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+import Mathlib
+import Nominal.Support
+import Nominal.Equivariance
+import Nominal.Category
+import Nominal.Permutations
+import Nominal.Instances.Basic
+
+/-!
+# Constructing Finitely Supported Sets
+
+We can restrict arbitrary `(FPerm 𝔸)`-sets to finitely supported subsets, which are nominal sets.
+
+Examples of this include finitely supported sets (`FinsuppSet`) and finitely supported functions (`FinsuppFunction`).
+-/
+
+variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
+
+namespace PermSet
+
+/-!
+## Finitely Supported Subsets
+
+We first define how every `PermSet` can be restricted to its finitely supported elements (`Finsupp`), which form a
+nominal set.
+-/
+section FSSubsets
+
+/-- Restrict a type to its finitely supported elements -/
+abbrev Finsupp (X : Type) [PermSet 𝔸 X] : Type :=
+  { x : X // ∃ S, PermSet.HasSupport 𝔸 x S ∧ S.Finite }
+
+namespace Finsupp
+
+variable {X : Type} [PermSet 𝔸 X]
+
+instance : PermSet 𝔸 (Finsupp 𝔸 X) where
+  smul π x := ⟨π • x.val, by
+    obtain ⟨S, ⟨hS₁, hS₂⟩⟩ := x.prop
+    use π • S; constructor
+    · rw [← Rel.equivariant 𝔸 (HasSupport 𝔸)]
+      exact hS₁
+    · exact Set.Finite.smul_set hS₂⟩
+  one_smul x := by
+    apply Subtype.ext
+    exact MulAction.one_smul x.val
+  mul_smul π τ x := by
+    apply Subtype.ext
+    exact mul_smul π τ x.val
+
+@[simp] lemma smul_val (π : FPerm 𝔸) (x : Finsupp 𝔸 X) : (π • x).val = π • x.val := rfl
+
+instance instValEquivariant : Function.Equivariant 𝔸 (Subtype.val : Finsupp 𝔸 X → X) := by
+  constructor; simp
+
+instance : NominalSet 𝔸 (Finsupp 𝔸 X) where
+  fin_supp x := by
+    obtain ⟨S, ⟨hS₁, hS₂⟩⟩ := x.prop
+    use S; constructor
+    · intro π hπ
+      apply Subtype.ext
+      exact hS₁ hπ
+    · exact hS₂
+
+end Finsupp
+
+end FSSubsets
+
+/-!
+## Finitely Supported Sets
+
+Since the powerset of an (FPerm 𝔸)-set is an (FPerm 𝔸)-set, we can restrict ourselves to finitely supported sets to
+construct a nominal set `FinsuppSet`.
+
+Furthermore, common binary operations such as union, intersection, and difference are equivariant functions.
+-/
+section FSSets
+
+abbrev FinsuppSet (X : Type) [PermSet 𝔸 X] : Type :=
+  Finsupp 𝔸 (Set X)
+
+namespace FinsuppSet
+
+variable (X Y : Type) [NominalSet 𝔸 X] [NominalSet 𝔸 Y]
+
+private def union' : (FinsuppSet 𝔸 X × FinsuppSet 𝔸 X) → Set X :=
+  fun (S₁, S₂) ↦ S₁.val ∪ S₂.val
+
+instance : Function.Equivariant 𝔸 (union' 𝔸 X) := by
+  constructor; intro (S₁, S₂) π
+  simp only [union', PermSet.Finsupp.smul_val]
+  exact Eq.symm Set.smul_set_union
+
+def union : (FinsuppSet 𝔸 X × FinsuppSet 𝔸 X) → FinsuppSet 𝔸 X :=
+  fun p ↦ ⟨union' 𝔸 X p, by
+    use NominalSet.supp 𝔸 p; constructor
+    · apply Function.Equivariant.preserve_support
+      apply NominalSet.supp_is_support
