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import Mathlib
import Nominal.Support
import Nominal.Equivariance
import Nominal.Category
import Nominal.Permutations
import Nominal.Instances.Basic
import Nominal.Instances.Abstraction
import Nominal.NameRestriction.Finsupp
variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
-- M₀ algebra structure: Equivariant join semilattices with name restriction
class AlgebraStructure (X : Type) [NominalSet 𝔸 X] where
bot : Unit →ₑ[𝔸] X
join : (X × X) →ₑ[𝔸] X
res : NameAbstraction 𝔸 X →ₑ[𝔸] X
ax_neutral : ∀ x, join (x, bot ()) = x
ax_idempotent : ∀ x, join (x, x) = x
ax_commutative : ∀ x y, join (x, y) = join (y, x)
ax_associative : ∀ x y z, join (join (x, y), z) = join (x, join (y, z))
ax_fresh : ∀ a x, a # x → res (⟪a⟫ x) = x
ax_swap : ∀ a b x, res (⟪a⟫ res (⟪b⟫ x)) = res (⟪b⟫ res (⟪a⟫ x))
ax_distributive : ∀ a x y, res (⟪a⟫ join (x, y)) = join (res (⟪a⟫ x), res (⟪a⟫ y))
ax_redundant : ∀ a x, x = join (x, res (⟪a⟫ x))
private abbrev AlgebraStructure.bot' {X : Type} [NominalSet 𝔸 X] [AlgebraStructure 𝔸 X] :=
(AlgebraStructure.bot : Unit →ₑ[𝔸] X)
private abbrev AlgebraStructure.join' {X : Type} [NominalSet 𝔸 X] [AlgebraStructure 𝔸 X] :=
(AlgebraStructure.join : (X × X) →ₑ[𝔸] X)
notation "⊥[" A "]" => AlgebraStructure.bot' A ()
notation:60 x " ∨[" A "] " y => AlgebraStructure.join' A (x, y)
infixl:60 " \\ " => fun a x => AlgebraStructure.res (⟪a⟫ x)
namespace AlgebraStructure
variable {X : Type} [NominalSet 𝔸 X] [AlgebraStructure 𝔸 X]
lemma bot_supp : NominalSet.supp 𝔸 (⊥[𝔸] : X) = ∅ := by
rw [← Set.subset_empty_iff]
show NominalSet.supp 𝔸 ((bot' 𝔸).toFun ()) ⊆ ∅
grw [Function.Equivariant.supp_subset]
simp
lemma join_equivariant {x y : X} {π : FPerm 𝔸} : π • (x ∨[𝔸] y) = (π • x) ∨[𝔸] (π • y) := by
show π • (join' 𝔸).toFun (x, y) = (join' 𝔸).toFun (π • (x, y))
rw [← Function.equivariant]
lemma res_equivariant {a : 𝔸} {x : X} {π : FPerm 𝔸} : π • (a \ x) = (π • a) \ (π • x) := by
show π • res.toFun (⟪a⟫ x) = res.toFun (π • (⟪a⟫ x))
rw [← Function.equivariant]
end AlgebraStructure
-- Base Model: M₀ X
def BaseModel (X : Type) [NominalSet 𝔸 X] : Type :=
PermSet.FinSet 𝔸 X
namespace BaseModel
variable {X : Type} [NominalSet 𝔸 X]
instance : NominalSet 𝔸 (BaseModel 𝔸 X) := PermSet.FinSet.instNominalSet 𝔸
