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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 96 97 98 99 100 | 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
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