File size: 4,217 Bytes
ea76aac | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | 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
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