)) (funext (λ _, rfl)) | |
lemma sum_elim_comp_inr (ψ : L'' →ᴸ L') : | |
(ϕ.sum_elim ψ) ∘ Lhom.sum_inr = ψ := | |
Lhom.funext (funext (λ _, rfl)) (funext (λ _, rfl)) | |
theorem sum_elim_inl_inr : | |
(Lhom.sum_inl).sum_elim (Lhom.sum_inr) = Lhom.id (L.sum L') := | |
Lhom.funext (funext (λ _, sum.elim_inl_inr)) (funext (λ _, sum.elim_inl_inr)) | |
theorem comp_sum_elim {L3 : language} (θ : L' →ᴸ L3) : | |
θ ∘ (ϕ.sum_elim ψ) = (θ ∘ ϕ).sum_elim (θ ∘ ψ) := | |
Lhom.funext (funext (λ n, sum.comp_elim _ _ _)) (funext (λ n, sum.comp_elim _ _ _)) | |
end sum_elim | |
section sum_map | |
variables {L₁ L₂ : language} (ψ : L₁ →ᴸ L₂) | |
/-- The map between two sum-languages induced by maps on the two factors. -/ | |
@[simps] def sum_map : L.sum L₁ →ᴸ L'.sum L₂ := | |
{ on_function := λ n, sum.map (λ f, ϕ.on_function f) (λ f, ψ.on_function f), | |
on_relation := λ n, sum.map (λ f, ϕ.on_relation f) (λ f, ψ.on_relation f) } | |
@[simp] lemma sum_map_comp_inl : | |
(ϕ.sum_map ψ) ∘ Lhom.sum_inl = Lhom.sum_inl ∘ ϕ := | |
Lhom.funext (funext (λ _, rfl)) (funext (λ _, rfl)) | |
@[simp] lemma sum_map_comp_inr : | |
(ϕ.sum_map ψ) ∘ Lhom.sum_inr = Lhom.sum_inr ∘ ψ := | |
Lhom.funext (funext (λ _, rfl)) (funext (λ _, rfl)) | |
end sum_map | |
/-- A language homomorphism is injective when all the maps between symbol types are. -/ | |
protected structure injective : Prop := | |
(on_function {n} : function.injective (on_function ϕ : L.functions n → L'.functions n)) | |
(on_relation {n} : function.injective (on_relation ϕ : L.relations n → L'.relations n)) | |
/-- A language homomorphism is an expansion on a structure if it commutes with the interpretation of | |
all symbols on that structure. -/ | |
class is_expansion_on (M : Type*) [L.Structure M] [L'.Structure M] : Prop := | |
(map_on_function : ∀ {n} (f : L.functions n) (x : fin n → M), | |
fun_map (ϕ.on_function f) x = fun_map f x) | |
(map_on_relation : ∀ {n} (R : L.relations n) (x : fin n → M), | |
rel_map (ϕ.on_relation R) x = rel_map R x) | |
@[simp] lemma map_on_function {M : Type*} | |
[L.Structure M] [L'.Structure M] [ϕ.is_expansion_on M] | |
{n} (f : L.functions n) (x : fin n → M) : | |
fun_map (ϕ.on_function f) x = fun_map f x := | |
is_expansion_on.map_on_function f x | |
@[simp] lemma map_on_relation {M : Type*} | |
[L.Structure M] [L'.Structure M] [ϕ.is_expansion_on M] | |
{n} (R : L.relations n) (x : fin n → M) : | |
rel_map (ϕ.on_relation R) x = rel_map R x := | |
is_expansion_on.map_on_relation R x | |
instance id_is_expansion_on (M : Type*) [L.Structure M] : is_expansion_on (Lhom.id L) M := | |
⟨λ _ _ _, rfl, λ _ _ _, rfl⟩ | |
instance of_is_empty_is_expansion_on (M : Type*) [L.Structure M] [L'.Structure M] | |
[L.is_algebraic] [L.is_relational] : | |
is_expansion_on (Lhom.of_is_empty L L') M := | |
⟨λ n, (is_relational.empty_functions n).elim, λ n, (is_algebraic.empty_relations n).elim⟩ | |
instance sum_elim_is_expansion_on {L'' : language} (ψ : L'' →ᴸ L') (M : Type*) | |
[L.Structure M] [L'.Structure M] [L''.Structure M] | |
