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import Mathlib
import Nominal.Support
import Nominal.Equivariance
import Nominal.Category
import Nominal.Permutations
import Nominal.Instances.Basic
/-!
# Name Abstractions
Given a nominal set X, we define the set `[𝔸]X` of abstractions as pairs in 𝔸 × X modulo alpha-equivalence.
-/
variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
/-!
## Raw Structure
We first define the raw type of name abstractions as a pair (𝔸, X).
This structure, along with the obvious permutation action, forms a nominal set where supp (a, x) = supp x ∪ {a}.
-/
section RawStructure
/-- The structure forming a name abstraction. -/
structure NameAbstraction' (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸] (X : Type) [NominalSet 𝔸 X] where
/-- The atom that is abstracted away. -/
atom : 𝔸
/-- The body from which the atom is abstracted. -/
body : X
variable (X : Type) [NominalSet 𝔸 X]
attribute [ext] NameAbstraction'
namespace NameAbstraction'
instance : SMul (FPerm 𝔸) (NameAbstraction' 𝔸 X) where
smul π x := .mk (π • x.atom) (π • x.body)
@[simp] lemma smul_atom (π : FPerm 𝔸) (x : NameAbstraction' 𝔸 X) : (π • x).atom = π • x.atom := rfl
@[simp] lemma smul_body (π : FPerm 𝔸) (x : NameAbstraction' 𝔸 X) : (π • x).body = π • x.body := rfl
instance : PermSet 𝔸 (NameAbstraction' 𝔸 X) where
one_smul := by intro _; ext <;> simp [one_smul]
mul_smul := by intro _ _ _; ext <;> simp [mul_smul]
private def ofPair (x : (𝔸 × X)) : NameAbstraction' 𝔸 X := ⟨x.fst, x.snd⟩
private lemma ofPair_injective : Function.Injective (ofPair 𝔸 X) := by
intro x y hxy
simp only [ofPair, mk.injEq] at hxy
ext
· exact hxy.left
· exact hxy.right
private lemma ofPair_surjective : Function.Surjective (ofPair 𝔸 X) := by
intro x
use ⟨x.atom, x.body⟩
simp [ofPair]
private lemma ofPair_equivariant : Equivariant 𝔸 (ofPair 𝔸 X) := by
intro x π
ext <;> simp [ofPair]
instance : NominalSet 𝔸 (NameAbstraction' 𝔸 X) :=
NominalSet.fromEquivariantSurjective 𝔸 (ofPair 𝔸 X) (ofPair_equivariant 𝔸 X) (ofPair_surjective 𝔸 X)
@[simp] lemma supp {x : NameAbstraction' 𝔸 X} : NominalSet.supp 𝔸 x = NominalSet.supp 𝔸 x.body ∪ {x.atom} := by
have h : NominalSet.supp 𝔸 (ofPair 𝔸 X (x.atom, x.body)) = NominalSet.supp 𝔸 (x.atom, x.body) :=
Equivariant.supp_eq_if_inj 𝔸 (ofPair_equivariant 𝔸 X) (ofPair_injective 𝔸 X)
simp only [Prod.supp, NominalSet.supp_atom, ofPair] at h
rwa [Set.union_comm] at h
@[simp] lemma fresh {a : 𝔸} {x : NameAbstraction' 𝔸 X} : a # x ↔ a ≠ x.atom ∧ a # x.body := by
simp [NominalSet.IsFresh]
end NameAbstraction'
end RawStructure
/-!
## Alpha-Equivalence
We continue to define alpha-equivalence on this structure.
-/
section AlphaEquivalence
namespace NameAbstraction
variable {𝔸 : Type} [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸] {X : Type} [NominalSet 𝔸 X]
/-- (a, x) ≈ (b, y) if (c a) x = (c b) y for some fresh c. -/
def AlphaEq (x y : NameAbstraction' 𝔸 X) : Prop :=
И c, ⸨c, x.atom⸩ • x.body = ⸨c, y.atom⸩ • y.body
/-- Alternatively define alpha equivalence as an equivariant relation.
