| import Mathlib |
| import OAI.Combinatorics.IndependentSets.PCP.LCTable |
| import OAI.Combinatorics.IndependentSets.Geometry.GeometryNaming |
|
|
| namespace OAI |
|
|
| namespace LargeIndependentSets.GeometryNames |
| open IndependentSetsCut.CounterMachine UniformLC |
| open scoped Classical BigOperators |
| noncomputable section |
|
|
| variable (L R : Type) [Fintype L] [Fintype R] (n D : ℕ) [NeZero D] |
|
|
| abbrev staticCount := Fintype.card (StaticPoint L R n D) |
| abbrev size (t : Table L R) := (t.nu^n*t.nv^n)*staticCount L R n D |
|
|
| lemma staticCount_pos : 0<staticCount L R n D := by |
| let : Nonempty (StaticPoint L R n D) := ⟨⟨⟨0,by omega⟩,fun _ => 0⟩⟩ |
| exact Fintype.card_pos |
| lemma size_pos (t : Table L R) : 0<size L R n D t := by |
| exact Nat.mul_pos (Nat.mul_pos (pow_pos t.hu _) (pow_pos t.hv _)) (staticCount_pos L R n D) |
|
|
| def sizeExpr : Expr := .mul (.mul (Expr.power nuExpr n) (Expr.power nvExpr n)) (.const (staticCount L R n D)) |
| def pointExpr (x : Expr) : Expr := Expr.remainder x (.const (staticCount L R n D)) |
| def uExpr (x : Expr) (k : Fin n) : Expr := |
| Expr.remainder (Expr.quotient |
| (Expr.quotient (Expr.quotient x (.const (staticCount L R n D))) (Expr.power nvExpr n)) |
| (Expr.power nuExpr k.val)) nuExpr |
|
|
| def vExpr (x : Expr) (k : Fin n) : Expr := |
| Expr.remainder (Expr.quotient |
| (Expr.remainder (Expr.quotient x (.const (staticCount L R n D))) (Expr.power nvExpr n)) |
| (Expr.power nvExpr k.val)) nvExpr |
|
|
| @[simp] lemma sizeExpr_eval (t : Table L R) (a : ℕ → ℕ) : |
| (sizeExpr L R n D).eval t.bits a=size L R n D t := by simp [sizeExpr,Expr.eval,size] |
|
|
| lemma uExpr_eval (t : Table L R) (x : Expr) (a : ℕ → ℕ) (i : Fin (size L R n D t)) |
| (hi : x.eval t.bits a=i.val) (k : Fin n) : |
| (uExpr L R n D x k).eval t.bits a=(((nodeEquiv L R n D t.nu t.nv).symm i).1.1 k).val := by |
| simp only [uExpr,Expr.remainder_eval,Expr.quotient_eval,Expr.power_eval,nuExpr_eval,nvExpr_eval,Expr.eval,hi] |
| rfl |
| lemma vExpr_eval (t : Table L R) (x : Expr) (a : ℕ → ℕ) (i : Fin (size L R n D t)) |
| (hi : x.eval t.bits a=i.val) (k : Fin n) : |
| (vExpr L R n D x k).eval t.bits a=(((nodeEquiv L R n D t.nu t.nv).symm i).1.2 k).val := by |
| simp only [vExpr,Expr.remainder_eval,Expr.quotient_eval,Expr.power_eval,nvExpr_eval,Expr.eval,hi] |
| rfl |
| lemma pointExpr_eval (t : Table L R) (x : Expr) (a : ℕ → ℕ) (i : Fin (size L R n D t)) |
| (hi : x.eval t.bits a=i.val) : |
| (pointExpr L R n D x).eval t.bits a= |
| (Fintype.equivFin (StaticPoint L R n D) ((nodeEquiv L R n D t.nu t.nv).symm i).2).val := by |
| simp only [pointExpr,Expr.remainder_eval,Expr.eval,hi] |
| change _=(Fintype.equivFin _ ((Fintype.equivFin _).symm _)).val |
| rw [Equiv.apply_symm_apply] |
| rfl |
|
|
| def sameQuestion {U V : Type} (u u' : Fin n → U) (v v' : Fin n → V) |
| (a b : StaticPoint L R n D) : Prop := |
| a.1=b.1 ∧ ∀ k, if k.val<a.1.val then v k=v' k else u k=u' k |
|
|
| omit [Fintype L] [Fintype R] [NeZero D] in |
| lemma sameQuestion_iff {U V : Type} (u u' : Fin n → U) (v v' : Fin n → V) |
| (a b : StaticPoint L R n D) : |
| sameQuestion L R n D u u' v v' a b ↔ (decode u v a).1=(decode u' v' b).1 := by |
| rcases a with ⟨i,a⟩ |
| rcases b with ⟨j,b⟩ |
| simp only [sameQuestion,decode] |
| constructor |
| · rintro ⟨rfl,h⟩ |
| congr 1 |
| apply Subtype.ext |
| funext k |
| have hk := h k |
| by_cases hl : k.val < i.val <;> simpa [mixed_val,hl] using hk |
| · intro h |
| have hij : i=j := congrArg Sigma.fst h |
| subst j |
| refine ⟨rfl,?_⟩ |
| have hq : mixed u v i.val=mixed u' v' i.val := Sigma.mk.inj_iff.mp h |>.2 |> eq_of_heq |
| intro k |
| have hh := congrFun (congrArg Subtype.val hq) k |
| by_cases hl : k.val < i.val <;> simpa [mixed_val,hl] using hh |
|
|
| end |
| end LargeIndependentSets.GeometryNames |
|
|
| end OAI |
|
|