[ϕ.is_expansion_on M] [ψ.is_expansion_on M] : | |
(ϕ.sum_elim ψ).is_expansion_on M := | |
⟨λ _ f _, sum.cases_on f (by simp) (by simp), λ _ R _, sum.cases_on R (by simp) (by simp)⟩ | |
instance sum_map_is_expansion_on {L₁ L₂ : language} (ψ : L₁ →ᴸ L₂) (M : Type*) | |
[L.Structure M] [L'.Structure M] [L₁.Structure M] [L₂.Structure M] | |
[ϕ.is_expansion_on M] [ψ.is_expansion_on M] : | |
(ϕ.sum_map ψ).is_expansion_on M := | |
⟨λ _ f _, sum.cases_on f (by simp) (by simp), λ _ R _, sum.cases_on R (by simp) (by simp)⟩ | |
instance sum_inl_is_expansion_on (M : Type*) | |
[L.Structure M] [L'.Structure M] : | |
(Lhom.sum_inl : L →ᴸ L.sum L').is_expansion_on M := | |
⟨λ _ f _, rfl, λ _ R _, rfl⟩ | |
instance sum_inr_is_expansion_on (M : Type*) | |
[L.Structure M] [L'.Structure M] : | |
(Lhom.sum_inr : L' →ᴸ L.sum L').is_expansion_on M := | |
⟨λ _ f _, rfl, λ _ R _, rfl⟩ | |
@[simp] lemma fun_map_sum_inl [(L.sum L').Structure M] | |
[(Lhom.sum_inl : L →ᴸ L.sum L').is_expansion_on M] | |
{n} {f : L.functions n} {x : fin n → M} : | |
@fun_map (L.sum L') M _ n (sum.inl f) x = fun_map f x := | |
(Lhom.sum_inl : L →ᴸ L.sum L').map_on_function f x | |
@[simp] lemma fun_map_sum_inr [(L'.sum L).Structure M] | |
[(Lhom.sum_inr : L →ᴸ L'.sum L).is_expansion_on M] | |
{n} {f : L.functions n} {x : fin n → M} : | |
@fun_map (L'.sum L) M _ n (sum.inr f) x = fun_map f x := | |
(Lhom.sum_inr : L →ᴸ L'.sum L).map_on_function f x | |
@[priority 100] instance is_expansion_on_reduct (ϕ : L →ᴸ L') (M : Type*) [L'.Structure M] : | |
@is_expansion_on L L' ϕ M (ϕ.reduct M) _ := | |
begin | |
letI := ϕ.reduct M, | |
exact ⟨λ _ f _, rfl, λ _ R _, rfl⟩, | |
end | |
end Lhom | |
/-- A language equivalence maps the symbols of one language to symbols of another bijectively. -/ | |
structure Lequiv (L L' : language) := | |
(to_Lhom : L →ᴸ L') | |
(inv_Lhom : L' →ᴸ L) | |
(left_inv : inv_Lhom.comp to_Lhom = Lhom.id L) | |
(right_inv : to_Lhom.comp inv_Lhom = Lhom.id L') | |
infix ` ≃ᴸ `:10 := Lequiv -- \^L | |
namespace Lequiv | |
variable (L) | |
/-- The identity equivalence from a first-order language to itself. -/ | |
@[simps] protected def refl : L ≃ᴸ L := | |
⟨Lhom.id L, Lhom.id L, Lhom.id_comp _, Lhom.id_comp _⟩ | |
variable {L} | |
instance : inhabited (L ≃ᴸ L) := ⟨Lequiv.refl L⟩ | |
variables {L'' : language} (e' : L' ≃ᴸ L'') (e : L ≃ᴸ L') | |
/-- The inverse of an equivalence of first-order languages. -/ | |
@[simps] protected def symm : L' ≃ᴸ L := | |
⟨e.inv_Lhom, e.to_Lhom, e.right_inv, e.left_inv⟩ | |
/-- The composition of equivalences of first-order languages. -/ | |
@[simps, trans] protected def trans (e : L ≃ᴸ L') (e' : L' ≃ᴸ L'') : L ≃ᴸ L'' := | |
⟨e'.to_Lhom.comp e.to_Lhom, e.inv_Lhom.comp e'.inv_Lhom, | |
by rw [Lhom.comp_assoc, ← Lhom.comp_assoc e'.inv_Lhom, e'.left_inv, Lhom.id_comp, e.left_inv], | |
by rw [Lhom.comp_assoc, ← Lhom.comp_assoc e.to_Lhom, e.right_inv, Lhom.id_comp, e'.right_inv]⟩ | |
end Lequiv | |
section constants_on | |
variables (α : Type u') | |
/-- A language with constants indexed by a type. -/ | |