instance : EmptyCollection (BaseModel 𝔸 X) := PermSet.FinSet.instEmptyCollection 𝔸
instance : Union (BaseModel 𝔸 X) := PermSet.FinSet.instUnion 𝔸
instance : Inter (BaseModel 𝔸 X) := PermSet.FinSet.instInter 𝔸
instance : SDiff (BaseModel 𝔸 X) := PermSet.FinSet.instSDiff 𝔸
@[simp] lemma empty_val : (∅ : BaseModel 𝔸 X).val = ∅ := rfl
@[simp] lemma union_val {S₁ S₂ : BaseModel 𝔸 X} : (S₁ ∪ S₂).val = S₁.val ∪ S₂.val := rfl
@[simp] lemma inter_val {S₁ S₂ : BaseModel 𝔸 X} : (S₁ ∩ S₂).val = S₁.val ∩ S₂.val := rfl
@[simp] lemma sdiff_val {S₁ S₂ : BaseModel 𝔸 X} : (S₁ \ S₂).val = S₁.val \ S₂.val := rfl
@[simp] lemma smul_val {L : BaseModel 𝔸 X} {π : FPerm 𝔸} : (π • L).val = π • L.val := rfl
def bot : Unit → BaseModel 𝔸 X :=
fun _ ↦ ∅
instance : Function.Equivariant 𝔸 (bot 𝔸 : Unit → BaseModel 𝔸 X) := by
constructor; intro _ π
apply Subtype.eq; simp [bot]
def union : (BaseModel 𝔸 X × BaseModel 𝔸 X) → BaseModel 𝔸 X := PermSet.FinSet.union 𝔸
instance : Function.Equivariant 𝔸 (union 𝔸 : (BaseModel 𝔸 X × BaseModel 𝔸 X) → BaseModel 𝔸 X) :=
PermSet.FinSet.instEquivariantProdUnion 𝔸
def restrict : NameAbstraction 𝔸 (BaseModel 𝔸 X) → BaseModel 𝔸 X := Quotient.lift
(fun ⟨a, S⟩ ↦ ⟨{x ∈ S.val | a # x}, by
apply Set.Finite.subset S.prop; simp⟩)
(by
intro ⟨a₁, S₁⟩ ⟨a₂, S₂⟩ haeq
apply Subtype.eq
simp only
rw [← NameAbstraction.eq_iff_alpha_eq 𝔸 (BaseModel 𝔸 X), NameAbstraction.eq_iff_cases] at haeq
cases haeq with
| inl h => rw [h.left, h.right]
| inr h =>
ext x; constructor
· intro ⟨hx, ha₁⟩
have hx' : ⸨a₁, a₂⸩ • x ∈ S₂.val := by
rw [← Set.mem_inv_smul_set_iff, FPerm.swap_inv]
show x ∈ (⸨a₁, a₂⸩ • S₂).val
rw [← h.right.right]; exact hx
have ha₂ := h.right.left
rw [PermSet.FinSet.fresh] at ha₂
specialize ha₂ (⸨a₁, a₂⸩ • x) hx'
nth_rw 1 [show a₁ = ⸨a₁, a₂⸩ • a₂ by simp] at ha₂
rw [← Rel.equivariant 𝔸 NominalSet.IsFresh] at ha₂
constructor
· rw [← show ⸨a₁, a₂⸩ • x = x from NominalSet.fresh_swap_fix 𝔸 ha₁ ha₂]
exact hx'
· exact ha₂
· intro ⟨hx, ha₂⟩
have hx' : ⸨a₁, a₂⸩ • x ∈ S₁.val := by
rw [← Set.mem_inv_smul_set_iff, FPerm.swap_inv]
show x ∈ (⸨a₁, a₂⸩ • S₁).val
rw [h.right.right, ← MulAction.mul_smul, FPerm.swap_mul_self, MulAction.one_smul]
exact hx
have ha₁ := h.right.left
rw [PermSet.FinSet.fresh] at ha₁
specialize ha₁ x hx
rw [← show ⸨a₁, a₂⸩ • x = x from NominalSet.fresh_swap_fix 𝔸 ha₁ ha₂] at hx'
rw [← MulAction.mul_smul, FPerm.swap_mul_self, MulAction.one_smul] at hx'
