Discrete Summation

These theories introduce basic concepts and proofs about discrete summation: shifts, formal summation, falling factorials and stirling numbers. As proof of concept, a simple summation conversion is provided. 1 Stirling numbers of first and second kind theory Stirling imports Main begin 1.1 Stirling numbers of the second kind fun Stirling :: nat ⇒ nat ⇒ nat where Stirling 0 0 = 1 | Stirling 0 (Suc k) = 0 | Stirling (Suc n) 0 = 0 | Stirling (Suc n) (Suc k) = Suc k ∗ Stirling n (Suc k) + Stirling n k lemma Stirling-1 [simp]: Stirling (Suc n) (Suc 0 ) = 1 〈proof 〉 lemma Stirling-less [simp]: n < k =⇒ Stirling n k = 0 〈proof 〉 lemma Stirling-same [simp]: Stirling n n = 1 〈proof 〉 lemma Stirling-2-2 : Stirling (Suc (Suc n)) (Suc (Suc 0 )) = 2 ˆ Suc n − 1 〈proof 〉