Proof step e-61dbccc566bad14f

theorem_check ma_wang_interval → concludes about Triangle read by rows: The n-th derivative of the logistic function written in terms of y, where y = 1/(1 + exp(-x)). JSON

Theorem

Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Lean (RealRooted): RealRooted.affineEndpointStep_closed, RealRooted.affineEndpointStep_roots_mem_Icc_iff, RealRooted.affineEndpointStep_rootMultiplicity_zero, RealRooted.affineEndpointStep_rootMultiplicity_neg_one, RealRooted.MaWang.endpointStep, RealRooted.MaWang.affineEndpointSequence, RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff

Operation

Theorem check of the node's own rule

{
  "operation": "theorem_check",
  "theorem": "ma_wang_interval"
}

Premises

None: this step is a direct theorem check.

Claims

• { "claim": "real_rooted", "from_index": 1, "region": { "lower": { "closed": false, "value": "0" }, "upper": { "closed": true, "value": "1" } } }
• { "claim": "interlacing", "from_index": 2, "kind": "weak", "orientation": "previous_interlaces_next" }

Verification and certificate

• polytool: replayed (the certificate replayed when the step was written)
• Lean: not attempted
• certificate: theorem_check, SHA-256 d738902a74bd88be069ff4f01cfa1d3a456528dd93224c6b359fa77f2ff84ad2, 3,960 bytes (1,311 compressed); download JSON

Cited by

• n-2ae6a2215aff79d4 (one_signed_coefficients)
• n-2ae6a2215aff79d4 (interlacing)
• n-2ae6a2215aff79d4 (real_rooted)

References

1. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741