| import Mathlib |
| import OAI.Computability.DeterministicSum.AxisMaps |
| import OAI.Computability.DeterministicSum.BilinearForms |
|
|
| namespace OAI |
|
|
| /-! Digit encodings for tensor transforms. -/ |
|
|
| namespace DeterministicThreeSum.Structured.Indexed.Axis |
| open Command Finset DeterministicThreeSum.Rectangular |
| open scoped BigOperators |
|
|
| def digitCode (q : ℕ) : (d : ℕ) → DigitBox q d → ℕ |
| | 0,_ => 0 |
| | d+1,u => u.1.val*q^d+digitCode q d u.2 |
|
|
| lemma digitCode_lt {q : ℕ} (_hq : 0<q) (d : ℕ) (u : DigitBox q d) : digitCode q d u<q^d := by |
| induction d with |
| | zero => simp [digitCode] |
| | succ d ih => |
| have hu:=u.1.isLt |
| have ht:=ih u.2 |
| have hl:=Nat.mul_le_mul_right (q^d) (show u.1.val+1≤q by omega) |
| simp only [digitCode,pow_succ'] |
| nlinarith only [hl,ht] |
|
|
| def digitCoefficient (T q r : ℕ) (c : ℕ → ℕ) : (d : ℕ) → DigitBox r d → DigitBox q d → ZMod T |
| | 0,_,_ => 1 |
| | d+1,v,u => (c (v.1.val*q+u.1.val):ZMod T)*digitCoefficient T q r c d v.2 u.2 |
|
|
| theorem tensorValue_eq_tensor {T q r : ℕ} (hq : 0<q) (hr : 0<r) |
| (c : ℕ → ℕ) (d a : ℕ) (x : ℕ → ℕ) (v : DigitBox r d) : |
| (tensorValue T q r c d x (a*r^d+digitCode r d v):ZMod T)= |
| ∑ u : DigitBox q d, digitCoefficient T q r c d v u * |
| (x (a*q^d+digitCode q d u):ZMod T) := by |
| induction d generalizing a x with |
| | zero => simp [tensorValue,digitCode,digitCoefficient] |
| | succ d ih => |
| have hidx : a*r^(d+1)+digitCode r (d+1) v= |
| (a*r+v.1.val)*r^d+digitCode r d v.2 := by |
| simp only [digitCode,pow_succ']; ring |
| rw [tensorValue,hidx,ih] |
| simp only [digitCoefficient,digitCode,Fintype.sum_prod_type] |
| calc |
| (∑ u : DigitBox q d, digitCoefficient T q r c d v.2 u * |
| (mapValue T q r (q^d) x c ((a*r+v.1.val)*q^d+digitCode q d u):ZMod T)) = |
| ∑ u : DigitBox q d, ∑ h : Fin q, digitCoefficient T q r c d v.2 u * |
| (c (v.1.val*q+h.val):ZMod T)*(x (boxIndex q (q^d) a h.val (digitCode q d u)):ZMod T) := by |
| apply sum_congr rfl |
| intro u _ |
| rw [show (a*r+v.1.val)*q^d+digitCode q d u=boxIndex r (q^d) a v.1.val (digitCode q d u) from rfl, |
| mapValue_box x c hr (pow_pos hq _) v.1.isLt (digitCode_lt hq d u),mul_sum] |
| apply sum_congr rfl |
| intro h _ |
| ring |
| _ = _ := by |
| rw [sum_comm] |
| apply sum_congr rfl |
| intro h _ |
| apply sum_congr rfl |
| intro u _ |
| have he : boxIndex q (q^d) a h.val (digitCode q d u)=a*q^(d+1)+(h.val*q^d+digitCode q d u) := by |
| simp only [boxIndex,pow_succ']; ring |
| rw [he] |
| ring |
|
|
| theorem tensorCommand_tensor_correct {w T q r L d A P S C : ℕ} (s : Data) (x c : ℕ → ℕ) |
| (hq : 1<q) (hr : 0<r) (hA : 0<A) (hT : 0<T) |
| (hadd : 2*T<wordModulus w) (hmul : T*T<wordModulus w) |
| (hL : A*(max q r)^d≤L) (hLw : L*r+L+3<wordModulus w) |
| (hP : P+L+1<wordModulus w) (hS : S+L+1<wordModulus w) |
| (hC : C+r*q<wordModulus w) |
| (hPS : S+L≤P ∨ P+L≤S) |
| (hCP : C+r*q≤P ∨ P+L≤C) (hCS : C+r*q≤S ∨ S+L≤C) |
| (hregs : ∀ a : Fin 8, s.registers a.val= |
| environment T (A*r*tensorStride q d) P S C (tensorStride q d) A 0 a) |
| (hx : ∀ i, i<A*q^d → s.memory (S+i)=some (x i) ∧ x i<T) |
| (hc : ∀ i, i<r*q → s.memory (C+i)=some (c i) ∧ c i<T) : |
| ∃ cost t, Eval w (tensorCommand q r) s cost t ∧ |
| cost≤(((row q r).cost+5)*L+11)*d+1 ∧ |
| (∀ a : Fin 8, t.registers a.val= |
| environment T 0 (banks d P S).1 (banks d P S).2 C 0 (A*r^d) 0 a) ∧ |
| (∀ (a : Fin A) (v : DigitBox r d), ∃ y, |
| t.memory ((banks d P S).2+a.val*r^d+digitCode r d v)=some y ∧ y<T ∧ |
| (y:ZMod T)=∑ u : DigitBox q d, digitCoefficient T q r c d v u * |
| (x (a.val*q^d+digitCode q d u):ZMod T)) ∧ |
| (∀ a, (a<P ∨ P+L≤a) → (a<S ∨ S+L≤a) → t.memory a=s.memory a) := by |
| obtain ⟨cost,t,he,hcost,hregs,hm,houtside⟩:=tensorCommand_correct s x c hq hr hA hT hadd hmul hL hLw hP hS hC hPS hCP hCS hregs hx hc |
| refine ⟨cost,t,he,hcost,hregs,?_,houtside⟩ |
| intro a v |
| have hv:=digitCode_lt hr d v |
| have hl:=Nat.mul_le_mul_right (r^d) (show a.val+1≤A by omega) |
| have hij : a.val*r^d+digitCode r d v<A*r^d := by nlinarith only [hv,hl] |
| refine ⟨_,?_,(hm _ hij).2,tensorValue_eq_tensor (by omega) hr c d a.val x v⟩ |
| simpa only [Nat.add_assoc] using (hm _ hij).1 |
|
|
| end DeterministicThreeSum.Structured.Indexed.Axis |
|
|
| end OAI |
|
|