This is used to show some properties using the Equivariance.for_fresh_* lemmas about the freshness quantifier. -/
private def AlphaEqPred (c : 𝔸) (xy : NameAbstraction' 𝔸 X × NameAbstraction' 𝔸 X) : Prop :=
⸨c, xy.fst.atom⸩ • xy.fst.body = ⸨c, xy.snd.atom⸩ • xy.snd.body
private lemma alpha_eq_iff_pred {x y : NameAbstraction' 𝔸 X} : AlphaEq x y = (И c, AlphaEqPred c (x, y)) := by
apply NominalSet.for_fresh_ext
simp [AlphaEqPred]
private lemma alpha_eq_pred_equivariant : EquivariantRelation 𝔸 (AlphaEqPred : 𝔸 → NameAbstraction' 𝔸 X × NameAbstraction' 𝔸 X → Prop) := by
intro a x π
rw [AlphaEqPred, AlphaEqPred]
intro h
have h' : π • ⸨a, x.fst.atom⸩ • x.fst.body = π • ⸨a, x.snd.atom⸩ • x.snd.body := by congr
simp only [smul_smul, FPerm.mul_swap_eq_swap_mul] at h'
simpa [smul_smul]
/-- The freshness quantifier in the definition is equivalent to "there exists some c # (x, y)". -/
lemma alpha_eq_iff_exists {x y : NameAbstraction' 𝔸 X} : AlphaEq x y ↔ ∃ c, c # x ∧ c # y ∧ ⸨c, x.atom⸩ • x.body = ⸨c, y.atom⸩ • y.body := by
suffices (∃ c, c # x ∧ c # y ∧ ⸨c, x.atom⸩ • x.body = ⸨c, y.atom⸩ • y.body) = (∃ c, c # (x, y) ∧ AlphaEqPred c (x, y)) by
rw [alpha_eq_iff_pred, this]
apply Equivariance.for_fresh_iff_exists
exact alpha_eq_pred_equivariant
simp [Prod.fresh, AlphaEqPred, @and_assoc]
/-- The freshness quantifier in the definition is equivalent to "for all c # (x, y)". -/
lemma alpha_eq_iff_all {x y : NameAbstraction' 𝔸 X} : AlphaEq x y ↔ ∀ c, c # x ∧ c # y → ⸨c, x.atom⸩ • x.body = ⸨c, y.atom⸩ • y.body := by
suffices (∀ c, c # x ∧ c # y → ⸨c, x.atom⸩ • x.body = ⸨c, y.atom⸩ • y.body) = (∀ c, c # (x, y) → AlphaEqPred c (x, y)) by
rw [alpha_eq_iff_pred, this]
apply Equivariance.for_fresh_iff_all
exact alpha_eq_pred_equivariant
simp [Prod.fresh, AlphaEqPred]
/-- An alternative definition of α-equivalence:
For ⟪a⟫ x = ⟪b⟫ y, we either have (a, x) = (b, y) or x is just (a b) y, requiring that a is not in the support of y. -/
lemma alpha_eq_iff_cases {x y : NameAbstraction' 𝔸 X} : AlphaEq x y ↔ (x.atom = y.atom ∧ x.body = y.body) ∨ (x.atom ≠ y.atom ∧ x.atom # y.body ∧ x.body = ⸨x.atom, y.atom⸩ • y.body) := by
constructor
· rw [alpha_eq_iff_exists]
intro h
obtain ⟨c, ⟨hcx, ⟨hcy, hc⟩⟩⟩ := h
by_cases h_eq : x.atom = y.atom
· left; constructor
· exact h_eq
· calc
x.body = ⸨c, x.atom⸩ • ⸨c, x.atom⸩ • x.body := by rw [← mul_smul, FPerm.swap_mul_self, one_smul]
_ = ⸨c, x.atom⸩ • ⸨c, y.atom⸩ • y.body := by rw [hc]
_ = ⸨c, y.atom⸩ • ⸨c, y.atom⸩ • y.body := by rw [h_eq]
_ = y.body := by rw [← mul_smul, FPerm.swap_mul_self, one_smul]
· right; constructor
· exact h_eq
· simp at hcx
simp at hcy
have h_atom : (⸨c, y.atom⸩ * ⸨c, x.atom⸩) • c # (⸨c, y.atom⸩ * ⸨c, x.atom⸩) • x.body :=