+    · apply NominalSet.supp_finite⟩
+
+instance : Function.Equivariant 𝔸 (union 𝔸 X) := by
+  constructor; intro p π
+  apply Subtype.ext
+  simp only [union, Finsupp.smul_val]
+  apply Function.equivariant
+
+instance : Union (FinsuppSet 𝔸 X) := ⟨Function.curry (union 𝔸 X)⟩
+
+@[simp] lemma union_val {S₁ S₂ : FinsuppSet 𝔸 X} :
+    (S₁ ∪ S₂).val = S₁.val ∪ S₂.val := rfl
+
+@[simp] lemma union_equivariant {S₁ S₂ : FinsuppSet 𝔸 X} {π : FPerm 𝔸} :
+    π • (S₁ ∪ S₂) = π • S₁ ∪ π • S₂ := by
+  show π • (union _ _ (S₁, S₂)) = union _ _ (π • (S₁, S₂))
+  rw [← Function.equivariant]
+
+private def inter' : (FinsuppSet 𝔸 X × FinsuppSet 𝔸 X) → Set X :=
+  fun (S₁, S₂) ↦ S₁.val ∩ S₂.val
+
+instance : Function.Equivariant 𝔸 (inter' 𝔸 X) := by
+  constructor; intro (S₁, S₂) π
+  simp only [inter', PermSet.Finsupp.smul_val]
+  exact Eq.symm Set.smul_set_inter
+
+def inter : (FinsuppSet 𝔸 X × FinsuppSet 𝔸 X) → FinsuppSet 𝔸 X :=
+  fun p ↦ ⟨inter' 𝔸 X p, by
+    use NominalSet.supp 𝔸 p; constructor
+    · apply Function.Equivariant.preserve_support
+      apply NominalSet.supp_is_support
+    · apply NominalSet.supp_finite⟩
+
+instance : Function.Equivariant 𝔸 (inter 𝔸 X) := by
+  constructor; intro p π
+  apply Subtype.ext
+  simp only [inter, Finsupp.smul_val]
+  apply Function.equivariant
+
+instance : Inter (FinsuppSet 𝔸 X) := ⟨Function.curry (inter 𝔸 X)⟩
+
+@[simp] lemma inter_val {S₁ S₂ : FinsuppSet 𝔸 X} :
+    (S₁ ∩ S₂).val = S₁.val ∩ S₂.val := rfl
+
+@[simp] lemma inter_equivariant {S₁ S₂ : FinsuppSet 𝔸 X} {π : FPerm 𝔸} :
+    π • (S₁ ∩ S₂) = π • S₁ ∩ π • S₂ := by
+  show π • (inter _ _ (S₁, S₂)) = inter _ _ (π • (S₁, S₂))
+  rw [← Function.equivariant]
+
+private def sdiff' : (FinsuppSet 𝔸 X × FinsuppSet 𝔸 X) → Set X :=
+  fun (S₁, S₂) ↦ S₁.val \ S₂.val
+
+instance : Function.Equivariant 𝔸 (sdiff' 𝔸 X) := by
+  constructor; intro (S₁, S₂) π
+  simp only [sdiff', PermSet.Finsupp.smul_val]
+  exact Eq.symm Set.smul_set_sdiff
+
+def sdiff : (FinsuppSet 𝔸 X × FinsuppSet 𝔸 X) → FinsuppSet 𝔸 X :=
+  fun p ↦ ⟨sdiff' 𝔸 X p, by
+    use NominalSet.supp 𝔸 p; constructor
+    · apply Function.Equivariant.preserve_support
+      apply NominalSet.supp_is_support
+    · apply NominalSet.supp_finite⟩
+
+instance : Function.Equivariant 𝔸 (sdiff 𝔸 X) := by
+  constructor; intro p π
+  apply Subtype.ext
+  simp only [sdiff, Finsupp.smul_val]
+  apply Function.equivariant
+
+instance : SDiff (FinsuppSet 𝔸 X) := ⟨Function.curry (sdiff 𝔸 X)⟩
+
+@[simp] lemma sdiff_val {S₁ S₂ : FinsuppSet 𝔸 X} :
+    (S₁ \ S₂).val = S₁.val \ S₂.val := rfl
+
+@[simp] lemma sdiff_equivariant {S₁ S₂ : FinsuppSet 𝔸 X} {π : FPerm 𝔸} :
+    π • (S₁ \ S₂) = π • S₁ \ π • S₂ := by
+  show π • (sdiff _ _ (S₁, S₂)) = sdiff _ _ (π • (S₁, S₂))
+  rw [← Function.equivariant]
+
+end FinsuppSet
+
+end FSSets
+
+/-!
