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import Mathlib
import Nominal.Permutations
import Nominal.Support
/-!
# Definitions and Properties for Equivariance
We define equivariance for functions, sets, and binary relations.
-/
variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
/-!
## Definitions
Basic definitions for equivariance.
-/
section Definition
variable {X Y : Type} [PermSet 𝔸 X] [PermSet 𝔸 Y]
/-- A function is equivariant if it is invariant to applying a permutation before or after the function. -/
def Equivariant (f : X → Y) : Prop :=
∀ (x : X) (π : FPerm 𝔸), f (π • x) = π • f x
/-- A set is equivariant if it is closed under permutations. -/
def EquivariantSet (S : Set X) : Prop :=
∀ (x : X) (π : FPerm 𝔸), x ∈ S → π • x ∈ S
/-- A binary relation is equivariant if it is closed under permutations. -/
def EquivariantRelation (R : Rel X Y) : Prop :=
∀ (x : X) (y : Y) (π : FPerm 𝔸), R x y → R (π • x) (π • y)
/-- For equivariant sets, we even have "if and only if". -/
lemma equivariant_set_iff {S : Set X} : EquivariantSet 𝔸 S ↔ ∀ (x : X) (π : FPerm 𝔸), x ∈ S ↔ π • x ∈ S := by
constructor
· intro h x π; constructor
· intro hx; exact h x π hx
· intro hx
apply (h (π • x) π⁻¹) at hx
simp only [inv_smul_smul] at hx
exact hx
· intro h x π hx
exact (h x π).mp hx
/-- For equivariant relations, we even have "if and only if". -/
lemma equivariant_relation_iff {R : Rel X Y} : EquivariantRelation 𝔸 R ↔ ∀ (x : X) (y : Y) (π : FPerm 𝔸), R x y ↔ R (π • x) (π • y) := by
constructor
· intro h x y π; constructor
· intro hxy; exact h x y π hxy
· intro hxy
apply (h (π • x) (π • y) π⁻¹) at hxy
simp only [inv_smul_smul] at hxy
exact hxy
· intro h x y π hxy
exact (h x y π).mp hxy
end Definition
/-!
## Trivial Instances
Some trivial instances of equivariant functions, sets, and relations.
-/
section Trivial
variable {X Y Z : Type} [PermSet 𝔸 X] [PermSet 𝔸 Y] [PermSet 𝔸 Z]
/-- The identity function is equivariant. -/
lemma id_equivariant : Equivariant 𝔸 (id : X → X) := by
simp [Equivariant]
/-- The composition of two equivariant functions is equivariant. -/
lemma comp_equivariant {f : Y → Z} {g : X → Y} : Equivariant 𝔸 f → Equivariant 𝔸 g → Equivariant 𝔸 (f ∘ g) := by
intro hf hg x π
calc
f (g (π • x)) = f (π • g x) := by rw [hg]
_ = π • (f (g x)) := by rw [hf]
/-- The subset relation is equivariant. -/
lemma subset_equivariant : EquivariantRelation 𝔸 (Set.Subset : Set X → Set X → Prop) := by
intro S T π hST x hx
rw [Set.mem_smul_set_iff_inv_smul_mem] at hx
rw [Set.mem_smul_set_iff_inv_smul_mem]
exact hST hx
end Trivial
/-!
## Support
Propositions relating equivariance to support in (FPerm 𝔸)-sets.