NominalSet.fresh_equivariant 𝔸 _ _ _ hcx.right
have h_swap : ⸨c, x.atom⸩ * ⸨c, y.atom⸩ = ⸨x.atom, y.atom⸩ * ⸨c, x.atom⸩ := by
ext a
by_cases a = c <;> by_cases a = x.atom <;> by_cases a = y.atom
all_goals simp_all [FPerm.swap_apply_of_ne_of_ne, hcy.left, hcx.left, Ne.symm hcy.left, Ne.symm hcx.left, Ne.symm h_eq]
rw [mul_smul, PermSet.atom_swap, PermSet.atom_swap, Equiv.swap_apply_left, Equiv.swap_apply_of_ne_of_ne (Ne.symm hcx.left) h_eq] at h_atom
rw [mul_smul, hc, ← mul_smul, FPerm.swap_mul_self, one_smul] at h_atom
constructor
· exact h_atom
· calc
x.body = ⸨c, x.atom⸩ • ⸨c, x.atom⸩ • x.body := by rw [← mul_smul, FPerm.swap_mul_self, one_smul]
_ = (⸨c, x.atom⸩ * ⸨c, y.atom⸩) • y.body := by rw [mul_smul, hc]
_ = ⸨x.atom, y.atom⸩ • ⸨c, x.atom⸩ • y.body := by rw [← mul_smul, h_swap]
_ = ⸨x.atom, y.atom⸩ • y.body := by rw [NominalSet.fresh_swap_fix 𝔸 hcy.right h_atom]
· rw [alpha_eq_iff_all]
intro h c ⟨hcx, hcy⟩
cases h with
| inl h =>
rw [h.left, h.right]
| inr h =>
simp at hcx
simp at hcy
have h_swap : ⸨c, x.atom⸩ * ⸨x.atom, y.atom⸩ = ⸨c, y.atom⸩ * ⸨c, x.atom⸩ := by
ext a
by_cases a = c <;> by_cases a = x.atom <;> by_cases a = y.atom
all_goals simp_all [FPerm.swap_apply_of_ne_of_ne, hcy.left, hcx.left, Ne.symm hcy.left, Ne.symm hcx.left, Ne.symm h.left]
rw [h.right.right, ← mul_smul, h_swap, mul_smul, NominalSet.fresh_swap_fix 𝔸 hcy.right h.right.left]
private lemma alpha_eq_refl (x : NameAbstraction' 𝔸 X) : AlphaEq x x := by
rw [alpha_eq_iff_cases]; left
trivial
private lemma alpha_eq_symm {x y : NameAbstraction' 𝔸 X} : AlphaEq x y → AlphaEq y x := by
rw [alpha_eq_iff_cases, alpha_eq_iff_cases]
intro h
cases h with
| inl h =>
left; exact ⟨h.left.symm, h.right.symm⟩
| inr h =>
right; constructor
· exact h.left.symm
· constructor
· have h : ⸨x.atom, y.atom⸩ • x.atom # x.body := by
rw [h.right.right]
exact NominalSet.fresh_equivariant 𝔸 _ _ _ h.right.left
simp at h; exact h
· rw [h.right.right, ← mul_smul, FPerm.swap_comm, FPerm.swap_mul_self, one_smul]
private lemma alpha_eq_trans {x y z : NameAbstraction' 𝔸 X} : AlphaEq x y → AlphaEq y z → AlphaEq x z := by
intro hxy hyz
have h : AlphaEq x y ∧ AlphaEq y z := ⟨hxy, hyz⟩
simp only [AlphaEq, AlphaEq, NominalSet.for_fresh_and] at h
apply NominalSet.for_fresh_imp
· exact h
· intro c ⟨hxy, hyz⟩
rw [hxy, hyz]
private lemma alpha_eq_equivariant : EquivariantRelation 𝔸 (AlphaEq : NameAbstraction' 𝔸 X → NameAbstraction' 𝔸 X → Prop) := by
intro x y π h
rw [alpha_eq_iff_exists] at *
obtain ⟨c, ⟨hcx, ⟨hcy, hc⟩⟩⟩ := h
use π • c
constructor
· apply NominalSet.fresh_equivariant; exact hcx
· constructor
· apply NominalSet.fresh_equivariant; exact hcy
· -- XXX: This is largely similar to alpha_eq_pred...
