| import Mathlib |
| import OAI.Computability.DeterministicSum.SparseCodes |
|
|
| namespace OAI |
|
|
| /-! Initialized evaluation grids and exact Vandermonde tables. -/ |
|
|
| namespace DeterministicThreeSum.Structured.Indexed |
| open Command |
|
|
| lemma integerTableMemory_outside (T base i : ℕ) (xs : List ℤ) (mem : ℕ → Option ℕ) (a : ℕ) |
| (ha : a<base+i ∨ base+i+xs.length≤a) : integerTableMemory T base i xs mem a=mem a := by |
| induction xs generalizing i mem with |
| | nil => rfl |
| | cons z xs ih => |
| rw [integerTableMemory,ih (i+1) _ (by simp only [List.length_cons] at ha; omega)] |
| exact Function.update_of_ne (by simp only [List.length_cons] at ha; omega) _ _ |
|
|
| lemma integerTableMemory_entry (T base i : ℕ) (xs : List ℤ) (mem : ℕ → Option ℕ) |
| (j : ℕ) (hj : j<xs.length) : |
| integerTableMemory T base i xs mem (base+i+j)=some (integerValue T xs[j]) := by |
| induction xs generalizing i mem j with |
| | nil => simp at hj |
| | cons a xs ih => |
| cases j with |
| | zero => |
| simp only [integerTableMemory,Nat.add_zero,List.getElem_cons_zero] |
| rw [integerTableMemory_outside _ _ _ _ _ _ (by omega)] |
| simp |
| | succ j => |
| have hj' : j<xs.length := by simpa only [List.length_cons,Nat.succ_lt_succ_iff] using hj |
| have he : base+i+(j+1)=base+(i+1)+j := by omega |
| simpa only [integerTableMemory,he,List.getElem_cons_succ] using |
| ih (i+1) (Function.update mem (base+i) (some (integerValue T a))) j hj' |
| end DeterministicThreeSum.Structured.Indexed |
| namespace DeterministicThreeSum.Structured.Indexed.Cache |
| open Command Axis SparseAxis DeterministicThreeSum.Rectangular Finset |
| open scoped BigOperators |
|
|
| def powerTable (T q i : ℕ) : ℕ := (i/q)^(i%q)%T |
|
|
| def powerIntegers (q : ℕ) : List ℤ := |
| List.ofFn (fun i : Fin (q*q) => (((i.val/q)^(i.val%q):ℕ):ℤ)) |
|
|
| def powerTableCommand (q : ℕ) : Command := integerTable 0 (powerIntegers q) |
|
|
| lemma powerTable_lt {T : ℕ} (hT : 0 < T) (q i : ℕ) : powerTable T q i < T := Nat.mod_lt _ hT |
|
|
| lemma powerTable_entry {T q : ℕ} (hq : 0 < q) (v u : Fin q) : |
| (powerTable T q (v.val*q+u.val):ZMod T)=(v.val:ZMod T)^u.val := by |
| have hd : (v.val*q+u.val)/q=v.val := by |
| rw [Nat.mul_comm v.val q,Nat.mul_add_div hq,Nat.div_eq_of_lt u.isLt,Nat.add_zero] |
| simp [powerTable,hd,Nat.add_mod,Nat.mod_eq_of_lt u.isLt] |
|
|
| theorem powerTableCommand_correct {w T base q : ℕ} (s : Data) |
| (hT : 0 < T) (hTw : T < wordModulus w) (hbase : base+q*q < wordModulus w) |
| (hbound : ∀ i, i < q*q → (i/q)^(i%q) < wordModulus w) |
| (h2 : s.registers 2=T) (h3 : s.registers 3=base) : |
| ∃ cost z, Eval w (powerTableCommand q) s cost z ∧ cost ≤ 6*(q*q) ∧ |
| (∀ i, i < q*q → z.memory (base+i)=some (powerTable T q i)) ∧ |
| (∀ a, a < base ∨ base+q*q ≤ a → z.memory a=s.memory a) ∧ |
| (∀ r, 2 ≤ r → z.registers r=s.registers r) := by |
| have hl : (powerIntegers q).length=q*q := by simp [powerIntegers] |
| obtain ⟨cost,z,he,hc,hzm,hzr⟩:=integerTable_correct (i:=0) s (powerIntegers q) hT hTw |
| (by simpa only [Nat.add_zero,hl] using hbase) (by |
| intro a ha |
| obtain ⟨i,hi⟩:=List.mem_ofFn.mp ha |
| subst a |
| exact hbound i.val i.isLt) h2 h3 |
| refine ⟨cost,z,he,by simpa only [hl] using hc,?_,?_,hzr⟩ |
| · intro i hi |
| rw [hzm] |
| have hh:=integerTableMemory_entry T base 0 (powerIntegers q) s.memory i (by rwa [hl]) |
| simpa only [Nat.add_zero,powerIntegers,List.getElem_ofFn,integerValue_nat,powerTable] using hh |
| · intro a ha |
| rw [hzm] |
| exact integerTableMemory_outside T base 0 _ _ a (by simpa only [Nat.add_zero,hl] using ha) |
|
|
| lemma tensorValue_evaluation {T q : ℕ} (hq : 0 < q) (d : ℕ) (x : ℕ → ℕ) |
| (v : DigitBox q d) : |
| (tensorValue T q q (powerTable T q) d x (lowCode q d v):ZMod T)= |
| ∑ u : DigitBox q d, (x (lowCode q d u):ZMod T)* |
| ∏ k : Fin d, ((digitFunction q d v k).val:ZMod T)^(digitFunction q d u k).val := by |
| have he:=NewtonPass.tensorValue_low (T:=T) hq hq (powerTable T q) d 0 x v |
| simp only [Nat.zero_mul,Nat.zero_add] at he |
| rw [he] |
| apply sum_congr rfl |
| intro u _ |
| rw [digitCoefficient_product] |
| simp_rw [powerTable_entry hq] |
| rw [mul_comm] |
| end DeterministicThreeSum.Structured.Indexed.Cache |
|
|
| end OAI |
|
|