-/
section Support
variable {X Y : Type} [PermSet 𝔸 X] [PermSet 𝔸 Y]
namespace Equivariant
/-- Equivariant functions preserve supports. -/
lemma preserve_support {x : X} {f : X → Y} {S : Set 𝔸} : Equivariant 𝔸 f → PermSet.HasSupport 𝔸 x S → PermSet.HasSupport 𝔸 (f x) S := by
intro hequi hsupp π hπ
calc
π • f x = f (π • x) := by rw [← hequi]
_ = f x := by rw [hsupp hπ]
/-- Equivariant injective functions exactly preserve supports. -/
lemma preserve_support_inj {x : X} {f : X → Y} {S : Set 𝔸} : Equivariant 𝔸 f → f.Injective → (PermSet.HasSupport 𝔸 x S ↔ PermSet.HasSupport 𝔸 (f x) S) := by
intro hequi hinj; constructor
· exact preserve_support 𝔸 hequi
· intro hsupp π hπ
by_contra h
have : π • f x ≠ f x := by rw [← hequi]; exact fun a => h (hinj a)
have : π • f x = f x := hsupp hπ
contradiction
end Equivariant
/-- HasSupport is an equivariant relation. -/
lemma PermSet.support_equivariant : EquivariantRelation 𝔸 (HasSupport 𝔸 : X → Set 𝔸 → Prop) := by
intro x S π hS σ hσ
have hfix : (π⁻¹ * σ * π) ∈ fix 𝔸 S := by
intro a ha
calc
(π⁻¹ * σ * π) • a = π⁻¹ • (σ • (π • a)) := rfl
_ = π⁻¹ • (π • a) := by rw [hσ (π • a) (Set.smul_mem_smul_set ha)]
_ = (π⁻¹ * π) • a := rfl
_ = (1 : FPerm 𝔸) • a := by rw [@Group.inv_mul_cancel]
_ = a := rfl
calc
σ • (π • x) = (σ * π) • x := by apply smul_smul
_ = (1 * σ * π) • x := rfl
_ = (π * π⁻¹ * σ * π) • x := by rw [@mul_inv_cancel]
_ = π • ((π⁻¹ * σ * π) • x) := by symm; apply smul_smul
_ = π • x := by rw [hS hfix]
end Support
/-!
## Nominal Sets
Properties about equivariance relating to nominal sets.
-/
section Nominal
variable {X Y : Type} [NominalSet 𝔸 X] [NominalSet 𝔸 Y]
namespace Equivariant
/-- If f is equivariant, then supp (f x) ⊆ supp x. -/
lemma supp_subset {x : X} {f : X → Y} : Equivariant 𝔸 f → NominalSet.supp 𝔸 (f x) ⊆ NominalSet.supp 𝔸 x := by
intro hequi
apply NominalSet.support_subset_supp
· exact preserve_support 𝔸 hequi (NominalSet.supp_is_support 𝔸)
· exact NominalSet.supp_finite 𝔸
/-- If f is equivariant and injective, then supp (f x) = supp x. -/
lemma supp_eq_if_inj {x : X} {f : X → Y} : Equivariant 𝔸 f → Function.Injective f → NominalSet.supp 𝔸 (f x) = NominalSet.supp 𝔸 x := by
intro hequi hinj
simp [NominalSet.supp, preserve_support_inj 𝔸 hequi hinj]
end Equivariant
/-- supp is an equivariant function. -/
lemma NominalSet.supp_equivariant : Equivariant 𝔸 (supp 𝔸 : X → Set 𝔸) := by
intro x π; ext a; constructor
· apply support_subset_supp 𝔸
· apply PermSet.support_equivariant
exact supp_is_support 𝔸
· apply Set.Finite.smul_set
exact supp_finite 𝔸
· intro ha S hS
have hsupp : PermSet.HasSupport 𝔸 x (π⁻¹ • S) := by
rw [show x = π⁻¹ • π • x by exact eq_inv_smul_iff.mpr rfl]
apply PermSet.support_equivariant
exact hS.out.left
rw [Set.mem_smul_set_iff_inv_smul_mem] at ha
suffices π⁻¹ • a ∈ π⁻¹ • S by rw [Set.smul_mem_smul_set_iff] at this; assumption
suffices supp 𝔸 x ⊆ π⁻¹ • S by exact this ha
apply support_subset_supp 𝔸
· exact hsupp
· apply Set.Finite.smul_set
exact hS.out.right
end Nominal
/-!
## Freshness
For propositions of the form `И a, a R x`, where `R` is an equivariant relation, we do not have to differentiate between "for any fresh a" and "for all fresh a".
