Proof step

Affine, reversal and Möbius substitutions (elementary)

Image under a real-rootedness preserver affine, shift 0

Concludes

shift_x_plus_one of shift_x_plus_one of chebyshev_t

Premises

shift_x_plus_one of chebyshev_t
sequence: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0)

Theorem

polytool ?

Lean

RealRooted Polynomial.Splits.comp_of_degree_le_one

Checked

polytool replayed · Lean not attempted

Technical details

Step id

e-bfb7184a4a2ca735 · JSON

Operation

Image under a real-rootedness preserver

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

Claims

{
  "claim": "real_rooted",
  "from_index": 0,
  "region": {
    "lower": {
      "closed": false,
      "value": "-3"
    },
    "upper": {
      "closed": false,
      "value": "-1"
    }
  }
}

{
  "claim": "interlacing",
  "from_index": 2,
  "kind": "weak",
  "orientation": "unordered"
}

Certificate

preserver_image, SHA-256 1d6ac57552782d89900633dad9ba2cecd379bf2616d37727b3b601cd4bea1392, 20,041 bytes (3,541 compressed); download

polytool replayed the certificate when the step was written.

Cited by

n-e77cd13e9e0a7e61 (real_rooted)
n-e77cd13e9e0a7e61 (one_signed_coefficients)
n-e77cd13e9e0a7e61 (hadamard_kernel)
n-e77cd13e9e0a7e61 (interlacing)