@[simp] def constants_on : language.{u' 0} := | |
language.mk₂ α pempty pempty pempty pempty | |
variables {α} | |
lemma constants_on_constants : (constants_on α).constants = α := rfl | |
instance is_algebraic_constants_on : is_algebraic (constants_on α) := | |
language.is_algebraic_mk₂ | |
instance is_relational_constants_on [ie : is_empty α] : is_relational (constants_on α) := | |
language.is_relational_mk₂ | |
instance is_empty_functions_constants_on_succ {n : ℕ} : | |
is_empty ((constants_on α).functions (n + 1)) := | |
nat.cases_on n pempty.is_empty (λ n, nat.cases_on n pempty.is_empty (λ _, pempty.is_empty)) | |
lemma card_constants_on : (constants_on α).card = # α := | |
by simp | |
/-- Gives a `constants_on α` structure to a type by assigning each constant a value. -/ | |
def constants_on.Structure (f : α → M) : (constants_on α).Structure M := | |
Structure.mk₂ f pempty.elim pempty.elim pempty.elim pempty.elim | |
variables {β : Type v'} | |
/-- A map between index types induces a map between constant languages. -/ | |
def Lhom.constants_on_map (f : α → β) : (constants_on α) →ᴸ (constants_on β) := | |
Lhom.mk₂ f pempty.elim pempty.elim pempty.elim pempty.elim | |
lemma constants_on_map_is_expansion_on {f : α → β} {fα : α → M} {fβ : β → M} | |
(h : fβ ∘ f = fα) : | |
@Lhom.is_expansion_on _ _ (Lhom.constants_on_map f) M | |
(constants_on.Structure fα) (constants_on.Structure fβ) := | |
begin | |
letI := constants_on.Structure fα, | |
letI := constants_on.Structure fβ, | |
exact ⟨λ n, nat.cases_on n (λ F x, (congr_fun h F : _)) (λ n F, is_empty_elim F), | |
λ _ R, is_empty_elim R⟩ | |
end | |
end constants_on | |
section with_constants | |
variable (L) | |
section | |
variables (α : Type w') | |
/-- Extends a language with a constant for each element of a parameter set in `M`. -/ | |
def with_constants : language.{(max u w') v} := L.sum (constants_on α) | |
localized "notation L`[[`:95 α`]]`:90 := L.with_constants α" in first_order | |
@[simp] lemma card_with_constants : | |
(L[[α]]).card = cardinal.lift.{w'} L.card + cardinal.lift.{max u v} (# α) := | |
by rw [with_constants, card_sum, card_constants_on] | |
/-- The language map adding constants. -/ | |
@[simps] def Lhom_with_constants : L →ᴸ L[[α]] := Lhom.sum_inl | |
variables {α} | |
/-- The constant symbol indexed by a particular element. -/ | |
protected def con (a : α) : L[[α]].constants := sum.inr a | |
variables {L} (α) | |
/-- Adds constants to a language map. -/ | |
def Lhom.add_constants {L' : language} (φ : L →ᴸ L') : | |
L[[α]] →ᴸ L'[[α]] := φ.sum_map (Lhom.id _) | |
instance params_Structure (A : set α) : (constants_on A).Structure α := constants_on.Structure coe | |
variables (L) (α) | |
/-- The language map removing an empty constant set. -/ | |
@[simps] def Lequiv.add_empty_constants [ie : is_empty α] : L ≃ᴸ L[[α]] := | |
{ to_Lhom := Lhom_with_constants L α, | |
inv_Lhom := Lhom.sum_elim (Lhom.id L) (Lhom.of_is_empty (constants_on α) L), | |
left_inv := by rw [Lhom_with_constants, Lhom.sum_elim_comp_inl], | |