+## Finitely Supported Functions
+
+Since functions between (FPerm 𝔸)-sets are an (FPerm 𝔸)-set, we can restrict ourselves to finitely supported functions
+to construct a nominal set `FinsuppFunction`.
+
+Furthermore, there is an isomorphism between finietly supported functions (X →fs Y) and sets 𝒫fs (X × Y).
+-/
+section FSFunctions
+
+abbrev FinsuppFunction (X Y : Type) [PermSet 𝔸 X] [PermSet 𝔸 Y] : Type :=
+  Finsupp 𝔸 (X → Y)
+
+lemma FinsuppFunction.supp_image {X Y : Type} [NominalSet 𝔸 X] [NominalSet 𝔸 Y] (f : PermSet.FinsuppFunction 𝔸 X Y) :
+    ∀ x, PermSet.HasSupport 𝔸 (f.val x) (NominalSet.supp 𝔸 f ∪ NominalSet.supp 𝔸 x) := by
+  intro x π hπ
+  have hπ₁ : π ∈ PermSet.fix 𝔸 (NominalSet.supp 𝔸 f) := by
+    apply PermSet.fix_antitone_mem 𝔸 hπ; exact Set.subset_union_left
+  have hπ₂ : π ∈ PermSet.fix 𝔸 (NominalSet.supp 𝔸 x) := by
+    apply PermSet.fix_antitone_mem 𝔸 hπ; exact Set.subset_union_right
+  suffices (π • f).val (π • x) = f.val x by
+    simp at this
+    exact this
+  rw [show π • f = f from NominalSet.supp_is_support 𝔸 hπ₁]
+  rw [show π • x = x from NominalSet.supp_is_support 𝔸 hπ₂]
+
+variable (X Y : Type) [NominalSet 𝔸 X] [NominalSet 𝔸 Y]
+
+namespace Finsupp
+
+private def flatten' (f : X → PermSet.FinsuppSet 𝔸 Y) : Set (X × Y) :=
+  {p : X × Y | p.snd ∈ (f p.fst).val}
+
+instance : Function.Equivariant 𝔸 (flatten' 𝔸 X Y) := by
+  constructor
+  intro f π; ext ⟨x, y⟩; constructor
+  · simp only [flatten', PermSet.smul_function, PermSet.Finsupp.smul_val, Set.mem_setOf_eq]
+    intro hy
+    have hy' : π⁻¹ • y ∈ π⁻¹ • π • (f (π⁻¹ • x)).val := by
+      rw [Set.smul_mem_smul_set_iff]; exact hy
+    rw [inv_smul_smul] at hy'
+    rw [show (x, y) = π • (π⁻¹ • x, π⁻¹ • y) by simp]
+    rw [Set.smul_mem_smul_set_iff]
+    exact hy'
+  · intro hxy
+    have hxy' : π⁻¹ • (x, y) ∈ π⁻¹ • π • flatten' 𝔸 X Y f := by
+      rw [Set.smul_mem_smul_set_iff]; exact hxy
+    simp only [inv_smul_smul, Prod.smul_mk, flatten', Set.mem_setOf_eq] at hxy'
+    simp only [flatten', PermSet.smul_function, PermSet.Finsupp.smul_val, Set.mem_setOf_eq]
+    have hy : π • π⁻¹ • y ∈ π • (f (π⁻¹ • x)).val := by
+      rw [Set.smul_mem_smul_set_iff]; exact hxy'
+    simp only [smul_inv_smul] at hy
+    exact hy
+
+def flatten (f : PermSet.FinsuppFunction 𝔸 X (PermSet.FinsuppSet 𝔸 Y)) :
+    PermSet.FinsuppSet 𝔸 (X × Y) :=
+  ⟨flatten' 𝔸 X Y f.val, by
+    obtain ⟨S, ⟨hS₁, hS₂⟩⟩ := f.prop
+    use S; constructor
+    · apply Function.Equivariant.preserve_support
+      exact hS₁
+    · exact hS₂⟩