-/
section Freshness
namespace Equivariance
variable {X : Type} [NominalSet 𝔸 X] {R: Rel 𝔸 X} {x : X}
-- We continue with a circular implication chain argument to show these equivalences.
private lemma for_fresh_iff₁ : EquivariantRelation 𝔸 R → (И a, R a x) → (∃ a, a # x ∧ R a x) := by
intro hequi h
have hinf : ({a | R a x} \ NominalSet.supp 𝔸 x).Infinite := Set.Infinite.diff h (NominalSet.supp_finite 𝔸)
obtain ⟨a, ha⟩ := hinf.nonempty
simp only [Set.mem_diff, Set.mem_setOf_eq] at ha
use a; exact ha.symm
private lemma for_fresh_iff₂ : EquivariantRelation 𝔸 R → (∃ a, a # x ∧ R a x) → (∀ a, a # x → R a x) := by
intro hequi h
obtain ⟨a, ⟨hfresha, ha⟩⟩ := h
intro b hfreshb
have hmem := hequi a x ⸨a, b⸩ ha
rw [PermSet.atom_swap, Equiv.swap_apply_left] at hmem
rw [NominalSet.fresh_swap_fix 𝔸 hfresha hfreshb] at hmem
exact hmem
private lemma for_fresh_iff₃ : EquivariantRelation 𝔸 R → (∀ a, a # x → R a x) → (И a, R a x) := by
intro hequi h
have hsub : {a : 𝔸 | R a x}ᶜ ⊆ NominalSet.supp 𝔸 x := by
rw [Set.compl_subset_comm]
intro a ha
rw [Set.mem_compl_iff] at ha
exact h a ha
simp only [NominalSet.ForFresh]
apply Set.infinite_of_finite_compl
apply Set.Finite.subset (NominalSet.supp_finite 𝔸) hsub
/-- The quantifier in `И a, a R x` is equivalent to "for any fresh `a`". -/
lemma for_fresh_iff_exists : EquivariantRelation 𝔸 R → ((И a, R a x) ↔ (∃ a, a # x ∧ R a x)) := by
intro hequi; constructor
· intro h
apply for_fresh_iff₁ 𝔸 hequi
exact h
· intro h
apply for_fresh_iff₃ 𝔸 hequi
apply for_fresh_iff₂ 𝔸 hequi
exact h
/-- The quantifier in `И a, a R x` is equivalent to "for all fresh `a`". -/
lemma for_fresh_iff_all : EquivariantRelation 𝔸 R → ((И a, R a x) ↔ (∀ a, a # x → R a x)) := by
intro hequi; constructor
· intro h
apply for_fresh_iff₂ 𝔸 hequi
apply for_fresh_iff₁ 𝔸 hequi
exact h
· intro h
apply for_fresh_iff₃ 𝔸 hequi
exact h
end Equivariance
end Freshness
/-!
## Bundled Functions
In the following, we also define a bundled function type for equivariant functions.
-/
section BundledFunction
/-- A function carrying a proof of its equivariance. -/
structure EquivariantMap (X Y : Type) [PermSet 𝔸 X] [PermSet 𝔸 Y] where
/-- The underlying function. -/
toFun : X → Y
/-- Proof of equivariance. -/
equivariant : Equivariant 𝔸 toFun
/-- We write `X →ₑ Y` or `X →ₑ[𝔸] Y` for an equivariant map X → Y. -/
infixr:25 " →ₑ " => EquivariantMap _
/-- We write `X →ₑ Y` or `X →ₑ[𝔸] Y` for an equivariant map X → Y. -/
notation:25 X " →ₑ[" A "] " Y => EquivariantMap A X Y
variable {X Y Z : Type} [PermSet 𝔸 X] [PermSet 𝔸 Y] [PermSet 𝔸 Z]
instance : FunLike (X →ₑ[𝔸] Y) X Y where
coe := EquivariantMap.toFun
coe_injective' f g h := by cases f; cases g; congr
namespace EquivariantMap
/-- The identity function. -/
def id : X →ₑ[𝔸] X := ⟨_root_.id, id_equivariant 𝔸⟩
/-- Function composition. -/
def comp (f : Y →ₑ[𝔸] Z) (g : X →ₑ[𝔸] Y) : X →ₑ[𝔸] Z := ⟨f ∘ g, comp_equivariant 𝔸 f.equivariant g.equivariant⟩
@[simp]
lemma toFun_eq_coe {f : X →ₑ[𝔸] Y} : f.toFun = (f : X → Y) :=
rfl
@[ext]
lemma ext {f g : X →ₑ[𝔸] Y} (h : ∀ a, f a = g a) : f = g :=
DFunLike.ext _ _ h
end EquivariantMap
end BundledFunction
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