Formal Verification of Discrete Integration Properties from First Principles

We define a recursive discrete integral over finite integer lists and verify its principal properties in Scala Stainless. At every valid position, the integral equals the initial value plus the corresponding prefix sum; its final value equals the initial value plus the total sum; and consecutive differences recover the corresponding input values. We prove pointwise, final-value, and length agreement between recursive lookup and the accumulated-list representation. We also verify that positive input values imply a strictly increasing integral, while a positive consecutive integral gap implies that the corresponding input value is positive. Together, these results characterize the discrete integral as a length-preserving cumulative-sum construction with verified representation agreement and value recovery.