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import Mathlib
import Nominal.Support
import Nominal.Equivariance
/-!
# Basic Nominal Sets
This file contains common instances of nominal sets and their smallest supports.
-/
variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
/-!
## Atoms
The set 𝔸 of atoms itself is a nominal set, where supp a = {a}.
-/
section Atoms
private lemma NominalSet.supp_atom' {a : 𝔸} : PermSet.HasSupport 𝔸 a {a} := by
simp [PermSet.HasSupport, PermSet.fix, PermSet.fix']
instance : NominalSet 𝔸 𝔸 where
fin_supp := by
intro a; use {a}
exact ⟨NominalSet.supp_atom' 𝔸, Set.finite_singleton a⟩
@[simp] lemma NominalSet.supp_atom {a : 𝔸} : supp 𝔸 a = {a} := by
ext b; constructor
· apply support_subset_supp 𝔸
· exact NominalSet.supp_atom' 𝔸
· exact Set.finite_singleton a
· suffices a ∈ supp 𝔸 a by
simp only [Set.mem_singleton_iff]
intro h
rwa [h]
intro S ⟨hsuppS, hfinS⟩
by_contra ha
-- Show that we can find a fresh c with (a c) ∈ fix S
obtain ⟨c, hc⟩ := Infinite.exists_notMem_finset ({a} ∪ hfinS.toFinset)
simp only [Finset.singleton_union, Finset.mem_insert, Set.Finite.mem_toFinset, not_or] at hc
have hfix : ⸨a, c⸩ ∈ PermSet.fix 𝔸 S := by
intro d hd
apply FPerm.swap_apply_of_ne_of_ne (ne_of_mem_of_not_mem hd ha) (ne_of_mem_of_not_mem hd hc.right)
have h₁ : ⸨a, c⸩ • a = c := by rw [PermSet.atom_swap, Equiv.swap_apply_left]
have h₂ : ⸨a, c⸩ • a = a := by apply hsuppS; assumption
rw [h₁] at h₂
exact hc.left h₂
@[simp] lemma NominalSet.fresh_atom {a b : 𝔸} : a # b ↔ a ≠ b := by
simp [IsFresh]
end Atoms
/-!
## Discrete Nominal Sets
Every set can be made into a discrete nominal set, where supp x = ∅.
-/
section Discrete
def NominalSet.mkDiscrete (X : Type) : NominalSet.Discrete 𝔸 X where
smul π x := x
one_smul := by intro; rfl
mul_smul := by intro _ _ _; rfl
is_discrete := by intro _ _; rfl
fin_supp := by
intro x; use {}
constructor
· intro π hπ; simp only [PermSet.fix', Set.mem_setOf_eq]; rfl
· exact Set.finite_empty
@[simp] lemma NominalSet.Discrete.supp {X : Type} [NominalSet.Discrete 𝔸 X] {x : X} : supp 𝔸 x = ∅ := by
rw [← Set.subset_empty_iff]
apply Set.sInter_subset_of_mem
constructor
· intro π hπ; simp [PermSet.fix', is_discrete]
· exact Set.finite_empty
@[simp] lemma NominalSet.Discrete.fresh {X : Type} [NominalSet.Discrete 𝔸 X] {a : 𝔸} {x : X} : a # x := by
simp [IsFresh]
/-- We consider Unit a discrete nominal set. -/
instance : NominalSet.Discrete 𝔸 Unit := NominalSet.mkDiscrete 𝔸 Unit
/-- We consider Bool a discrete nominal set. -/
instance : NominalSet.Discrete 𝔸 Bool := NominalSet.mkDiscrete 𝔸 Bool
end Discrete
/-!
## Products
Pairs of two nominal sets also form a nominal set, where supp (x, y) = supp x ∪ supp y.