exact ⟨hx', ha₁⟩)
instance : Function.Equivariant 𝔸 (restrict 𝔸 : NameAbstraction 𝔸 (BaseModel 𝔸 X) → BaseModel 𝔸 X) := by
constructor
apply Quotient.ind
intro ⟨a, S⟩ π
apply Subtype.eq
simp only [restrict, NameAbstraction.smul, Quotient.lift_mk, smul_val]
ext x; constructor
· intro ⟨hx, hfresh⟩
rw [Set.mem_smul_set_iff_inv_smul_mem]; constructor
· rw [← Set.mem_smul_set_iff_inv_smul_mem]; exact hx
· rw [show a = π⁻¹ • π • a by simp, ← Rel.equivariant 𝔸 NominalSet.IsFresh]
exact hfresh
· rw [Set.mem_smul_set_iff_inv_smul_mem]
intro ⟨hx, hfresh⟩; constructor
· rw [Set.mem_smul_set_iff_inv_smul_mem]; exact hx
· rw [show x = π • π⁻¹ • x by simp, ← Rel.equivariant 𝔸 NominalSet.IsFresh]
exact hfresh
lemma restrict_eq {a : 𝔸} {S : BaseModel 𝔸 X} :
(restrict 𝔸 (⟪a⟫ S)).val = {x ∈ S.val | a # x} := rfl
instance : AlgebraStructure 𝔸 (BaseModel 𝔸 X) where
bot := ⟨bot 𝔸, inferInstance⟩
join := ⟨union 𝔸, inferInstance⟩
res := ⟨restrict 𝔸, inferInstance⟩
ax_neutral x := by
apply Subtype.eq
show x.val ∪ ∅ = x.val
simp
ax_idempotent x := by
apply Subtype.eq
show x.val ∪ x.val = x.val
simp
ax_commutative x y := by
apply Subtype.eq
show x.val ∪ y.val = y.val ∪ x.val
apply Set.union_comm
ax_associative x y z := by
apply Subtype.eq
show (x.val ∪ y.val) ∪ z.val = x.val ∪ (y.val ∪ z.val)
apply Set.union_assoc
ax_fresh a S := by
intro hfresh
show restrict 𝔸 (⟪a⟫ S) = S
apply Subtype.eq
rw [restrict_eq]
rw [PermSet.FinSet.fresh] at hfresh
ext x; constructor
· intro ⟨hS, _⟩; exact hS
· intro hS; exact ⟨hS, hfresh x hS⟩
ax_swap a b S := by
show restrict 𝔸 (⟪a⟫ restrict 𝔸 (⟪b⟫ S)) = restrict 𝔸 (⟪b⟫ restrict 𝔸 (⟪a⟫ S))
apply Subtype.eq
ext x
simp only [restrict_eq, Set.mem_setOf_eq]
ac_nf
ax_distributive a S₁ S₂ := by
show restrict 𝔸 (⟪a⟫ S₁ ∪ S₂) = restrict 𝔸 (⟪a⟫ S₁) ∪ restrict 𝔸 (⟪a⟫ S₂)
apply Subtype.eq
simp [restrict_eq]
ax_redundant a S := by
show S = S ∪ restrict 𝔸 (⟪a⟫ S)
apply Subtype.eq; ext x
simp only [union_val, restrict_eq, Set.mem_union, Set.mem_setOf_eq, iff_self_or, and_imp]
intro h _; exact h
@[simp] lemma bot_eq : (⊥[𝔸] : BaseModel 𝔸 X) = ∅ := rfl
@[simp] lemma join_eq {S₁ S₂ : BaseModel 𝔸 X} : (S₁ ∨[𝔸] S₂) = S₁ ∪ S₂ := rfl
@[simp] lemma res_eq {a : 𝔸} {S : BaseModel 𝔸 X} : (a \ S) = restrict 𝔸 (⟪a⟫ S) := rfl
end BaseModel
-- Step Model: \bar{M₁} X