have h' : π • ⸨c, x.atom⸩ • x.body = π • ⸨c, y.atom⸩ • y.body := by congr
simp only [smul_smul, FPerm.mul_swap_eq_swap_mul] at h'
simpa [smul_smul]
end NameAbstraction
end AlphaEquivalence
/-!
## Quotient Type
We then define `NameAbstraction` as the quotient type of this structure modulo alpha-equivalence.
-/
section Quotient
variable (X : Type) [NominalSet 𝔸 X]
/-- Associate the name abstraction structure with alpha-equivalence. -/
instance NameAbstraction.setoid : EquivariantSetoid 𝔸 (NameAbstraction' 𝔸 X) where
r := AlphaEq
iseqv := ⟨alpha_eq_refl, alpha_eq_symm, alpha_eq_trans⟩
equivariant := alpha_eq_equivariant
/-- Define name abstractions as the quotient set (𝔸 × X)/≈. -/
def NameAbstraction := Quotient (NameAbstraction.setoid 𝔸 X).toSetoid
/-- The equivalence class of a name abstraction (a, x), often written `⟪a⟫ x` -/
def NameAbstraction.mk {𝔸 : Type} [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸] {X : Type} [NominalSet 𝔸 X] (a : 𝔸) (x : X) : NameAbstraction 𝔸 X :=
Quotient.mk (setoid 𝔸 X).toSetoid (.mk a x)
/-- We write `⟪a⟫ x` instead of the usual `⟨a⟩ x` because `⟨⟩` is already heavily used in Lean for constructors. -/
notation "⟪" a "⟫ " t => NameAbstraction.mk a t
instance : NominalSet 𝔸 (NameAbstraction 𝔸 X) := Quotient.instNominalSet 𝔸
@[simp]
lemma NameAbstraction.smul {π : FPerm 𝔸} {a : 𝔸} {x : X } : π • (⟪a⟫ x) = ⟪π • a⟫ π • x := by congr
@[simp]
lemma NameAbstraction.supp {a : 𝔸} {x : X} : NominalSet.supp 𝔸 (⟪a⟫ x) = NominalSet.supp 𝔸 x \ {a} := by
sorry
-- ext b; constructor
-- · apply NominalSet.support_subset_supp 𝔸
-- · intro π hπ
-- simp [PermSet.fix']
-- sorry
-- · exact Set.Finite.diff (NominalSet.supp_finite 𝔸)
-- · rw [mk, Quotient.supp]
-- intro hb x' hx'
-- sorry
end Quotient
/-!
## Nominal Set Instance
The equivalence classes of name abstractions form a nominal set where supp (⟪a⟫ x) = supp x \ {a}.
-/
section NominalSet
end NominalSet
/-!
## Abstraction Functor
Name abstraction can also be considered as a functor `Nom → Nom`.
-/
section Functor
end Functor
|
