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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 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 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 : X → Y} {g : Y → Z} : Equivariant 𝔸 f → Equivariant 𝔸 g → Equivariant 𝔸 (g ∘ f) := by
intro hf hg x π
calc
g (f (π • x)) = g (π • f x) := by rw [hf]
_ = π • (g (f x)) := by rw [hg]
/-- 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
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