import Mathlib.Tactic import Nominal.Permutations import Nominal.Helpers.Finset /-! # Support This file provides basic definitions for support of (FPerm 𝔸)-sets and nominal sets. -/ variable (𝔸 : Type) /-! ## Support Definitions We can define the notion of "supports" on elements of (FPerm 𝔸)-Sets as the atoms which need to be left unchanged to also fix the element. -/ section Support variable {X : Type} [PermSet 𝔸 X] namespace PermSet /-- The permutations that leave an element x unchanged. -/ def fix' (x : X) : Set (FPerm 𝔸) := { π : FPerm 𝔸 | π • x = x } /-- The permutations that leave all elements in A unchanged. -/ def fix (A : Set X) : Set (FPerm 𝔸) := { π : FPerm 𝔸 | ∀ x ∈ A, π • x = x } /-- A set S ⊆ 𝔸 is a support for x if every permutation that fixes S also leaves x unchanged. -/ def HasSupport (x : X) (S : Set 𝔸) : Prop := fix 𝔸 S ⊆ fix' 𝔸 x /-- Alternative definition with fix 𝔸 S unfolded. -/ lemma support_iff {x : X} {S : Set 𝔸} : HasSupport 𝔸 x S ↔ ∀ (π : FPerm 𝔸), (∀ x ∈ S, π x = x) → π ∈ fix' 𝔸 x := by constructor · intro hsupp π hπ exact hsupp hπ · intro hsupp π hπ exact hsupp π hπ /-- If a support is a subset of another set, that set is also a support. -/ lemma support_superset {x : X} {S T : Set 𝔸} : HasSupport 𝔸 x S → S ⊆ T → HasSupport 𝔸 x T := by intro hS hsub π hπ apply hS intro a ha exact hπ a (hsub ha) /-- For two finite supports, the intersection is also a support. -/ lemma support_inter [Infinite 𝔸] {x : X} {S T : Set 𝔸} : HasSupport 𝔸 x S → S.Finite → HasSupport 𝔸 x T → T.Finite → HasSupport 𝔸 x (S ∩ T) := by intro hsuppS hfinS hsuppT hfinT π hfixπ have h_cod_inf : ((π.val.supp ∪ S ∪ T)ᶜ).Infinite := by apply Set.Finite.infinite_compl simp; exact ⟨⟨π.val_prop, hfinS⟩, hfinT⟩ have h_dom_fin : (S \ T).Finite := Set.Finite.diff hfinS have hle : (S \ T).encard ≤ ((π.val.supp ∪ S ∪ T)ᶜ).encard := by rw [Set.Infinite.encard_eq h_cod_inf, Set.Finite.encard_eq_coe h_dom_fin] simp obtain ⟨σ, ⟨hpreσ, hinjσ⟩⟩ := Set.Finite.exists_injOn_of_encard_le h_dom_fin hle have himσ {a : 𝔸} : a ∈ S \ T → σ a ∉ S \ T := by intro ha suffices : σ a ∈ (π.val.supp ∪ S ∪ T)ᶜ · have h : S \ T ⊆ π.val.supp ∪ S ∪ T := by calc S \ T ⊆ S := Set.diff_subset _ ⊆ π.val.supp ∪ S := Set.subset_union_right _ ⊆ π.val.supp ∪ S ∪ T := Set.subset_union_left exact Set.notMem_subset h (hpreσ ha) · rw [← Set.image_subset_iff] at hpreσ exact hpreσ (Set.mem_image_of_mem σ ha) have himσ' {a : 𝔸} : a ∈ S \ T → σ a ∉ π.val.supp := by intro ha suffices : σ a ∈ (π.val.supp ∪ S ∪ T)ᶜ · have h : π.val.supp ⊆ π.val.supp ∪ S ∪ T := by calc π.val.supp ⊆ π.val.supp ∪ S := Set.subset_union_left _ ⊆ π.val.supp ∪ S ∪ T := Set.subset_union_left exact Set.notMem_subset h (hpreσ ha) · rw [← Set.image_subset_iff] at hpreσ exact hpreσ (Set.mem_image_of_mem σ ha) rw [Set.injOn_iff_injective] at hinjσ obtain hσinv := Function.partialInv_of_injective hinjσ have {a : 𝔸} : Decidable (a ∈ S \ T) := Classical.propDecidable (a ∈ S \ T) let τ : 𝔸 → 𝔸 := fun a => if h : a ∈ S \ T then (S \ T).restrict σ ⟨a, h⟩ else if let .some b := ((S \ T).restrict σ).partialInv a then b else a have hselfinvτ : ∀ a, τ (τ a) = a := by intro a by_cases h₁ : a ∈ S \ T · have h₂ : ((S \ T).restrict σ).partialInv (((S \ T).restrict σ) ⟨a, h₁⟩) = .some ⟨a, h₁⟩ := Function.partialInv_left hinjσ ⟨a, h₁⟩ simp at h₂ simp [τ, h₁, himσ h₁, h₂] · cases h₂ : ((S \ T).restrict σ).partialInv a with | some b => simp [τ, h₂, h₁] rwa [← Set.restrict_apply σ (S \ T) b, ← hσinv] | none => simp [τ, h₂, h₁] have hfinsuppτ : {a | τ a ≠ a}.Finite := by apply Set.Finite.subset · exact Set.finite_union.mpr ⟨h_dom_fin, Set.Finite.image σ h_dom_fin⟩ · intro a ha₁; by_contra ha₂ rw [← Set.mem_compl_iff, Set.compl_union, Set.mem_inter_iff] at ha₂ suffices τ a = a by contradiction have ha₃ : a ∉ S \ T := by rw [← Set.mem_compl_iff]; exact ha₂.left cases ha₄ : ((S \ T).restrict σ).partialInv a with | some b => rw [hσinv, Set.restrict_apply] at ha₄ have : a ∈ (σ '' (S \ T)) := by rw [← ha₄, Set.mem_image]; use b; simp have : a ∉ (σ '' (S \ T)) := Set.notMem_of_mem_diff ha₂ contradiction | none => simp [τ, ha₃, ha₄] let τ' : FPerm 𝔸 := ⟨⟨τ, τ, hselfinvτ, hselfinvτ⟩, hfinsuppτ⟩ have h₁ : τ' ∈ fix 𝔸 T := by intro a haT; show τ a = a have ha₁ : a ∉ S \ T := Set.notMem_diff_of_mem haT have h₁ : σ '' (S \ T) ⊆ (π.val.supp ∪ S ∪ T)ᶜ := by rwa [Set.image_subset_iff] have h₂ : a ∉ (π.val.supp ∪ S ∪ T)ᶜ := by rw [Set.notMem_compl_iff] apply Set.mem_union_right; assumption cases ha₂ : ((S \ T).restrict σ).partialInv a with | some b => rw [hσinv, Set.restrict_apply] at ha₂ have : a ∈ (σ '' (S \ T)) := by rw [← ha₂, Set.mem_image]; use b; simp have : a ∉ (σ '' (S \ T)) := fun x => h₂ (h₁ x) contradiction | none => simp [τ, ha₁, ha₂] have h₂ : (τ' * π * τ') ∈ fix 𝔸 S := by intro a haS; show τ (π (τ a)) = a by_cases haT : a ∈ T · have ht : τ a = a := by -- XXX: copied from above have ha₁ : a ∉ S \ T := Set.notMem_diff_of_mem haT have h₁ : σ '' (S \ T) ⊆ (π.val.supp ∪ S ∪ T)ᶜ := by rwa [Set.image_subset_iff] have h₂ : a ∉ (π.val.supp ∪ S ∪ T)ᶜ := by rw [Set.notMem_compl_iff] apply Set.mem_union_right; assumption cases ha₂ : ((S \ T).restrict σ).partialInv a with | some b => rw [hσinv, Set.restrict_apply] at ha₂ have : a ∈ (σ '' (S \ T)) := by rw [← ha₂, Set.mem_image]; use b; simp have : a ∉ (σ '' (S \ T)) := fun x => h₂ (h₁ x) contradiction | none => simp [τ, ha₁, ha₂] calc τ (π (τ a)) = τ (π a) := by rw [ht] _ = τ (π • a) := rfl _ = τ a := by rw [hfixπ a (Set.mem_inter haS haT)] _ = a := by rw [ht] · have h₁ : a ∈ S \ T := Set.mem_diff_of_mem haS haT have h₂ : π (τ a) = τ a := by apply Equiv.Perm.supp_notin rw [show τ a = σ a by simp [τ, h₁]] exact himσ' h₁ rw [h₂] exact hselfinvτ a calc π • x = (1 * π * 1) • x := rfl _ = ((τ' * τ'⁻¹) * π * (τ' * τ'⁻¹)) • x := by rw [@mul_inv_cancel] _ = ((τ' * τ' * π * τ') * τ') • x := rfl _ = (τ' * τ' * π * τ') • x := by rw [@mul_smul, hsuppT h₁] _ = (τ' * (τ' * π * τ')) • x := rfl _ = τ' • x := by rw [@mul_smul, hsuppS h₂] _ = x := by rw [hsuppT h₁] /-- For a finite number of finite supports, the intersection is also a support. -/ lemma support_sInter [Infinite 