Proof step e-1b4d12a0914b0a40

preserver_image affine, shift 0 → concludes about shift_x_plus_one of Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n). JSON

Theorem

Affine, reversal and Möbius substitutions (elementary) polytool ?

Lean (RealRooted): Polynomial.Splits.comp_of_degree_le_one

Operation

Image under a real-rootedness preserver

{
  "index_shift": 0,
  "operation": "preserver_image",
  "operator": "{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"
}

Premises

• Triangle read by rows: T(n,k) = binomial(n+1-k,k) (n >= 0, 0 <= k <= n). (sequence)
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]

Claims

• { "claim": "real_rooted", "from_index": 0, "region": { "upper": { "closed": true, "value": "-1" } } }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: preserver_image, SHA-256 e40942fbd1c721e68013ef27cde3aa73d844a2337d047e021c15f214992620c1, 8,766 bytes (2,423 compressed); download JSON

Cited by

• n-aeb67fdaad2bc42e (real_rooted)
• n-aeb67fdaad2bc42e (one_signed_coefficients)
• n-aeb67fdaad2bc42e (hadamard_kernel)