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import Mathlib
import Nominal.Support
import Nominal.Equivariance
/-!
# Instances
This file contains common instances of (FPerm 𝔸)-sets and nominal sets.
-/
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; 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 at hc
have hfix : ⸨a, c⸩ ∈ PermSet.fix 𝔸 S := by
intro d hd
apply Equiv.Perm.supp_notin
simp [FPerm.swap, 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 h₁ : ⸨a, c⸩ • a = c := by simp [show ⸨a,c⸩ • a = (Equiv.swap a c) a from rfl]
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.discrete (X : Type) : DiscreteNominalSet 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 [PermSet.fix']; rfl
· exact Set.finite_empty
@[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
constructor
· intro π hπ; simp [PermSet.fix', DiscreteNominalSet.is_discrete]
· exact Set.finite_empty
@[simp] lemma NominalSet.fresh_discrete {X : Type} [DiscreteNominalSet 𝔸 X] {a : 𝔸} {x : X} : a # x := by
simp [IsFresh]
/-- 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 Discrete
/-!
## Products
Pairs of two nominal sets also form a nominal set, where supp (x, y) = supp x ∪ supp y.
-/
section Products
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 NominalSet.supp_product' {x : X} {y : Y} : PermSet.HasSupport 𝔸 (x, y) (supp 𝔸 x ∪ supp 𝔸 y) := by
intro π hπ
obtain ⟨hπ₁, hπ₂⟩ := PermSet.fix_union 𝔸 hπ
exact Prod.mk_inj.mpr ⟨supp_is_support 𝔸 hπ₁, 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 ⟨NominalSet.supp_product' 𝔸, Set.Finite.union (NominalSet.supp_finite 𝔸) (NominalSet.supp_finite 𝔸)⟩
@[simp] lemma NominalSet.supp_product {x : X} {y : Y} : supp 𝔸 (x, y) = supp 𝔸 x ∪ supp 𝔸 y := by
ext b; constructor
· apply support_subset_supp
· exact supp_product' 𝔸
· exact Set.Finite.union (supp_finite 𝔸) (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 [PermSet.fix'] 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 [PermSet.fix'] at this; exact this.right
exact (hS.out.left hπ).out
@[simp] lemma NominalSet.fresh_product {a : 𝔸} {x : X} {y : Y} : a # (x, y) ↔ a # x ∧ a # y := by
simp [IsFresh]
lemma NominalSet.supp_product_comm {x : X} {y : Y} : supp 𝔸 (x, y) = supp 𝔸 (y, x) := by
simp; apply Set.union_comm
lemma Prod.fst_equivariant : Equivariant 𝔸 (Prod.fst : X × Y → X) := by
simp [Equivariant]
lemma Prod.snd_equivariant : Equivariant 𝔸 (Prod.snd : X × Y → Y) := by
simp [Equivariant]
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
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 NominalSet.supp_coproduct₁' {x : X} : PermSet.HasSupport 𝔸 (.inl x : X ⊕ Y) (supp 𝔸 x) := by
intro π hπ
simp [PermSet.fix']
exact supp_is_support 𝔸 hπ
private lemma NominalSet.supp_coproduct₂' {y : Y} : PermSet.HasSupport 𝔸 (.inr y : X ⊕ Y) (supp 𝔸 y) := by
intro π hπ
simp [PermSet.fix']
exact 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 ⟨NominalSet.supp_coproduct₁' 𝔸, NominalSet.supp_finite 𝔸⟩
| inr y => use NominalSet.supp 𝔸 y; exact ⟨NominalSet.supp_coproduct₂' 𝔸, NominalSet.supp_finite 𝔸⟩
@[simp] lemma NominalSet.supp_coproduct₁ {x : X} : supp 𝔸 (.inl x : X ⊕ Y) = supp 𝔸 x := by
ext b; constructor
· apply support_subset_supp
· exact supp_coproduct₁' 𝔸
· exact 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 [PermSet.fix'] at this; exact this
exact (hS.out.left hπ).out
@[simp] lemma NominalSet.supp_coproduct₂ {y : Y} : supp 𝔸 (.inr y : X ⊕ Y) = supp 𝔸 y := by
ext b; constructor
· apply support_subset_supp
· exact supp_coproduct₂' 𝔸
· exact 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 [PermSet.fix'] at this; exact this
exact (hS.out.left hπ).out
@[simp] lemma NominalSet.fresh_coproduct₁ {a : 𝔸} {x : X} : a # (.inl x : X ⊕ Y) ↔ a # x := by
simp [IsFresh]
@[simp] lemma NominalSet.fresh_coproduct₂ {a : 𝔸} {y : Y} : a # (.inr y : X ⊕ Y) ↔ a # y := by
simp [IsFresh]
lemma Sum.inl_equivariant : Equivariant 𝔸 (Sum.inl : X → X ⊕ Y) := by
simp [Equivariant]
lemma Sum.inr_equivariant : Equivariant 𝔸 (Sum.inr : Y → X ⊕ Y) := by
simp [Equivariant]
end Coproducts
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