-/
section Products
namespace Prod
instance {X Y : Type} [PermSet 𝔸 X] [PermSet 𝔸 Y] : PermSet 𝔸 (X × Y) where
variable {X Y : Type} [NominalSet 𝔸 X] [NominalSet 𝔸 Y]
private lemma supp' {x : X} {y : Y} : PermSet.HasSupport 𝔸 (x, y) (NominalSet.supp 𝔸 x ∪ NominalSet.supp 𝔸 y) := by
intro π hπ
obtain ⟨hπ₁, hπ₂⟩ := PermSet.fix_union 𝔸 hπ
exact Prod.mk_inj.mpr ⟨NominalSet.supp_is_support 𝔸 hπ₁, NominalSet.supp_is_support 𝔸 hπ₂⟩
instance : NominalSet 𝔸 (X × Y) where
fin_supp := by
intro ⟨x, y⟩; use NominalSet.supp 𝔸 x ∪ NominalSet.supp 𝔸 y
exact ⟨supp' 𝔸, Set.Finite.union (NominalSet.supp_finite 𝔸) (NominalSet.supp_finite 𝔸)⟩
@[simp] lemma supp {x : X} {y : Y} : NominalSet.supp 𝔸 (x, y) = NominalSet.supp 𝔸 x ∪ NominalSet.supp 𝔸 y := by
ext b; constructor
· apply NominalSet.support_subset_supp
· exact supp' 𝔸
· exact Set.Finite.union (NominalSet.supp_finite 𝔸) (NominalSet.supp_finite 𝔸)
· intro hb; cases hb with
| inl hx =>
intro S hS
suffices PermSet.HasSupport 𝔸 x S by apply hx; exact ⟨this, hS.out.right⟩
intro π hπ
suffices π ∈ PermSet.fix' 𝔸 (x, y) by
simp only [PermSet.fix', smul_mk, mk.injEq, Set.mem_setOf_eq] at this
exact this.left
exact (hS.out.left hπ).out
| inr hy =>
intro S hS
suffices PermSet.HasSupport 𝔸 y S by apply hy; exact ⟨this, hS.out.right⟩
intro π hπ
suffices π ∈ PermSet.fix' 𝔸 (x, y) by
simp only [PermSet.fix', smul_mk, mk.injEq, Set.mem_setOf_eq] at this
exact this.right
exact (hS.out.left hπ).out
@[simp] lemma fresh {a : 𝔸} {x : X} {y : Y} : a # (x, y) ↔ a # x ∧ a # y := by
simp [NominalSet.IsFresh]
lemma supp_comm {x : X} {y : Y} : NominalSet.supp 𝔸 (x, y) = NominalSet.supp 𝔸 (y, x) := by
simp; apply Set.union_comm
lemma fst_equivariant : Function.Equivariant 𝔸 (Prod.fst : X × Y → X) := by
simp [Function.Equivariant]
lemma snd_equivariant : Function.Equivariant 𝔸 (Prod.snd : X × Y → Y) := by
simp [Function.Equivariant]
end Prod
end Products
/-!
## Coproducts
Sums of two nominal sets also form a nominal set, where supp (inl x) = supp x and supp (inr y) = supp y.
-/
section Coproducts
namespace Sum
instance {X Y : Type} [PermSet 𝔸 X] [PermSet 𝔸 Y] : PermSet 𝔸 (X ⊕ Y) where
variable {X Y : Type} [NominalSet 𝔸 X] [NominalSet 𝔸 Y]
private lemma supp_inl' {x : X} : PermSet.HasSupport 𝔸 (.inl x : X ⊕ Y) (NominalSet.supp 𝔸 x) := by
intro π hπ
simp only [PermSet.fix', smul_inl, inl.injEq, Set.mem_setOf_eq]
exact NominalSet.supp_is_support 𝔸 hπ
private lemma supp_inr' {y : Y} : PermSet.HasSupport 𝔸 (.inr y : X ⊕ Y) (NominalSet.supp 𝔸 y) := by
intro π hπ
simp only [PermSet.fix', smul_inr, inr.injEq, Set.mem_setOf_eq]
exact NominalSet.supp_is_support 𝔸 hπ
instance : NominalSet 𝔸 (X ⊕ Y) where
fin_supp := by
intro xy
cases xy with
| inl x => use NominalSet.supp 𝔸 x; exact ⟨supp_inl' 𝔸, NominalSet.supp_finite 𝔸⟩
| inr y => use NominalSet.supp 𝔸 y; exact ⟨supp_inr' 𝔸, NominalSet.supp_finite 𝔸⟩
@[simp] lemma supp_inl {x : X} : NominalSet.supp 𝔸 (.inl x : X ⊕ Y) = NominalSet.supp 𝔸 x := by
ext b; constructor
· apply NominalSet.support_subset_supp
· exact supp_inl' 𝔸
· exact NominalSet.supp_finite 𝔸
· intro hb S hS
suffices PermSet.HasSupport 𝔸 x S by apply hb; exact ⟨this, hS.out.right⟩
intro π hπ
suffices π ∈ PermSet.fix' 𝔸 (.inl x : X ⊕ Y) by
simp only [PermSet.fix', smul_inl, inl.injEq, Set.mem_setOf_eq] at this
exact this
exact (hS.out.left hπ).out
@[simp] lemma supp_inr {y : Y} : NominalSet.supp 𝔸 (.inr y : X ⊕ Y) = NominalSet.supp 𝔸 y := by
ext b; constructor
· apply NominalSet.support_subset_supp
· exact supp_inr' 𝔸
· exact NominalSet.supp_finite 𝔸
· intro hb S hS
suffices PermSet.HasSupport 𝔸 y S by apply hb; exact ⟨this, hS.out.right⟩
intro π hπ
suffices π ∈ PermSet.fix' 𝔸 (.inr y : X ⊕ Y) by
simp only [PermSet.fix', smul_inr, inr.injEq, Set.mem_setOf_eq] at this
exact this
exact (hS.out.left hπ).out
@[simp] lemma fresh_inl {a : 𝔸} {x : X} : a # (.inl x : X ⊕ Y) ↔ a # x := by
simp [NominalSet.IsFresh]
@[simp] lemma fresh_inr {a : 𝔸} {y : Y} : a # (.inr y : X ⊕ Y) ↔ a # y := by
simp [NominalSet.IsFresh]
lemma inl_equivariant : Function.Equivariant 𝔸 (Sum.inl : X → X ⊕ Y) := by
simp [Function.Equivariant]
lemma inr_equivariant : Function.Equivariant 𝔸 (Sum.inr : Y → X ⊕ Y) := by
simp [Function.Equivariant]
end Sum
end Coproducts
/-!