def permissibleFunctions (X : Type) [NominalSet 𝔸 X] [AlgebraStructure 𝔸 X] :=
{ f : PermSet.FinsuppFunction 𝔸 𝔸 X |
∀ a,
PermSet.HasSupport 𝔸 (f.val a) (NominalSet.supp 𝔸 f ∪ {a}) ∧
∀ c, c # f → c ≠ a → f.val a = f.val a ∨[𝔸] (c \ (⸨a, c⸩ • f).val a) }
def StepModel (X : Type) [NominalSet 𝔸 X] [AlgebraStructure 𝔸 X] : Type :=
permissibleFunctions 𝔸 X
namespace StepModel
variable {X : Type} [NominalSet 𝔸 X] [AlgebraStructure 𝔸 X]
instance : Set.Equivariant 𝔸 (permissibleFunctions 𝔸 X) := by
constructor; intro f π hf a
simp only [PermSet.Finsupp.smul_val, PermSet.smul_function, FPerm.swap_inv,
PermSet.atom_swap, Equiv.swap_apply_left]
constructor
· have hf₁ := (hf (π⁻¹ • a)).left
rw [show NominalSet.supp 𝔸 (π • f) = π • NominalSet.supp 𝔸 f from Function.equivariant _ _ _ _]
rw [show ({a} : Set 𝔸) = π • {π⁻¹ • a} by simp]
rw [← Set.smul_set_union]
rw [← Rel.equivariant 𝔸 (PermSet.HasSupport 𝔸)]
exact hf₁
· intro c hcf hca
have hf₂ := (hf (π⁻¹ • a)).right (π⁻¹ • c)
specialize hf₂ (by
rw [show f = π⁻¹ • π • f by simp]
rw [← Rel.equivariant 𝔸 NominalSet.IsFresh]
exact hcf)
specialize hf₂ (by
intro hcontra
have : π • π⁻¹ • c = π • π⁻¹ • a := by rw [hcontra]
simp only [smul_inv_smul] at this
exact hca this)
nth_rw 1 [hf₂]
rw [AlgebraStructure.join_equivariant, AlgebraStructure.res_equivariant]
simp only [smul_inv_smul, PermSet.Finsupp.smul_val, PermSet.smul_function, FPerm.swap_inv,
PermSet.atom_swap, Equiv.swap_apply_left]
suffices π • ⸨π⁻¹ • a,π⁻¹ • c⸩ • f.val (π⁻¹ • c) = ⸨a, c⸩ • π • f.val (π⁻¹ • c) by rw [this]
rw [← MulAction.mul_smul, ← MulAction.mul_smul]
suffices π * ⸨π⁻¹ • a,π⁻¹ • c⸩ = ⸨a,c⸩ * π by rw [this]
rw [FPerm.mul_swap_eq_swap_mul, show π⁻¹ • a = π⁻¹ a from rfl, show π⁻¹ • c = π⁻¹ c from rfl]
simp
instance : NominalSet 𝔸 (StepModel 𝔸 X) := Subtype.instNominalSetElem 𝔸
@[simp] lemma smul_val (π : FPerm 𝔸) (f : StepModel 𝔸 X) : (π • f).val = π • f.val :=
Subtype.smul_val 𝔸 π f
instance instValEquivariant : Function.Equivariant 𝔸 (Subtype.val : StepModel 𝔸 X → PermSet.FinsuppFunction 𝔸 𝔸 X) :=
Subtype.instValEquivariant 𝔸
@[simp] lemma supp {f : StepModel 𝔸 X} : NominalSet.supp 𝔸 f = NominalSet.supp 𝔸 f.val :=
Subtype.supp 𝔸
@[simp] lemma fresh {a : 𝔸} {f : StepModel 𝔸 X} : a # f ↔ a # f.val :=
Subtype.fresh 𝔸
end StepModel
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