openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Numerical /AllFieldCertificatesReplayIntegration.lean
| 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 | |