{"fqn":"KLean.Bio.PcrAmplification.copies_pos","short":"copies_pos","module":"KLean.Bio.PcrAmplification.CopiesPos","path":"KLean/Bio/PcrAmplification/CopiesPos.lean","claim":null,"verified":true,"axiom_level":"kernel-standard","olean_sha256":"78b117b02aa7b17c11dd7bafdaf841a80d9d80bee7c1a6f35f37580e1278ab80","quarantined":false,"statement":"theorem copies_pos (N₀ E : ℝ) (n : ℕ) (hN₀ : 0 < N₀) (hE : 0 < E) : 0 < copies N₀ E n","topics":["biology"],"refs":[],"depends_on":["KLean.Bio.PcrAmplification.Base","KLean.Misc.Basic"],"used_by":["KLean.Bio.PcrAmplification","KLean.Bio.PcrAmplification.CopiesDoublesPerCycle","KLean.Bio.PcrAmplification.CopiesIncreasingInCycles"],"compute":"/api/v1/klean/compute/pcr_copies","seal":{"receipt_id":"d017cc7dffff572a7adc3c1d","digest":"ff9469cecfb2bfdf74174682d2499bf3dcc86c0d62e7197d6640fab11f61bd22","sealed_at":"2026-08-10T03:30:24.740384+00:00","time_source":"roughtime_chain"},"commit":"4b8413e1b8"}