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f18e7de | 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 | import OAI.LinearAlgebra.MatrixMultiplication.Numerical.AllFieldCertificatesReplaySemantics
import OAI.LinearAlgebra.MatrixMultiplication.Numerical.AllFieldCertificatesReplayStatic
namespace OAI
/-! Exact rational intervals, logarithm bounds and arithmetic circuit soundness. -/
namespace MatrixMultiplication.AllFieldCertificates.Replay
theorem Op.before_refs (op : Op) (n : ℕ) (h : op.before n = true) :
∀ j ∈ op.refs, j < n := by
cases op <;> simp_all [Op.before, Op.refs]
theorem checkedProgramValues_sound {two : Ball} (htwo : two.Encloses (Real.log 2))
(p : Program) (ops : ℕ → Op) (bs : ℕ → Ball) (count : ℕ)
(hget : ∀ i < count, p[i]? = some (ops i))
(hc : RangeChecked two ops bs 0 count) :
∀ i < count, (bs i).Encloses (p.eval[i]?.getD 0) := by
apply checkedValues_sound htwo ops bs _ count
· intro i hi
obtain ⟨w, hw⟩ := hc i (Nat.zero_le i) hi
simp only [checkAt, Bool.and_eq_true] at hw
exact p.eval_get i (ops i) (hget i hi) ((ops i).before_refs i hw.1.2)
· intro i hi
exact hc i (Nat.zero_le i) hi
def programOfLookup (ops : ℕ → Op) (count : ℕ) : Program :=
Array.ofFn (fun i : Fin count => ops i)
theorem programOfLookup_get (ops : ℕ → Op) (count i : ℕ) (hi : i < count) :
(programOfLookup ops count)[i]? = some (ops i) := by
have hh : i < (programOfLookup ops count).size := by simpa [programOfLookup] using hi
rw [Array.getElem?_eq_getElem hh]
simp [programOfLookup]
theorem checkedLookupValues_sound {two : Ball} (htwo : two.Encloses (Real.log 2))
(ops : ℕ → Op) (bs : ℕ → Ball) (count : ℕ)
(hc : RangeChecked two ops bs 0 count) :
∀ i < count, (bs i).Encloses ((programOfLookup ops count).eval[i]?.getD 0) :=
checkedProgramValues_sound htwo _ ops bs count (fun i hi => programOfLookup_get ops count i hi) hc
theorem checkedLookup_positive {two : Ball} (htwo : two.Encloses (Real.log 2))
(ops : ℕ → Op) (bs : ℕ → Ball) (count : ℕ)
(hc : RangeChecked two ops bs 0 count) (i : ℕ) (hi : i < count)
(hpos : 0 < (bs i).lower) : 0 < (programOfLookup ops count).eval[i]?.getD 0 := by
have h := (Ball.bounds (checkedLookupValues_sound htwo ops bs count hc i hi)).1
have hp : (0 : ℝ) < (bs i).lower := by exact_mod_cast hpos
exact hp.trans_le h
theorem checkedLookup_upper {two : Ball} (htwo : two.Encloses (Real.log 2))
(ops : ℕ → Op) (bs : ℕ → Ball) (count : ℕ)
(hc : RangeChecked two ops bs 0 count) (i : ℕ) (hi : i < count) (bound : ℚ)
(hu : (bs i).upper < bound) : (programOfLookup ops count).eval[i]?.getD 0 < (bound : ℝ) := by
have h := (Ball.bounds (checkedLookupValues_sound htwo ops bs count hc i hi)).2
have hb : ((bs i).upper : ℝ) < (bound : ℝ) := Rat.cast_lt.mpr hu
exact h.trans_lt hb
end MatrixMultiplication.AllFieldCertificates.Replay
end OAI
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