+
+instance : Function.Equivariant 𝔸 (flatten 𝔸 X Y) := by
+  constructor; intro p π
+  apply Subtype.ext
+  simp only [flatten, Finsupp.smul_val]
+  apply Function.equivariant
+
+private def unflatten' (S : PermSet.FinsuppSet 𝔸 (X × Y)) : X → (PermSet.FinsuppSet 𝔸 Y) :=
+  fun x ↦ ⟨{y : Y | (x, y) ∈ S.val}, by
+    use NominalSet.supp 𝔸 S ∪ NominalSet.supp 𝔸 x; constructor
+    · intro π hπ
+      have hfixx : π⁻¹ • x = x := by
+        apply NominalSet.supp_is_support
+        rw [← PermSet.fix_inv]
+        exact PermSet.fix_antitone_mem 𝔸 hπ Set.subset_union_right
+      have hfixS : π⁻¹ • S = S := by
+        apply NominalSet.supp_is_support
+        rw [← PermSet.fix_inv]
+        exact PermSet.fix_antitone_mem 𝔸 hπ Set.subset_union_left
+      ext y; constructor
+      · intro hy
+        simp only [Set.mem_smul_set_iff_inv_smul_mem, Set.mem_setOf_eq] at hy
+        rw [← hfixx, ← hfixS, smul_val, Set.mem_inv_smul_set_iff, Prod.smul_mk, smul_inv_smul, smul_inv_smul] at hy
+        exact hy
+      · intro hy
+        simp only [Set.mem_smul_set_iff_inv_smul_mem, Set.mem_setOf_eq]
+        simp only [Set.mem_setOf_eq] at hy
+        have hy' : (π⁻¹ • x, π⁻¹ • y) ∈ π⁻¹ • S.val := Set.smul_mem_smul_set hy
+        rw [hfixx, ← smul_val, hfixS] at hy'
+        exact hy'
+    · simp only [Set.finite_union]; constructor
+      · exact NominalSet.supp_finite 𝔸
+      · exact NominalSet.supp_finite 𝔸⟩
+
+instance : Function.Equivariant 𝔸 (unflatten' 𝔸 X Y) := by
+  constructor
+  intro S π
+  ext x y
+  simp only [unflatten', smul_val, Set.mem_setOf_eq, smul_function]
+  simp only [Set.mem_smul_set_iff_inv_smul_mem, Set.mem_setOf_eq, Prod.smul_mk]
+
+def unflatten (S : PermSet.FinsuppSet 𝔸 (X × Y)) : PermSet.FinsuppFunction 𝔸 X (PermSet.FinsuppSet 𝔸 Y) :=
+  ⟨unflatten' 𝔸 X Y S, by
+    obtain ⟨A, ⟨hS₁, hS₂⟩⟩ := S.prop
+    use A; constructor
+    · apply Function.Equivariant.preserve_support
+      rw [← Function.Equivariant.preserve_support_inj 𝔸 Subtype.val_injective] at hS₁
+      exact hS₁
+    · exact hS₂⟩
+
+instance : Function.Equivariant 𝔸 (unflatten 𝔸 X Y) := by
+  constructor; intro p π
+  apply Subtype.ext
+  simp only [unflatten, Finsupp.smul_val]
+  apply Function.equivariant
+
+def flattenEquiv  :
+    EquivariantEquiv 𝔸
+      (PermSet.FinsuppFunction 𝔸 X (PermSet.FinsuppSet 𝔸 Y))
+      (PermSet.FinsuppSet 𝔸 (X × Y)) where
+  toFun := flatten 𝔸 X Y
+  invFun := unflatten 𝔸 X Y
+  toFun_equivariant := inferInstance
+
+end Finsupp
+
+end FSFunctions
+
+end PermSet