𝔸] {x : X} {C : Set (Set 𝔸)}: C.Finite → C.Nonempty → (∀ S ∈ C, HasSupport 𝔸 x S ∧ S.Finite) → HasSupport 𝔸 x (⋂₀ C) := by suffices (C : Finset (Set 𝔸)) : C.Nonempty → (∀ S ∈ C, HasSupport 𝔸 x S ∧ S.Finite) → HasSupport 𝔸 x (⋂₀ C.toSet) ∧ (⋂₀ C.toSet).Finite · intro hfin rw [show C = hfin.toFinset.toSet by simp] intro hne h exact (this hfin.toFinset hne h).left · have : DecidableEq (Set 𝔸) := Classical.typeDecidableEq (Set 𝔸) apply Finset.Nonempty.induction · intro S hS rw [show ⋂₀ ({S} : Finset (Set 𝔸)).toSet = S by simp] apply hS simp · intro S C' hS hC' hsing rw [Finset.coe_insert S C', Set.sInter_insert S ↑C'] have hother : (∀ S ∈ C', HasSupport 𝔸 x S ∧ S.Finite) := by intro S' hS' apply hsing exact Finset.mem_insert_of_mem hS' specialize hC' hother specialize hsing S (by simp) constructor · apply support_inter 𝔸 hsing.left hsing.right hC'.left hC'.right · apply Set.Finite.inter_of_left exact hsing.right end PermSet end Support /-! ## Nominal Sets Nominal sets are then just (FPerm 𝔸)-Sets where all elements are finitely supported. -/ section NominalSet variable (𝔸 : Type) /-- An (FPerm 𝔸)-Set X is nominal if every x ∈ X is finitely supported. -/ class NominalSet (X : Type) extends PermSet 𝔸 X where fin_supp : ∀ (x : X), ∃ (S : Set 𝔸), PermSet.HasSupport 𝔸 x S ∧ S.Finite /-- A discrete nominal set where permutations do not change anything. -/ class DiscreteNominalSet (X : Type) extends NominalSet 𝔸 X where is_discrete : ∀ (x : X) (π : FPerm 𝔸), π • x = x namespace NominalSet variable {X : Type} [nom : NominalSet 𝔸 X] /-- For every element x of a nominal set, there exists a least finite support, denoted "supp x". --/ def supp (x : X) : Set 𝔸 := ⋂₀ { S : Set 𝔸 | PermSet.HasSupport 𝔸 x S ∧ S.Finite } /-- The least finite support in a nominal set is always finite. -/ lemma supp_finite {x : X} : (supp 𝔸 x).Finite := by unfold supp obtain ⟨S, ⟨hsuppS, hfinS⟩⟩ := nom.fin_supp x apply Set.Finite.subset · exact hfinS · grind /-- The least finite support is itself a support. -/ lemma supp_is_support [Infinite 𝔸] {x : X} : PermSet.HasSupport 𝔸 x (supp 𝔸 x) := by obtain ⟨S, ⟨hsuppS, hfinS⟩⟩ := nom.fin_supp x have suppf : (supp 𝔸 x = ⋂₀ { T : Set 𝔸 | T ⊆ S ∧ PermSet.HasSupport 𝔸 x T ∧ T.Finite }) := by unfold supp; ext a constructor · intro h T hT apply h exact hT.out.right · intro h T hT suffices : a ∈ S ∩ T · apply Set.mem_of_mem_inter_right; assumption · apply h constructor · exact Set.inter_subset_left · constructor · exact PermSet.support_inter 𝔸 hsuppS hfinS hT.out.left hT.out.right · apply Set.Finite.inter_of_left; assumption rw [suppf] apply PermSet.support_sInter · apply Set.Finite.subset · apply Set.Finite.powerset exact hfinS · intro T hT exact hT.out.left · use S exact ⟨by rfl, ⟨hsuppS, hfinS⟩⟩ · intro T hT exact hT.right /-- If S is a finite support for x, then supp x is a subset of it. -/ lemma support_subset_supp {x : X} {S : Set 𝔸} : PermSet.HasSupport 𝔸 x S → S.Finite → supp 𝔸 x ⊆ S := by intro hsupp hfin apply Set.sInter_subset_of_mem exact ⟨hsupp, hfin⟩ end NominalSet end NominalSet