right_inv := by { simp only [Lhom.comp_sum_elim, Lhom_with_constants, Lhom.comp_id], | |
exact trans (congr rfl (subsingleton.elim _ _)) Lhom.sum_elim_inl_inr } } | |
variables {α} {β : Type*} | |
@[simp] lemma with_constants_fun_map_sum_inl [L[[α]].Structure M] | |
[(Lhom_with_constants L α).is_expansion_on M] | |
{n} {f : L.functions n} {x : fin n → M} : | |
@fun_map (L[[α]]) M _ n (sum.inl f) x = fun_map f x := | |
(Lhom_with_constants L α).map_on_function f x | |
@[simp] lemma with_constants_rel_map_sum_inl [L[[α]].Structure M] | |
[(Lhom_with_constants L α).is_expansion_on M] | |
{n} {R : L.relations n} {x : fin n → M} : | |
@rel_map (L[[α]]) M _ n (sum.inl R) x = rel_map R x := | |
(Lhom_with_constants L α).map_on_relation R x | |
/-- The language map extending the constant set. -/ | |
def Lhom_with_constants_map (f : α → β) : L[[α]] →ᴸ L[[β]] := | |
Lhom.sum_map (Lhom.id L) (Lhom.constants_on_map f) | |
@[simp] lemma Lhom.map_constants_comp_sum_inl {f : α → β} : | |
(L.Lhom_with_constants_map f).comp (Lhom.sum_inl) = L.Lhom_with_constants β := | |
by ext n f R; refl | |
end | |
open_locale first_order | |
instance constants_on_self_Structure : (constants_on M).Structure M := | |
constants_on.Structure id | |
instance with_constants_self_Structure : L[[M]].Structure M := | |
language.sum_Structure _ _ M | |
instance with_constants_self_expansion : (Lhom_with_constants L M).is_expansion_on M := | |
⟨λ _ _ _, rfl, λ _ _ _, rfl⟩ | |
variables (α : Type*) [(constants_on α).Structure M] | |
instance with_constants_Structure : L[[α]].Structure M := | |
language.sum_Structure _ _ _ | |
instance with_constants_expansion : (L.Lhom_with_constants α).is_expansion_on M := | |
⟨λ _ _ _, rfl, λ _ _ _, rfl⟩ | |
instance add_empty_constants_is_expansion_on' : | |
(Lequiv.add_empty_constants L (∅ : set M)).to_Lhom.is_expansion_on M := | |
L.with_constants_expansion _ | |
instance add_empty_constants_symm_is_expansion_on : | |
(Lequiv.add_empty_constants L (∅ : set M)).symm.to_Lhom.is_expansion_on M := | |
Lhom.sum_elim_is_expansion_on _ _ _ | |
instance add_constants_expansion {L' : language} [L'.Structure M] (φ : L →ᴸ L') | |
[φ.is_expansion_on M] : | |
(φ.add_constants α).is_expansion_on M := | |
Lhom.sum_map_is_expansion_on _ _ M | |
@[simp] lemma with_constants_fun_map_sum_inr {a : α} {x : fin 0 → M} : | |
@fun_map (L[[α]]) M _ 0 (sum.inr a : L[[α]].functions 0) x = L.con a := | |
begin | |
rw unique.eq_default x, | |
exact (Lhom.sum_inr : (constants_on α) →ᴸ L.sum _).map_on_function _ _, | |
end | |
variables {α} (A : set M) | |
@[simp] lemma coe_con {a : A} : ((L.con a) : M) = a := rfl | |
variables {A} {B : set M} (h : A ⊆ B) | |
instance constants_on_map_inclusion_is_expansion_on : | |
(Lhom.constants_on_map (set.inclusion h)).is_expansion_on M := | |
constants_on_map_is_expansion_on rfl | |
instance map_constants_inclusion_is_expansion_on : | |
(L.Lhom_with_constants_map (set.inclusion h)).is_expansion_on M := | |
Lhom.sum_map_is_expansion_on _ _ _ | |
end with_constants | |
end language | |
end first_order | |