Proof step e-bab1139f7615a265

basis_map_theorem → concludes about Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ... JSON

Theorem

Affine-current basis map theorem polytool RealRooted orderTwoConstants_nonneg_iff ?

Operation

Basis map theorem for the node's own rule

{
  "operation": "basis_map_theorem"
}

Premises

None: this step is a direct theorem check.

Claims

• { "claim": "basis_map_preserver", "input_interval": "[-1 - sqrt(3), -1 + sqrt(3)]", "output_interval": "(-inf, +inf)", "statement": "every real polynomial with all roots in E = [-1 - sqrt(3), -1 + sqrt(3)] has a real-rooted image" }
• { "claim": "basis_map_preserver", "input_interval": "[-1 - sqrt(3), -1 + sqrt(3)]", "output_interval": "[-2 - sqrt(3), -2 + sqrt(3)]", "statement": "every real polynomial with all roots in E = [-1 - sqrt(3), -1 + sqrt(3)] has an image with all roots in I = [-2 - sqrt(3), -2 + sqrt(3)]" }
• { "claim": "basis_map_preserver", "input_interval": "[-1 - sqrt(3), 0]", "output_interval": "[-2 - sqrt(3), -2 + sqrt(3)]", "statement": "every PF input with all roots in [-1 - sqrt(3), 0] has a PF image; no larger [-R, 0] works" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: basis_map_theorem, SHA-256 df659f5eeec72393579edc9794ce118585945bd75bf5dc5fa8068548cc7c0ef6, 5,866 bytes (1,500 compressed); download JSON

Cited by

• n-1bffaa6198344f93 (basis_map_preserver)