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import Mathlib.Tactic
import Nominal.Permutations
import Nominal.Helpers.Finset
/-!
# Basic Definitions
This file provides basic definitions for nominal sets.
-/
/-!
## (FPerm 𝔸)-Sets
Sets equipped with a permutation operation.
-/
section PermSet
variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
/-- A set X equipped with a group action •: (FPerm 𝔸) × X → X. -/
class PermSet (X : Type) extends MulAction (FPerm 𝔸) X
/-- The set of atoms itself forms a (FPerm 𝔸)-Set. -/
instance : PermSet 𝔸 𝔸 where
smul π a := π a
one_smul := by intro; rfl
mul_smul := by intro _ _ _; rfl
/-- The set of finite permutations forms a (FPerm 𝔸)-Set. -/
instance : PermSet 𝔸 (FPerm 𝔸) where
smul π τ := π * τ
one_smul := by intro; rfl
mul_smul := by intro _ _ _; rfl
class DiscretePermSet (X : Type) extends PermSet 𝔸 X where
is_discrete : ∀ (x : X) (π : FPerm 𝔸), π • x = x
/-- A discrete (FPerm 𝔸)-Set where permutations do not change anything. -/
def PermSet.discrete (X : Type) : DiscretePermSet 𝔸 X where
smul π x := x
one_smul := by intro; rfl
mul_smul := by intro _ _ _; rfl
is_discrete := by intro _ _; rfl
/-- We consider Unit a discrete (FPerm 𝔸)-Set. -/
instance : DiscretePermSet 𝔸 Unit := PermSet.discrete 𝔸 Unit
/-- We consider Bool a discrete (FPerm 𝔸)-Set. -/
instance : DiscretePermSet 𝔸 Bool := PermSet.discrete 𝔸 Bool
end PermSet
/-!
## 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 (𝔸 : Type) {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
@[simp]
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
lemma support_subset {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)
#check Set.Finite.exists_injOn_of_encard_le
#check Set.injOn_iff_injective
#check Function.partialInv_of_injective
lemma support_inter {x : X} {S T : Set 𝔸} : HasSupport 𝔸 x S → S.Finite → HasSupport 𝔸 x T → T.Finite → HasSupport 𝔸 x (S ∩ T) := by
sorry
lemma support_sInter {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 [show (⋂₀ (insert S C' : Finset (Set 𝔸))) = S ∩ (⋂₀ C') by aesop]
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
/-!
## Basic Nominal Set Definition
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
finsupp : ∀ (x : X), ∃ (S : Set 𝔸), PermSet.HasSupport 𝔸 x S ∧ S.Finite
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.finsupp x
apply Set.Finite.subset
· exact hfinS
· grind
/-- The least finite support is itself a support. -/
lemma supp_is_support {x : X} : PermSet.HasSupport 𝔸 x (supp 𝔸 x) := by
obtain ⟨S, ⟨hsuppS, hfinS⟩⟩ := nom.finsupp 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
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
private lemma NominalSet.supp_atoms' {a : 𝔸} : PermSet.HasSupport 𝔸 a {a} := by
simp [PermSet.fix']
/-- The set of atoms is a nominal set, where supp a = {a}. -/
instance : NominalSet 𝔸 𝔸 where
finsupp := by
intro a
use {a}
constructor
· exact NominalSet.supp_atoms' 𝔸
· simp
lemma NominalSet.supp_atoms [Infinite 𝔸] [DecidableEq 𝔸] {a : 𝔸} : supp 𝔸 a = {a} := by
ext b; constructor
· apply support_subset_supp
· apply NominalSet.supp_atoms'
· simp
· suffices : a ∈ supp 𝔸 a
· simp; intro h; rw [h]; assumption
· intro S ⟨hsuppS, hfinS⟩
by_contra ha
obtain ⟨c, hc⟩ := Infinite.exists_notMem_finset ({a} ∪ hfinS.toFinset)
simp at hc
have hfix : ⸨a, c⸩ ∈ PermSet.fix 𝔸 S := by
intro d hd
rw [show ⸨a,c⸩ • d = (Equiv.swap a c) d from rfl]
apply Equiv.Perm.supp_notin
simp [Equiv.Perm.supp_swap, Ne.symm hc.left]
exact ⟨ne_of_mem_of_not_mem hd ha, ne_of_mem_of_not_mem hd hc.right⟩
have ha₁ : ⸨a, c⸩ • a = c := by
simp [show ⸨a,c⸩ • a = (Equiv.swap a c) a from rfl]
have ha₂ : ⸨a, c⸩ • a = a := by
specialize hsuppS hfix
simpa [PermSet.fix']
rw [ha₁] at ha₂
exact hc.left ha₂
class DiscreteNominalSet (X : Type) extends NominalSet 𝔸 X, DiscretePermSet 𝔸 X
private lemma NominalSet.supp_discrete' {X : Type} [disc : DiscretePermSet 𝔸 X] {x : X} : PermSet.HasSupport 𝔸 x ∅ := by
intro π hπ
simp [PermSet.fix']
exact DiscretePermSet.is_discrete x π
/-- Every set can be made into a discrete nominal set where every element has empty support. -/
def NominalSet.discrete (X : Type) : DiscreteNominalSet 𝔸 X :=
{ PermSet.discrete 𝔸 X with
finsupp := by
intro x
use {}
constructor
· intro π hπ; simp [PermSet.fix']; rfl
· simp
}
lemma NominalSet.supp_discrete {X : Type} [DiscreteNominalSet 𝔸 X] {x : X} : supp 𝔸 x = ∅ := by
rw [← Set.subset_empty_iff]
apply Set.sInter_subset_of_mem
exact ⟨NominalSet.supp_discrete' 𝔸, Set.finite_empty⟩
/-- We consider Unit a discrete nominal set. -/
instance : DiscreteNominalSet 𝔸 Unit := NominalSet.discrete 𝔸 Unit
/-- We consider Bool a discrete nominal set. -/
instance : DiscreteNominalSet 𝔸 Bool := NominalSet.discrete 𝔸 Bool
end NominalSet
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