Proof step e-28cf6d258736730b

basis_map_theorem → concludes about Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1). 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(2), -1 + sqrt(2)]", "output_interval": "(-inf, +inf)", "statement": "every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] has a real-rooted image" }
• { "claim": "basis_map_preserver", "input_interval": "[-1 - sqrt(2), -1 + sqrt(2)]", "output_interval": "[-3 - 2 sqrt(2), -3 + 2 sqrt(2)]", "statement": "every real polynomial with all roots in E = [-1 - sqrt(2), -1 + sqrt(2)] has an image with all roots in I = [-3 - 2 sqrt(2), -3 + 2 sqrt(2)]" }
• { "claim": "basis_map_preserver", "input_interval": "[-1 - sqrt(2), 0]", "output_interval": "[-3 - 2 sqrt(2), -3 + 2 sqrt(2)]", "statement": "every PF input with all roots in [-1 - sqrt(2), 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 108f74e95a83f20278ebcfb6318ab6e8a446475ffdadfc3989408f30f55434e8, 5,825 bytes (1,495 compressed); download JSON

Cited by

• n-14dfd91463cae288 (basis_map_preserver)