## Quotient Types
If ≈ is an equivariant relation on X × X, then X/≈ is a nominal set with `supp [x] = ⋂₀ {supp y | y ∈ [x]}`.
-/
section Quotient
namespace Quotient
variable {X : Type} [PermSet 𝔸 X] {setoid : EquivariantSetoid 𝔸 X}
instance : SMul (FPerm 𝔸) (Quotient setoid.toSetoid) where
smul π := Quotient.lift (fun x => ⟦π • x⟧) (by
intro x y hxy
simp only [Quotient.eq]
apply setoid.equivariant x y π hxy)
@[simp] lemma smul_quotient (π : FPerm 𝔸) (x : X) : (π • Quotient.mk setoid.toSetoid x) = ⟦π • x⟧ := rfl
instance : PermSet 𝔸 (Quotient setoid.toSetoid) where
one_smul := by
apply Quotient.ind
intro x
simp
mul_smul := by
intro π τ
apply Quotient.ind
simp [mul_smul]
lemma mk_equivariant : Function.Equivariant 𝔸 (Quotient.mk setoid.toSetoid) := by
simp [Function.Equivariant]
variable {X : Type} [NominalSet 𝔸 X] {setoid : EquivariantSetoid 𝔸 X}
instance : NominalSet 𝔸 (Quotient setoid.toSetoid) :=
NominalSet.fromEquivariantSurjective 𝔸 (Quotient.mk setoid.toSetoid) (mk_equivariant 𝔸) mk_surjective
lemma supp {x : X} : NominalSet.supp 𝔸 (Quotient.mk setoid.toSetoid x) = {a | ∀ (x' : X), x' ≈ x → a ∈ NominalSet.supp 𝔸 x'} := by
ext a
constructor
· intro ha x' hx'
rw [← Quotient.sound hx'] at ha
have hsub : NominalSet.supp 𝔸 (Quotient.mk setoid.toSetoid x') ⊆ NominalSet.supp 𝔸 x' :=
Function.Equivariant.supp_subset 𝔸 (mk_equivariant 𝔸)
exact hsub ha
· contrapose; intro ha
simp only [Set.mem_setOf_eq, Classical.not_imp, not_forall]
have hfin₁ : (NominalSet.supp 𝔸 (Quotient.mk setoid.toSetoid x)).Finite := NominalSet.supp_finite 𝔸
have hfin₂ : (NominalSet.supp 𝔸 x).Finite := NominalSet.supp_finite 𝔸
-- Show that we can find a fresh c for which (a c) • ⟦x⟧ = ⟦x⟧
obtain ⟨c, hc⟩ := Infinite.exists_notMem_finset (hfin₁.toFinset ∪ hfin₂.toFinset)
simp only [Finset.mem_union, Set.Finite.mem_toFinset, not_or] at hc
use ⸨a, c⸩ • x; constructor
· apply Quotient.exact
rw [← smul_quotient]
exact NominalSet.fresh_swap_fix 𝔸 ha hc.left
· rw [NominalSet.supp_equivariant, Set.mem_smul_set_iff_inv_smul_mem]
simp only [FPerm.swap_inv, PermSet.atom_swap, Equiv.swap_apply_left]
exact hc.right
end Quotient
end Quotient
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