| import Mathlib |
| import OAI.GroupTheory.SimpleOvergroups.ArrayNames |
| import OAI.GroupTheory.SimpleOvergroups.EnlargementCore |
|
|
| namespace OAI |
|
|
| namespace SimpleFInftyOvergroups.EffectiveNames.Array |
| noncomputable section |
| open scoped Classical |
| variable {A G X D V Y : Type*} |
|
|
| def product (cmul : A → A → A) (op : G → X → X) (p : Name A G) (q : Name A X) : Name A X := |
| p.flatMap fun a => q.map fun b => (cmul a.1 b.1,op a.2 b.2) |
|
|
| @[simp] theorem product_nil (cmul : A → A → A) (op : G → X → X) (q : Name A X) : |
| product cmul op [] q = [] := rfl |
| @[simp] theorem product_cons (cmul : A → A → A) (op : G → X → X) |
| (a : A × G) (p : Name A G) (q : Name A X) : |
| product cmul op (a::p) q = (q.map fun b => (cmul a.1 b.1,op a.2 b.2)) ++ product cmul op p q := rfl |
|
|
| theorem product_bound (cmul : A → A → A) (op : G → X → X) |
| (gc : G → ℕ) (xc : X → ℕ) (hop : ∀ g x, xc (op g x) = gc g + xc x) |
| {p : Name A G} {q : Name A X} {b c : ℕ} |
| (hp : ∀ a ∈ p, gc a.2 ≤ b) (hq : ∀ a ∈ q, xc a.2 ≤ c) : |
| ∀ a ∈ product cmul op p q, xc a.2 ≤ b+c := by |
| intro a ha |
| obtain ⟨u,hu,hv⟩ := List.mem_flatMap.mp ha |
| obtain ⟨w,hw,rfl⟩ := List.mem_map.mp hv |
| simpa only [hop] using Nat.add_le_add (hp u hu) (hq w hw) |
|
|
| variable [Ring D] [Group V] |
| variable (coeff : A → D) (group : G → V) |
|
|
| def algebraValue (p : Name A G) : MonoidAlgebra D V := MonoidAlgebra.ofCoeff (value coeff group p) |
|
|
| @[simp] theorem algebraValue_nil : algebraValue coeff group [] = 0 := rfl |
| @[simp] theorem algebraValue_cons (a : A × G) (p : Name A G) : |
| algebraValue coeff group (a::p) = MonoidAlgebra.single (group a.2) (coeff a.1) + algebraValue coeff group p := rfl |
| @[simp] theorem algebraValue_append (p q : Name A G) : |
| algebraValue coeff group (p++q) = algebraValue coeff group p + algebraValue coeff group q := by |
| simp only [algebraValue,value_append,MonoidAlgebra.ofCoeff_add] |
|
|
| theorem algebraValue_product (cmul : A → A → A) (mul : G → G → G) |
| (hc : ∀ a b, coeff (cmul a b) = coeff a * coeff b) |
| (hg : ∀ g h, group (mul g h) = group g * group h) (p q : Name A G) : |
| algebraValue coeff group (product cmul mul p q) = algebraValue coeff group p * algebraValue coeff group q := by |
| induction p with |
| | nil => simp |
| | cons a p ih => |
| rw [product_cons,algebraValue_append,ih,algebraValue_cons,add_mul] |
| congr 1 |
| clear ih |
| induction q with |
| | nil => simp |
| | cons b q ih => |
| simp only [List.map_cons,algebraValue_cons,ih,hc,hg,mul_add,MonoidAlgebra.single_mul_single] |
|
|
| variable [MulAction V Y] (point : X → Y) |
|
|
| theorem value_action (cmul : A → A → A) (act : G → X → X) |
| (hc : ∀ a b, coeff (cmul a b) = coeff a * coeff b) |
| (ha : ∀ g x, point (act g x) = group g • point x) (p : Name A G) (q : Name A X) : |
| value coeff point (product cmul act p q) = |
| SourceEnlargement.ringAction (algebraValue coeff group p) (value coeff point q) := by |
| induction p with |
| | nil => simp |
| | cons a p ih => |
| rw [product_cons,value_append,ih,algebraValue_cons,map_add] |
| change _ = SourceEnlargement.ringAction (MonoidAlgebra.single (group a.2) (coeff a.1)) |
| (value coeff point q) + SourceEnlargement.ringAction (algebraValue coeff group p) (value coeff point q) |
| congr 1 |
| clear ih |
| induction q with |
| | nil => simp |
| | cons b q ih => |
| simp only [List.map_cons,value_cons,ih,hc,ha,map_add,SourceEnlargement.ringAction_single] |
|
|
| variable [Primcodable A] [Primcodable G] [Primcodable X] |
| theorem primrec_product {cmul : A → A → A} {op : G → X → X} |
| (hc : Primrec₂ cmul) (ho : Primrec₂ op) : Primrec₂ (product cmul op) := by |
| unfold product |
| apply Primrec.list_flatMap Primrec.fst |
| change Primrec (fun p : (Name A G × Name A X) × (A × G) => |
| p.1.2.map fun b => (cmul p.2.1 b.1,op p.2.2 b.2)) |
| apply Primrec.list_map (Primrec.snd.comp Primrec.fst) |
| exact ((hc.comp (Primrec.fst.comp (Primrec.snd.comp Primrec.fst)) |
| (Primrec.fst.comp Primrec.snd)).pair |
| (ho.comp (Primrec.snd.comp (Primrec.snd.comp Primrec.fst)) |
| (Primrec.snd.comp Primrec.snd))).to₂ |
|
|
| end |
| end SimpleFInftyOvergroups.EffectiveNames.Array |
|
|
| end OAI |
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|
